1,1,33,33,0.0085708,"\int x^2 \left(a+b x^3\right) \left(A+B x^3\right) \, dx","Integrate[x^2*(a + b*x^3)*(A + B*x^3),x]","\frac{1}{6} x^6 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{9} b B x^9","\frac{1}{6} x^6 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{9} b B x^9",1,"(a*A*x^3)/3 + ((A*b + a*B)*x^6)/6 + (b*B*x^9)/9","A",1
2,1,33,33,0.0067937,"\int x \left(a+b x^3\right) \left(A+B x^3\right) \, dx","Integrate[x*(a + b*x^3)*(A + B*x^3),x]","\frac{1}{5} x^5 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{8} b B x^8","\frac{1}{5} x^5 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{8} b B x^8",1,"(a*A*x^2)/2 + ((A*b + a*B)*x^5)/5 + (b*B*x^8)/8","A",1
3,1,28,28,0.0069719,"\int \left(a+b x^3\right) \left(A+B x^3\right) \, dx","Integrate[(a + b*x^3)*(A + B*x^3),x]","\frac{1}{4} x^4 (a B+A b)+a A x+\frac{1}{7} b B x^7","\frac{1}{4} x^4 (a B+A b)+a A x+\frac{1}{7} b B x^7",1,"a*A*x + ((A*b + a*B)*x^4)/4 + (b*B*x^7)/7","A",1
4,1,29,29,0.0088642,"\int \frac{\left(a+b x^3\right) \left(A+B x^3\right)}{x} \, dx","Integrate[((a + b*x^3)*(A + B*x^3))/x,x]","\frac{1}{3} x^3 (a B+A b)+a A \log (x)+\frac{1}{6} b B x^6","\frac{1}{3} x^3 (a B+A b)+a A \log (x)+\frac{1}{6} b B x^6",1,"((A*b + a*B)*x^3)/3 + (b*B*x^6)/6 + a*A*Log[x]","A",1
5,1,31,31,0.0114242,"\int \frac{\left(a+b x^3\right) \left(A+B x^3\right)}{x^2} \, dx","Integrate[((a + b*x^3)*(A + B*x^3))/x^2,x]","\frac{1}{2} x^2 (a B+A b)-\frac{a A}{x}+\frac{1}{5} b B x^5","\frac{1}{2} x^2 (a B+A b)-\frac{a A}{x}+\frac{1}{5} b B x^5",1,"-((a*A)/x) + ((A*b + a*B)*x^2)/2 + (b*B*x^5)/5","A",1
6,1,28,28,0.0091831,"\int \frac{\left(a+b x^3\right) \left(A+B x^3\right)}{x^3} \, dx","Integrate[((a + b*x^3)*(A + B*x^3))/x^3,x]","x (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{4} b B x^4","x (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{4} b B x^4",1,"-1/2*(a*A)/x^2 + (A*b + a*B)*x + (b*B*x^4)/4","A",1
7,1,29,29,0.0126368,"\int \frac{\left(a+b x^3\right) \left(A+B x^3\right)}{x^4} \, dx","Integrate[((a + b*x^3)*(A + B*x^3))/x^4,x]","\log (x) (a B+A b)-\frac{a A}{3 x^3}+\frac{1}{3} b B x^3","\log (x) (a B+A b)-\frac{a A}{3 x^3}+\frac{1}{3} b B x^3",1,"-1/3*(a*A)/x^3 + (b*B*x^3)/3 + (A*b + a*B)*Log[x]","A",1
8,1,32,31,0.0134578,"\int \frac{\left(a+b x^3\right) \left(A+B x^3\right)}{x^5} \, dx","Integrate[((a + b*x^3)*(A + B*x^3))/x^5,x]","\frac{-a B-A b}{x}-\frac{a A}{4 x^4}+\frac{1}{2} b B x^2","-\frac{a B+A b}{x}-\frac{a A}{4 x^4}+\frac{1}{2} b B x^2",1,"-1/4*(a*A)/x^4 + (-(A*b) - a*B)/x + (b*B*x^2)/2","A",1
9,1,30,28,0.0121736,"\int \frac{\left(a+b x^3\right) \left(A+B x^3\right)}{x^6} \, dx","Integrate[((a + b*x^3)*(A + B*x^3))/x^6,x]","\frac{-a B-A b}{2 x^2}-\frac{a A}{5 x^5}+b B x","-\frac{a B+A b}{2 x^2}-\frac{a A}{5 x^5}+b B x",1,"-1/5*(a*A)/x^5 + (-(A*b) - a*B)/(2*x^2) + b*B*x","A",1
10,1,31,29,0.0176392,"\int \frac{\left(a+b x^3\right) \left(A+B x^3\right)}{x^7} \, dx","Integrate[((a + b*x^3)*(A + B*x^3))/x^7,x]","\frac{-a B-A b}{3 x^3}-\frac{a A}{6 x^6}+b B \log (x)","-\frac{a B+A b}{3 x^3}-\frac{a A}{6 x^6}+b B \log (x)",1,"-1/6*(a*A)/x^6 + (-(A*b) - a*B)/(3*x^3) + b*B*Log[x]","A",1
11,1,51,42,0.0161828,"\int x^2 \left(a+b x^3\right)^2 \left(A+B x^3\right) \, dx","Integrate[x^2*(a + b*x^3)^2*(A + B*x^3),x]","\frac{1}{36} x^3 \left(12 a^2 A+4 b x^6 (2 a B+A b)+6 a x^3 (a B+2 A b)+3 b^2 B x^9\right)","\frac{\left(a+b x^3\right)^3 (A b-a B)}{9 b^2}+\frac{B \left(a+b x^3\right)^4}{12 b^2}",1,"(x^3*(12*a^2*A + 6*a*(2*A*b + a*B)*x^3 + 4*b*(A*b + 2*a*B)*x^6 + 3*b^2*B*x^9))/36","A",1
12,1,55,55,0.0076679,"\int x \left(a+b x^3\right)^2 \left(A+B x^3\right) \, dx","Integrate[x*(a + b*x^3)^2*(A + B*x^3),x]","\frac{1}{2} a^2 A x^2+\frac{1}{8} b x^8 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{11} b^2 B x^{11}","\frac{1}{2} a^2 A x^2+\frac{1}{8} b x^8 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{11} b^2 B x^{11}",1,"(a^2*A*x^2)/2 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^8)/8 + (b^2*B*x^11)/11","A",1
13,1,50,50,0.0386369,"\int \left(a+b x^3\right)^2 \left(A+B x^3\right) \, dx","Integrate[(a + b*x^3)^2*(A + B*x^3),x]","a^2 A x+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{4} a x^4 (a B+2 A b)+\frac{1}{10} b^2 B x^{10}","a^2 A x+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{4} a x^4 (a B+2 A b)+\frac{1}{10} b^2 B x^{10}",1,"a^2*A*x + (a*(2*A*b + a*B)*x^4)/4 + (b*(A*b + 2*a*B)*x^7)/7 + (b^2*B*x^10)/10","A",1
14,1,51,46,0.0382377,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x,x]","a^2 A \log (x)+\frac{1}{6} b x^6 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{9} b^2 B x^9","a^2 A \log (x)+\frac{2}{3} a A b x^3+\frac{B \left(a+b x^3\right)^3}{9 b}+\frac{1}{6} A b^2 x^6",1,"(a*(2*A*b + a*B)*x^3)/3 + (b*(A*b + 2*a*B)*x^6)/6 + (b^2*B*x^9)/9 + a^2*A*Log[x]","A",1
15,1,53,53,0.0215145,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x^2} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x^2,x]","-\frac{a^2 A}{x}+\frac{1}{5} b x^5 (2 a B+A b)+\frac{1}{2} a x^2 (a B+2 A b)+\frac{1}{8} b^2 B x^8","-\frac{a^2 A}{x}+\frac{1}{5} b x^5 (2 a B+A b)+\frac{1}{2} a x^2 (a B+2 A b)+\frac{1}{8} b^2 B x^8",1,"-((a^2*A)/x) + (a*(2*A*b + a*B)*x^2)/2 + (b*(A*b + 2*a*B)*x^5)/5 + (b^2*B*x^8)/8","A",1
16,1,50,50,0.0156787,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x^3} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x^3,x]","-\frac{a^2 A}{2 x^2}+\frac{1}{4} b x^4 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{7} b^2 B x^7","-\frac{a^2 A}{2 x^2}+\frac{1}{4} b x^4 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{7} b^2 B x^7",1,"-1/2*(a^2*A)/x^2 + a*(2*A*b + a*B)*x + (b*(A*b + 2*a*B)*x^4)/4 + (b^2*B*x^7)/7","A",1
17,1,49,51,0.0263006,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x^4} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x^4,x]","\frac{1}{6} \left(-\frac{2 a^2 A}{x^3}+2 b x^3 (2 a B+A b)+6 a \log (x) (a B+2 A b)+b^2 B x^6\right)","-\frac{a^2 A}{3 x^3}+\frac{1}{3} b x^3 (2 a B+A b)+a \log (x) (a B+2 A b)+\frac{1}{6} b^2 B x^6",1,"((-2*a^2*A)/x^3 + 2*b*(A*b + 2*a*B)*x^3 + b^2*B*x^6 + 6*a*(2*A*b + a*B)*Log[x])/6","A",1
18,1,51,53,0.0168435,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x^5} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x^5,x]","\frac{-5 a^2 A+10 b x^6 (2 a B+A b)-20 a x^3 (a B+2 A b)+4 b^2 B x^9}{20 x^4}","-\frac{a^2 A}{4 x^4}+\frac{1}{2} b x^2 (2 a B+A b)-\frac{a (a B+2 A b)}{x}+\frac{1}{5} b^2 B x^5",1,"(-5*a^2*A - 20*a*(2*A*b + a*B)*x^3 + 10*b*(A*b + 2*a*B)*x^6 + 4*b^2*B*x^9)/(20*x^4)","A",1
19,1,50,50,0.0197846,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x^6} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x^6,x]","-\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{2 x^2}+b x (2 a B+A b)+\frac{1}{4} b^2 B x^4","-\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{2 x^2}+b x (2 a B+A b)+\frac{1}{4} b^2 B x^4",1,"-1/5*(a^2*A)/x^5 - (a*(2*A*b + a*B))/(2*x^2) + b*(A*b + 2*a*B)*x + (b^2*B*x^4)/4","A",1
20,1,51,51,0.0217533,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x^7} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x^7,x]","\frac{1}{6} \left(-\frac{a^2 \left(A+2 B x^3\right)}{x^6}+6 b \log (x) (2 a B+A b)-\frac{4 a A b}{x^3}+2 b^2 B x^3\right)","-\frac{a^2 A}{6 x^6}-\frac{a (a B+2 A b)}{3 x^3}+b \log (x) (2 a B+A b)+\frac{1}{3} b^2 B x^3",1,"((-4*a*A*b)/x^3 + 2*b^2*B*x^3 - (a^2*(A + 2*B*x^3))/x^6 + 6*b*(A*b + 2*a*B)*Log[x])/6","A",1
21,1,54,53,0.0168962,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x^8} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x^8,x]","-\frac{a^2 \left(4 A+7 B x^3\right)+14 a b x^3 \left(A+4 B x^3\right)-14 b^2 x^6 \left(B x^3-2 A\right)}{28 x^7}","-\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{4 x^4}-\frac{b (2 a B+A b)}{x}+\frac{1}{2} b^2 B x^2",1,"-1/28*(-14*b^2*x^6*(-2*A + B*x^3) + 14*a*b*x^3*(A + 4*B*x^3) + a^2*(4*A + 7*B*x^3))/x^7","A",1
22,1,50,50,0.031119,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x^9} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x^9,x]","-\frac{a^2 A}{8 x^8}-\frac{a (a B+2 A b)}{5 x^5}-\frac{b (2 a B+A b)}{2 x^2}+b^2 B x","-\frac{a^2 A}{8 x^8}-\frac{a (a B+2 A b)}{5 x^5}-\frac{b (2 a B+A b)}{2 x^2}+b^2 B x",1,"-1/8*(a^2*A)/x^8 - (a*(2*A*b + a*B))/(5*x^5) - (b*(A*b + 2*a*B))/(2*x^2) + b^2*B*x","A",1
23,1,117,117,0.02177,"\int x^9 \left(a+b x^3\right)^5 \left(A+B x^3\right) \, dx","Integrate[x^9*(a + b*x^3)^5*(A + B*x^3),x]","\frac{1}{10} a^5 A x^{10}+\frac{1}{13} a^4 x^{13} (a B+5 A b)+\frac{5}{16} a^3 b x^{16} (a B+2 A b)+\frac{10}{19} a^2 b^2 x^{19} (a B+A b)+\frac{1}{25} b^4 x^{25} (5 a B+A b)+\frac{5}{22} a b^3 x^{22} (2 a B+A b)+\frac{1}{28} b^5 B x^{28}","\frac{1}{10} a^5 A x^{10}+\frac{1}{13} a^4 x^{13} (a B+5 A b)+\frac{5}{16} a^3 b x^{16} (a B+2 A b)+\frac{10}{19} a^2 b^2 x^{19} (a B+A b)+\frac{1}{25} b^4 x^{25} (5 a B+A b)+\frac{5}{22} a b^3 x^{22} (2 a B+A b)+\frac{1}{28} b^5 B x^{28}",1,"(a^5*A*x^10)/10 + (a^4*(5*A*b + a*B)*x^13)/13 + (5*a^3*b*(2*A*b + a*B)*x^16)/16 + (10*a^2*b^2*(A*b + a*B)*x^19)/19 + (5*a*b^3*(A*b + 2*a*B)*x^22)/22 + (b^4*(A*b + 5*a*B)*x^25)/25 + (b^5*B*x^28)/28","A",1
24,1,107,95,0.0288756,"\int x^8 \left(a+b x^3\right)^5 \left(A+B x^3\right) \, dx","Integrate[x^8*(a + b*x^3)^5*(A + B*x^3),x]","\frac{x^9 \left(168 a^5 A+126 a^4 x^3 (a B+5 A b)+504 a^3 b x^6 (a B+2 A b)+840 a^2 b^2 x^9 (a B+A b)+63 b^4 x^{15} (5 a B+A b)+360 a b^3 x^{12} (2 a B+A b)+56 b^5 B x^{18}\right)}{1512}","\frac{a^2 \left(a+b x^3\right)^6 (A b-a B)}{18 b^4}+\frac{\left(a+b x^3\right)^8 (A b-3 a B)}{24 b^4}-\frac{a \left(a+b x^3\right)^7 (2 A b-3 a B)}{21 b^4}+\frac{B \left(a+b x^3\right)^9}{27 b^4}",1,"(x^9*(168*a^5*A + 126*a^4*(5*A*b + a*B)*x^3 + 504*a^3*b*(2*A*b + a*B)*x^6 + 840*a^2*b^2*(A*b + a*B)*x^9 + 360*a*b^3*(A*b + 2*a*B)*x^12 + 63*b^4*(A*b + 5*a*B)*x^15 + 56*b^5*B*x^18))/1512","A",1
25,1,117,117,0.0196922,"\int x^7 \left(a+b x^3\right)^5 \left(A+B x^3\right) \, dx","Integrate[x^7*(a + b*x^3)^5*(A + B*x^3),x]","\frac{1}{8} a^5 A x^8+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{5}{14} a^3 b x^{14} (a B+2 A b)+\frac{10}{17} a^2 b^2 x^{17} (a B+A b)+\frac{1}{23} b^4 x^{23} (5 a B+A b)+\frac{1}{4} a b^3 x^{20} (2 a B+A b)+\frac{1}{26} b^5 B x^{26}","\frac{1}{8} a^5 A x^8+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{5}{14} a^3 b x^{14} (a B+2 A b)+\frac{10}{17} a^2 b^2 x^{17} (a B+A b)+\frac{1}{23} b^4 x^{23} (5 a B+A b)+\frac{1}{4} a b^3 x^{20} (2 a B+A b)+\frac{1}{26} b^5 B x^{26}",1,"(a^5*A*x^8)/8 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^14)/14 + (10*a^2*b^2*(A*b + a*B)*x^17)/17 + (a*b^3*(A*b + 2*a*B)*x^20)/4 + (b^4*(A*b + 5*a*B)*x^23)/23 + (b^5*B*x^26)/26","A",1
26,1,117,117,0.0199317,"\int x^6 \left(a+b x^3\right)^5 \left(A+B x^3\right) \, dx","Integrate[x^6*(a + b*x^3)^5*(A + B*x^3),x]","\frac{1}{7} a^5 A x^7+\frac{1}{10} a^4 x^{10} (a B+5 A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{5}{8} a^2 b^2 x^{16} (a B+A b)+\frac{1}{22} b^4 x^{22} (5 a B+A b)+\frac{5}{19} a b^3 x^{19} (2 a B+A b)+\frac{1}{25} b^5 B x^{25}","\frac{1}{7} a^5 A x^7+\frac{1}{10} a^4 x^{10} (a B+5 A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{5}{8} a^2 b^2 x^{16} (a B+A b)+\frac{1}{22} b^4 x^{22} (5 a B+A b)+\frac{5}{19} a b^3 x^{19} (2 a B+A b)+\frac{1}{25} b^5 B x^{25}",1,"(a^5*A*x^7)/7 + (a^4*(5*A*b + a*B)*x^10)/10 + (5*a^3*b*(2*A*b + a*B)*x^13)/13 + (5*a^2*b^2*(A*b + a*B)*x^16)/8 + (5*a*b^3*(A*b + 2*a*B)*x^19)/19 + (b^4*(A*b + 5*a*B)*x^22)/22 + (b^5*B*x^25)/25","A",1
27,1,107,67,0.0250762,"\int x^5 \left(a+b x^3\right)^5 \left(A+B x^3\right) \, dx","Integrate[x^5*(a + b*x^3)^5*(A + B*x^3),x]","\frac{1}{504} x^6 \left(84 a^5 A+56 a^4 x^3 (a B+5 A b)+210 a^3 b x^6 (a B+2 A b)+336 a^2 b^2 x^9 (a B+A b)+24 b^4 x^{15} (5 a B+A b)+140 a b^3 x^{12} (2 a B+A b)+21 b^5 B x^{18}\right)","\frac{\left(a+b x^3\right)^7 (A b-2 a B)}{21 b^3}-\frac{a \left(a+b x^3\right)^6 (A b-a B)}{18 b^3}+\frac{B \left(a+b x^3\right)^8}{24 b^3}",1,"(x^6*(84*a^5*A + 56*a^4*(5*A*b + a*B)*x^3 + 210*a^3*b*(2*A*b + a*B)*x^6 + 336*a^2*b^2*(A*b + a*B)*x^9 + 140*a*b^3*(A*b + 2*a*B)*x^12 + 24*b^4*(A*b + 5*a*B)*x^15 + 21*b^5*B*x^18))/504","A",1
28,1,117,117,0.0214271,"\int x^4 \left(a+b x^3\right)^5 \left(A+B x^3\right) \, dx","Integrate[x^4*(a + b*x^3)^5*(A + B*x^3),x]","\frac{1}{5} a^5 A x^5+\frac{1}{8} a^4 x^8 (a B+5 A b)+\frac{5}{11} a^3 b x^{11} (a B+2 A b)+\frac{5}{7} a^2 b^2 x^{14} (a B+A b)+\frac{1}{20} b^4 x^{20} (5 a B+A b)+\frac{5}{17} a b^3 x^{17} (2 a B+A b)+\frac{1}{23} b^5 B x^{23}","\frac{1}{5} a^5 A x^5+\frac{1}{8} a^4 x^8 (a B+5 A b)+\frac{5}{11} a^3 b x^{11} (a B+2 A b)+\frac{5}{7} a^2 b^2 x^{14} (a B+A b)+\frac{1}{20} b^4 x^{20} (5 a B+A b)+\frac{5}{17} a b^3 x^{17} (2 a B+A b)+\frac{1}{23} b^5 B x^{23}",1,"(a^5*A*x^5)/5 + (a^4*(5*A*b + a*B)*x^8)/8 + (5*a^3*b*(2*A*b + a*B)*x^11)/11 + (5*a^2*b^2*(A*b + a*B)*x^14)/7 + (5*a*b^3*(A*b + 2*a*B)*x^17)/17 + (b^4*(A*b + 5*a*B)*x^20)/20 + (b^5*B*x^23)/23","A",1
29,1,117,117,0.0181179,"\int x^3 \left(a+b x^3\right)^5 \left(A+B x^3\right) \, dx","Integrate[x^3*(a + b*x^3)^5*(A + B*x^3),x]","\frac{1}{4} a^5 A x^4+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{1}{2} a^3 b x^{10} (a B+2 A b)+\frac{10}{13} a^2 b^2 x^{13} (a B+A b)+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{16} a b^3 x^{16} (2 a B+A b)+\frac{1}{22} b^5 B x^{22}","\frac{1}{4} a^5 A x^4+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{1}{2} a^3 b x^{10} (a B+2 A b)+\frac{10}{13} a^2 b^2 x^{13} (a B+A b)+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{16} a b^3 x^{16} (2 a B+A b)+\frac{1}{22} b^5 B x^{22}",1,"(a^5*A*x^4)/4 + (a^4*(5*A*b + a*B)*x^7)/7 + (a^3*b*(2*A*b + a*B)*x^10)/2 + (10*a^2*b^2*(A*b + a*B)*x^13)/13 + (5*a*b^3*(A*b + 2*a*B)*x^16)/16 + (b^4*(A*b + 5*a*B)*x^19)/19 + (b^5*B*x^22)/22","A",1
30,1,107,42,0.0296082,"\int x^2 \left(a+b x^3\right)^5 \left(A+B x^3\right) \, dx","Integrate[x^2*(a + b*x^3)^5*(A + B*x^3),x]","\frac{1}{126} x^3 \left(42 a^5 A+21 a^4 x^3 (a B+5 A b)+70 a^3 b x^6 (a B+2 A b)+105 a^2 b^2 x^9 (a B+A b)+7 b^4 x^{15} (5 a B+A b)+42 a b^3 x^{12} (2 a B+A b)+6 b^5 B x^{18}\right)","\frac{\left(a+b x^3\right)^6 (A b-a B)}{18 b^2}+\frac{B \left(a+b x^3\right)^7}{21 b^2}",1,"(x^3*(42*a^5*A + 21*a^4*(5*A*b + a*B)*x^3 + 70*a^3*b*(2*A*b + a*B)*x^6 + 105*a^2*b^2*(A*b + a*B)*x^9 + 42*a*b^3*(A*b + 2*a*B)*x^12 + 7*b^4*(A*b + 5*a*B)*x^15 + 6*b^5*B*x^18))/126","B",1
31,1,117,117,0.0178922,"\int x \left(a+b x^3\right)^5 \left(A+B x^3\right) \, dx","Integrate[x*(a + b*x^3)^5*(A + B*x^3),x]","\frac{1}{2} a^5 A x^2+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{5}{8} a^3 b x^8 (a B+2 A b)+\frac{10}{11} a^2 b^2 x^{11} (a B+A b)+\frac{1}{17} b^4 x^{17} (5 a B+A b)+\frac{5}{14} a b^3 x^{14} (2 a B+A b)+\frac{1}{20} b^5 B x^{20}","\frac{1}{2} a^5 A x^2+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{5}{8} a^3 b x^8 (a B+2 A b)+\frac{10}{11} a^2 b^2 x^{11} (a B+A b)+\frac{1}{17} b^4 x^{17} (5 a B+A b)+\frac{5}{14} a b^3 x^{14} (2 a B+A b)+\frac{1}{20} b^5 B x^{20}",1,"(a^5*A*x^2)/2 + (a^4*(5*A*b + a*B)*x^5)/5 + (5*a^3*b*(2*A*b + a*B)*x^8)/8 + (10*a^2*b^2*(A*b + a*B)*x^11)/11 + (5*a*b^3*(A*b + 2*a*B)*x^14)/14 + (b^4*(A*b + 5*a*B)*x^17)/17 + (b^5*B*x^20)/20","A",1
32,1,109,109,0.0167158,"\int \left(a+b x^3\right)^5 \left(A+B x^3\right) \, dx","Integrate[(a + b*x^3)^5*(A + B*x^3),x]","a^5 A x+\frac{1}{4} a^4 x^4 (a B+5 A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+a^2 b^2 x^{10} (a B+A b)+\frac{1}{16} b^4 x^{16} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{19} b^5 B x^{19}","a^5 A x+\frac{1}{4} a^4 x^4 (a B+5 A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+a^2 b^2 x^{10} (a B+A b)+\frac{1}{16} b^4 x^{16} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{19} b^5 B x^{19}",1,"a^5*A*x + (a^4*(5*A*b + a*B)*x^4)/4 + (5*a^3*b*(2*A*b + a*B)*x^7)/7 + a^2*b^2*(A*b + a*B)*x^10 + (5*a*b^3*(A*b + 2*a*B)*x^13)/13 + (b^4*(A*b + 5*a*B)*x^16)/16 + (b^5*B*x^19)/19","A",1
33,1,113,88,0.0292216,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x,x]","a^5 A \log (x)+\frac{1}{3} a^4 x^3 (a B+5 A b)+\frac{5}{6} a^3 b x^6 (a B+2 A b)+\frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{1}{15} b^4 x^{15} (5 a B+A b)+\frac{5}{12} a b^3 x^{12} (2 a B+A b)+\frac{1}{18} b^5 B x^{18}","a^5 A \log (x)+\frac{5}{3} a^4 A b x^3+\frac{5}{3} a^3 A b^2 x^6+\frac{10}{9} a^2 A b^3 x^9+\frac{5}{12} a A b^4 x^{12}+\frac{B \left(a+b x^3\right)^6}{18 b}+\frac{1}{15} A b^5 x^{15}",1,"(a^4*(5*A*b + a*B)*x^3)/3 + (5*a^3*b*(2*A*b + a*B)*x^6)/6 + (10*a^2*b^2*(A*b + a*B)*x^9)/9 + (5*a*b^3*(A*b + 2*a*B)*x^12)/12 + (b^4*(A*b + 5*a*B)*x^15)/15 + (b^5*B*x^18)/18 + a^5*A*Log[x]","A",1
34,1,112,112,0.0347876,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^2} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^2,x]","-\frac{a^5 A}{x}+\frac{1}{2} a^4 x^2 (a B+5 A b)+a^3 b x^5 (a B+2 A b)+\frac{5}{4} a^2 b^2 x^8 (a B+A b)+\frac{1}{14} b^4 x^{14} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{17} b^5 B x^{17}","-\frac{a^5 A}{x}+\frac{1}{2} a^4 x^2 (a B+5 A b)+a^3 b x^5 (a B+2 A b)+\frac{5}{4} a^2 b^2 x^8 (a B+A b)+\frac{1}{14} b^4 x^{14} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{17} b^5 B x^{17}",1,"-((a^5*A)/x) + (a^4*(5*A*b + a*B)*x^2)/2 + a^3*b*(2*A*b + a*B)*x^5 + (5*a^2*b^2*(A*b + a*B)*x^8)/4 + (5*a*b^3*(A*b + 2*a*B)*x^11)/11 + (b^4*(A*b + 5*a*B)*x^14)/14 + (b^5*B*x^17)/17","A",1
35,1,112,112,0.0321547,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^3} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^3,x]","-\frac{a^5 A}{2 x^2}+a^4 x (a B+5 A b)+\frac{5}{4} a^3 b x^4 (a B+2 A b)+\frac{10}{7} a^2 b^2 x^7 (a B+A b)+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{1}{2} a b^3 x^{10} (2 a B+A b)+\frac{1}{16} b^5 B x^{16}","-\frac{a^5 A}{2 x^2}+a^4 x (a B+5 A b)+\frac{5}{4} a^3 b x^4 (a B+2 A b)+\frac{10}{7} a^2 b^2 x^7 (a B+A b)+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{1}{2} a b^3 x^{10} (2 a B+A b)+\frac{1}{16} b^5 B x^{16}",1,"-1/2*(a^5*A)/x^2 + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^4)/4 + (10*a^2*b^2*(A*b + a*B)*x^7)/7 + (a*b^3*(A*b + 2*a*B)*x^10)/2 + (b^4*(A*b + 5*a*B)*x^13)/13 + (b^5*B*x^16)/16","A",1
36,1,115,113,0.0445352,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^4} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^4,x]","-\frac{a^5 A}{3 x^3}+\frac{5}{3} a^3 b x^3 (a B+2 A b)+\frac{5}{3} a^2 b^2 x^6 (a B+A b)+\log (x) \left(a^5 B+5 a^4 A b\right)+\frac{1}{12} b^4 x^{12} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{15} b^5 B x^{15}","-\frac{a^5 A}{3 x^3}+a^4 \log (x) (a B+5 A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+\frac{5}{3} a^2 b^2 x^6 (a B+A b)+\frac{1}{12} b^4 x^{12} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{15} b^5 B x^{15}",1,"-1/3*(a^5*A)/x^3 + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + (5*a^2*b^2*(A*b + a*B)*x^6)/3 + (5*a*b^3*(A*b + 2*a*B)*x^9)/9 + (b^4*(A*b + 5*a*B)*x^12)/12 + (b^5*B*x^15)/15 + (5*a^4*A*b + a^5*B)*Log[x]","A",1
37,1,115,113,0.063398,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^5} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^5,x]","-\frac{a^5 A}{4 x^4}+\frac{5}{2} a^3 b x^2 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac{a^5 (-B)-5 a^4 A b}{x}+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{5}{8} a b^3 x^8 (2 a B+A b)+\frac{1}{14} b^5 B x^{14}","-\frac{a^5 A}{4 x^4}-\frac{a^4 (a B+5 A b)}{x}+\frac{5}{2} a^3 b x^2 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{5}{8} a b^3 x^8 (2 a B+A b)+\frac{1}{14} b^5 B x^{14}",1,"-1/4*(a^5*A)/x^4 + (-5*a^4*A*b - a^5*B)/x + (5*a^3*b*(2*A*b + a*B)*x^2)/2 + 2*a^2*b^2*(A*b + a*B)*x^5 + (5*a*b^3*(A*b + 2*a*B)*x^8)/8 + (b^4*(A*b + 5*a*B)*x^11)/11 + (b^5*B*x^14)/14","A",1
38,1,113,113,0.0566669,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^6} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^6,x]","-\frac{a^5 A}{5 x^5}-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b x (a B+2 A b)+\frac{5}{2} a^2 b^2 x^4 (a B+A b)+\frac{1}{10} b^4 x^{10} (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{13} b^5 B x^{13}","-\frac{a^5 A}{5 x^5}-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b x (a B+2 A b)+\frac{5}{2} a^2 b^2 x^4 (a B+A b)+\frac{1}{10} b^4 x^{10} (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{13} b^5 B x^{13}",1,"-1/5*(a^5*A)/x^5 - (a^4*(5*A*b + a*B))/(2*x^2) + 5*a^3*b*(2*A*b + a*B)*x + (5*a^2*b^2*(A*b + a*B)*x^4)/2 + (5*a*b^3*(A*b + 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)*x^10)/10 + (b^5*B*x^13)/13","A",1
39,1,106,114,0.0571611,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^7} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^7,x]","\frac{1}{36} \left(-\frac{6 a^5 A}{x^6}-\frac{12 a^4 (a B+5 A b)}{x^3}+180 a^3 b \log (x) (a B+2 A b)+120 a^2 b^2 x^3 (a B+A b)+4 b^4 x^9 (5 a B+A b)+30 a b^3 x^6 (2 a B+A b)+3 b^5 B x^{12}\right)","-\frac{a^5 A}{6 x^6}-\frac{a^4 (a B+5 A b)}{3 x^3}+5 a^3 b \log (x) (a B+2 A b)+\frac{10}{3} a^2 b^2 x^3 (a B+A b)+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{6} a b^3 x^6 (2 a B+A b)+\frac{1}{12} b^5 B x^{12}",1,"((-6*a^5*A)/x^6 - (12*a^4*(5*A*b + a*B))/x^3 + 120*a^2*b^2*(A*b + a*B)*x^3 + 30*a*b^3*(A*b + 2*a*B)*x^6 + 4*b^4*(A*b + 5*a*B)*x^9 + 3*b^5*B*x^12 + 180*a^3*b*(2*A*b + a*B)*Log[x])/36","A",1
40,1,110,110,0.0379125,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^8} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^8,x]","-\frac{a^5 A}{7 x^7}-\frac{a^4 (a B+5 A b)}{4 x^4}-\frac{5 a^3 b (a B+2 A b)}{x}+5 a^2 b^2 x^2 (a B+A b)+\frac{1}{8} b^4 x^8 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{11} b^5 B x^{11}","-\frac{a^5 A}{7 x^7}-\frac{a^4 (a B+5 A b)}{4 x^4}-\frac{5 a^3 b (a B+2 A b)}{x}+5 a^2 b^2 x^2 (a B+A b)+\frac{1}{8} b^4 x^8 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{11} b^5 B x^{11}",1,"-1/7*(a^5*A)/x^7 - (a^4*(5*A*b + a*B))/(4*x^4) - (5*a^3*b*(2*A*b + a*B))/x + 5*a^2*b^2*(A*b + a*B)*x^2 + a*b^3*(A*b + 2*a*B)*x^5 + (b^4*(A*b + 5*a*B)*x^8)/8 + (b^5*B*x^11)/11","A",1
41,1,113,113,0.0378825,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^9} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^9,x]","-\frac{a^5 A}{8 x^8}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{5 a^3 b (a B+2 A b)}{2 x^2}+10 a^2 b^2 x (a B+A b)+\frac{1}{7} b^4 x^7 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{10} b^5 B x^{10}","-\frac{a^5 A}{8 x^8}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{5 a^3 b (a B+2 A b)}{2 x^2}+10 a^2 b^2 x (a B+A b)+\frac{1}{7} b^4 x^7 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{10} b^5 B x^{10}",1,"-1/8*(a^5*A)/x^8 - (a^4*(5*A*b + a*B))/(5*x^5) - (5*a^3*b*(2*A*b + a*B))/(2*x^2) + 10*a^2*b^2*(A*b + a*B)*x + (5*a*b^3*(A*b + 2*a*B)*x^4)/4 + (b^4*(A*b + 5*a*B)*x^7)/7 + (b^5*B*x^10)/10","A",1
42,1,106,114,0.055305,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{10}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^10,x]","\frac{1}{18} \left(-\frac{2 a^5 A}{x^9}-\frac{3 a^4 (a B+5 A b)}{x^6}-\frac{30 a^3 b (a B+2 A b)}{x^3}+180 a^2 b^2 \log (x) (a B+A b)+3 b^4 x^6 (5 a B+A b)+30 a b^3 x^3 (2 a B+A b)+2 b^5 B x^9\right)","-\frac{a^5 A}{9 x^9}-\frac{a^4 (a B+5 A b)}{6 x^6}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}+10 a^2 b^2 \log (x) (a B+A b)+\frac{1}{6} b^4 x^6 (5 a B+A b)+\frac{5}{3} a b^3 x^3 (2 a B+A b)+\frac{1}{9} b^5 B x^9",1,"((-2*a^5*A)/x^9 - (3*a^4*(5*A*b + a*B))/x^6 - (30*a^3*b*(2*A*b + a*B))/x^3 + 30*a*b^3*(A*b + 2*a*B)*x^3 + 3*b^4*(A*b + 5*a*B)*x^6 + 2*b^5*B*x^9 + 180*a^2*b^2*(A*b + a*B)*Log[x])/18","A",1
43,1,118,115,0.0244969,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{11}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^11,x]","\frac{-4 a^5 \left(7 A+10 B x^3\right)-50 a^4 b x^3 \left(4 A+7 B x^3\right)-700 a^3 b^2 x^6 \left(A+4 B x^3\right)+1400 a^2 b^3 x^9 \left(B x^3-2 A\right)+140 a b^4 x^{12} \left(5 A+2 B x^3\right)+7 b^5 x^{15} \left(8 A+5 B x^3\right)}{280 x^{10}}","-\frac{a^5 A}{10 x^{10}}-\frac{a^4 (a B+5 A b)}{7 x^7}-\frac{5 a^3 b (a B+2 A b)}{4 x^4}-\frac{10 a^2 b^2 (a B+A b)}{x}+\frac{1}{5} b^4 x^5 (5 a B+A b)+\frac{5}{2} a b^3 x^2 (2 a B+A b)+\frac{1}{8} b^5 B x^8",1,"(1400*a^2*b^3*x^9*(-2*A + B*x^3) + 140*a*b^4*x^12*(5*A + 2*B*x^3) - 700*a^3*b^2*x^6*(A + 4*B*x^3) + 7*b^5*x^15*(8*A + 5*B*x^3) - 50*a^4*b*x^3*(4*A + 7*B*x^3) - 4*a^5*(7*A + 10*B*x^3))/(280*x^10)","A",1
44,1,109,109,0.0416366,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{12}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^12,x]","-\frac{a^5 A}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{a^3 b (a B+2 A b)}{x^5}-\frac{5 a^2 b^2 (a B+A b)}{x^2}+\frac{1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{7} b^5 B x^7","-\frac{a^5 A}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{a^3 b (a B+2 A b)}{x^5}-\frac{5 a^2 b^2 (a B+A b)}{x^2}+\frac{1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{7} b^5 B x^7",1,"-1/11*(a^5*A)/x^11 - (a^4*(5*A*b + a*B))/(8*x^8) - (a^3*b*(2*A*b + a*B))/x^5 - (5*a^2*b^2*(A*b + a*B))/x^2 + 5*a*b^3*(A*b + 2*a*B)*x + (b^4*(A*b + 5*a*B)*x^4)/4 + (b^5*B*x^7)/7","A",1
45,1,118,114,0.0478399,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{13}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^13,x]","-\frac{a^5 \left(3 A+4 B x^3\right)+10 a^4 b x^3 \left(2 A+3 B x^3\right)+60 a^3 b^2 x^6 \left(A+2 B x^3\right)+120 a^2 A b^3 x^9-180 a b^3 x^{12} \log (x) (2 a B+A b)-60 a b^4 B x^{15}-6 b^5 x^{15} \left(2 A+B x^3\right)}{36 x^{12}}","-\frac{a^5 A}{12 x^{12}}-\frac{a^4 (a B+5 A b)}{9 x^9}-\frac{5 a^3 b (a B+2 A b)}{6 x^6}-\frac{10 a^2 b^2 (a B+A b)}{3 x^3}+\frac{1}{3} b^4 x^3 (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac{1}{6} b^5 B x^6",1,"-1/36*(120*a^2*A*b^3*x^9 - 60*a*b^4*B*x^15 - 6*b^5*x^15*(2*A + B*x^3) + 60*a^3*b^2*x^6*(A + 2*B*x^3) + 10*a^4*b*x^3*(2*A + 3*B*x^3) + a^5*(3*A + 4*B*x^3) - 180*a*b^3*(A*b + 2*a*B)*x^12*Log[x])/x^12","A",1
46,1,117,115,0.0512465,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{14}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^14,x]","-\frac{a^5 \left(70 A+91 B x^3\right)+65 a^4 b x^3 \left(7 A+10 B x^3\right)+325 a^3 b^2 x^6 \left(4 A+7 B x^3\right)+2275 a^2 b^3 x^9 \left(A+4 B x^3\right)-2275 a b^4 x^{12} \left(B x^3-2 A\right)-91 b^5 x^{15} \left(5 A+2 B x^3\right)}{910 x^{13}}","-\frac{a^5 A}{13 x^{13}}-\frac{a^4 (a B+5 A b)}{10 x^{10}}-\frac{5 a^3 b (a B+2 A b)}{7 x^7}-\frac{5 a^2 b^2 (a B+A b)}{2 x^4}+\frac{1}{2} b^4 x^2 (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{x}+\frac{1}{5} b^5 B x^5",1,"-1/910*(-2275*a*b^4*x^12*(-2*A + B*x^3) - 91*b^5*x^15*(5*A + 2*B*x^3) + 2275*a^2*b^3*x^9*(A + 4*B*x^3) + 325*a^3*b^2*x^6*(4*A + 7*B*x^3) + 65*a^4*b*x^3*(7*A + 10*B*x^3) + a^5*(70*A + 91*B*x^3))/x^13","A",1
47,1,110,110,0.0707222,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{15}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^15,x]","-\frac{a^5 A}{14 x^{14}}-\frac{a^4 (a B+5 A b)}{11 x^{11}}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{2 a^2 b^2 (a B+A b)}{x^5}+b^4 x (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+\frac{1}{4} b^5 B x^4","-\frac{a^5 A}{14 x^{14}}-\frac{a^4 (a B+5 A b)}{11 x^{11}}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{2 a^2 b^2 (a B+A b)}{x^5}+b^4 x (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+\frac{1}{4} b^5 B x^4",1,"-1/14*(a^5*A)/x^14 - (a^4*(5*A*b + a*B))/(11*x^11) - (5*a^3*b*(2*A*b + a*B))/(8*x^8) - (2*a^2*b^2*(A*b + a*B))/x^5 - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + b^4*(A*b + 5*a*B)*x + (b^5*B*x^4)/4","A",1
48,1,116,113,0.0675558,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{16}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^16,x]","b^4 \log (x) (5 a B+A b)-\frac{3 a^5 \left(4 A+5 B x^3\right)+25 a^4 b x^3 \left(3 A+4 B x^3\right)+100 a^3 b^2 x^6 \left(2 A+3 B x^3\right)+300 a^2 b^3 x^9 \left(A+2 B x^3\right)+300 a A b^4 x^{12}-60 b^5 B x^{18}}{180 x^{15}}","-\frac{a^5 A}{15 x^{15}}-\frac{a^4 (a B+5 A b)}{12 x^{12}}-\frac{5 a^3 b (a B+2 A b)}{9 x^9}-\frac{5 a^2 b^2 (a B+A b)}{3 x^6}+b^4 \log (x) (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{3 x^3}+\frac{1}{3} b^5 B x^3",1,"-1/180*(300*a*A*b^4*x^12 - 60*b^5*B*x^18 + 300*a^2*b^3*x^9*(A + 2*B*x^3) + 100*a^3*b^2*x^6*(2*A + 3*B*x^3) + 25*a^4*b*x^3*(3*A + 4*B*x^3) + 3*a^5*(4*A + 5*B*x^3))/x^15 + b^4*(A*b + 5*a*B)*Log[x]","A",1
49,1,118,115,0.032942,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{17}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^17,x]","-\frac{7 a^5 \left(13 A+16 B x^3\right)+56 a^4 b x^3 \left(10 A+13 B x^3\right)+208 a^3 b^2 x^6 \left(7 A+10 B x^3\right)+520 a^2 b^3 x^9 \left(4 A+7 B x^3\right)+1820 a b^4 x^{12} \left(A+4 B x^3\right)-728 b^5 x^{15} \left(B x^3-2 A\right)}{1456 x^{16}}","-\frac{a^5 A}{16 x^{16}}-\frac{a^4 (a B+5 A b)}{13 x^{13}}-\frac{a^3 b (a B+2 A b)}{2 x^{10}}-\frac{10 a^2 b^2 (a B+A b)}{7 x^7}-\frac{b^4 (5 a B+A b)}{x}-\frac{5 a b^3 (2 a B+A b)}{4 x^4}+\frac{1}{2} b^5 B x^2",1,"-1/1456*(-728*b^5*x^15*(-2*A + B*x^3) + 1820*a*b^4*x^12*(A + 4*B*x^3) + 520*a^2*b^3*x^9*(4*A + 7*B*x^3) + 208*a^3*b^2*x^6*(7*A + 10*B*x^3) + 56*a^4*b*x^3*(10*A + 13*B*x^3) + 7*a^5*(13*A + 16*B*x^3))/x^16","A",1
50,1,110,110,0.0533265,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{18}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^18,x]","-\frac{a^5 A}{17 x^{17}}-\frac{a^4 (a B+5 A b)}{14 x^{14}}-\frac{5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac{5 a^2 b^2 (a B+A b)}{4 x^8}-\frac{b^4 (5 a B+A b)}{2 x^2}-\frac{a b^3 (2 a B+A b)}{x^5}+b^5 B x","-\frac{a^5 A}{17 x^{17}}-\frac{a^4 (a B+5 A b)}{14 x^{14}}-\frac{5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac{5 a^2 b^2 (a B+A b)}{4 x^8}-\frac{b^4 (5 a B+A b)}{2 x^2}-\frac{a b^3 (2 a B+A b)}{x^5}+b^5 B x",1,"-1/17*(a^5*A)/x^17 - (a^4*(5*A*b + a*B))/(14*x^14) - (5*a^3*b*(2*A*b + a*B))/(11*x^11) - (5*a^2*b^2*(A*b + a*B))/(4*x^8) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b + 5*a*B))/(2*x^2) + b^5*B*x","A",1
51,1,121,91,0.0454242,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{19}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^19,x]","-\frac{2 a^5 \left(5 A+6 B x^3\right)+15 a^4 b x^3 \left(4 A+5 B x^3\right)+50 a^3 b^2 x^6 \left(3 A+4 B x^3\right)+100 a^2 b^3 x^9 \left(2 A+3 B x^3\right)+150 a b^4 x^{12} \left(A+2 B x^3\right)+60 A b^5 x^{15}-180 b^5 B x^{18} \log (x)}{180 x^{18}}","-\frac{a^5 B}{15 x^{15}}-\frac{5 a^4 b B}{12 x^{12}}-\frac{10 a^3 b^2 B}{9 x^9}-\frac{5 a^2 b^3 B}{3 x^6}-\frac{A \left(a+b x^3\right)^6}{18 a x^{18}}-\frac{5 a b^4 B}{3 x^3}+b^5 B \log (x)",1,"-1/180*(60*A*b^5*x^15 + 150*a*b^4*x^12*(A + 2*B*x^3) + 100*a^2*b^3*x^9*(2*A + 3*B*x^3) + 50*a^3*b^2*x^6*(3*A + 4*B*x^3) + 15*a^4*b*x^3*(4*A + 5*B*x^3) + 2*a^5*(5*A + 6*B*x^3) - 180*b^5*B*x^18*Log[x])/x^18","A",1
52,1,119,113,0.0321639,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{20}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^20,x]","-\frac{91 a^5 \left(16 A+19 B x^3\right)+665 a^4 b x^3 \left(13 A+16 B x^3\right)+2128 a^3 b^2 x^6 \left(10 A+13 B x^3\right)+3952 a^2 b^3 x^9 \left(7 A+10 B x^3\right)+4940 a b^4 x^{12} \left(4 A+7 B x^3\right)+6916 b^5 x^{15} \left(A+4 B x^3\right)}{27664 x^{19}}","-\frac{a^5 A}{19 x^{19}}-\frac{a^4 (a B+5 A b)}{16 x^{16}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{a^2 b^2 (a B+A b)}{x^{10}}-\frac{b^4 (5 a B+A b)}{4 x^4}-\frac{5 a b^3 (2 a B+A b)}{7 x^7}-\frac{b^5 B}{x}",1,"-1/27664*(6916*b^5*x^15*(A + 4*B*x^3) + 4940*a*b^4*x^12*(4*A + 7*B*x^3) + 3952*a^2*b^3*x^9*(7*A + 10*B*x^3) + 2128*a^3*b^2*x^6*(10*A + 13*B*x^3) + 665*a^4*b*x^3*(13*A + 16*B*x^3) + 91*a^5*(16*A + 19*B*x^3))/x^19","A",1
53,1,121,117,0.0366192,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{21}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^21,x]","-\frac{154 a^5 \left(17 A+20 B x^3\right)+1100 a^4 b x^3 \left(14 A+17 B x^3\right)+3400 a^3 b^2 x^6 \left(11 A+14 B x^3\right)+5950 a^2 b^3 x^9 \left(8 A+11 B x^3\right)+6545 a b^4 x^{12} \left(5 A+8 B x^3\right)+5236 b^5 x^{15} \left(2 A+5 B x^3\right)}{52360 x^{20}}","-\frac{a^5 A}{20 x^{20}}-\frac{a^4 (a B+5 A b)}{17 x^{17}}-\frac{5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{5 a b^3 (2 a B+A b)}{8 x^8}-\frac{b^5 B}{2 x^2}",1,"-1/52360*(5236*b^5*x^15*(2*A + 5*B*x^3) + 6545*a*b^4*x^12*(5*A + 8*B*x^3) + 5950*a^2*b^3*x^9*(8*A + 11*B*x^3) + 3400*a^3*b^2*x^6*(11*A + 14*B*x^3) + 1100*a^4*b*x^3*(14*A + 17*B*x^3) + 154*a^5*(17*A + 20*B*x^3))/x^20","A",1
54,1,118,48,0.0360891,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{22}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^22,x]","-\frac{a^5 \left(6 A+7 B x^3\right)+7 a^4 b x^3 \left(5 A+6 B x^3\right)+21 a^3 b^2 x^6 \left(4 A+5 B x^3\right)+35 a^2 b^3 x^9 \left(3 A+4 B x^3\right)+35 a b^4 x^{12} \left(2 A+3 B x^3\right)+21 b^5 x^{15} \left(A+2 B x^3\right)}{126 x^{21}}","\frac{\left(a+b x^3\right)^6 (A b-7 a B)}{126 a^2 x^{18}}-\frac{A \left(a+b x^3\right)^6}{21 a x^{21}}",1,"-1/126*(21*b^5*x^15*(A + 2*B*x^3) + 35*a*b^4*x^12*(2*A + 3*B*x^3) + 35*a^2*b^3*x^9*(3*A + 4*B*x^3) + 21*a^3*b^2*x^6*(4*A + 5*B*x^3) + 7*a^4*b*x^3*(5*A + 6*B*x^3) + a^5*(6*A + 7*B*x^3))/x^21","B",1
55,1,117,117,0.0495531,"\int \frac{\left(a+b x^3\right)^5 \left(A+B x^3\right)}{x^{23}} \, dx","Integrate[((a + b*x^3)^5*(A + B*x^3))/x^23,x]","-\frac{a^5 A}{22 x^{22}}-\frac{a^4 (a B+5 A b)}{19 x^{19}}-\frac{5 a^3 b (a B+2 A b)}{16 x^{16}}-\frac{10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{a b^3 (2 a B+A b)}{2 x^{10}}-\frac{b^5 B}{4 x^4}","-\frac{a^5 A}{22 x^{22}}-\frac{a^4 (a B+5 A b)}{19 x^{19}}-\frac{5 a^3 b (a B+2 A b)}{16 x^{16}}-\frac{10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{a b^3 (2 a B+A b)}{2 x^{10}}-\frac{b^5 B}{4 x^4}",1,"-1/22*(a^5*A)/x^22 - (a^4*(5*A*b + a*B))/(19*x^19) - (5*a^3*b*(2*A*b + a*B))/(16*x^16) - (10*a^2*b^2*(A*b + a*B))/(13*x^13) - (a*b^3*(A*b + 2*a*B))/(2*x^10) - (b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(4*x^4)","A",1
56,1,171,183,0.1348508,"\int \frac{x^6 \left(A+B x^3\right)}{a+b x^3} \, dx","Integrate[(x^6*(A + B*x^3))/(a + b*x^3),x]","\frac{14 a^{4/3} (a B-A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-28 a^{4/3} (a B-A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+28 \sqrt{3} a^{4/3} (a B-A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+21 b^{4/3} x^4 (A b-a B)+84 a \sqrt[3]{b} x (a B-A b)+12 b^{7/3} B x^7}{84 b^{10/3}}","-\frac{a^{4/3} (A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 b^{10/3}}+\frac{a^{4/3} (A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 b^{10/3}}-\frac{a^{4/3} (A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^{10/3}}-\frac{a x (A b-a B)}{b^3}+\frac{x^4 (A b-a B)}{4 b^2}+\frac{B x^7}{7 b}",1,"(84*a*b^(1/3)*(-(A*b) + a*B)*x + 21*b^(4/3)*(A*b - a*B)*x^4 + 12*b^(7/3)*B*x^7 + 28*Sqrt[3]*a^(4/3)*(-(A*b) + a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 28*a^(4/3)*(-(A*b) + a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*a^(4/3)*(-(A*b) + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(84*b^(10/3))","A",1
57,1,47,54,0.0221154,"\int \frac{x^5 \left(A+B x^3\right)}{a+b x^3} \, dx","Integrate[(x^5*(A + B*x^3))/(a + b*x^3),x]","\frac{b x^3 \left(-2 a B+2 A b+b B x^3\right)+2 a (a B-A b) \log \left(a+b x^3\right)}{6 b^3}","-\frac{a (A b-a B) \log \left(a+b x^3\right)}{3 b^3}+\frac{x^3 (A b-a B)}{3 b^2}+\frac{B x^6}{6 b}",1,"(b*x^3*(2*A*b - 2*a*B + b*B*x^3) + 2*a*(-(A*b) + a*B)*Log[a + b*x^3])/(6*b^3)","A",1
58,1,154,167,0.0925644,"\int \frac{x^4 \left(A+B x^3\right)}{a+b x^3} \, dx","Integrate[(x^4*(A + B*x^3))/(a + b*x^3),x]","\frac{5 a^{2/3} (a B-A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-10 a^{2/3} (a B-A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-10 \sqrt{3} a^{2/3} (a B-A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+15 b^{2/3} x^2 (A b-a B)+6 b^{5/3} B x^5}{30 b^{8/3}}","-\frac{a^{2/3} (A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 b^{8/3}}+\frac{a^{2/3} (A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 b^{8/3}}+\frac{a^{2/3} (A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^{8/3}}+\frac{x^2 (A b-a B)}{2 b^2}+\frac{B x^5}{5 b}",1,"(15*b^(2/3)*(A*b - a*B)*x^2 + 6*b^(5/3)*B*x^5 - 10*Sqrt[3]*a^(2/3)*(-(A*b) + a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 10*a^(2/3)*(-(A*b) + a*B)*Log[a^(1/3) + b^(1/3)*x] + 5*a^(2/3)*(-(A*b) + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(30*b^(8/3))","A",1
59,1,152,162,0.0926734,"\int \frac{x^3 \left(A+B x^3\right)}{a+b x^3} \, dx","Integrate[(x^3*(A + B*x^3))/(a + b*x^3),x]","\frac{-2 \sqrt[3]{a} (a B-A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+12 \sqrt[3]{b} x (A b-a B)+4 \sqrt[3]{a} (a B-A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-4 \sqrt{3} \sqrt[3]{a} (a B-A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+3 b^{4/3} B x^4}{12 b^{7/3}}","\frac{\sqrt[3]{a} (A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 b^{7/3}}-\frac{\sqrt[3]{a} (A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 b^{7/3}}+\frac{\sqrt[3]{a} (A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^{7/3}}+\frac{x (A b-a B)}{b^2}+\frac{B x^4}{4 b}",1,"(12*b^(1/3)*(A*b - a*B)*x + 3*b^(4/3)*B*x^4 - 4*Sqrt[3]*a^(1/3)*(-(A*b) + a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 4*a^(1/3)*(-(A*b) + a*B)*Log[a^(1/3) + b^(1/3)*x] - 2*a^(1/3)*(-(A*b) + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(12*b^(7/3))","A",1
60,1,31,35,0.0289822,"\int \frac{x^2 \left(A+B x^3\right)}{a+b x^3} \, dx","Integrate[(x^2*(A + B*x^3))/(a + b*x^3),x]","\frac{(A b-a B) \log \left(a+b x^3\right)+b B x^3}{3 b^2}","\frac{(A b-a B) \log \left(a+b x^3\right)}{3 b^2}+\frac{B x^3}{3 b}",1,"(b*B*x^3 + (A*b - a*B)*Log[a + b*x^3])/(3*b^2)","A",1
61,1,152,150,0.0566019,"\int \frac{x \left(A+B x^3\right)}{a+b x^3} \, dx","Integrate[(x*(A + B*x^3))/(a + b*x^3),x]","-\frac{(a B-A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 \sqrt[3]{a} b^{5/3}}+\frac{(a B-A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 \sqrt[3]{a} b^{5/3}}-\frac{(a B-A b) \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} b^{5/3}}+\frac{B x^2}{2 b}","\frac{(A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 \sqrt[3]{a} b^{5/3}}-\frac{(A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 \sqrt[3]{a} b^{5/3}}-\frac{(A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} b^{5/3}}+\frac{B x^2}{2 b}",1,"(B*x^2)/(2*b) - ((-(A*b) + a*B)*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*b^(5/3)) + ((-(A*b) + a*B)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(1/3)*b^(5/3)) - ((-(A*b) + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(1/3)*b^(5/3))","A",1
62,1,129,145,0.0890263,"\int \frac{A+B x^3}{a+b x^3} \, dx","Integrate[(A + B*x^3)/(a + b*x^3),x]","\frac{-(A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+6 a^{2/3} \sqrt[3]{b} B x+2 (A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} (A b-a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{6 a^{2/3} b^{4/3}}","-\frac{(A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} b^{4/3}}+\frac{(A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} b^{4/3}}-\frac{(A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} b^{4/3}}+\frac{B x}{b}",1,"(6*a^(2/3)*b^(1/3)*B*x - 2*Sqrt[3]*(A*b - a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x] - (A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(4/3))","A",1
63,1,34,34,0.0140435,"\int \frac{A+B x^3}{x \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(x*(a + b*x^3)),x]","\frac{(a B-A b) \log \left(a+b x^3\right)}{3 a b}+\frac{A \log (x)}{a}","\frac{A \log (x)}{a}-\frac{(A b-a B) \log \left(a+b x^3\right)}{3 a b}",1,"(A*Log[x])/a + ((-(A*b) + a*B)*Log[a + b*x^3])/(3*a*b)","A",1
64,1,134,147,0.094378,"\int \frac{A+B x^3}{x^2 \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(x^2*(a + b*x^3)),x]","\frac{-x (A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-6 \sqrt[3]{a} A b^{2/3}+2 x (A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+2 \sqrt{3} x (A b-a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{6 a^{4/3} b^{2/3} x}","-\frac{(A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{4/3} b^{2/3}}+\frac{(A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{4/3} b^{2/3}}+\frac{(A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{4/3} b^{2/3}}-\frac{A}{a x}",1,"(-6*a^(1/3)*A*b^(2/3) + 2*Sqrt[3]*(A*b - a*B)*x*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(A*b - a*B)*x*Log[a^(1/3) + b^(1/3)*x] - (A*b - a*B)*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(4/3)*b^(2/3)*x)","A",1
65,1,135,149,0.1032964,"\int \frac{A+B x^3}{x^3 \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(x^3*(a + b*x^3)),x]","\frac{\frac{(A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{b}}-\frac{3 a^{2/3} A}{x^2}+\frac{2 (a B-A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}+\frac{2 \sqrt{3} (A b-a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt[3]{b}}}{6 a^{5/3}}","\frac{(A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{5/3} \sqrt[3]{b}}-\frac{(A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{5/3} \sqrt[3]{b}}+\frac{(A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{5/3} \sqrt[3]{b}}-\frac{A}{2 a x^2}",1,"((-3*a^(2/3)*A)/x^2 + (2*Sqrt[3]*(A*b - a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (2*(-(A*b) + a*B)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + ((A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3))/(6*a^(5/3))","A",1
66,1,49,50,0.022368,"\int \frac{A+B x^3}{x^4 \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(x^4*(a + b*x^3)),x]","\frac{(A b-a B) \log \left(a+b x^3\right)}{3 a^2}+\frac{\log (x) (a B-A b)}{a^2}-\frac{A}{3 a x^3}","\frac{(A b-a B) \log \left(a+b x^3\right)}{3 a^2}-\frac{\log (x) (A b-a B)}{a^2}-\frac{A}{3 a x^3}",1,"-1/3*A/(a*x^3) + ((-(A*b) + a*B)*Log[x])/a^2 + ((A*b - a*B)*Log[a + b*x^3])/(3*a^2)","A",1
67,1,154,165,0.1409071,"\int \frac{A+B x^3}{x^5 \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(x^5*(a + b*x^3)),x]","\frac{2 \sqrt[3]{b} (A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-\frac{3 a^{4/3} A}{x^4}+\frac{12 \sqrt[3]{a} (A b-a B)}{x}+4 \sqrt[3]{b} (a B-A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-4 \sqrt{3} \sqrt[3]{b} (A b-a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{12 a^{7/3}}","\frac{\sqrt[3]{b} (A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{7/3}}-\frac{\sqrt[3]{b} (A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{7/3}}-\frac{\sqrt[3]{b} (A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{7/3}}+\frac{A b-a B}{a^2 x}-\frac{A}{4 a x^4}",1,"((-3*a^(4/3)*A)/x^4 + (12*a^(1/3)*(A*b - a*B))/x - 4*Sqrt[3]*b^(1/3)*(A*b - a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 4*b^(1/3)*(-(A*b) + a*B)*Log[a^(1/3) + b^(1/3)*x] + 2*b^(1/3)*(A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(12*a^(7/3))","A",1
68,1,154,168,0.1343366,"\int \frac{A+B x^3}{x^6 \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(x^6*(a + b*x^3)),x]","\frac{5 b^{2/3} (a B-A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+\frac{15 a^{2/3} (A b-a B)}{x^2}-\frac{6 a^{5/3} A}{x^5}+10 b^{2/3} (A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-10 \sqrt{3} b^{2/3} (A b-a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{30 a^{8/3}}","-\frac{b^{2/3} (A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{8/3}}+\frac{b^{2/3} (A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{8/3}}-\frac{b^{2/3} (A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{8/3}}+\frac{A b-a B}{2 a^2 x^2}-\frac{A}{5 a x^5}",1,"((-6*a^(5/3)*A)/x^5 + (15*a^(2/3)*(A*b - a*B))/x^2 - 10*Sqrt[3]*b^(2/3)*(A*b - a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 10*b^(2/3)*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x] + 5*b^(2/3)*(-(A*b) + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(30*a^(8/3))","A",1
69,1,70,69,0.0329784,"\int \frac{A+B x^3}{x^7 \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(x^7*(a + b*x^3)),x]","\frac{6 b x^6 \log (x) (A b-a B)-a \left(a A+2 a B x^3-2 A b x^3\right)+2 b x^6 (a B-A b) \log \left(a+b x^3\right)}{6 a^3 x^6}","-\frac{b (A b-a B) \log \left(a+b x^3\right)}{3 a^3}+\frac{b \log (x) (A b-a B)}{a^3}+\frac{A b-a B}{3 a^2 x^3}-\frac{A}{6 a x^6}",1,"(-(a*(a*A - 2*A*b*x^3 + 2*a*B*x^3)) + 6*b*(A*b - a*B)*x^6*Log[x] + 2*b*(-(A*b) + a*B)*x^6*Log[a + b*x^3])/(6*a^3*x^6)","A",1
70,1,173,184,0.1440448,"\int \frac{A+B x^3}{x^8 \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(x^8*(a + b*x^3)),x]","\frac{14 b^{4/3} (a B-A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+\frac{21 a^{4/3} (A b-a B)}{x^4}-\frac{12 a^{7/3} A}{x^7}+28 b^{4/3} (A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+28 \sqrt{3} b^{4/3} (A b-a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+\frac{84 \sqrt[3]{a} b (a B-A b)}{x}}{84 a^{10/3}}","-\frac{b^{4/3} (A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{10/3}}+\frac{b^{4/3} (A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{10/3}}+\frac{b^{4/3} (A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{10/3}}-\frac{b (A b-a B)}{a^3 x}+\frac{A b-a B}{4 a^2 x^4}-\frac{A}{7 a x^7}",1,"((-12*a^(7/3)*A)/x^7 + (21*a^(4/3)*(A*b - a*B))/x^4 + (84*a^(1/3)*b*(-(A*b) + a*B))/x + 28*Sqrt[3]*b^(4/3)*(A*b - a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 28*b^(4/3)*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*b^(4/3)*(-(A*b) + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(84*a^(10/3))","A",1
71,1,203,233,0.1692255,"\int \frac{x^9 \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^9*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{14 a^{4/3} (10 a B-7 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-28 a^{4/3} (10 a B-7 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+28 \sqrt{3} a^{4/3} (10 a B-7 A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+\frac{84 a^2 \sqrt[3]{b} x (a B-A b)}{a+b x^3}+63 b^{4/3} x^4 (A b-2 a B)+252 a \sqrt[3]{b} x (3 a B-2 A b)+36 b^{7/3} B x^7}{252 b^{13/3}}","-\frac{a^{4/3} (7 A b-10 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 b^{13/3}}+\frac{a^{4/3} (7 A b-10 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 b^{13/3}}-\frac{a^{4/3} (7 A b-10 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} b^{13/3}}-\frac{a x (7 A b-10 a B)}{3 b^4}+\frac{x^4 (7 A b-10 a B)}{12 b^3}-\frac{x^7 (7 A b-10 a B)}{21 a b^2}+\frac{x^{10} (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"(252*a*b^(1/3)*(-2*A*b + 3*a*B)*x + 63*b^(4/3)*(A*b - 2*a*B)*x^4 + 36*b^(7/3)*B*x^7 + (84*a^2*b^(1/3)*(-(A*b) + a*B)*x)/(a + b*x^3) + 28*Sqrt[3]*a^(4/3)*(-7*A*b + 10*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 28*a^(4/3)*(-7*A*b + 10*a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*a^(4/3)*(-7*A*b + 10*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(252*b^(13/3))","A",1
72,1,72,82,0.1013124,"\int \frac{x^8 \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^8*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{\frac{2 a^2 (a B-A b)}{a+b x^3}+2 b x^3 (A b-2 a B)+2 a (3 a B-2 A b) \log \left(a+b x^3\right)+b^2 B x^6}{6 b^4}","-\frac{a^2 (A b-a B)}{3 b^4 \left(a+b x^3\right)}-\frac{a (2 A b-3 a B) \log \left(a+b x^3\right)}{3 b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^6}{6 b^2}",1,"(2*b*(A*b - 2*a*B)*x^3 + b^2*B*x^6 + (2*a^2*(-(A*b) + a*B))/(a + b*x^3) + 2*a*(-2*A*b + 3*a*B)*Log[a + b*x^3])/(6*b^4)","A",1
73,1,185,215,0.1687807,"\int \frac{x^7 \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^7*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{5 a^{2/3} (8 a B-5 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-10 a^{2/3} (8 a B-5 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-10 \sqrt{3} a^{2/3} (8 a B-5 A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+45 b^{2/3} x^2 (A b-2 a B)+\frac{30 a b^{2/3} x^2 (A b-a B)}{a+b x^3}+18 b^{5/3} B x^5}{90 b^{11/3}}","-\frac{a^{2/3} (5 A b-8 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} b^{11/3}}+\frac{x^2 (5 A b-8 a B)}{6 b^3}-\frac{x^5 (5 A b-8 a B)}{15 a b^2}+\frac{x^8 (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"(45*b^(2/3)*(A*b - 2*a*B)*x^2 + 18*b^(5/3)*B*x^5 + (30*a*b^(2/3)*(A*b - a*B)*x^2)/(a + b*x^3) - 10*Sqrt[3]*a^(2/3)*(-5*A*b + 8*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 10*a^(2/3)*(-5*A*b + 8*a*B)*Log[a^(1/3) + b^(1/3)*x] + 5*a^(2/3)*(-5*A*b + 8*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(90*b^(11/3))","A",1
74,1,181,213,0.1496415,"\int \frac{x^6 \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^6*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{-2 \sqrt[3]{a} (7 a B-4 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+\frac{12 a \sqrt[3]{b} x (A b-a B)}{a+b x^3}+36 \sqrt[3]{b} x (A b-2 a B)+4 \sqrt[3]{a} (7 a B-4 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-4 \sqrt{3} \sqrt[3]{a} (7 a B-4 A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+9 b^{4/3} B x^4}{36 b^{10/3}}","\frac{\sqrt[3]{a} (4 A b-7 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 b^{10/3}}-\frac{\sqrt[3]{a} (4 A b-7 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 b^{10/3}}+\frac{\sqrt[3]{a} (4 A b-7 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} b^{10/3}}+\frac{x (4 A b-7 a B)}{3 b^3}-\frac{x^4 (4 A b-7 a B)}{12 a b^2}+\frac{x^7 (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"(36*b^(1/3)*(A*b - 2*a*B)*x + 9*b^(4/3)*B*x^4 + (12*a*b^(1/3)*(A*b - a*B)*x)/(a + b*x^3) - 4*Sqrt[3]*a^(1/3)*(-4*A*b + 7*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 4*a^(1/3)*(-4*A*b + 7*a*B)*Log[a^(1/3) + b^(1/3)*x] - 2*a^(1/3)*(-4*A*b + 7*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(36*b^(10/3))","A",1
75,1,50,60,0.0629442,"\int \frac{x^5 \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^5*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{\frac{a (A b-a B)}{a+b x^3}+(A b-2 a B) \log \left(a+b x^3\right)+b B x^3}{3 b^3}","\frac{a (A b-a B)}{3 b^3 \left(a+b x^3\right)}+\frac{(A b-2 a B) \log \left(a+b x^3\right)}{3 b^3}+\frac{B x^3}{3 b^2}",1,"(b*B*x^3 + (a*(A*b - a*B))/(a + b*x^3) + (A*b - 2*a*B)*Log[a + b*x^3])/(3*b^3)","A",1
76,1,165,196,0.1398629,"\int \frac{x^4 \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^4*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{\frac{(2 A b-5 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{a}}-\frac{6 b^{2/3} x^2 (A b-a B)}{a+b x^3}+\frac{2 (5 a B-2 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a}}+\frac{2 \sqrt{3} (5 a B-2 A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt[3]{a}}+9 b^{2/3} B x^2}{18 b^{8/3}}","\frac{(2 A b-5 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 \sqrt[3]{a} b^{8/3}}-\frac{(2 A b-5 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 \sqrt[3]{a} b^{8/3}}-\frac{(2 A b-5 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} \sqrt[3]{a} b^{8/3}}-\frac{x^2 (2 A b-5 a B)}{6 a b^2}+\frac{x^5 (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"(9*b^(2/3)*B*x^2 - (6*b^(2/3)*(A*b - a*B)*x^2)/(a + b*x^3) + (2*Sqrt[3]*(-2*A*b + 5*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (2*(-2*A*b + 5*a*B)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + ((2*A*b - 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3))/(18*b^(8/3))","A",1
77,1,160,190,0.1699048,"\int \frac{x^3 \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^3*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{\frac{(4 a B-A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{2/3}}+\frac{2 (A b-4 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{2/3}}+\frac{2 \sqrt{3} (4 a B-A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{2/3}}-\frac{6 \sqrt[3]{b} x (A b-a B)}{a+b x^3}+18 \sqrt[3]{b} B x}{18 b^{7/3}}","-\frac{(A b-4 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{2/3} b^{7/3}}+\frac{(A b-4 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{2/3} b^{7/3}}-\frac{(A b-4 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{2/3} b^{7/3}}-\frac{x (A b-4 a B)}{3 a b^2}+\frac{x^4 (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"(18*b^(1/3)*B*x - (6*b^(1/3)*(A*b - a*B)*x)/(a + b*x^3) + (2*Sqrt[3]*(-(A*b) + 4*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*(A*b - 4*a*B)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + ((-(A*b) + 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(18*b^(7/3))","A",1
78,1,41,41,0.0319328,"\int \frac{x^2 \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^2*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{a B-A b}{3 b^2 \left(a+b x^3\right)}+\frac{B \log \left(a+b x^3\right)}{3 b^2}","\frac{a B-A b}{3 b^2 \left(a+b x^3\right)}+\frac{B \log \left(a+b x^3\right)}{3 b^2}",1,"(-(A*b) + a*B)/(3*b^2*(a + b*x^3)) + (B*Log[a + b*x^3])/(3*b^2)","A",1
79,1,146,171,0.1338846,"\int \frac{x \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{(2 a B+A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-\frac{6 \sqrt[3]{a} b^{2/3} x^2 (a B-A b)}{a+b x^3}-2 (2 a B+A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} (2 a B+A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{18 a^{4/3} b^{5/3}}","\frac{(2 a B+A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{4/3} b^{5/3}}-\frac{(2 a B+A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{4/3} b^{5/3}}-\frac{(2 a B+A b) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{4/3} b^{5/3}}+\frac{x^2 (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"((-6*a^(1/3)*b^(2/3)*(-(A*b) + a*B)*x^2)/(a + b*x^3) - 2*Sqrt[3]*(A*b + 2*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*(A*b + 2*a*B)*Log[a^(1/3) + b^(1/3)*x] + (A*b + 2*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(4/3)*b^(5/3))","A",1
80,1,145,169,0.1391325,"\int \frac{A+B x^3}{\left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(a + b*x^3)^2,x]","\frac{-(a B+2 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-\frac{6 a^{2/3} \sqrt[3]{b} x (a B-A b)}{a+b x^3}+2 (a B+2 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} (a B+2 A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{18 a^{5/3} b^{4/3}}","-\frac{(a B+2 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{5/3} b^{4/3}}+\frac{(a B+2 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} b^{4/3}}-\frac{(a B+2 A b) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} b^{4/3}}+\frac{x (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"((-6*a^(2/3)*b^(1/3)*(-(A*b) + a*B)*x)/(a + b*x^3) - 2*Sqrt[3]*(2*A*b + a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(2*A*b + a*B)*Log[a^(1/3) + b^(1/3)*x] - (2*A*b + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(5/3)*b^(4/3))","A",1
81,1,46,51,0.034267,"\int \frac{A+B x^3}{x \left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(x*(a + b*x^3)^2),x]","\frac{\frac{a (A b-a B)}{b \left(a+b x^3\right)}-A \log \left(a+b x^3\right)+3 A \log (x)}{3 a^2}","-\frac{A \log \left(a+b x^3\right)}{3 a^2}+\frac{A \log (x)}{a^2}+\frac{A b-a B}{3 a b \left(a+b x^3\right)}",1,"((a*(A*b - a*B))/(b*(a + b*x^3)) + 3*A*Log[x] - A*Log[a + b*x^3])/(3*a^2)","A",1
82,1,164,195,0.1436665,"\int \frac{A+B x^3}{x^2 \left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(x^2*(a + b*x^3)^2),x]","\frac{\frac{(a B-4 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+\frac{2 (4 A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}+\frac{2 \sqrt{3} (4 A b-a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{2/3}}+\frac{6 \sqrt[3]{a} x^2 (a B-A b)}{a+b x^3}-\frac{18 \sqrt[3]{a} A}{x}}{18 a^{7/3}}","-\frac{(4 A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{7/3} b^{2/3}}+\frac{(4 A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{7/3} b^{2/3}}+\frac{(4 A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{7/3} b^{2/3}}+\frac{a B-4 A b}{3 a^2 b x}+\frac{A b-a B}{3 a b x \left(a+b x^3\right)}",1,"((-18*a^(1/3)*A)/x + (6*a^(1/3)*(-(A*b) + a*B)*x^2)/(a + b*x^3) + (2*Sqrt[3]*(4*A*b - a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (2*(4*A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((-4*A*b + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(18*a^(7/3))","A",1
83,1,163,196,0.1556825,"\int \frac{A+B x^3}{x^3 \left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(x^3*(a + b*x^3)^2),x]","\frac{\frac{(5 A b-2 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{b}}+\frac{6 a^{2/3} x (a B-A b)}{a+b x^3}-\frac{9 a^{2/3} A}{x^2}+\frac{2 (2 a B-5 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}+\frac{2 \sqrt{3} (5 A b-2 a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt[3]{b}}}{18 a^{8/3}}","\frac{(5 A b-2 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{8/3} \sqrt[3]{b}}-\frac{(5 A b-2 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{8/3} \sqrt[3]{b}}+\frac{(5 A b-2 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{8/3} \sqrt[3]{b}}+\frac{2 a B-5 A b}{6 a^2 b x^2}+\frac{A b-a B}{3 a b x^2 \left(a+b x^3\right)}",1,"((-9*a^(2/3)*A)/x^2 + (6*a^(2/3)*(-(A*b) + a*B)*x)/(a + b*x^3) + (2*Sqrt[3]*(5*A*b - 2*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (2*(-5*A*b + 2*a*B)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + ((5*A*b - 2*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3))/(18*a^(8/3))","A",1
84,1,64,76,0.0720101,"\int \frac{A+B x^3}{x^4 \left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(x^4*(a + b*x^3)^2),x]","\frac{\frac{a (a B-A b)}{a+b x^3}+(2 A b-a B) \log \left(a+b x^3\right)+3 \log (x) (a B-2 A b)-\frac{a A}{x^3}}{3 a^3}","\frac{(2 A b-a B) \log \left(a+b x^3\right)}{3 a^3}-\frac{\log (x) (2 A b-a B)}{a^3}-\frac{A b-a B}{3 a^2 \left(a+b x^3\right)}-\frac{A}{3 a^2 x^3}",1,"(-((a*A)/x^3) + (a*(-(A*b) + a*B))/(a + b*x^3) + 3*(-2*A*b + a*B)*Log[x] + (2*A*b - a*B)*Log[a + b*x^3])/(3*a^3)","A",1
85,1,185,215,0.2379186,"\int \frac{A+B x^3}{x^5 \left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(x^5*(a + b*x^3)^2),x]","\frac{2 \sqrt[3]{b} (7 A b-4 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-\frac{9 a^{4/3} A}{x^4}-\frac{12 \sqrt[3]{a} b x^2 (a B-A b)}{a+b x^3}-\frac{36 \sqrt[3]{a} (a B-2 A b)}{x}+4 \sqrt[3]{b} (4 a B-7 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-4 \sqrt{3} \sqrt[3]{b} (7 A b-4 a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{36 a^{10/3}}","\frac{\sqrt[3]{b} (7 A b-4 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{10/3}}-\frac{\sqrt[3]{b} (7 A b-4 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{10/3}}-\frac{\sqrt[3]{b} (7 A b-4 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{10/3}}+\frac{7 A b-4 a B}{3 a^3 x}+\frac{4 a B-7 A b}{12 a^2 b x^4}+\frac{A b-a B}{3 a b x^4 \left(a+b x^3\right)}",1,"((-9*a^(4/3)*A)/x^4 - (36*a^(1/3)*(-2*A*b + a*B))/x - (12*a^(1/3)*b*(-(A*b) + a*B)*x^2)/(a + b*x^3) - 4*Sqrt[3]*b^(1/3)*(7*A*b - 4*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 4*b^(1/3)*(-7*A*b + 4*a*B)*Log[a^(1/3) + b^(1/3)*x] + 2*b^(1/3)*(7*A*b - 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(36*a^(10/3))","A",1
86,1,183,215,0.1800253,"\int \frac{A+B x^3}{x^6 \left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(x^6*(a + b*x^3)^2),x]","\frac{5 b^{2/3} (5 a B-8 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-\frac{30 a^{2/3} b x (a B-A b)}{a+b x^3}-\frac{45 a^{2/3} (a B-2 A b)}{x^2}-\frac{18 a^{5/3} A}{x^5}+10 b^{2/3} (8 A b-5 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-10 \sqrt{3} b^{2/3} (8 A b-5 a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{90 a^{11/3}}","-\frac{b^{2/3} (8 A b-5 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{11/3}}+\frac{b^{2/3} (8 A b-5 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{11/3}}-\frac{b^{2/3} (8 A b-5 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{11/3}}+\frac{8 A b-5 a B}{6 a^3 x^2}+\frac{5 a B-8 A b}{15 a^2 b x^5}+\frac{A b-a B}{3 a b x^5 \left(a+b x^3\right)}",1,"((-18*a^(5/3)*A)/x^5 - (45*a^(2/3)*(-2*A*b + a*B))/x^2 - (30*a^(2/3)*b*(-(A*b) + a*B)*x)/(a + b*x^3) - 10*Sqrt[3]*b^(2/3)*(8*A*b - 5*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 10*b^(2/3)*(8*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x] + 5*b^(2/3)*(-8*A*b + 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(90*a^(11/3))","A",1
87,1,85,97,0.1145912,"\int \frac{A+B x^3}{x^7 \left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(x^7*(a + b*x^3)^2),x]","-\frac{\frac{a^2 A}{x^6}+\frac{2 a b (a B-A b)}{a+b x^3}+\frac{2 a (a B-2 A b)}{x^3}+2 b (3 A b-2 a B) \log \left(a+b x^3\right)-6 b \log (x) (3 A b-2 a B)}{6 a^4}","-\frac{b (3 A b-2 a B) \log \left(a+b x^3\right)}{3 a^4}+\frac{b \log (x) (3 A b-2 a B)}{a^4}+\frac{b (A b-a B)}{3 a^3 \left(a+b x^3\right)}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{A}{6 a^2 x^6}",1,"-1/6*((a^2*A)/x^6 + (2*a*(-2*A*b + a*B))/x^3 + (2*a*b*(-(A*b) + a*B))/(a + b*x^3) - 6*b*(3*A*b - 2*a*B)*Log[x] + 2*b*(3*A*b - 2*a*B)*Log[a + b*x^3])/a^4","A",1
88,1,94,107,0.0724356,"\int \frac{x^{11} \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^11*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{\frac{a^3 (A b-a B)}{\left(a+b x^3\right)^2}+\frac{2 a^2 (4 a B-3 A b)}{a+b x^3}+2 b x^3 (A b-3 a B)+6 a (2 a B-A b) \log \left(a+b x^3\right)+b^2 B x^6}{6 b^5}","\frac{a^3 (A b-a B)}{6 b^5 \left(a+b x^3\right)^2}-\frac{a^2 (3 A b-4 a B)}{3 b^5 \left(a+b x^3\right)}-\frac{a (A b-2 a B) \log \left(a+b x^3\right)}{b^5}+\frac{x^3 (A b-3 a B)}{3 b^4}+\frac{B x^6}{6 b^3}",1,"(2*b*(A*b - 3*a*B)*x^3 + b^2*B*x^6 + (a^3*(A*b - a*B))/(a + b*x^3)^2 + (2*a^2*(-3*A*b + 4*a*B))/(a + b*x^3) + 6*a*(-(A*b) + 2*a*B)*Log[a + b*x^3])/(6*b^5)","A",1
89,1,92,88,0.0398336,"\int \frac{x^8 \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^8*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{2 a A b-3 a^2 B}{3 b^4 \left(a+b x^3\right)}+\frac{a^3 B-a^2 A b}{6 b^4 \left(a+b x^3\right)^2}+\frac{(A b-3 a B) \log \left(a+b x^3\right)}{3 b^4}+\frac{B x^3}{3 b^3}","-\frac{a^2 (A b-a B)}{6 b^4 \left(a+b x^3\right)^2}+\frac{a (2 A b-3 a B)}{3 b^4 \left(a+b x^3\right)}+\frac{(A b-3 a B) \log \left(a+b x^3\right)}{3 b^4}+\frac{B x^3}{3 b^3}",1,"(B*x^3)/(3*b^3) + (-(a^2*A*b) + a^3*B)/(6*b^4*(a + b*x^3)^2) + (2*a*A*b - 3*a^2*B)/(3*b^4*(a + b*x^3)) + ((A*b - 3*a*B)*Log[a + b*x^3])/(3*b^4)","A",1
90,1,64,66,0.0254574,"\int \frac{x^5 \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^5*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{3 a^2 B-a b \left(A-4 B x^3\right)+2 B \left(a+b x^3\right)^2 \log \left(a+b x^3\right)-2 A b^2 x^3}{6 b^3 \left(a+b x^3\right)^2}","-\frac{A b-2 a B}{3 b^3 \left(a+b x^3\right)}+\frac{a (A b-a B)}{6 b^3 \left(a+b x^3\right)^2}+\frac{B \log \left(a+b x^3\right)}{3 b^3}",1,"(3*a^2*B - 2*A*b^2*x^3 - a*b*(A - 4*B*x^3) + 2*B*(a + b*x^3)^2*Log[a + b*x^3])/(6*b^3*(a + b*x^3)^2)","A",1
91,1,30,32,0.0154557,"\int \frac{x^2 \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^2*(A + B*x^3))/(a + b*x^3)^3,x]","-\frac{B \left(a+2 b x^3\right)+A b}{6 b^2 \left(a+b x^3\right)^2}","-\frac{\left(A+B x^3\right)^2}{6 \left(a+b x^3\right)^2 (A b-a B)}",1,"-1/6*(A*b + B*(a + 2*b*x^3))/(b^2*(a + b*x^3)^2)","A",1
92,1,59,68,0.0507541,"\int \frac{A+B x^3}{x \left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(x*(a + b*x^3)^3),x]","\frac{\frac{a \left(-a^2 B+3 a A b+2 A b^2 x^3\right)}{b \left(a+b x^3\right)^2}-2 A \log \left(a+b x^3\right)+6 A \log (x)}{6 a^3}","-\frac{A \log \left(a+b x^3\right)}{3 a^3}+\frac{A \log (x)}{a^3}+\frac{A}{3 a^2 \left(a+b x^3\right)}+\frac{A b-a B}{6 a b \left(a+b x^3\right)^2}",1,"((a*(3*a*A*b - a^2*B + 2*A*b^2*x^3))/(b*(a + b*x^3)^2) + 6*A*Log[x] - 2*A*Log[a + b*x^3])/(6*a^3)","A",1
93,1,87,101,0.0723279,"\int \frac{A+B x^3}{x^4 \left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(x^4*(a + b*x^3)^3),x]","\frac{\frac{a^2 (a B-A b)}{\left(a+b x^3\right)^2}+\frac{2 a (a B-2 A b)}{a+b x^3}+2 (3 A b-a B) \log \left(a+b x^3\right)+6 \log (x) (a B-3 A b)-\frac{2 a A}{x^3}}{6 a^4}","\frac{(3 A b-a B) \log \left(a+b x^3\right)}{3 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{2 A b-a B}{3 a^3 \left(a+b x^3\right)}-\frac{A}{3 a^3 x^3}-\frac{A b-a B}{6 a^2 \left(a+b x^3\right)^2}",1,"((-2*a*A)/x^3 + (a^2*(-(A*b) + a*B))/(a + b*x^3)^2 + (2*a*(-2*A*b + a*B))/(a + b*x^3) + 6*(-3*A*b + a*B)*Log[x] + 2*(3*A*b - a*B)*Log[a + b*x^3])/(6*a^4)","A",1
94,1,108,122,0.109541,"\int \frac{A+B x^3}{x^7 \left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(x^7*(a + b*x^3)^3),x]","\frac{\frac{a^2 b (A b-a B)}{\left(a+b x^3\right)^2}-\frac{a^2 A}{x^6}+\frac{2 a b (3 A b-2 a B)}{a+b x^3}-\frac{2 a (a B-3 A b)}{x^3}+6 b (a B-2 A b) \log \left(a+b x^3\right)+18 b \log (x) (2 A b-a B)}{6 a^5}","-\frac{b (2 A b-a B) \log \left(a+b x^3\right)}{a^5}+\frac{3 b \log (x) (2 A b-a B)}{a^5}+\frac{b (3 A b-2 a B)}{3 a^4 \left(a+b x^3\right)}+\frac{3 A b-a B}{3 a^4 x^3}+\frac{b (A b-a B)}{6 a^3 \left(a+b x^3\right)^2}-\frac{A}{6 a^3 x^6}",1,"(-((a^2*A)/x^6) - (2*a*(-3*A*b + a*B))/x^3 + (a^2*b*(A*b - a*B))/(a + b*x^3)^2 + (2*a*b*(3*A*b - 2*a*B))/(a + b*x^3) + 18*b*(2*A*b - a*B)*Log[x] + 6*b*(-2*A*b + a*B)*Log[a + b*x^3])/(6*a^5)","A",1
95,1,216,246,0.3230769,"\int \frac{x^{10} \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^10*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{20 a^{2/3} (11 a B-5 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-40 a^{2/3} (11 a B-5 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-40 \sqrt{3} a^{2/3} (11 a B-5 A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+\frac{45 a^2 b^{2/3} x^2 (a B-A b)}{\left(a+b x^3\right)^2}+135 b^{2/3} x^2 (A b-3 a B)+\frac{30 a b^{2/3} x^2 (7 A b-10 a B)}{a+b x^3}+54 b^{5/3} B x^5}{270 b^{14/3}}","-\frac{2 a^{2/3} (5 A b-11 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{27 b^{14/3}}+\frac{4 a^{2/3} (5 A b-11 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 b^{14/3}}+\frac{4 a^{2/3} (5 A b-11 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} b^{14/3}}+\frac{2 x^2 (5 A b-11 a B)}{9 b^4}-\frac{4 x^5 (5 A b-11 a B)}{45 a b^3}+\frac{x^8 (5 A b-11 a B)}{18 a b^2 \left(a+b x^3\right)}+\frac{x^{11} (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"(135*b^(2/3)*(A*b - 3*a*B)*x^2 + 54*b^(5/3)*B*x^5 + (45*a^2*b^(2/3)*(-(A*b) + a*B)*x^2)/(a + b*x^3)^2 + (30*a*b^(2/3)*(7*A*b - 10*a*B)*x^2)/(a + b*x^3) - 40*Sqrt[3]*a^(2/3)*(-5*A*b + 11*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 40*a^(2/3)*(-5*A*b + 11*a*B)*Log[a^(1/3) + b^(1/3)*x] + 20*a^(2/3)*(-5*A*b + 11*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(270*b^(14/3))","A",1
96,1,210,244,0.2442707,"\int \frac{x^9 \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^9*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{-14 \sqrt[3]{a} (5 a B-2 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+\frac{18 a^2 \sqrt[3]{b} x (a B-A b)}{\left(a+b x^3\right)^2}+\frac{6 a \sqrt[3]{b} x (13 A b-19 a B)}{a+b x^3}+108 \sqrt[3]{b} x (A b-3 a B)+28 \sqrt[3]{a} (5 a B-2 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-28 \sqrt{3} \sqrt[3]{a} (5 a B-2 A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+27 b^{4/3} B x^4}{108 b^{13/3}}","\frac{7 \sqrt[3]{a} (2 A b-5 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 b^{13/3}}-\frac{7 \sqrt[3]{a} (2 A b-5 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 b^{13/3}}+\frac{7 \sqrt[3]{a} (2 A b-5 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} b^{13/3}}+\frac{7 x (2 A b-5 a B)}{9 b^4}-\frac{7 x^4 (2 A b-5 a B)}{36 a b^3}+\frac{x^7 (2 A b-5 a B)}{9 a b^2 \left(a+b x^3\right)}+\frac{x^{10} (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"(108*b^(1/3)*(A*b - 3*a*B)*x + 27*b^(4/3)*B*x^4 + (18*a^2*b^(1/3)*(-(A*b) + a*B)*x)/(a + b*x^3)^2 + (6*a*b^(1/3)*(13*A*b - 19*a*B)*x)/(a + b*x^3) - 28*Sqrt[3]*a^(1/3)*(-2*A*b + 5*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 28*a^(1/3)*(-2*A*b + 5*a*B)*Log[a^(1/3) + b^(1/3)*x] - 14*a^(1/3)*(-2*A*b + 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(108*b^(13/3))","A",1
97,1,194,222,0.2373704,"\int \frac{x^7 \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^7*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{\frac{5 (A b-4 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{a}}-\frac{6 b^{2/3} x^2 (4 A b-7 a B)}{a+b x^3}+\frac{9 a b^{2/3} x^2 (A b-a B)}{\left(a+b x^3\right)^2}+\frac{10 (4 a B-A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a}}+\frac{10 \sqrt{3} (4 a B-A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt[3]{a}}+27 b^{2/3} B x^2}{54 b^{11/3}}","\frac{5 (A b-4 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 \sqrt[3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 \sqrt[3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} \sqrt[3]{a} b^{11/3}}-\frac{5 x^2 (A b-4 a B)}{18 a b^3}+\frac{x^5 (A b-4 a B)}{9 a b^2 \left(a+b x^3\right)}+\frac{x^8 (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"(27*b^(2/3)*B*x^2 + (9*a*b^(2/3)*(A*b - a*B)*x^2)/(a + b*x^3)^2 - (6*b^(2/3)*(4*A*b - 7*a*B)*x^2)/(a + b*x^3) + (10*Sqrt[3]*(-(A*b) + 4*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (10*(-(A*b) + 4*a*B)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (5*(A*b - 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3))/(54*b^(11/3))","A",1
98,1,188,220,0.2725939,"\int \frac{x^6 \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^6*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{\frac{2 (7 a B-A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{2/3}}+\frac{4 (A b-7 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{2/3}}+\frac{4 \sqrt{3} (7 a B-A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{2/3}}-\frac{3 \sqrt[3]{b} x (7 A b-13 a B)}{a+b x^3}+\frac{9 a \sqrt[3]{b} x (A b-a B)}{\left(a+b x^3\right)^2}+54 \sqrt[3]{b} B x}{54 b^{10/3}}","-\frac{(A b-7 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{27 a^{2/3} b^{10/3}}+\frac{2 (A b-7 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{2/3} b^{10/3}}-\frac{2 (A b-7 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{2/3} b^{10/3}}-\frac{2 x (A b-7 a B)}{9 a b^3}+\frac{x^4 (A b-7 a B)}{18 a b^2 \left(a+b x^3\right)}+\frac{x^7 (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"(54*b^(1/3)*B*x + (9*a*b^(1/3)*(A*b - a*B)*x)/(a + b*x^3)^2 - (3*b^(1/3)*(7*A*b - 13*a*B)*x)/(a + b*x^3) + (4*Sqrt[3]*(-(A*b) + 7*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (4*(A*b - 7*a*B)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + (2*(-(A*b) + 7*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(54*b^(10/3))","A",1
99,1,181,201,0.2897701,"\int \frac{x^4 \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^4*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{\frac{(5 a B+A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{4/3}}-\frac{2 (5 a B+A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{4/3}}-\frac{2 \sqrt{3} (5 a B+A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{4/3}}+\frac{6 b^{2/3} x^2 (A b-4 a B)}{a \left(a+b x^3\right)}-\frac{9 b^{2/3} x^2 (A b-a B)}{\left(a+b x^3\right)^2}}{54 b^{8/3}}","\frac{(5 a B+A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{4/3} b^{8/3}}-\frac{(5 a B+A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{4/3} b^{8/3}}-\frac{(5 a B+A b) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{4/3} b^{8/3}}-\frac{x^2 (5 a B+A b)}{18 a b^2 \left(a+b x^3\right)}+\frac{x^5 (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"((-9*b^(2/3)*(A*b - a*B)*x^2)/(a + b*x^3)^2 + (6*b^(2/3)*(A*b - 4*a*B)*x^2)/(a*(a + b*x^3)) - (2*Sqrt[3]*(A*b + 5*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) - (2*(A*b + 5*a*B)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + ((A*b + 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3))/(54*b^(8/3))","A",1
100,1,178,199,0.2435045,"\int \frac{x^3 \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^3*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{-\frac{(2 a B+A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{5/3}}+\frac{2 (2 a B+A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{5/3}}-\frac{2 \sqrt{3} (2 a B+A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{5/3}}+\frac{3 \sqrt[3]{b} x (A b-7 a B)}{a \left(a+b x^3\right)}-\frac{9 \sqrt[3]{b} x (A b-a B)}{\left(a+b x^3\right)^2}}{54 b^{7/3}}","-\frac{(2 a B+A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{5/3} b^{7/3}}+\frac{(2 a B+A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{5/3} b^{7/3}}-\frac{(2 a B+A b) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{5/3} b^{7/3}}-\frac{x (2 a B+A b)}{9 a b^2 \left(a+b x^3\right)}+\frac{x^4 (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"((-9*b^(1/3)*(A*b - a*B)*x)/(a + b*x^3)^2 + (3*b^(1/3)*(A*b - 7*a*B)*x)/(a*(a + b*x^3)) - (2*Sqrt[3]*(A*b + 2*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*(A*b + 2*a*B)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - ((A*b + 2*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(54*b^(7/3))","A",1
101,1,178,201,0.2405938,"\int \frac{x \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{(a B+2 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-\frac{9 a^{4/3} b^{2/3} x^2 (a B-A b)}{\left(a+b x^3\right)^2}+\frac{6 \sqrt[3]{a} b^{2/3} x^2 (a B+2 A b)}{a+b x^3}-2 (a B+2 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} (a B+2 A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{54 a^{7/3} b^{5/3}}","\frac{(a B+2 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{7/3} b^{5/3}}-\frac{(a B+2 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{7/3} b^{5/3}}-\frac{(a B+2 A b) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{7/3} b^{5/3}}+\frac{x^2 (a B+2 A b)}{9 a^2 b \left(a+b x^3\right)}+\frac{x^2 (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"((-9*a^(4/3)*b^(2/3)*(-(A*b) + a*B)*x^2)/(a + b*x^3)^2 + (6*a^(1/3)*b^(2/3)*(2*A*b + a*B)*x^2)/(a + b*x^3) - 2*Sqrt[3]*(2*A*b + a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*(2*A*b + a*B)*Log[a^(1/3) + b^(1/3)*x] + (2*A*b + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(5/3))","A",1
102,1,175,197,0.1850058,"\int \frac{A+B x^3}{\left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(a + b*x^3)^3,x]","\frac{-(a B+5 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-\frac{9 a^{5/3} \sqrt[3]{b} x (a B-A b)}{\left(a+b x^3\right)^2}+\frac{3 a^{2/3} \sqrt[3]{b} x (a B+5 A b)}{a+b x^3}+2 (a B+5 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} (a B+5 A b) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{54 a^{8/3} b^{4/3}}","-\frac{(a B+5 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{8/3} b^{4/3}}+\frac{(a B+5 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{8/3} b^{4/3}}-\frac{(a B+5 A b) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{8/3} b^{4/3}}+\frac{x (a B+5 A b)}{18 a^2 b \left(a+b x^3\right)}+\frac{x (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"((-9*a^(5/3)*b^(1/3)*(-(A*b) + a*B)*x)/(a + b*x^3)^2 + (3*a^(2/3)*b^(1/3)*(5*A*b + a*B)*x)/(a + b*x^3) - 2*Sqrt[3]*(5*A*b + a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(5*A*b + a*B)*Log[a^(1/3) + b^(1/3)*x] - (5*A*b + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(4/3))","A",1
103,1,193,227,0.279547,"\int \frac{A+B x^3}{x^2 \left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(x^2*(a + b*x^3)^3),x]","\frac{\frac{2 (a B-7 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+\frac{9 a^{4/3} x^2 (a B-A b)}{\left(a+b x^3\right)^2}+\frac{4 (7 A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}+\frac{4 \sqrt{3} (7 A b-a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{2/3}}+\frac{6 \sqrt[3]{a} x^2 (2 a B-5 A b)}{a+b x^3}-\frac{54 \sqrt[3]{a} A}{x}}{54 a^{10/3}}","-\frac{(7 A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{27 a^{10/3} b^{2/3}}+\frac{2 (7 A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{10/3} b^{2/3}}+\frac{2 (7 A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{10/3} b^{2/3}}-\frac{2 (7 A b-a B)}{9 a^3 b x}+\frac{7 A b-a B}{18 a^2 b x \left(a+b x^3\right)}+\frac{A b-a B}{6 a b x \left(a+b x^3\right)^2}",1,"((-54*a^(1/3)*A)/x + (9*a^(4/3)*(-(A*b) + a*B)*x^2)/(a + b*x^3)^2 + (6*a^(1/3)*(-5*A*b + 2*a*B)*x^2)/(a + b*x^3) + (4*Sqrt[3]*(7*A*b - a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (4*(7*A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (2*(-7*A*b + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(54*a^(10/3))","A",1
104,1,189,227,0.2614432,"\int \frac{A+B x^3}{x^3 \left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(x^3*(a + b*x^3)^3),x]","\frac{\frac{5 (4 A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{b}}+\frac{9 a^{5/3} x (a B-A b)}{\left(a+b x^3\right)^2}+\frac{3 a^{2/3} x (5 a B-11 A b)}{a+b x^3}-\frac{27 a^{2/3} A}{x^2}+\frac{10 (a B-4 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}+\frac{10 \sqrt{3} (4 A b-a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt[3]{b}}}{54 a^{11/3}}","\frac{5 (4 A b-a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{11/3} \sqrt[3]{b}}-\frac{5 (4 A b-a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{11/3} \sqrt[3]{b}}+\frac{5 (4 A b-a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{11/3} \sqrt[3]{b}}-\frac{5 (4 A b-a B)}{18 a^3 b x^2}+\frac{4 A b-a B}{9 a^2 b x^2 \left(a+b x^3\right)}+\frac{A b-a B}{6 a b x^2 \left(a+b x^3\right)^2}",1,"((-27*a^(2/3)*A)/x^2 + (9*a^(5/3)*(-(A*b) + a*B)*x)/(a + b*x^3)^2 + (3*a^(2/3)*(-11*A*b + 5*a*B)*x)/(a + b*x^3) + (10*Sqrt[3]*(4*A*b - a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (10*(-4*A*b + a*B)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (5*(4*A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3))/(54*a^(11/3))","A",1
105,1,214,246,0.2738256,"\int \frac{A+B x^3}{x^5 \left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(x^5*(a + b*x^3)^3),x]","\frac{14 \sqrt[3]{b} (5 A b-2 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-\frac{18 a^{4/3} b x^2 (a B-A b)}{\left(a+b x^3\right)^2}-\frac{27 a^{4/3} A}{x^4}-\frac{12 \sqrt[3]{a} b x^2 (5 a B-8 A b)}{a+b x^3}-\frac{108 \sqrt[3]{a} (a B-3 A b)}{x}+28 \sqrt[3]{b} (2 a B-5 A b) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-28 \sqrt{3} \sqrt[3]{b} (5 A b-2 a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{108 a^{13/3}}","\frac{7 \sqrt[3]{b} (5 A b-2 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{13/3}}-\frac{7 \sqrt[3]{b} (5 A b-2 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{13/3}}-\frac{7 \sqrt[3]{b} (5 A b-2 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{13/3}}+\frac{7 (5 A b-2 a B)}{9 a^4 x}-\frac{7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left(a+b x^3\right)}+\frac{A b-a B}{6 a b x^4 \left(a+b x^3\right)^2}",1,"((-27*a^(4/3)*A)/x^4 - (108*a^(1/3)*(-3*A*b + a*B))/x - (18*a^(4/3)*b*(-(A*b) + a*B)*x^2)/(a + b*x^3)^2 - (12*a^(1/3)*b*(-8*A*b + 5*a*B)*x^2)/(a + b*x^3) - 28*Sqrt[3]*b^(1/3)*(5*A*b - 2*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 28*b^(1/3)*(-5*A*b + 2*a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*b^(1/3)*(5*A*b - 2*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(108*a^(13/3))","A",1
106,1,210,246,0.3167535,"\int \frac{A+B x^3}{x^6 \left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(x^6*(a + b*x^3)^3),x]","\frac{20 b^{2/3} (5 a B-11 A b) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-\frac{45 a^{5/3} b x (a B-A b)}{\left(a+b x^3\right)^2}-\frac{15 a^{2/3} b x (11 a B-17 A b)}{a+b x^3}-\frac{135 a^{2/3} (a B-3 A b)}{x^2}-\frac{54 a^{5/3} A}{x^5}+40 b^{2/3} (11 A b-5 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-40 \sqrt{3} b^{2/3} (11 A b-5 a B) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{270 a^{14/3}}","-\frac{2 b^{2/3} (11 A b-5 a B) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{27 a^{14/3}}+\frac{4 b^{2/3} (11 A b-5 a B) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{14/3}}-\frac{4 b^{2/3} (11 A b-5 a B) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{14/3}}+\frac{2 (11 A b-5 a B)}{9 a^4 x^2}-\frac{4 (11 A b-5 a B)}{45 a^3 b x^5}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left(a+b x^3\right)}+\frac{A b-a B}{6 a b x^5 \left(a+b x^3\right)^2}",1,"((-54*a^(5/3)*A)/x^5 - (135*a^(2/3)*(-3*A*b + a*B))/x^2 - (45*a^(5/3)*b*(-(A*b) + a*B)*x)/(a + b*x^3)^2 - (15*a^(2/3)*b*(-17*A*b + 11*a*B)*x)/(a + b*x^3) - 40*Sqrt[3]*b^(2/3)*(11*A*b - 5*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 40*b^(2/3)*(11*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x] + 20*b^(2/3)*(-11*A*b + 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(270*a^(14/3))","A",1
107,1,66,70,0.0673029,"\int \frac{x^8}{\left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[x^8/((a + b*x^3)*(c + d*x^3)),x]","\frac{a^2 d^2 \log \left(a+b x^3\right)-b \left(d x^3 (a d-b c)+b c^2 \log \left(c+d x^3\right)\right)}{3 b^2 d^2 (b c-a d)}","\frac{a^2 \log \left(a+b x^3\right)}{3 b^2 (b c-a d)}-\frac{c^2 \log \left(c+d x^3\right)}{3 d^2 (b c-a d)}+\frac{x^3}{3 b d}",1,"(a^2*d^2*Log[a + b*x^3] - b*(d*(-(b*c) + a*d)*x^3 + b*c^2*Log[c + d*x^3]))/(3*b^2*d^2*(b*c - a*d))","A",1
108,1,242,301,0.2397979,"\int \frac{x^7}{\left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[x^7/((a + b*x^3)*(c + d*x^3)),x]","\frac{\frac{a^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{5/3}}-\frac{2 a^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{5/3}}-\frac{2 \sqrt{3} a^{5/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{5/3}}-\frac{3 a x^2}{b}-\frac{c^{5/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{d^{5/3}}+\frac{2 c^{5/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{d^{5/3}}+\frac{2 \sqrt{3} c^{5/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{d^{5/3}}+\frac{3 c x^2}{d}}{6 b c-6 a d}","\frac{a^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 b^{5/3} (b c-a d)}-\frac{a^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 b^{5/3} (b c-a d)}-\frac{a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^{5/3} (b c-a d)}-\frac{c^{5/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 d^{5/3} (b c-a d)}+\frac{c^{5/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 d^{5/3} (b c-a d)}+\frac{c^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} d^{5/3} (b c-a d)}+\frac{x^2}{2 b d}",1,"((-3*a*x^2)/b + (3*c*x^2)/d - (2*Sqrt[3]*a^(5/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(5/3) + (2*Sqrt[3]*c^(5/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/d^(5/3) - (2*a^(5/3)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3) + (2*c^(5/3)*Log[c^(1/3) + d^(1/3)*x])/d^(5/3) + (a^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(5/3) - (c^(5/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/d^(5/3))/(6*b*c - 6*a*d)","A",1
109,1,238,296,0.1654742,"\int \frac{x^6}{\left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[x^6/((a + b*x^3)*(c + d*x^3)),x]","\frac{-\frac{a^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{4/3}}+\frac{2 a^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{4/3}}-\frac{2 \sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{4/3}}-\frac{6 a x}{b}+\frac{c^{4/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{d^{4/3}}-\frac{2 c^{4/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{d^{4/3}}+\frac{2 \sqrt{3} c^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{d^{4/3}}+\frac{6 c x}{d}}{6 b c-6 a d}","-\frac{a^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 b^{4/3} (b c-a d)}+\frac{a^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 b^{4/3} (b c-a d)}-\frac{a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^{4/3} (b c-a d)}+\frac{c^{4/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 d^{4/3} (b c-a d)}-\frac{c^{4/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 d^{4/3} (b c-a d)}+\frac{c^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} d^{4/3} (b c-a d)}+\frac{x}{b d}",1,"((-6*a*x)/b + (6*c*x)/d - (2*Sqrt[3]*a^(4/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(4/3) + (2*Sqrt[3]*c^(4/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/d^(4/3) + (2*a^(4/3)*Log[a^(1/3) + b^(1/3)*x])/b^(4/3) - (2*c^(4/3)*Log[c^(1/3) + d^(1/3)*x])/d^(4/3) - (a^(4/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(4/3) + (c^(4/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/d^(4/3))/(6*b*c - 6*a*d)","A",1
110,1,43,53,0.0610598,"\int \frac{x^5}{\left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[x^5/((a + b*x^3)*(c + d*x^3)),x]","-\frac{a d \log \left(a+b x^3\right)-b c \log \left(c+d x^3\right)}{3 b^2 c d-3 a b d^2}","\frac{c \log \left(c+d x^3\right)}{3 d (b c-a d)}-\frac{a \log \left(a+b x^3\right)}{3 b (b c-a d)}",1,"-((a*d*Log[a + b*x^3] - b*c*Log[c + d*x^3])/(3*b^2*c*d - 3*a*b*d^2))","A",1
111,1,224,288,0.1268218,"\int \frac{x^4}{\left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[x^4/((a + b*x^3)*(c + d*x^3)),x]","\frac{-\frac{a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+\frac{2 a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}+\frac{2 \sqrt{3} a^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{b^{2/3}}+\frac{c^{2/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{d^{2/3}}-\frac{2 c^{2/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{d^{2/3}}-\frac{2 \sqrt{3} c^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{d^{2/3}}}{6 b c-6 a d}","-\frac{a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 b^{2/3} (b c-a d)}+\frac{a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 b^{2/3} (b c-a d)}+\frac{a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^{2/3} (b c-a d)}+\frac{c^{2/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 d^{2/3} (b c-a d)}-\frac{c^{2/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 d^{2/3} (b c-a d)}-\frac{c^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} d^{2/3} (b c-a d)}",1,"((2*Sqrt[3]*a^(2/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) - (2*Sqrt[3]*c^(2/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/d^(2/3) + (2*a^(2/3)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) - (2*c^(2/3)*Log[c^(1/3) + d^(1/3)*x])/d^(2/3) - (a^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + (c^(2/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/d^(2/3))/(6*b*c - 6*a*d)","A",1
112,1,224,288,0.1243378,"\int \frac{x^3}{\left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[x^3/((a + b*x^3)*(c + d*x^3)),x]","\frac{\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{b}}-\frac{2 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}+\frac{2 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt[3]{b}}-\frac{\sqrt[3]{c} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{\sqrt[3]{d}}+\frac{2 \sqrt[3]{c} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt[3]{d}}-\frac{2 \sqrt{3} \sqrt[3]{c} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{\sqrt[3]{d}}}{6 b c-6 a d}","\frac{\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 \sqrt[3]{b} (b c-a d)}-\frac{\sqrt[3]{c} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 \sqrt[3]{d} (b c-a d)}-\frac{\sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 \sqrt[3]{b} (b c-a d)}+\frac{\sqrt[3]{c} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 \sqrt[3]{d} (b c-a d)}+\frac{\sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{b} (b c-a d)}-\frac{\sqrt[3]{c} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} \sqrt[3]{d} (b c-a d)}",1,"((2*Sqrt[3]*a^(1/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) - (2*Sqrt[3]*c^(1/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/d^(1/3) - (2*a^(1/3)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (2*c^(1/3)*Log[c^(1/3) + d^(1/3)*x])/d^(1/3) + (a^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3) - (c^(1/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/d^(1/3))/(6*b*c - 6*a*d)","A",1
113,1,31,45,0.0200183,"\int \frac{x^2}{\left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[x^2/((a + b*x^3)*(c + d*x^3)),x]","\frac{\log \left(a+b x^3\right)-\log \left(c+d x^3\right)}{3 b c-3 a d}","\frac{\log \left(a+b x^3\right)}{3 (b c-a d)}-\frac{\log \left(c+d x^3\right)}{3 (b c-a d)}",1,"(Log[a + b*x^3] - Log[c + d*x^3])/(3*b*c - 3*a*d)","A",1
114,1,224,288,0.1439769,"\int \frac{x}{\left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[x/((a + b*x^3)*(c + d*x^3)),x]","\frac{-\frac{\sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{a}}+\frac{2 \sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a}}+\frac{2 \sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{\sqrt[3]{a}}+\frac{\sqrt[3]{d} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{\sqrt[3]{c}}-\frac{2 \sqrt[3]{d} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt[3]{c}}-\frac{2 \sqrt{3} \sqrt[3]{d} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{\sqrt[3]{c}}}{6 a d-6 b c}","\frac{\sqrt[3]{b} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 \sqrt[3]{a} (b c-a d)}-\frac{\sqrt[3]{d} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 \sqrt[3]{c} (b c-a d)}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 \sqrt[3]{a} (b c-a d)}+\frac{\sqrt[3]{d} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 \sqrt[3]{c} (b c-a d)}-\frac{\sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} (b c-a d)}+\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} \sqrt[3]{c} (b c-a d)}",1,"((2*Sqrt[3]*b^(1/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) - (2*Sqrt[3]*d^(1/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(1/3) + (2*b^(1/3)*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) - (2*d^(1/3)*Log[c^(1/3) + d^(1/3)*x])/c^(1/3) - (b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3) + (d^(1/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(1/3))/(-6*b*c + 6*a*d)","A",1
115,1,224,288,0.1823869,"\int \frac{1}{\left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[1/((a + b*x^3)*(c + d*x^3)),x]","\frac{\frac{b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{2/3}}-\frac{2 b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{2/3}}+\frac{2 \sqrt{3} b^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{2/3}}-\frac{d^{2/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{c^{2/3}}+\frac{2 d^{2/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{c^{2/3}}-\frac{2 \sqrt{3} d^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{c^{2/3}}}{6 a d-6 b c}","-\frac{b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} (b c-a d)}+\frac{b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} (b c-a d)}-\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} (b c-a d)}+\frac{d^{2/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c^{2/3} (b c-a d)}-\frac{d^{2/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{2/3} (b c-a d)}+\frac{d^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{2/3} (b c-a d)}",1,"((2*Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) - (2*Sqrt[3]*d^(2/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(2/3) - (2*b^(2/3)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + (2*d^(2/3)*Log[c^(1/3) + d^(1/3)*x])/c^(2/3) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3) - (d^(2/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(2/3))/(-6*b*c + 6*a*d)","A",1
116,1,54,62,0.0500118,"\int \frac{1}{x \left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[1/(x*(a + b*x^3)*(c + d*x^3)),x]","\frac{-b c \log \left(a+b x^3\right)+a d \log \left(c+d x^3\right)-3 a d \log (x)+3 b c \log (x)}{3 a b c^2-3 a^2 c d}","-\frac{b \log \left(a+b x^3\right)}{3 a (b c-a d)}+\frac{d \log \left(c+d x^3\right)}{3 c (b c-a d)}+\frac{\log (x)}{a c}",1,"(3*b*c*Log[x] - 3*a*d*Log[x] - b*c*Log[a + b*x^3] + a*d*Log[c + d*x^3])/(3*a*b*c^2 - 3*a^2*c*d)","A",1
117,1,244,299,0.2131395,"\int \frac{1}{x^2 \left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[1/(x^2*(a + b*x^3)*(c + d*x^3)),x]","\frac{\frac{b^{4/3} x \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{4/3}}-\frac{2 b^{4/3} x \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{4/3}}-\frac{2 \sqrt{3} b^{4/3} x \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{4/3}}+\frac{6 b}{a}-\frac{d^{4/3} x \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{c^{4/3}}+\frac{2 d^{4/3} x \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{c^{4/3}}+\frac{2 \sqrt{3} d^{4/3} x \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{c^{4/3}}-\frac{6 d}{c}}{6 a d x-6 b c x}","-\frac{b^{4/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{4/3} (b c-a d)}+\frac{b^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{4/3} (b c-a d)}+\frac{b^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{4/3} (b c-a d)}+\frac{d^{4/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c^{4/3} (b c-a d)}-\frac{d^{4/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{4/3} (b c-a d)}-\frac{d^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{4/3} (b c-a d)}-\frac{1}{a c x}",1,"((6*b)/a - (6*d)/c - (2*Sqrt[3]*b^(4/3)*x*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) + (2*Sqrt[3]*d^(4/3)*x*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(4/3) - (2*b^(4/3)*x*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + (2*d^(4/3)*x*Log[c^(1/3) + d^(1/3)*x])/c^(4/3) + (b^(4/3)*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3) - (d^(4/3)*x*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(4/3))/(-6*b*c*x + 6*a*d*x)","A",1
118,1,259,301,0.2244087,"\int \frac{1}{x^3 \left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[1/(x^3*(a + b*x^3)*(c + d*x^3)),x]","\frac{\frac{2 b^{5/3} x^2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{5/3}}-\frac{b^{5/3} x^2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{5/3}}-\frac{2 \sqrt{3} b^{5/3} x^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{5/3}}+\frac{3 b}{a}-\frac{2 d^{5/3} x^2 \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{c^{5/3}}+\frac{d^{5/3} x^2 \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{c^{5/3}}+\frac{2 \sqrt{3} d^{5/3} x^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{c^{5/3}}-\frac{3 d}{c}}{6 x^2 (a d-b c)}","\frac{b^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{5/3} (b c-a d)}-\frac{b^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{5/3} (b c-a d)}+\frac{b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{5/3} (b c-a d)}-\frac{d^{5/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c^{5/3} (b c-a d)}+\frac{d^{5/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{5/3} (b c-a d)}-\frac{d^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{5/3} (b c-a d)}-\frac{1}{2 a c x^2}",1,"((3*b)/a - (3*d)/c - (2*Sqrt[3]*b^(5/3)*x^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*Sqrt[3]*d^(5/3)*x^2*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(5/3) + (2*b^(5/3)*x^2*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (2*d^(5/3)*x^2*Log[c^(1/3) + d^(1/3)*x])/c^(5/3) - (b^(5/3)*x^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3) + (d^(5/3)*x^2*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(5/3))/(6*(-(b*c) + a*d)*x^2)","A",1
119,1,88,87,0.1061112,"\int \frac{1}{x^4 \left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[1/(x^4*(a + b*x^3)*(c + d*x^3)),x]","-\frac{b^2 \log \left(a+b x^3\right)}{3 a^2 (a d-b c)}+\frac{\log (x) (-a d-b c)}{a^2 c^2}-\frac{d^2 \log \left(c+d x^3\right)}{3 c^2 (b c-a d)}-\frac{1}{3 a c x^3}","\frac{b^2 \log \left(a+b x^3\right)}{3 a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log \left(c+d x^3\right)}{3 c^2 (b c-a d)}-\frac{1}{3 a c x^3}",1,"-1/3*1/(a*c*x^3) + ((-(b*c) - a*d)*Log[x])/(a^2*c^2) - (b^2*Log[a + b*x^3])/(3*a^2*(-(b*c) + a*d)) - (d^2*Log[c + d*x^3])/(3*c^2*(b*c - a*d))","A",1
120,1,282,318,0.264912,"\int \frac{1}{x^5 \left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[1/(x^5*(a + b*x^3)*(c + d*x^3)),x]","\frac{\frac{4 b^{7/3} x^4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{7/3}}+\frac{4 \sqrt{3} b^{7/3} x^4 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{7/3}}-\frac{2 b^{7/3} x^4 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{7/3}}-\frac{12 b^2 x^3}{a^2}+\frac{3 b}{a}-\frac{4 d^{7/3} x^4 \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{c^{7/3}}-\frac{4 \sqrt{3} d^{7/3} x^4 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{c^{7/3}}+\frac{2 d^{7/3} x^4 \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{c^{7/3}}+\frac{12 d^2 x^3}{c^2}-\frac{3 d}{c}}{12 x^4 (a d-b c)}","\frac{b^{7/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{7/3} (b c-a d)}-\frac{b^{7/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{7/3} (b c-a d)}-\frac{b^{7/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{7/3} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{7/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c^{7/3} (b c-a d)}+\frac{d^{7/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{7/3} (b c-a d)}+\frac{d^{7/3} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{7/3} (b c-a d)}-\frac{1}{4 a c x^4}",1,"((3*b)/a - (3*d)/c - (12*b^2*x^3)/a^2 + (12*d^2*x^3)/c^2 + (4*Sqrt[3]*b^(7/3)*x^4*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(7/3) - (4*Sqrt[3]*d^(7/3)*x^4*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(7/3) + (4*b^(7/3)*x^4*Log[a^(1/3) + b^(1/3)*x])/a^(7/3) - (4*d^(7/3)*x^4*Log[c^(1/3) + d^(1/3)*x])/c^(7/3) - (2*b^(7/3)*x^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(7/3) + (2*d^(7/3)*x^4*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(7/3))/(12*(-(b*c) + a*d)*x^4)","A",1
121,1,282,321,0.2212683,"\int \frac{1}{x^6 \left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[1/(x^6*(a + b*x^3)*(c + d*x^3)),x]","\frac{-\frac{10 b^{8/3} x^5 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{8/3}}+\frac{10 \sqrt{3} b^{8/3} x^5 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{8/3}}+\frac{5 b^{8/3} x^5 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{8/3}}-\frac{15 b^2 x^3}{a^2}+\frac{6 b}{a}+\frac{10 d^{8/3} x^5 \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{c^{8/3}}-\frac{10 \sqrt{3} d^{8/3} x^5 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{c^{8/3}}-\frac{5 d^{8/3} x^5 \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{c^{8/3}}+\frac{15 d^2 x^3}{c^2}-\frac{6 d}{c}}{30 x^5 (a d-b c)}","-\frac{b^{8/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{8/3} (b c-a d)}+\frac{b^{8/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{8/3} (b c-a d)}-\frac{b^{8/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{8/3} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{d^{8/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c^{8/3} (b c-a d)}-\frac{d^{8/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{8/3} (b c-a d)}+\frac{d^{8/3} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{8/3} (b c-a d)}-\frac{1}{5 a c x^5}",1,"((6*b)/a - (6*d)/c - (15*b^2*x^3)/a^2 + (15*d^2*x^3)/c^2 + (10*Sqrt[3]*b^(8/3)*x^5*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(8/3) - (10*Sqrt[3]*d^(8/3)*x^5*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(8/3) - (10*b^(8/3)*x^5*Log[a^(1/3) + b^(1/3)*x])/a^(8/3) + (10*d^(8/3)*x^5*Log[c^(1/3) + d^(1/3)*x])/c^(8/3) + (5*b^(8/3)*x^5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(8/3) - (5*d^(8/3)*x^5*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(8/3))/(30*(-(b*c) + a*d)*x^5)","A",1
122,1,119,119,0.0630633,"\int \frac{1}{x^7 \left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[1/(x^7*(a + b*x^3)*(c + d*x^3)),x]","\frac{b^3 \log \left(a+b x^3\right)}{3 a^3 (a d-b c)}+\frac{a d+b c}{3 a^2 c^2 x^3}+\frac{\log (x) \left(a^2 d^2+a b c d+b^2 c^2\right)}{a^3 c^3}+\frac{d^3 \log \left(c+d x^3\right)}{3 c^3 (b c-a d)}-\frac{1}{6 a c x^6}","-\frac{b^3 \log \left(a+b x^3\right)}{3 a^3 (b c-a d)}+\frac{a d+b c}{3 a^2 c^2 x^3}+\frac{\log (x) \left(a^2 d^2+a b c d+b^2 c^2\right)}{a^3 c^3}+\frac{d^3 \log \left(c+d x^3\right)}{3 c^3 (b c-a d)}-\frac{1}{6 a c x^6}",1,"-1/6*1/(a*c*x^6) + (b*c + a*d)/(3*a^2*c^2*x^3) + ((b^2*c^2 + a*b*c*d + a^2*d^2)*Log[x])/(a^3*c^3) + (b^3*Log[a + b*x^3])/(3*a^3*(-(b*c) + a*d)) + (d^3*Log[c + d*x^3])/(3*c^3*(b*c - a*d))","A",1
123,1,304,352,0.2447405,"\int \frac{1}{x^8 \left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[1/(x^8*(a + b*x^3)*(c + d*x^3)),x]","\frac{-\frac{28 b^{10/3} x^7 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{10/3}}-\frac{28 \sqrt{3} b^{10/3} x^7 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{10/3}}+\frac{14 b^{10/3} x^7 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{10/3}}+\frac{84 b^3 x^6}{a^3}-\frac{21 b^2 x^3}{a^2}+\frac{12 b}{a}+\frac{28 d^{10/3} x^7 \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{c^{10/3}}+\frac{28 \sqrt{3} d^{10/3} x^7 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{c^{10/3}}-\frac{14 d^{10/3} x^7 \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{c^{10/3}}-\frac{84 d^3 x^6}{c^3}+\frac{21 d^2 x^3}{c^2}-\frac{12 d}{c}}{84 x^7 (a d-b c)}","-\frac{b^{10/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{10/3} (b c-a d)}+\frac{b^{10/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{10/3} (b c-a d)}+\frac{b^{10/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{10/3} (b c-a d)}+\frac{a d+b c}{4 a^2 c^2 x^4}-\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}+\frac{d^{10/3} \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c^{10/3} (b c-a d)}-\frac{d^{10/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{10/3} (b c-a d)}-\frac{d^{10/3} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{10/3} (b c-a d)}-\frac{1}{7 a c x^7}",1,"((12*b)/a - (12*d)/c - (21*b^2*x^3)/a^2 + (21*d^2*x^3)/c^2 + (84*b^3*x^6)/a^3 - (84*d^3*x^6)/c^3 - (28*Sqrt[3]*b^(10/3)*x^7*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(10/3) + (28*Sqrt[3]*d^(10/3)*x^7*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(10/3) - (28*b^(10/3)*x^7*Log[a^(1/3) + b^(1/3)*x])/a^(10/3) + (28*d^(10/3)*x^7*Log[c^(1/3) + d^(1/3)*x])/c^(10/3) + (14*b^(10/3)*x^7*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(10/3) - (14*d^(10/3)*x^7*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(10/3))/(84*(-(b*c) + a*d)*x^7)","A",1
124,1,137,148,0.2642288,"\int x^m \left(a+b x^3\right)^5 \left(A+B x^3\right) \, dx","Integrate[x^m*(a + b*x^3)^5*(A + B*x^3),x]","x^{m+1} \left(\frac{a^5 A}{m+1}+\frac{a^4 x^3 (a B+5 A b)}{m+4}+\frac{5 a^3 b x^6 (a B+2 A b)}{m+7}+\frac{10 a^2 b^2 x^9 (a B+A b)}{m+10}+\frac{b^4 x^{15} (5 a B+A b)}{m+16}+\frac{5 a b^3 x^{12} (2 a B+A b)}{m+13}+\frac{b^5 B x^{18}}{m+19}\right)","\frac{a^5 A x^{m+1}}{m+1}+\frac{a^4 x^{m+4} (a B+5 A b)}{m+4}+\frac{5 a^3 b x^{m+7} (a B+2 A b)}{m+7}+\frac{10 a^2 b^2 x^{m+10} (a B+A b)}{m+10}+\frac{b^4 x^{m+16} (5 a B+A b)}{m+16}+\frac{5 a b^3 x^{m+13} (2 a B+A b)}{m+13}+\frac{b^5 B x^{m+19}}{m+19}",1,"x^(1 + m)*((a^5*A)/(1 + m) + (a^4*(5*A*b + a*B)*x^3)/(4 + m) + (5*a^3*b*(2*A*b + a*B)*x^6)/(7 + m) + (10*a^2*b^2*(A*b + a*B)*x^9)/(10 + m) + (5*a*b^3*(A*b + 2*a*B)*x^12)/(13 + m) + (b^4*(A*b + 5*a*B)*x^15)/(16 + m) + (b^5*B*x^18)/(19 + m))","A",1
125,1,66,71,0.0579695,"\int x^m \left(a+b x^3\right)^2 \left(A+B x^3\right) \, dx","Integrate[x^m*(a + b*x^3)^2*(A + B*x^3),x]","x^{m+1} \left(\frac{a^2 A}{m+1}+\frac{b x^6 (2 a B+A b)}{m+7}+\frac{a x^3 (a B+2 A b)}{m+4}+\frac{b^2 B x^9}{m+10}\right)","\frac{a^2 A x^{m+1}}{m+1}+\frac{a x^{m+4} (a B+2 A b)}{m+4}+\frac{b x^{m+7} (2 a B+A b)}{m+7}+\frac{b^2 B x^{m+10}}{m+10}",1,"x^(1 + m)*((a^2*A)/(1 + m) + (a*(2*A*b + a*B)*x^3)/(4 + m) + (b*(A*b + 2*a*B)*x^6)/(7 + m) + (b^2*B*x^9)/(10 + m))","A",1
126,1,42,45,0.047458,"\int x^m \left(a+b x^3\right) \left(A+B x^3\right) \, dx","Integrate[x^m*(a + b*x^3)*(A + B*x^3),x]","x^{m+1} \left(\frac{x^3 (a B+A b)}{m+4}+\frac{a A}{m+1}+\frac{b B x^6}{m+7}\right)","\frac{x^{m+4} (a B+A b)}{m+4}+\frac{a A x^{m+1}}{m+1}+\frac{b B x^{m+7}}{m+7}",1,"x^(1 + m)*((a*A)/(1 + m) + ((A*b + a*B)*x^3)/(4 + m) + (b*B*x^6)/(7 + m))","A",1
127,1,55,66,0.0709606,"\int \frac{x^m \left(A+B x^3\right)}{a+b x^3} \, dx","Integrate[(x^m*(A + B*x^3))/(a + b*x^3),x]","\frac{x^{m+1} \left((A b-a B) \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)+a B\right)}{a b (m+1)}","\frac{x^{m+1} (A b-a B) \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{a b (m+1)}+\frac{B x^{m+1}}{b (m+1)}",1,"(x^(1 + m)*(a*B + (A*b - a*B)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)]))/(a*b*(1 + m))","A",1
128,1,80,93,0.0812694,"\int \frac{x^m \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^m*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{x^{m+1} \left((A b-a B) \, _2F_1\left(2,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)+a B \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)\right)}{a^2 b (m+1)}","\frac{x^{m+1} (a B (m+1)+A b (2-m)) \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{3 a^2 b (m+1)}+\frac{x^{m+1} (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"(x^(1 + m)*(a*B*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)] + (A*b - a*B)*Hypergeometric2F1[2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)]))/(a^2*b*(1 + m))","A",1
129,1,80,93,0.0820871,"\int \frac{x^m \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^m*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{x^{m+1} \left((A b-a B) \, _2F_1\left(3,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)+a B \, _2F_1\left(2,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)\right)}{a^3 b (m+1)}","\frac{x^{m+1} (a B (m+1)+A b (5-m)) \, _2F_1\left(2,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{6 a^3 b (m+1)}+\frac{x^{m+1} (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"(x^(1 + m)*(a*B*Hypergeometric2F1[2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)] + (A*b - a*B)*Hypergeometric2F1[3, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)]))/(a^3*b*(1 + m))","A",1
130,1,86,112,0.0923536,"\int \frac{(e x)^m}{\left(a+b x^3\right) \left(c+d x^3\right)} \, dx","Integrate[(e*x)^m/((a + b*x^3)*(c + d*x^3)),x]","\frac{x (e x)^m \left(a d \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{d x^3}{c}\right)-b c \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)\right)}{a c (m+1) (a d-b c)}","\frac{b (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{a e (m+1) (b c-a d)}-\frac{d (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{d x^3}{c}\right)}{c e (m+1) (b c-a d)}",1,"(x*(e*x)^m*(-(b*c*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)]) + a*d*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((d*x^3)/c)]))/(a*c*(-(b*c) + a*d)*(1 + m))","A",1
131,1,33,39,0.0170456,"\int x^{7/2} \left(a+b x^3\right) \left(A+B x^3\right) \, dx","Integrate[x^(7/2)*(a + b*x^3)*(A + B*x^3),x]","\frac{2}{315} x^{9/2} \left(21 x^3 (a B+A b)+35 a A+15 b B x^6\right)","\frac{2}{15} x^{15/2} (a B+A b)+\frac{2}{9} a A x^{9/2}+\frac{2}{21} b B x^{21/2}",1,"(2*x^(9/2)*(35*a*A + 21*(A*b + a*B)*x^3 + 15*b*B*x^6))/315","A",1
132,1,33,39,0.0162516,"\int x^{5/2} \left(a+b x^3\right) \left(A+B x^3\right) \, dx","Integrate[x^(5/2)*(a + b*x^3)*(A + B*x^3),x]","\frac{2 x^{7/2} \left(133 x^3 (a B+A b)+247 a A+91 b B x^6\right)}{1729}","\frac{2}{13} x^{13/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{19} b B x^{19/2}",1,"(2*x^(7/2)*(247*a*A + 133*(A*b + a*B)*x^3 + 91*b*B*x^6))/1729","A",1
133,1,33,39,0.0147874,"\int x^{3/2} \left(a+b x^3\right) \left(A+B x^3\right) \, dx","Integrate[x^(3/2)*(a + b*x^3)*(A + B*x^3),x]","\frac{2}{935} x^{5/2} \left(85 x^3 (a B+A b)+187 a A+55 b B x^6\right)","\frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{17} b B x^{17/2}",1,"(2*x^(5/2)*(187*a*A + 85*(A*b + a*B)*x^3 + 55*b*B*x^6))/935","A",1
134,1,33,39,0.0230878,"\int \sqrt{x} \left(a+b x^3\right) \left(A+B x^3\right) \, dx","Integrate[Sqrt[x]*(a + b*x^3)*(A + B*x^3),x]","\frac{2}{45} x^{3/2} \left(5 x^3 (a B+A b)+15 a A+3 b B x^6\right)","\frac{2}{9} x^{9/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{15} b B x^{15/2}",1,"(2*x^(3/2)*(15*a*A + 5*(A*b + a*B)*x^3 + 3*b*B*x^6))/45","A",1
135,1,33,37,0.0699989,"\int \frac{\left(a+b x^3\right) \left(A+B x^3\right)}{\sqrt{x}} \, dx","Integrate[((a + b*x^3)*(A + B*x^3))/Sqrt[x],x]","\frac{2}{91} \sqrt{x} \left(13 x^3 (a B+A b)+91 a A+7 b B x^6\right)","\frac{2}{7} x^{7/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{13} b B x^{13/2}",1,"(2*Sqrt[x]*(91*a*A + 13*(A*b + a*B)*x^3 + 7*b*B*x^6))/91","A",1
136,1,35,37,0.0108965,"\int \frac{\left(a+b x^3\right) \left(A+B x^3\right)}{x^{3/2}} \, dx","Integrate[((a + b*x^3)*(A + B*x^3))/x^(3/2),x]","\frac{2 \left(-55 a A+11 a B x^3+11 A b x^3+5 b B x^6\right)}{55 \sqrt{x}}","\frac{2}{5} x^{5/2} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{11} b B x^{11/2}",1,"(2*(-55*a*A + 11*A*b*x^3 + 11*a*B*x^3 + 5*b*B*x^6))/(55*Sqrt[x])","A",1
137,1,34,39,0.0125118,"\int \frac{\left(a+b x^3\right) \left(A+B x^3\right)}{x^{5/2}} \, dx","Integrate[((a + b*x^3)*(A + B*x^3))/x^(5/2),x]","\frac{2 \left(-3 a A+3 a B x^3+3 A b x^3+b B x^6\right)}{9 x^{3/2}}","\frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{9} b B x^{9/2}",1,"(2*(-3*a*A + 3*A*b*x^3 + 3*a*B*x^3 + b*B*x^6))/(9*x^(3/2))","A",1
138,1,36,37,0.0119979,"\int \frac{\left(a+b x^3\right) \left(A+B x^3\right)}{x^{7/2}} \, dx","Integrate[((a + b*x^3)*(A + B*x^3))/x^(7/2),x]","\frac{2 \left(5 b x^3 \left(7 A+B x^3\right)-7 a \left(A-5 B x^3\right)\right)}{35 x^{5/2}}","2 \sqrt{x} (a B+A b)-\frac{2 a A}{5 x^{5/2}}+\frac{2}{7} b B x^{7/2}",1,"(2*(-7*a*(A - 5*B*x^3) + 5*b*x^3*(7*A + B*x^3)))/(35*x^(5/2))","A",1
139,1,53,63,0.0644416,"\int x^{7/2} \left(a+b x^3\right)^2 \left(A+B x^3\right) \, dx","Integrate[x^(7/2)*(a + b*x^3)^2*(A + B*x^3),x]","\frac{2}{945} x^{9/2} \left(105 a^2 A+45 b x^6 (2 a B+A b)+63 a x^3 (a B+2 A b)+35 b^2 B x^9\right)","\frac{2}{9} a^2 A x^{9/2}+\frac{2}{21} b x^{21/2} (2 a B+A b)+\frac{2}{15} a x^{15/2} (a B+2 A b)+\frac{2}{27} b^2 B x^{27/2}",1,"(2*x^(9/2)*(105*a^2*A + 63*a*(2*A*b + a*B)*x^3 + 45*b*(A*b + 2*a*B)*x^6 + 35*b^2*B*x^9))/945","A",1
140,1,63,63,0.0463311,"\int x^{5/2} \left(a+b x^3\right)^2 \left(A+B x^3\right) \, dx","Integrate[x^(5/2)*(a + b*x^3)^2*(A + B*x^3),x]","\frac{2}{7} a^2 A x^{7/2}+\frac{2}{19} b x^{19/2} (2 a B+A b)+\frac{2}{13} a x^{13/2} (a B+2 A b)+\frac{2}{25} b^2 B x^{25/2}","\frac{2}{7} a^2 A x^{7/2}+\frac{2}{19} b x^{19/2} (2 a B+A b)+\frac{2}{13} a x^{13/2} (a B+2 A b)+\frac{2}{25} b^2 B x^{25/2}",1,"(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(13/2))/13 + (2*b*(A*b + 2*a*B)*x^(19/2))/19 + (2*b^2*B*x^(25/2))/25","A",1
141,1,53,63,0.0599621,"\int x^{3/2} \left(a+b x^3\right)^2 \left(A+B x^3\right) \, dx","Integrate[x^(3/2)*(a + b*x^3)^2*(A + B*x^3),x]","\frac{2 x^{5/2} \left(4301 a^2 A+1265 b x^6 (2 a B+A b)+1955 a x^3 (a B+2 A b)+935 b^2 B x^9\right)}{21505}","\frac{2}{5} a^2 A x^{5/2}+\frac{2}{17} b x^{17/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{23} b^2 B x^{23/2}",1,"(2*x^(5/2)*(4301*a^2*A + 1955*a*(2*A*b + a*B)*x^3 + 1265*b*(A*b + 2*a*B)*x^6 + 935*b^2*B*x^9))/21505","A",1
142,1,53,63,0.1100487,"\int \sqrt{x} \left(a+b x^3\right)^2 \left(A+B x^3\right) \, dx","Integrate[Sqrt[x]*(a + b*x^3)^2*(A + B*x^3),x]","\frac{2}{315} x^{3/2} \left(105 a^2 A+21 b x^6 (2 a B+A b)+35 a x^3 (a B+2 A b)+15 b^2 B x^9\right)","\frac{2}{3} a^2 A x^{3/2}+\frac{2}{15} b x^{15/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{21} b^2 B x^{21/2}",1,"(2*x^(3/2)*(105*a^2*A + 35*a*(2*A*b + a*B)*x^3 + 21*b*(A*b + 2*a*B)*x^6 + 15*b^2*B*x^9))/315","A",1
143,1,53,61,0.0667457,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{\sqrt{x}} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/Sqrt[x],x]","\frac{2 \sqrt{x} \left(1729 a^2 A+133 b x^6 (2 a B+A b)+247 a x^3 (a B+2 A b)+91 b^2 B x^9\right)}{1729}","2 a^2 A \sqrt{x}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{19} b^2 B x^{19/2}",1,"(2*Sqrt[x]*(1729*a^2*A + 247*a*(2*A*b + a*B)*x^3 + 133*b*(A*b + 2*a*B)*x^6 + 91*b^2*B*x^9))/1729","A",1
144,1,60,61,0.0696597,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x^{3/2}} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x^(3/2),x]","\frac{-374 a^2 \left(5 A-B x^3\right)+68 a b x^3 \left(11 A+5 B x^3\right)+10 b^2 x^6 \left(17 A+11 B x^3\right)}{935 \sqrt{x}}","-\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{17} b^2 B x^{17/2}",1,"(-374*a^2*(5*A - B*x^3) + 68*a*b*x^3*(11*A + 5*B*x^3) + 10*b^2*x^6*(17*A + 11*B*x^3))/(935*Sqrt[x])","A",1
145,1,57,63,0.0673388,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x^{5/2}} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x^(5/2),x]","\frac{-30 a^2 \left(A-B x^3\right)+20 a b x^3 \left(3 A+B x^3\right)+2 b^2 x^6 \left(5 A+3 B x^3\right)}{45 x^{3/2}}","-\frac{2 a^2 A}{3 x^{3/2}}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2}",1,"(-30*a^2*(A - B*x^3) + 20*a*b*x^3*(3*A + B*x^3) + 2*b^2*x^6*(5*A + 3*B*x^3))/(45*x^(3/2))","A",1
146,1,57,61,0.0181585,"\int \frac{\left(a+b x^3\right)^2 \left(A+B x^3\right)}{x^{7/2}} \, dx","Integrate[((a + b*x^3)^2*(A + B*x^3))/x^(7/2),x]","\frac{2 \left(-91 a^2 \left(A-5 B x^3\right)+130 a b x^3 \left(7 A+B x^3\right)+5 b^2 x^6 \left(13 A+7 B x^3\right)\right)}{455 x^{5/2}}","-\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2}",1,"(2*(-91*a^2*(A - 5*B*x^3) + 130*a*b*x^3*(7*A + B*x^3) + 5*b^2*x^6*(13*A + 7*B*x^3)))/(455*x^(5/2))","A",1
147,1,71,85,0.0624233,"\int x^{7/2} \left(a+b x^3\right)^3 \left(A+B x^3\right) \, dx","Integrate[x^(7/2)*(a + b*x^3)^3*(A + B*x^3),x]","\frac{2 x^{9/2} \left(1155 a^3 A+693 a^2 x^3 (a B+3 A b)+385 b^2 x^9 (3 a B+A b)+1485 a b x^6 (a B+A b)+315 b^3 B x^{12}\right)}{10395}","\frac{2}{9} a^3 A x^{9/2}+\frac{2}{15} a^2 x^{15/2} (a B+3 A b)+\frac{2}{27} b^2 x^{27/2} (3 a B+A b)+\frac{2}{7} a b x^{21/2} (a B+A b)+\frac{2}{33} b^3 B x^{33/2}",1,"(2*x^(9/2)*(1155*a^3*A + 693*a^2*(3*A*b + a*B)*x^3 + 1485*a*b*(A*b + a*B)*x^6 + 385*b^2*(A*b + 3*a*B)*x^9 + 315*b^3*B*x^12))/10395","A",1
148,1,85,85,0.100497,"\int x^{5/2} \left(a+b x^3\right)^3 \left(A+B x^3\right) \, dx","Integrate[x^(5/2)*(a + b*x^3)^3*(A + B*x^3),x]","\frac{2}{7} a^3 A x^{7/2}+\frac{2}{13} a^2 x^{13/2} (a B+3 A b)+\frac{2}{25} b^2 x^{25/2} (3 a B+A b)+\frac{6}{19} a b x^{19/2} (a B+A b)+\frac{2}{31} b^3 B x^{31/2}","\frac{2}{7} a^3 A x^{7/2}+\frac{2}{13} a^2 x^{13/2} (a B+3 A b)+\frac{2}{25} b^2 x^{25/2} (3 a B+A b)+\frac{6}{19} a b x^{19/2} (a B+A b)+\frac{2}{31} b^3 B x^{31/2}",1,"(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(19/2))/19 + (2*b^2*(A*b + 3*a*B)*x^(25/2))/25 + (2*b^3*B*x^(31/2))/31","A",1
149,1,85,85,0.0834829,"\int x^{3/2} \left(a+b x^3\right)^3 \left(A+B x^3\right) \, dx","Integrate[x^(3/2)*(a + b*x^3)^3*(A + B*x^3),x]","\frac{2}{5} a^3 A x^{5/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{23} b^2 x^{23/2} (3 a B+A b)+\frac{6}{17} a b x^{17/2} (a B+A b)+\frac{2}{29} b^3 B x^{29/2}","\frac{2}{5} a^3 A x^{5/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{23} b^2 x^{23/2} (3 a B+A b)+\frac{6}{17} a b x^{17/2} (a B+A b)+\frac{2}{29} b^3 B x^{29/2}",1,"(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*b*(A*b + a*B)*x^(17/2))/17 + (2*b^2*(A*b + 3*a*B)*x^(23/2))/23 + (2*b^3*B*x^(29/2))/29","A",1
150,1,71,85,0.0662167,"\int \sqrt{x} \left(a+b x^3\right)^3 \left(A+B x^3\right) \, dx","Integrate[Sqrt[x]*(a + b*x^3)^3*(A + B*x^3),x]","\frac{2}{945} x^{3/2} \left(315 a^3 A+105 a^2 x^3 (a B+3 A b)+45 b^2 x^9 (3 a B+A b)+189 a b x^6 (a B+A b)+35 b^3 B x^{12}\right)","\frac{2}{3} a^3 A x^{3/2}+\frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{21} b^2 x^{21/2} (3 a B+A b)+\frac{2}{5} a b x^{15/2} (a B+A b)+\frac{2}{27} b^3 B x^{27/2}",1,"(2*x^(3/2)*(315*a^3*A + 105*a^2*(3*A*b + a*B)*x^3 + 189*a*b*(A*b + a*B)*x^6 + 45*b^2*(A*b + 3*a*B)*x^9 + 35*b^3*B*x^12))/945","A",1
151,1,83,83,0.0608577,"\int \frac{\left(a+b x^3\right)^3 \left(A+B x^3\right)}{\sqrt{x}} \, dx","Integrate[((a + b*x^3)^3*(A + B*x^3))/Sqrt[x],x]","2 a^3 A \sqrt{x}+\frac{2}{7} a^2 x^{7/2} (a B+3 A b)+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{25} b^3 B x^{25/2}","2 a^3 A \sqrt{x}+\frac{2}{7} a^2 x^{7/2} (a B+3 A b)+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{25} b^3 B x^{25/2}",1,"2*a^3*A*Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(7/2))/7 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*B)*x^(19/2))/19 + (2*b^3*B*x^(25/2))/25","A",1
152,1,81,83,0.0442672,"\int \frac{\left(a+b x^3\right)^3 \left(A+B x^3\right)}{x^{3/2}} \, dx","Integrate[((a + b*x^3)^3*(A + B*x^3))/x^(3/2),x]","\frac{-8602 a^3 \left(5 A-B x^3\right)+2346 a^2 b x^3 \left(11 A+5 B x^3\right)+690 a b^2 x^6 \left(17 A+11 B x^3\right)+110 b^3 x^9 \left(23 A+17 B x^3\right)}{21505 \sqrt{x}}","-\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{5} a^2 x^{5/2} (a B+3 A b)+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{23} b^3 B x^{23/2}",1,"(-8602*a^3*(5*A - B*x^3) + 2346*a^2*b*x^3*(11*A + 5*B*x^3) + 690*a*b^2*x^6*(17*A + 11*B*x^3) + 110*b^3*x^9*(23*A + 17*B*x^3))/(21505*Sqrt[x])","A",1
153,1,77,85,0.0395351,"\int \frac{\left(a+b x^3\right)^3 \left(A+B x^3\right)}{x^{5/2}} \, dx","Integrate[((a + b*x^3)^3*(A + B*x^3))/x^(5/2),x]","\frac{2 \left(-35 a^3 \left(A-B x^3\right)+35 a^2 b x^3 \left(3 A+B x^3\right)+7 a b^2 x^6 \left(5 A+3 B x^3\right)+b^3 x^9 \left(7 A+5 B x^3\right)\right)}{105 x^{3/2}}","-\frac{2 a^3 A}{3 x^{3/2}}+\frac{2}{3} a^2 x^{3/2} (a B+3 A b)+\frac{2}{15} b^2 x^{15/2} (3 a B+A b)+\frac{2}{3} a b x^{9/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2}",1,"(2*(-35*a^3*(A - B*x^3) + 35*a^2*b*x^3*(3*A + B*x^3) + 7*a*b^2*x^6*(5*A + 3*B*x^3) + b^3*x^9*(7*A + 5*B*x^3)))/(105*x^(3/2))","A",1
154,1,78,83,0.0850643,"\int \frac{\left(a+b x^3\right)^3 \left(A+B x^3\right)}{x^{7/2}} \, dx","Integrate[((a + b*x^3)^3*(A + B*x^3))/x^(7/2),x]","\frac{-3458 a^3 \left(A-5 B x^3\right)+7410 a^2 b x^3 \left(7 A+B x^3\right)+570 a b^2 x^6 \left(13 A+7 B x^3\right)+70 b^3 x^9 \left(19 A+13 B x^3\right)}{8645 x^{5/2}}","-\frac{2 a^3 A}{5 x^{5/2}}+2 a^2 \sqrt{x} (a B+3 A b)+\frac{2}{13} b^2 x^{13/2} (3 a B+A b)+\frac{6}{7} a b x^{7/2} (a B+A b)+\frac{2}{19} b^3 B x^{19/2}",1,"(-3458*a^3*(A - 5*B*x^3) + 7410*a^2*b*x^3*(7*A + B*x^3) + 570*a*b^2*x^6*(13*A + 7*B*x^3) + 70*b^3*x^9*(19*A + 13*B*x^3))/(8645*x^(5/2))","A",1
155,1,67,73,0.1241053,"\int \frac{x^{7/2} \left(A+B x^3\right)}{a+b x^3} \, dx","Integrate[(x^(7/2)*(A + B*x^3))/(a + b*x^3),x]","\frac{2 \sqrt{a} (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{3 b^{5/2}}+\frac{2 x^{3/2} \left(-3 a B+3 A b+b B x^3\right)}{9 b^2}","-\frac{2 \sqrt{a} (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{3 b^{5/2}}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{9/2}}{9 b}",1,"(2*x^(3/2)*(3*A*b - 3*a*B + b*B*x^3))/(9*b^2) + (2*Sqrt[a]*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(3*b^(5/2))","A",1
156,1,54,288,0.0925984,"\int \frac{x^{5/2} \left(A+B x^3\right)}{a+b x^3} \, dx","Integrate[(x^(5/2)*(A + B*x^3))/(a + b*x^3),x]","\frac{2 \sqrt{x} \left((7 a B-7 A b) \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right)-7 a B+7 A b+b B x^3\right)}{7 b^2}","\frac{\sqrt[6]{a} (A b-a B) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{2 \sqrt{3} b^{13/6}}-\frac{\sqrt[6]{a} (A b-a B) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{2 \sqrt{3} b^{13/6}}+\frac{\sqrt[6]{a} (A b-a B) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{3 b^{13/6}}-\frac{\sqrt[6]{a} (A b-a B) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{3 b^{13/6}}-\frac{2 \sqrt[6]{a} (A b-a B) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{3 b^{13/6}}+\frac{2 \sqrt{x} (A b-a B)}{b^2}+\frac{2 B x^{7/2}}{7 b}",1,"(2*Sqrt[x]*(7*A*b - 7*a*B + b*B*x^3 + (-7*A*b + 7*a*B)*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/a)]))/(7*b^2)","C",1
157,1,45,270,0.0711411,"\int \frac{x^{3/2} \left(A+B x^3\right)}{a+b x^3} \, dx","Integrate[(x^(3/2)*(A + B*x^3))/(a + b*x^3),x]","\frac{2 x^{5/2} \left((A b-a B) \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};-\frac{b x^3}{a}\right)+a B\right)}{5 a b}","\frac{(A b-a B) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{2 \sqrt{3} \sqrt[6]{a} b^{11/6}}-\frac{(A b-a B) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{2 \sqrt{3} \sqrt[6]{a} b^{11/6}}-\frac{(A b-a B) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{3 \sqrt[6]{a} b^{11/6}}+\frac{(A b-a B) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{3 \sqrt[6]{a} b^{11/6}}+\frac{2 (A b-a B) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{3 \sqrt[6]{a} b^{11/6}}+\frac{2 B x^{5/2}}{5 b}",1,"(2*x^(5/2)*(a*B + (A*b - a*B)*Hypergeometric2F1[5/6, 1, 11/6, -((b*x^3)/a)]))/(5*a*b)","C",1
158,1,52,53,0.0803278,"\int \frac{\sqrt{x} \left(A+B x^3\right)}{a+b x^3} \, dx","Integrate[(Sqrt[x]*(A + B*x^3))/(a + b*x^3),x]","\frac{2}{3} \left(\frac{B x^{3/2}}{b}-\frac{(a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}\right)","\frac{2 (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{3 \sqrt{a} b^{3/2}}+\frac{2 B x^{3/2}}{3 b}",1,"(2*((B*x^(3/2))/b - ((-(A*b) + a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(Sqrt[a]*b^(3/2))))/3","A",1
159,1,43,268,0.0291137,"\int \frac{A+B x^3}{\sqrt{x} \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(Sqrt[x]*(a + b*x^3)),x]","\frac{2 \sqrt{x} \left((A b-a B) \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right)+a B\right)}{a b}","-\frac{(A b-a B) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{2 \sqrt{3} a^{5/6} b^{7/6}}+\frac{(A b-a B) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{2 \sqrt{3} a^{5/6} b^{7/6}}-\frac{(A b-a B) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{3 a^{5/6} b^{7/6}}+\frac{(A b-a B) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{3 a^{5/6} b^{7/6}}+\frac{2 (A b-a B) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{3 a^{5/6} b^{7/6}}+\frac{2 B \sqrt{x}}{b}",1,"(2*Sqrt[x]*(a*B + (A*b - a*B)*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/a)]))/(a*b)","C",1
160,1,46,268,0.0140637,"\int \frac{A+B x^3}{x^{3/2} \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(x^(3/2)*(a + b*x^3)),x]","\frac{2 \left(x^3 (a B-A b) \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};-\frac{b x^3}{a}\right)-5 a A\right)}{5 a^2 \sqrt{x}}","-\frac{(A b-a B) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{2 \sqrt{3} a^{7/6} b^{5/6}}+\frac{(A b-a B) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{2 \sqrt{3} a^{7/6} b^{5/6}}+\frac{(A b-a B) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{3 a^{7/6} b^{5/6}}-\frac{(A b-a B) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{3 a^{7/6} b^{5/6}}-\frac{2 (A b-a B) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{3 a^{7/6} b^{5/6}}-\frac{2 A}{a \sqrt{x}}",1,"(2*(-5*a*A + (-(A*b) + a*B)*x^3*Hypergeometric2F1[5/6, 1, 11/6, -((b*x^3)/a)]))/(5*a^2*Sqrt[x])","C",1
161,1,53,53,0.0504935,"\int \frac{A+B x^3}{x^{5/2} \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(x^(5/2)*(a + b*x^3)),x]","\frac{2 (a B-A b) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{3 a^{3/2} \sqrt{b}}-\frac{2 A}{3 a x^{3/2}}","-\frac{2 (A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{3 a^{3/2} \sqrt{b}}-\frac{2 A}{3 a x^{3/2}}",1,"(-2*A)/(3*a*x^(3/2)) + (2*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(3*a^(3/2)*Sqrt[b])","A",1
162,1,47,270,0.0190398,"\int \frac{A+B x^3}{x^{7/2} \left(a+b x^3\right)} \, dx","Integrate[(A + B*x^3)/(x^(7/2)*(a + b*x^3)),x]","\frac{10 x^3 (a B-A b) \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right)-2 a A}{5 a^2 x^{5/2}}","\frac{(A b-a B) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{2 \sqrt{3} a^{11/6} \sqrt[6]{b}}-\frac{(A b-a B) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{2 \sqrt{3} a^{11/6} \sqrt[6]{b}}+\frac{(A b-a B) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{3 a^{11/6} \sqrt[6]{b}}-\frac{(A b-a B) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{3 a^{11/6} \sqrt[6]{b}}-\frac{2 (A b-a B) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{3 a^{11/6} \sqrt[6]{b}}-\frac{2 A}{5 a x^{5/2}}",1,"(-2*a*A + 10*(-(A*b) + a*B)*x^3*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/a)])/(5*a^2*x^(5/2))","C",1
163,1,77,95,0.1790809,"\int \frac{x^{7/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^(7/2)*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{\frac{(A b-3 a B) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{\sqrt{b} x^{3/2} \left(3 a B-A b+2 b B x^3\right)}{a+b x^3}}{3 b^{5/2}}","\frac{(A b-3 a B) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{3 \sqrt{a} b^{5/2}}-\frac{x^{3/2} (A b-3 a B)}{3 a b^2}+\frac{x^{9/2} (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"((Sqrt[b]*x^(3/2)*(-(A*b) + 3*a*B + 2*b*B*x^3))/(a + b*x^3) + ((A*b - 3*a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/Sqrt[a])/(3*b^(5/2))","A",1
164,1,76,312,0.0867646,"\int \frac{x^{5/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^(5/2)*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{\sqrt{x} \left(\left(a+b x^3\right) (A b-7 a B) \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right)+a \left(7 a B-A b+6 b B x^3\right)\right)}{3 a b^2 \left(a+b x^3\right)}","-\frac{(A b-7 a B) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{12 \sqrt{3} a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{12 \sqrt{3} a^{5/6} b^{13/6}}-\frac{(A b-7 a B) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{9 a^{5/6} b^{13/6}}-\frac{\sqrt{x} (A b-7 a B)}{3 a b^2}+\frac{x^{7/2} (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"(Sqrt[x]*(a*(-(A*b) + 7*a*B + 6*b*B*x^3) + (A*b - 7*a*B)*(a + b*x^3)*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/a)]))/(3*a*b^2*(a + b*x^3))","C",1
165,1,62,289,0.0977146,"\int \frac{x^{3/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^(3/2)*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{2 x^{5/2} \left((A b-a B) \, _2F_1\left(\frac{5}{6},2;\frac{11}{6};-\frac{b x^3}{a}\right)+a B \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};-\frac{b x^3}{a}\right)\right)}{5 a^2 b}","\frac{(5 a B+A b) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{12 \sqrt{3} a^{7/6} b^{11/6}}-\frac{(5 a B+A b) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{12 \sqrt{3} a^{7/6} b^{11/6}}-\frac{(5 a B+A b) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{18 a^{7/6} b^{11/6}}+\frac{(5 a B+A b) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{18 a^{7/6} b^{11/6}}+\frac{(5 a B+A b) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{9 a^{7/6} b^{11/6}}+\frac{x^{5/2} (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"(2*x^(5/2)*(a*B*Hypergeometric2F1[5/6, 1, 11/6, -((b*x^3)/a)] + (A*b - a*B)*Hypergeometric2F1[5/6, 2, 11/6, -((b*x^3)/a)]))/(5*a^2*b)","C",1
166,1,71,71,0.0552729,"\int \frac{\sqrt{x} \left(A+B x^3\right)}{\left(a+b x^3\right)^2} \, dx","Integrate[(Sqrt[x]*(A + B*x^3))/(a + b*x^3)^2,x]","\frac{(a B+A b) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{3 a^{3/2} b^{3/2}}+\frac{x^{3/2} (A b-a B)}{3 a b \left(a+b x^3\right)}","\frac{(a B+A b) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{3 a^{3/2} b^{3/2}}+\frac{x^{3/2} (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"((A*b - a*B)*x^(3/2))/(3*a*b*(a + b*x^3)) + ((A*b + a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(3*a^(3/2)*b^(3/2))","A",1
167,1,68,289,0.0321434,"\int \frac{A+B x^3}{\sqrt{x} \left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(Sqrt[x]*(a + b*x^3)^2),x]","\frac{\sqrt{x} \left(\left(a+b x^3\right) (a B+5 A b) \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right)+a (A b-a B)\right)}{3 a^2 b \left(a+b x^3\right)}","-\frac{(a B+5 A b) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{12 \sqrt{3} a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{12 \sqrt{3} a^{11/6} b^{7/6}}-\frac{(a B+5 A b) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{18 a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{18 a^{11/6} b^{7/6}}+\frac{(a B+5 A b) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{9 a^{11/6} b^{7/6}}+\frac{\sqrt{x} (A b-a B)}{3 a b \left(a+b x^3\right)}",1,"(Sqrt[x]*(a*(A*b - a*B) + (5*A*b + a*B)*(a + b*x^3)*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/a)]))/(3*a^2*b*(a + b*x^3))","C",1
168,1,70,318,0.1367423,"\int \frac{A+B x^3}{x^{3/2} \left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(x^(3/2)*(a + b*x^3)^2),x]","\frac{2 \left(x^3 (a B-A b) \, _2F_1\left(\frac{5}{6},2;\frac{11}{6};-\frac{b x^3}{a}\right)-A b x^3 \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};-\frac{b x^3}{a}\right)-5 a A\right)}{5 a^3 \sqrt{x}}","-\frac{(7 A b-a B) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{12 \sqrt{3} a^{13/6} b^{5/6}}+\frac{(7 A b-a B) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{12 \sqrt{3} a^{13/6} b^{5/6}}+\frac{(7 A b-a B) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{18 a^{13/6} b^{5/6}}-\frac{(7 A b-a B) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{18 a^{13/6} b^{5/6}}-\frac{(7 A b-a B) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{9 a^{13/6} b^{5/6}}-\frac{7 A b-a B}{3 a^2 b \sqrt{x}}+\frac{A b-a B}{3 a b \sqrt{x} \left(a+b x^3\right)}",1,"(2*(-5*a*A - A*b*x^3*Hypergeometric2F1[5/6, 1, 11/6, -((b*x^3)/a)] + (-(A*b) + a*B)*x^3*Hypergeometric2F1[5/6, 2, 11/6, -((b*x^3)/a)]))/(5*a^3*Sqrt[x])","C",1
169,1,79,96,0.1295321,"\int \frac{A+B x^3}{x^{5/2} \left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(x^(5/2)*(a + b*x^3)^2),x]","\frac{\frac{(a B-3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{\sqrt{b}}+\frac{\sqrt{a} \left(-2 a A+a B x^3-3 A b x^3\right)}{x^{3/2} \left(a+b x^3\right)}}{3 a^{5/2}}","-\frac{(3 A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{3 a^{5/2} \sqrt{b}}+\frac{a B-3 A b}{3 a^2 b x^{3/2}}+\frac{A b-a B}{3 a b x^{3/2} \left(a+b x^3\right)}",1,"((Sqrt[a]*(-2*a*A - 3*A*b*x^3 + a*B*x^3))/(x^(3/2)*(a + b*x^3)) + ((-3*A*b + a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/Sqrt[b])/(3*a^(5/2))","A",1
170,1,74,318,0.1112579,"\int \frac{A+B x^3}{x^{7/2} \left(a+b x^3\right)^2} \, dx","Integrate[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^2),x]","\frac{5 x^3 (5 a B-11 A b) \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right)+\frac{a \left(-6 a A+5 a B x^3-11 A b x^3\right)}{a+b x^3}}{15 a^3 x^{5/2}}","\frac{(11 A b-5 a B) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{12 \sqrt{3} a^{17/6} \sqrt[6]{b}}-\frac{(11 A b-5 a B) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{12 \sqrt{3} a^{17/6} \sqrt[6]{b}}+\frac{(11 A b-5 a B) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{18 a^{17/6} \sqrt[6]{b}}-\frac{(11 A b-5 a B) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{18 a^{17/6} \sqrt[6]{b}}-\frac{(11 A b-5 a B) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{9 a^{17/6} \sqrt[6]{b}}-\frac{11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac{A b-a B}{3 a b x^{5/2} \left(a+b x^3\right)}",1,"((a*(-6*a*A - 11*A*b*x^3 + 5*a*B*x^3))/(a + b*x^3) + 5*(-11*A*b + 5*a*B)*x^3*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/a)])/(15*a^3*x^(5/2))","C",1
171,1,93,104,0.1938005,"\int \frac{x^{7/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^(7/2)*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{\frac{\sqrt{a} \sqrt{b} x^{3/2} \left(-3 a^2 B-a b \left(A+5 B x^3\right)+A b^2 x^3\right)}{\left(a+b x^3\right)^2}+(3 a B+A b) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{12 a^{3/2} b^{5/2}}","\frac{(3 a B+A b) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{12 a^{3/2} b^{5/2}}-\frac{x^{3/2} (3 a B+A b)}{12 a b^2 \left(a+b x^3\right)}+\frac{x^{9/2} (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"((Sqrt[a]*Sqrt[b]*x^(3/2)*(-3*a^2*B + A*b^2*x^3 - a*b*(A + 5*B*x^3)))/(a + b*x^3)^2 + (A*b + 3*a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(12*a^(3/2)*b^(5/2))","A",1
172,1,92,327,0.1102443,"\int \frac{x^{5/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^(5/2)*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{\sqrt{x} \left(a \left(-7 a^2 B-a b \left(5 A+13 B x^3\right)+A b^2 x^3\right)+\left(a+b x^3\right)^2 (7 a B+5 A b) \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right)\right)}{36 a^2 b^2 \left(a+b x^3\right)^2}","-\frac{(7 a B+5 A b) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{144 \sqrt{3} a^{11/6} b^{13/6}}+\frac{(7 a B+5 A b) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{144 \sqrt{3} a^{11/6} b^{13/6}}-\frac{(7 a B+5 A b) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{216 a^{11/6} b^{13/6}}+\frac{(7 a B+5 A b) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{216 a^{11/6} b^{13/6}}+\frac{(7 a B+5 A b) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{108 a^{11/6} b^{13/6}}-\frac{\sqrt{x} (7 a B+5 A b)}{36 a b^2 \left(a+b x^3\right)}+\frac{x^{7/2} (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"(Sqrt[x]*(a*(-7*a^2*B + A*b^2*x^3 - a*b*(5*A + 13*B*x^3)) + (5*A*b + 7*a*B)*(a + b*x^3)^2*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/a)]))/(36*a^2*b^2*(a + b*x^3)^2)","C",1
173,1,62,327,0.0647402,"\int \frac{x^{3/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^(3/2)*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{2 x^{5/2} \left((A b-a B) \, _2F_1\left(\frac{5}{6},3;\frac{11}{6};-\frac{b x^3}{a}\right)+a B \, _2F_1\left(\frac{5}{6},2;\frac{11}{6};-\frac{b x^3}{a}\right)\right)}{5 a^3 b}","\frac{(5 a B+7 A b) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{144 \sqrt{3} a^{13/6} b^{11/6}}-\frac{(5 a B+7 A b) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{144 \sqrt{3} a^{13/6} b^{11/6}}-\frac{(5 a B+7 A b) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{216 a^{13/6} b^{11/6}}+\frac{(5 a B+7 A b) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{216 a^{13/6} b^{11/6}}+\frac{(5 a B+7 A b) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{108 a^{13/6} b^{11/6}}+\frac{x^{5/2} (5 a B+7 A b)}{36 a^2 b \left(a+b x^3\right)}+\frac{x^{5/2} (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"(2*x^(5/2)*(a*B*Hypergeometric2F1[5/6, 2, 11/6, -((b*x^3)/a)] + (A*b - a*B)*Hypergeometric2F1[5/6, 3, 11/6, -((b*x^3)/a)]))/(5*a^3*b)","C",1
174,1,94,104,0.1075822,"\int \frac{\sqrt{x} \left(A+B x^3\right)}{\left(a+b x^3\right)^3} \, dx","Integrate[(Sqrt[x]*(A + B*x^3))/(a + b*x^3)^3,x]","\frac{\frac{\sqrt{a} \sqrt{b} x^{3/2} \left(-a^2 B+a b \left(5 A+B x^3\right)+3 A b^2 x^3\right)}{\left(a+b x^3\right)^2}+(a B+3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{12 a^{5/2} b^{3/2}}","\frac{(a B+3 A b) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{12 a^{5/2} b^{3/2}}+\frac{x^{3/2} (a B+3 A b)}{12 a^2 b \left(a+b x^3\right)}+\frac{x^{3/2} (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"((Sqrt[a]*Sqrt[b]*x^(3/2)*(-(a^2*B) + 3*A*b^2*x^3 + a*b*(5*A + B*x^3)))/(a + b*x^3)^2 + (3*A*b + a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(12*a^(5/2)*b^(3/2))","A",1
175,1,91,321,0.0479311,"\int \frac{A+B x^3}{\sqrt{x} \left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(Sqrt[x]*(a + b*x^3)^3),x]","\frac{\sqrt{x} \left(a \left(-5 a^2 B+a b \left(17 A+B x^3\right)+11 A b^2 x^3\right)+5 \left(a+b x^3\right)^2 (a B+11 A b) \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right)\right)}{36 a^3 b \left(a+b x^3\right)^2}","-\frac{5 (a B+11 A b) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{144 \sqrt{3} a^{17/6} b^{7/6}}+\frac{5 (a B+11 A b) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{144 \sqrt{3} a^{17/6} b^{7/6}}-\frac{5 (a B+11 A b) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{216 a^{17/6} b^{7/6}}+\frac{5 (a B+11 A b) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{216 a^{17/6} b^{7/6}}+\frac{5 (a B+11 A b) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{108 a^{17/6} b^{7/6}}+\frac{\sqrt{x} (a B+11 A b)}{36 a^2 b \left(a+b x^3\right)}+\frac{\sqrt{x} (A b-a B)}{6 a b \left(a+b x^3\right)^2}",1,"(Sqrt[x]*(a*(-5*a^2*B + 11*A*b^2*x^3 + a*b*(17*A + B*x^3)) + 5*(11*A*b + a*B)*(a + b*x^3)^2*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/a)]))/(36*a^3*b*(a + b*x^3)^2)","C",1
176,1,113,351,0.1328181,"\int \frac{A+B x^3}{x^{3/2} \left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(x^(3/2)*(a + b*x^3)^3),x]","2 \left(-\frac{x^{5/2} (A b-a B) \, _2F_1\left(\frac{5}{6},3;\frac{11}{6};-\frac{b x^3}{a}\right)}{5 a^4}-\frac{A b x^{5/2} \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};-\frac{b x^3}{a}\right)}{5 a^4}-\frac{A b x^{5/2} \, _2F_1\left(\frac{5}{6},2;\frac{11}{6};-\frac{b x^3}{a}\right)}{5 a^4}-\frac{A}{a^3 \sqrt{x}}\right)","-\frac{7 (13 A b-a B) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{144 \sqrt{3} a^{19/6} b^{5/6}}+\frac{7 (13 A b-a B) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{144 \sqrt{3} a^{19/6} b^{5/6}}+\frac{7 (13 A b-a B) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{216 a^{19/6} b^{5/6}}-\frac{7 (13 A b-a B) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{216 a^{19/6} b^{5/6}}-\frac{7 (13 A b-a B) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{108 a^{19/6} b^{5/6}}-\frac{7 (13 A b-a B)}{36 a^3 b \sqrt{x}}+\frac{13 A b-a B}{36 a^2 b \sqrt{x} \left(a+b x^3\right)}+\frac{A b-a B}{6 a b \sqrt{x} \left(a+b x^3\right)^2}",1,"2*(-(A/(a^3*Sqrt[x])) - (A*b*x^(5/2)*Hypergeometric2F1[5/6, 1, 11/6, -((b*x^3)/a)])/(5*a^4) - (A*b*x^(5/2)*Hypergeometric2F1[5/6, 2, 11/6, -((b*x^3)/a)])/(5*a^4) - ((A*b - a*B)*x^(5/2)*Hypergeometric2F1[5/6, 3, 11/6, -((b*x^3)/a)])/(5*a^4))","C",1
177,1,102,129,0.1854747,"\int \frac{A+B x^3}{x^{5/2} \left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(x^(5/2)*(a + b*x^3)^3),x]","\frac{(a B-5 A b) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{4 a^{7/2} \sqrt{b}}+\frac{a^2 \left(5 B x^3-8 A\right)+a \left(3 b B x^6-25 A b x^3\right)-15 A b^2 x^6}{12 a^3 x^{3/2} \left(a+b x^3\right)^2}","-\frac{(5 A b-a B) \tan ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{4 a^{7/2} \sqrt{b}}+\frac{a B-5 A b}{4 a^3 b x^{3/2}}+\frac{5 A b-a B}{12 a^2 b x^{3/2} \left(a+b x^3\right)}+\frac{A b-a B}{6 a b x^{3/2} \left(a+b x^3\right)^2}",1,"(-15*A*b^2*x^6 + a^2*(-8*A + 5*B*x^3) + a*(-25*A*b*x^3 + 3*b*B*x^6))/(12*a^3*x^(3/2)*(a + b*x^3)^2) + ((-5*A*b + a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(4*a^(7/2)*Sqrt[b])","A",1
178,1,96,351,0.1216553,"\int \frac{A+B x^3}{x^{7/2} \left(a+b x^3\right)^3} \, dx","Integrate[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^3),x]","\frac{\frac{a \left(a^2 \left(85 B x^3-72 A\right)+a \left(55 b B x^6-289 A b x^3\right)-187 A b^2 x^6\right)}{\left(a+b x^3\right)^2}+55 x^3 (5 a B-17 A b) \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right)}{180 a^4 x^{5/2}}","\frac{11 (17 A b-5 a B) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{144 \sqrt{3} a^{23/6} \sqrt[6]{b}}-\frac{11 (17 A b-5 a B) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{b} \sqrt{x}+\sqrt[3]{a}+\sqrt[3]{b} x\right)}{144 \sqrt{3} a^{23/6} \sqrt[6]{b}}+\frac{11 (17 A b-5 a B) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{216 a^{23/6} \sqrt[6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left(\frac{2 \sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}+\sqrt{3}\right)}{216 a^{23/6} \sqrt[6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt{x}}{\sqrt[6]{a}}\right)}{108 a^{23/6} \sqrt[6]{b}}-\frac{11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left(a+b x^3\right)}+\frac{A b-a B}{6 a b x^{5/2} \left(a+b x^3\right)^2}",1,"((a*(-187*A*b^2*x^6 + a^2*(-72*A + 85*B*x^3) + a*(-289*A*b*x^3 + 55*b*B*x^6)))/(a + b*x^3)^2 + 55*(-17*A*b + 5*a*B)*x^3*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/a)])/(180*a^4*x^(5/2))","C",1
179,1,75,103,0.062169,"\int x^8 \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[x^8*Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{2 \left(a+b x^3\right)^{3/2} \left(-16 a^3 B+24 a^2 b \left(A+B x^3\right)-6 a b^2 x^3 \left(6 A+5 B x^3\right)+5 b^3 x^6 \left(9 A+7 B x^3\right)\right)}{945 b^4}","\frac{2 a^2 \left(a+b x^3\right)^{3/2} (A b-a B)}{9 b^4}+\frac{2 \left(a+b x^3\right)^{7/2} (A b-3 a B)}{21 b^4}-\frac{2 a \left(a+b x^3\right)^{5/2} (2 A b-3 a B)}{15 b^4}+\frac{2 B \left(a+b x^3\right)^{9/2}}{27 b^4}",1,"(2*(a + b*x^3)^(3/2)*(-16*a^3*B + 24*a^2*b*(A + B*x^3) - 6*a*b^2*x^3*(6*A + 5*B*x^3) + 5*b^3*x^6*(9*A + 7*B*x^3)))/(945*b^4)","A",1
180,1,57,73,0.0444462,"\int x^5 \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[x^5*Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{2 \left(a+b x^3\right)^{3/2} \left(8 a^2 B-2 a b \left(7 A+6 B x^3\right)+3 b^2 x^3 \left(7 A+5 B x^3\right)\right)}{315 b^3}","\frac{2 \left(a+b x^3\right)^{5/2} (A b-2 a B)}{15 b^3}-\frac{2 a \left(a+b x^3\right)^{3/2} (A b-a B)}{9 b^3}+\frac{2 B \left(a+b x^3\right)^{7/2}}{21 b^3}",1,"(2*(a + b*x^3)^(3/2)*(8*a^2*B + 3*b^2*x^3*(7*A + 5*B*x^3) - 2*a*b*(7*A + 6*B*x^3)))/(315*b^3)","A",1
181,1,34,46,0.0307088,"\int x^2 \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[x^2*Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{2 \left(a+b x^3\right)^{3/2} \left(-2 a B+5 A b+3 b B x^3\right)}{45 b^2}","\frac{2 \left(a+b x^3\right)^{3/2} (A b-a B)}{9 b^2}+\frac{2 B \left(a+b x^3\right)^{5/2}}{15 b^2}",1,"(2*(a + b*x^3)^(3/2)*(5*A*b - 2*a*B + 3*b*B*x^3))/(45*b^2)","A",1
182,1,60,64,0.0512183,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x,x]","\frac{2}{9} \left(\frac{\sqrt{a+b x^3} \left(B \left(a+b x^3\right)+3 A b\right)}{b}-3 \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)","\frac{2}{3} A \sqrt{a+b x^3}-\frac{2}{3} \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)+\frac{2 B \left(a+b x^3\right)^{3/2}}{9 b}",1,"(2*((Sqrt[a + b*x^3]*(3*A*b + B*(a + b*x^3)))/b - 3*Sqrt[a]*A*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]))/9","A",1
183,1,63,84,0.0459233,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^4} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^4,x]","\frac{1}{3} \left(\frac{\sqrt{a+b x^3} \left(2 B x^3-A\right)}{x^3}-\frac{(2 a B+A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{\sqrt{a}}\right)","\frac{\sqrt{a+b x^3} (2 a B+A b)}{3 a}-\frac{(2 a B+A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}-\frac{A \left(a+b x^3\right)^{3/2}}{3 a x^3}",1,"((Sqrt[a + b*x^3]*(-A + 2*B*x^3))/x^3 - ((A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/Sqrt[a])/3","A",1
184,1,93,88,0.0754223,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^7} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^7,x]","\frac{-\left(a+b x^3\right) \left(2 a \left(A+2 B x^3\right)+A b x^3\right)-b x^6 \sqrt{\frac{b x^3}{a}+1} (4 a B-A b) \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)}{12 a x^6 \sqrt{a+b x^3}}","\frac{b (A b-4 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{12 a^{3/2}}+\frac{\sqrt{a+b x^3} (A b-4 a B)}{12 a x^3}-\frac{A \left(a+b x^3\right)^{3/2}}{6 a x^6}",1,"(-((a + b*x^3)*(A*b*x^3 + 2*a*(A + 2*B*x^3))) - b*(-(A*b) + 4*a*B)*x^6*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1 + (b*x^3)/a]])/(12*a*x^6*Sqrt[a + b*x^3])","A",1
185,1,89,303,0.1655453,"\int x^3 \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[x^3*Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{2 x \sqrt{a+b x^3} \left(\frac{a (8 a B-17 A b) \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-\left(a+b x^3\right) \left(8 a B-17 A b-11 b B x^3\right)\right)}{187 b^2}","-\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (17 A b-8 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{935 b^{7/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{6 a x \sqrt{a+b x^3} (17 A b-8 a B)}{935 b^2}+\frac{2 x^4 \sqrt{a+b x^3} (17 A b-8 a B)}{187 b}+\frac{2 B x^4 \left(a+b x^3\right)^{3/2}}{17 b}",1,"(2*x*Sqrt[a + b*x^3]*(-((a + b*x^3)*(-17*A*b + 8*a*B - 11*b*B*x^3)) + (a*(-17*A*b + 8*a*B)*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(187*b^2)","C",1
186,1,75,268,0.0920997,"\int \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{2 x \sqrt{a+b x^3} \left(\frac{(11 A b-2 a B) \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{\frac{b x^3}{a}+1}}+B \left(a+b x^3\right)\right)}{11 b}","\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (11 A b-2 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{55 b^{4/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x \sqrt{a+b x^3} (11 A b-2 a B)}{55 b}+\frac{2 B x \left(a+b x^3\right)^{3/2}}{11 b}",1,"(2*x*Sqrt[a + b*x^3]*(B*(a + b*x^3) + ((11*A*b - 2*a*B)*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)])/(2*Sqrt[1 + (b*x^3)/a])))/(11*b)","C",1
187,1,81,269,0.0749236,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^3} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^3,x]","\frac{\sqrt{a+b x^3} \left(\frac{x^3 (4 a B+5 A b) \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{\frac{b x^3}{a}+1}}-A \left(a+b x^3\right)\right)}{2 a x^2}","\frac{3^{3/4} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (4 a B+5 A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{10 \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{x \sqrt{a+b x^3} (4 a B+5 A b)}{10 a}-\frac{A \left(a+b x^3\right)^{3/2}}{2 a x^2}",1,"(Sqrt[a + b*x^3]*(-(A*(a + b*x^3)) + ((5*A*b + 4*a*B)*x^3*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)])/(2*Sqrt[1 + (b*x^3)/a])))/(2*a*x^2)","C",1
188,1,80,272,0.179357,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^6} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^6,x]","\frac{\sqrt{a+b x^3} \left(\frac{x^3 \left(\frac{A b}{2}-5 a B\right) \, _2F_1\left(-\frac{2}{3},-\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-2 A \left(a+b x^3\right)\right)}{10 a x^5}","-\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (A b-10 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{20 a \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{a+b x^3} (A b-10 a B)}{20 a x^2}-\frac{A \left(a+b x^3\right)^{3/2}}{5 a x^5}",1,"(Sqrt[a + b*x^3]*(-2*A*(a + b*x^3) + (((A*b)/2 - 5*a*B)*x^3*Hypergeometric2F1[-2/3, -1/2, 1/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(10*a*x^5)","C",1
189,1,80,305,0.1565838,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^9} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^9,x]","\frac{\sqrt{a+b x^3} \left(\frac{x^3 \left(\frac{7 A b}{2}-8 a B\right) \, _2F_1\left(-\frac{5}{3},-\frac{1}{2};-\frac{2}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-5 A \left(a+b x^3\right)\right)}{40 a x^8}","\frac{3 b \sqrt{a+b x^3} (7 A b-16 a B)}{320 a^2 x^2}+\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (7 A b-16 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{320 a^2 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{a+b x^3} (7 A b-16 a B)}{80 a x^5}-\frac{A \left(a+b x^3\right)^{3/2}}{8 a x^8}",1,"(Sqrt[a + b*x^3]*(-5*A*(a + b*x^3) + (((7*A*b)/2 - 8*a*B)*x^3*Hypergeometric2F1[-5/3, -1/2, -2/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(40*a*x^8)","C",1
190,1,91,581,0.1515343,"\int x^4 \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[x^4*Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{2 x^2 \sqrt{a+b x^3} \left(\frac{a (10 a B-19 A b) \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-\left(a+b x^3\right) \left(10 a B-19 A b-13 b B x^3\right)\right)}{247 b^2}","-\frac{8 \sqrt{2} 3^{3/4} a^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (19 A b-10 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{1729 b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{12 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (19 A b-10 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{1729 b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{24 a^2 \sqrt{a+b x^3} (19 A b-10 a B)}{1729 b^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{6 a x^2 \sqrt{a+b x^3} (19 A b-10 a B)}{1729 b^2}+\frac{2 x^5 \sqrt{a+b x^3} (19 A b-10 a B)}{247 b}+\frac{2 B x^5 \left(a+b x^3\right)^{3/2}}{19 b}",1,"(2*x^2*Sqrt[a + b*x^3]*(-((a + b*x^3)*(-19*A*b + 10*a*B - 13*b*B*x^3)) + (a*(-19*A*b + 10*a*B)*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(247*b^2)","C",1
191,1,75,548,0.1098947,"\int x \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[x*Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{x^2 \sqrt{a+b x^3} \left(\frac{(13 A b-4 a B) \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}+4 B \left(a+b x^3\right)\right)}{26 b}","\frac{2 \sqrt{2} 3^{3/4} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (13 A b-4 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{91 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (13 A b-4 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{91 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{6 a \sqrt{a+b x^3} (13 A b-4 a B)}{91 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x^2 \sqrt{a+b x^3} (13 A b-4 a B)}{91 b}+\frac{2 B x^2 \left(a+b x^3\right)^{3/2}}{13 b}",1,"(x^2*Sqrt[a + b*x^3]*(4*B*(a + b*x^3) + ((13*A*b - 4*a*B)*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(26*b)","C",1
192,1,81,545,0.0777306,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^2} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^2,x]","\frac{\sqrt{a+b x^3} \left(\frac{x^3 (2 a B+7 A b) \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{\frac{b x^3}{a}+1}}-2 A \left(a+b x^3\right)\right)}{2 a x}","\frac{\sqrt{2} 3^{3/4} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a B+7 A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{7 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a B+7 A b) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{14 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{3 \sqrt{a+b x^3} (2 a B+7 A b)}{7 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{x^2 \sqrt{a+b x^3} (2 a B+7 A b)}{7 a}-\frac{A \left(a+b x^3\right)^{3/2}}{a x}",1,"(Sqrt[a + b*x^3]*(-2*A*(a + b*x^3) + ((7*A*b + 2*a*B)*x^3*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)])/(2*Sqrt[1 + (b*x^3)/a])))/(2*a*x)","C",1
193,1,80,546,0.0949845,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^5} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^5,x]","\frac{\sqrt{a+b x^3} \left(-\frac{x^3 (8 a B+A b) \, _2F_1\left(-\frac{1}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{\frac{b x^3}{a}+1}}-A \left(a+b x^3\right)\right)}{4 a x^4}","\frac{3^{3/4} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (8 a B+A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt{2} a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (8 a B+A b) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{16 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt{a+b x^3} (8 a B+A b)}{8 a x}+\frac{3 \sqrt[3]{b} \sqrt{a+b x^3} (8 a B+A b)}{8 a \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{A \left(a+b x^3\right)^{3/2}}{4 a x^4}",1,"(Sqrt[a + b*x^3]*(-(A*(a + b*x^3)) - ((A*b + 8*a*B)*x^3*Hypergeometric2F1[-1/2, -1/3, 2/3, -((b*x^3)/a)])/(2*Sqrt[1 + (b*x^3)/a])))/(4*a*x^4)","C",1
194,1,80,581,0.1013342,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^8} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^8,x]","\frac{\sqrt{a+b x^3} \left(\frac{x^3 \left(\frac{5 A b}{2}-7 a B\right) \, _2F_1\left(-\frac{4}{3},-\frac{1}{2};-\frac{1}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-4 A \left(a+b x^3\right)\right)}{28 a x^7}","-\frac{3^{3/4} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 A b-14 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{56 \sqrt{2} a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 A b-14 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{224 a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 b^{4/3} \sqrt{a+b x^3} (5 A b-14 a B)}{112 a^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{3 b \sqrt{a+b x^3} (5 A b-14 a B)}{112 a^2 x}+\frac{\sqrt{a+b x^3} (5 A b-14 a B)}{56 a x^4}-\frac{A \left(a+b x^3\right)^{3/2}}{7 a x^7}",1,"(Sqrt[a + b*x^3]*(-4*A*(a + b*x^3) + (((5*A*b)/2 - 7*a*B)*x^3*Hypergeometric2F1[-4/3, -1/2, -1/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(28*a*x^7)","C",1
195,1,80,614,0.1037695,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^{11}} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^11,x]","\frac{\sqrt{a+b x^3} \left(\frac{x^3 \left(\frac{11 A b}{2}-10 a B\right) \, _2F_1\left(-\frac{7}{3},-\frac{1}{2};-\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-7 A \left(a+b x^3\right)\right)}{70 a x^{10}}","\frac{3^{3/4} b^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (11 A b-20 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{224 \sqrt{2} a^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (11 A b-20 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{896 a^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{3 b^{7/3} \sqrt{a+b x^3} (11 A b-20 a B)}{448 a^3 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{3 b^2 \sqrt{a+b x^3} (11 A b-20 a B)}{448 a^3 x}+\frac{3 b \sqrt{a+b x^3} (11 A b-20 a B)}{1120 a^2 x^4}+\frac{\sqrt{a+b x^3} (11 A b-20 a B)}{140 a x^7}-\frac{A \left(a+b x^3\right)^{3/2}}{10 a x^{10}}",1,"(Sqrt[a + b*x^3]*(-7*A*(a + b*x^3) + (((11*A*b)/2 - 10*a*B)*x^3*Hypergeometric2F1[-7/3, -1/2, -4/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(70*a*x^10)","C",1
196,1,78,103,0.0918879,"\int x^8 \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[x^8*(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{2 \left(a+b x^3\right)^{5/2} \left(-48 a^3 B+8 a^2 b \left(11 A+15 B x^3\right)-10 a b^2 x^3 \left(22 A+21 B x^3\right)+35 b^3 x^6 \left(11 A+9 B x^3\right)\right)}{10395 b^4}","\frac{2 a^2 \left(a+b x^3\right)^{5/2} (A b-a B)}{15 b^4}+\frac{2 \left(a+b x^3\right)^{9/2} (A b-3 a B)}{27 b^4}-\frac{2 a \left(a+b x^3\right)^{7/2} (2 A b-3 a B)}{21 b^4}+\frac{2 B \left(a+b x^3\right)^{11/2}}{33 b^4}",1,"(2*(a + b*x^3)^(5/2)*(-48*a^3*B + 35*b^3*x^6*(11*A + 9*B*x^3) + 8*a^2*b*(11*A + 15*B*x^3) - 10*a*b^2*x^3*(22*A + 21*B*x^3)))/(10395*b^4)","A",1
197,1,57,73,0.0549803,"\int x^5 \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[x^5*(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{2 \left(a+b x^3\right)^{5/2} \left(8 a^2 B-2 a b \left(9 A+10 B x^3\right)+5 b^2 x^3 \left(9 A+7 B x^3\right)\right)}{945 b^3}","\frac{2 \left(a+b x^3\right)^{7/2} (A b-2 a B)}{21 b^3}-\frac{2 a \left(a+b x^3\right)^{5/2} (A b-a B)}{15 b^3}+\frac{2 B \left(a+b x^3\right)^{9/2}}{27 b^3}",1,"(2*(a + b*x^3)^(5/2)*(8*a^2*B + 5*b^2*x^3*(9*A + 7*B*x^3) - 2*a*b*(9*A + 10*B*x^3)))/(945*b^3)","A",1
198,1,34,46,0.0277704,"\int x^2 \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[x^2*(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{2 \left(a+b x^3\right)^{5/2} \left(-2 a B+7 A b+5 b B x^3\right)}{105 b^2}","\frac{2 \left(a+b x^3\right)^{5/2} (A b-a B)}{15 b^2}+\frac{2 B \left(a+b x^3\right)^{7/2}}{21 b^2}",1,"(2*(a + b*x^3)^(5/2)*(7*A*b - 2*a*B + 5*b*B*x^3))/(105*b^2)","A",1
199,1,80,81,0.0820781,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{x} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x,x]","\frac{2 \left(-15 a^{3/2} A b \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)+5 A b \left(a+b x^3\right)^{3/2}+15 a A b \sqrt{a+b x^3}+3 B \left(a+b x^3\right)^{5/2}\right)}{45 b}","-\frac{2}{3} a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)+\frac{2}{9} A \left(a+b x^3\right)^{3/2}+\frac{2}{3} a A \sqrt{a+b x^3}+\frac{2 B \left(a+b x^3\right)^{5/2}}{15 b}",1,"(2*(15*a*A*b*Sqrt[a + b*x^3] + 5*A*b*(a + b*x^3)^(3/2) + 3*B*(a + b*x^3)^(5/2) - 15*a^(3/2)*A*b*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]))/(45*b)","A",1
200,1,80,110,0.0896151,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{x^4} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^4,x]","\frac{1}{9} \left(\frac{\sqrt{a+b x^3} \left(-3 a A+8 a B x^3+6 A b x^3+2 b B x^6\right)}{x^3}-3 \sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)\right)","\frac{\left(a+b x^3\right)^{3/2} (2 a B+3 A b)}{9 a}+\frac{1}{3} \sqrt{a+b x^3} (2 a B+3 A b)-\frac{1}{3} \sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)-\frac{A \left(a+b x^3\right)^{5/2}}{3 a x^3}",1,"((Sqrt[a + b*x^3]*(-3*a*A + 6*A*b*x^3 + 8*a*B*x^3 + 2*b*B*x^6))/x^3 - 3*Sqrt[a]*(3*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/9","A",1
201,1,59,115,0.0363792,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{x^7} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^7,x]","\frac{\left(a+b x^3\right)^{5/2} \left(b x^6 (4 a B+A b) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^3}{a}+1\right)-5 a^2 A\right)}{30 a^3 x^6}","-\frac{\left(a+b x^3\right)^{3/2} (4 a B+A b)}{12 a x^3}+\frac{b \sqrt{a+b x^3} (4 a B+A b)}{4 a}-\frac{b (4 a B+A b) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{4 \sqrt{a}}-\frac{A \left(a+b x^3\right)^{5/2}}{6 a x^6}",1,"((a + b*x^3)^(5/2)*(-5*a^2*A + b*(A*b + 4*a*B)*x^6*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^3)/a]))/(30*a^3*x^6)","C",1
202,1,93,336,0.1743257,"\int x^3 \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[x^3*(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{2 x \sqrt{a+b x^3} \left(\frac{a^2 (8 a B-23 A b) \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-\left(a+b x^3\right)^2 \left(8 a B-23 A b-17 b B x^3\right)\right)}{391 b^2}","\frac{54 a^2 x \sqrt{a+b x^3} (23 A b-8 a B)}{21505 b^2}-\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (23 A b-8 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{21505 b^{7/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x^4 \left(a+b x^3\right)^{3/2} (23 A b-8 a B)}{391 b}+\frac{18 a x^4 \sqrt{a+b x^3} (23 A b-8 a B)}{4301 b}+\frac{2 B x^4 \left(a+b x^3\right)^{5/2}}{23 b}",1,"(2*x*Sqrt[a + b*x^3]*(-((a + b*x^3)^2*(-23*A*b + 8*a*B - 17*b*B*x^3)) + (a^2*(-23*A*b + 8*a*B)*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(391*b^2)","C",1
203,1,77,299,0.0752629,"\int \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{2 x \sqrt{a+b x^3} \left(B \left(a+b x^3\right)^2-\frac{a \left(a B-\frac{17 A b}{2}\right) \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}\right)}{17 b}","\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (17 A b-2 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{935 b^{4/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x \left(a+b x^3\right)^{3/2} (17 A b-2 a B)}{187 b}+\frac{18 a x \sqrt{a+b x^3} (17 A b-2 a B)}{935 b}+\frac{2 B x \left(a+b x^3\right)^{5/2}}{17 b}",1,"(2*x*Sqrt[a + b*x^3]*(B*(a + b*x^3)^2 - (a*((-17*A*b)/2 + a*B)*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(17*b)","C",1
204,1,83,295,0.0552556,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{x^3} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^3,x]","-\frac{x \sqrt{a+b x^3} \left(-2 a B-\frac{11 A b}{2}\right) \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{\frac{b x^3}{a}+1}}-\frac{A \left(a+b x^3\right)^{5/2}}{2 a x^2}","\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (4 a B+11 A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{110 \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{x \left(a+b x^3\right)^{3/2} (4 a B+11 A b)}{22 a}+\frac{9}{110} x \sqrt{a+b x^3} (4 a B+11 A b)-\frac{A \left(a+b x^3\right)^{5/2}}{2 a x^2}",1,"-1/2*(A*(a + b*x^3)^(5/2))/(a*x^2) - (((-11*A*b)/2 - 2*a*B)*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)])/(2*Sqrt[1 + (b*x^3)/a])","C",1
205,1,82,297,0.0788104,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{x^6} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^6,x]","\frac{\sqrt{a+b x^3} \left(-\frac{5 x^3 (2 a B+A b) \, _2F_1\left(-\frac{3}{2},-\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{\frac{b x^3}{a}+1}}-\frac{2 A \left(a+b x^3\right)^2}{a}\right)}{10 x^5}","\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a B+A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{20 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{9 b x \sqrt{a+b x^3} (2 a B+A b)}{20 a}-\frac{\left(a+b x^3\right)^{3/2} (2 a B+A b)}{4 a x^2}-\frac{A \left(a+b x^3\right)^{5/2}}{5 a x^5}",1,"(Sqrt[a + b*x^3]*((-2*A*(a + b*x^3)^2)/a - (5*(A*b + 2*a*B)*x^3*Hypergeometric2F1[-3/2, -2/3, 1/3, -((b*x^3)/a)])/(2*Sqrt[1 + (b*x^3)/a])))/(10*x^5)","C",1
206,1,82,302,0.146507,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{x^9} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^9,x]","\frac{\sqrt{a+b x^3} \left(\frac{x^3 \left(\frac{A b}{2}-8 a B\right) \, _2F_1\left(-\frac{5}{3},-\frac{3}{2};-\frac{2}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-\frac{5 A \left(a+b x^3\right)^2}{a}\right)}{40 x^8}","-\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (A b-16 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{320 a \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\left(a+b x^3\right)^{3/2} (A b-16 a B)}{80 a x^5}+\frac{9 b \sqrt{a+b x^3} (A b-16 a B)}{320 a x^2}-\frac{A \left(a+b x^3\right)^{5/2}}{8 a x^8}",1,"(Sqrt[a + b*x^3]*((-5*A*(a + b*x^3)^2)/a + (((A*b)/2 - 8*a*B)*x^3*Hypergeometric2F1[-5/3, -3/2, -2/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(40*x^8)","C",1
207,1,96,614,0.242735,"\int x^4 \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[x^4*(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{2 x^2 \sqrt{a+b x^3} \left(\frac{5 a^2 (2 a B-5 A b) \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-\left(a+b x^3\right)^2 \left(10 a B-25 A b-19 b B x^3\right)\right)}{475 b^2}","-\frac{72 \sqrt{2} 3^{3/4} a^{10/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 A b-2 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{8645 b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{108 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 A b-2 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{8645 b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{216 a^3 \sqrt{a+b x^3} (5 A b-2 a B)}{8645 b^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{54 a^2 x^2 \sqrt{a+b x^3} (5 A b-2 a B)}{8645 b^2}+\frac{2 x^5 \left(a+b x^3\right)^{3/2} (5 A b-2 a B)}{95 b}+\frac{18 a x^5 \sqrt{a+b x^3} (5 A b-2 a B)}{1235 b}+\frac{2 B x^5 \left(a+b x^3\right)^{5/2}}{25 b}",1,"(2*x^2*Sqrt[a + b*x^3]*(-((a + b*x^3)^2*(-25*A*b + 10*a*B - 19*b*B*x^3)) + (5*a^2*(-5*A*b + 2*a*B)*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(475*b^2)","C",1
208,1,78,581,0.2488765,"\int x \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[x*(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{x^2 \sqrt{a+b x^3} \left(\frac{a (19 A b-4 a B) \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}+4 B \left(a+b x^3\right)^2\right)}{38 b}","\frac{18 \sqrt{2} 3^{3/4} a^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (19 A b-4 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{1729 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (19 A b-4 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{1729 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{54 a^2 \sqrt{a+b x^3} (19 A b-4 a B)}{1729 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x^2 \left(a+b x^3\right)^{3/2} (19 A b-4 a B)}{247 b}+\frac{18 a x^2 \sqrt{a+b x^3} (19 A b-4 a B)}{1729 b}+\frac{2 B x^2 \left(a+b x^3\right)^{5/2}}{19 b}",1,"(x^2*Sqrt[a + b*x^3]*(4*B*(a + b*x^3)^2 + (a*(19*A*b - 4*a*B)*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(38*b)","C",1
209,1,83,573,0.0389495,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{x^2} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^2,x]","-\frac{x^2 \sqrt{a+b x^3} \left(-a B-\frac{13 A b}{2}\right) \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{\frac{b x^3}{a}+1}}-\frac{A \left(a+b x^3\right)^{5/2}}{a x}","\frac{9 \sqrt{2} 3^{3/4} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a B+13 A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{91 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a B+13 A b) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{182 b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{27 a \sqrt{a+b x^3} (2 a B+13 A b)}{91 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{x^2 \left(a+b x^3\right)^{3/2} (2 a B+13 A b)}{13 a}+\frac{9}{91} x^2 \sqrt{a+b x^3} (2 a B+13 A b)-\frac{A \left(a+b x^3\right)^{5/2}}{a x}",1,"-((A*(a + b*x^3)^(5/2))/(a*x)) - (((-13*A*b)/2 - a*B)*x^2*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)])/(2*Sqrt[1 + (b*x^3)/a])","C",1
210,1,85,578,0.0968414,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{x^5} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^5,x]","\frac{\sqrt{a+b x^3} \left(-4 a B-\frac{7 A b}{2}\right) \, _2F_1\left(-\frac{3}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right)}{4 x \sqrt{\frac{b x^3}{a}+1}}-\frac{A \left(a+b x^3\right)^{5/2}}{4 a x^4}","\frac{9\ 3^{3/4} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (8 a B+7 A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{28 \sqrt{2} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (8 a B+7 A b) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{112 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\left(a+b x^3\right)^{3/2} (8 a B+7 A b)}{8 a x}+\frac{27 \sqrt[3]{b} \sqrt{a+b x^3} (8 a B+7 A b)}{56 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{9 b x^2 \sqrt{a+b x^3} (8 a B+7 A b)}{56 a}-\frac{A \left(a+b x^3\right)^{5/2}}{4 a x^4}",1,"-1/4*(A*(a + b*x^3)^(5/2))/(a*x^4) + (((-7*A*b)/2 - 4*a*B)*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -1/3, 2/3, -((b*x^3)/a)])/(4*x*Sqrt[1 + (b*x^3)/a])","C",1
211,1,82,576,0.0814128,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{x^8} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^8,x]","\frac{\sqrt{a+b x^3} \left(-\frac{x^3 (14 a B+A b) \, _2F_1\left(-\frac{3}{2},-\frac{4}{3};-\frac{1}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{\frac{b x^3}{a}+1}}-\frac{4 A \left(a+b x^3\right)^2}{a}\right)}{28 x^7}","\frac{9\ 3^{3/4} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (14 a B+A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{56 \sqrt{2} a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (14 a B+A b) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{224 a^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{27 b^{4/3} \sqrt{a+b x^3} (14 a B+A b)}{112 a \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{9 b \sqrt{a+b x^3} (14 a B+A b)}{112 a x}-\frac{\left(a+b x^3\right)^{3/2} (14 a B+A b)}{56 a x^4}-\frac{A \left(a+b x^3\right)^{5/2}}{7 a x^7}",1,"(Sqrt[a + b*x^3]*((-4*A*(a + b*x^3)^2)/a - ((A*b + 14*a*B)*x^3*Hypergeometric2F1[-3/2, -4/3, -1/3, -((b*x^3)/a)])/(2*Sqrt[1 + (b*x^3)/a])))/(28*x^7)","C",1
212,1,82,608,0.0830542,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{x^{11}} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^11,x]","\frac{\sqrt{a+b x^3} \left(\frac{5 x^3 (A b-4 a B) \, _2F_1\left(-\frac{7}{3},-\frac{3}{2};-\frac{4}{3};-\frac{b x^3}{a}\right)}{2 \sqrt{\frac{b x^3}{a}+1}}-\frac{7 A \left(a+b x^3\right)^2}{a}\right)}{70 x^{10}}","-\frac{9\ 3^{3/4} b^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (A b-4 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{224 \sqrt{2} a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{7/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (A b-4 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{896 a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 b^{7/3} \sqrt{a+b x^3} (A b-4 a B)}{448 a^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{27 b^2 \sqrt{a+b x^3} (A b-4 a B)}{448 a^2 x}+\frac{\left(a+b x^3\right)^{3/2} (A b-4 a B)}{28 a x^7}+\frac{9 b \sqrt{a+b x^3} (A b-4 a B)}{224 a x^4}-\frac{A \left(a+b x^3\right)^{5/2}}{10 a x^{10}}",1,"(Sqrt[a + b*x^3]*((-7*A*(a + b*x^3)^2)/a + (5*(A*b - 4*a*B)*x^3*Hypergeometric2F1[-7/3, -3/2, -4/3, -((b*x^3)/a)])/(2*Sqrt[1 + (b*x^3)/a])))/(70*x^10)","C",1
213,1,78,103,0.0792813,"\int \frac{x^8 \left(A+B x^3\right)}{\sqrt{a+b x^3}} \, dx","Integrate[(x^8*(A + B*x^3))/Sqrt[a + b*x^3],x]","\frac{2 \sqrt{a+b x^3} \left(-48 a^3 B+8 a^2 b \left(7 A+3 B x^3\right)-2 a b^2 x^3 \left(14 A+9 B x^3\right)+3 b^3 x^6 \left(7 A+5 B x^3\right)\right)}{315 b^4}","\frac{2 a^2 \sqrt{a+b x^3} (A b-a B)}{3 b^4}+\frac{2 \left(a+b x^3\right)^{5/2} (A b-3 a B)}{15 b^4}-\frac{2 a \left(a+b x^3\right)^{3/2} (2 A b-3 a B)}{9 b^4}+\frac{2 B \left(a+b x^3\right)^{7/2}}{21 b^4}",1,"(2*Sqrt[a + b*x^3]*(-48*a^3*B + 8*a^2*b*(7*A + 3*B*x^3) + 3*b^3*x^6*(7*A + 5*B*x^3) - 2*a*b^2*x^3*(14*A + 9*B*x^3)))/(315*b^4)","A",1
214,1,56,73,0.0580315,"\int \frac{x^5 \left(A+B x^3\right)}{\sqrt{a+b x^3}} \, dx","Integrate[(x^5*(A + B*x^3))/Sqrt[a + b*x^3],x]","\frac{2 \sqrt{a+b x^3} \left(8 a^2 B-2 a b \left(5 A+2 B x^3\right)+b^2 x^3 \left(5 A+3 B x^3\right)\right)}{45 b^3}","\frac{2 \left(a+b x^3\right)^{3/2} (A b-2 a B)}{9 b^3}-\frac{2 a \sqrt{a+b x^3} (A b-a B)}{3 b^3}+\frac{2 B \left(a+b x^3\right)^{5/2}}{15 b^3}",1,"(2*Sqrt[a + b*x^3]*(8*a^2*B - 2*a*b*(5*A + 2*B*x^3) + b^2*x^3*(5*A + 3*B*x^3)))/(45*b^3)","A",1
215,1,33,46,0.0346486,"\int \frac{x^2 \left(A+B x^3\right)}{\sqrt{a+b x^3}} \, dx","Integrate[(x^2*(A + B*x^3))/Sqrt[a + b*x^3],x]","\frac{2 \sqrt{a+b x^3} \left(-2 a B+3 A b+b B x^3\right)}{9 b^2}","\frac{2 \sqrt{a+b x^3} (A b-a B)}{3 b^2}+\frac{2 B \left(a+b x^3\right)^{3/2}}{9 b^2}",1,"(2*Sqrt[a + b*x^3]*(3*A*b - 2*a*B + b*B*x^3))/(9*b^2)","A",1
216,1,48,48,0.046937,"\int \frac{A+B x^3}{x \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/(x*Sqrt[a + b*x^3]),x]","\frac{2 B \sqrt{a+b x^3}}{3 b}-\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}","\frac{2 B \sqrt{a+b x^3}}{3 b}-\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a}}",1,"(2*B*Sqrt[a + b*x^3])/(3*b) - (2*A*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*Sqrt[a])","A",1
217,1,57,58,0.0736324,"\int \frac{A+B x^3}{x^4 \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/(x^4*Sqrt[a + b*x^3]),x]","\frac{1}{3} \left(\frac{(A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{A \sqrt{a+b x^3}}{a x^3}\right)","\frac{(A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 a^{3/2}}-\frac{A \sqrt{a+b x^3}}{3 a x^3}",1,"(-((A*Sqrt[a + b*x^3])/(a*x^3)) + ((A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/a^(3/2))/3","A",1
218,1,81,90,0.2861706,"\int \frac{A+B x^3}{x^7 \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/(x^7*Sqrt[a + b*x^3]),x]","\frac{\sqrt{a+b x^3} \left(b \left(2 a B-\frac{3 A b}{2}\right) \left(\frac{\tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)}{\sqrt{\frac{b x^3}{a}+1}}-\frac{a}{b x^3}\right)-\frac{a^2 A}{x^6}\right)}{6 a^3}","-\frac{b (3 A b-4 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{12 a^{5/2}}+\frac{\sqrt{a+b x^3} (3 A b-4 a B)}{12 a^2 x^3}-\frac{A \sqrt{a+b x^3}}{6 a x^6}",1,"(Sqrt[a + b*x^3]*(-((a^2*A)/x^6) + b*((-3*A*b)/2 + 2*a*B)*(-(a/(b*x^3)) + ArcTanh[Sqrt[1 + (b*x^3)/a]]/Sqrt[1 + (b*x^3)/a])))/(6*a^3)","A",1
219,1,89,270,0.127718,"\int \frac{x^3 \left(A+B x^3\right)}{\sqrt{a+b x^3}} \, dx","Integrate[(x^3*(A + B*x^3))/Sqrt[a + b*x^3],x]","\frac{2 x \left(a \sqrt{\frac{b x^3}{a}+1} (8 a B-11 A b) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)-\left(a+b x^3\right) \left(8 a B-11 A b-5 b B x^3\right)\right)}{55 b^2 \sqrt{a+b x^3}}","-\frac{4 \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (11 A b-8 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{55 \sqrt[4]{3} b^{7/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x \sqrt{a+b x^3} (11 A b-8 a B)}{55 b^2}+\frac{2 B x^4 \sqrt{a+b x^3}}{11 b}",1,"(2*x*(-((a + b*x^3)*(-11*A*b + 8*a*B - 5*b*B*x^3)) + a*(-11*A*b + 8*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)]))/(55*b^2*Sqrt[a + b*x^3])","C",1
220,1,74,239,0.0457748,"\int \frac{A+B x^3}{\sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/Sqrt[a + b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} (5 A b-2 a B) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+2 B x \left(a+b x^3\right)}{5 b \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 A b-2 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{5 \sqrt[4]{3} b^{4/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 B x \sqrt{a+b x^3}}{5 b}",1,"(2*B*x*(a + b*x^3) + (5*A*b - 2*a*B)*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)])/(5*b*Sqrt[a + b*x^3])","C",1
221,1,78,243,0.0457563,"\int \frac{A+B x^3}{x^3 \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/(x^3*Sqrt[a + b*x^3]),x]","\frac{x^3 \sqrt{\frac{b x^3}{a}+1} (4 a B-A b) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)-2 A \left(a+b x^3\right)}{4 a x^2 \sqrt{a+b x^3}}","-\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (A b-4 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{2 \sqrt[4]{3} a \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{A \sqrt{a+b x^3}}{2 a x^2}",1,"(-2*A*(a + b*x^3) + (-(A*b) + 4*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)])/(4*a*x^2*Sqrt[a + b*x^3])","C",1
222,1,78,274,0.069409,"\int \frac{A+B x^3}{x^6 \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/(x^6*Sqrt[a + b*x^3]),x]","\frac{x^3 \sqrt{\frac{b x^3}{a}+1} (7 A b-10 a B) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a}\right)-4 A \left(a+b x^3\right)}{20 a x^5 \sqrt{a+b x^3}}","\frac{\sqrt{a+b x^3} (7 A b-10 a B)}{20 a^2 x^2}+\frac{\sqrt{2+\sqrt{3}} b^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (7 A b-10 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{20 \sqrt[4]{3} a^2 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{A \sqrt{a+b x^3}}{5 a x^5}",1,"(-4*A*(a + b*x^3) + (7*A*b - 10*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-2/3, 1/2, 1/3, -((b*x^3)/a)])/(20*a*x^5*Sqrt[a + b*x^3])","C",1
223,1,91,548,0.1219357,"\int \frac{x^4 \left(A+B x^3\right)}{\sqrt{a+b x^3}} \, dx","Integrate[(x^4*(A + B*x^3))/Sqrt[a + b*x^3],x]","\frac{2 x^2 \left(a \sqrt{\frac{b x^3}{a}+1} (10 a B-13 A b) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)-\left(a+b x^3\right) \left(10 a B-13 A b-7 b B x^3\right)\right)}{91 b^2 \sqrt{a+b x^3}}","-\frac{8 \sqrt{2} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (13 A b-10 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{91 \sqrt[4]{3} b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (13 A b-10 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{91 b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{8 a \sqrt{a+b x^3} (13 A b-10 a B)}{91 b^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x^2 \sqrt{a+b x^3} (13 A b-10 a B)}{91 b^2}+\frac{2 B x^5 \sqrt{a+b x^3}}{13 b}",1,"(2*x^2*(-((a + b*x^3)*(-13*A*b + 10*a*B - 7*b*B*x^3)) + a*(-13*A*b + 10*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(91*b^2*Sqrt[a + b*x^3])","C",1
224,1,75,517,0.0581492,"\int \frac{x \left(A+B x^3\right)}{\sqrt{a+b x^3}} \, dx","Integrate[(x*(A + B*x^3))/Sqrt[a + b*x^3],x]","\frac{x^2 \left(\sqrt{\frac{b x^3}{a}+1} (7 A b-4 a B) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+4 B \left(a+b x^3\right)\right)}{14 b \sqrt{a+b x^3}}","\frac{2 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (7 A b-4 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{7 \sqrt[4]{3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (7 A b-4 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{7 b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3} (7 A b-4 a B)}{7 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 B x^2 \sqrt{a+b x^3}}{7 b}",1,"(x^2*(4*B*(a + b*x^3) + (7*A*b - 4*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)]))/(14*b*Sqrt[a + b*x^3])","C",1
225,1,77,509,0.0422019,"\int \frac{A+B x^3}{x^2 \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/(x^2*Sqrt[a + b*x^3]),x]","\frac{x^3 \sqrt{\frac{b x^3}{a}+1} (2 a B+A b) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)-4 A \left(a+b x^3\right)}{4 a x \sqrt{a+b x^3}}","\frac{\sqrt{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a B+A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} a^{2/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a B+A b) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{2 a^{2/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{a+b x^3} (2 a B+A b)}{a b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{A \sqrt{a+b x^3}}{a x}",1,"(-4*A*(a + b*x^3) + (A*b + 2*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a)])/(4*a*x*Sqrt[a + b*x^3])","C",1
226,1,78,550,0.0548072,"\int \frac{A+B x^3}{x^5 \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/(x^5*Sqrt[a + b*x^3]),x]","\frac{x^3 \sqrt{\frac{b x^3}{a}+1} (5 A b-8 a B) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a}\right)-2 A \left(a+b x^3\right)}{8 a x^4 \sqrt{a+b x^3}}","-\frac{\sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 A b-8 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt{2} \sqrt[4]{3} a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 A b-8 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{16 a^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{a+b x^3} (5 A b-8 a B)}{8 a^2 x}-\frac{\sqrt[3]{b} \sqrt{a+b x^3} (5 A b-8 a B)}{8 a^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{A \sqrt{a+b x^3}}{4 a x^4}",1,"(-2*A*(a + b*x^3) + (5*A*b - 8*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-1/3, 1/2, 2/3, -((b*x^3)/a)])/(8*a*x^4*Sqrt[a + b*x^3])","C",1
227,1,78,581,0.0384926,"\int \frac{A+B x^3}{x^8 \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/(x^8*Sqrt[a + b*x^3]),x]","\frac{x^3 \sqrt{\frac{b x^3}{a}+1} (11 A b-14 a B) \, _2F_1\left(-\frac{4}{3},\frac{1}{2};-\frac{1}{3};-\frac{b x^3}{a}\right)-8 A \left(a+b x^3\right)}{56 a x^7 \sqrt{a+b x^3}}","\frac{5 b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (11 A b-14 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{56 \sqrt{2} \sqrt[4]{3} a^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{5 \sqrt[4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (11 A b-14 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{224 a^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{5 b^{4/3} \sqrt{a+b x^3} (11 A b-14 a B)}{112 a^3 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{5 b \sqrt{a+b x^3} (11 A b-14 a B)}{112 a^3 x}+\frac{\sqrt{a+b x^3} (11 A b-14 a B)}{56 a^2 x^4}-\frac{A \sqrt{a+b x^3}}{7 a x^7}",1,"(-8*A*(a + b*x^3) + (11*A*b - 14*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-4/3, 1/2, -1/3, -((b*x^3)/a)])/(56*a*x^7*Sqrt[a + b*x^3])","C",1
228,1,77,103,0.0528524,"\int \frac{x^8 \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x^8*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{2 \left(48 a^3 B-8 a^2 b \left(5 A-3 B x^3\right)-2 a b^2 x^3 \left(10 A+3 B x^3\right)+b^3 x^6 \left(5 A+3 B x^3\right)\right)}{45 b^4 \sqrt{a+b x^3}}","-\frac{2 a^2 (A b-a B)}{3 b^4 \sqrt{a+b x^3}}+\frac{2 \left(a+b x^3\right)^{3/2} (A b-3 a B)}{9 b^4}-\frac{2 a \sqrt{a+b x^3} (2 A b-3 a B)}{3 b^4}+\frac{2 B \left(a+b x^3\right)^{5/2}}{15 b^4}",1,"(2*(48*a^3*B - 8*a^2*b*(5*A - 3*B*x^3) + b^3*x^6*(5*A + 3*B*x^3) - 2*a*b^2*x^3*(10*A + 3*B*x^3)))/(45*b^4*Sqrt[a + b*x^3])","A",1
229,1,55,73,0.0444424,"\int \frac{x^5 \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x^5*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{2 \left(-8 a^2 B+a \left(6 A b-4 b B x^3\right)+b^2 x^3 \left(3 A+B x^3\right)\right)}{9 b^3 \sqrt{a+b x^3}}","\frac{2 \sqrt{a+b x^3} (A b-2 a B)}{3 b^3}+\frac{2 a (A b-a B)}{3 b^3 \sqrt{a+b x^3}}+\frac{2 B \left(a+b x^3\right)^{3/2}}{9 b^3}",1,"(2*(-8*a^2*B + b^2*x^3*(3*A + B*x^3) + a*(6*A*b - 4*b*B*x^3)))/(9*b^3*Sqrt[a + b*x^3])","A",1
230,1,33,46,0.0271142,"\int \frac{x^2 \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x^2*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{2 \left(2 a B-A b+b B x^3\right)}{3 b^2 \sqrt{a+b x^3}}","\frac{2 B \sqrt{a+b x^3}}{3 b^2}-\frac{2 (A b-a B)}{3 b^2 \sqrt{a+b x^3}}",1,"(2*(-(A*b) + 2*a*B + b*B*x^3))/(3*b^2*Sqrt[a + b*x^3])","A",1
231,1,58,58,0.0656758,"\int \frac{A+B x^3}{x \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/(x*(a + b*x^3)^(3/2)),x]","\frac{1}{3} \left(\frac{2 (A b-a B)}{a b \sqrt{a+b x^3}}-\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{a^{3/2}}\right)","\frac{2 (A b-a B)}{3 a b \sqrt{a+b x^3}}-\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 a^{3/2}}",1,"((2*(A*b - a*B))/(a*b*Sqrt[a + b*x^3]) - (2*A*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/a^(3/2))/3","A",1
232,1,57,86,0.0196089,"\int \frac{A+B x^3}{x^4 \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/(x^4*(a + b*x^3)^(3/2)),x]","\frac{x^3 (2 a B-3 A b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^3}{a}+1\right)-a A}{3 a^2 x^3 \sqrt{a+b x^3}}","\frac{(3 A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 a^{5/2}}+\frac{2 a B-3 A b}{3 a^2 \sqrt{a+b x^3}}-\frac{A}{3 a x^3 \sqrt{a+b x^3}}",1,"(-(a*A) + (-3*A*b + 2*a*B)*x^3*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^3)/a])/(3*a^2*x^3*Sqrt[a + b*x^3])","C",1
233,1,60,118,0.0230669,"\int \frac{A+B x^3}{x^7 \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/(x^7*(a + b*x^3)^(3/2)),x]","\frac{b x^6 (5 A b-4 a B) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{b x^3}{a}+1\right)-a^2 A}{6 a^3 x^6 \sqrt{a+b x^3}}","-\frac{b (5 A b-4 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{4 a^{7/2}}+\frac{b (5 A b-4 a B)}{4 a^3 \sqrt{a+b x^3}}+\frac{5 A b-4 a B}{12 a^2 x^3 \sqrt{a+b x^3}}-\frac{A}{6 a x^6 \sqrt{a+b x^3}}",1,"(-(a^2*A) + b*(5*A*b - 4*a*B)*x^6*Hypergeometric2F1[-1/2, 2, 1/2, 1 + (b*x^3)/a])/(6*a^3*x^6*Sqrt[a + b*x^3])","C",1
234,1,103,300,0.1504452,"\int \frac{x^6 \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x^6*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{2 x \left(-112 a^2 B+8 a \sqrt{\frac{b x^3}{a}+1} (14 a B-11 A b) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+a \left(88 A b-42 b B x^3\right)+3 b^2 x^3 \left(11 A+5 B x^3\right)\right)}{165 b^3 \sqrt{a+b x^3}}","-\frac{32 \sqrt{2+\sqrt{3}} a \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (11 A b-14 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{165 \sqrt[4]{3} b^{10/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{16 x \sqrt{a+b x^3} (11 A b-14 a B)}{165 b^3}-\frac{2 x^4 (11 A b-14 a B)}{33 b^2 \sqrt{a+b x^3}}+\frac{2 B x^7}{11 b \sqrt{a+b x^3}}",1,"(2*x*(-112*a^2*B + 3*b^2*x^3*(11*A + 5*B*x^3) + a*(88*A*b - 42*b*B*x^3) + 8*a*(-11*A*b + 14*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)]))/(165*b^3*Sqrt[a + b*x^3])","C",1
235,1,78,269,0.1013667,"\int \frac{x^3 \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x^3*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{2 x \left(\sqrt{\frac{b x^3}{a}+1} (5 A b-8 a B) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+8 a B-5 A b+3 b B x^3\right)}{15 b^2 \sqrt{a+b x^3}}","\frac{4 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 A b-8 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{15 \sqrt[4]{3} b^{7/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 x (5 A b-8 a B)}{15 b^2 \sqrt{a+b x^3}}+\frac{2 B x^4}{5 b \sqrt{a+b x^3}}",1,"(2*x*(-5*A*b + 8*a*B + 3*b*B*x^3 + (5*A*b - 8*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)]))/(15*b^2*Sqrt[a + b*x^3])","C",1
236,1,73,251,0.0419942,"\int \frac{A+B x^3}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/(a + b*x^3)^(3/2),x]","\frac{x \left(\sqrt{\frac{b x^3}{a}+1} (2 a B+A b) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)-2 a B+2 A b\right)}{3 a b \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a B+A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} a b^{4/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x (A b-a B)}{3 a b \sqrt{a+b x^3}}",1,"(x*(2*A*b - 2*a*B + (A*b + 2*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)]))/(3*a*b*Sqrt[a + b*x^3])","C",1
237,1,86,272,0.0631976,"\int \frac{A+B x^3}{x^3 \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/(x^3*(a + b*x^3)^(3/2)),x]","\frac{x^3 \sqrt{\frac{b x^3}{a}+1} (4 a B-7 A b) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)-6 a A+8 a B x^3-14 A b x^3}{12 a^2 x^2 \sqrt{a+b x^3}}","-\frac{x (7 A b-4 a B)}{6 a^2 \sqrt{a+b x^3}}-\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (7 A b-4 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{6 \sqrt[4]{3} a^2 \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{A}{2 a x^2 \sqrt{a+b x^3}}",1,"(-6*a*A - 14*A*b*x^3 + 8*a*B*x^3 + (-7*A*b + 4*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)])/(12*a^2*x^2*Sqrt[a + b*x^3])","C",1
238,1,72,304,0.0591184,"\int \frac{A+B x^3}{x^6 \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/(x^6*(a + b*x^3)^(3/2)),x]","\frac{x^3 \sqrt{\frac{b x^3}{a}+1} (13 A b-10 a B) \, _2F_1\left(-\frac{2}{3},\frac{3}{2};\frac{1}{3};-\frac{b x^3}{a}\right)-4 a A}{20 a^2 x^5 \sqrt{a+b x^3}}","\frac{7 \sqrt{a+b x^3} (13 A b-10 a B)}{60 a^3 x^2}-\frac{13 A b-10 a B}{15 a^2 x^2 \sqrt{a+b x^3}}+\frac{7 \sqrt{2+\sqrt{3}} b^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (13 A b-10 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{60 \sqrt[4]{3} a^3 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{A}{5 a x^5 \sqrt{a+b x^3}}",1,"(-4*a*A + (13*A*b - 10*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-2/3, 3/2, 1/3, -((b*x^3)/a)])/(20*a^2*x^5*Sqrt[a + b*x^3])","C",1
239,1,79,547,0.0799262,"\int \frac{x^4 \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x^4*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{2 x^2 \left(\sqrt{\frac{b x^3}{a}+1} (10 a B-7 A b) \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)-10 a B+7 A b+b B x^3\right)}{7 b^2 \sqrt{a+b x^3}}","\frac{8 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (7 A b-10 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{21 \sqrt[4]{3} b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{4 \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (7 A b-10 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{7\ 3^{3/4} b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{8 \sqrt{a+b x^3} (7 A b-10 a B)}{21 b^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{2 x^2 (7 A b-10 a B)}{21 b^2 \sqrt{a+b x^3}}+\frac{2 B x^5}{7 b \sqrt{a+b x^3}}",1,"(2*x^2*(7*A*b - 10*a*B + b*B*x^3 + (-7*A*b + 10*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)]))/(7*b^2*Sqrt[a + b*x^3])","C",1
240,1,71,524,0.0642316,"\int \frac{x \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(x*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{x^2 \left(\sqrt{\frac{b x^3}{a}+1} (A b-4 a B) \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)+4 a B\right)}{2 a b \sqrt{a+b x^3}}","-\frac{2 \sqrt{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (A b-4 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} a^{2/3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (A b-4 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} a^{2/3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} (A b-4 a B)}{3 a b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x^2 (A b-a B)}{3 a b \sqrt{a+b x^3}}",1,"(x^2*(4*a*B + (A*b - 4*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)]))/(2*a*b*Sqrt[a + b*x^3])","C",1
241,1,72,548,0.0367945,"\int \frac{A+B x^3}{x^2 \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/(x^2*(a + b*x^3)^(3/2)),x]","\frac{x^3 \sqrt{\frac{b x^3}{a}+1} (2 a B-5 A b) \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right)-4 a A}{4 a^2 x \sqrt{a+b x^3}}","\frac{\sqrt{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 A b-2 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} a^{5/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 A b-2 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{2\ 3^{3/4} a^{5/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{a+b x^3} (5 A b-2 a B)}{3 a^2 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{x^2 (5 A b-2 a B)}{3 a^2 \sqrt{a+b x^3}}-\frac{A}{a x \sqrt{a+b x^3}}",1,"(-4*a*A + (-5*A*b + 2*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)])/(4*a^2*x*Sqrt[a + b*x^3])","C",1
242,1,72,580,0.0392607,"\int \frac{A+B x^3}{x^5 \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/(x^5*(a + b*x^3)^(3/2)),x]","\frac{x^3 \sqrt{\frac{b x^3}{a}+1} (11 A b-8 a B) \, _2F_1\left(-\frac{1}{3},\frac{3}{2};\frac{2}{3};-\frac{b x^3}{a}\right)-2 a A}{8 a^2 x^4 \sqrt{a+b x^3}}","-\frac{5 \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (11 A b-8 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{12 \sqrt{2} \sqrt[4]{3} a^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{5 \sqrt{2-\sqrt{3}} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (11 A b-8 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{16\ 3^{3/4} a^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{5 \sqrt{a+b x^3} (11 A b-8 a B)}{24 a^3 x}-\frac{5 \sqrt[3]{b} \sqrt{a+b x^3} (11 A b-8 a B)}{24 a^3 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{11 A b-8 a B}{12 a^2 x \sqrt{a+b x^3}}-\frac{A}{4 a x^4 \sqrt{a+b x^3}}",1,"(-2*a*A + (11*A*b - 8*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-1/3, 3/2, 2/3, -((b*x^3)/a)])/(8*a^2*x^4*Sqrt[a + b*x^3])","C",1
243,1,72,611,0.0388583,"\int \frac{A+B x^3}{x^8 \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/(x^8*(a + b*x^3)^(3/2)),x]","\frac{x^3 \sqrt{\frac{b x^3}{a}+1} (17 A b-14 a B) \, _2F_1\left(-\frac{4}{3},\frac{3}{2};-\frac{1}{3};-\frac{b x^3}{a}\right)-8 a A}{56 a^2 x^7 \sqrt{a+b x^3}}","\frac{55 b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (17 A b-14 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{168 \sqrt{2} \sqrt[4]{3} a^{11/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{55 \sqrt{2-\sqrt{3}} b^{4/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (17 A b-14 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{224\ 3^{3/4} a^{11/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{55 b^{4/3} \sqrt{a+b x^3} (17 A b-14 a B)}{336 a^4 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{55 b \sqrt{a+b x^3} (17 A b-14 a B)}{336 a^4 x}+\frac{11 \sqrt{a+b x^3} (17 A b-14 a B)}{168 a^3 x^4}-\frac{17 A b-14 a B}{21 a^2 x^4 \sqrt{a+b x^3}}-\frac{A}{7 a x^7 \sqrt{a+b x^3}}",1,"(-8*a*A + (17*A*b - 14*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-4/3, 3/2, -1/3, -((b*x^3)/a)])/(56*a^2*x^7*Sqrt[a + b*x^3])","C",1
244,1,73,103,0.0670847,"\int \frac{x^8 \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(x^8*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{2 \left(-16 a^3 B+8 a^2 b \left(A-3 B x^3\right)-6 a b^2 x^3 \left(B x^3-2 A\right)+b^3 x^6 \left(3 A+B x^3\right)\right)}{9 b^4 \left(a+b x^3\right)^{3/2}}","-\frac{2 a^2 (A b-a B)}{9 b^4 \left(a+b x^3\right)^{3/2}}+\frac{2 a (2 A b-3 a B)}{3 b^4 \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3} (A b-3 a B)}{3 b^4}+\frac{2 B \left(a+b x^3\right)^{3/2}}{9 b^4}",1,"(2*(-16*a^3*B + 8*a^2*b*(A - 3*B*x^3) - 6*a*b^2*x^3*(-2*A + B*x^3) + b^3*x^6*(3*A + B*x^3)))/(9*b^4*(a + b*x^3)^(3/2))","A",1
245,1,54,73,0.047119,"\int \frac{x^5 \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(x^5*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{16 a^2 B-4 a b \left(A-6 B x^3\right)+6 b^2 x^3 \left(B x^3-A\right)}{9 b^3 \left(a+b x^3\right)^{3/2}}","-\frac{2 (A b-2 a B)}{3 b^3 \sqrt{a+b x^3}}+\frac{2 a (A b-a B)}{9 b^3 \left(a+b x^3\right)^{3/2}}+\frac{2 B \sqrt{a+b x^3}}{3 b^3}",1,"(16*a^2*B - 4*a*b*(A - 6*B*x^3) + 6*b^2*x^3*(-A + B*x^3))/(9*b^3*(a + b*x^3)^(3/2))","A",1
246,1,33,46,0.0287585,"\int \frac{x^2 \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(x^2*(A + B*x^3))/(a + b*x^3)^(5/2),x]","-\frac{2 \left(2 a B+A b+3 b B x^3\right)}{9 b^2 \left(a+b x^3\right)^{3/2}}","-\frac{2 (A b-a B)}{9 b^2 \left(a+b x^3\right)^{3/2}}-\frac{2 B}{3 b^2 \sqrt{a+b x^3}}",1,"(-2*(A*b + 2*a*B + 3*b*B*x^3))/(9*b^2*(a + b*x^3)^(3/2))","A",1
247,1,62,77,0.0470757,"\int \frac{A+B x^3}{x \left(a+b x^3\right)^{5/2}} \, dx","Integrate[(A + B*x^3)/(x*(a + b*x^3)^(5/2)),x]","\frac{2 a (A b-a B)+6 A b \left(a+b x^3\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^3}{a}+1\right)}{9 a^2 b \left(a+b x^3\right)^{3/2}}","-\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 a^{5/2}}+\frac{2 A}{3 a^2 \sqrt{a+b x^3}}+\frac{2 (A b-a B)}{9 a b \left(a+b x^3\right)^{3/2}}",1,"(2*a*(A*b - a*B) + 6*A*b*(a + b*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^3)/a])/(9*a^2*b*(a + b*x^3)^(3/2))","C",1
248,1,57,113,0.0294524,"\int \frac{A+B x^3}{x^4 \left(a+b x^3\right)^{5/2}} \, dx","Integrate[(A + B*x^3)/(x^4*(a + b*x^3)^(5/2)),x]","\frac{x^3 (2 a B-5 A b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b x^3}{a}+1\right)-3 a A}{9 a^2 x^3 \left(a+b x^3\right)^{3/2}}","\frac{(5 A b-2 a B) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 a^{7/2}}-\frac{5 A b-2 a B}{3 a^3 \sqrt{a+b x^3}}+\frac{2 a B-5 A b}{9 a^2 \left(a+b x^3\right)^{3/2}}-\frac{A}{3 a x^3 \left(a+b x^3\right)^{3/2}}",1,"(-3*a*A + (-5*A*b + 2*a*B)*x^3*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*x^3)/a])/(9*a^2*x^3*(a + b*x^3)^(3/2))","C",1
249,1,108,299,0.166244,"\int \frac{x^6 \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(x^6*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{2 x \left(112 a^2 B+8 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (5 A b-14 a B) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+a \left(154 b B x^3-40 A b\right)+b^2 x^3 \left(27 B x^3-55 A\right)\right)}{135 b^3 \left(a+b x^3\right)^{3/2}}","\frac{32 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (5 A b-14 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{135 \sqrt[4]{3} b^{10/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{16 x (5 A b-14 a B)}{135 b^3 \sqrt{a+b x^3}}-\frac{2 x^4 (5 A b-14 a B)}{45 b^2 \left(a+b x^3\right)^{3/2}}+\frac{2 B x^7}{5 b \left(a+b x^3\right)^{3/2}}",1,"(2*x*(112*a^2*B + b^2*x^3*(-55*A + 27*B*x^3) + a*(-40*A*b + 154*b*B*x^3) + 8*(5*A*b - 14*a*B)*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)]))/(135*b^3*(a + b*x^3)^(3/2))","C",1
250,1,99,283,0.1856634,"\int \frac{x^3 \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(x^3*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{2 x \left(-8 a^2 B+\left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (8 a B+A b) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)-a b \left(A+11 B x^3\right)+2 A b^2 x^3\right)}{27 a b^2 \left(a+b x^3\right)^{3/2}}","\frac{4 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (8 a B+A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} a b^{7/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 x (8 a B+A b)}{27 a b^2 \sqrt{a+b x^3}}+\frac{2 x^4 (A b-a B)}{9 a b \left(a+b x^3\right)^{3/2}}",1,"(2*x*(-8*a^2*B + 2*A*b^2*x^3 - a*b*(A + 11*B*x^3) + (A*b + 8*a*B)*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)]))/(27*a*b^2*(a + b*x^3)^(3/2))","C",1
251,1,103,283,0.0650678,"\int \frac{A+B x^3}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(A + B*x^3)/(a + b*x^3)^(5/2),x]","\frac{-2 a^2 B x+x \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (2 a B+7 A b) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+4 a b x \left(5 A+B x^3\right)+14 A b^2 x^4}{27 a^2 b \left(a+b x^3\right)^{3/2}}","\frac{2 x (2 a B+7 A b)}{27 a^2 b \sqrt{a+b x^3}}+\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (2 a B+7 A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} a^2 b^{4/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 x (A b-a B)}{9 a b \left(a+b x^3\right)^{3/2}}",1,"(-2*a^2*B*x + 14*A*b^2*x^4 + 4*a*b*x*(5*A + B*x^3) + (7*A*b + 2*a*B)*x*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)])/(27*a^2*b*(a + b*x^3)^(3/2))","C",1
252,1,116,300,0.0943721,"\int \frac{A+B x^3}{x^3 \left(a+b x^3\right)^{5/2}} \, dx","Integrate[(A + B*x^3)/(x^3*(a + b*x^3)^(5/2)),x]","\frac{a^2 \left(80 B x^3-54 A\right)+7 x^3 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (4 a B-13 A b) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right)+a \left(56 b B x^6-260 A b x^3\right)-182 A b^2 x^6}{108 a^3 x^2 \left(a+b x^3\right)^{3/2}}","-\frac{7 x (13 A b-4 a B)}{54 a^3 \sqrt{a+b x^3}}-\frac{x (13 A b-4 a B)}{18 a^2 \left(a+b x^3\right)^{3/2}}-\frac{7 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (13 A b-4 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{54 \sqrt[4]{3} a^3 \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{A}{2 a x^2 \left(a+b x^3\right)^{3/2}}",1,"(-182*A*b^2*x^6 + a^2*(-54*A + 80*B*x^3) + a*(-260*A*b*x^3 + 56*b*B*x^6) + 7*(-13*A*b + 4*a*B)*x^3*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)])/(108*a^3*x^2*(a + b*x^3)^(3/2))","C",1
253,1,83,334,0.056464,"\int \frac{A+B x^3}{x^6 \left(a+b x^3\right)^{5/2}} \, dx","Integrate[(A + B*x^3)/(x^6*(a + b*x^3)^(5/2)),x]","\frac{x^3 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \left(\frac{19 A b}{2}-5 a B\right) \, _2F_1\left(-\frac{2}{3},\frac{5}{2};\frac{1}{3};-\frac{b x^3}{a}\right)-2 a^2 A}{10 a^3 x^5 \left(a+b x^3\right)^{3/2}}","\frac{91 \sqrt{a+b x^3} (19 A b-10 a B)}{540 a^4 x^2}-\frac{13 (19 A b-10 a B)}{135 a^3 x^2 \sqrt{a+b x^3}}-\frac{19 A b-10 a B}{45 a^2 x^2 \left(a+b x^3\right)^{3/2}}+\frac{91 \sqrt{2+\sqrt{3}} b^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (19 A b-10 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{540 \sqrt[4]{3} a^4 \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{A}{5 a x^5 \left(a+b x^3\right)^{3/2}}",1,"(-2*a^2*A + ((19*A*b)/2 - 5*a*B)*x^3*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-2/3, 5/2, 1/3, -((b*x^3)/a)])/(10*a^3*x^5*(a + b*x^3)^(3/2))","C",1
254,1,109,577,0.1123652,"\int \frac{x^7 \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(x^7*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{2 x^2 \left(-32 a^2 B+2 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (16 a B-7 A b) \, _2F_1\left(\frac{2}{3},\frac{5}{2};\frac{5}{3};-\frac{b x^3}{a}\right)+2 a b \left(7 A-8 B x^3\right)+b^2 x^3 \left(7 A+B x^3\right)\right)}{7 b^3 \left(a+b x^3\right)^{3/2}}","\frac{80 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (7 A b-16 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{189 \sqrt[4]{3} b^{11/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{40 \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (7 A b-16 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{63\ 3^{3/4} b^{11/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{80 \sqrt{a+b x^3} (7 A b-16 a B)}{189 b^{11/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{20 x^2 (7 A b-16 a B)}{189 b^3 \sqrt{a+b x^3}}-\frac{2 x^5 (7 A b-16 a B)}{63 b^2 \left(a+b x^3\right)^{3/2}}+\frac{2 B x^8}{7 b \left(a+b x^3\right)^{3/2}}",1,"(2*x^2*(-32*a^2*B + 2*a*b*(7*A - 8*B*x^3) + b^2*x^3*(7*A + B*x^3) + 2*(-7*A*b + 16*a*B)*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 5/2, 5/3, -((b*x^3)/a)]))/(7*b^3*(a + b*x^3)^(3/2))","C",1
255,1,92,559,0.0974768,"\int \frac{x^4 \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(x^4*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{2 x^2 \left(\left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (A b-10 a B) \, _2F_1\left(\frac{2}{3},\frac{5}{2};\frac{5}{3};-\frac{b x^3}{a}\right)-a A b+5 a B \left(2 a+b x^3\right)\right)}{5 a b^2 \left(a+b x^3\right)^{3/2}}","-\frac{8 \sqrt{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (A b-10 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} a^{2/3} b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{4 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (A b-10 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{9\ 3^{3/4} a^{2/3} b^{8/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{8 \sqrt{a+b x^3} (A b-10 a B)}{27 a b^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x^2 (A b-10 a B)}{27 a b^2 \sqrt{a+b x^3}}+\frac{2 x^5 (A b-a B)}{9 a b \left(a+b x^3\right)^{3/2}}",1,"(2*x^2*(-(a*A*b) + 5*a*B*(2*a + b*x^3) + (A*b - 10*a*B)*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 5/2, 5/3, -((b*x^3)/a)]))/(5*a*b^2*(a + b*x^3)^(3/2))","C",1
256,1,81,563,0.0760307,"\int \frac{x \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(x*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{x^2 \left(\left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (4 a B+5 A b) \, _2F_1\left(\frac{2}{3},\frac{5}{2};\frac{5}{3};-\frac{b x^3}{a}\right)-4 a^2 B\right)}{10 a^2 b \left(a+b x^3\right)^{3/2}}","-\frac{2 \sqrt{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (4 a B+5 A b) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} a^{5/3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (4 a B+5 A b) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{9\ 3^{3/4} a^{5/3} b^{5/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} (4 a B+5 A b)}{27 a^2 b^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{2 x^2 (4 a B+5 A b)}{27 a^2 b \sqrt{a+b x^3}}+\frac{2 x^2 (A b-a B)}{9 a b \left(a+b x^3\right)^{3/2}}",1,"(x^2*(-4*a^2*B + (5*A*b + 4*a*B)*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 5/2, 5/3, -((b*x^3)/a)]))/(10*a^2*b*(a + b*x^3)^(3/2))","C",1
257,1,86,578,0.0405955,"\int \frac{A+B x^3}{x^2 \left(a+b x^3\right)^{5/2}} \, dx","Integrate[(A + B*x^3)/(x^2*(a + b*x^3)^(5/2)),x]","-\frac{x^2 \sqrt{\frac{b x^3}{a}+1} \left(\frac{11 A b}{2}-a B\right) \, _2F_1\left(\frac{2}{3},\frac{5}{2};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 a^3 \sqrt{a+b x^3}}-\frac{A}{a x \left(a+b x^3\right)^{3/2}}","\frac{5 \sqrt{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (11 A b-2 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} a^{8/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{5 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (11 A b-2 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{18\ 3^{3/4} a^{8/3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{5 \sqrt{a+b x^3} (11 A b-2 a B)}{27 a^3 b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{5 x^2 (11 A b-2 a B)}{27 a^3 \sqrt{a+b x^3}}-\frac{x^2 (11 A b-2 a B)}{9 a^2 \left(a+b x^3\right)^{3/2}}-\frac{A}{a x \left(a+b x^3\right)^{3/2}}",1,"-(A/(a*x*(a + b*x^3)^(3/2))) - (((11*A*b)/2 - a*B)*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 5/2, 5/3, -((b*x^3)/a)])/(2*a^3*Sqrt[a + b*x^3])","C",1
258,1,83,610,0.058265,"\int \frac{A+B x^3}{x^5 \left(a+b x^3\right)^{5/2}} \, dx","Integrate[(A + B*x^3)/(x^5*(a + b*x^3)^(5/2)),x]","\frac{x^3 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \left(\frac{17 A b}{2}-4 a B\right) \, _2F_1\left(-\frac{1}{3},\frac{5}{2};\frac{2}{3};-\frac{b x^3}{a}\right)-a^2 A}{4 a^3 x^4 \left(a+b x^3\right)^{3/2}}","-\frac{55 \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (17 A b-8 a B) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{108 \sqrt{2} \sqrt[4]{3} a^{11/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{55 \sqrt{2-\sqrt{3}} \sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} (17 A b-8 a B) E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{144\ 3^{3/4} a^{11/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{55 \sqrt{a+b x^3} (17 A b-8 a B)}{216 a^4 x}-\frac{55 \sqrt[3]{b} \sqrt{a+b x^3} (17 A b-8 a B)}{216 a^4 \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{11 (17 A b-8 a B)}{108 a^3 x \sqrt{a+b x^3}}-\frac{17 A b-8 a B}{36 a^2 x \left(a+b x^3\right)^{3/2}}-\frac{A}{4 a x^4 \left(a+b x^3\right)^{3/2}}",1,"(-(a^2*A) + ((17*A*b)/2 - 4*a*B)*x^3*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-1/3, 5/2, 2/3, -((b*x^3)/a)])/(4*a^3*x^4*(a + b*x^3)^(3/2))","C",1
259,1,77,97,0.0873138,"\int \frac{x^8 \sqrt{c+d x^3}}{4 c+d x^3} \, dx","Integrate[(x^8*Sqrt[c + d*x^3])/(4*c + d*x^3),x]","\frac{2 \sqrt{c+d x^3} \left(218 c^2-19 c d x^3+3 d^2 x^6\right)-480 \sqrt{3} c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{45 d^3}","-\frac{32 c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{\sqrt{3} d^3}+\frac{32 c^2 \sqrt{c+d x^3}}{3 d^3}-\frac{10 c \left(c+d x^3\right)^{3/2}}{9 d^3}+\frac{2 \left(c+d x^3\right)^{5/2}}{15 d^3}",1,"(2*Sqrt[c + d*x^3]*(218*c^2 - 19*c*d*x^3 + 3*d^2*x^6) - 480*Sqrt[3]*c^(5/2)*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(45*d^3)","A",1
260,1,65,76,0.0414981,"\int \frac{x^5 \sqrt{c+d x^3}}{4 c+d x^3} \, dx","Integrate[(x^5*Sqrt[c + d*x^3])/(4*c + d*x^3),x]","\frac{24 \sqrt{3} c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)+2 \left(d x^3-11 c\right) \sqrt{c+d x^3}}{9 d^2}","\frac{8 c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{\sqrt{3} d^2}-\frac{8 c \sqrt{c+d x^3}}{3 d^2}+\frac{2 \left(c+d x^3\right)^{3/2}}{9 d^2}",1,"(2*(-11*c + d*x^3)*Sqrt[c + d*x^3] + 24*Sqrt[3]*c^(3/2)*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(9*d^2)","A",1
261,1,54,57,0.0160149,"\int \frac{x^2 \sqrt{c+d x^3}}{4 c+d x^3} \, dx","Integrate[(x^2*Sqrt[c + d*x^3])/(4*c + d*x^3),x]","\frac{2 \left(\sqrt{c+d x^3}-\sqrt{3} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)\right)}{3 d}","\frac{2 \sqrt{c+d x^3}}{3 d}-\frac{2 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{\sqrt{3} d}",1,"(2*(Sqrt[c + d*x^3] - Sqrt[3]*Sqrt[c]*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])]))/(3*d)","A",1
262,1,59,65,0.0182422,"\int \frac{\sqrt{c+d x^3}}{x \left(4 c+d x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x*(4*c + d*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)-\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{6 \sqrt{c}}","\frac{\tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{2 \sqrt{3} \sqrt{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{6 \sqrt{c}}",1,"(Sqrt[3]*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])] - ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(6*Sqrt[c])","A",1
263,1,88,88,0.0434973,"\int \frac{\sqrt{c+d x^3}}{x^4 \left(4 c+d x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x^4*(4*c + d*x^3)),x]","-\frac{d \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{8 \sqrt{3} c^{3/2}}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{24 c^{3/2}}-\frac{\sqrt{c+d x^3}}{12 c x^3}","-\frac{d \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{8 \sqrt{3} c^{3/2}}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{24 c^{3/2}}-\frac{\sqrt{c+d x^3}}{12 c x^3}",1,"-1/12*Sqrt[c + d*x^3]/(c*x^3) - (d*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(8*Sqrt[3]*c^(3/2)) - (d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(24*c^(3/2))","A",1
264,1,133,689,0.0844466,"\int \frac{x^4 \sqrt{c+d x^3}}{4 c+d x^3} \, dx","Integrate[(x^4*Sqrt[c + d*x^3])/(4*c + d*x^3),x]","\frac{-5 d x^5 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)-8 c x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)+8 x^2 \left(c+d x^3\right)}{28 d \sqrt{c+d x^3}}","-\frac{2 \sqrt[3]{2} c^{7/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{\sqrt{3} d^{5/3}}+\frac{2 \sqrt[3]{2} c^{7/6} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{\sqrt{3} d^{5/3}}-\frac{2 \sqrt[3]{2} c^{7/6} \tanh ^{-1}\left(\frac{\sqrt[6]{c} \left(\sqrt[3]{c}-\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{d^{5/3}}+\frac{2 \sqrt[3]{2} c^{7/6} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 d^{5/3}}-\frac{50 \sqrt{2} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{7 \sqrt[4]{3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{25 \sqrt[4]{3} \sqrt{2-\sqrt{3}} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{7 d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{50 c \sqrt{c+d x^3}}{7 d^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{2 x^2 \sqrt{c+d x^3}}{7 d}",1,"(8*x^2*(c + d*x^3) - 8*c*x^2*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -1/4*(d*x^3)/c] - 5*d*x^5*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(28*d*Sqrt[c + d*x^3])","C",0
265,1,63,659,0.0347733,"\int \frac{x \sqrt{c+d x^3}}{4 c+d x^3} \, dx","Integrate[(x*Sqrt[c + d*x^3])/(4*c + d*x^3),x]","\frac{x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)}{8 \sqrt{c+d x^3}}","\frac{2 \sqrt{2} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{c+d x^3}}{d^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{\sqrt[6]{c} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{2^{2/3} \sqrt{3} d^{2/3}}-\frac{\sqrt[6]{c} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{2^{2/3} \sqrt{3} d^{2/3}}+\frac{\sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt[6]{c} \left(\sqrt[3]{c}-\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{2^{2/3} d^{2/3}}-\frac{\sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3\ 2^{2/3} d^{2/3}}",1,"(x^2*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, -1/2, 1, 5/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(8*Sqrt[c + d*x^3])","C",0
266,1,136,697,0.0800446,"\int \frac{\sqrt{c+d x^3}}{x^2 \left(4 c+d x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x^2*(4*c + d*x^3)),x]","\frac{d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)+25 c d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)-40 c \left(c+d x^3\right)}{160 c^2 x \sqrt{c+d x^3}}","\frac{\sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{2 \sqrt{2} \sqrt[4]{3} c^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{8 c^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{4\ 2^{2/3} \sqrt{3} c^{5/6}}+\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{4\ 2^{2/3} \sqrt{3} c^{5/6}}-\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\sqrt[6]{c} \left(\sqrt[3]{c}-\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{4\ 2^{2/3} c^{5/6}}+\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{12\ 2^{2/3} c^{5/6}}-\frac{\sqrt{c+d x^3}}{4 c x}+\frac{\sqrt[3]{d} \sqrt{c+d x^3}}{4 c \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}",1,"(-40*c*(c + d*x^3) + 25*c*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(160*c^2*x*Sqrt[c + d*x^3])","C",0
267,1,236,66,0.3631766,"\int \frac{x^3 \sqrt{c+d x^3}}{4 c+d x^3} \, dx","Integrate[(x^3*Sqrt[c + d*x^3])/(4*c + d*x^3),x]","\frac{x \left(32 \left(\frac{64 c^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)}{d \left(4 c+d x^3\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)+2 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)\right)-16 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)\right)}+\frac{c}{d}+x^3\right)-17 x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)\right)}{80 \sqrt{c+d x^3}}","\frac{x^4 \sqrt{c+d x^3} F_1\left(\frac{4}{3};1,-\frac{1}{2};\frac{7}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right)}{16 c \sqrt{\frac{d x^3}{c}+1}}",1,"(x*(-17*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + 32*(c/d + x^3 + (64*c^3*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(d*(4*c + d*x^3)*(-16*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + 2*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c]))))))/(80*Sqrt[c + d*x^3])","B",0
268,1,165,64,0.1956239,"\int \frac{\sqrt{c+d x^3}}{4 c+d x^3} \, dx","Integrate[Sqrt[c + d*x^3]/(4*c + d*x^3),x]","\frac{16 c x \sqrt{c+d x^3} F_1\left(\frac{1}{3};-\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)}{\left(4 c+d x^3\right) \left(16 c F_1\left(\frac{1}{3};-\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)-3 d x^3 \left(F_1\left(\frac{4}{3};-\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)-2 F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)\right)\right)}","\frac{x \sqrt{c+d x^3} F_1\left(\frac{1}{3};1,-\frac{1}{2};\frac{4}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right)}{4 c \sqrt{\frac{d x^3}{c}+1}}",1,"(16*c*x*Sqrt[c + d*x^3]*AppellF1[1/3, -1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/((4*c + d*x^3)*(16*c*AppellF1[1/3, -1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c] - 3*d*x^3*(AppellF1[4/3, -1/2, 2, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c] - 2*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c])))","B",0
269,1,244,66,0.1362774,"\int \frac{\sqrt{c+d x^3}}{x^3 \left(4 c+d x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x^3*(4*c + d*x^3)),x]","\frac{\frac{2048 c^3 d x^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)}{\left(4 c+d x^3\right) \left(16 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)-3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)+2 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)\right)\right)}-d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)-32 c \left(c+d x^3\right)}{256 c^2 x^2 \sqrt{c+d x^3}}","-\frac{\sqrt{c+d x^3} F_1\left(-\frac{2}{3};1,-\frac{1}{2};\frac{1}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right)}{8 c x^2 \sqrt{\frac{d x^3}{c}+1}}",1,"(-32*c*(c + d*x^3) - d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + (2048*c^3*d*x^3*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/((4*c + d*x^3)*(16*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c] - 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + 2*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c]))))/(256*c^2*x^2*Sqrt[c + d*x^3])","B",0
270,1,65,78,0.0701686,"\int \frac{x^8}{\sqrt{c+d x^3} \left(4 c+d x^3\right)} \, dx","Integrate[x^8/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]","\frac{32 \sqrt{3} c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)+2 \left(d x^3-14 c\right) \sqrt{c+d x^3}}{9 d^3}","\frac{32 c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{3 \sqrt{3} d^3}-\frac{10 c \sqrt{c+d x^3}}{3 d^3}+\frac{2 \left(c+d x^3\right)^{3/2}}{9 d^3}",1,"(2*(-14*c + d*x^3)*Sqrt[c + d*x^3] + 32*Sqrt[3]*c^(3/2)*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(9*d^3)","A",1
271,1,56,59,0.0219389,"\int \frac{x^5}{\sqrt{c+d x^3} \left(4 c+d x^3\right)} \, dx","Integrate[x^5/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]","\frac{6 \sqrt{c+d x^3}-8 \sqrt{3} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{9 d^2}","\frac{2 \sqrt{c+d x^3}}{3 d^2}-\frac{8 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{3 \sqrt{3} d^2}",1,"(6*Sqrt[c + d*x^3] - 8*Sqrt[3]*Sqrt[c]*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(9*d^2)","A",1
272,1,40,40,0.0102656,"\int \frac{x^2}{\sqrt{c+d x^3} \left(4 c+d x^3\right)} \, dx","Integrate[x^2/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{3 \sqrt{3} \sqrt{c} d}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{3 \sqrt{3} \sqrt{c} d}",1,"(2*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(3*Sqrt[3]*Sqrt[c]*d)","A",1
273,1,59,65,0.0206445,"\int \frac{1}{x \sqrt{c+d x^3} \left(4 c+d x^3\right)} \, dx","Integrate[1/(x*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)+3 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{18 c^{3/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{6 \sqrt{3} c^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{6 c^{3/2}}",1,"-1/18*(Sqrt[3]*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])] + 3*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/c^(3/2)","A",1
274,1,88,88,0.0350471,"\int \frac{1}{x^4 \sqrt{c+d x^3} \left(4 c+d x^3\right)} \, dx","Integrate[1/(x^4*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]","\frac{d \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{24 \sqrt{3} c^{5/2}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{8 c^{5/2}}-\frac{\sqrt{c+d x^3}}{12 c^2 x^3}","\frac{d \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{24 \sqrt{3} c^{5/2}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{8 c^{5/2}}-\frac{\sqrt{c+d x^3}}{12 c^2 x^3}",1,"-1/12*Sqrt[c + d*x^3]/(c^2*x^3) + (d*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(24*Sqrt[3]*c^(5/2)) + (d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(8*c^(5/2))","A",1
275,1,67,667,0.0367546,"\int \frac{x^4}{\sqrt{c+d x^3} \left(4 c+d x^3\right)} \, dx","Integrate[x^4/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]","\frac{x^5 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)}{20 c \sqrt{c+d x^3}}","\frac{2 \sqrt{2} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{c+d x^3}}{d^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{2 \sqrt[3]{2} \sqrt[6]{c} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{3 \sqrt{3} d^{5/3}}-\frac{2 \sqrt[3]{2} \sqrt[6]{c} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{3 \sqrt{3} d^{5/3}}+\frac{2 \sqrt[3]{2} \sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt[6]{c} \left(\sqrt[3]{c}-\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{3 d^{5/3}}-\frac{2 \sqrt[3]{2} \sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{9 d^{5/3}}",1,"(x^5*Sqrt[(c + d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(20*c*Sqrt[c + d*x^3])","C",0
276,1,67,206,0.0259164,"\int \frac{x}{\sqrt{c+d x^3} \left(4 c+d x^3\right)} \, dx","Integrate[x/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]","\frac{x^2 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)}{8 c \sqrt{c+d x^3}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{3\ 2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{3\ 2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{c} \left(\sqrt[3]{c}-\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{3\ 2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{9\ 2^{2/3} c^{5/6} d^{2/3}}",1,"(x^2*Sqrt[(c + d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(8*c*Sqrt[c + d*x^3])","C",0
277,1,136,697,0.0973007,"\int \frac{1}{x^2 \sqrt{c+d x^3} \left(4 c+d x^3\right)} \, dx","Integrate[1/(x^2*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]","\frac{d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)+5 c d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)-40 c \left(c+d x^3\right)}{160 c^3 x \sqrt{c+d x^3}}","\frac{\sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{2 \sqrt{2} \sqrt[4]{3} c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{8 c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{12\ 2^{2/3} \sqrt{3} c^{11/6}}-\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right)}{12\ 2^{2/3} \sqrt{3} c^{11/6}}+\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\sqrt[6]{c} \left(\sqrt[3]{c}-\sqrt[3]{2} \sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{12\ 2^{2/3} c^{11/6}}-\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{36\ 2^{2/3} c^{11/6}}-\frac{\sqrt{c+d x^3}}{4 c^2 x}+\frac{\sqrt[3]{d} \sqrt{c+d x^3}}{4 c^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}",1,"(-40*c*(c + d*x^3) + 5*c*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(160*c^3*x*Sqrt[c + d*x^3])","C",0
278,1,67,66,0.061054,"\int \frac{x^3}{\sqrt{c+d x^3} \left(4 c+d x^3\right)} \, dx","Integrate[x^3/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]","\frac{x^4 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)}{16 c \sqrt{c+d x^3}}","\frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};1,\frac{1}{2};\frac{7}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right)}{16 c \sqrt{c+d x^3}}",1,"(x^4*Sqrt[(c + d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(16*c*Sqrt[c + d*x^3])","A",0
279,1,165,64,0.0685202,"\int \frac{1}{\sqrt{c+d x^3} \left(4 c+d x^3\right)} \, dx","Integrate[1/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]","\frac{16 c x F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)}{\sqrt{c+d x^3} \left(4 c+d x^3\right) \left(16 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)-3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)+2 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)\right)\right)}","\frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{1}{3};1,\frac{1}{2};\frac{4}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right)}{4 c \sqrt{c+d x^3}}",1,"(16*c*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(Sqrt[c + d*x^3]*(4*c + d*x^3)*(16*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c] - 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + 2*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c])))","B",0
280,1,243,66,0.1829667,"\int \frac{1}{x^3 \sqrt{c+d x^3} \left(4 c+d x^3\right)} \, dx","Integrate[1/(x^3*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]","\frac{-\frac{d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)}{c^3}+\frac{2048 d x^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)}{\left(4 c+d x^3\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)+2 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)\right)-16 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right)\right)}-\frac{32 \left(c+d x^3\right)}{c^2}}{256 x^2 \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{2}{3};1,\frac{1}{2};\frac{1}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right)}{8 c x^2 \sqrt{c+d x^3}}",1,"((-32*(c + d*x^3))/c^2 - (d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/c^3 + (2048*d*x^3*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/((4*c + d*x^3)*(-16*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + 2*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c]))))/(256*x^2*Sqrt[c + d*x^3])","B",0
281,1,28,127,0.0291732,"\int \frac{x}{\sqrt{1-x^3} \left(4-x^3\right)} \, dx","Integrate[x/(Sqrt[1 - x^3]*(4 - x^3)),x]","\frac{1}{8} x^2 F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};x^3,\frac{x^3}{4}\right)","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} x\right)}{\sqrt{1-x^3}}\right)}{3\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\sqrt{1-x^3}}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[3]{2} x+1}{\sqrt{1-x^3}}\right)}{3\ 2^{2/3}}+\frac{\tanh ^{-1}\left(\sqrt{1-x^3}\right)}{9\ 2^{2/3}}",1,"(x^2*AppellF1[2/3, 1/2, 1, 5/3, x^3, x^3/4])/8","C",0
282,1,81,111,0.1108925,"\int \frac{x^{11} \sqrt{c+d x^3}}{8 c-d x^3} \, dx","Integrate[(x^11*Sqrt[c + d*x^3])/(8*c - d*x^3),x]","\frac{107520 c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-2 \sqrt{c+d x^3} \left(18632 c^3+764 c^2 d x^3+57 c d^2 x^6+5 d^3 x^9\right)}{105 d^4}","\frac{1024 c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^4}-\frac{1024 c^3 \sqrt{c+d x^3}}{3 d^4}-\frac{38 c^2 \left(c+d x^3\right)^{3/2}}{3 d^4}-\frac{4 c \left(c+d x^3\right)^{5/2}}{5 d^4}-\frac{2 \left(c+d x^3\right)^{7/2}}{21 d^4}",1,"(-2*Sqrt[c + d*x^3]*(18632*c^3 + 764*c^2*d*x^3 + 57*c*d^2*x^6 + 5*d^3*x^9) + 107520*c^(7/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(105*d^4)","A",1
283,1,70,90,0.0637312,"\int \frac{x^8 \sqrt{c+d x^3}}{8 c-d x^3} \, dx","Integrate[(x^8*Sqrt[c + d*x^3])/(8*c - d*x^3),x]","\frac{5760 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-2 \sqrt{c+d x^3} \left(998 c^2+41 c d x^3+3 d^2 x^6\right)}{45 d^3}","\frac{128 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^3}-\frac{128 c^2 \sqrt{c+d x^3}}{3 d^3}-\frac{14 c \left(c+d x^3\right)^{3/2}}{9 d^3}-\frac{2 \left(c+d x^3\right)^{5/2}}{15 d^3}",1,"(-2*Sqrt[c + d*x^3]*(998*c^2 + 41*c*d*x^3 + 3*d^2*x^6) + 5760*c^(5/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(45*d^3)","A",1
284,1,58,69,0.0397351,"\int \frac{x^5 \sqrt{c+d x^3}}{8 c-d x^3} \, dx","Integrate[(x^5*Sqrt[c + d*x^3])/(8*c - d*x^3),x]","\frac{144 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-2 \sqrt{c+d x^3} \left(25 c+d x^3\right)}{9 d^2}","\frac{16 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^2}-\frac{16 c \sqrt{c+d x^3}}{3 d^2}-\frac{2 \left(c+d x^3\right)^{3/2}}{9 d^2}",1,"(-2*Sqrt[c + d*x^3]*(25*c + d*x^3) + 144*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*d^2)","A",1
285,1,47,50,0.027364,"\int \frac{x^2 \sqrt{c+d x^3}}{8 c-d x^3} \, dx","Integrate[(x^2*Sqrt[c + d*x^3])/(8*c - d*x^3),x]","-\frac{2 \left(\sqrt{c+d x^3}-3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)\right)}{3 d}","\frac{2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d}-\frac{2 \sqrt{c+d x^3}}{3 d}",1,"(-2*(Sqrt[c + d*x^3] - 3*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]))/(3*d)","A",1
286,1,53,58,0.0183996,"\int \frac{\sqrt{c+d x^3}}{x \left(8 c-d x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x*(8*c - d*x^3)),x]","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{12 \sqrt{c}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{4 \sqrt{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{12 \sqrt{c}}",1,"(3*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] - ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(12*Sqrt[c])","A",1
287,1,81,81,0.0292746,"\int \frac{\sqrt{c+d x^3}}{x^4 \left(8 c-d x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x^4*(8*c - d*x^3)),x]","\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{32 c^{3/2}}-\frac{5 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{96 c^{3/2}}-\frac{\sqrt{c+d x^3}}{24 c x^3}","\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{32 c^{3/2}}-\frac{5 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{96 c^{3/2}}-\frac{\sqrt{c+d x^3}}{24 c x^3}",1,"-1/24*Sqrt[c + d*x^3]/(c*x^3) + (d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(32*c^(3/2)) - (5*d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(96*c^(3/2))","A",1
288,1,96,107,0.0598673,"\int \frac{\sqrt{c+d x^3}}{x^7 \left(8 c-d x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x^7*(8*c - d*x^3)),x]","\frac{3 d^2 x^6 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)+3 d^2 x^6 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)-4 \sqrt{c} \sqrt{c+d x^3} \left(4 c+3 d x^3\right)}{768 c^{5/2} x^6}","\frac{d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{256 c^{5/2}}+\frac{d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{256 c^{5/2}}-\frac{d \sqrt{c+d x^3}}{64 c^2 x^3}-\frac{\sqrt{c+d x^3}}{48 c x^6}",1,"(-4*Sqrt[c]*Sqrt[c + d*x^3]*(4*c + 3*d*x^3) + 3*d^2*x^6*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] + 3*d^2*x^6*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(768*c^(5/2)*x^6)","A",1
289,1,150,648,0.0950096,"\int \frac{x^7 \sqrt{c+d x^3}}{8 c-d x^3} \, dx","Integrate[(x^7*Sqrt[c + d*x^3])/(8*c - d*x^3),x]","\frac{2140 c^2 x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+1531 c d x^5 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-20 \left(107 c^2 x^2+114 c d x^5+7 d^2 x^8\right)}{910 d^2 \sqrt{c+d x^3}}","-\frac{32 \sqrt{3} c^{13/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{d^{8/3}}+\frac{32 c^{13/6} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{d^{8/3}}-\frac{32 c^{13/6} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^{8/3}}-\frac{12248 \sqrt{2} c^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{91 \sqrt[4]{3} d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{6124 \sqrt[4]{3} \sqrt{2-\sqrt{3}} c^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{91 d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{12248 c^2 \sqrt{c+d x^3}}{91 d^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{214 c x^2 \sqrt{c+d x^3}}{91 d^2}-\frac{2 x^5 \sqrt{c+d x^3}}{13 d}",1,"(-20*(107*c^2*x^2 + 114*c*d*x^5 + 7*d^2*x^8) + 2140*c^2*x^2*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 1531*c*d*x^5*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(910*d^2*Sqrt[c + d*x^3])","C",0
290,1,130,624,0.0817194,"\int \frac{x^4 \sqrt{c+d x^3}}{8 c-d x^3} \, dx","Integrate[(x^4*Sqrt[c + d*x^3])/(8*c - d*x^3),x]","\frac{x^2 \left(59 d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+80 c \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-80 \left(c+d x^3\right)\right)}{280 d \sqrt{c+d x^3}}","-\frac{4 \sqrt{3} c^{7/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{d^{5/3}}+\frac{4 c^{7/6} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{d^{5/3}}-\frac{4 c^{7/6} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^{5/3}}-\frac{118 \sqrt{2} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{7 \sqrt[4]{3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{59 \sqrt[4]{3} \sqrt{2-\sqrt{3}} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{7 d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{118 c \sqrt{c+d x^3}}{7 d^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{2 x^2 \sqrt{c+d x^3}}{7 d}",1,"(x^2*(-80*(c + d*x^3) + 80*c*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 59*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(280*d*Sqrt[c + d*x^3])","C",0
291,1,63,601,0.0397322,"\int \frac{x \sqrt{c+d x^3}}{8 c-d x^3} \, dx","Integrate[(x*Sqrt[c + d*x^3])/(8*c - d*x^3),x]","\frac{x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{16 \sqrt{c+d x^3}}","-\frac{2 \sqrt{2} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{2 \sqrt{c+d x^3}}{d^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{\sqrt{3} \sqrt[6]{c} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{2 d^{2/3}}+\frac{\sqrt[6]{c} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{2 d^{2/3}}-\frac{\sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{2 d^{2/3}}",1,"(x^2*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, -1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(16*Sqrt[c + d*x^3])","C",0
292,1,137,632,0.1105741,"\int \frac{\sqrt{c+d x^3}}{x^2 \left(8 c-d x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x^2*(8*c - d*x^3)),x]","\frac{-d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+65 c d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-80 c \left(c+d x^3\right)}{640 c^2 x \sqrt{c+d x^3}}","\frac{\sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt{2} \sqrt[4]{3} c^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{16 c^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{3} \sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{16 c^{5/6}}+\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{16 c^{5/6}}-\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{16 c^{5/6}}-\frac{\sqrt{c+d x^3}}{8 c x}+\frac{\sqrt[3]{d} \sqrt{c+d x^3}}{8 c \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}",1,"(-80*c*(c + d*x^3) + 65*c*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(640*c^2*x*Sqrt[c + d*x^3])","C",0
293,1,153,654,0.1401747,"\int \frac{\sqrt{c+d x^3}}{x^5 \left(8 c-d x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x^5*(8*c - d*x^3)),x]","\frac{125 c d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 \left(d^3 x^9 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+40 c \left(c^2+3 c d x^3+2 d^2 x^6\right)\right)}{5120 c^3 x^4 \sqrt{c+d x^3}}","-\frac{\sqrt{3} d^{4/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{128 c^{11/6}}+\frac{d^{4/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{128 c^{11/6}}-\frac{d^{4/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{128 c^{11/6}}+\frac{d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{8 \sqrt{2} \sqrt[4]{3} c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{32 c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{d^{4/3} \sqrt{c+d x^3}}{16 c^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{d \sqrt{c+d x^3}}{16 c^2 x}-\frac{\sqrt{c+d x^3}}{32 c x^4}",1,"(125*c*d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*(40*c*(c^2 + 3*c*d*x^3 + 2*d^2*x^6) + d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(5120*c^3*x^4*Sqrt[c + d*x^3])","C",0
294,1,164,678,0.1112251,"\int \frac{\sqrt{c+d x^3}}{x^8 \left(8 c-d x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x^8*(8*c - d*x^3)),x]","\frac{32 d^4 x^{12} \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-325 c d^3 x^9 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-160 c \left(32 c^3+51 c^2 d x^3+3 c d^2 x^6-16 d^3 x^9\right)}{286720 c^4 x^7 \sqrt{c+d x^3}}","-\frac{\sqrt{3} d^{7/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{1024 c^{17/6}}+\frac{d^{7/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{1024 c^{17/6}}-\frac{d^{7/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{1024 c^{17/6}}-\frac{d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{56 \sqrt{2} \sqrt[4]{3} c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{224 c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{d^{7/3} \sqrt{c+d x^3}}{112 c^3 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{d^2 \sqrt{c+d x^3}}{112 c^3 x}-\frac{19 d \sqrt{c+d x^3}}{1792 c^2 x^4}-\frac{\sqrt{c+d x^3}}{56 c x^7}",1,"(-160*c*(32*c^3 + 51*c^2*d*x^3 + 3*c*d^2*x^6 - 16*d^3*x^9) - 325*c*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 32*d^4*x^12*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(286720*c^4*x^7*Sqrt[c + d*x^3])","C",0
295,1,93,130,0.1334841,"\int \frac{x^{11} \left(c+d x^3\right)^{3/2}}{8 c-d x^3} \, dx","Integrate[(x^11*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]","\frac{9216 c^{9/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^4}-\frac{2 \sqrt{c+d x^3} \left(1509176 c^4+61892 c^3 d x^3+4611 c^2 d^2 x^6+410 c d^3 x^9+35 d^4 x^{12}\right)}{945 d^4}","\frac{9216 c^{9/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^4}-\frac{3072 c^4 \sqrt{c+d x^3}}{d^4}-\frac{1024 c^3 \left(c+d x^3\right)^{3/2}}{9 d^4}-\frac{38 c^2 \left(c+d x^3\right)^{5/2}}{5 d^4}-\frac{4 c \left(c+d x^3\right)^{7/2}}{7 d^4}-\frac{2 \left(c+d x^3\right)^{9/2}}{27 d^4}",1,"(-2*Sqrt[c + d*x^3]*(1509176*c^4 + 61892*c^3*d*x^3 + 4611*c^2*d^2*x^6 + 410*c*d^3*x^9 + 35*d^4*x^12))/(945*d^4) + (9216*c^(9/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^4","A",1
296,1,81,109,0.0872895,"\int \frac{x^8 \left(c+d x^3\right)^{3/2}}{8 c-d x^3} \, dx","Integrate[(x^8*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]","\frac{362880 c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-2 \sqrt{c+d x^3} \left(62882 c^3+2579 c^2 d x^3+192 c d^2 x^6+15 d^3 x^9\right)}{315 d^3}","\frac{1152 c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^3}-\frac{384 c^3 \sqrt{c+d x^3}}{d^3}-\frac{128 c^2 \left(c+d x^3\right)^{3/2}}{9 d^3}-\frac{14 c \left(c+d x^3\right)^{5/2}}{15 d^3}-\frac{2 \left(c+d x^3\right)^{7/2}}{21 d^3}",1,"(-2*Sqrt[c + d*x^3]*(62882*c^3 + 2579*c^2*d*x^3 + 192*c*d^2*x^6 + 15*d^3*x^9) + 362880*c^(7/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(315*d^3)","A",1
297,1,70,88,0.0385987,"\int \frac{x^5 \left(c+d x^3\right)^{3/2}}{8 c-d x^3} \, dx","Integrate[(x^5*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]","\frac{6480 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-2 \sqrt{c+d x^3} \left(1123 c^2+46 c d x^3+3 d^2 x^6\right)}{45 d^2}","\frac{144 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^2}-\frac{48 c^2 \sqrt{c+d x^3}}{d^2}-\frac{16 c \left(c+d x^3\right)^{3/2}}{9 d^2}-\frac{2 \left(c+d x^3\right)^{5/2}}{15 d^2}",1,"(-2*Sqrt[c + d*x^3]*(1123*c^2 + 46*c*d*x^3 + 3*d^2*x^6) + 6480*c^(5/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(45*d^2)","A",1
298,1,58,67,0.0378769,"\int \frac{x^2 \left(c+d x^3\right)^{3/2}}{8 c-d x^3} \, dx","Integrate[(x^2*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]","\frac{162 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-2 \sqrt{c+d x^3} \left(28 c+d x^3\right)}{9 d}","\frac{18 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d}-\frac{6 c \sqrt{c+d x^3}}{d}-\frac{2 \left(c+d x^3\right)^{3/2}}{9 d}",1,"(-2*Sqrt[c + d*x^3]*(28*c + d*x^3) + 162*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*d)","A",1
299,1,73,73,0.0477325,"\int \frac{\left(c+d x^3\right)^{3/2}}{x \left(8 c-d x^3\right)} \, dx","Integrate[(c + d*x^3)^(3/2)/(x*(8*c - d*x^3)),x]","-\frac{2}{3} \sqrt{c+d x^3}+\frac{9}{4} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-\frac{1}{12} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)","-\frac{2}{3} \sqrt{c+d x^3}+\frac{9}{4} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-\frac{1}{12} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)",1,"(-2*Sqrt[c + d*x^3])/3 + (9*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/4 - (Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/12","A",1
300,1,78,78,0.0461185,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^4 \left(8 c-d x^3\right)} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^4*(8*c - d*x^3)),x]","-\frac{\sqrt{c+d x^3}}{24 x^3}+\frac{9 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{32 \sqrt{c}}-\frac{13 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{96 \sqrt{c}}","-\frac{\sqrt{c+d x^3}}{24 x^3}+\frac{9 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{32 \sqrt{c}}-\frac{13 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{96 \sqrt{c}}",1,"-1/24*Sqrt[c + d*x^3]/x^3 + (9*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(32*Sqrt[c]) - (13*d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(96*Sqrt[c])","A",1
301,1,96,104,0.106045,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^7 \left(8 c-d x^3\right)} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^7*(8*c - d*x^3)),x]","\frac{27 d^2 x^6 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-37 d^2 x^6 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)-4 \sqrt{c} \sqrt{c+d x^3} \left(4 c+11 d x^3\right)}{768 c^{3/2} x^6}","\frac{9 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{256 c^{3/2}}-\frac{37 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{768 c^{3/2}}-\frac{11 d \sqrt{c+d x^3}}{192 c x^3}-\frac{\sqrt{c+d x^3}}{48 x^6}",1,"(-4*Sqrt[c]*Sqrt[c + d*x^3]*(4*c + 11*d*x^3) + 27*d^2*x^6*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] - 37*d^2*x^6*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(768*c^(3/2)*x^6)","A",1
302,1,163,669,0.116321,"\int \frac{x^7 \left(c+d x^3\right)^{3/2}}{8 c-d x^3} \, dx","Integrate[(x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]","\frac{365340 c^3 x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+261831 c^2 d x^5 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-20 x^2 \left(18267 c^3+19485 c^2 d x^3+1309 c d^2 x^6+91 d^3 x^9\right)}{17290 d^2 \sqrt{c+d x^3}}","-\frac{288 \sqrt{3} c^{19/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{d^{8/3}}+\frac{288 c^{19/6} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{d^{8/3}}-\frac{288 c^{19/6} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^{8/3}}-\frac{698216 \sqrt{2} 3^{3/4} c^{10/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{1729 d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{1047324 \sqrt[4]{3} \sqrt{2-\sqrt{3}} c^{10/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{1729 d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{2094648 c^3 \sqrt{c+d x^3}}{1729 d^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{36534 c^2 x^2 \sqrt{c+d x^3}}{1729 d^2}-\frac{2}{19} x^8 \sqrt{c+d x^3}-\frac{348 c x^5 \sqrt{c+d x^3}}{247 d}",1,"(-20*x^2*(18267*c^3 + 19485*c^2*d*x^3 + 1309*c*d^2*x^6 + 91*d^3*x^9) + 365340*c^3*x^2*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 261831*c^2*d*x^5*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(17290*d^2*Sqrt[c + d*x^3])","C",0
303,1,150,645,0.11347,"\int \frac{x^4 \left(c+d x^3\right)^{3/2}}{8 c-d x^3} \, dx","Integrate[(x^4*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]","\frac{9600 c^2 x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+6891 c d x^5 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-80 \left(120 c^2 x^2+127 c d x^5+7 d^2 x^8\right)}{3640 d \sqrt{c+d x^3}}","-\frac{36 \sqrt{3} c^{13/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{d^{5/3}}+\frac{36 c^{13/6} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{d^{5/3}}-\frac{36 c^{13/6} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^{5/3}}-\frac{4594 \sqrt{2} 3^{3/4} c^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{91 d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{6891 \sqrt[4]{3} \sqrt{2-\sqrt{3}} c^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{91 d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{13782 c^2 \sqrt{c+d x^3}}{91 d^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{2}{13} x^5 \sqrt{c+d x^3}-\frac{240 c x^2 \sqrt{c+d x^3}}{91 d}",1,"(-80*(120*c^2*x^2 + 127*c*d*x^5 + 7*d^2*x^8) + 9600*c^2*x^2*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 6891*c*d*x^5*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(3640*d*Sqrt[c + d*x^3])","C",0
304,1,127,627,0.1181515,"\int \frac{x \left(c+d x^3\right)^{3/2}}{8 c-d x^3} \, dx","Integrate[(x*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]","\frac{x^2 \left(132 d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+195 c \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-160 \left(c+d x^3\right)\right)}{560 \sqrt{c+d x^3}}","-\frac{9 \sqrt{3} c^{7/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{2 d^{2/3}}+\frac{9 c^{7/6} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{2 d^{2/3}}-\frac{9 c^{7/6} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{2 d^{2/3}}-\frac{44 \sqrt{2} 3^{3/4} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{7 d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{66 \sqrt[4]{3} \sqrt{2-\sqrt{3}} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{7 d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{132 c \sqrt{c+d x^3}}{7 d^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{2}{7} x^2 \sqrt{c+d x^3}",1,"(x^2*(-160*(c + d*x^3) + 195*c*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 132*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(560*Sqrt[c + d*x^3])","C",0
305,1,137,626,0.1023499,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^2 \left(8 c-d x^3\right)} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^2*(8*c - d*x^3)),x]","\frac{3 d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+21 c d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-16 c \left(c+d x^3\right)}{128 c x \sqrt{c+d x^3}}","-\frac{5\ 3^{3/4} \sqrt[3]{c} \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt{2} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{15 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{c} \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{16 \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{c+d x^3}}{8 x}-\frac{15 \sqrt[3]{d} \sqrt{c+d x^3}}{8 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{9}{16} \sqrt{3} \sqrt[6]{c} \sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)+\frac{9}{16} \sqrt[6]{c} \sqrt[3]{d} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)-\frac{9}{16} \sqrt[6]{c} \sqrt[3]{d} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)",1,"(-16*c*(c + d*x^3) + 21*c*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(128*c*x*Sqrt[c + d*x^3])","C",0
306,1,154,651,0.1112297,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^5 \left(8 c-d x^3\right)} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^5*(8*c - d*x^3)),x]","\frac{645 c d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 \left(3 d^3 x^9 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+40 c \left(c^2+7 c d x^3+6 d^2 x^6\right)\right)}{5120 c^2 x^4 \sqrt{c+d x^3}}","-\frac{9 \sqrt{3} d^{4/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{128 c^{5/6}}+\frac{9 d^{4/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{128 c^{5/6}}-\frac{9 d^{4/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{128 c^{5/6}}+\frac{3^{3/4} d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{8 \sqrt{2} c^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{32 c^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{3 d^{4/3} \sqrt{c+d x^3}}{16 c \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{3 d \sqrt{c+d x^3}}{16 c x}-\frac{\sqrt{c+d x^3}}{32 x^4}",1,"(645*c*d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*(40*c*(c^2 + 7*c*d*x^3 + 6*d^2*x^6) + 3*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(5120*c^2*x^4*Sqrt[c + d*x^3])","C",0
307,1,167,675,0.1083298,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^8 \left(8 c-d x^3\right)} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^8*(8*c - d*x^3)),x]","\frac{6675 c d^3 x^9 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-32 \left(6 d^4 x^{12} \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+5 c \left(32 c^3+107 c^2 d x^3+171 c d^2 x^6+96 d^3 x^9\right)\right)}{286720 c^3 x^7 \sqrt{c+d x^3}}","-\frac{9 \sqrt{3} d^{7/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{1024 c^{11/6}}+\frac{9 d^{7/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{1024 c^{11/6}}-\frac{9 d^{7/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{1024 c^{11/6}}+\frac{3^{3/4} d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{28 \sqrt{2} c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{112 c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{3 d^{7/3} \sqrt{c+d x^3}}{56 c^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{3 d^2 \sqrt{c+d x^3}}{56 c^2 x}-\frac{\sqrt{c+d x^3}}{56 x^7}-\frac{75 d \sqrt{c+d x^3}}{1792 c x^4}",1,"(6675*c*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 32*(5*c*(32*c^3 + 107*c^2*d*x^3 + 171*c*d^2*x^6 + 96*d^3*x^9) + 6*d^4*x^12*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(286720*c^3*x^7*Sqrt[c + d*x^3])","C",0
308,1,69,90,0.1064591,"\int \frac{x^{11}}{\left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x^11/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{5120 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-6 \sqrt{c+d x^3} \left(296 c^2+12 c d x^3+d^2 x^6\right)}{45 d^4}","\frac{1024 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 d^4}-\frac{38 c^2 \sqrt{c+d x^3}}{d^4}-\frac{4 c \left(c+d x^3\right)^{3/2}}{3 d^4}-\frac{2 \left(c+d x^3\right)^{5/2}}{15 d^4}",1,"(-6*Sqrt[c + d*x^3]*(296*c^2 + 12*c*d*x^3 + d^2*x^6) + 5120*c^(5/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(45*d^4)","A",1
309,1,58,71,0.0685977,"\int \frac{x^8}{\left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x^8/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{128 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-2 \sqrt{c+d x^3} \left(22 c+d x^3\right)}{9 d^3}","\frac{128 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 d^3}-\frac{14 c \sqrt{c+d x^3}}{3 d^3}-\frac{2 \left(c+d x^3\right)^{3/2}}{9 d^3}",1,"(-2*Sqrt[c + d*x^3]*(22*c + d*x^3) + 128*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*d^3)","A",1
310,1,49,52,0.0197834,"\int \frac{x^5}{\left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x^5/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]","-\frac{2 \left(3 \sqrt{c+d x^3}-8 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)\right)}{9 d^2}","\frac{16 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 d^2}-\frac{2 \sqrt{c+d x^3}}{3 d^2}",1,"(-2*(3*Sqrt[c + d*x^3] - 8*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]))/(9*d^2)","A",1
311,1,33,33,0.0109432,"\int \frac{x^2}{\left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x^2/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 \sqrt{c} d}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 \sqrt{c} d}",1,"(2*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*Sqrt[c]*d)","A",1
312,1,51,58,0.0288716,"\int \frac{1}{x \left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-3 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{36 c^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{36 c^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{12 c^{3/2}}",1,"(ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] - 3*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(36*c^(3/2))","A",1
313,1,81,81,0.0522625,"\int \frac{1}{x^4 \left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x^4*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{288 c^{5/2}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{32 c^{5/2}}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3}","\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{288 c^{5/2}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{32 c^{5/2}}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3}",1,"-1/24*Sqrt[c + d*x^3]/(c^2*x^3) + (d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(288*c^(5/2)) + (d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(32*c^(5/2))","A",1
314,1,95,107,0.0709181,"\int \frac{1}{x^7 \left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x^7*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{d^2 x^6 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-63 d^2 x^6 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)+12 \sqrt{c} \sqrt{c+d x^3} \left(5 d x^3-4 c\right)}{2304 c^{7/2} x^6}","\frac{d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{2304 c^{7/2}}-\frac{7 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{256 c^{7/2}}+\frac{5 d \sqrt{c+d x^3}}{192 c^3 x^3}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6}",1,"(12*Sqrt[c]*Sqrt[c + d*x^3]*(-4*c + 5*d*x^3) + d^2*x^6*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] - 63*d^2*x^6*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(2304*c^(7/2)*x^6)","A",1
315,1,130,630,0.0869754,"\int \frac{x^7}{\left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x^7/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{x^2 \left(13 d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+20 c \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-20 \left(c+d x^3\right)\right)}{70 d^2 \sqrt{c+d x^3}}","-\frac{32 c^{7/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{3 \sqrt{3} d^{8/3}}+\frac{32 c^{7/6} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{9 d^{8/3}}-\frac{32 c^{7/6} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 d^{8/3}}-\frac{104 \sqrt{2} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{7 \sqrt[4]{3} d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{52 \sqrt[4]{3} \sqrt{2-\sqrt{3}} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{7 d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{104 c \sqrt{c+d x^3}}{7 d^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{2 x^2 \sqrt{c+d x^3}}{7 d^2}",1,"(x^2*(-20*(c + d*x^3) + 20*c*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 13*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(70*d^2*Sqrt[c + d*x^3])","C",0
316,1,67,601,0.0336024,"\int \frac{x^4}{\left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x^4/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{x^5 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{40 c \sqrt{c+d x^3}}","-\frac{2 \sqrt{2} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{2 \sqrt{c+d x^3}}{d^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{4 \sqrt[6]{c} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{3 \sqrt{3} d^{5/3}}+\frac{4 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{9 d^{5/3}}-\frac{4 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 d^{5/3}}",1,"(x^5*Sqrt[(c + d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(40*c*Sqrt[c + d*x^3])","C",0
317,1,67,141,0.0609827,"\int \frac{x}{\left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{x^2 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{16 c \sqrt{c+d x^3}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{6 \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{18 c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{18 c^{5/6} d^{2/3}}",1,"(x^2*Sqrt[(c + d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(16*c*Sqrt[c + d*x^3])","C",0
318,1,137,632,0.0905695,"\int \frac{1}{x^2 \left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x^2*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{-d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+25 c d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-80 c \left(c+d x^3\right)}{640 c^3 x \sqrt{c+d x^3}}","\frac{\sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt{2} \sqrt[4]{3} c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{16 c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{48 \sqrt{3} c^{11/6}}+\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{144 c^{11/6}}-\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{144 c^{11/6}}-\frac{\sqrt{c+d x^3}}{8 c^2 x}+\frac{\sqrt[3]{d} \sqrt{c+d x^3}}{8 c^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}",1,"(-80*c*(c + d*x^3) + 25*c*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(640*c^3*x*Sqrt[c + d*x^3])","C",0
319,1,152,654,0.1020062,"\int \frac{1}{x^5 \left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x^5*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{4 d^3 x^9 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-75 c d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+160 c \left(-c^2+c d x^3+2 d^2 x^6\right)}{5120 c^4 x^4 \sqrt{c+d x^3}}","-\frac{d^{4/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{384 \sqrt{3} c^{17/6}}+\frac{d^{4/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{1152 c^{17/6}}-\frac{d^{4/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{1152 c^{17/6}}-\frac{d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{8 \sqrt{2} \sqrt[4]{3} c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{32 c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{d^{4/3} \sqrt{c+d x^3}}{16 c^3 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{d \sqrt{c+d x^3}}{16 c^3 x}-\frac{\sqrt{c+d x^3}}{32 c^2 x^4}",1,"(160*c*(-c^2 + c*d*x^3 + 2*d^2*x^6) - 75*c*d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 4*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(5120*c^4*x^4*Sqrt[c + d*x^3])","C",0
320,1,167,678,0.1557927,"\int \frac{1}{x^8 \left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x^8*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{3875 c d^3 x^9 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-32 \left(6 d^4 x^{12} \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+5 c \left(32 c^3-5 c^2 d x^3+59 c d^2 x^6+96 d^3 x^9\right)\right)}{286720 c^5 x^7 \sqrt{c+d x^3}}","-\frac{d^{7/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{3072 \sqrt{3} c^{23/6}}+\frac{d^{7/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{9216 c^{23/6}}-\frac{d^{7/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9216 c^{23/6}}+\frac{3^{3/4} d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{28 \sqrt{2} c^{11/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{112 c^{11/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{3 d^{7/3} \sqrt{c+d x^3}}{56 c^4 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{3 d^2 \sqrt{c+d x^3}}{56 c^4 x}+\frac{37 d \sqrt{c+d x^3}}{1792 c^3 x^4}-\frac{\sqrt{c+d x^3}}{56 c^2 x^7}",1,"(3875*c*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 32*(5*c*(32*c^3 - 5*c^2*d*x^3 + 59*c*d^2*x^6 + 96*d^3*x^9) + 6*d^4*x^12*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(286720*c^5*x^7*Sqrt[c + d*x^3])","C",0
321,1,67,66,0.0397037,"\int \frac{x^3}{\left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x^3/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{x^4 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{32 c \sqrt{c+d x^3}}","\frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};1,\frac{1}{2};\frac{7}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{32 c \sqrt{c+d x^3}}",1,"(x^4*Sqrt[(c + d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(32*c*Sqrt[c + d*x^3])","A",0
322,1,166,64,0.1718518,"\int \frac{1}{\left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{32 c x F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{\left(8 c-d x^3\right) \sqrt{c+d x^3} \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}","\frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{1}{3};1,\frac{1}{2};\frac{4}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{8 c \sqrt{c+d x^3}}",1,"(32*c*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x^3)*Sqrt[c + d*x^3]*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])))","B",0
323,1,242,66,0.2444103,"\int \frac{1}{x^3 \left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x^3*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{\frac{d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c^3}+\frac{4096 d x^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{\left(d x^3-8 c\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}-\frac{64 \left(c+d x^3\right)}{c^2}}{1024 x^2 \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{2}{3};1,\frac{1}{2};\frac{1}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{16 c x^2 \sqrt{c+d x^3}}",1,"((-64*(c + d*x^3))/c^2 + (d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])/c^3 + (4096*d*x^3*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((-8*c + d*x^3)*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)]))))/(1024*x^2*Sqrt[c + d*x^3])","B",0
324,1,261,66,0.2462828,"\int \frac{1}{x^6 \left(8 c-d x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x^6*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]","\frac{64 c \left(\frac{3264 c^2 d^2 x^6 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{\left(8 c-d x^3\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}-16 c^2+7 c d x^3+23 d^2 x^6\right)-23 d^3 x^9 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{40960 c^4 x^5 \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{5}{3};1,\frac{1}{2};-\frac{2}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{40 c x^5 \sqrt{c+d x^3}}",1,"(-23*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 64*c*(-16*c^2 + 7*c*d*x^3 + 23*d^2*x^6 + (3264*c^2*d^2*x^6*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x^3)*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])))))/(40960*c^4*x^5*Sqrt[c + d*x^3])","B",0
325,1,66,90,0.0570159,"\int \frac{x^{11}}{\left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^11/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","-\frac{2 \left(512 c^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right)-456 c^2+60 c d x^3+3 d^2 x^6\right)}{27 d^4 \sqrt{c+d x^3}}","\frac{1024 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 d^4}+\frac{2 c^2}{27 d^4 \sqrt{c+d x^3}}-\frac{4 c \sqrt{c+d x^3}}{d^4}-\frac{2 \left(c+d x^3\right)^{3/2}}{9 d^4}",1,"(-2*(-456*c^2 + 60*c*d*x^3 + 3*d^2*x^6 + 512*c^2*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)]))/(27*d^4*Sqrt[c + d*x^3])","C",1
326,1,53,71,0.0546847,"\int \frac{x^8}{\left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^8/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","-\frac{2 \left(64 c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right)-54 c+9 d x^3\right)}{27 d^3 \sqrt{c+d x^3}}","-\frac{2 c}{27 d^3 \sqrt{c+d x^3}}-\frac{2 \sqrt{c+d x^3}}{3 d^3}+\frac{128 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 d^3}",1,"(-2*(-54*c + 9*d*x^3 + 64*c*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)]))/(27*d^3*Sqrt[c + d*x^3])","C",1
327,1,49,52,0.034601,"\int \frac{x^5}{\left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^5/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{2 \left(\frac{3}{\sqrt{c+d x^3}}+\frac{8 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{\sqrt{c}}\right)}{81 d^2}","\frac{2}{27 d^2 \sqrt{c+d x^3}}+\frac{16 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 \sqrt{c} d^2}",1,"(2*(3/Sqrt[c + d*x^3] + (8*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/Sqrt[c]))/(81*d^2)","A",1
328,1,43,55,0.0122837,"\int \frac{x^2}{\left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^2/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","-\frac{2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right)}{27 c d \sqrt{c+d x^3}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 c^{3/2} d}-\frac{2}{27 c d \sqrt{c+d x^3}}",1,"(-2*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)])/(27*c*d*Sqrt[c + d*x^3])","C",1
329,1,63,76,0.027778,"\int \frac{1}{x \left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{9 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right)-\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right)}{108 c^2 \sqrt{c+d x^3}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{324 c^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{12 c^{5/2}}+\frac{2}{27 c^2 \sqrt{c+d x^3}}",1,"(-Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)] + 9*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^3)/c])/(108*c^2*Sqrt[c + d*x^3])","C",1
330,1,77,100,0.0359404,"\int \frac{1}{x^4 \left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^4*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{-d x^3 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right)-99 d x^3 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right)-36 c}{864 c^3 x^3 \sqrt{c+d x^3}}","\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{2592 c^{7/2}}+\frac{11 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{96 c^{7/2}}-\frac{25 d}{216 c^3 \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \sqrt{c+d x^3}}",1,"(-36*c - d*x^3*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)] - 99*d*x^3*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^3)/c])/(864*c^3*x^3*Sqrt[c + d*x^3])","C",1
331,1,91,128,0.0789924,"\int \frac{1}{x^7 \left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^7*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{-d^2 x^6 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right)+981 d^2 x^6 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right)+36 c \left(9 d x^3-4 c\right)}{6912 c^4 x^6 \sqrt{c+d x^3}}","\frac{d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{20736 c^{9/2}}-\frac{109 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{768 c^{9/2}}+\frac{245 d^2}{1728 c^4 \sqrt{c+d x^3}}+\frac{3 d}{64 c^3 x^3 \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \sqrt{c+d x^3}}",1,"(36*c*(-4*c + 9*d*x^3) - d^2*x^6*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)] + 981*d^2*x^6*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^3)/c])/(6912*c^4*x^6*Sqrt[c + d*x^3])","C",1
332,1,127,629,0.1239444,"\int \frac{x^7}{\left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^7/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{x^2 \left(7 d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-20 c \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+20 c\right)}{270 c d^2 \sqrt{c+d x^3}}","-\frac{56 \sqrt{2} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{28 \sqrt{2-\sqrt{3}} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{9\ 3^{3/4} d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{56 \sqrt{c+d x^3}}{27 d^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{32 \sqrt[6]{c} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{27 \sqrt{3} d^{8/3}}+\frac{32 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{81 d^{8/3}}-\frac{32 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 d^{8/3}}+\frac{2 x^2}{27 d^2 \sqrt{c+d x^3}}",1,"(x^2*(20*c - 20*c*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 7*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(270*c*d^2*Sqrt[c + d*x^3])","C",0
333,1,126,635,0.1109723,"\int \frac{x^4}{\left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^4/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","-\frac{x^2 \left(d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-80 c \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+80 c\right)}{1080 c^2 d \sqrt{c+d x^3}}","-\frac{4 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{27 \sqrt{3} c^{5/6} d^{5/3}}+\frac{4 \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{81 c^{5/6} d^{5/3}}-\frac{4 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 c^{5/6} d^{5/3}}+\frac{2 \sqrt{2} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} c^{2/3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{9\ 3^{3/4} c^{2/3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{c+d x^3}}{27 c d^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{2 x^2}{27 c d \sqrt{c+d x^3}}",1,"-1/1080*(x^2*(80*c - 80*c*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(c^2*d*Sqrt[c + d*x^3])","C",0
334,1,124,632,0.0987518,"\int \frac{x}{\left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{x^2 \left(2 d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-25 c \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+160 c\right)}{2160 c^3 \sqrt{c+d x^3}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{54 \sqrt{3} c^{11/6} d^{2/3}}+\frac{\tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{162 c^{11/6} d^{2/3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{162 c^{11/6} d^{2/3}}-\frac{2 \sqrt{2} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} c^{5/3} d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{9\ 3^{3/4} c^{5/3} d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{2 \sqrt{c+d x^3}}{27 c^2 d^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{2 x^2}{27 c^2 \sqrt{c+d x^3}}",1,"(x^2*(160*c - 25*c*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 2*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(2160*c^3*Sqrt[c + d*x^3])","C",0
335,1,140,653,0.0991016,"\int \frac{1}{x^2 \left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^2*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{-43 d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+875 c d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-80 c \left(27 c+43 d x^3\right)}{17280 c^4 x \sqrt{c+d x^3}}","\frac{43 \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{108 \sqrt{2} \sqrt[4]{3} c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{43 \sqrt{2-\sqrt{3}} \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{144\ 3^{3/4} c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{432 \sqrt{3} c^{17/6}}+\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{1296 c^{17/6}}-\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{1296 c^{17/6}}-\frac{43 \sqrt{c+d x^3}}{216 c^3 x}+\frac{43 \sqrt[3]{d} \sqrt{c+d x^3}}{216 c^3 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{2}{27 c^2 x \sqrt{c+d x^3}}",1,"(-80*c*(27*c + 43*d*x^3) + 875*c*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 43*d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(17280*c^4*x*Sqrt[c + d*x^3])","C",0
336,1,153,675,0.1000563,"\int \frac{1}{x^5 \left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^5*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{452 d^3 x^9 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-9025 c d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+160 c \left(-27 c^2+135 c d x^3+226 d^2 x^6\right)}{138240 c^5 x^4 \sqrt{c+d x^3}}","-\frac{d^{4/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{3456 \sqrt{3} c^{23/6}}+\frac{d^{4/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{10368 c^{23/6}}-\frac{d^{4/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{10368 c^{23/6}}-\frac{113 d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{216 \sqrt{2} \sqrt[4]{3} c^{11/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{113 \sqrt{2-\sqrt{3}} d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{288\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{113 d^{4/3} \sqrt{c+d x^3}}{432 c^4 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{113 d \sqrt{c+d x^3}}{432 c^4 x}-\frac{91 \sqrt{c+d x^3}}{864 c^3 x^4}+\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}}",1,"(160*c*(-27*c^2 + 135*c*d*x^3 + 226*d^2*x^6) - 9025*c*d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 452*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(138240*c^5*x^4*Sqrt[c + d*x^3])","C",0
337,1,167,699,0.1193279,"\int \frac{1}{x^8 \left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^8*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{610025 c d^3 x^9 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-32 \left(953 d^4 x^{12} \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+5 c \left(864 c^3-1647 c^2 d x^3+9153 c d^2 x^6+15248 d^3 x^9\right)\right)}{7741440 c^6 x^7 \sqrt{c+d x^3}}","-\frac{d^{7/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{27648 \sqrt{3} c^{29/6}}+\frac{d^{7/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{82944 c^{29/6}}-\frac{d^{7/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{82944 c^{29/6}}+\frac{953 d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{1512 \sqrt{2} \sqrt[4]{3} c^{14/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{953 \sqrt{2-\sqrt{3}} d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{2016\ 3^{3/4} c^{14/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{953 d^{7/3} \sqrt{c+d x^3}}{3024 c^5 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{953 d^2 \sqrt{c+d x^3}}{3024 c^5 x}+\frac{6095 d \sqrt{c+d x^3}}{48384 c^4 x^4}-\frac{139 \sqrt{c+d x^3}}{1512 c^3 x^7}+\frac{2}{27 c^2 x^7 \sqrt{c+d x^3}}",1,"(610025*c*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 32*(5*c*(864*c^3 - 1647*c^2*d*x^3 + 9153*c*d^2*x^6 + 15248*d^3*x^9) + 953*d^4*x^12*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(7741440*c^6*x^7*Sqrt[c + d*x^3])","C",0
338,1,233,66,0.20935,"\int \frac{x^3}{\left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^3/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{x \left(\frac{64 c \left(\frac{256 c^2 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{\left(8 c-d x^3\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}-1\right)}{d}+x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}{864 c^2 \sqrt{c+d x^3}}","\frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};1,\frac{3}{2};\frac{7}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{32 c^2 \sqrt{c+d x^3}}",1,"(x*(x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] + (64*c*(-1 + (256*c^2*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x^3)*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])))))/d))/(864*c^2*Sqrt[c + d*x^3])","B",0
339,1,230,64,0.1578696,"\int \frac{1}{\left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{x \left(64 \left(\frac{176 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{\left(8 c-d x^3\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}+\frac{1}{c^2}\right)-\frac{d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c^3}\right)}{864 \sqrt{c+d x^3}}","\frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{1}{3};1,\frac{3}{2};\frac{4}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{8 c^2 \sqrt{c+d x^3}}",1,"(x*(-((d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])/c^3) + 64*(c^(-2) + (176*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x^3)*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)]))))))/(864*Sqrt[c + d*x^3])","B",0
340,1,248,66,0.1489843,"\int \frac{1}{x^3 \left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^3*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{64 c \left(-\frac{7360 c^2 d x^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{\left(8 c-d x^3\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}-27 c-59 d x^3\right)+59 d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{27648 c^4 x^2 \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{2}{3};1,\frac{3}{2};\frac{1}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{16 c^2 x^2 \sqrt{c+d x^3}}",1,"(59*d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 64*c*(-27*c - 59*d*x^3 - (7360*c^2*d*x^3*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x^3)*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])))))/(27648*c^4*x^2*Sqrt[c + d*x^3])","B",0
341,1,261,66,0.2380858,"\int \frac{1}{x^6 \left(8 c-d x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^6*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]","\frac{64 c \left(\frac{382528 c^2 d^2 x^6 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{\left(8 c-d x^3\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}-432 c^2+1269 c d x^3+2981 d^2 x^6\right)-2981 d^3 x^9 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{1105920 c^5 x^5 \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{5}{3};1,\frac{3}{2};-\frac{2}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{40 c^2 x^5 \sqrt{c+d x^3}}",1,"(-2981*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 64*c*(-432*c^2 + 1269*c*d*x^3 + 2981*d^2*x^6 + (382528*c^2*d^2*x^6*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x^3)*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])))))/(1105920*c^5*x^5*Sqrt[c + d*x^3])","B",0
342,1,80,737,0.0935423,"\int \frac{x \sqrt{a+b x^3}}{2 \left(5+3 \sqrt{3}\right) a+b x^3} \, dx","Integrate[(x*Sqrt[a + b*x^3])/(2*(5 + 3*Sqrt[3])*a + b*x^3),x]","\frac{x^2 \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)}{\left(20+12 \sqrt{3}\right) \sqrt{a+b x^3}}","\frac{2 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{3^{3/4} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt[6]{a} \tan ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt{a+b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{\sqrt{2} \sqrt[4]{3} b^{2/3}}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{\sqrt{2} b^{2/3}}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{2 \sqrt{2} b^{2/3}}",1,"(x^2*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, -1/2, 1, 5/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))])/((20 + 12*Sqrt[3])*Sqrt[a + b*x^3])","C",0
343,1,80,757,0.0848379,"\int \frac{x \sqrt{a-b x^3}}{2 \left(5+3 \sqrt{3}\right) a-b x^3} \, dx","Integrate[(x*Sqrt[a - b*x^3])/(2*(5 + 3*Sqrt[3])*a - b*x^3),x]","\frac{x^2 \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)}{\left(20+12 \sqrt{3}\right) \sqrt{a-b x^3}}","\frac{2 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}+\frac{2 \sqrt{a-b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}+\frac{3^{3/4} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt[6]{a} \tan ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{\sqrt{2} \sqrt[4]{3} b^{2/3}}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{\sqrt{2} b^{2/3}}",1,"(x^2*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, -1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)])/((20 + 12*Sqrt[3])*Sqrt[a - b*x^3])","C",0
344,1,87,774,0.0847008,"\int \frac{x \sqrt{-a+b x^3}}{-2 \left(5+3 \sqrt{3}\right) a+b x^3} \, dx","Integrate[(x*Sqrt[-a + b*x^3])/(-2*(5 + 3*Sqrt[3])*a + b*x^3),x]","-\frac{x^2 \sqrt{b x^3-a} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)}{4 \left(5+3 \sqrt{3}\right) a \sqrt{1-\frac{b x^3}{a}}}","-\frac{2 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}-\frac{2 \sqrt{b x^3-a}}{b^{2/3} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt[6]{a} \tanh ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt{b x^3-a}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{\sqrt{2} \sqrt[4]{3} b^{2/3}}",1,"-1/4*(x^2*Sqrt[-a + b*x^3]*AppellF1[2/3, -1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)])/((5 + 3*Sqrt[3])*a*Sqrt[1 - (b*x^3)/a])","C",0
345,1,90,768,0.083253,"\int \frac{x \sqrt{-a-b x^3}}{-2 \left(5+3 \sqrt{3}\right) a-b x^3} \, dx","Integrate[(x*Sqrt[-a - b*x^3])/(-2*(5 + 3*Sqrt[3])*a - b*x^3),x]","-\frac{x^2 \sqrt{-a-b x^3} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)}{4 \left(5+3 \sqrt{3}\right) a \sqrt{\frac{a+b x^3}{a}}}","-\frac{2 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}-\frac{2 \sqrt{-a-b x^3}}{b^{2/3} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{\sqrt{2} b^{2/3}}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{2 \sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt[6]{a} \tanh ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt{-a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{\sqrt{2} \sqrt[4]{3} b^{2/3}}",1,"-1/4*(x^2*Sqrt[-a - b*x^3]*AppellF1[2/3, -1/2, 1, 5/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))])/((5 + 3*Sqrt[3])*a*Sqrt[(a + b*x^3)/a])","C",0
346,1,80,738,0.1930261,"\int \frac{x \sqrt{a+b x^3}}{2 \left(5-3 \sqrt{3}\right) a+b x^3} \, dx","Integrate[(x*Sqrt[a + b*x^3])/(2*(5 - 3*Sqrt[3])*a + b*x^3),x]","\frac{x^2 \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)}{\left(20-12 \sqrt{3}\right) \sqrt{a+b x^3}}","\frac{2 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{\sqrt[4]{3} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{\sqrt{2} b^{2/3}}-\frac{\sqrt[4]{3} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{2 \sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt[6]{a} \tanh ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt{a+b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{\sqrt{2} \sqrt[4]{3} b^{2/3}}",1,"(x^2*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, -1/2, 1, 5/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))])/((20 - 12*Sqrt[3])*Sqrt[a + b*x^3])","C",0
347,1,80,758,0.0822304,"\int \frac{x \sqrt{a-b x^3}}{2 \left(5-3 \sqrt{3}\right) a-b x^3} \, dx","Integrate[(x*Sqrt[a - b*x^3])/(2*(5 - 3*Sqrt[3])*a - b*x^3),x]","\frac{x^2 \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)}{\left(20-12 \sqrt{3}\right) \sqrt{a-b x^3}}","\frac{2 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}+\frac{2 \sqrt{a-b x^3}}{b^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}-\frac{\sqrt[4]{3} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{2 \sqrt{2} b^{2/3}}-\frac{\sqrt[4]{3} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{\sqrt{2} b^{2/3}}+\frac{3^{3/4} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt[6]{a} \tanh ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{\sqrt{2} \sqrt[4]{3} b^{2/3}}",1,"(x^2*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, -1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)])/((20 - 12*Sqrt[3])*Sqrt[a - b*x^3])","C",0
348,1,89,774,0.0484209,"\int \frac{x \sqrt{-a+b x^3}}{2 \left(5-3 \sqrt{3}\right) a-b x^3} \, dx","Integrate[(x*Sqrt[-a + b*x^3])/(2*(5 - 3*Sqrt[3])*a - b*x^3),x]","-\frac{x^2 \sqrt{b x^3-a} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a-10 a}\right)}{4 \left(3 \sqrt{3}-5\right) a \sqrt{\frac{a-b x^3}{a}}}","\frac{2 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}-\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}+\frac{2 \sqrt{b x^3-a}}{b^{2/3} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)}-\frac{3^{3/4} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt[6]{a} \tan ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt{b x^3-a}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{\sqrt{2} \sqrt[4]{3} b^{2/3}}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{\sqrt{2} b^{2/3}}",1,"-1/4*(x^2*Sqrt[-a + b*x^3]*AppellF1[2/3, -1/2, 1, 5/3, (b*x^3)/a, -((b*x^3)/(-10*a + 6*Sqrt[3]*a))])/((-5 + 3*Sqrt[3])*a*Sqrt[(a - b*x^3)/a])","C",0
349,1,89,768,0.0751829,"\int \frac{x \sqrt{-a-b x^3}}{2 \left(5-3 \sqrt{3}\right) a+b x^3} \, dx","Integrate[(x*Sqrt[-a - b*x^3])/(2*(5 - 3*Sqrt[3])*a + b*x^3),x]","-\frac{x^2 \sqrt{-a-b x^3} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)}{4 \left(3 \sqrt{3}-5\right) a \sqrt{\frac{b x^3}{a}+1}}","\frac{2 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}-\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}+\frac{2 \sqrt{-a-b x^3}}{b^{2/3} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)}-\frac{3^{3/4} \sqrt[6]{a} \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{2 \sqrt{2} b^{2/3}}+\frac{\sqrt[6]{a} \tan ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt{-a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{\sqrt{2} \sqrt[4]{3} b^{2/3}}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{\sqrt{2} b^{2/3}}+\frac{\sqrt[4]{3} \sqrt[6]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{2 \sqrt{2} b^{2/3}}",1,"-1/4*(x^2*Sqrt[-a - b*x^3]*AppellF1[2/3, -1/2, 1, 5/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))])/((-5 + 3*Sqrt[3])*a*Sqrt[1 + (b*x^3)/a])","C",0
350,1,83,318,0.1043452,"\int \frac{x}{\sqrt{a+b x^3} \left(2 \left(5+3 \sqrt{3}\right) a+b x^3\right)} \, dx","Integrate[x/(Sqrt[a + b*x^3]*(2*(5 + 3*Sqrt[3])*a + b*x^3)),x]","\frac{x^2 \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)}{\left(12 \sqrt{3} a+20 a\right) \sqrt{a+b x^3}}","-\frac{\left(2-\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left(2-\sqrt{3}\right) \tan ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt{a+b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left(2-\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{3 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}-\frac{\left(2-\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{6 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}",1,"(x^2*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))])/((20*a + 12*Sqrt[3]*a)*Sqrt[a + b*x^3])","C",0
351,1,83,324,0.0757202,"\int \frac{x}{\sqrt{a-b x^3} \left(2 \left(5+3 \sqrt{3}\right) a-b x^3\right)} \, dx","Integrate[x/(Sqrt[a - b*x^3]*(2*(5 + 3*Sqrt[3])*a - b*x^3)),x]","\frac{x^2 \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)}{\left(12 \sqrt{3} a+20 a\right) \sqrt{a-b x^3}}","-\frac{\left(2-\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left(2-\sqrt{3}\right) \tan ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left(2-\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{6 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}-\frac{\left(2-\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{3 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}",1,"(x^2*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)])/((20*a + 12*Sqrt[3]*a)*Sqrt[a - b*x^3])","C",0
352,1,85,328,0.1232466,"\int \frac{x}{\sqrt{-a+b x^3} \left(-2 \left(5+3 \sqrt{3}\right) a+b x^3\right)} \, dx","Integrate[x/(Sqrt[-a + b*x^3]*(-2*(5 + 3*Sqrt[3])*a + b*x^3)),x]","-\frac{x^2 \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)}{\left(12 \sqrt{3} a+20 a\right) \sqrt{b x^3-a}}","\frac{\left(2-\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{6 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}+\frac{\left(2-\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{3 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}+\frac{\left(2-\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left(2-\sqrt{3}\right) \tanh ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt{b x^3-a}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}",1,"-((x^2*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)])/((20*a + 12*Sqrt[3]*a)*Sqrt[-a + b*x^3]))","C",0
353,1,87,330,0.1097521,"\int \frac{x}{\sqrt{-a-b x^3} \left(-2 \left(5+3 \sqrt{3}\right) a-b x^3\right)} \, dx","Integrate[x/(Sqrt[-a - b*x^3]*(-2*(5 + 3*Sqrt[3])*a - b*x^3)),x]","-\frac{x^2 \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)}{\left(12 \sqrt{3} a+20 a\right) \sqrt{-a-b x^3}}","\frac{\left(2-\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{3 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}+\frac{\left(2-\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{6 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}+\frac{\left(2-\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left(2-\sqrt{3}\right) \tanh ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt{-a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}",1,"-((x^2*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))])/((20*a + 12*Sqrt[3]*a)*Sqrt[-a - b*x^3]))","C",0
354,1,83,310,0.1492832,"\int \frac{x}{\sqrt{a+b x^3} \left(2 \left(5-3 \sqrt{3}\right) a+b x^3\right)} \, dx","Integrate[x/(Sqrt[a + b*x^3]*(2*(5 - 3*Sqrt[3])*a + b*x^3)),x]","\frac{x^2 \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)}{\left(20 a-12 \sqrt{3} a\right) \sqrt{a+b x^3}}","-\frac{\left(2+\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{3 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}-\frac{\left(2+\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{6 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}+\frac{\left(2+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a+b x^3}}\right)}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}+\frac{\left(2+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt{a+b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}",1,"(x^2*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))])/((20*a - 12*Sqrt[3]*a)*Sqrt[a + b*x^3])","C",0
355,1,83,316,0.0990601,"\int \frac{x}{\sqrt{a-b x^3} \left(2 \left(5-3 \sqrt{3}\right) a-b x^3\right)} \, dx","Integrate[x/(Sqrt[a - b*x^3]*(2*(5 - 3*Sqrt[3])*a - b*x^3)),x]","\frac{x^2 \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)}{\left(20 a-12 \sqrt{3} a\right) \sqrt{a-b x^3}}","-\frac{\left(2+\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{6 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}-\frac{\left(2+\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{3 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}+\frac{\left(2+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{a-b x^3}}\right)}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}+\frac{\left(2+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt{a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}",1,"(x^2*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)])/((20*a - 12*Sqrt[3]*a)*Sqrt[a - b*x^3])","C",0
356,1,84,320,0.1056851,"\int \frac{x}{\left(2 \left(5-3 \sqrt{3}\right) a-b x^3\right) \sqrt{-a+b x^3}} \, dx","Integrate[x/((2*(5 - 3*Sqrt[3])*a - b*x^3)*Sqrt[-a + b*x^3]),x]","\frac{x^2 \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)}{\left(20 a-12 \sqrt{3} a\right) \sqrt{b x^3-a}}","\frac{\left(2+\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left(2+\sqrt{3}\right) \tan ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt{b x^3-a}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left(2+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{6 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}-\frac{\left(2+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{b x^3-a}}\right)}{3 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}",1,"(x^2*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)])/((20*a - 12*Sqrt[3]*a)*Sqrt[-a + b*x^3])","C",0
357,1,86,322,0.1321786,"\int \frac{x}{\sqrt{-a-b x^3} \left(2 \left(5-3 \sqrt{3}\right) a+b x^3\right)} \, dx","Integrate[x/(Sqrt[-a - b*x^3]*(2*(5 - 3*Sqrt[3])*a + b*x^3)),x]","\frac{x^2 \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)}{\left(20 a-12 \sqrt{3} a\right) \sqrt{-a-b x^3}}","\frac{\left(2+\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt[4]{3} \left(1-\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{2 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left(2+\sqrt{3}\right) \tan ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt{-a-b x^3}}{\sqrt{2} 3^{3/4} \sqrt{a}}\right)}{3 \sqrt{2} 3^{3/4} a^{5/6} b^{2/3}}-\frac{\left(2+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[6]{a} \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-2 \sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{3 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}-\frac{\left(2+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{3} \left(1+\sqrt{3}\right) \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{2} \sqrt{-a-b x^3}}\right)}{6 \sqrt{2} \sqrt[4]{3} a^{5/6} b^{2/3}}",1,"(x^2*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))])/((20*a - 12*Sqrt[3]*a)*Sqrt[-a - b*x^3])","C",0
358,1,121,125,0.3073952,"\int \frac{x^8 \sqrt{c+d x^3}}{a+b x^3} \, dx","Integrate[(x^8*Sqrt[c + d*x^3])/(a + b*x^3),x]","\frac{2 \sqrt{c+d x^3} \left(15 a^2 d^2-5 a b d \left(c+d x^3\right)+b^2 \left(-2 c^2+c d x^3+3 d^2 x^6\right)\right)}{45 b^3 d^2}-\frac{2 a^2 \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{7/2}}","-\frac{2 a^2 \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{7/2}}+\frac{2 a^2 \sqrt{c+d x^3}}{3 b^3}-\frac{2 \left(c+d x^3\right)^{3/2} (a d+b c)}{9 b^2 d^2}+\frac{2 \left(c+d x^3\right)^{5/2}}{15 b d^2}",1,"(2*Sqrt[c + d*x^3]*(15*a^2*d^2 - 5*a*b*d*(c + d*x^3) + b^2*(-2*c^2 + c*d*x^3 + 3*d^2*x^6)))/(45*b^3*d^2) - (2*a^2*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(7/2))","A",1
359,1,88,93,0.0931194,"\int \frac{x^5 \sqrt{c+d x^3}}{a+b x^3} \, dx","Integrate[(x^5*Sqrt[c + d*x^3])/(a + b*x^3),x]","\frac{2 a \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{5/2}}+\frac{2 \sqrt{c+d x^3} \left(b \left(c+d x^3\right)-3 a d\right)}{9 b^2 d}","\frac{2 a \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{5/2}}-\frac{2 a \sqrt{c+d x^3}}{3 b^2}+\frac{2 \left(c+d x^3\right)^{3/2}}{9 b d}",1,"(2*Sqrt[c + d*x^3]*(-3*a*d + b*(c + d*x^3)))/(9*b^2*d) + (2*a*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(5/2))","A",1
360,1,70,70,0.0367754,"\int \frac{x^2 \sqrt{c+d x^3}}{a+b x^3} \, dx","Integrate[(x^2*Sqrt[c + d*x^3])/(a + b*x^3),x]","\frac{1}{3} \left(\frac{2 \sqrt{c+d x^3}}{b}-\frac{2 \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{b^{3/2}}\right)","\frac{2 \sqrt{c+d x^3}}{3 b}-\frac{2 \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{3/2}}",1,"((2*Sqrt[c + d*x^3])/b - (2*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/b^(3/2))/3","A",1
361,1,81,85,0.0524898,"\int \frac{\sqrt{c+d x^3}}{x \left(a+b x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x*(a + b*x^3)),x]","\frac{2 \left(\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{\sqrt{b}}-\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)\right)}{3 a}","\frac{2 \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a \sqrt{b}}-\frac{2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a}",1,"(2*(-(Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]]) + (Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/Sqrt[b]))/(3*a)","A",1
362,1,107,115,0.1783953,"\int \frac{\sqrt{c+d x^3}}{x^4 \left(a+b x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x^4*(a + b*x^3)),x]","\frac{\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{\sqrt{c}}-2 \sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)-\frac{a \sqrt{c+d x^3}}{x^3}}{3 a^2}","\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^2 \sqrt{c}}-\frac{2 \sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^2}-\frac{\sqrt{c+d x^3}}{3 a x^3}",1,"(-((a*Sqrt[c + d*x^3])/x^3) + ((2*b*c - a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/Sqrt[c] - 2*Sqrt[b]*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*a^2)","A",1
363,1,241,64,0.6187013,"\int \frac{x^3 \sqrt{c+d x^3}}{a+b x^3} \, dx","Integrate[(x^3*Sqrt[c + d*x^3])/(a + b*x^3),x]","\frac{x \left(8 \left(\frac{8 a^2 c^2 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}+c+d x^3\right)+\frac{x^3 \sqrt{\frac{d x^3}{c}+1} (3 b c-5 a d) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{a}\right)}{20 b \sqrt{c+d x^3}}","\frac{x^4 \sqrt{c+d x^3} F_1\left(\frac{4}{3};1,-\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 a \sqrt{\frac{d x^3}{c}+1}}",1,"(x*(((3*b*c - 5*a*d)*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/a + 8*(c + d*x^3 + (8*a^2*c^2*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)])/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))))/(20*b*Sqrt[c + d*x^3])","B",0
364,1,65,64,0.0590176,"\int \frac{x \sqrt{c+d x^3}}{a+b x^3} \, dx","Integrate[(x*Sqrt[c + d*x^3])/(a + b*x^3),x]","\frac{x^2 \sqrt{c+d x^3} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{2 a \sqrt{\frac{c+d x^3}{c}}}","\frac{x^2 \sqrt{c+d x^3} F_1\left(\frac{2}{3};1,-\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a \sqrt{\frac{d x^3}{c}+1}}",1,"(x^2*Sqrt[c + d*x^3]*AppellF1[2/3, -1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)])/(2*a*Sqrt[(c + d*x^3)/c])","A",0
365,1,161,59,0.1865807,"\int \frac{\sqrt{c+d x^3}}{a+b x^3} \, dx","Integrate[Sqrt[c + d*x^3]/(a + b*x^3),x]","\frac{8 a c x \sqrt{c+d x^3} F_1\left(\frac{1}{3};-\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{\left(a+b x^3\right) \left(3 x^3 \left(a d F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-2 b c F_1\left(\frac{4}{3};-\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)+8 a c F_1\left(\frac{1}{3};-\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}","\frac{x \sqrt{c+d x^3} F_1\left(\frac{1}{3};1,-\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a \sqrt{\frac{d x^3}{c}+1}}",1,"(8*a*c*x*Sqrt[c + d*x^3]*AppellF1[1/3, -1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)])/((a + b*x^3)*(8*a*c*AppellF1[1/3, -1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(-2*b*c*AppellF1[4/3, -1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))","B",0
366,1,139,62,0.1151912,"\int \frac{\sqrt{c+d x^3}}{x^2 \left(a+b x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x^2*(a + b*x^3)),x]","\frac{5 x^3 \sqrt{\frac{d x^3}{c}+1} (3 a d-2 b c) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+2 b d x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-20 a \left(c+d x^3\right)}{20 a^2 x \sqrt{c+d x^3}}","-\frac{\sqrt{c+d x^3} F_1\left(-\frac{1}{3};1,-\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a x \sqrt{\frac{d x^3}{c}+1}}",1,"(-20*a*(c + d*x^3) + 5*(-2*b*c + 3*a*d)*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 2*b*d*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/(20*a^2*x*Sqrt[c + d*x^3])","B",0
367,1,335,64,0.2526138,"\int \frac{\sqrt{c+d x^3}}{x^3 \left(a+b x^3\right)} \, dx","Integrate[Sqrt[c + d*x^3]/(x^3*(a + b*x^3)),x]","\frac{\frac{a \left(32 a c \left(2 a c-a d x^3+6 b c x^3+2 b d x^6\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-24 x^3 \left(a+b x^3\right) \left(c+d x^3\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}-b d x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{16 a^2 x^2 \sqrt{c+d x^3}}","-\frac{\sqrt{c+d x^3} F_1\left(-\frac{2}{3};1,-\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a x^2 \sqrt{\frac{d x^3}{c}+1}}",1,"(-(b*d*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]) + (a*(32*a*c*(2*a*c + 6*b*c*x^3 - a*d*x^3 + 2*b*d*x^6)*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 24*x^3*(a + b*x^3)*(c + d*x^3)*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(16*a^2*x^2*Sqrt[c + d*x^3])","B",0
368,1,145,154,0.2359479,"\int \frac{x^8 \left(c+d x^3\right)^{3/2}}{a+b x^3} \, dx","Integrate[(x^8*(c + d*x^3)^(3/2))/(a + b*x^3),x]","\frac{2 \left(105 a^2 (b c-a d) \left(\frac{\sqrt{c+d x^3}}{b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{b^{3/2}}\right)+35 a^2 \left(c+d x^3\right)^{3/2}-\frac{21 b \left(c+d x^3\right)^{5/2} (a d+b c)}{d^2}+\frac{15 b^2 \left(c+d x^3\right)^{7/2}}{d^2}\right)}{315 b^3}","-\frac{2 a^2 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{9/2}}+\frac{2 a^2 \sqrt{c+d x^3} (b c-a d)}{3 b^4}+\frac{2 a^2 \left(c+d x^3\right)^{3/2}}{9 b^3}-\frac{2 \left(c+d x^3\right)^{5/2} (a d+b c)}{15 b^2 d^2}+\frac{2 \left(c+d x^3\right)^{7/2}}{21 b d^2}",1,"(2*(35*a^2*(c + d*x^3)^(3/2) - (21*b*(b*c + a*d)*(c + d*x^3)^(5/2))/d^2 + (15*b^2*(c + d*x^3)^(7/2))/d^2 + 105*a^2*(b*c - a*d)*(Sqrt[c + d*x^3]/b - (Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/b^(3/2))))/(315*b^3)","A",1
369,1,111,120,0.1205056,"\int \frac{x^5 \left(c+d x^3\right)^{3/2}}{a+b x^3} \, dx","Integrate[(x^5*(c + d*x^3)^(3/2))/(a + b*x^3),x]","\frac{2 \sqrt{c+d x^3} \left(15 a^2 d^2-5 a b d \left(4 c+d x^3\right)+3 b^2 \left(c+d x^3\right)^2\right)}{45 b^3 d}+\frac{2 a (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{7/2}}","\frac{2 a (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{7/2}}-\frac{2 a \sqrt{c+d x^3} (b c-a d)}{3 b^3}-\frac{2 a \left(c+d x^3\right)^{3/2}}{9 b^2}+\frac{2 \left(c+d x^3\right)^{5/2}}{15 b d}",1,"(2*Sqrt[c + d*x^3]*(15*a^2*d^2 + 3*b^2*(c + d*x^3)^2 - 5*a*b*d*(4*c + d*x^3)))/(45*b^3*d) + (2*a*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(7/2))","A",1
370,1,85,96,0.0858415,"\int \frac{x^2 \left(c+d x^3\right)^{3/2}}{a+b x^3} \, dx","Integrate[(x^2*(c + d*x^3)^(3/2))/(a + b*x^3),x]","\frac{2 \sqrt{c+d x^3} \left(-3 a d+4 b c+b d x^3\right)}{9 b^2}-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{5/2}}","-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{5/2}}+\frac{2 \sqrt{c+d x^3} (b c-a d)}{3 b^2}+\frac{2 \left(c+d x^3\right)^{3/2}}{9 b}",1,"(2*Sqrt[c + d*x^3]*(4*b*c - 3*a*d + b*d*x^3))/(9*b^2) - (2*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(5/2))","A",1
371,1,105,104,0.0966762,"\int \frac{\left(c+d x^3\right)^{3/2}}{x \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3)^(3/2)/(x*(a + b*x^3)),x]","\frac{2 \left(a \sqrt{b} d \sqrt{c+d x^3}+(b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)-b^{3/2} c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)\right)}{3 a b^{3/2}}","\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a b^{3/2}}-\frac{2 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a}+\frac{2 d \sqrt{c+d x^3}}{3 b}",1,"(2*(a*Sqrt[b]*d*Sqrt[c + d*x^3] - b^(3/2)*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]] + (b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]]))/(3*a*b^(3/2))","A",1
372,1,108,116,0.1326414,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^4 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^4*(a + b*x^3)),x]","\frac{-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{\sqrt{b}}+\sqrt{c} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)-\frac{a c \sqrt{c+d x^3}}{x^3}}{3 a^2}","-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^2 \sqrt{b}}+\frac{\sqrt{c} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^2}-\frac{c \sqrt{c+d x^3}}{3 a x^3}",1,"(-((a*c*Sqrt[c + d*x^3])/x^3) + Sqrt[c]*(2*b*c - 3*a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]] - (2*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/Sqrt[b])/(3*a^2)","A",1
373,1,280,65,0.5697827,"\int \frac{x^3 \left(c+d x^3\right)^{3/2}}{a+b x^3} \, dx","Integrate[(x^3*(c + d*x^3)^(3/2))/(a + b*x^3),x]","\frac{x \left(\frac{x^3 \sqrt{\frac{d x^3}{c}+1} \left(55 a^2 d^2-88 a b c d+27 b^2 c^2\right) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{a}-\frac{64 a^2 c^2 (11 a d-14 b c) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}+8 \left(c+d x^3\right) \left(-11 a d+14 b c+5 b d x^3\right)\right)}{220 b^2 \sqrt{c+d x^3}}","\frac{c x^4 \sqrt{c+d x^3} F_1\left(\frac{4}{3};1,-\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 a \sqrt{\frac{d x^3}{c}+1}}",1,"(x*(8*(c + d*x^3)*(14*b*c - 11*a*d + 5*b*d*x^3) + ((27*b^2*c^2 - 88*a*b*c*d + 55*a^2*d^2)*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/a - (64*a^2*c^2*(-14*b*c + 11*a*d)*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)])/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))))/(220*b^2*Sqrt[c + d*x^3])","B",0
374,1,149,65,0.2075033,"\int \frac{x \left(c+d x^3\right)^{3/2}}{a+b x^3} \, dx","Integrate[(x*(c + d*x^3)^(3/2))/(a + b*x^3),x]","\frac{x^2 \left(2 d x^3 \sqrt{\frac{d x^3}{c}+1} (10 b c-7 a d) F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+5 c \sqrt{\frac{d x^3}{c}+1} (7 b c-4 a d) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+20 a d \left(c+d x^3\right)\right)}{70 a b \sqrt{c+d x^3}}","\frac{c x^2 \sqrt{c+d x^3} F_1\left(\frac{2}{3};1,-\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a \sqrt{\frac{d x^3}{c}+1}}",1,"(x^2*(20*a*d*(c + d*x^3) + 5*c*(7*b*c - 4*a*d)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 2*d*(10*b*c - 7*a*d)*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)]))/(70*a*b*Sqrt[c + d*x^3])","B",0
375,1,351,60,0.360584,"\int \frac{\left(c+d x^3\right)^{3/2}}{a+b x^3} \, dx","Integrate[(c + d*x^3)^(3/2)/(a + b*x^3),x]","\frac{x \left(\frac{8 \left(3 d x^3 \left(a+b x^3\right) \left(c+d x^3\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-4 a c \left(2 a d^2 x^3+b \left(5 c^2+2 c d x^3+2 d^2 x^6\right)\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}+\frac{d x^3 \sqrt{\frac{d x^3}{c}+1} (8 b c-5 a d) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{a}\right)}{20 b \sqrt{c+d x^3}}","\frac{c x \sqrt{c+d x^3} F_1\left(\frac{1}{3};1,-\frac{3}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a \sqrt{\frac{d x^3}{c}+1}}",1,"(x*((d*(8*b*c - 5*a*d)*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/a + (8*(-4*a*c*(2*a*d^2*x^3 + b*(5*c^2 + 2*c*d*x^3 + 2*d^2*x^6))*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*d*x^3*(a + b*x^3)*(c + d*x^3)*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))))/(20*b*Sqrt[c + d*x^3])","B",0
376,1,148,63,0.1473121,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^2 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^2*(a + b*x^3)),x]","\frac{5 c x^3 \sqrt{\frac{d x^3}{c}+1} (5 a d-2 b c) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+2 d x^6 \sqrt{\frac{d x^3}{c}+1} (2 a d+b c) F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-20 a c \left(c+d x^3\right)}{20 a^2 x \sqrt{c+d x^3}}","-\frac{c \sqrt{c+d x^3} F_1\left(-\frac{1}{3};1,-\frac{3}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a x \sqrt{\frac{d x^3}{c}+1}}",1,"(-20*a*c*(c + d*x^3) + 5*c*(-2*b*c + 5*a*d)*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 2*d*(b*c + 2*a*d)*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/(20*a^2*x*Sqrt[c + d*x^3])","B",0
377,1,343,65,0.3259777,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^3 \left(a+b x^3\right)} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^3*(a + b*x^3)),x]","-\frac{d x^6 \sqrt{\frac{d x^3}{c}+1} (b c-4 a d) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+\frac{8 a c \left(3 x^3 \left(a+b x^3\right) \left(c+d x^3\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-4 a c \left(2 a c-5 a d x^3+6 b c x^3+2 b d x^6\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}}{16 a^2 x^2 \sqrt{c+d x^3}}","-\frac{c \sqrt{c+d x^3} F_1\left(-\frac{2}{3};1,-\frac{3}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a x^2 \sqrt{\frac{d x^3}{c}+1}}",1,"-1/16*(d*(b*c - 4*a*d)*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + (8*a*c*(-4*a*c*(2*a*c + 6*b*c*x^3 - 5*a*d*x^3 + 2*b*d*x^6)*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(a + b*x^3)*(c + d*x^3)*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(a^2*x^2*Sqrt[c + d*x^3])","B",0
378,1,91,104,0.2164734,"\int \frac{x^8}{\left(a+b x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x^8/((a + b*x^3)*Sqrt[c + d*x^3]),x]","\frac{2 \sqrt{c+d x^3} \left(-3 a d-2 b c+b d x^3\right)}{9 b^2 d^2}-\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{5/2} \sqrt{b c-a d}}","-\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{5/2} \sqrt{b c-a d}}-\frac{2 \sqrt{c+d x^3} (a d+b c)}{3 b^2 d^2}+\frac{2 \left(c+d x^3\right)^{3/2}}{9 b d^2}",1,"(2*Sqrt[c + d*x^3]*(-2*b*c - 3*a*d + b*d*x^3))/(9*b^2*d^2) - (2*a^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(5/2)*Sqrt[b*c - a*d])","A",1
379,1,74,74,0.1159534,"\int \frac{x^5}{\left(a+b x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x^5/((a + b*x^3)*Sqrt[c + d*x^3]),x]","\frac{2 \left(\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}+\frac{\sqrt{b} \sqrt{c+d x^3}}{d}\right)}{3 b^{3/2}}","\frac{2 a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{3/2} \sqrt{b c-a d}}+\frac{2 \sqrt{c+d x^3}}{3 b d}",1,"(2*((Sqrt[b]*Sqrt[c + d*x^3])/d + (a*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d]))/(3*b^(3/2))","A",1
380,1,51,51,0.0165069,"\int \frac{x^2}{\left(a+b x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x^2/((a + b*x^3)*Sqrt[c + d*x^3]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 \sqrt{b} \sqrt{b c-a d}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 \sqrt{b} \sqrt{b c-a d}}",1,"(-2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*Sqrt[b]*Sqrt[b*c - a*d])","A",1
381,1,81,85,0.100413,"\int \frac{1}{x \left(a+b x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x*(a + b*x^3)*Sqrt[c + d*x^3]),x]","\frac{2 \left(\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{\sqrt{c}}\right)}{3 a}","\frac{2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a \sqrt{b c-a d}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a \sqrt{c}}",1,"(2*(-(ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]]/Sqrt[c]) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d]))/(3*a)","A",1
382,1,151,117,0.1313775,"\int \frac{1}{x^4 \left(a+b x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x^4*(a + b*x^3)*Sqrt[c + d*x^3]),x]","\frac{2 b^{3/2} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^2 (a d-b c)}+\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^2 \sqrt{c}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a c^{3/2}}-\frac{\sqrt{c+d x^3}}{3 a c x^3}","-\frac{2 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^2 c^{3/2}}-\frac{\sqrt{c+d x^3}}{3 a c x^3}",1,"-1/3*Sqrt[c + d*x^3]/(a*c*x^3) + (2*b*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^2*Sqrt[c]) + (d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a*c^(3/2)) + (2*b^(3/2)*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*a^2*(-(b*c) + a*d))","A",1
383,1,65,64,0.0384548,"\int \frac{x^3}{\left(a+b x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x^3/((a + b*x^3)*Sqrt[c + d*x^3]),x]","\frac{x^4 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{4 a \sqrt{c+d x^3}}","\frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};1,\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 a \sqrt{c+d x^3}}",1,"(x^4*Sqrt[(c + d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/(4*a*Sqrt[c + d*x^3])","A",0
384,1,65,64,0.0313464,"\int \frac{x}{\left(a+b x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[x/((a + b*x^3)*Sqrt[c + d*x^3]),x]","\frac{x^2 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{2 a \sqrt{c+d x^3}}","\frac{x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};1,\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a \sqrt{c+d x^3}}",1,"(x^2*Sqrt[(c + d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)])/(2*a*Sqrt[c + d*x^3])","A",0
385,1,161,59,0.0470594,"\int \frac{1}{\left(a+b x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/((a + b*x^3)*Sqrt[c + d*x^3]),x]","-\frac{8 a c x F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{\left(a+b x^3\right) \sqrt{c+d x^3} \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}","\frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{1}{3};1,\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a \sqrt{c+d x^3}}",1,"(-8*a*c*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)])/((a + b*x^3)*Sqrt[c + d*x^3]*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))","B",0
386,1,141,62,0.1184356,"\int \frac{1}{x^2 \left(a+b x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x^2*(a + b*x^3)*Sqrt[c + d*x^3]),x]","\frac{5 x^3 \sqrt{\frac{d x^3}{c}+1} (a d-2 b c) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+2 b d x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-20 a \left(c+d x^3\right)}{20 a^2 c x \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{1}{3};1,\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a x \sqrt{c+d x^3}}",1,"(-20*a*(c + d*x^3) + 5*(-2*b*c + a*d)*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 2*b*d*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/(20*a^2*c*x*Sqrt[c + d*x^3])","B",0
387,1,339,64,0.3203661,"\int \frac{1}{x^3 \left(a+b x^3\right) \sqrt{c+d x^3}} \, dx","Integrate[1/(x^3*(a + b*x^3)*Sqrt[c + d*x^3]),x]","\frac{\frac{8 a \left(3 x^3 \left(a+b x^3\right) \left(c+d x^3\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-4 a c \left(2 a c+3 a d x^3+6 b c x^3+2 b d x^6\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}{\left(a+b x^3\right) \left(8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)\right)}-b d x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{16 a^2 c x^2 \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{2}{3};1,\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a x^2 \sqrt{c+d x^3}}",1,"(-(b*d*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]) + (8*a*(-4*a*c*(2*a*c + 6*b*c*x^3 + 3*a*d*x^3 + 2*b*d*x^6)*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(a + b*x^3)*(c + d*x^3)*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(16*a^2*c*x^2*Sqrt[c + d*x^3])","B",0
388,1,100,107,0.0801269,"\int \frac{x^8}{\left(a+b x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^8/((a + b*x^3)*(c + d*x^3)^(3/2)),x]","\frac{2 \left(-a^2 d^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^3+c\right)}{b c-a d}\right)+a^2 d^2+a b d \left(c+d x^3\right)+b^2 (-c) \left(2 c+d x^3\right)\right)}{3 b^2 d^2 \sqrt{c+d x^3} (a d-b c)}","-\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{3/2} (b c-a d)^{3/2}}+\frac{2 c^2}{3 d^2 \sqrt{c+d x^3} (b c-a d)}+\frac{2 \sqrt{c+d x^3}}{3 b d^2}",1,"(2*(a^2*d^2 + a*b*d*(c + d*x^3) - b^2*c*(2*c + d*x^3) - a^2*d^2*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^3))/(b*c - a*d)]))/(3*b^2*d^2*(-(b*c) + a*d)*Sqrt[c + d*x^3])","C",1
389,1,88,82,0.1214545,"\int \frac{x^5}{\left(a+b x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^5/((a + b*x^3)*(c + d*x^3)^(3/2)),x]","\frac{2 \left(\frac{c (a d-b c)}{d \sqrt{c+d x^3}}+\frac{a \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{\sqrt{b}}\right)}{3 (b c-a d)^2}","\frac{2 a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 \sqrt{b} (b c-a d)^{3/2}}-\frac{2 c}{3 d \sqrt{c+d x^3} (b c-a d)}",1,"(2*((c*(-(b*c) + a*d))/(d*Sqrt[c + d*x^3]) + (a*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/Sqrt[b]))/(3*(b*c - a*d)^2)","A",1
390,1,52,77,0.025805,"\int \frac{x^2}{\left(a+b x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^2/((a + b*x^3)*(c + d*x^3)^(3/2)),x]","-\frac{2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^3+c\right)}{b c-a d}\right)}{3 \sqrt{c+d x^3} (a d-b c)}","\frac{2}{3 \sqrt{c+d x^3} (b c-a d)}-\frac{2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 (b c-a d)^{3/2}}",1,"(-2*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^3))/(b*c - a*d)])/(3*(-(b*c) + a*d)*Sqrt[c + d*x^3])","C",1
391,1,89,114,0.0477481,"\int \frac{1}{x \left(a+b x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x*(a + b*x^3)*(c + d*x^3)^(3/2)),x]","-\frac{2 \left(b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^3+c\right)}{b c-a d}\right)+(a d-b c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right)\right)}{3 a c \sqrt{c+d x^3} (b c-a d)}","\frac{2 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a (b c-a d)^{3/2}}-\frac{2 d}{3 c \sqrt{c+d x^3} (b c-a d)}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a c^{3/2}}",1,"(-2*(b*c*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^3))/(b*c - a*d)] + (-(b*c) + a*d)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^3)/c]))/(3*a*c*(b*c - a*d)*Sqrt[c + d*x^3])","C",1
392,1,117,158,0.0583625,"\int \frac{1}{x^4 \left(a+b x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^4*(a + b*x^3)*(c + d*x^3)^(3/2)),x]","\frac{2 b^2 c^2 x^3 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^3+c\right)}{b c-a d}\right)+(a d-b c) \left(x^3 (3 a d+2 b c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right)+a c\right)}{3 a^2 c^2 x^3 \sqrt{c+d x^3} (b c-a d)}","-\frac{2 b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^2 (b c-a d)^{3/2}}+\frac{(3 a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^2 c^{5/2}}-\frac{d (b c-3 a d)}{3 a c^2 \sqrt{c+d x^3} (b c-a d)}-\frac{1}{3 a c x^3 \sqrt{c+d x^3}}",1,"(2*b^2*c^2*x^3*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^3))/(b*c - a*d)] + (-(b*c) + a*d)*(a*c + (2*b*c + 3*a*d)*x^3*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^3)/c]))/(3*a^2*c^2*(b*c - a*d)*x^3*Sqrt[c + d*x^3])","C",1
393,1,231,67,0.2097445,"\int \frac{x^3}{\left(a+b x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^3/((a + b*x^3)*(c + d*x^3)^(3/2)),x]","\frac{x \left(-\frac{64 a^2 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}-\frac{b x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{a}-8\right)}{12 \sqrt{c+d x^3} (a d-b c)}","\frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};1,\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 a c \sqrt{c+d x^3}}",1,"(x*(-8 - (b*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/a - (64*a^2*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)])/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))))/(12*(-(b*c) + a*d)*Sqrt[c + d*x^3])","B",0
394,1,142,67,0.1231742,"\int \frac{x}{\left(a+b x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x/((a + b*x^3)*(c + d*x^3)^(3/2)),x]","\frac{x^2 \left(2 b d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+5 \sqrt{\frac{d x^3}{c}+1} (a d+3 b c) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-20 a d\right)}{30 a c \sqrt{c+d x^3} (b c-a d)}","\frac{x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};1,\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a c \sqrt{c+d x^3}}",1,"(x^2*(-20*a*d + 5*(3*b*c + a*d)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 2*b*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)]))/(30*a*c*(b*c - a*d)*Sqrt[c + d*x^3])","B",0
395,1,338,62,0.3268922,"\int \frac{1}{\left(a+b x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/((a + b*x^3)*(c + d*x^3)^(3/2)),x]","\frac{x \left(\frac{b d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{a (a d-b c)}+\frac{32 a c \left(3 a d-3 b c+2 b d x^3\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-24 d x^3 \left(a+b x^3\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}{\left(a+b x^3\right) (b c-a d) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}\right)}{12 c \sqrt{c+d x^3}}","\frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{1}{3};1,\frac{3}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a c \sqrt{c+d x^3}}",1,"(x*((b*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/(a*(-(b*c) + a*d)) + (32*a*c*(-3*b*c + 3*a*d + 2*b*d*x^3)*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 24*d*x^3*(a + b*x^3)*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))/((b*c - a*d)*(a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))))/(12*c*Sqrt[c + d*x^3])","B",0
396,1,193,65,0.2369533,"\int \frac{1}{x^2 \left(a+b x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^2*(a + b*x^3)*(c + d*x^3)^(3/2)),x]","\frac{-5 x^3 \sqrt{\frac{d x^3}{c}+1} \left(5 a^2 d^2-3 a b c d+6 b^2 c^2\right) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+2 b d x^6 \sqrt{\frac{d x^3}{c}+1} (3 b c-5 a d) F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+20 a \left(a d \left(3 c+5 d x^3\right)-3 b c \left(c+d x^3\right)\right)}{60 a^2 c^2 x \sqrt{c+d x^3} (b c-a d)}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{1}{3};1,\frac{3}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a c x \sqrt{c+d x^3}}",1,"(20*a*(-3*b*c*(c + d*x^3) + a*d*(3*c + 5*d*x^3)) - 5*(6*b^2*c^2 - 3*a*b*c*d + 5*a^2*d^2)*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 2*b*d*(3*b*c - 5*a*d)*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/(60*a^2*c^2*(b*c - a*d)*x*Sqrt[c + d*x^3])","B",0
397,1,425,67,0.7256785,"\int \frac{1}{x^3 \left(a+b x^3\right) \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^3*(a + b*x^3)*(c + d*x^3)^(3/2)),x]","\frac{\frac{8 a \left(3 x^3 \left(a^2 d \left(3 c+7 d x^3\right)+a b \left(7 d^2 x^6-3 c^2\right)-3 b^2 c x^3 \left(c+d x^3\right)\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-4 a c \left(3 a^2 d \left(2 c+7 d x^3\right)+a b \left(-6 c^2-3 c d x^3+14 d^2 x^6\right)-6 b^2 c x^3 \left(3 c+d x^3\right)\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}{\left(a+b x^3\right) \left(8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)\right)}+b d x^6 \sqrt{\frac{d x^3}{c}+1} (3 b c-7 a d) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{48 a^2 c^2 x^2 \sqrt{c+d x^3} (a d-b c)}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{2}{3};1,\frac{3}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a c x^2 \sqrt{c+d x^3}}",1,"(b*d*(3*b*c - 7*a*d)*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + (8*a*(-4*a*c*(-6*b^2*c*x^3*(3*c + d*x^3) + 3*a^2*d*(2*c + 7*d*x^3) + a*b*(-6*c^2 - 3*c*d*x^3 + 14*d^2*x^6))*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(-3*b^2*c*x^3*(c + d*x^3) + a^2*d*(3*c + 7*d*x^3) + a*b*(-3*c^2 + 7*d^2*x^6))*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(48*a^2*c^2*(-(b*c) + a*d)*x^2*Sqrt[c + d*x^3])","B",0
398,1,101,117,0.1057255,"\int \frac{x^{11} \sqrt{c+d x^3}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^11*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]","\frac{19840 c^{5/2} \left(8 c-d x^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)+6 \sqrt{c+d x^3} \left(-9168 c^3+770 c^2 d x^3+19 c d^2 x^6+d^3 x^9\right)}{45 d^4 \left(d x^3-8 c\right)}","-\frac{3968 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 d^4}+\frac{2 c \sqrt{c+d x^3} \left(1146 c+47 d x^3\right)}{15 d^4}+\frac{7 x^6 \sqrt{c+d x^3}}{15 d^2}+\frac{x^9 \sqrt{c+d x^3}}{3 d \left(8 c-d x^3\right)}",1,"(6*Sqrt[c + d*x^3]*(-9168*c^3 + 770*c^2*d*x^3 + 19*c*d^2*x^6 + d^3*x^9) + 19840*c^(5/2)*(8*c - d*x^3)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(45*d^4*(-8*c + d*x^3))","A",1
399,1,90,102,0.08638,"\int \frac{x^8 \sqrt{c+d x^3}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^8*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]","\frac{352 c^{3/2} \left(8 c-d x^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)+2 \sqrt{c+d x^3} \left(-488 c^2+41 c d x^3+d^2 x^6\right)}{9 d^3 \left(d x^3-8 c\right)}","-\frac{352 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 d^3}+\frac{64 c \left(c+d x^3\right)^{3/2}}{27 d^3 \left(8 c-d x^3\right)}+\frac{352 c \sqrt{c+d x^3}}{27 d^3}+\frac{2 \left(c+d x^3\right)^{3/2}}{9 d^3}",1,"(2*Sqrt[c + d*x^3]*(-488*c^2 + 41*c*d*x^3 + d^2*x^6) + 352*c^(3/2)*(8*c - d*x^3)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*d^3*(-8*c + d*x^3))","A",1
400,1,79,82,0.041764,"\int \frac{x^5 \sqrt{c+d x^3}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^5*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]","\frac{6 \sqrt{c+d x^3} \left(d x^3-12 c\right)+26 \sqrt{c} \left(8 c-d x^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 d^2 \left(d x^3-8 c\right)}","\frac{8 \left(c+d x^3\right)^{3/2}}{27 d^2 \left(8 c-d x^3\right)}+\frac{26 \sqrt{c+d x^3}}{27 d^2}-\frac{26 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 d^2}",1,"(6*(-12*c + d*x^3)*Sqrt[c + d*x^3] + 26*Sqrt[c]*(8*c - d*x^3)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*d^2*(-8*c + d*x^3))","A",1
401,1,61,64,0.0620861,"\int \frac{x^2 \sqrt{c+d x^3}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^2*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]","\frac{\frac{3 \sqrt{c+d x^3}}{8 c-d x^3}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{\sqrt{c}}}{9 d}","\frac{\sqrt{c+d x^3}}{3 d \left(8 c-d x^3\right)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 \sqrt{c} d}",1,"((3*Sqrt[c + d*x^3])/(8*c - d*x^3) - ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/Sqrt[c])/(9*d)","A",1
402,1,102,88,0.0488165,"\int \frac{\sqrt{c+d x^3}}{x \left(8 c-d x^3\right)^2} \, dx","Integrate[Sqrt[c + d*x^3]/(x*(8*c - d*x^3)^2),x]","\frac{12 \sqrt{c} \sqrt{c+d x^3}+5 \left(8 c-d x^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)+3 \left(d x^3-8 c\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{288 c^{3/2} \left(8 c-d x^3\right)}","\frac{5 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{288 c^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{96 c^{3/2}}+\frac{\sqrt{c+d x^3}}{24 c \left(8 c-d x^3\right)}",1,"(12*Sqrt[c]*Sqrt[c + d*x^3] + 5*(8*c - d*x^3)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] + 3*(-8*c + d*x^3)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(288*c^(3/2)*(8*c - d*x^3))","A",1
403,1,97,124,0.1505534,"\int \frac{\sqrt{c+d x^3}}{x^4 \left(8 c-d x^3\right)^2} \, dx","Integrate[Sqrt[c + d*x^3]/(x^4*(8*c - d*x^3)^2),x]","\frac{7 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-9 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)+\frac{12 \sqrt{c} \sqrt{c+d x^3} \left(4 c-d x^3\right)}{d x^6-8 c x^3}}{1152 c^{5/2}}","\frac{7 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{1152 c^{5/2}}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{128 c^{5/2}}+\frac{d \sqrt{c+d x^3}}{96 c^2 \left(8 c-d x^3\right)}-\frac{\sqrt{c+d x^3}}{24 c x^3 \left(8 c-d x^3\right)}",1,"((12*Sqrt[c]*(4*c - d*x^3)*Sqrt[c + d*x^3])/(-8*c*x^3 + d*x^6) + 7*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] - 9*d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(1152*c^(5/2))","A",1
404,1,112,164,0.2107577,"\int \frac{\sqrt{c+d x^3}}{x^7 \left(8 c-d x^3\right)^2} \, dx","Integrate[Sqrt[c + d*x^3]/(x^7*(8*c - d*x^3)^2),x]","\frac{\frac{12 \sqrt{c} \sqrt{c+d x^3} \left(32 c^2+28 c d x^3-5 d^2 x^6\right)}{d x^9-8 c x^6}+23 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-9 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{18432 c^{7/2}}","\frac{23 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{18432 c^{7/2}}-\frac{d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{2048 c^{7/2}}+\frac{5 d^2 \sqrt{c+d x^3}}{1536 c^3 \left(8 c-d x^3\right)}-\frac{7 d \sqrt{c+d x^3}}{384 c^2 x^3 \left(8 c-d x^3\right)}-\frac{\sqrt{c+d x^3}}{48 c x^6 \left(8 c-d x^3\right)}",1,"((12*Sqrt[c]*Sqrt[c + d*x^3]*(32*c^2 + 28*c*d*x^3 - 5*d^2*x^6))/(-8*c*x^6 + d*x^9) + 23*d^2*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] - 9*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(18432*c^(7/2))","A",1
405,1,176,663,0.1968623,"\int \frac{x^7 \sqrt{c+d x^3}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^7*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]","-\frac{373 d x^5 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+520 c x^2 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+80 \left(52 c^2 x^2+49 c d x^5-3 d^2 x^8\right)}{840 d^2 \left(d x^3-8 c\right) \sqrt{c+d x^3}}","\frac{76 c^{7/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{3 \sqrt{3} d^{8/3}}-\frac{76 c^{7/6} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{9 d^{8/3}}+\frac{76 c^{7/6} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 d^{8/3}}+\frac{746 \sqrt{2} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{21 \sqrt[4]{3} d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{373 \sqrt{2-\sqrt{3}} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{7\ 3^{3/4} d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{746 c \sqrt{c+d x^3}}{21 d^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{13 x^2 \sqrt{c+d x^3}}{21 d^2}+\frac{x^5 \sqrt{c+d x^3}}{3 d \left(8 c-d x^3\right)}",1,"-1/840*(80*(52*c^2*x^2 + 49*c*d*x^5 - 3*d^2*x^8) + 520*c*x^2*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 373*d*x^5*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(d^2*(-8*c + d*x^3)*Sqrt[c + d*x^3])","C",0
406,1,167,641,0.150725,"\int \frac{x^4 \sqrt{c+d x^3}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^4*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]","\frac{7 d x^5 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+10 c x^2 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+80 c x^2 \left(c+d x^3\right)}{240 c d \left(8 c-d x^3\right) \sqrt{c+d x^3}}","\frac{7 \sqrt{2} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{7 \sqrt{2-\sqrt{3}} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{2\ 3^{3/4} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{7 \sqrt{c+d x^3}}{3 d^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{5 \sqrt[6]{c} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{3 \sqrt{3} d^{5/3}}-\frac{5 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{9 d^{5/3}}+\frac{5 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9 d^{5/3}}+\frac{x^2 \sqrt{c+d x^3}}{3 d \left(8 c-d x^3\right)}",1,"(80*c*x^2*(c + d*x^3) + 10*c*x^2*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 7*d*x^5*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(240*c*d*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
407,1,164,644,0.0953244,"\int \frac{x \sqrt{c+d x^3}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]","\frac{d x^5 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+5 c x^2 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+80 c x^2 \left(c+d x^3\right)}{1920 c^2 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{48 \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{144 c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{144 c^{5/6} d^{2/3}}+\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{12 \sqrt{2} \sqrt[4]{3} c^{2/3} d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{16\ 3^{3/4} c^{2/3} d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{\sqrt{c+d x^3}}{24 c d^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{x^2 \sqrt{c+d x^3}}{24 c \left(8 c-d x^3\right)}",1,"(80*c*x^2*(c + d*x^3) + 5*c*x^2*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + d*x^5*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(1920*c^2*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
408,1,179,665,0.1659266,"\int \frac{\sqrt{c+d x^3}}{x^2 \left(8 c-d x^3\right)^2} \, dx","Integrate[Sqrt[c + d*x^3]/(x^2*(8*c - d*x^3)^2),x]","\frac{d^2 x^6 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+50 c d x^3 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-80 c \left(6 c^2+5 c d x^3-d^2 x^6\right)}{3840 c^3 \sqrt{c+d x^3} \left(8 c x-d x^4\right)}","\frac{\sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{24 \sqrt{2} \sqrt[4]{3} c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{2-\sqrt{3}} \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{32\ 3^{3/4} c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{48 \sqrt{3} c^{11/6}}+\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{144 c^{11/6}}-\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{144 c^{11/6}}-\frac{\sqrt{c+d x^3}}{48 c^2 x}+\frac{\sqrt[3]{d} \sqrt{c+d x^3}}{48 c^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{\sqrt{c+d x^3}}{24 c x \left(8 c-d x^3\right)}",1,"(-80*c*(6*c^2 + 5*c*d*x^3 - d^2*x^6) + 50*c*d*x^3*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + d^2*x^6*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(3840*c^3*Sqrt[c + d*x^3]*(8*c*x - d*x^4))","C",0
409,1,199,687,0.140527,"\int \frac{\sqrt{c+d x^3}}{x^5 \left(8 c-d x^3\right)^2} \, dx","Integrate[Sqrt[c + d*x^3]/(x^5*(8*c - d*x^3)^2),x]","-\frac{d^3 x^5 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{7680 c^4 \sqrt{c+d x^3}}+\frac{115 d^2 x^2 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{24576 c^3 \sqrt{c+d x^3}}+\sqrt{c+d x^3} \left(-\frac{d^2 x^2}{1536 c^3 \left(d x^3-8 c\right)}-\frac{5 d}{512 c^3 x}-\frac{1}{256 c^2 x^4}\right)","-\frac{17 d^{4/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{3072 \sqrt{3} c^{17/6}}+\frac{17 d^{4/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{9216 c^{17/6}}-\frac{17 d^{4/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{9216 c^{17/6}}+\frac{d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{48 \sqrt{2} \sqrt[4]{3} c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{2-\sqrt{3}} d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{64\ 3^{3/4} c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{d^{4/3} \sqrt{c+d x^3}}{96 c^3 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{d \sqrt{c+d x^3}}{96 c^3 x}-\frac{7 \sqrt{c+d x^3}}{768 c^2 x^4}+\frac{\sqrt{c+d x^3}}{24 c x^4 \left(8 c-d x^3\right)}",1,"Sqrt[c + d*x^3]*(-1/256*1/(c^2*x^4) - (5*d)/(512*c^3*x) - (d^2*x^2)/(1536*c^3*(-8*c + d*x^3))) + (115*d^2*x^2*Sqrt[(c + d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(24576*c^3*Sqrt[c + d*x^3]) - (d^3*x^5*Sqrt[(c + d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(7680*c^4*Sqrt[c + d*x^3])","C",0
410,1,209,711,0.1654489,"\int \frac{\sqrt{c+d x^3}}{x^8 \left(8 c-d x^3\right)^2} \, dx","Integrate[Sqrt[c + d*x^3]/(x^8*(8*c - d*x^3)^2),x]","\frac{1525 c d^3 x^9 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-8 \left(d^4 x^{12} \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+20 c \left(384 c^4+648 c^3 d x^3+243 c^2 d^2 x^6-25 c d^3 x^9-4 d^4 x^{12}\right)\right)}{3440640 c^5 x^7 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","-\frac{13 d^{7/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{12288 \sqrt{3} c^{23/6}}+\frac{13 d^{7/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{36864 c^{23/6}}-\frac{13 d^{7/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{36864 c^{23/6}}+\frac{d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{2688 \sqrt{2} \sqrt[4]{3} c^{11/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{2-\sqrt{3}} d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{3584\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{d^{7/3} \sqrt{c+d x^3}}{5376 c^4 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{d^2 \sqrt{c+d x^3}}{5376 c^4 x}-\frac{53 d \sqrt{c+d x^3}}{21504 c^3 x^4}-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left(8 c-d x^3\right)}",1,"(1525*c*d^3*x^9*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 8*(20*c*(384*c^4 + 648*c^3*d*x^3 + 243*c^2*d^2*x^6 - 25*c*d^3*x^9 - 4*d^4*x^12) + d^4*x^12*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(3440640*c^5*x^7*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
411,1,111,134,0.0820673,"\int \frac{x^{11} \left(c+d x^3\right)^{3/2}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^11*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]","\frac{2 \left(52416 c^{7/2} \left(8 c-d x^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)+\sqrt{c+d x^3} \left(-145328 c^4+12206 c^3 d x^3+301 c^2 d^2 x^6+16 c d^3 x^9+d^4 x^{12}\right)\right)}{21 d^4 \left(d x^3-8 c\right)}","-\frac{4992 c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^4}+\frac{1664 c^3 \sqrt{c+d x^3}}{d^4}+\frac{2 c \left(c+d x^3\right)^{3/2} \left(694 c+51 d x^3\right)}{21 d^4}+\frac{3 x^6 \left(c+d x^3\right)^{3/2}}{7 d^2}+\frac{x^9 \left(c+d x^3\right)^{3/2}}{3 d \left(8 c-d x^3\right)}",1,"(2*(Sqrt[c + d*x^3]*(-145328*c^4 + 12206*c^3*d*x^3 + 301*c^2*d^2*x^6 + 16*c*d^3*x^9 + d^4*x^12) + 52416*c^(7/2)*(8*c - d*x^3)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]))/(21*d^4*(-8*c + d*x^3))","A",1
412,1,102,119,0.1257308,"\int \frac{x^8 \left(c+d x^3\right)^{3/2}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^8*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]","\frac{21600 c^{5/2} \left(8 c-d x^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)+2 \sqrt{c+d x^3} \left(-29944 c^3+2515 c^2 d x^3+62 c d^2 x^6+3 d^3 x^9\right)}{45 d^3 \left(d x^3-8 c\right)}","-\frac{480 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^3}+\frac{160 c^2 \sqrt{c+d x^3}}{d^3}+\frac{64 c \left(c+d x^3\right)^{5/2}}{27 d^3 \left(8 c-d x^3\right)}+\frac{160 c \left(c+d x^3\right)^{3/2}}{27 d^3}+\frac{2 \left(c+d x^3\right)^{5/2}}{15 d^3}",1,"(2*Sqrt[c + d*x^3]*(-29944*c^3 + 2515*c^2*d*x^3 + 62*c*d^2*x^6 + 3*d^3*x^9) + 21600*c^(5/2)*(8*c - d*x^3)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(45*d^3*(-8*c + d*x^3))","A",1
413,1,90,97,0.0606092,"\int \frac{x^5 \left(c+d x^3\right)^{3/2}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^5*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]","\frac{378 c^{3/2} \left(8 c-d x^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)+2 \sqrt{c+d x^3} \left(-524 c^2+44 c d x^3+d^2 x^6\right)}{9 d^2 \left(d x^3-8 c\right)}","-\frac{42 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^2}+\frac{8 \left(c+d x^3\right)^{5/2}}{27 d^2 \left(8 c-d x^3\right)}+\frac{14 \left(c+d x^3\right)^{3/2}}{27 d^2}+\frac{14 c \sqrt{c+d x^3}}{d^2}",1,"(2*Sqrt[c + d*x^3]*(-524*c^2 + 44*c*d*x^3 + d^2*x^6) + 378*c^(3/2)*(8*c - d*x^3)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*d^2*(-8*c + d*x^3))","A",1
414,1,43,77,0.0148492,"\int \frac{x^2 \left(c+d x^3\right)^{3/2}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^2*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]","\frac{2 \left(c+d x^3\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{d x^3+c}{9 c}\right)}{1215 c^2 d}","\frac{\left(c+d x^3\right)^{3/2}}{3 d \left(8 c-d x^3\right)}+\frac{\sqrt{c+d x^3}}{d}-\frac{3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d}",1,"(2*(c + d*x^3)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, (c + d*x^3)/(9*c)])/(1215*c^2*d)","C",1
415,1,100,85,0.0548918,"\int \frac{\left(c+d x^3\right)^{3/2}}{x \left(8 c-d x^3\right)^2} \, dx","Integrate[(c + d*x^3)^(3/2)/(x*(8*c - d*x^3)^2),x]","\frac{36 \sqrt{c} \sqrt{c+d x^3}+\left(9 d x^3-72 c\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)+\left(d x^3-8 c\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{96 \sqrt{c} \left(8 c-d x^3\right)}","\frac{3 \sqrt{c+d x^3}}{8 \left(8 c-d x^3\right)}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{32 \sqrt{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{96 \sqrt{c}}",1,"(36*Sqrt[c]*Sqrt[c + d*x^3] + (-72*c + 9*d*x^3)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] + (-8*c + d*x^3)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(96*Sqrt[c]*(8*c - d*x^3))","A",1
416,1,97,121,0.1555035,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^4 \left(8 c-d x^3\right)^2} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^4*(8*c - d*x^3)^2),x]","\frac{9 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-7 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)+\frac{4 \sqrt{c} \sqrt{c+d x^3} \left(4 c-5 d x^3\right)}{d x^6-8 c x^3}}{384 c^{3/2}}","\frac{3 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{128 c^{3/2}}-\frac{7 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{384 c^{3/2}}+\frac{5 d \sqrt{c+d x^3}}{96 c \left(8 c-d x^3\right)}-\frac{\sqrt{c+d x^3}}{24 x^3 \left(8 c-d x^3\right)}",1,"((4*Sqrt[c]*(4*c - 5*d*x^3)*Sqrt[c + d*x^3])/(-8*c*x^3 + d*x^6) + 9*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] - 7*d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(384*c^(3/2))","A",1
417,1,112,161,0.2401952,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^7 \left(8 c-d x^3\right)^2} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^7*(8*c - d*x^3)^2),x]","\frac{\frac{4 \sqrt{c} \sqrt{c+d x^3} \left(32 c^2+92 c d x^3-21 d^2 x^6\right)}{d x^9-8 c x^6}+45 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-51 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{6144 c^{5/2}}","\frac{15 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{2048 c^{5/2}}-\frac{17 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{2048 c^{5/2}}+\frac{7 d^2 \sqrt{c+d x^3}}{512 c^2 \left(8 c-d x^3\right)}-\frac{23 d \sqrt{c+d x^3}}{384 c x^3 \left(8 c-d x^3\right)}-\frac{\sqrt{c+d x^3}}{48 x^6 \left(8 c-d x^3\right)}",1,"((4*Sqrt[c]*Sqrt[c + d*x^3]*(32*c^2 + 92*c*d*x^3 - 21*d^2*x^6))/(-8*c*x^6 + d*x^9) + 45*d^2*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] - 51*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(6144*c^(5/2))","A",1
418,1,191,681,0.1697168,"\int \frac{x^7 \left(c+d x^3\right)^{3/2}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]","\frac{4120 c^2 x^2 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+2953 c d x^5 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+80 x^2 \left(-412 c^3-388 c^2 d x^3+25 c d^2 x^6+d^3 x^9\right)}{520 d^2 \left(d x^3-8 c\right) \sqrt{c+d x^3}}","\frac{108 \sqrt{3} c^{13/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{d^{8/3}}-\frac{108 c^{13/6} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{d^{8/3}}+\frac{108 c^{13/6} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^{8/3}}+\frac{5906 \sqrt{2} c^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{13 \sqrt[4]{3} d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{2953 \sqrt[4]{3} \sqrt{2-\sqrt{3}} c^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{13 d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{5906 c^2 \sqrt{c+d x^3}}{13 d^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{103 c x^2 \sqrt{c+d x^3}}{13 d^2}+\frac{x^5 \left(c+d x^3\right)^{3/2}}{3 d \left(8 c-d x^3\right)}+\frac{19 x^5 \sqrt{c+d x^3}}{39 d}",1,"(80*x^2*(-412*c^3 - 388*c^2*d*x^3 + 25*c*d^2*x^6 + d^3*x^9) + 4120*c^2*x^2*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 2953*c*d*x^5*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(520*d^2*(-8*c + d*x^3)*Sqrt[c + d*x^3])","C",0
419,1,176,657,0.183475,"\int \frac{x^4 \left(c+d x^3\right)^{3/2}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x^4*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]","-\frac{53 d x^5 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+74 c x^2 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+16 x^2 \left(37 c^2+35 c d x^3-2 d^2 x^6\right)}{112 d \left(d x^3-8 c\right) \sqrt{c+d x^3}}","\frac{9 \sqrt{3} c^{7/6} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{d^{5/3}}-\frac{9 c^{7/6} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{d^{5/3}}+\frac{9 c^{7/6} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{d^{5/3}}+\frac{265 \sqrt{2} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{7 \sqrt[4]{3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{265 \sqrt[4]{3} \sqrt{2-\sqrt{3}} c^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{14 d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{265 c \sqrt{c+d x^3}}{7 d^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{x^2 \left(c+d x^3\right)^{3/2}}{3 d \left(8 c-d x^3\right)}+\frac{13 x^2 \sqrt{c+d x^3}}{21 d}",1,"-1/112*(16*x^2*(37*c^2 + 35*c*d*x^3 - 2*d^2*x^6) + 74*c*x^2*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 53*d*x^5*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(d*(-8*c + d*x^3)*Sqrt[c + d*x^3])","C",0
420,1,141,638,0.2038293,"\int \frac{x \left(c+d x^3\right)^{3/2}}{\left(8 c-d x^3\right)^2} \, dx","Integrate[(x*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]","\frac{x^2 \left(-\frac{19 d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c}-25 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+\frac{240 \left(c+d x^3\right)}{8 c-d x^3}\right)}{640 \sqrt{c+d x^3}}","\frac{19 \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt{2} \sqrt[4]{3} d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{19 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{16 d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{19 \sqrt{c+d x^3}}{8 d^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{9 \sqrt{3} \sqrt[6]{c} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{16 d^{2/3}}-\frac{9 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{16 d^{2/3}}+\frac{9 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{16 d^{2/3}}+\frac{3 x^2 \sqrt{c+d x^3}}{8 \left(8 c-d x^3\right)}",1,"(x^2*((240*(c + d*x^3))/(8*c - d*x^3) - 25*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - (19*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/c))/(640*Sqrt[c + d*x^3])","C",0
421,1,242,522,0.699223,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^2 \left(8 c-d x^3\right)^2} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^2*(8*c - d*x^3)^2),x]","\frac{\left(2 c-d x^3\right) \sqrt{c+d x^3}}{16 c x \left(d x^3-8 c\right)}-\frac{\sqrt[6]{-1} \sqrt[3]{-d} \sqrt{(-1)^{5/6} \left(\frac{\sqrt[3]{-d} x}{\sqrt[3]{c}}-1\right)} \sqrt{\frac{(-d)^{2/3} x^2}{c^{2/3}}+\frac{\sqrt[3]{-d} x}{\sqrt[3]{c}}+1} \left(\sqrt[3]{-1} F\left(\sin ^{-1}\left(\frac{\sqrt{-\frac{i \sqrt[3]{-d} x}{\sqrt[3]{c}}-(-1)^{5/6}}}{\sqrt[4]{3}}\right)|\sqrt[3]{-1}\right)-i \sqrt{3} E\left(\sin ^{-1}\left(\frac{\sqrt{-\frac{i \sqrt[3]{-d} x}{\sqrt[3]{c}}-(-1)^{5/6}}}{\sqrt[4]{3}}\right)|\sqrt[3]{-1}\right)\right)}{16 \sqrt[4]{3} \sqrt[3]{c} \sqrt{c+d x^3}}","\frac{\sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{8 \sqrt{2} \sqrt[4]{3} c^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{32 c^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{c+d x^3}}{16 c x}+\frac{\sqrt[3]{d} \sqrt{c+d x^3}}{16 c \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{3 \sqrt{c+d x^3}}{8 x \left(8 c-d x^3\right)}",1,"((2*c - d*x^3)*Sqrt[c + d*x^3])/(16*c*x*(-8*c + d*x^3)) - ((-1)^(1/6)*(-d)^(1/3)*Sqrt[(-1)^(5/6)*(-1 + ((-d)^(1/3)*x)/c^(1/3))]*Sqrt[1 + ((-d)^(1/3)*x)/c^(1/3) + ((-d)^(2/3)*x^2)/c^(2/3)]*((-I)*Sqrt[3]*EllipticE[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-d)^(1/3)*x)/c^(1/3)]/3^(1/4)], (-1)^(1/3)] + (-1)^(1/3)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-d)^(1/3)*x)/c^(1/3)]/3^(1/4)], (-1)^(1/3)]))/(16*3^(1/4)*c^(1/3)*Sqrt[c + d*x^3])","C",0
422,1,199,684,0.1531233,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^5 \left(8 c-d x^3\right)^2} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^5*(8*c - d*x^3)^2),x]","-\frac{d^3 x^5 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{2560 c^3 \sqrt{c+d x^3}}+\frac{145 d^2 x^2 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{8192 c^2 \sqrt{c+d x^3}}+\sqrt{c+d x^3} \left(-\frac{3 d^2 x^2}{512 c^2 \left(d x^3-8 c\right)}-\frac{13 d}{512 c^2 x}-\frac{1}{256 c x^4}\right)","-\frac{9 \sqrt{3} d^{4/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{1024 c^{11/6}}+\frac{9 d^{4/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{1024 c^{11/6}}-\frac{9 d^{4/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{1024 c^{11/6}}+\frac{d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{16 \sqrt{2} \sqrt[4]{3} c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{64 c^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{d^{4/3} \sqrt{c+d x^3}}{32 c^2 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{d \sqrt{c+d x^3}}{32 c^2 x}+\frac{3 \sqrt{c+d x^3}}{8 x^4 \left(8 c-d x^3\right)}-\frac{13 \sqrt{c+d x^3}}{256 c x^4}",1,"Sqrt[c + d*x^3]*(-1/256*1/(c*x^4) - (13*d)/(512*c^2*x) - (3*d^2*x^2)/(512*c^2*(-8*c + d*x^3))) + (145*d^2*x^2*Sqrt[(c + d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(8192*c^2*Sqrt[c + d*x^3]) - (d^3*x^5*Sqrt[(c + d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(2560*c^3*Sqrt[c + d*x^3])","C",0
423,1,212,708,0.1529581,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^8 \left(8 c-d x^3\right)^2} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^8*(8*c - d*x^3)^2),x]","-\frac{19 d^4 x^5 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{143360 c^4 \sqrt{c+d x^3}}+\frac{1175 d^3 x^2 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{229376 c^3 \sqrt{c+d x^3}}+\sqrt{c+d x^3} \left(-\frac{3 d^3 x^2}{4096 c^3 \left(d x^3-8 c\right)}-\frac{283 d^2}{28672 c^3 x}-\frac{41 d}{7168 c^2 x^4}-\frac{1}{448 c x^7}\right)","-\frac{9 \sqrt{3} d^{7/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{4096 c^{17/6}}+\frac{9 d^{7/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{4096 c^{17/6}}-\frac{9 d^{7/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{4096 c^{17/6}}+\frac{19 d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{896 \sqrt{2} \sqrt[4]{3} c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{19 \sqrt[4]{3} \sqrt{2-\sqrt{3}} d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{3584 c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{19 d^{7/3} \sqrt{c+d x^3}}{1792 c^3 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{19 d^2 \sqrt{c+d x^3}}{1792 c^3 x}-\frac{83 d \sqrt{c+d x^3}}{7168 c^2 x^4}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left(8 c-d x^3\right)}-\frac{11 \sqrt{c+d x^3}}{224 c x^7}",1,"Sqrt[c + d*x^3]*(-1/448*1/(c*x^7) - (41*d)/(7168*c^2*x^4) - (283*d^2)/(28672*c^3*x) - (3*d^3*x^2)/(4096*c^3*(-8*c + d*x^3))) + (1175*d^3*x^2*Sqrt[(c + d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(229376*c^3*Sqrt[c + d*x^3]) - (19*d^4*x^5*Sqrt[(c + d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(143360*c^4*Sqrt[c + d*x^3])","C",0
424,1,91,95,0.0477092,"\int \frac{x^{11}}{\left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^11/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{2944 c^{3/2} \left(8 c-d x^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)+6 \sqrt{c+d x^3} \left(-1360 c^2+114 c d x^3+3 d^2 x^6\right)}{81 d^4 \left(d x^3-8 c\right)}","-\frac{2944 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 d^4}+\frac{2 \sqrt{c+d x^3} \left(170 c+7 d x^3\right)}{27 d^4}+\frac{8 x^6 \sqrt{c+d x^3}}{27 d^2 \left(8 c-d x^3\right)}",1,"(6*Sqrt[c + d*x^3]*(-1360*c^2 + 114*c*d*x^3 + 3*d^2*x^6) + 2944*c^(3/2)*(8*c - d*x^3)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*d^4*(-8*c + d*x^3))","A",1
425,1,82,83,0.0526293,"\int \frac{x^8}{\left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^8/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","-\frac{64 c \sqrt{c+d x^3}}{27 d^3 \left(d x^3-8 c\right)}+\frac{2 \sqrt{c+d x^3}}{3 d^3}-\frac{224 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 d^3}","\frac{64 c \sqrt{c+d x^3}}{27 d^3 \left(8 c-d x^3\right)}+\frac{2 \sqrt{c+d x^3}}{3 d^3}-\frac{224 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 d^3}",1,"(2*Sqrt[c + d*x^3])/(3*d^3) - (64*c*Sqrt[c + d*x^3])/(27*d^3*(-8*c + d*x^3)) - (224*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*d^3)","A",1
426,1,63,64,0.052491,"\int \frac{x^5}{\left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^5/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","-\frac{8 \sqrt{c+d x^3}}{27 d^2 \left(d x^3-8 c\right)}-\frac{10 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 \sqrt{c} d^2}","\frac{8 \sqrt{c+d x^3}}{27 d^2 \left(8 c-d x^3\right)}-\frac{10 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 \sqrt{c} d^2}",1,"(-8*Sqrt[c + d*x^3])/(27*d^2*(-8*c + d*x^3)) - (10*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*Sqrt[c]*d^2)","A",1
427,1,64,67,0.0367895,"\int \frac{x^2}{\left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^2/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{\frac{3 \sqrt{c} \sqrt{c+d x^3}}{8 c-d x^3}+\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 c^{3/2} d}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 c^{3/2} d}+\frac{\sqrt{c+d x^3}}{27 c d \left(8 c-d x^3\right)}",1,"((3*Sqrt[c]*Sqrt[c + d*x^3])/(8*c - d*x^3) + ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*c^(3/2)*d)","A",1
428,1,83,88,0.095819,"\int \frac{1}{x \left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{\frac{12 \sqrt{c} \sqrt{c+d x^3}}{8 c-d x^3}+13 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-27 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{2592 c^{5/2}}","\frac{13 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{2592 c^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{96 c^{5/2}}+\frac{\sqrt{c+d x^3}}{216 c^2 \left(8 c-d x^3\right)}",1,"((12*Sqrt[c]*Sqrt[c + d*x^3])/(8*c - d*x^3) + 13*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] - 27*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(2592*c^(5/2))","A",1
429,1,97,124,0.1488662,"\int \frac{1}{x^4 \left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x^4*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{11 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)+27 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)+\frac{12 \sqrt{c} \sqrt{c+d x^3} \left(36 c-5 d x^3\right)}{d x^6-8 c x^3}}{10368 c^{7/2}}","\frac{11 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{10368 c^{7/2}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{384 c^{7/2}}+\frac{5 d \sqrt{c+d x^3}}{864 c^3 \left(8 c-d x^3\right)}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3 \left(8 c-d x^3\right)}",1,"((12*Sqrt[c]*(36*c - 5*d*x^3)*Sqrt[c + d*x^3])/(-8*c*x^3 + d*x^6) + 11*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] + 27*d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(10368*c^(7/2))","A",1
430,1,112,164,0.2084165,"\int \frac{1}{x^7 \left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x^7*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{\frac{12 \sqrt{c} \sqrt{c+d x^3} \left(288 c^2-324 c d x^3+35 d^2 x^6\right)}{d x^9-8 c x^6}+31 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)-513 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{165888 c^{9/2}}","\frac{31 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{165888 c^{9/2}}-\frac{19 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{6144 c^{9/2}}-\frac{35 d^2 \sqrt{c+d x^3}}{13824 c^4 \left(8 c-d x^3\right)}+\frac{3 d \sqrt{c+d x^3}}{128 c^3 x^3 \left(8 c-d x^3\right)}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6 \left(8 c-d x^3\right)}",1,"((12*Sqrt[c]*Sqrt[c + d*x^3]*(288*c^2 - 324*c*d*x^3 + 35*d^2*x^6))/(-8*c*x^6 + d*x^9) + 31*d^2*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] - 513*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(165888*c^(9/2))","A",1
431,1,167,641,0.1854241,"\int \frac{x^7}{\left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^7/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{31 d x^5 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+40 c x^2 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+320 c x^2 \left(c+d x^3\right)}{1080 c d^2 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","\frac{62 \sqrt{2} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{31 \sqrt{2-\sqrt{3}} \sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{9\ 3^{3/4} d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{62 \sqrt{c+d x^3}}{27 d^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{44 \sqrt[6]{c} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{27 \sqrt{3} d^{8/3}}-\frac{44 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{81 d^{8/3}}+\frac{44 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 d^{8/3}}+\frac{8 x^2 \sqrt{c+d x^3}}{27 d^2 \left(8 c-d x^3\right)}",1,"(320*c*x^2*(c + d*x^3) + 40*c*x^2*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 31*d*x^5*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(1080*c*d^2*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
432,1,166,647,0.1039463,"\int \frac{x^4}{\left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^4/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{d x^5 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+10 c x^2 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+80 c x^2 \left(c+d x^3\right)}{2160 c^2 d \left(8 c-d x^3\right) \sqrt{c+d x^3}}","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{27 \sqrt{3} c^{5/6} d^{5/3}}-\frac{\tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{81 c^{5/6} d^{5/3}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{81 c^{5/6} d^{5/3}}+\frac{\sqrt{2} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} c^{2/3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{18\ 3^{3/4} c^{2/3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{\sqrt{c+d x^3}}{27 c d^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{x^2 \sqrt{c+d x^3}}{27 c d \left(8 c-d x^3\right)}",1,"(80*c*x^2*(c + d*x^3) + 10*c*x^2*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + d*x^5*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(2160*c^2*d*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
433,1,164,644,0.0910286,"\int \frac{x}{\left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{d x^5 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+125 c x^2 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+80 c x^2 \left(c+d x^3\right)}{17280 c^3 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","-\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{432 \sqrt{3} c^{11/6} d^{2/3}}+\frac{7 \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{1296 c^{11/6} d^{2/3}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{1296 c^{11/6} d^{2/3}}+\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{108 \sqrt{2} \sqrt[4]{3} c^{5/3} d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{144\ 3^{3/4} c^{5/3} d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{\sqrt{c+d x^3}}{216 c^2 d^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{x^2 \sqrt{c+d x^3}}{216 c^2 \left(8 c-d x^3\right)}",1,"(80*c*x^2*(c + d*x^3) + 125*c*x^2*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + d*x^5*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(17280*c^3*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
434,1,180,665,0.1730327,"\int \frac{1}{x^2 \left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x^2*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{7 d^2 x^6 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+200 c d x^3 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-80 c \left(54 c^2+47 c d x^3-7 d^2 x^6\right)}{34560 c^4 \sqrt{c+d x^3} \left(8 c x-d x^4\right)}","\frac{7 \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{216 \sqrt{2} \sqrt[4]{3} c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{7 \sqrt{2-\sqrt{3}} \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{288\ 3^{3/4} c^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{216 \sqrt{3} c^{17/6}}+\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{648 c^{17/6}}-\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{648 c^{17/6}}-\frac{7 \sqrt{c+d x^3}}{432 c^3 x}+\frac{7 \sqrt[3]{d} \sqrt{c+d x^3}}{432 c^3 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{\sqrt{c+d x^3}}{216 c^2 x \left(8 c-d x^3\right)}",1,"(-80*c*(54*c^2 + 47*c*d*x^3 - 7*d^2*x^6) + 200*c*d*x^3*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 7*d^2*x^6*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(34560*c^4*Sqrt[c + d*x^3]*(8*c*x - d*x^4))","C",0
435,1,196,687,0.197573,"\int \frac{1}{x^5 \left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x^5*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{245 c d^2 x^6 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-16 \left(d^3 x^9 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+2 c \left(216 c^3-135 c^2 d x^3-311 c d^2 x^6+40 d^3 x^9\right)\right)}{221184 c^5 x^4 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","-\frac{25 d^{4/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{27648 \sqrt{3} c^{23/6}}+\frac{25 d^{4/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{82944 c^{23/6}}-\frac{25 d^{4/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{82944 c^{23/6}}-\frac{5 d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{432 \sqrt{2} \sqrt[4]{3} c^{11/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{5 \sqrt{2-\sqrt{3}} d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{576\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{5 d^{4/3} \sqrt{c+d x^3}}{864 c^4 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{5 d \sqrt{c+d x^3}}{864 c^4 x}-\frac{31 \sqrt{c+d x^3}}{6912 c^3 x^4}+\frac{\sqrt{c+d x^3}}{216 c^2 x^4 \left(8 c-d x^3\right)}",1,"(245*c*d^2*x^6*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 16*(2*c*(216*c^3 - 135*c^2*d*x^3 - 311*c*d^2*x^6 + 40*d^3*x^9) + d^3*x^9*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(221184*c^5*x^4*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
436,1,212,711,0.2317142,"\int \frac{1}{x^8 \left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x^8*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","-\frac{289 d^4 x^5 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{3870720 c^6 \sqrt{c+d x^3}}+\frac{9605 d^3 x^2 \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{6193152 c^5 \sqrt{c+d x^3}}+\sqrt{c+d x^3} \left(-\frac{d^3 x^2}{110592 c^5 \left(d x^3-8 c\right)}-\frac{171 d^2}{28672 c^5 x}+\frac{15 d}{7168 c^4 x^4}-\frac{1}{448 c^3 x^7}\right)","-\frac{17 d^{7/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{110592 \sqrt{3} c^{29/6}}+\frac{17 d^{7/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{331776 c^{29/6}}-\frac{17 d^{7/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{331776 c^{29/6}}+\frac{289 d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{24192 \sqrt{2} \sqrt[4]{3} c^{14/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{289 \sqrt{2-\sqrt{3}} d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{32256\ 3^{3/4} c^{14/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{289 d^{7/3} \sqrt{c+d x^3}}{48384 c^5 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{289 d^2 \sqrt{c+d x^3}}{48384 c^5 x}+\frac{391 d \sqrt{c+d x^3}}{193536 c^4 x^4}-\frac{17 \sqrt{c+d x^3}}{6048 c^3 x^7}+\frac{\sqrt{c+d x^3}}{216 c^2 x^7 \left(8 c-d x^3\right)}",1,"Sqrt[c + d*x^3]*(-1/448*1/(c^3*x^7) + (15*d)/(7168*c^4*x^4) - (171*d^2)/(28672*c^5*x) - (d^3*x^2)/(110592*c^5*(-8*c + d*x^3))) + (9605*d^3*x^2*Sqrt[(c + d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(6193152*c^5*Sqrt[c + d*x^3]) - (289*d^4*x^5*Sqrt[(c + d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(3870720*c^6*Sqrt[c + d*x^3])","C",0
437,1,239,66,0.3308311,"\int \frac{x^6}{\left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^6/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{x \left(\frac{256 \left(-\frac{32 c^2 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}+c+d x^3\right)}{8 c-d x^3}-\frac{23 d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c}\right)}{864 d^2 \sqrt{c+d x^3}}","\frac{x^7 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{7}{3};2,\frac{1}{2};\frac{10}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{448 c^2 \sqrt{c+d x^3}}",1,"(x*((-23*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])/c + (256*(c + d*x^3 - (32*c^2*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)]))))/(8*c - d*x^3)))/(864*d^2*Sqrt[c + d*x^3])","B",0
438,1,237,66,0.2328056,"\int \frac{x^3}{\left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^3/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{x \left(x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-\frac{64 c \left(-\frac{32 c^2 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}+c+d x^3\right)}{d \left(d x^3-8 c\right)}\right)}{1728 c^2 \sqrt{c+d x^3}}","\frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};2,\frac{1}{2};\frac{7}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{256 c^2 \sqrt{c+d x^3}}",1,"(x*(x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - (64*c*(c + d*x^3 - (32*c^2*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)]))))/(d*(-8*c + d*x^3))))/(1728*c^2*Sqrt[c + d*x^3])","B",0
439,1,237,64,0.1883181,"\int \frac{1}{\left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{x \left(\frac{d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c^3}+\frac{64 \left(\frac{832 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}+\frac{c+d x^3}{c^2}\right)}{8 c-d x^3}\right)}{13824 \sqrt{c+d x^3}}","\frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{1}{3};2,\frac{1}{2};\frac{4}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{64 c^2 \sqrt{c+d x^3}}",1,"(x*((d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])/c^3 + (64*((c + d*x^3)/c^2 + (832*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)]))))/(8*c - d*x^3)))/(13824*Sqrt[c + d*x^3])","B",0
440,1,266,66,0.2054863,"\int \frac{1}{x^3 \left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x^3*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{\frac{29 d^2 x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c^4}-\frac{4096 d x F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c \left(8 c-d x^3\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}-\frac{64 \left(c+d x^3\right) \left(29 d x^3-216 c\right)}{c^3 x^2 \left(d x^3-8 c\right)}}{221184 \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{2}{3};2,\frac{1}{2};\frac{1}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{128 c^2 x^2 \sqrt{c+d x^3}}",1,"((-64*(c + d*x^3)*(-216*c + 29*d*x^3))/(c^3*x^2*(-8*c + d*x^3)) + (29*d^2*x^4*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])/c^4 - (4096*d*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(c*(8*c - d*x^3)*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)]))))/(221184*Sqrt[c + d*x^3])","B",0
441,1,279,66,0.2944606,"\int \frac{1}{x^6 \left(8 c-d x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x^6*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{-\frac{119 d^3 x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c^5}+\frac{1404928 d^2 x F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c^2 \left(8 c-d x^3\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}+\frac{64 \left(c+d x^3\right) \left(864 c^2-1080 c d x^3+119 d^2 x^6\right)}{c^4 x^5 \left(d x^3-8 c\right)}}{2211840 \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{5}{3};2,\frac{1}{2};-\frac{2}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{320 c^2 x^5 \sqrt{c+d x^3}}",1,"((64*(c + d*x^3)*(864*c^2 - 1080*c*d*x^3 + 119*d^2*x^6))/(c^4*x^5*(-8*c + d*x^3)) - (119*d^3*x^4*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])/c^5 + (1404928*d^2*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(c^2*(8*c - d*x^3)*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)]))))/(2211840*Sqrt[c + d*x^3])","B",0
442,1,88,95,0.0459044,"\int \frac{x^{11}}{\left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^11/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{6 \left(752 c^2-198 c d x^3+9 d^2 x^6\right)-640 c \left(8 c-d x^3\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right)}{81 d^4 \left(d x^3-8 c\right) \sqrt{c+d x^3}}","\frac{2 \left(38 c+39 d x^3\right)}{81 d^4 \sqrt{c+d x^3}}-\frac{640 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{243 d^4}+\frac{8 x^6}{27 d^2 \left(8 c-d x^3\right) \sqrt{c+d x^3}}",1,"(6*(752*c^2 - 198*c*d*x^3 + 9*d^2*x^6) - 640*c*(8*c - d*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)])/(81*d^4*(-8*c + d*x^3)*Sqrt[c + d*x^3])","C",1
443,1,71,83,0.1054231,"\int \frac{x^8}{\left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^8/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{2 \left(\frac{3 \left(8 c+11 d x^3\right)}{\left(8 c-d x^3\right) \sqrt{c+d x^3}}-\frac{16 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{\sqrt{c}}\right)}{243 d^3}","\frac{64 c}{27 d^3 \left(8 c-d x^3\right) \sqrt{c+d x^3}}-\frac{22}{81 d^3 \sqrt{c+d x^3}}-\frac{32 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{243 \sqrt{c} d^3}",1,"(2*((3*(8*c + 11*d*x^3))/((8*c - d*x^3)*Sqrt[c + d*x^3]) - (16*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/Sqrt[c]))/(243*d^3)","A",1
444,1,68,85,0.0253746,"\int \frac{x^5}{\left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^5/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","-\frac{2 \left(\left(d x^3-8 c\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right)+12 c\right)}{81 c d^2 \left(d x^3-8 c\right) \sqrt{c+d x^3}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{243 c^{3/2} d^2}+\frac{8}{27 d^2 \left(8 c-d x^3\right) \sqrt{c+d x^3}}-\frac{2}{81 c d^2 \sqrt{c+d x^3}}",1,"(-2*(12*c + (-8*c + d*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)]))/(81*c*d^2*(-8*c + d*x^3)*Sqrt[c + d*x^3])","C",1
445,1,43,88,0.0129578,"\int \frac{x^2}{\left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^2/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","-\frac{2 \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{d x^3+c}{9 c}\right)}{243 c^2 d \sqrt{c+d x^3}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{243 c^{5/2} d}-\frac{1}{81 c^2 d \sqrt{c+d x^3}}+\frac{1}{27 c d \left(8 c-d x^3\right) \sqrt{c+d x^3}}",1,"(-2*Hypergeometric2F1[-1/2, 2, 1/2, (c + d*x^3)/(9*c)])/(243*c^2*d*Sqrt[c + d*x^3])","C",1
446,1,97,106,0.0450665,"\int \frac{1}{x \left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{\left(7 d x^3-56 c\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right)+27 \left(8 c-d x^3\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right)+12 c}{2592 c^3 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","\frac{7 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{7776 c^{7/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{96 c^{7/2}}+\frac{5}{648 c^3 \sqrt{c+d x^3}}+\frac{1}{216 c^2 \left(8 c-d x^3\right) \sqrt{c+d x^3}}",1,"(12*c + (-56*c + 7*d*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)] + 27*(8*c - d*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^3)/c])/(2592*c^3*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",1
447,1,117,143,0.0644763,"\int \frac{1}{x^4 \left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^4*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{5 d x^3 \left(d x^3-8 c\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right)+135 d x^3 \left(d x^3-8 c\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right)+12 c \left(5 d x^3-36 c\right)}{10368 c^4 x^3 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","\frac{5 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{31104 c^{9/2}}+\frac{5 d \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{384 c^{9/2}}-\frac{35 d}{2592 c^4 \sqrt{c+d x^3}}+\frac{5 d}{864 c^3 \left(8 c-d x^3\right) \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \left(8 c-d x^3\right) \sqrt{c+d x^3}}",1,"(12*c*(-36*c + 5*d*x^3) + 5*d*x^3*(-8*c + d*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)] + 135*d*x^3*(-8*c + d*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^3)/c])/(10368*c^4*x^3*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",1
448,1,135,185,0.0643327,"\int \frac{1}{x^7 \left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^7*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{13 d^2 x^6 \left(d x^3-8 c\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right)-3 \left(4 c \left(288 c^2-612 c d x^3+71 d^2 x^6\right)+891 d^2 x^6 \left(d x^3-8 c\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right)\right)}{165888 c^5 x^6 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","\frac{13 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{497664 c^{11/2}}-\frac{33 d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{2048 c^{11/2}}+\frac{665 d^2}{41472 c^5 \sqrt{c+d x^3}}-\frac{71 d^2}{13824 c^4 \left(8 c-d x^3\right) \sqrt{c+d x^3}}+\frac{17 d}{384 c^3 x^3 \left(8 c-d x^3\right) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left(8 c-d x^3\right) \sqrt{c+d x^3}}",1,"(13*d^2*x^6*(-8*c + d*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*x^3)/(9*c)] - 3*(4*c*(288*c^2 - 612*c*d*x^3 + 71*d^2*x^6) + 891*d^2*x^6*(-8*c + d*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^3)/c]))/(165888*c^5*x^6*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",1
449,1,168,668,0.1374766,"\int \frac{x^7}{\left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^7/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{d x^5 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+40 c x^2 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+80 c x^2 \left(4 c+d x^3\right)}{3240 c^2 d^2 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","\frac{4 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{81 \sqrt{3} c^{5/6} d^{8/3}}-\frac{4 \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{243 c^{5/6} d^{8/3}}+\frac{4 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{243 c^{5/6} d^{8/3}}+\frac{2 \sqrt{2} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{81 \sqrt[4]{3} c^{2/3} d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{27\ 3^{3/4} c^{2/3} d^{8/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{c+d x^3}}{81 c d^{8/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{8 x^2}{27 d^2 \left(8 c-d x^3\right) \sqrt{c+d x^3}}-\frac{2 x^2}{81 c d^2 \sqrt{c+d x^3}}",1,"(80*c*x^2*(4*c + d*x^3) + 40*c*x^2*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + d*x^5*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(3240*c^2*d^2*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
450,1,169,671,0.1305433,"\int \frac{x^4}{\left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^4/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{d x^5 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+50 c x^2 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+80 c x^2 \left(d x^3-5 c\right)}{6480 c^3 d \left(8 c-d x^3\right) \sqrt{c+d x^3}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{81 \sqrt{3} c^{11/6} d^{5/3}}+\frac{\tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{243 c^{11/6} d^{5/3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{243 c^{11/6} d^{5/3}}+\frac{\sqrt{2} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{81 \sqrt[4]{3} c^{5/3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{54\ 3^{3/4} c^{5/3} d^{5/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{\sqrt{c+d x^3}}{81 c^2 d^{5/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{x^2}{81 c^2 d \sqrt{c+d x^3}}+\frac{x^2}{27 c d \left(8 c-d x^3\right) \sqrt{c+d x^3}}",1,"(80*c*x^2*(-5*c + d*x^3) + 50*c*x^2*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + d*x^5*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(6480*c^3*d*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
451,1,167,665,0.1393771,"\int \frac{x}{\left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{d x^5 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+5 c x^2 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+16 c x^2 \left(43 c-5 d x^3\right)}{10368 c^4 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{1296 \sqrt{3} c^{17/6} d^{2/3}}+\frac{5 \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{3888 c^{17/6} d^{2/3}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{3888 c^{17/6} d^{2/3}}-\frac{5 \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{324 \sqrt{2} \sqrt[4]{3} c^{8/3} d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{5 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{432\ 3^{3/4} c^{8/3} d^{2/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{5 \sqrt{c+d x^3}}{648 c^3 d^{2/3} \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{5 x^2}{648 c^3 \sqrt{c+d x^3}}+\frac{x^2}{216 c^2 \left(8 c-d x^3\right) \sqrt{c+d x^3}}",1,"(16*c*x^2*(43*c - 5*d*x^3) + 5*c*x^2*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + d*x^5*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(10368*c^4*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
452,1,180,686,0.1440533,"\int \frac{1}{x^2 \left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^2*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{31 d^2 x^6 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+650 c d x^3 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-80 c \left(162 c^2+227 c d x^3-31 d^2 x^6\right)}{103680 c^5 \sqrt{c+d x^3} \left(8 c x-d x^4\right)}","\frac{31 \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{648 \sqrt{2} \sqrt[4]{3} c^{11/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{31 \sqrt{2-\sqrt{3}} \sqrt[3]{d} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{864\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{1296 \sqrt{3} c^{23/6}}+\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{3888 c^{23/6}}-\frac{\sqrt[3]{d} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{3888 c^{23/6}}-\frac{31 \sqrt{c+d x^3}}{1296 c^4 x}+\frac{31 \sqrt[3]{d} \sqrt{c+d x^3}}{1296 c^4 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{5}{648 c^3 x \sqrt{c+d x^3}}+\frac{1}{216 c^2 x \left(8 c-d x^3\right) \sqrt{c+d x^3}}",1,"(-80*c*(162*c^2 + 227*c*d*x^3 - 31*d^2*x^6) + 650*c*d*x^3*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 31*d^2*x^6*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(103680*c^5*Sqrt[c + d*x^3]*(8*c*x - d*x^4))","C",0
453,1,198,708,0.198391,"\int \frac{1}{x^5 \left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^5*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{-16 \left(77 d^3 x^9 \left(d x^3-8 c\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+10 c \left(648 c^3-2997 c^2 d x^3-4565 c d^2 x^6+616 d^3 x^9\right)\right)-24475 c d^2 x^6 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{3317760 c^6 x^4 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","-\frac{11 d^{4/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{82944 \sqrt{3} c^{29/6}}+\frac{11 d^{4/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{248832 c^{29/6}}-\frac{11 d^{4/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{248832 c^{29/6}}-\frac{77 d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{1296 \sqrt{2} \sqrt[4]{3} c^{14/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{77 \sqrt{2-\sqrt{3}} d^{4/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{1728\ 3^{3/4} c^{14/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{77 d^{4/3} \sqrt{c+d x^3}}{2592 c^5 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}+\frac{77 d \sqrt{c+d x^3}}{2592 c^5 x}-\frac{253 \sqrt{c+d x^3}}{20736 c^4 x^4}+\frac{5}{648 c^3 x^4 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^4 \left(8 c-d x^3\right) \sqrt{c+d x^3}}",1,"(-24475*c*d^2*x^6*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 16*(10*c*(648*c^3 - 2997*c^2*d*x^3 - 4565*c*d^2*x^6 + 616*d^3*x^9) + 77*d^3*x^9*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(3317760*c^6*x^4*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
454,1,210,732,0.2340128,"\int \frac{1}{x^8 \left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^8*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{829375 c d^3 x^9 \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-8 \left(5179 d^4 x^{12} \left(8 c-d x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)+20 c \left(10368 c^4-18792 c^3 d x^3+101817 c^2 d^2 x^6+153269 c d^3 x^9-20716 d^4 x^{12}\right)\right)}{92897280 c^7 x^7 \left(8 c-d x^3\right) \sqrt{c+d x^3}}","-\frac{7 d^{7/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\sqrt{c+d x^3}}\right)}{331776 \sqrt{3} c^{35/6}}+\frac{7 d^{7/3} \tanh ^{-1}\left(\frac{\left(\sqrt[3]{c}+\sqrt[3]{d} x\right)^2}{3 \sqrt[6]{c} \sqrt{c+d x^3}}\right)}{995328 c^{35/6}}-\frac{7 d^{7/3} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right)}{995328 c^{35/6}}+\frac{5179 d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{72576 \sqrt{2} \sqrt[4]{3} c^{17/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}-\frac{5179 \sqrt{2-\sqrt{3}} d^{7/3} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{96768\ 3^{3/4} c^{17/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}+\frac{5179 d^{7/3} \sqrt{c+d x^3}}{145152 c^6 \left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)}-\frac{5179 d^2 \sqrt{c+d x^3}}{145152 c^6 x}+\frac{8257 d \sqrt{c+d x^3}}{580608 c^5 x^4}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^7 \left(8 c-d x^3\right) \sqrt{c+d x^3}}",1,"(829375*c*d^3*x^9*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 8*(20*c*(10368*c^4 - 18792*c^3*d*x^3 + 101817*c^2*d^2*x^6 + 153269*c*d^3*x^9 - 20716*d^4*x^12) + 5179*d^4*x^12*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(92897280*c^7*x^7*(8*c - d*x^3)*Sqrt[c + d*x^3])","C",0
455,1,189,256,0.6731675,"\int \frac{x^6}{\left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^6/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","-\frac{6 \sqrt[3]{-d} x \left(4 c+d x^3\right)+2 i 3^{3/4} \sqrt[3]{c} \sqrt{\frac{(-1)^{5/6} \left(\sqrt[3]{-d} x-\sqrt[3]{c}\right)}{\sqrt[3]{c}}} \sqrt{\frac{(-d)^{2/3} x^2}{c^{2/3}}+\frac{\sqrt[3]{-d} x}{\sqrt[3]{c}}+1} \left(d x^3-8 c\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-\frac{i \sqrt[3]{-d} x}{\sqrt[3]{c}}-(-1)^{5/6}}}{\sqrt[4]{3}}\right)|\sqrt[3]{-1}\right)}{243 c (-d)^{7/3} \left(d x^3-8 c\right) \sqrt{c+d x^3}}","\frac{2 x \left(4 c+d x^3\right)}{81 c d^2 \left(8 c-d x^3\right) \sqrt{c+d x^3}}-\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right) \sqrt{\frac{c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{81 \sqrt[4]{3} c d^{7/3} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{c}+\sqrt[3]{d} x\right)^2}} \sqrt{c+d x^3}}",1,"-1/243*(6*(-d)^(1/3)*x*(4*c + d*x^3) + (2*I)*3^(3/4)*c^(1/3)*Sqrt[((-1)^(5/6)*(-c^(1/3) + (-d)^(1/3)*x))/c^(1/3)]*Sqrt[1 + ((-d)^(1/3)*x)/c^(1/3) + ((-d)^(2/3)*x^2)/c^(2/3)]*(-8*c + d*x^3)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-d)^(1/3)*x)/c^(1/3)]/3^(1/4)], (-1)^(1/3)])/(c*(-d)^(7/3)*(-8*c + d*x^3)*Sqrt[c + d*x^3])","C",0
456,1,242,66,0.3170425,"\int \frac{x^3}{\left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^3/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{x \left(\frac{3 x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c^3}+\frac{192 \left(\frac{160 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}+\frac{d x^3-5 c}{c^2}\right)}{d \left(8 c-d x^3\right)}\right)}{15552 \sqrt{c+d x^3}}","\frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};2,\frac{3}{2};\frac{7}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{256 c^3 \sqrt{c+d x^3}}",1,"(x*((3*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])/c^3 + (192*((-5*c + d*x^3)/c^2 + (160*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)]))))/(d*(8*c - d*x^3))))/(15552*Sqrt[c + d*x^3])","B",0
457,1,253,64,0.3324323,"\int \frac{1}{\left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{x \left(192 c \left(\frac{1216 c^2 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{\left(8 c-d x^3\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}+\frac{5 d x^3-43 c}{d x^3-8 c}\right)-15 d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}{124416 c^4 \sqrt{c+d x^3}}","\frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{1}{3};2,\frac{3}{2};\frac{4}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{64 c^3 \sqrt{c+d x^3}}",1,"(x*(-15*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 192*c*((-43*c + 5*d*x^3)/(-8*c + d*x^3) + (1216*c^2*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x^3)*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)]))))))/(124416*c^4*Sqrt[c + d*x^3])","B",0
458,1,259,66,0.2119584,"\int \frac{1}{x^3 \left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^3*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{\frac{64 c \left(-\frac{19648 c^2 d x^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}-648 c^2-1249 c d x^3+167 d^2 x^6\right)}{8 c-d x^3}+167 d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{663552 c^5 x^2 \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{2}{3};2,\frac{3}{2};\frac{1}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{128 c^3 x^2 \sqrt{c+d x^3}}",1,"(167*d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] + (64*c*(-648*c^2 - 1249*c*d*x^3 + 167*d^2*x^6 - (19648*c^2*d*x^3*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)]))))/(8*c - d*x^3))/(663552*c^5*x^2*Sqrt[c + d*x^3])","B",0
459,1,283,66,0.2842172,"\int \frac{1}{x^6 \left(8 c-d x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^6*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{-\frac{2027 d^3 x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c^6}+\frac{16789504 d^2 x F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)}{c^3 \left(8 c-d x^3\right) \left(3 d x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)-4 F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)+32 c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right)\right)}+\frac{64 \left(2592 c^3-7128 c^2 d x^3-15373 c d^2 x^6+2027 d^3 x^9\right)}{c^5 x^5 \left(d x^3-8 c\right)}}{6635520 \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{5}{3};2,\frac{3}{2};-\frac{2}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right)}{320 c^3 x^5 \sqrt{c+d x^3}}",1,"((64*(2592*c^3 - 7128*c^2*d*x^3 - 15373*c*d^2*x^6 + 2027*d^3*x^9))/(c^5*x^5*(-8*c + d*x^3)) - (2027*d^3*x^4*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])/c^6 + (16789504*d^2*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(c^3*(8*c - d*x^3)*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)]))))/(6635520*Sqrt[c + d*x^3])","B",0
460,1,147,161,0.2908132,"\int \frac{x^8 \sqrt{c+d x^3}}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^8*Sqrt[c + d*x^3])/(a + b*x^3)^2,x]","\frac{-\frac{a^2 \left(c+d x^3\right)^{3/2}}{a+b x^3}+\frac{a (5 a d-4 b c) \left(\sqrt{b} \sqrt{c+d x^3}-\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)\right)}{b^{3/2}}+\frac{2 \left(c+d x^3\right)^{3/2} (b c-a d)}{3 d}}{3 b^2 (b c-a d)}","-\frac{a^2 \left(c+d x^3\right)^{3/2}}{3 b^2 \left(a+b x^3\right) (b c-a d)}+\frac{a (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{7/2} \sqrt{b c-a d}}-\frac{a \sqrt{c+d x^3} (4 b c-5 a d)}{3 b^3 (b c-a d)}+\frac{2 \left(c+d x^3\right)^{3/2}}{9 b^2 d}",1,"((2*(b*c - a*d)*(c + d*x^3)^(3/2))/(3*d) - (a^2*(c + d*x^3)^(3/2))/(a + b*x^3) + (a*(-4*b*c + 5*a*d)*(Sqrt[b]*Sqrt[c + d*x^3] - Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]]))/b^(3/2))/(3*b^2*(b*c - a*d))","A",1
461,1,117,136,0.1535152,"\int \frac{x^5 \sqrt{c+d x^3}}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^5*Sqrt[c + d*x^3])/(a + b*x^3)^2,x]","\frac{\frac{(2 b c-3 a d) \left(\sqrt{b} \sqrt{c+d x^3}-\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)\right)}{b^{3/2}}+\frac{a \left(c+d x^3\right)^{3/2}}{a+b x^3}}{3 b (b c-a d)}","-\frac{(2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^3} (2 b c-3 a d)}{3 b^2 (b c-a d)}+\frac{a \left(c+d x^3\right)^{3/2}}{3 b \left(a+b x^3\right) (b c-a d)}",1,"((a*(c + d*x^3)^(3/2))/(a + b*x^3) + ((2*b*c - 3*a*d)*(Sqrt[b]*Sqrt[c + d*x^3] - Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]]))/b^(3/2))/(3*b*(b*c - a*d))","A",1
462,1,80,80,0.114097,"\int \frac{x^2 \sqrt{c+d x^3}}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^2*Sqrt[c + d*x^3])/(a + b*x^3)^2,x]","\frac{d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{a d-b c}}\right)}{3 b^{3/2} \sqrt{a d-b c}}-\frac{\sqrt{c+d x^3}}{3 b \left(a+b x^3\right)}","-\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^3}}{3 b \left(a+b x^3\right)}",1,"-1/3*Sqrt[c + d*x^3]/(b*(a + b*x^3)) + (d*ArcTan[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[-(b*c) + a*d]])/(3*b^(3/2)*Sqrt[-(b*c) + a*d])","A",1
463,1,112,121,0.2941495,"\int \frac{\sqrt{c+d x^3}}{x \left(a+b x^3\right)^2} \, dx","Integrate[Sqrt[c + d*x^3]/(x*(a + b*x^3)^2),x]","\frac{\frac{a \sqrt{c+d x^3}}{a+b x^3}+\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{\sqrt{b} \sqrt{b c-a d}}-2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^2}","\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^2 \sqrt{b} \sqrt{b c-a d}}-\frac{2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^2}+\frac{\sqrt{c+d x^3}}{3 a \left(a+b x^3\right)}",1,"((a*Sqrt[c + d*x^3])/(a + b*x^3) - 2*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]] + ((2*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(Sqrt[b]*Sqrt[b*c - a*d]))/(3*a^2)","A",1
464,1,190,161,0.2825139,"\int \frac{\sqrt{c+d x^3}}{x^4 \left(a+b x^3\right)^2} \, dx","Integrate[Sqrt[c + d*x^3]/(x^4*(a + b*x^3)^2),x]","\frac{\sqrt{c} \left(a \left(a+2 b x^3\right) \sqrt{c+d x^3} (b c-a d)+\sqrt{b} x^3 \left(a+b x^3\right) (4 b c-3 a d) \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)\right)-x^3 \left(a+b x^3\right) \left(a^2 d^2-5 a b c d+4 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^3 \sqrt{c} x^3 \left(a+b x^3\right) (a d-b c)}","-\frac{\sqrt{b} (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^3 \sqrt{b c-a d}}+\frac{(4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^3 \sqrt{c}}-\frac{2 b \sqrt{c+d x^3}}{3 a^2 \left(a+b x^3\right)}-\frac{\sqrt{c+d x^3}}{3 a x^3 \left(a+b x^3\right)}",1,"(-((4*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*x^3*(a + b*x^3)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]]) + Sqrt[c]*(a*(b*c - a*d)*(a + 2*b*x^3)*Sqrt[c + d*x^3] + Sqrt[b]*(4*b*c - 3*a*d)*Sqrt[b*c - a*d]*x^3*(a + b*x^3)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]]))/(3*a^3*Sqrt[c]*(-(b*c) + a*d)*x^3*(a + b*x^3))","A",1
465,1,235,64,0.271666,"\int \frac{x^3 \sqrt{c+d x^3}}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^3*Sqrt[c + d*x^3])/(a + b*x^3)^2,x]","\frac{x \left(\frac{8 \left(\frac{8 a c^2 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}-c-d x^3\right)}{a+b x^3}+\frac{5 d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{a}\right)}{24 b \sqrt{c+d x^3}}","\frac{x^4 \sqrt{c+d x^3} F_1\left(\frac{4}{3};2,-\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 a^2 \sqrt{\frac{d x^3}{c}+1}}",1,"(x*((5*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/a + (8*(-c - d*x^3 + (8*a*c^2*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)])/(8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(a + b*x^3)))/(24*b*Sqrt[c + d*x^3])","B",0
466,1,153,64,0.1302868,"\int \frac{x \sqrt{c+d x^3}}{\left(a+b x^3\right)^2} \, dx","Integrate[(x*Sqrt[c + d*x^3])/(a + b*x^3)^2,x]","\frac{-d x^5 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+5 c x^2 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+10 a x^2 \left(c+d x^3\right)}{30 a^2 \left(a+b x^3\right) \sqrt{c+d x^3}}","\frac{x^2 \sqrt{c+d x^3} F_1\left(\frac{2}{3};2,-\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a^2 \sqrt{\frac{d x^3}{c}+1}}",1,"(10*a*x^2*(c + d*x^3) + 5*c*x^2*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] - d*x^5*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/(30*a^2*(a + b*x^3)*Sqrt[c + d*x^3])","B",0
467,1,232,59,0.25085,"\int \frac{\sqrt{c+d x^3}}{\left(a+b x^3\right)^2} \, dx","Integrate[Sqrt[c + d*x^3]/(a + b*x^3)^2,x]","\frac{x \left(\frac{d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{a^2}+\frac{8 \left(\frac{16 c^2 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}+\frac{c+d x^3}{a}\right)}{a+b x^3}\right)}{24 \sqrt{c+d x^3}}","\frac{x \sqrt{c+d x^3} F_1\left(\frac{1}{3};2,-\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2 \sqrt{\frac{d x^3}{c}+1}}",1,"(x*((d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/a^2 + (8*((c + d*x^3)/a + (16*c^2*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)])/(8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(a + b*x^3)))/(24*Sqrt[c + d*x^3])","B",0
468,1,172,62,0.1629106,"\int \frac{\sqrt{c+d x^3}}{x^2 \left(a+b x^3\right)^2} \, dx","Integrate[Sqrt[c + d*x^3]/(x^2*(a + b*x^3)^2),x]","\frac{5 x^3 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} (9 a d-8 b c) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+8 b d x^6 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-20 a \left(3 a+4 b x^3\right) \left(c+d x^3\right)}{60 a^3 x \left(a+b x^3\right) \sqrt{c+d x^3}}","-\frac{\sqrt{c+d x^3} F_1\left(-\frac{1}{3};2,-\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2 x \sqrt{\frac{d x^3}{c}+1}}",1,"(-20*a*(3*a + 4*b*x^3)*(c + d*x^3) + 5*(-8*b*c + 9*a*d)*x^3*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 8*b*d*x^6*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/(60*a^3*x*(a + b*x^3)*Sqrt[c + d*x^3])","B",0
469,1,338,64,0.3585037,"\int \frac{\sqrt{c+d x^3}}{x^3 \left(a+b x^3\right)^2} \, dx","Integrate[Sqrt[c + d*x^3]/(x^3*(a + b*x^3)^2),x]","\frac{\frac{a \left(32 a c \left(6 a c-3 a d x^3+30 b c x^3+10 b d x^6\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-24 x^3 \left(3 a+5 b x^3\right) \left(c+d x^3\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}-5 b d x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{48 a^3 x^2 \sqrt{c+d x^3}}","-\frac{\sqrt{c+d x^3} F_1\left(-\frac{2}{3};2,-\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a^2 x^2 \sqrt{\frac{d x^3}{c}+1}}",1,"(-5*b*d*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + (a*(32*a*c*(6*a*c + 30*b*c*x^3 - 3*a*d*x^3 + 10*b*d*x^6)*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 24*x^3*(3*a + 5*b*x^3)*(c + d*x^3)*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(48*a^3*x^2*Sqrt[c + d*x^3])","B",0
470,1,162,189,0.1853411,"\int \frac{x^8 \left(c+d x^3\right)^{3/2}}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^8*(c + d*x^3)^(3/2))/(a + b*x^3)^2,x]","\frac{\sqrt{c+d x^3} \left(105 a^3 d^2+5 a^2 b d \left(14 d x^3-19 c\right)+2 a b^2 \left(3 c^2-34 c d x^3-7 d^2 x^6\right)+6 b^3 x^3 \left(c+d x^3\right)^2\right)}{45 b^4 d \left(a+b x^3\right)}+\frac{a (4 b c-7 a d) \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{9/2}}","-\frac{a^2 \left(c+d x^3\right)^{5/2}}{3 b^2 \left(a+b x^3\right) (b c-a d)}+\frac{a (4 b c-7 a d) \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{9/2}}-\frac{a \sqrt{c+d x^3} (4 b c-7 a d)}{3 b^4}-\frac{a \left(c+d x^3\right)^{3/2} (4 b c-7 a d)}{9 b^3 (b c-a d)}+\frac{2 \left(c+d x^3\right)^{5/2}}{15 b^2 d}",1,"(Sqrt[c + d*x^3]*(105*a^3*d^2 + 6*b^3*x^3*(c + d*x^3)^2 + 5*a^2*b*d*(-19*c + 14*d*x^3) + 2*a*b^2*(3*c^2 - 34*c*d*x^3 - 7*d^2*x^6)))/(45*b^4*d*(a + b*x^3)) + (a*(4*b*c - 7*a*d)*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(9/2))","A",1
471,1,125,163,0.1359271,"\int \frac{x^5 \left(c+d x^3\right)^{3/2}}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^5*(c + d*x^3)^(3/2))/(a + b*x^3)^2,x]","\frac{\sqrt{c+d x^3} \left(-15 a^2 d+a b \left(11 c-10 d x^3\right)+2 b^2 x^3 \left(4 c+d x^3\right)\right)}{9 b^3 \left(a+b x^3\right)}-\frac{(2 b c-5 a d) \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{7/2}}","-\frac{(2 b c-5 a d) \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{7/2}}+\frac{\sqrt{c+d x^3} (2 b c-5 a d)}{3 b^3}+\frac{\left(c+d x^3\right)^{3/2} (2 b c-5 a d)}{9 b^2 (b c-a d)}+\frac{a \left(c+d x^3\right)^{5/2}}{3 b \left(a+b x^3\right) (b c-a d)}",1,"(Sqrt[c + d*x^3]*(-15*a^2*d + a*b*(11*c - 10*d*x^3) + 2*b^2*x^3*(4*c + d*x^3)))/(9*b^3*(a + b*x^3)) - ((2*b*c - 5*a*d)*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(7/2))","A",1
472,1,54,94,0.0236637,"\int \frac{x^2 \left(c+d x^3\right)^{3/2}}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^2*(c + d*x^3)^(3/2))/(a + b*x^3)^2,x]","\frac{2 d \left(c+d x^3\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\frac{b \left(d x^3+c\right)}{a d-b c}\right)}{15 (a d-b c)^2}","-\frac{d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{b^{5/2}}-\frac{\left(c+d x^3\right)^{3/2}}{3 b \left(a+b x^3\right)}+\frac{d \sqrt{c+d x^3}}{b^2}",1,"(2*d*(c + d*x^3)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, -((b*(c + d*x^3))/(-(b*c) + a*d))])/(15*(-(b*c) + a*d)^2)","C",1
473,1,122,131,0.1862921,"\int \frac{\left(c+d x^3\right)^{3/2}}{x \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3)^(3/2)/(x*(a + b*x^3)^2),x]","\frac{\frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{b^{3/2}}+\frac{a \sqrt{c+d x^3} (b c-a d)}{b \left(a+b x^3\right)}-2 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^2}","\frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^2 b^{3/2}}-\frac{2 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^2}+\frac{\sqrt{c+d x^3} (b c-a d)}{3 a b \left(a+b x^3\right)}",1,"((a*(b*c - a*d)*Sqrt[c + d*x^3])/(b*(a + b*x^3)) - 2*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]] + (Sqrt[b*c - a*d]*(2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/b^(3/2))/(3*a^2)","A",1
474,1,142,170,0.3327881,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^4 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^4*(a + b*x^3)^2),x]","\frac{\frac{a \sqrt{c+d x^3} \left(-a c+a d x^3-2 b c x^3\right)}{x^3 \left(a+b x^3\right)}+\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{\sqrt{b}}}{3 a^3}","\frac{\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^3}-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^3 \sqrt{b}}-\frac{\sqrt{c+d x^3} (2 b c-a d)}{3 a^2 \left(a+b x^3\right)}-\frac{c \sqrt{c+d x^3}}{3 a x^3 \left(a+b x^3\right)}",1,"((a*Sqrt[c + d*x^3]*(-(a*c) - 2*b*c*x^3 + a*d*x^3))/(x^3*(a + b*x^3)) + Sqrt[c]*(4*b*c - 3*a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]] - (Sqrt[b*c - a*d]*(4*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/Sqrt[b])/(3*a^3)","A",1
475,1,338,65,0.4792591,"\int \frac{x^3 \left(c+d x^3\right)^{3/2}}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^3*(c + d*x^3)^(3/2))/(a + b*x^3)^2,x]","\frac{x^4 \left(\frac{d \sqrt{\frac{d x^3}{c}+1} (43 b c-55 a d) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{a}+\frac{8 \left(-8 a c d \left(11 a d+b \left(c+6 d x^3\right)\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-3 \left(c+d x^3\right) \left(-11 a d+5 b c-6 b d x^3\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}\right)}{120 b^2 \sqrt{c+d x^3}}","\frac{c x^4 \sqrt{c+d x^3} F_1\left(\frac{4}{3};2,-\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 a^2 \sqrt{\frac{d x^3}{c}+1}}",1,"(x^4*((d*(43*b*c - 55*a*d)*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/a + (8*(-8*a*c*d*(11*a*d + b*(c + 6*d*x^3))*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 3*(c + d*x^3)*(5*b*c - 11*a*d - 6*b*d*x^3)*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))))/(120*b^2*Sqrt[c + d*x^3])","B",0
476,1,177,65,0.2625361,"\int \frac{x \left(c+d x^3\right)^{3/2}}{\left(a+b x^3\right)^2} \, dx","Integrate[(x*(c + d*x^3)^(3/2))/(a + b*x^3)^2,x]","\frac{x^2 \left(-d x^3 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} (b c-7 a d) F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+5 c \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} (2 a d+b c) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-10 a \left(c+d x^3\right) (a d-b c)\right)}{30 a^2 b \left(a+b x^3\right) \sqrt{c+d x^3}}","\frac{c x^2 \sqrt{c+d x^3} F_1\left(\frac{2}{3};2,-\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a^2 \sqrt{\frac{d x^3}{c}+1}}",1,"(x^2*(-10*a*(-(b*c) + a*d)*(c + d*x^3) + 5*c*(b*c + 2*a*d)*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] - d*(b*c - 7*a*d)*x^3*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)]))/(30*a^2*b*(a + b*x^3)*Sqrt[c + d*x^3])","B",0
477,1,339,60,0.4255877,"\int \frac{\left(c+d x^3\right)^{3/2}}{\left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3)^(3/2)/(a + b*x^3)^2,x]","\frac{x \left(\frac{a \left(24 x^3 \left(c+d x^3\right) (b c-a d) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-64 a c \left(b c \left(3 c+d x^3\right)-a d^2 x^3\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}+d x^3 \sqrt{\frac{d x^3}{c}+1} (5 a d+b c) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}{24 a^2 b \sqrt{c+d x^3}}","\frac{c x \sqrt{c+d x^3} F_1\left(\frac{1}{3};2,-\frac{3}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2 \sqrt{\frac{d x^3}{c}+1}}",1,"(x*(d*(b*c + 5*a*d)*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + (a*(-64*a*c*(-(a*d^2*x^3) + b*c*(3*c + d*x^3))*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 24*(b*c - a*d)*x^3*(c + d*x^3)*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))))/(24*a^2*b*Sqrt[c + d*x^3])","B",0
478,1,190,63,0.2060823,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^2 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^2*(a + b*x^3)^2),x]","\frac{5 c x^3 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} (11 a d-8 b c) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+2 d x^6 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} (4 b c-a d) F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-20 a \left(c+d x^3\right) \left(3 a c-a d x^3+4 b c x^3\right)}{60 a^3 x \left(a+b x^3\right) \sqrt{c+d x^3}}","-\frac{c \sqrt{c+d x^3} F_1\left(-\frac{1}{3};2,-\frac{3}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2 x \sqrt{\frac{d x^3}{c}+1}}",1,"(-20*a*(c + d*x^3)*(3*a*c + 4*b*c*x^3 - a*d*x^3) + 5*c*(-8*b*c + 11*a*d)*x^3*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 2*d*(4*b*c - a*d)*x^6*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/(60*a^3*x*(a + b*x^3)*Sqrt[c + d*x^3])","B",0
479,1,370,65,0.3782826,"\int \frac{\left(c+d x^3\right)^{3/2}}{x^3 \left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3)^(3/2)/(x^3*(a + b*x^3)^2),x]","\frac{\frac{8 a \left(4 a c \left(a \left(6 c^2-15 c d x^3-4 d^2 x^6\right)+10 b c x^3 \left(3 c+d x^3\right)\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-3 x^3 \left(c+d x^3\right) \left(3 a c-2 a d x^3+5 b c x^3\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}-d x^6 \sqrt{\frac{d x^3}{c}+1} (5 b c-2 a d) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{48 a^3 x^2 \sqrt{c+d x^3}}","-\frac{c \sqrt{c+d x^3} F_1\left(-\frac{2}{3};2,-\frac{3}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a^2 x^2 \sqrt{\frac{d x^3}{c}+1}}",1,"(-(d*(5*b*c - 2*a*d)*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]) + (8*a*(4*a*c*(10*b*c*x^3*(3*c + d*x^3) + a*(6*c^2 - 15*c*d*x^3 - 4*d^2*x^6))*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 3*x^3*(c + d*x^3)*(3*a*c + 5*b*c*x^3 - 2*a*d*x^3)*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(48*a^3*x^2*Sqrt[c + d*x^3])","B",0
480,1,107,123,0.2756654,"\int \frac{x^8}{\left(a+b x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^8/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{1}{3} \left(\frac{\sqrt{c+d x^3} \left(\frac{a^2}{\left(a+b x^3\right) (a d-b c)}+\frac{2}{d}\right)}{b^2}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{b^{5/2} (b c-a d)^{3/2}}\right)","-\frac{a^2 \sqrt{c+d x^3}}{3 b^2 \left(a+b x^3\right) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{5/2} (b c-a d)^{3/2}}+\frac{2 \sqrt{c+d x^3}}{3 b^2 d}",1,"((Sqrt[c + d*x^3]*(2/d + a^2/((-(b*c) + a*d)*(a + b*x^3))))/b^2 + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(3/2)))/3","A",1
481,1,98,99,0.0970332,"\int \frac{x^5}{\left(a+b x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^5/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{\frac{a \sqrt{b} \sqrt{c+d x^3}}{\left(a+b x^3\right) (b c-a d)}+\frac{(a d-2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}}{3 b^{3/2}}","\frac{a \sqrt{c+d x^3}}{3 b \left(a+b x^3\right) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{3/2} (b c-a d)^{3/2}}",1,"((a*Sqrt[b]*Sqrt[c + d*x^3])/((b*c - a*d)*(a + b*x^3)) + ((-2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(3*b^(3/2))","A",1
482,1,85,87,0.1036732,"\int \frac{x^2}{\left(a+b x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^2/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{1}{3} \left(\frac{\sqrt{c+d x^3}}{\left(a+b x^3\right) (a d-b c)}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{a d-b c}}\right)}{\sqrt{b} (a d-b c)^{3/2}}\right)","\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^3}}{3 \left(a+b x^3\right) (b c-a d)}",1,"(Sqrt[c + d*x^3]/((-(b*c) + a*d)*(a + b*x^3)) + (d*ArcTan[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[-(b*c) + a*d]])/(Sqrt[b]*(-(b*c) + a*d)^(3/2)))/3","A",1
483,1,123,132,0.2647668,"\int \frac{1}{x \left(a+b x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x*(a + b*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{\frac{a b \sqrt{c+d x^3}}{\left(a+b x^3\right) (b c-a d)}+\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{\sqrt{c}}}{3 a^2}","\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^2 (b c-a d)^{3/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^3}}{3 a \left(a+b x^3\right) (b c-a d)}",1,"((a*b*Sqrt[c + d*x^3])/((b*c - a*d)*(a + b*x^3)) - (2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/Sqrt[c] + (Sqrt[b]*(2*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(3*a^2)","A",1
484,1,163,185,0.6112586,"\int \frac{1}{x^4 \left(a+b x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x^4*(a + b*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{\frac{a \sqrt{c+d x^3} \left(a^2 d+a b \left(d x^3-c\right)-2 b^2 c x^3\right)}{x^3 \left(a+b x^3\right) (b c-a d)}+\frac{b^{3/2} c (5 a d-4 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{\sqrt{c}}}{3 a^3 c}","-\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^3} (2 b c-a d)}{3 a^2 c \left(a+b x^3\right) (b c-a d)}-\frac{\sqrt{c+d x^3}}{3 a c x^3 \left(a+b x^3\right)}",1,"((a*Sqrt[c + d*x^3]*(a^2*d - 2*b^2*c*x^3 + a*b*(-c + d*x^3)))/((b*c - a*d)*x^3*(a + b*x^3)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/Sqrt[c] + (b^(3/2)*c*(-4*b*c + 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(3*a^3*c)","A",1
485,1,238,64,0.2094643,"\int \frac{x^3}{\left(a+b x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x^3/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{x \left(\frac{8 \left(\frac{8 a c^2 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}+c+d x^3\right)}{a+b x^3}+\frac{d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{a}\right)}{24 \sqrt{c+d x^3} (a d-b c)}","\frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};2,\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 a^2 \sqrt{c+d x^3}}",1,"(x*((d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/a + (8*(c + d*x^3 + (8*a*c^2*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)])/(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(a + b*x^3)))/(24*(-(b*c) + a*d)*Sqrt[c + d*x^3])","B",0
486,1,172,64,0.1633699,"\int \frac{x}{\left(a+b x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[x/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{-b d x^5 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+5 x^2 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} (b c-3 a d) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+10 a b x^2 \left(c+d x^3\right)}{30 a^2 \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)}","\frac{x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a^2 \sqrt{c+d x^3}}",1,"(10*a*b*x^2*(c + d*x^3) + 5*(b*c - 3*a*d)*x^2*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] - b*d*x^5*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/(30*a^2*(b*c - a*d)*(a + b*x^3)*Sqrt[c + d*x^3])","B",0
487,1,392,59,0.3065839,"\int \frac{1}{\left(a+b x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{3 b x^4 \left(d x^3 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+8 a \left(c+d x^3\right)\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c x F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right) \left(b d x^3 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+8 a \left(-3 a d+3 b c+b d x^3\right)\right)}{24 a^2 \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}","\frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{1}{3};2,\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2 \sqrt{c+d x^3}}",1,"(-8*a*c*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)]*(8*a*(3*b*c - 3*a*d + b*d*x^3) + b*d*x^3*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]) + 3*b*x^4*(8*a*(c + d*x^3) + d*x^3*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))/(24*a^2*(b*c - a*d)*(a + b*x^3)*Sqrt[c + d*x^3]*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))","B",0
488,1,226,62,0.2547302,"\int \frac{1}{x^2 \left(a+b x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x^2*(a + b*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{-5 x^3 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} \left(3 a^2 d^2-15 a b c d+8 b^2 c^2\right) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+20 a \left(c+d x^3\right) \left(3 a^2 d-3 a b \left(c-d x^3\right)-4 b^2 c x^3\right)+2 b d x^6 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} (4 b c-3 a d) F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{60 a^3 c x \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{1}{3};2,\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2 x \sqrt{c+d x^3}}",1,"(20*a*(c + d*x^3)*(3*a^2*d - 4*b^2*c*x^3 - 3*a*b*(c - d*x^3)) - 5*(8*b^2*c^2 - 15*a*b*c*d + 3*a^2*d^2)*x^3*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 2*b*d*(4*b*c - 3*a*d)*x^6*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/(60*a^3*c*(b*c - a*d)*x*(a + b*x^3)*Sqrt[c + d*x^3])","B",0
489,1,411,64,0.5610315,"\int \frac{1}{x^3 \left(a+b x^3\right)^2 \sqrt{c+d x^3}} \, dx","Integrate[1/(x^3*(a + b*x^3)^2*Sqrt[c + d*x^3]),x]","\frac{\frac{a \left(32 a c \left(3 a^2 d \left(2 c+3 d x^3\right)+3 a b \left(-2 c^2+7 c d x^3+2 d^2 x^6\right)-10 b^2 c x^3 \left(3 c+d x^3\right)\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+24 x^3 \left(c+d x^3\right) \left(-3 a^2 d+3 a b \left(c-d x^3\right)+5 b^2 c x^3\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}+b d x^6 \sqrt{\frac{d x^3}{c}+1} (5 b c-3 a d) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{48 a^3 c x^2 \sqrt{c+d x^3} (a d-b c)}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{2}{3};2,\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a^2 x^2 \sqrt{c+d x^3}}",1,"(b*d*(5*b*c - 3*a*d)*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + (a*(32*a*c*(-10*b^2*c*x^3*(3*c + d*x^3) + 3*a^2*d*(2*c + 3*d*x^3) + 3*a*b*(-2*c^2 + 7*c*d*x^3 + 2*d^2*x^6))*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 24*x^3*(c + d*x^3)*(-3*a^2*d + 5*b^2*c*x^3 + 3*a*b*(c - d*x^3))*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(48*a^3*c*(-(b*c) + a*d)*x^2*Sqrt[c + d*x^3])","B",0
490,1,134,150,0.2981765,"\int \frac{x^8}{\left(a+b x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^8/((a + b*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{\frac{a (4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{5/2}}-\frac{\sqrt{b} \left(a^2 d \left(c+d x^3\right)+2 a b c^2+2 b^2 c^2 x^3\right)}{d \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)^2}}{3 b^{3/2}}","\frac{-a^2 d^2-2 b^2 c^2}{3 b^2 d \sqrt{c+d x^3} (b c-a d)^2}-\frac{a^2}{3 b^2 \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)}+\frac{a (4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 b^{3/2} (b c-a d)^{5/2}}",1,"(-((Sqrt[b]*(2*a*b*c^2 + 2*b^2*c^2*x^3 + a^2*d*(c + d*x^3)))/(d*(b*c - a*d)^2*(a + b*x^3)*Sqrt[c + d*x^3])) + (a*(4*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(b*c - a*d)^(5/2))/(3*b^(3/2))","A",1
491,1,91,134,0.0325763,"\int \frac{x^5}{\left(a+b x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^5/((a + b*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{\left(a+b x^3\right) (a d+2 b c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^3+c\right)}{b c-a d}\right)+a (b c-a d)}{3 b \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)^2}","\frac{a}{3 b \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)}+\frac{a d+2 b c}{3 b \sqrt{c+d x^3} (b c-a d)^2}-\frac{(a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 \sqrt{b} (b c-a d)^{5/2}}",1,"(a*(b*c - a*d) + (2*b*c + a*d)*(a + b*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^3))/(b*c - a*d)])/(3*b*(b*c - a*d)^2*(a + b*x^3)*Sqrt[c + d*x^3])","C",1
492,1,54,108,0.0183569,"\int \frac{x^2}{\left(a+b x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^2/((a + b*x^3)^2*(c + d*x^3)^(3/2)),x]","-\frac{2 d \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};-\frac{b \left(d x^3+c\right)}{a d-b c}\right)}{3 \sqrt{c+d x^3} (a d-b c)^2}","-\frac{d}{\sqrt{c+d x^3} (b c-a d)^2}-\frac{1}{3 \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)}+\frac{\sqrt{b} d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{5/2}}",1,"(-2*d*Hypergeometric2F1[-1/2, 2, 1/2, -((b*(c + d*x^3))/(-(b*c) + a*d))])/(3*(-(b*c) + a*d)^2*Sqrt[c + d*x^3])","C",1
493,1,123,172,0.1274426,"\int \frac{1}{x \left(a+b x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{\frac{b (2 b c-5 a d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^3+c\right)}{b c-a d}\right)}{a (a d-b c)}+\left(\frac{2 b}{a}-\frac{2 d}{c}\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right)+\frac{b}{a+b x^3}}{3 a \sqrt{c+d x^3} (b c-a d)}","\frac{b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^2 (b c-a d)^{5/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^2 c^{3/2}}+\frac{b}{3 a \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)}+\frac{d (2 a d+b c)}{3 a c \sqrt{c+d x^3} (b c-a d)^2}",1,"(b/(a + b*x^3) + (b*(2*b*c - 5*a*d)*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^3))/(b*c - a*d)])/(a*(-(b*c) + a*d)) + ((2*b)/a - (2*d)/c)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^3)/c])/(3*a*(b*c - a*d)*Sqrt[c + d*x^3])","C",1
494,1,189,241,0.1557569,"\int \frac{1}{x^4 \left(a+b x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^4*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{b^2 c^2 x^3 \left(a+b x^3\right) (4 b c-7 a d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \left(d x^3+c\right)}{b c-a d}\right)-(a d-b c) \left(x^3 \left(a+b x^3\right) \left(3 a^2 d^2+a b c d-4 b^2 c^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right)+a c \left(a^2 d+a b \left(d x^3-c\right)-2 b^2 c x^3\right)\right)}{3 a^3 c^2 x^3 \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)^2}","-\frac{b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right)}{3 a^3 (b c-a d)^{5/2}}+\frac{(3 a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right)}{3 a^3 c^{5/2}}-\frac{d \left(3 a^2 d^2-2 a b c d+2 b^2 c^2\right)}{3 a^2 c^2 \sqrt{c+d x^3} (b c-a d)^2}-\frac{b (2 b c-a d)}{3 a^2 c \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)}-\frac{1}{3 a c x^3 \left(a+b x^3\right) \sqrt{c+d x^3}}",1,"(b^2*c^2*(4*b*c - 7*a*d)*x^3*(a + b*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^3))/(b*c - a*d)] - (-(b*c) + a*d)*(a*c*(a^2*d - 2*b^2*c*x^3 + a*b*(-c + d*x^3)) + (-4*b^2*c^2 + a*b*c*d + 3*a^2*d^2)*x^3*(a + b*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^3)/c]))/(3*a^3*c^2*(b*c - a*d)^2*x^3*(a + b*x^3)*Sqrt[c + d*x^3])","C",1
495,1,381,67,0.3585555,"\int \frac{x^3}{\left(a+b x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x^3/((a + b*x^3)^2*(c + d*x^3)^(3/2)),x]","-\frac{x^4 \left(\left(3 b d x^3 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+8 a \left(2 a d+b \left(c+3 d x^3\right)\right)\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a b c d F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right) \left(\left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+8 a\right)\right)}{8 a \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)^2 \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}","\frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};2,\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 a^2 c \sqrt{c+d x^3}}",1,"-1/8*(x^4*(-8*a*b*c*d*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)]*(8*a + (a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]) + (8*a*(2*a*d + b*(c + 3*d*x^3)) + 3*b*d*x^3*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/(a*(b*c - a*d)^2*(a + b*x^3)*Sqrt[c + d*x^3]*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))","B",0
496,1,216,67,0.2980101,"\int \frac{x}{\left(a+b x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[x/((a + b*x^3)^2*(c + d*x^3)^(3/2)),x]","-\frac{x^2 \left(5 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} \left(a^2 d^2+6 a b c d-b^2 c^2\right) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-10 a \left(2 a^2 d^2+2 a b d^2 x^3+b^2 c \left(c+d x^3\right)\right)+b d x^3 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} (2 a d+b c) F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}{30 a^2 c \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)^2}","\frac{x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};2,\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a^2 c \sqrt{c+d x^3}}",1,"-1/30*(x^2*(-10*a*(2*a^2*d^2 + 2*a*b*d^2*x^3 + b^2*c*(c + d*x^3)) + 5*(-(b^2*c^2) + 6*a*b*c*d + a^2*d^2)*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + b*d*(b*c + 2*a*d)*x^3*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)]))/(a^2*c*(b*c - a*d)^2*(a + b*x^3)*Sqrt[c + d*x^3])","B",0
497,1,381,62,0.5480524,"\int \frac{1}{\left(a+b x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/((a + b*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{x \left(\frac{a \left(64 a c \left(3 a^2 d^2+2 a b d \left(d x^3-3 c\right)+b^2 c \left(3 c+d x^3\right)\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-24 x^3 \left(2 a^2 d^2+2 a b d^2 x^3+b^2 c \left(c+d x^3\right)\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)\right)}{\left(a+b x^3\right) \left(8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)\right)}+b d x^3 \sqrt{\frac{d x^3}{c}+1} (2 a d+b c) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}{24 a^2 c \sqrt{c+d x^3} (b c-a d)^2}","\frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{1}{3};2,\frac{3}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2 c \sqrt{c+d x^3}}",1,"(x*(b*d*(b*c + 2*a*d)*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + (a*(64*a*c*(3*a^2*d^2 + 2*a*b*d*(-3*c + d*x^3) + b^2*c*(3*c + d*x^3))*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 24*x^3*(2*a^2*d^2 + 2*a*b*d^2*x^3 + b^2*c*(c + d*x^3))*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))))/(24*a^2*c*(b*c - a*d)^2*Sqrt[c + d*x^3])","B",0
498,1,308,65,0.4328792,"\int \frac{1}{x^2 \left(a+b x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^2*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{2 b d x^6 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} \left(5 a^2 d^2-6 a b c d+4 b^2 c^2\right) F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+5 x^3 \left(a+b x^3\right) \sqrt{\frac{d x^3}{c}+1} \left(5 a^3 d^3-6 a^2 b c d^2+21 a b^2 c^2 d-8 b^3 c^3\right) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-20 a \left(a^3 d^2 \left(3 c+5 d x^3\right)+a^2 b d \left(-6 c^2-3 c d x^3+5 d^2 x^6\right)+3 a b^2 c \left(c^2-c d x^3-2 d^2 x^6\right)+4 b^3 c^2 x^3 \left(c+d x^3\right)\right)}{60 a^3 c^2 x \left(a+b x^3\right) \sqrt{c+d x^3} (b c-a d)^2}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{1}{3};2,\frac{3}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2 c x \sqrt{c+d x^3}}",1,"(-20*a*(4*b^3*c^2*x^3*(c + d*x^3) + a^3*d^2*(3*c + 5*d*x^3) + 3*a*b^2*c*(c^2 - c*d*x^3 - 2*d^2*x^6) + a^2*b*d*(-6*c^2 - 3*c*d*x^3 + 5*d^2*x^6)) + 5*(-8*b^3*c^3 + 21*a*b^2*c^2*d - 6*a^2*b*c*d^2 + 5*a^3*d^3)*x^3*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 2*b*d*(4*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*x^6*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/(60*a^3*c^2*(b*c - a*d)^2*x*(a + b*x^3)*Sqrt[c + d*x^3])","B",0
499,1,515,67,0.9636678,"\int \frac{1}{x^3 \left(a+b x^3\right)^2 \left(c+d x^3\right)^{3/2}} \, dx","Integrate[1/(x^3*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x]","\frac{\frac{a \left(32 a c \left(3 a^3 d^2 \left(2 c+7 d x^3\right)+2 a^2 b d \left(-6 c^2-6 c d x^3+7 d^2 x^6\right)+3 a b^2 c \left(2 c^2-13 c d x^3-4 d^2 x^6\right)+10 b^3 c^2 x^3 \left(3 c+d x^3\right)\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-24 x^3 \left(a^3 d^2 \left(3 c+7 d x^3\right)+a^2 b d \left(-6 c^2-3 c d x^3+7 d^2 x^6\right)+3 a b^2 c \left(c^2-c d x^3-2 d^2 x^6\right)+5 b^3 c^2 x^3 \left(c+d x^3\right)\right) \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)\right)}{\left(a+b x^3\right) \left(3 x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)+a d F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}-b d x^6 \sqrt{\frac{d x^3}{c}+1} \left(7 a^2 d^2-6 a b c d+5 b^2 c^2\right) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)}{48 a^3 c^2 x^2 \sqrt{c+d x^3} (b c-a d)^2}","-\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{2}{3};2,\frac{3}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a^2 c x^2 \sqrt{c+d x^3}}",1,"(-(b*d*(5*b^2*c^2 - 6*a*b*c*d + 7*a^2*d^2)*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]) + (a*(32*a*c*(10*b^3*c^2*x^3*(3*c + d*x^3) + 3*a^3*d^2*(2*c + 7*d*x^3) + 3*a*b^2*c*(2*c^2 - 13*c*d*x^3 - 4*d^2*x^6) + 2*a^2*b*d*(-6*c^2 - 6*c*d*x^3 + 7*d^2*x^6))*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 24*x^3*(5*b^3*c^2*x^3*(c + d*x^3) + a^3*d^2*(3*c + 7*d*x^3) + 3*a*b^2*c*(c^2 - c*d*x^3 - 2*d^2*x^6) + a^2*b*d*(-6*c^2 - 3*c*d*x^3 + 7*d^2*x^6))*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(48*a^3*c^2*(b*c - a*d)^2*x^2*Sqrt[c + d*x^3])","B",0
500,1,113,134,0.124143,"\int (e x)^m \left(a+b x^3\right)^{5/2} \left(A+B x^3\right) \, dx","Integrate[(e*x)^m*(a + b*x^3)^(5/2)*(A + B*x^3),x]","\frac{a^2 x \sqrt{a+b x^3} (e x)^m \left(A (m+4) \, _2F_1\left(-\frac{5}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)+B (m+1) x^3 \, _2F_1\left(-\frac{5}{2},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right)\right)}{(m+1) (m+4) \sqrt{\frac{b x^3}{a}+1}}","\frac{2 B \left(a+b x^3\right)^{7/2} (e x)^{m+1}}{b e (2 m+23)}-\frac{a^2 \sqrt{a+b x^3} (e x)^{m+1} (2 a B (m+1)-A b (2 m+23)) \, _2F_1\left(-\frac{5}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{b e (m+1) (2 m+23) \sqrt{\frac{b x^3}{a}+1}}",1,"(a^2*x*(e*x)^m*Sqrt[a + b*x^3]*(A*(4 + m)*Hypergeometric2F1[-5/2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)] + B*(1 + m)*x^3*Hypergeometric2F1[-5/2, (4 + m)/3, (7 + m)/3, -((b*x^3)/a)]))/((1 + m)*(4 + m)*Sqrt[1 + (b*x^3)/a])","A",1
501,1,111,132,0.089914,"\int (e x)^m \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[(e*x)^m*(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{a x \sqrt{a+b x^3} (e x)^m \left(A (m+4) \, _2F_1\left(-\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)+B (m+1) x^3 \, _2F_1\left(-\frac{3}{2},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right)\right)}{(m+1) (m+4) \sqrt{\frac{b x^3}{a}+1}}","\frac{2 B \left(a+b x^3\right)^{5/2} (e x)^{m+1}}{b e (2 m+17)}-\frac{a \sqrt{a+b x^3} (e x)^{m+1} (2 a B (m+1)-A b (2 m+17)) \, _2F_1\left(-\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{b e (m+1) (2 m+17) \sqrt{\frac{b x^3}{a}+1}}",1,"(a*x*(e*x)^m*Sqrt[a + b*x^3]*(A*(4 + m)*Hypergeometric2F1[-3/2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)] + B*(1 + m)*x^3*Hypergeometric2F1[-3/2, (4 + m)/3, (7 + m)/3, -((b*x^3)/a)]))/((1 + m)*(4 + m)*Sqrt[1 + (b*x^3)/a])","A",1
502,1,110,131,0.0904192,"\int (e x)^m \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[(e*x)^m*Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{x \sqrt{a+b x^3} (e x)^m \left(A (m+4) \, _2F_1\left(-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)+B (m+1) x^3 \, _2F_1\left(-\frac{1}{2},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right)\right)}{(m+1) (m+4) \sqrt{\frac{b x^3}{a}+1}}","\frac{2 B \left(a+b x^3\right)^{3/2} (e x)^{m+1}}{b e (2 m+11)}-\frac{\sqrt{a+b x^3} (e x)^{m+1} (2 a B (m+1)-A b (2 m+11)) \, _2F_1\left(-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{b e (m+1) (2 m+11) \sqrt{\frac{b x^3}{a}+1}}",1,"(x*(e*x)^m*Sqrt[a + b*x^3]*(A*(4 + m)*Hypergeometric2F1[-1/2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)] + B*(1 + m)*x^3*Hypergeometric2F1[-1/2, (4 + m)/3, (7 + m)/3, -((b*x^3)/a)]))/((1 + m)*(4 + m)*Sqrt[1 + (b*x^3)/a])","A",1
503,1,110,131,0.1064106,"\int \frac{(e x)^m \left(A+B x^3\right)}{\sqrt{a+b x^3}} \, dx","Integrate[((e*x)^m*(A + B*x^3))/Sqrt[a + b*x^3],x]","\frac{x \sqrt{\frac{b x^3}{a}+1} (e x)^m \left(A (m+4) \, _2F_1\left(\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)+B (m+1) x^3 \, _2F_1\left(\frac{1}{2},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right)\right)}{(m+1) (m+4) \sqrt{a+b x^3}}","\frac{2 B \sqrt{a+b x^3} (e x)^{m+1}}{b e (2 m+5)}-\frac{\sqrt{\frac{b x^3}{a}+1} (e x)^{m+1} (2 a B (m+1)-A b (2 m+5)) \, _2F_1\left(\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{b e (m+1) (2 m+5) \sqrt{a+b x^3}}",1,"(x*(e*x)^m*Sqrt[1 + (b*x^3)/a]*(A*(4 + m)*Hypergeometric2F1[1/2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)] + B*(1 + m)*x^3*Hypergeometric2F1[1/2, (4 + m)/3, (7 + m)/3, -((b*x^3)/a)]))/((1 + m)*(4 + m)*Sqrt[a + b*x^3])","A",1
504,1,113,133,0.1075133,"\int \frac{(e x)^m \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[((e*x)^m*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{x \sqrt{\frac{b x^3}{a}+1} (e x)^m \left(A (m+4) \, _2F_1\left(\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)+B (m+1) x^3 \, _2F_1\left(\frac{3}{2},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right)\right)}{a (m+1) (m+4) \sqrt{a+b x^3}}","\frac{\sqrt{\frac{b x^3}{a}+1} (e x)^{m+1} (2 a B (m+1)+A (b-2 b m)) \, _2F_1\left(\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{3 a b e (m+1) \sqrt{a+b x^3}}+\frac{2 (e x)^{m+1} (A b-a B)}{3 a b e \sqrt{a+b x^3}}",1,"(x*(e*x)^m*Sqrt[1 + (b*x^3)/a]*(A*(4 + m)*Hypergeometric2F1[3/2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)] + B*(1 + m)*x^3*Hypergeometric2F1[3/2, (4 + m)/3, (7 + m)/3, -((b*x^3)/a)]))/(a*(1 + m)*(4 + m)*Sqrt[a + b*x^3])","A",1
505,1,113,133,0.1335748,"\int \frac{(e x)^m \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[((e*x)^m*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{x \sqrt{\frac{b x^3}{a}+1} (e x)^m \left(A (m+4) \, _2F_1\left(\frac{5}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)+B (m+1) x^3 \, _2F_1\left(\frac{5}{2},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right)\right)}{a^2 (m+1) (m+4) \sqrt{a+b x^3}}","\frac{\sqrt{\frac{b x^3}{a}+1} (e x)^{m+1} (2 a B (m+1)+A b (7-2 m)) \, _2F_1\left(\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{9 a^2 b e (m+1) \sqrt{a+b x^3}}+\frac{2 (e x)^{m+1} (A b-a B)}{9 a b e \left(a+b x^3\right)^{3/2}}",1,"(x*(e*x)^m*Sqrt[1 + (b*x^3)/a]*(A*(4 + m)*Hypergeometric2F1[5/2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)] + B*(1 + m)*x^3*Hypergeometric2F1[5/2, (4 + m)/3, (7 + m)/3, -((b*x^3)/a)]))/(a^2*(1 + m)*(4 + m)*Sqrt[a + b*x^3])","A",1
506,1,123,88,0.2658747,"\int \frac{x^5}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx","Integrate[x^5/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]","\frac{b \sqrt{d} \sqrt{a+b x^3} \left(c+d x^3\right)-\sqrt{b c-a d} (a d+b c) \sqrt{\frac{b \left(c+d x^3\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b c-a d}}\right)}{3 b^2 d^{3/2} \sqrt{c+d x^3}}","\frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 b d}-\frac{(a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right)}{3 b^{3/2} d^{3/2}}",1,"(b*Sqrt[d]*Sqrt[a + b*x^3]*(c + d*x^3) - Sqrt[b*c - a*d]*(b*c + a*d)*Sqrt[(b*(c + d*x^3))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^3])/Sqrt[b*c - a*d]])/(3*b^2*d^(3/2)*Sqrt[c + d*x^3])","A",1
507,1,85,48,0.096293,"\int \frac{x^2}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx","Integrate[x^2/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]","\frac{2 \sqrt{c+d x^3} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b c-a d}}\right)}{3 \sqrt{d} \sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^3\right)}{b c-a d}}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right)}{3 \sqrt{b} \sqrt{d}}",1,"(2*Sqrt[c + d*x^3]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^3])/Sqrt[b*c - a*d]])/(3*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^3))/(b*c - a*d)])","A",1
508,1,48,48,0.0186297,"\int \frac{1}{x \sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx","Integrate[1/(x*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^3}}{\sqrt{a} \sqrt{c+d x^3}}\right)}{3 \sqrt{a} \sqrt{c}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^3}}{\sqrt{a} \sqrt{c+d x^3}}\right)}{3 \sqrt{a} \sqrt{c}}",1,"(-2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^3])/(Sqrt[a]*Sqrt[c + d*x^3])])/(3*Sqrt[a]*Sqrt[c])","A",1
509,1,91,91,0.0650046,"\int \frac{1}{x^4 \sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx","Integrate[1/(x^4*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]","\frac{(a d+b c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^3}}{\sqrt{a} \sqrt{c+d x^3}}\right)}{3 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 a c x^3}","\frac{(a d+b c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^3}}{\sqrt{a} \sqrt{c+d x^3}}\right)}{3 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 a c x^3}",1,"-1/3*(Sqrt[a + b*x^3]*Sqrt[c + d*x^3])/(a*c*x^3) + ((b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^3])/(Sqrt[a]*Sqrt[c + d*x^3])])/(3*a^(3/2)*c^(3/2))","A",1
510,1,90,88,0.0591524,"\int \frac{x^4}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx","Integrate[x^4/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]","\frac{x^5 \sqrt{\frac{a+b x^3}{a}} \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{5 \sqrt{a+b x^3} \sqrt{c+d x^3}}","\frac{x^5 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{5 \sqrt{a+b x^3} \sqrt{c+d x^3}}",1,"(x^5*Sqrt[(a + b*x^3)/a]*Sqrt[(c + d*x^3)/c]*AppellF1[5/3, 1/2, 1/2, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(5*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])","A",1
511,1,90,88,0.0542856,"\int \frac{x^3}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx","Integrate[x^3/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]","\frac{x^4 \sqrt{\frac{a+b x^3}{a}} \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 \sqrt{a+b x^3} \sqrt{c+d x^3}}","\frac{x^4 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 \sqrt{a+b x^3} \sqrt{c+d x^3}}",1,"(x^4*Sqrt[(a + b*x^3)/a]*Sqrt[(c + d*x^3)/c]*AppellF1[4/3, 1/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/(4*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])","A",1
512,1,90,88,0.0493991,"\int \frac{x}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx","Integrate[x/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]","\frac{x^2 \sqrt{\frac{a+b x^3}{a}} \sqrt{\frac{c+d x^3}{c}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 \sqrt{a+b x^3} \sqrt{c+d x^3}}","\frac{x^2 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 \sqrt{a+b x^3} \sqrt{c+d x^3}}",1,"(x^2*Sqrt[(a + b*x^3)/a]*Sqrt[(c + d*x^3)/c]*AppellF1[2/3, 1/2, 1/2, 5/3, -((b*x^3)/a), -((d*x^3)/c)])/(2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])","A",1
513,1,170,83,0.3267618,"\int \frac{1}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx","Integrate[1/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]","-\frac{8 a c x F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\sqrt{a+b x^3} \sqrt{c+d x^3} \left(3 x^3 \left(a d F_1\left(\frac{4}{3};\frac{1}{2},\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+b c F_1\left(\frac{4}{3};\frac{3}{2},\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-8 a c F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}","\frac{x \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\sqrt{a+b x^3} \sqrt{c+d x^3}}",1,"(-8*a*c*x*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]*(-8*a*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + 3*x^3*(a*d*AppellF1[4/3, 1/2, 3/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + b*c*AppellF1[4/3, 3/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))","B",0
514,1,189,86,0.2562748,"\int \frac{1}{x^2 \sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx","Integrate[1/(x^2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]","\frac{5 x^3 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} (a d+b c) F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+8 b d x^6 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-20 \left(a+b x^3\right) \left(c+d x^3\right)}{20 a c x \sqrt{a+b x^3} \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{x \sqrt{a+b x^3} \sqrt{c+d x^3}}",1,"(-20*(a + b*x^3)*(c + d*x^3) + 5*(b*c + a*d)*x^3*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1/2, 5/3, -((b*x^3)/a), -((d*x^3)/c)] + 8*b*d*x^6*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1/2, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(20*a*c*x*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])","B",0
515,1,365,88,0.2719112,"\int \frac{1}{x^3 \sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx","Integrate[1/(x^3*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]","\frac{b d x^6 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+\frac{4 \left(3 x^3 \left(a+b x^3\right) \left(c+d x^3\right) \left(a d F_1\left(\frac{4}{3};\frac{1}{2},\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+b c F_1\left(\frac{4}{3};\frac{3}{2},\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c \left(2 a c+3 a d x^3+3 b c x^3+2 b d x^6\right) F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}{8 a c F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-3 x^3 \left(a d F_1\left(\frac{4}{3};\frac{1}{2},\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+b c F_1\left(\frac{4}{3};\frac{3}{2},\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}}{8 a c x^2 \sqrt{a+b x^3} \sqrt{c+d x^3}}","-\frac{\sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left(-\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 x^2 \sqrt{a+b x^3} \sqrt{c+d x^3}}",1,"(b*d*x^6*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + (4*(-4*a*c*(2*a*c + 3*b*c*x^3 + 3*a*d*x^3 + 2*b*d*x^6)*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + 3*x^3*(a + b*x^3)*(c + d*x^3)*(a*d*AppellF1[4/3, 1/2, 3/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + b*c*AppellF1[4/3, 3/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))/(8*a*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)] - 3*x^3*(a*d*AppellF1[4/3, 1/2, 3/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + b*c*AppellF1[4/3, 3/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))/(8*a*c*x^2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])","B",0
516,1,145,161,0.2422581,"\int (e x)^{7/2} \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[(e*x)^(7/2)*Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{e^3 \sqrt{e x} \sqrt{a+b x^3} \left(3 a^{3/2} (a B-2 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)+\sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1} \left(-3 a^2 B+2 a b \left(3 A+B x^3\right)+4 b^2 x^3 \left(3 A+2 B x^3\right)\right)\right)}{72 b^{5/2} \sqrt{x} \sqrt{\frac{b x^3}{a}+1}}","-\frac{a^2 e^{7/2} (2 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{24 b^{5/2}}+\frac{a e^2 (e x)^{3/2} \sqrt{a+b x^3} (2 A b-a B)}{24 b^2}+\frac{(e x)^{9/2} \sqrt{a+b x^3} (2 A b-a B)}{12 b e}+\frac{B (e x)^{9/2} \left(a+b x^3\right)^{3/2}}{9 b e}",1,"(e^3*Sqrt[e*x]*Sqrt[a + b*x^3]*(Sqrt[b]*x^(3/2)*Sqrt[1 + (b*x^3)/a]*(-3*a^2*B + 2*a*b*(3*A + B*x^3) + 4*b^2*x^3*(3*A + 2*B*x^3)) + 3*a^(3/2)*(-2*A*b + a*B)*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(72*b^(5/2)*Sqrt[x]*Sqrt[1 + (b*x^3)/a])","A",1
517,1,112,324,0.1554252,"\int (e x)^{5/2} \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[(e*x)^(5/2)*Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{e^2 \sqrt{e x} \sqrt{a+b x^3} \left(a (7 a B-16 A b) \, _2F_1\left(-\frac{1}{2},\frac{1}{6};\frac{7}{6};-\frac{b x^3}{a}\right)-\left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \left(7 a B-16 A b-10 b B x^3\right)\right)}{80 b^2 \sqrt{\frac{b x^3}{a}+1}}","-\frac{3^{3/4} a^{5/3} e^2 \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (16 A b-7 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{640 b^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{3 a e^2 \sqrt{e x} \sqrt{a+b x^3} (16 A b-7 a B)}{320 b^2}+\frac{(e x)^{7/2} \sqrt{a+b x^3} (16 A b-7 a B)}{80 b e}+\frac{B (e x)^{7/2} \left(a+b x^3\right)^{3/2}}{8 b e}",1,"(e^2*Sqrt[e*x]*Sqrt[a + b*x^3]*(-((a + b*x^3)*Sqrt[1 + (b*x^3)/a]*(-16*A*b + 7*a*B - 10*b*B*x^3)) + a*(-16*A*b + 7*a*B)*Hypergeometric2F1[-1/2, 1/6, 7/6, -((b*x^3)/a)]))/(80*b^2*Sqrt[1 + (b*x^3)/a])","C",1
518,1,94,581,0.1126594,"\int (e x)^{3/2} \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[(e*x)^(3/2)*Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{x (e x)^{3/2} \sqrt{a+b x^3} \left((14 A b-5 a B) \, _2F_1\left(-\frac{1}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right)+5 B \sqrt{\frac{b x^3}{a}+1} \left(a+b x^3\right)\right)}{35 b \sqrt{\frac{b x^3}{a}+1}}","-\frac{3^{3/4} \left(1-\sqrt{3}\right) a^{4/3} e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (14 A b-5 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{224 b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} a^{4/3} e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (14 A b-5 a B) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{112 b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{3 \left(1+\sqrt{3}\right) a e \sqrt{e x} \sqrt{a+b x^3} (14 A b-5 a B)}{112 b^{5/3} \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}+\frac{(e x)^{5/2} \sqrt{a+b x^3} (14 A b-5 a B)}{56 b e}+\frac{B (e x)^{5/2} \left(a+b x^3\right)^{3/2}}{7 b e}",1,"(x*(e*x)^(3/2)*Sqrt[a + b*x^3]*(5*B*(a + b*x^3)*Sqrt[1 + (b*x^3)/a] + (14*A*b - 5*a*B)*Hypergeometric2F1[-1/2, 5/6, 11/6, -((b*x^3)/a)]))/(35*b*Sqrt[1 + (b*x^3)/a])","C",1
519,1,119,121,0.1450726,"\int \sqrt{e x} \sqrt{a+b x^3} \left(A+B x^3\right) \, dx","Integrate[Sqrt[e*x]*Sqrt[a + b*x^3]*(A + B*x^3),x]","\frac{\sqrt{e x} \sqrt{a+b x^3} \left(\sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1} \left(B \left(a+2 b x^3\right)+4 A b\right)-\sqrt{a} (a B-4 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)\right)}{12 b^{3/2} \sqrt{x} \sqrt{\frac{b x^3}{a}+1}}","\frac{a \sqrt{e} (4 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{12 b^{3/2}}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (4 A b-a B)}{12 b e}+\frac{B (e x)^{3/2} \left(a+b x^3\right)^{3/2}}{6 b e}",1,"(Sqrt[e*x]*Sqrt[a + b*x^3]*(Sqrt[b]*x^(3/2)*Sqrt[1 + (b*x^3)/a]*(4*A*b + B*(a + 2*b*x^3)) - Sqrt[a]*(-4*A*b + a*B)*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(12*b^(3/2)*Sqrt[x]*Sqrt[1 + (b*x^3)/a])","A",1
520,1,93,286,0.0667397,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{\sqrt{e x}} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/Sqrt[e*x],x]","\frac{x \sqrt{a+b x^3} \left((10 A b-a B) \, _2F_1\left(-\frac{1}{2},\frac{1}{6};\frac{7}{6};-\frac{b x^3}{a}\right)+B \sqrt{\frac{b x^3}{a}+1} \left(a+b x^3\right)\right)}{5 b \sqrt{e x} \sqrt{\frac{b x^3}{a}+1}}","\frac{3^{3/4} a^{2/3} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (10 A b-a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{40 b e \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{e x} \sqrt{a+b x^3} (10 A b-a B)}{20 b e}+\frac{B \sqrt{e x} \left(a+b x^3\right)^{3/2}}{5 b e}",1,"(x*Sqrt[a + b*x^3]*(B*(a + b*x^3)*Sqrt[1 + (b*x^3)/a] + (10*A*b - a*B)*Hypergeometric2F1[-1/2, 1/6, 7/6, -((b*x^3)/a)]))/(5*b*Sqrt[e*x]*Sqrt[1 + (b*x^3)/a])","C",1
521,1,98,580,0.0697205,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{(e x)^{3/2}} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/(e*x)^(3/2),x]","-\frac{4 x^4 \sqrt{a+b x^3} \left(-\frac{a B}{2}-4 A b\right) \, _2F_1\left(-\frac{1}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right)}{5 a (e x)^{3/2} \sqrt{\frac{b x^3}{a}+1}}-\frac{2 A x \left(a+b x^3\right)^{3/2}}{a (e x)^{3/2}}","-\frac{3^{3/4} \left(1-\sqrt{3}\right) \sqrt[3]{a} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (a B+8 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{16 b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt[3]{a} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (a B+8 A b) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{8 b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{3 \left(1+\sqrt{3}\right) \sqrt{e x} \sqrt{a+b x^3} (a B+8 A b)}{8 b^{2/3} e^2 \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}+\frac{(e x)^{5/2} \sqrt{a+b x^3} (a B+8 A b)}{4 a e^4}-\frac{2 A \left(a+b x^3\right)^{3/2}}{a e \sqrt{e x}}",1,"(-2*A*x*(a + b*x^3)^(3/2))/(a*(e*x)^(3/2)) - (4*(-4*A*b - (a*B)/2)*x^4*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 5/6, 11/6, -((b*x^3)/a)])/(5*a*(e*x)^(3/2)*Sqrt[1 + (b*x^3)/a])","C",1
522,1,87,118,0.2092923,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{(e x)^{5/2}} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/(e*x)^(5/2),x]","\frac{x \sqrt{a+b x^3} \left(\frac{x^{3/2} (a B+2 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{\frac{b x^3}{a}+1}}-2 A+B x^3\right)}{3 (e x)^{5/2}}","\frac{(a B+2 A b) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{3 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+2 A b)}{3 a e^4}-\frac{2 A \left(a+b x^3\right)^{3/2}}{3 a e (e x)^{3/2}}",1,"(x*Sqrt[a + b*x^3]*(-2*A + B*x^3 + ((2*A*b + a*B)*x^(3/2)*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Sqrt[1 + (b*x^3)/a])))/(3*(e*x)^(5/2))","A",1
523,1,97,283,0.0655643,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{(e x)^{7/2}} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/(e*x)^(7/2),x]","\frac{2 x \sqrt{a+b x^3} \left(x^3 (5 a B+4 A b) \, _2F_1\left(-\frac{1}{2},\frac{1}{6};\frac{7}{6};-\frac{b x^3}{a}\right)-A \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1}\right)}{5 a (e x)^{7/2} \sqrt{\frac{b x^3}{a}+1}}","\frac{3^{3/4} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (5 a B+4 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{20 \sqrt[3]{a} e^4 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{e x} \sqrt{a+b x^3} (5 a B+4 A b)}{10 a e^4}-\frac{2 A \left(a+b x^3\right)^{3/2}}{5 a e (e x)^{5/2}}",1,"(2*x*Sqrt[a + b*x^3]*(-(A*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]) + (4*A*b + 5*a*B)*x^3*Hypergeometric2F1[-1/2, 1/6, 7/6, -((b*x^3)/a)]))/(5*a*(e*x)^(7/2)*Sqrt[1 + (b*x^3)/a])","C",1
524,1,81,564,0.104697,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^{9/2}} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^(9/2),x]","\frac{2 \sqrt{a+b x^3} \left(-\frac{x^3 (7 a B+2 A b) \, _2F_1\left(-\frac{1}{2},-\frac{1}{6};\frac{5}{6};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-A \left(a+b x^3\right)\right)}{7 a x^{7/2}}","-\frac{3^{3/4} \left(1-\sqrt{3}\right) \sqrt[3]{b} \sqrt{x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (7 a B+2 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{14 a^{2/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{3 \sqrt[4]{3} \sqrt[3]{b} \sqrt{x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (7 a B+2 A b) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a^{2/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} (7 a B+2 A b)}{7 a \sqrt{x}}+\frac{3 \left(1+\sqrt{3}\right) \sqrt[3]{b} \sqrt{x} \sqrt{a+b x^3} (7 a B+2 A b)}{7 a \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}-\frac{2 A \left(a+b x^3\right)^{3/2}}{7 a x^{7/2}}",1,"(2*Sqrt[a + b*x^3]*(-(A*(a + b*x^3)) - ((2*A*b + 7*a*B)*x^3*Hypergeometric2F1[-1/2, -1/6, 5/6, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(7*a*x^(7/2))","C",1
525,1,87,79,0.1810824,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^{11/2}} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^(11/2),x]","\frac{2 \sqrt{a+b x^3} \left(\frac{3 \sqrt{a} \sqrt{b} B \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)}{\sqrt{\frac{b x^3}{a}+1}}-\frac{a \left(A+3 B x^3\right)+A b x^3}{x^{9/2}}\right)}{9 a}","-\frac{2 A \left(a+b x^3\right)^{3/2}}{9 a x^{9/2}}-\frac{2 B \sqrt{a+b x^3}}{3 x^{3/2}}+\frac{2}{3} \sqrt{b} B \tanh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a+b x^3}}\right)",1,"(2*Sqrt[a + b*x^3]*(-((A*b*x^3 + a*(A + 3*B*x^3))/x^(9/2)) + (3*Sqrt[a]*Sqrt[b]*B*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]])/Sqrt[1 + (b*x^3)/a]))/(9*a)","A",1
526,1,80,269,0.090359,"\int \frac{\sqrt{a+b x^3} \left(A+B x^3\right)}{x^{13/2}} \, dx","Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^(13/2),x]","\frac{2 \sqrt{a+b x^3} \left(\frac{x^3 (2 A b-11 a B) \, _2F_1\left(-\frac{5}{6},-\frac{1}{2};\frac{1}{6};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-5 A \left(a+b x^3\right)\right)}{55 a x^{11/2}}","-\frac{3^{3/4} b \sqrt{x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (2 A b-11 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{55 a^{4/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3} (2 A b-11 a B)}{55 a x^{5/2}}-\frac{2 A \left(a+b x^3\right)^{3/2}}{11 a x^{11/2}}",1,"(2*Sqrt[a + b*x^3]*(-5*A*(a + b*x^3) + ((2*A*b - 11*a*B)*x^3*Hypergeometric2F1[-5/6, -1/2, 1/6, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(55*a*x^(11/2))","C",1
527,1,167,201,0.3345799,"\int (e x)^{7/2} \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[(e*x)^(7/2)*(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{e^3 \sqrt{e x} \sqrt{a+b x^3} \left(3 a^{5/2} (3 a B-8 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)+\sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1} \left(-9 a^3 B+6 a^2 b \left(4 A+B x^3\right)+8 a b^2 x^3 \left(14 A+9 B x^3\right)+16 b^3 x^6 \left(4 A+3 B x^3\right)\right)\right)}{576 b^{5/2} \sqrt{x} \sqrt{\frac{b x^3}{a}+1}}","-\frac{a^3 e^{7/2} (8 A b-3 a B) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{192 b^{5/2}}+\frac{a^2 e^2 (e x)^{3/2} \sqrt{a+b x^3} (8 A b-3 a B)}{192 b^2}+\frac{(e x)^{9/2} \left(a+b x^3\right)^{3/2} (8 A b-3 a B)}{72 b e}+\frac{a (e x)^{9/2} \sqrt{a+b x^3} (8 A b-3 a B)}{96 b e}+\frac{B (e x)^{9/2} \left(a+b x^3\right)^{5/2}}{12 b e}",1,"(e^3*Sqrt[e*x]*Sqrt[a + b*x^3]*(Sqrt[b]*x^(3/2)*Sqrt[1 + (b*x^3)/a]*(-9*a^3*B + 6*a^2*b*(4*A + B*x^3) + 16*b^3*x^6*(4*A + 3*B*x^3) + 8*a*b^2*x^3*(14*A + 9*B*x^3)) + 3*a^(5/2)*(-8*A*b + 3*a*B)*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(576*b^(5/2)*Sqrt[x]*Sqrt[1 + (b*x^3)/a])","A",1
528,1,116,364,0.1743629,"\int (e x)^{5/2} \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[(e*x)^(5/2)*(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{e^2 \sqrt{e x} \sqrt{a+b x^3} \left(a^2 (7 a B-22 A b) \, _2F_1\left(-\frac{3}{2},\frac{1}{6};\frac{7}{6};-\frac{b x^3}{a}\right)-\left(a+b x^3\right)^2 \sqrt{\frac{b x^3}{a}+1} \left(7 a B-22 A b-16 b B x^3\right)\right)}{176 b^2 \sqrt{\frac{b x^3}{a}+1}}","-\frac{9\ 3^{3/4} a^{8/3} e^2 \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (22 A b-7 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{14080 b^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{27 a^2 e^2 \sqrt{e x} \sqrt{a+b x^3} (22 A b-7 a B)}{7040 b^2}+\frac{(e x)^{7/2} \left(a+b x^3\right)^{3/2} (22 A b-7 a B)}{176 b e}+\frac{9 a (e x)^{7/2} \sqrt{a+b x^3} (22 A b-7 a B)}{1760 b e}+\frac{B (e x)^{7/2} \left(a+b x^3\right)^{5/2}}{11 b e}",1,"(e^2*Sqrt[e*x]*Sqrt[a + b*x^3]*(-((a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*(-22*A*b + 7*a*B - 16*b*B*x^3)) + a^2*(-22*A*b + 7*a*B)*Hypergeometric2F1[-3/2, 1/6, 7/6, -((b*x^3)/a)]))/(176*b^2*Sqrt[1 + (b*x^3)/a])","C",1
529,1,96,621,0.1336527,"\int (e x)^{3/2} \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[(e*x)^(3/2)*(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{x (e x)^{3/2} \sqrt{a+b x^3} \left(a (4 A b-a B) \, _2F_1\left(-\frac{3}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right)+B \sqrt{\frac{b x^3}{a}+1} \left(a+b x^3\right)^2\right)}{10 b \sqrt{\frac{b x^3}{a}+1}}","-\frac{9\ 3^{3/4} \left(1-\sqrt{3}\right) a^{7/3} e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (4 A b-a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{896 b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} a^{7/3} e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (4 A b-a B) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{448 b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{27 \left(1+\sqrt{3}\right) a^2 e \sqrt{e x} \sqrt{a+b x^3} (4 A b-a B)}{448 b^{5/3} \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}+\frac{(e x)^{5/2} \left(a+b x^3\right)^{3/2} (4 A b-a B)}{28 b e}+\frac{9 a (e x)^{5/2} \sqrt{a+b x^3} (4 A b-a B)}{224 b e}+\frac{B (e x)^{5/2} \left(a+b x^3\right)^{5/2}}{10 b e}",1,"(x*(e*x)^(3/2)*Sqrt[a + b*x^3]*(B*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a] + a*(4*A*b - a*B)*Hypergeometric2F1[-3/2, 5/6, 11/6, -((b*x^3)/a)]))/(10*b*Sqrt[1 + (b*x^3)/a])","C",1
530,1,143,161,0.1824734,"\int \sqrt{e x} \left(a+b x^3\right)^{3/2} \left(A+B x^3\right) \, dx","Integrate[Sqrt[e*x]*(a + b*x^3)^(3/2)*(A + B*x^3),x]","\frac{\sqrt{e x} \sqrt{a+b x^3} \left(\sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1} \left(3 a^2 B+2 a b \left(15 A+7 B x^3\right)+4 b^2 x^3 \left(3 A+2 B x^3\right)\right)-3 a^{3/2} (a B-6 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)\right)}{72 b^{3/2} \sqrt{x} \sqrt{\frac{b x^3}{a}+1}}","\frac{a^2 \sqrt{e} (6 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{24 b^{3/2}}+\frac{(e x)^{3/2} \left(a+b x^3\right)^{3/2} (6 A b-a B)}{36 b e}+\frac{a (e x)^{3/2} \sqrt{a+b x^3} (6 A b-a B)}{24 b e}+\frac{B (e x)^{3/2} \left(a+b x^3\right)^{5/2}}{9 b e}",1,"(Sqrt[e*x]*Sqrt[a + b*x^3]*(Sqrt[b]*x^(3/2)*Sqrt[1 + (b*x^3)/a]*(3*a^2*B + 4*b^2*x^3*(3*A + 2*B*x^3) + 2*a*b*(15*A + 7*B*x^3)) - 3*a^(3/2)*(-6*A*b + a*B)*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(72*b^(3/2)*Sqrt[x]*Sqrt[1 + (b*x^3)/a])","A",1
531,1,96,324,0.076762,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{\sqrt{e x}} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/Sqrt[e*x],x]","\frac{x \sqrt{a+b x^3} \left(a (16 A b-a B) \, _2F_1\left(-\frac{3}{2},\frac{1}{6};\frac{7}{6};-\frac{b x^3}{a}\right)+B \sqrt{\frac{b x^3}{a}+1} \left(a+b x^3\right)^2\right)}{8 b \sqrt{e x} \sqrt{\frac{b x^3}{a}+1}}","\frac{9\ 3^{3/4} a^{5/3} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (16 A b-a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{640 b e \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{e x} \left(a+b x^3\right)^{3/2} (16 A b-a B)}{80 b e}+\frac{9 a \sqrt{e x} \sqrt{a+b x^3} (16 A b-a B)}{320 b e}+\frac{B \sqrt{e x} \left(a+b x^3\right)^{5/2}}{8 b e}",1,"(x*Sqrt[a + b*x^3]*(B*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a] + a*(16*A*b - a*B)*Hypergeometric2F1[-3/2, 1/6, 7/6, -((b*x^3)/a)]))/(8*b*Sqrt[e*x]*Sqrt[1 + (b*x^3)/a])","C",1
532,1,84,614,0.12109,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{(e x)^{3/2}} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/(e*x)^(3/2),x]","\frac{2 x \sqrt{a+b x^3} \left(\frac{x^3 (a B+14 A b) \, _2F_1\left(-\frac{3}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-\frac{5 A \left(a+b x^3\right)^2}{a}\right)}{5 (e x)^{3/2}}","-\frac{9\ 3^{3/4} \left(1-\sqrt{3}\right) a^{4/3} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (a B+14 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{224 b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt[4]{3} a^{4/3} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (a B+14 A b) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{112 b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{27 \left(1+\sqrt{3}\right) a \sqrt{e x} \sqrt{a+b x^3} (a B+14 A b)}{112 b^{2/3} e^2 \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}+\frac{(e x)^{5/2} \left(a+b x^3\right)^{3/2} (a B+14 A b)}{7 a e^4}+\frac{9 (e x)^{5/2} \sqrt{a+b x^3} (a B+14 A b)}{56 e^4}-\frac{2 A \left(a+b x^3\right)^{5/2}}{a e \sqrt{e x}}",1,"(2*x*Sqrt[a + b*x^3]*((-5*A*(a + b*x^3)^2)/a + ((14*A*b + a*B)*x^3*Hypergeometric2F1[-3/2, 5/6, 11/6, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(5*(e*x)^(3/2))","C",1
533,1,126,152,0.1539423,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{(e x)^{5/2}} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/(e*x)^(5/2),x]","\frac{x \sqrt{a+b x^3} \left(3 \sqrt{a} x^{3/2} (a B+4 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)+\sqrt{b} \sqrt{\frac{b x^3}{a}+1} \left(-8 a A+5 a B x^3+4 A b x^3+2 b B x^6\right)\right)}{12 \sqrt{b} (e x)^{5/2} \sqrt{\frac{b x^3}{a}+1}}","\frac{a (a B+4 A b) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{4 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \left(a+b x^3\right)^{3/2} (a B+4 A b)}{6 a e^4}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+4 A b)}{4 e^4}-\frac{2 A \left(a+b x^3\right)^{5/2}}{3 a e (e x)^{3/2}}",1,"(x*Sqrt[a + b*x^3]*(Sqrt[b]*Sqrt[1 + (b*x^3)/a]*(-8*a*A + 4*A*b*x^3 + 5*a*B*x^3 + 2*b*B*x^6) + 3*Sqrt[a]*(4*A*b + a*B)*x^(3/2)*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(12*Sqrt[b]*(e*x)^(5/2)*Sqrt[1 + (b*x^3)/a])","A",1
534,1,85,314,0.0777034,"\int \frac{\left(a+b x^3\right)^{3/2} \left(A+B x^3\right)}{(e x)^{7/2}} \, dx","Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/(e*x)^(7/2),x]","\frac{2 x \sqrt{a+b x^3} \left(\frac{5 x^3 (a B+2 A b) \, _2F_1\left(-\frac{3}{2},\frac{1}{6};\frac{7}{6};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-\frac{A \left(a+b x^3\right)^2}{a}\right)}{5 (e x)^{7/2}}","\frac{9\ 3^{3/4} a^{2/3} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (a B+2 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{40 e^4 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{e x} \left(a+b x^3\right)^{3/2} (a B+2 A b)}{5 a e^4}+\frac{9 \sqrt{e x} \sqrt{a+b x^3} (a B+2 A b)}{20 e^4}-\frac{2 A \left(a+b x^3\right)^{5/2}}{5 a e (e x)^{5/2}}",1,"(2*x*Sqrt[a + b*x^3]*(-((A*(a + b*x^3)^2)/a) + (5*(2*A*b + a*B)*x^3*Hypergeometric2F1[-3/2, 1/6, 7/6, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(5*(e*x)^(7/2))","C",1
535,1,188,241,0.3137624,"\int (e x)^{7/2} \left(a+b x^3\right)^{5/2} \left(A+B x^3\right) \, dx","Integrate[(e*x)^(7/2)*(a + b*x^3)^(5/2)*(A + B*x^3),x]","\frac{e^3 \sqrt{e x} \sqrt{a+b x^3} \left(15 a^{7/2} (3 a B-10 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)+\sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1} \left(-45 a^4 B+30 a^3 b \left(5 A+B x^3\right)+4 a^2 b^2 x^3 \left(295 A+186 B x^3\right)+16 a b^3 x^6 \left(85 A+63 B x^3\right)+96 b^4 x^9 \left(5 A+4 B x^3\right)\right)\right)}{5760 b^{5/2} \sqrt{x} \sqrt{\frac{b x^3}{a}+1}}","-\frac{a^4 e^{7/2} (10 A b-3 a B) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{384 b^{5/2}}+\frac{a^3 e^2 (e x)^{3/2} \sqrt{a+b x^3} (10 A b-3 a B)}{384 b^2}+\frac{a^2 (e x)^{9/2} \sqrt{a+b x^3} (10 A b-3 a B)}{192 b e}+\frac{(e x)^{9/2} \left(a+b x^3\right)^{5/2} (10 A b-3 a B)}{120 b e}+\frac{a (e x)^{9/2} \left(a+b x^3\right)^{3/2} (10 A b-3 a B)}{144 b e}+\frac{B (e x)^{9/2} \left(a+b x^3\right)^{7/2}}{15 b e}",1,"(e^3*Sqrt[e*x]*Sqrt[a + b*x^3]*(Sqrt[b]*x^(3/2)*Sqrt[1 + (b*x^3)/a]*(-45*a^4*B + 30*a^3*b*(5*A + B*x^3) + 96*b^4*x^9*(5*A + 4*B*x^3) + 16*a*b^3*x^6*(85*A + 63*B*x^3) + 4*a^2*b^2*x^3*(295*A + 186*B*x^3)) + 15*a^(7/2)*(-10*A*b + 3*a*B)*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(5760*b^(5/2)*Sqrt[x]*Sqrt[1 + (b*x^3)/a])","A",1
536,1,116,404,0.1935639,"\int (e x)^{5/2} \left(a+b x^3\right)^{5/2} \left(A+B x^3\right) \, dx","Integrate[(e*x)^(5/2)*(a + b*x^3)^(5/2)*(A + B*x^3),x]","\frac{e^2 \sqrt{e x} \sqrt{a+b x^3} \left(7 a^3 (a B-4 A b) \, _2F_1\left(-\frac{5}{2},\frac{1}{6};\frac{7}{6};-\frac{b x^3}{a}\right)-\left(a+b x^3\right)^3 \sqrt{\frac{b x^3}{a}+1} \left(7 a B-28 A b-22 b B x^3\right)\right)}{308 b^2 \sqrt{\frac{b x^3}{a}+1}}","-\frac{27\ 3^{3/4} a^{11/3} e^2 \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (4 A b-a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{11264 b^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{81 a^3 e^2 \sqrt{e x} \sqrt{a+b x^3} (4 A b-a B)}{5632 b^2}+\frac{27 a^2 (e x)^{7/2} \sqrt{a+b x^3} (4 A b-a B)}{1408 b e}+\frac{15 a (e x)^{7/2} \left(a+b x^3\right)^{3/2} (4 A b-a B)}{704 b e}+\frac{(e x)^{7/2} \left(a+b x^3\right)^{5/2} (4 A b-a B)}{44 b e}+\frac{B (e x)^{7/2} \left(a+b x^3\right)^{7/2}}{14 b e}",1,"(e^2*Sqrt[e*x]*Sqrt[a + b*x^3]*(-((a + b*x^3)^3*Sqrt[1 + (b*x^3)/a]*(-28*A*b + 7*a*B - 22*b*B*x^3)) + 7*a^3*(-4*A*b + a*B)*Hypergeometric2F1[-5/2, 1/6, 7/6, -((b*x^3)/a)]))/(308*b^2*Sqrt[1 + (b*x^3)/a])","C",1
537,1,99,661,0.1804781,"\int (e x)^{3/2} \left(a+b x^3\right)^{5/2} \left(A+B x^3\right) \, dx","Integrate[(e*x)^(3/2)*(a + b*x^3)^(5/2)*(A + B*x^3),x]","\frac{x (e x)^{3/2} \sqrt{a+b x^3} \left(a^2 (26 A b-5 a B) \, _2F_1\left(-\frac{5}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right)+5 B \sqrt{\frac{b x^3}{a}+1} \left(a+b x^3\right)^3\right)}{65 b \sqrt{\frac{b x^3}{a}+1}}","-\frac{27\ 3^{3/4} \left(1-\sqrt{3}\right) a^{10/3} e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (26 A b-5 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{23296 b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{81 \sqrt[4]{3} a^{10/3} e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (26 A b-5 a B) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{11648 b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{81 \left(1+\sqrt{3}\right) a^3 e \sqrt{e x} \sqrt{a+b x^3} (26 A b-5 a B)}{11648 b^{5/3} \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}+\frac{27 a^2 (e x)^{5/2} \sqrt{a+b x^3} (26 A b-5 a B)}{5824 b e}+\frac{(e x)^{5/2} \left(a+b x^3\right)^{5/2} (26 A b-5 a B)}{260 b e}+\frac{3 a (e x)^{5/2} \left(a+b x^3\right)^{3/2} (26 A b-5 a B)}{728 b e}+\frac{B (e x)^{5/2} \left(a+b x^3\right)^{7/2}}{13 b e}",1,"(x*(e*x)^(3/2)*Sqrt[a + b*x^3]*(5*B*(a + b*x^3)^3*Sqrt[1 + (b*x^3)/a] + a^2*(26*A*b - 5*a*B)*Hypergeometric2F1[-5/2, 5/6, 11/6, -((b*x^3)/a)]))/(65*b*Sqrt[1 + (b*x^3)/a])","C",1
538,1,146,201,0.425912,"\int \sqrt{e x} \left(a+b x^3\right)^{5/2} \left(A+B x^3\right) \, dx","Integrate[Sqrt[e*x]*(a + b*x^3)^(5/2)*(A + B*x^3),x]","\frac{x \sqrt{e x} \sqrt{a+b x^3} \left(\frac{(8 A b-a B) \left(15 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)+\sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1} \left(33 a^2+26 a b x^3+8 b^2 x^6\right)\right)}{48 \sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1}}+B \left(a+b x^3\right)^3\right)}{12 b}","\frac{5 a^3 \sqrt{e} (8 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{192 b^{3/2}}+\frac{5 a^2 (e x)^{3/2} \sqrt{a+b x^3} (8 A b-a B)}{192 b e}+\frac{(e x)^{3/2} \left(a+b x^3\right)^{5/2} (8 A b-a B)}{72 b e}+\frac{5 a (e x)^{3/2} \left(a+b x^3\right)^{3/2} (8 A b-a B)}{288 b e}+\frac{B (e x)^{3/2} \left(a+b x^3\right)^{7/2}}{12 b e}",1,"(x*Sqrt[e*x]*Sqrt[a + b*x^3]*(B*(a + b*x^3)^3 + ((8*A*b - a*B)*(Sqrt[b]*x^(3/2)*Sqrt[1 + (b*x^3)/a]*(33*a^2 + 26*a*b*x^3 + 8*b^2*x^6) + 15*a^(5/2)*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(48*Sqrt[b]*x^(3/2)*Sqrt[1 + (b*x^3)/a])))/(12*b)","A",1
539,1,84,364,0.0674286,"\int \frac{\left(a+b x^3\right)^{5/2} \left(A+B x^3\right)}{\sqrt{e x}} \, dx","Integrate[((a + b*x^3)^(5/2)*(A + B*x^3))/Sqrt[e*x],x]","\frac{x \sqrt{a+b x^3} \left(B \left(a+b x^3\right)^3-\frac{a^2 (a B-22 A b) \, _2F_1\left(-\frac{5}{2},\frac{1}{6};\frac{7}{6};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}\right)}{11 b \sqrt{e x}}","\frac{27\ 3^{3/4} a^{8/3} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (22 A b-a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2816 b e \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{27 a^2 \sqrt{e x} \sqrt{a+b x^3} (22 A b-a B)}{1408 b e}+\frac{\sqrt{e x} \left(a+b x^3\right)^{5/2} (22 A b-a B)}{176 b e}+\frac{3 a \sqrt{e x} \left(a+b x^3\right)^{3/2} (22 A b-a B)}{352 b e}+\frac{B \sqrt{e x} \left(a+b x^3\right)^{7/2}}{11 b e}",1,"(x*Sqrt[a + b*x^3]*(B*(a + b*x^3)^3 - (a^2*(-22*A*b + a*B)*Hypergeometric2F1[-5/2, 1/6, 7/6, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(11*b*Sqrt[e*x])","C",1
540,1,87,650,0.0463354,"\int \frac{\left(a+b x^3\right)^{5/2} \left(A+B x^3\right)}{(e x)^{3/2}} \, dx","Integrate[((a + b*x^3)^(5/2)*(A + B*x^3))/(e*x)^(3/2),x]","\frac{2 x \sqrt{a+b x^3} \left(\frac{a^2 x^3 (a B+20 A b) \, _2F_1\left(-\frac{5}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-5 A \left(a+b x^3\right)^3\right)}{5 a (e x)^{3/2}}","-\frac{27\ 3^{3/4} \left(1-\sqrt{3}\right) a^{7/3} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (a B+20 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{896 b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{81 \sqrt[4]{3} a^{7/3} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (a B+20 A b) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{448 b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{81 \left(1+\sqrt{3}\right) a^2 \sqrt{e x} \sqrt{a+b x^3} (a B+20 A b)}{448 b^{2/3} e^2 \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}+\frac{(e x)^{5/2} \left(a+b x^3\right)^{5/2} (a B+20 A b)}{10 a e^4}+\frac{3 (e x)^{5/2} \left(a+b x^3\right)^{3/2} (a B+20 A b)}{28 e^4}+\frac{27 a (e x)^{5/2} \sqrt{a+b x^3} (a B+20 A b)}{224 e^4}-\frac{2 A \left(a+b x^3\right)^{7/2}}{a e \sqrt{e x}}",1,"(2*x*Sqrt[a + b*x^3]*(-5*A*(a + b*x^3)^3 + (a^2*(20*A*b + a*B)*x^3*Hypergeometric2F1[-5/2, 5/6, 11/6, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(5*a*(e*x)^(3/2))","C",1
541,1,150,188,0.2132195,"\int \frac{\left(a+b x^3\right)^{5/2} \left(A+B x^3\right)}{(e x)^{5/2}} \, dx","Integrate[((a + b*x^3)^(5/2)*(A + B*x^3))/(e*x)^(5/2),x]","\frac{x \sqrt{a+b x^3} \left(15 a^{3/2} x^{3/2} (a B+6 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)+\sqrt{b} \sqrt{\frac{b x^3}{a}+1} \left(a^2 \left(33 B x^3-48 A\right)+a \left(54 A b x^3+26 b B x^6\right)+4 b^2 x^6 \left(3 A+2 B x^3\right)\right)\right)}{72 \sqrt{b} (e x)^{5/2} \sqrt{\frac{b x^3}{a}+1}}","\frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{24 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \left(a+b x^3\right)^{5/2} (a B+6 A b)}{9 a e^4}+\frac{5 (e x)^{3/2} \left(a+b x^3\right)^{3/2} (a B+6 A b)}{36 e^4}+\frac{5 a (e x)^{3/2} \sqrt{a+b x^3} (a B+6 A b)}{24 e^4}-\frac{2 A \left(a+b x^3\right)^{7/2}}{3 a e (e x)^{3/2}}",1,"(x*Sqrt[a + b*x^3]*(Sqrt[b]*Sqrt[1 + (b*x^3)/a]*(4*b^2*x^6*(3*A + 2*B*x^3) + a^2*(-48*A + 33*B*x^3) + a*(54*A*b*x^3 + 26*b*B*x^6)) + 15*a^(3/2)*(6*A*b + a*B)*x^(3/2)*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(72*Sqrt[b]*(e*x)^(5/2)*Sqrt[1 + (b*x^3)/a])","A",1
542,1,88,352,0.045248,"\int \frac{\left(a+b x^3\right)^{5/2} \left(A+B x^3\right)}{(e x)^{7/2}} \, dx","Integrate[((a + b*x^3)^(5/2)*(A + B*x^3))/(e*x)^(7/2),x]","\frac{2 x \sqrt{a+b x^3} \left(\frac{a^2 x^3 (5 a B+16 A b) \, _2F_1\left(-\frac{5}{2},\frac{1}{6};\frac{7}{6};-\frac{b x^3}{a}\right)}{\sqrt{\frac{b x^3}{a}+1}}-A \left(a+b x^3\right)^3\right)}{5 a (e x)^{7/2}}","\frac{27\ 3^{3/4} a^{5/3} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (5 a B+16 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{640 e^4 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\sqrt{e x} \left(a+b x^3\right)^{5/2} (5 a B+16 A b)}{40 a e^4}+\frac{3 \sqrt{e x} \left(a+b x^3\right)^{3/2} (5 a B+16 A b)}{80 e^4}+\frac{27 a \sqrt{e x} \sqrt{a+b x^3} (5 a B+16 A b)}{320 e^4}-\frac{2 A \left(a+b x^3\right)^{7/2}}{5 a e (e x)^{5/2}}",1,"(2*x*Sqrt[a + b*x^3]*(-(A*(a + b*x^3)^3) + (a^2*(16*A*b + 5*a*B)*x^3*Hypergeometric2F1[-5/2, 1/6, 7/6, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(5*a*(e*x)^(7/2))","C",1
543,1,97,121,0.1689931,"\int \frac{(e x)^{7/2} \left(A+B x^3\right)}{\sqrt{a+b x^3}} \, dx","Integrate[((e*x)^(7/2)*(A + B*x^3))/Sqrt[a + b*x^3],x]","\frac{e^3 \sqrt{e x} \left(\sqrt{b} x^{3/2} \sqrt{a+b x^3} \left(-3 a B+4 A b+2 b B x^3\right)+a (3 a B-4 A b) \tanh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a+b x^3}}\right)\right)}{12 b^{5/2} \sqrt{x}}","-\frac{a e^{7/2} (4 A b-3 a B) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{12 b^{5/2}}+\frac{e^2 (e x)^{3/2} \sqrt{a+b x^3} (4 A b-3 a B)}{12 b^2}+\frac{B (e x)^{9/2} \sqrt{a+b x^3}}{6 b e}",1,"(e^3*Sqrt[e*x]*(Sqrt[b]*x^(3/2)*Sqrt[a + b*x^3]*(4*A*b - 3*a*B + 2*b*B*x^3) + a*(-4*A*b + 3*a*B)*ArcTanh[(Sqrt[b]*x^(3/2))/Sqrt[a + b*x^3]]))/(12*b^(5/2)*Sqrt[x])","A",1
544,1,98,286,0.1490599,"\int \frac{(e x)^{5/2} \left(A+B x^3\right)}{\sqrt{a+b x^3}} \, dx","Integrate[((e*x)^(5/2)*(A + B*x^3))/Sqrt[a + b*x^3],x]","\frac{e^2 \sqrt{e x} \left(a \sqrt{\frac{b x^3}{a}+1} (7 a B-10 A b) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{b x^3}{a}\right)-\left(a+b x^3\right) \left(7 a B-10 A b-4 b B x^3\right)\right)}{20 b^2 \sqrt{a+b x^3}}","-\frac{a^{2/3} e^2 \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (10 A b-7 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{40 \sqrt[4]{3} b^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{e^2 \sqrt{e x} \sqrt{a+b x^3} (10 A b-7 a B)}{20 b^2}+\frac{B (e x)^{7/2} \sqrt{a+b x^3}}{5 b e}",1,"(e^2*Sqrt[e*x]*(-((a + b*x^3)*(-10*A*b + 7*a*B - 4*b*B*x^3)) + a*(-10*A*b + 7*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/6, 1/2, 7/6, -((b*x^3)/a)]))/(20*b^2*Sqrt[a + b*x^3])","C",1
545,1,80,543,0.1303552,"\int \frac{(e x)^{3/2} \left(A+B x^3\right)}{\sqrt{a+b x^3}} \, dx","Integrate[((e*x)^(3/2)*(A + B*x^3))/Sqrt[a + b*x^3],x]","\frac{x (e x)^{3/2} \left(\sqrt{\frac{b x^3}{a}+1} (8 A b-5 a B) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right)+5 B \left(a+b x^3\right)\right)}{20 b \sqrt{a+b x^3}}","-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a} e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (8 A b-5 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{16 \sqrt[4]{3} b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt[3]{a} e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (8 A b-5 a B) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{8 b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\left(1+\sqrt{3}\right) e \sqrt{e x} \sqrt{a+b x^3} (8 A b-5 a B)}{8 b^{5/3} \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}+\frac{B (e x)^{5/2} \sqrt{a+b x^3}}{4 b e}",1,"(x*(e*x)^(3/2)*(5*B*(a + b*x^3) + (8*A*b - 5*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 5/6, 11/6, -((b*x^3)/a)]))/(20*b*Sqrt[a + b*x^3])","C",1
546,1,78,83,0.0591412,"\int \frac{\sqrt{e x} \left(A+B x^3\right)}{\sqrt{a+b x^3}} \, dx","Integrate[(Sqrt[e*x]*(A + B*x^3))/Sqrt[a + b*x^3],x]","\frac{\sqrt{e x} \left((2 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a+b x^3}}\right)+\sqrt{b} B x^{3/2} \sqrt{a+b x^3}\right)}{3 b^{3/2} \sqrt{x}}","\frac{\sqrt{e} (2 A b-a B) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{3 b^{3/2}}+\frac{B (e x)^{3/2} \sqrt{a+b x^3}}{3 b e}",1,"(Sqrt[e*x]*(Sqrt[b]*B*x^(3/2)*Sqrt[a + b*x^3] + (2*A*b - a*B)*ArcTanh[(Sqrt[b]*x^(3/2))/Sqrt[a + b*x^3]]))/(3*b^(3/2)*Sqrt[x])","A",1
547,1,80,249,0.0996358,"\int \frac{A+B x^3}{\sqrt{e x} \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/(Sqrt[e*x]*Sqrt[a + b*x^3]),x]","\frac{x \sqrt{\frac{b x^3}{a}+1} (4 A b-a B) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{b x^3}{a}\right)+B x \left(a+b x^3\right)}{2 b \sqrt{e x} \sqrt{a+b x^3}}","\frac{\sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (4 A b-a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{4 \sqrt[4]{3} \sqrt[3]{a} b e \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{B \sqrt{e x} \sqrt{a+b x^3}}{2 b e}",1,"(B*x*(a + b*x^3) + (4*A*b - a*B)*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/6, 1/2, 7/6, -((b*x^3)/a)])/(2*b*Sqrt[e*x]*Sqrt[a + b*x^3])","C",1
548,1,83,542,0.0535897,"\int \frac{A+B x^3}{(e x)^{3/2} \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/((e*x)^(3/2)*Sqrt[a + b*x^3]),x]","\frac{x \left(2 x^3 \sqrt{\frac{b x^3}{a}+1} (a B+2 A b) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right)-10 A \left(a+b x^3\right)\right)}{5 a (e x)^{3/2} \sqrt{a+b x^3}}","-\frac{\left(1-\sqrt{3}\right) \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (a B+2 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 \sqrt[4]{3} a^{2/3} b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt[4]{3} \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (a B+2 A b) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{a^{2/3} b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{\left(1+\sqrt{3}\right) \sqrt{e x} \sqrt{a+b x^3} (a B+2 A b)}{a b^{2/3} e^2 \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}-\frac{2 A \sqrt{a+b x^3}}{a e \sqrt{e x}}",1,"(x*(-10*A*(a + b*x^3) + 2*(2*A*b + a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 5/6, 11/6, -((b*x^3)/a)]))/(5*a*(e*x)^(3/2)*Sqrt[a + b*x^3])","C",1
549,1,65,75,0.0643526,"\int \frac{A+B x^3}{(e x)^{5/2} \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/((e*x)^(5/2)*Sqrt[a + b*x^3]),x]","\frac{2 x \left(\frac{B x^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a+b x^3}}\right)}{\sqrt{b}}-\frac{A \sqrt{a+b x^3}}{a}\right)}{3 (e x)^{5/2}}","\frac{2 B \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{3 \sqrt{b} e^{5/2}}-\frac{2 A \sqrt{a+b x^3}}{3 a e (e x)^{3/2}}",1,"(2*x*(-((A*Sqrt[a + b*x^3])/a) + (B*x^(3/2)*ArcTanh[(Sqrt[b]*x^(3/2))/Sqrt[a + b*x^3]])/Sqrt[b]))/(3*(e*x)^(5/2))","A",1
550,1,82,246,0.0568086,"\int \frac{A+B x^3}{(e x)^{7/2} \sqrt{a+b x^3}} \, dx","Integrate[(A + B*x^3)/((e*x)^(7/2)*Sqrt[a + b*x^3]),x]","-\frac{2 x \left(x^3 \sqrt{\frac{b x^3}{a}+1} (2 A b-5 a B) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{b x^3}{a}\right)+A \left(a+b x^3\right)\right)}{5 a (e x)^{7/2} \sqrt{a+b x^3}}","-\frac{\sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (2 A b-5 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{5 \sqrt[4]{3} a^{4/3} e^4 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 A \sqrt{a+b x^3}}{5 a e (e x)^{5/2}}",1,"(-2*x*(A*(a + b*x^3) + (2*A*b - 5*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/6, 1/2, 7/6, -((b*x^3)/a)]))/(5*a*(e*x)^(7/2)*Sqrt[a + b*x^3])","C",1
551,1,109,120,0.1763371,"\int \frac{(e x)^{7/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[((e*x)^(7/2)*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{e^3 \sqrt{e x} \left(\sqrt{b} x^{3/2} \left(3 a B-2 A b+b B x^3\right)-\sqrt{a} \sqrt{\frac{b x^3}{a}+1} (3 a B-2 A b) \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)\right)}{3 b^{5/2} \sqrt{x} \sqrt{a+b x^3}}","\frac{e^{7/2} (2 A b-3 a B) \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{3 b^{5/2}}-\frac{e^2 (e x)^{3/2} (2 A b-3 a B)}{3 b^2 \sqrt{a+b x^3}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}}",1,"(e^3*Sqrt[e*x]*(Sqrt[b]*x^(3/2)*(-2*A*b + 3*a*B + b*B*x^3) - Sqrt[a]*(-2*A*b + 3*a*B)*Sqrt[1 + (b*x^3)/a]*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(3*b^(5/2)*Sqrt[x]*Sqrt[a + b*x^3])","A",1
552,1,87,286,0.168799,"\int \frac{(e x)^{5/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[((e*x)^(5/2)*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{e^2 \sqrt{e x} \left(\sqrt{\frac{b x^3}{a}+1} (4 A b-7 a B) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{b x^3}{a}\right)+7 a B-4 A b+3 b B x^3\right)}{6 b^2 \sqrt{a+b x^3}}","\frac{e^2 \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (4 A b-7 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{12 \sqrt[4]{3} \sqrt[3]{a} b^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{e^2 \sqrt{e x} (4 A b-7 a B)}{6 b^2 \sqrt{a+b x^3}}+\frac{B (e x)^{7/2}}{2 b e \sqrt{a+b x^3}}",1,"(e^2*Sqrt[e*x]*(-4*A*b + 7*a*B + 3*b*B*x^3 + (4*A*b - 7*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/6, 1/2, 7/6, -((b*x^3)/a)]))/(6*b^2*Sqrt[a + b*x^3])","C",1
553,1,77,553,0.1186256,"\int \frac{(e x)^{3/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[((e*x)^(3/2)*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{x (e x)^{3/2} \left(\sqrt{\frac{b x^3}{a}+1} (2 A b-5 a B) \, _2F_1\left(\frac{5}{6},\frac{3}{2};\frac{11}{6};-\frac{b x^3}{a}\right)+5 a B\right)}{5 a b \sqrt{a+b x^3}}","\frac{\left(1-\sqrt{3}\right) e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (2 A b-5 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{6 \sqrt[4]{3} a^{2/3} b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (2 A b-5 a B) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{3^{3/4} a^{2/3} b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\left(1+\sqrt{3}\right) e \sqrt{e x} \sqrt{a+b x^3} (2 A b-5 a B)}{3 a b^{5/3} \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}+\frac{2 (e x)^{5/2} (A b-a B)}{3 a b e \sqrt{a+b x^3}}",1,"(x*(e*x)^(3/2)*(5*a*B + (2*A*b - 5*a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[5/6, 3/2, 11/6, -((b*x^3)/a)]))/(5*a*b*Sqrt[a + b*x^3])","C",1
554,1,93,85,0.0985456,"\int \frac{\sqrt{e x} \left(A+B x^3\right)}{\left(a+b x^3\right)^{3/2}} \, dx","Integrate[(Sqrt[e*x]*(A + B*x^3))/(a + b*x^3)^(3/2),x]","\frac{2 \sqrt{e x} \left(a^{3/2} B \sqrt{\frac{b x^3}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)+\sqrt{b} x^{3/2} (A b-a B)\right)}{3 a b^{3/2} \sqrt{x} \sqrt{a+b x^3}}","\frac{2 (e x)^{3/2} (A b-a B)}{3 a b e \sqrt{a+b x^3}}+\frac{2 B \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{3 b^{3/2}}",1,"(2*Sqrt[e*x]*(Sqrt[b]*(A*b - a*B)*x^(3/2) + a^(3/2)*B*Sqrt[1 + (b*x^3)/a]*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(3*a*b^(3/2)*Sqrt[x]*Sqrt[a + b*x^3])","A",1
555,1,79,258,0.0633654,"\int \frac{A+B x^3}{\sqrt{e x} \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/(Sqrt[e*x]*(a + b*x^3)^(3/2)),x]","\frac{2 x \left(\sqrt{\frac{b x^3}{a}+1} (a B+2 A b) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{b x^3}{a}\right)-a B+A b\right)}{3 a b \sqrt{e x} \sqrt{a+b x^3}}","\frac{\sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (a B+2 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{3 \sqrt[4]{3} a^{4/3} b e \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{e x} (A b-a B)}{3 a b e \sqrt{a+b x^3}}",1,"(2*x*(A*b - a*B + (2*A*b + a*B)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/6, 1/2, 7/6, -((b*x^3)/a)]))/(3*a*b*Sqrt[e*x]*Sqrt[a + b*x^3])","C",1
556,1,77,585,0.0512604,"\int \frac{A+B x^3}{(e x)^{3/2} \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/((e*x)^(3/2)*(a + b*x^3)^(3/2)),x]","\frac{x \left(2 x^3 \sqrt{\frac{b x^3}{a}+1} (a B-4 A b) \, _2F_1\left(\frac{5}{6},\frac{3}{2};\frac{11}{6};-\frac{b x^3}{a}\right)-10 a A\right)}{5 a^2 (e x)^{3/2} \sqrt{a+b x^3}}","-\frac{\left(1-\sqrt{3}\right) \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (4 A b-a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{3 \sqrt[4]{3} a^{5/3} b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (4 A b-a B) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{3^{3/4} a^{5/3} b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \left(1+\sqrt{3}\right) \sqrt{e x} \sqrt{a+b x^3} (4 A b-a B)}{3 a^2 b^{2/3} e^2 \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}-\frac{2 (e x)^{5/2} (4 A b-a B)}{3 a^2 e^4 \sqrt{a+b x^3}}-\frac{2 A}{a e \sqrt{e x} \sqrt{a+b x^3}}",1,"(x*(-10*a*A + 2*(-4*A*b + a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[5/6, 3/2, 11/6, -((b*x^3)/a)]))/(5*a^2*(e*x)^(3/2)*Sqrt[a + b*x^3])","C",1
557,1,45,67,0.0229171,"\int \frac{A+B x^3}{(e x)^{5/2} \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/((e*x)^(5/2)*(a + b*x^3)^(3/2)),x]","\frac{x \left(-2 a A+2 a B x^3-4 A b x^3\right)}{3 a^2 (e x)^{5/2} \sqrt{a+b x^3}}","-\frac{2 (e x)^{3/2} (2 A b-a B)}{3 a^2 e^4 \sqrt{a+b x^3}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^3}}",1,"(x*(-2*a*A - 4*A*b*x^3 + 2*a*B*x^3))/(3*a^2*(e*x)^(5/2)*Sqrt[a + b*x^3])","A",1
558,1,95,283,0.0579499,"\int \frac{A+B x^3}{(e x)^{7/2} \left(a+b x^3\right)^{3/2}} \, dx","Integrate[(A + B*x^3)/((e*x)^(7/2)*(a + b*x^3)^(3/2)),x]","\frac{x \left(4 x^3 \sqrt{\frac{b x^3}{a}+1} (5 a B-8 A b) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{b x^3}{a}\right)-2 \left(3 a A-5 a B x^3+8 A b x^3\right)\right)}{15 a^2 (e x)^{7/2} \sqrt{a+b x^3}}","-\frac{2 \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (8 A b-5 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{15 \sqrt[4]{3} a^{7/3} e^4 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{e x} (8 A b-5 a B)}{15 a^2 e^4 \sqrt{a+b x^3}}-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^3}}",1,"(x*(-2*(3*a*A + 8*A*b*x^3 - 5*a*B*x^3) + 4*(-8*A*b + 5*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/6, 1/2, 7/6, -((b*x^3)/a)]))/(15*a^2*(e*x)^(7/2)*Sqrt[a + b*x^3])","C",1
559,1,119,114,0.3034769,"\int \frac{(e x)^{7/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[((e*x)^(7/2)*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{2 e^3 \sqrt{e x} \left(3 a^{3/2} B \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right)+\sqrt{b} x^{3/2} \left(-3 a^2 B-4 a b B x^3+A b^2 x^3\right)\right)}{9 a b^{5/2} \sqrt{x} \left(a+b x^3\right)^{3/2}}","\frac{2 (e x)^{9/2} (A b-a B)}{9 a b e \left(a+b x^3\right)^{3/2}}+\frac{2 B e^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right)}{3 b^{5/2}}-\frac{2 B e^2 (e x)^{3/2}}{3 b^2 \sqrt{a+b x^3}}",1,"(2*e^3*Sqrt[e*x]*(Sqrt[b]*x^(3/2)*(-3*a^2*B + A*b^2*x^3 - 4*a*b*B*x^3) + 3*a^(3/2)*B*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*ArcSinh[(Sqrt[b]*x^(3/2))/Sqrt[a]]))/(9*a*b^(5/2)*Sqrt[x]*(a + b*x^3)^(3/2))","A",1
560,1,108,299,0.2251657,"\int \frac{(e x)^{5/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[((e*x)^(5/2)*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{2 e^2 \sqrt{e x} \left(-7 a^2 B+\left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (7 a B+2 A b) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{b x^3}{a}\right)-2 a b \left(A+5 B x^3\right)+A b^2 x^3\right)}{27 a b^2 \left(a+b x^3\right)^{3/2}}","\frac{e^2 \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (7 a B+2 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{27 \sqrt[4]{3} a^{4/3} b^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 e^2 \sqrt{e x} (7 a B+2 A b)}{27 a b^2 \sqrt{a+b x^3}}+\frac{2 (e x)^{7/2} (A b-a B)}{9 a b e \left(a+b x^3\right)^{3/2}}",1,"(2*e^2*Sqrt[e*x]*(-7*a^2*B + A*b^2*x^3 - 2*a*b*(A + 5*B*x^3) + (2*A*b + 7*a*B)*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/6, 1/2, 7/6, -((b*x^3)/a)]))/(27*a*b^2*(a + b*x^3)^(3/2))","C",1
561,1,86,596,0.1235182,"\int \frac{(e x)^{3/2} \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[((e*x)^(3/2)*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{x (e x)^{3/2} \left(\left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (5 a B+4 A b) \, _2F_1\left(\frac{5}{6},\frac{5}{2};\frac{11}{6};-\frac{b x^3}{a}\right)-5 a^2 B\right)}{10 a^2 b \left(a+b x^3\right)^{3/2}}","\frac{\left(1-\sqrt{3}\right) e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (5 a B+4 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{27 \sqrt[4]{3} a^{5/3} b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 e \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (5 a B+4 A b) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{9\ 3^{3/4} a^{5/3} b^{5/3} \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2 \left(1+\sqrt{3}\right) e \sqrt{e x} \sqrt{a+b x^3} (5 a B+4 A b)}{27 a^2 b^{5/3} \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}+\frac{2 (e x)^{5/2} (5 a B+4 A b)}{27 a^2 b e \sqrt{a+b x^3}}+\frac{2 (e x)^{5/2} (A b-a B)}{9 a b e \left(a+b x^3\right)^{3/2}}",1,"(x*(e*x)^(3/2)*(-5*a^2*B + (4*A*b + 5*a*B)*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[5/6, 5/2, 11/6, -((b*x^3)/a)]))/(10*a^2*b*(a + b*x^3)^(3/2))","C",1
562,1,44,79,0.038768,"\int \frac{\sqrt{e x} \left(A+B x^3\right)}{\left(a+b x^3\right)^{5/2}} \, dx","Integrate[(Sqrt[e*x]*(A + B*x^3))/(a + b*x^3)^(5/2),x]","\frac{2 x \sqrt{e x} \left(3 a A+a B x^3+2 A b x^3\right)}{9 a^2 \left(a+b x^3\right)^{3/2}}","\frac{2 (e x)^{3/2} (a B+2 A b)}{9 a^2 b e \sqrt{a+b x^3}}+\frac{2 (e x)^{3/2} (A b-a B)}{9 a b e \left(a+b x^3\right)^{3/2}}",1,"(2*x*Sqrt[e*x]*(3*a*A + 2*A*b*x^3 + a*B*x^3))/(9*a^2*(a + b*x^3)^(3/2))","A",1
563,1,107,297,0.1068535,"\int \frac{A+B x^3}{\sqrt{e x} \left(a+b x^3\right)^{5/2}} \, dx","Integrate[(A + B*x^3)/(Sqrt[e*x]*(a + b*x^3)^(5/2)),x]","\frac{2 x \left(-2 a^2 B+2 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (a B+8 A b) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{b x^3}{a}\right)+a b \left(11 A+B x^3\right)+8 A b^2 x^3\right)}{27 a^2 b \sqrt{e x} \left(a+b x^3\right)^{3/2}}","\frac{2 \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (a B+8 A b) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{27 \sqrt[4]{3} a^{7/3} b e \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{e x} (a B+8 A b)}{27 a^2 b e \sqrt{a+b x^3}}+\frac{2 \sqrt{e x} (A b-a B)}{9 a b e \left(a+b x^3\right)^{3/2}}",1,"(2*x*(-2*a^2*B + 8*A*b^2*x^3 + a*b*(11*A + B*x^3) + 2*(8*A*b + a*B)*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/6, 1/2, 7/6, -((b*x^3)/a)]))/(27*a^2*b*Sqrt[e*x]*(a + b*x^3)^(3/2))","C",1
564,1,85,624,0.0488546,"\int \frac{A+B x^3}{(e x)^{3/2} \left(a+b x^3\right)^{5/2}} \, dx","Integrate[(A + B*x^3)/((e*x)^(3/2)*(a + b*x^3)^(5/2)),x]","\frac{2 x \left(x^3 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (a B-10 A b) \, _2F_1\left(\frac{5}{6},\frac{5}{2};\frac{11}{6};-\frac{b x^3}{a}\right)-5 a^2 A\right)}{5 a^3 (e x)^{3/2} \left(a+b x^3\right)^{3/2}}","-\frac{4 \left(1-\sqrt{3}\right) \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (10 A b-a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{27 \sqrt[4]{3} a^{8/3} b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{8 \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (10 A b-a B) E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{9\ 3^{3/4} a^{8/3} b^{2/3} e^2 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{8 \left(1+\sqrt{3}\right) \sqrt{e x} \sqrt{a+b x^3} (10 A b-a B)}{27 a^3 b^{2/3} e^2 \left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)}-\frac{8 (e x)^{5/2} (10 A b-a B)}{27 a^3 e^4 \sqrt{a+b x^3}}-\frac{2 (e x)^{5/2} (10 A b-a B)}{9 a^2 e^4 \left(a+b x^3\right)^{3/2}}-\frac{2 A}{a e \sqrt{e x} \left(a+b x^3\right)^{3/2}}",1,"(2*x*(-5*a^2*A + (-10*A*b + a*B)*x^3*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[5/6, 5/2, 11/6, -((b*x^3)/a)]))/(5*a^3*(e*x)^(3/2)*(a + b*x^3)^(3/2))","C",1
565,1,65,104,0.0363865,"\int \frac{A+B x^3}{(e x)^{5/2} \left(a+b x^3\right)^{5/2}} \, dx","Integrate[(A + B*x^3)/((e*x)^(5/2)*(a + b*x^3)^(5/2)),x]","\frac{x \left(-6 a^2 \left(A-B x^3\right)+4 a b x^3 \left(B x^3-6 A\right)-16 A b^2 x^6\right)}{9 a^3 (e x)^{5/2} \left(a+b x^3\right)^{3/2}}","-\frac{4 (e x)^{3/2} (4 A b-a B)}{9 a^3 e^4 \sqrt{a+b x^3}}-\frac{2 (e x)^{3/2} (4 A b-a B)}{9 a^2 e^4 \left(a+b x^3\right)^{3/2}}-\frac{2 A}{3 a e (e x)^{3/2} \left(a+b x^3\right)^{3/2}}",1,"(x*(-16*A*b^2*x^6 - 6*a^2*(A - B*x^3) + 4*a*b*x^3*(-6*A + B*x^3)))/(9*a^3*(e*x)^(5/2)*(a + b*x^3)^(3/2))","A",1
566,1,121,320,0.0876837,"\int \frac{A+B x^3}{(e x)^{7/2} \left(a+b x^3\right)^{5/2}} \, dx","Integrate[(A + B*x^3)/((e*x)^(7/2)*(a + b*x^3)^(5/2)),x]","\frac{x \left(a^2 \left(110 B x^3-54 A\right)+32 x^3 \left(a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} (5 a B-14 A b) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{b x^3}{a}\right)+a \left(80 b B x^6-308 A b x^3\right)-224 A b^2 x^6\right)}{135 a^3 (e x)^{7/2} \left(a+b x^3\right)^{3/2}}","-\frac{16 \sqrt{e x} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} (14 A b-5 a B) F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt{3}\right) \sqrt[3]{b} x+\sqrt[3]{a}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{135 \sqrt[4]{3} a^{10/3} e^4 \sqrt{\frac{\sqrt[3]{b} x \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\sqrt[3]{a}+\left(1+\sqrt{3}\right) \sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{16 \sqrt{e x} (14 A b-5 a B)}{135 a^3 e^4 \sqrt{a+b x^3}}-\frac{2 \sqrt{e x} (14 A b-5 a B)}{45 a^2 e^4 \left(a+b x^3\right)^{3/2}}-\frac{2 A}{5 a e (e x)^{5/2} \left(a+b x^3\right)^{3/2}}",1,"(x*(-224*A*b^2*x^6 + a^2*(-54*A + 110*B*x^3) + a*(-308*A*b*x^3 + 80*b*B*x^6) + 32*(-14*A*b + 5*a*B)*x^3*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/6, 1/2, 7/6, -((b*x^3)/a)]))/(135*a^3*(e*x)^(7/2)*(a + b*x^3)^(3/2))","C",1
567,1,230,220,0.2593412,"\int \frac{x^{11} \sqrt[3]{a+b x^3}}{a d-b d x^3} \, dx","Integrate[(x^11*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]","-\frac{140 \sqrt[3]{2} a^{10/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)-70 \sqrt[3]{2} a^{10/3} \log \left(2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-140 \sqrt[3]{2} \sqrt{3} a^{10/3} \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)+507 a^3 \sqrt[3]{a+b x^3}+111 a^2 b x^3 \sqrt[3]{a+b x^3}+42 b^3 x^9 \sqrt[3]{a+b x^3}+66 a b^2 x^6 \sqrt[3]{a+b x^3}}{420 b^4 d}","\frac{a^{10/3} \log \left(a-b x^3\right)}{3\ 2^{2/3} b^4 d}-\frac{a^{10/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2^{2/3} b^4 d}+\frac{\sqrt[3]{2} a^{10/3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^4 d}-\frac{a^3 \sqrt[3]{a+b x^3}}{b^4 d}-\frac{a^2 \left(a+b x^3\right)^{4/3}}{4 b^4 d}+\frac{a \left(a+b x^3\right)^{7/3}}{7 b^4 d}-\frac{\left(a+b x^3\right)^{10/3}}{10 b^4 d}",1,"-1/420*(507*a^3*(a + b*x^3)^(1/3) + 111*a^2*b*x^3*(a + b*x^3)^(1/3) + 66*a*b^2*x^6*(a + b*x^3)^(1/3) + 42*b^3*x^9*(a + b*x^3)^(1/3) - 140*2^(1/3)*Sqrt[3]*a^(10/3)*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] + 140*2^(1/3)*a^(10/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)] - 70*2^(1/3)*a^(10/3)*Log[2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(b^4*d)","A",1
568,1,207,174,0.1386037,"\int \frac{x^8 \sqrt[3]{a+b x^3}}{a d-b d x^3} \, dx","Integrate[(x^8*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]","\frac{1}{3} \left(-\frac{\frac{2 \sqrt[3]{2} a^{7/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{b d}-\frac{\sqrt[3]{2} a^{7/3} \left(\log \left(2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)\right)}{b d}}{2 b^2}-\frac{3 a^2 \sqrt[3]{a+b x^3}}{b^3 d}-\frac{3 \left(a+b x^3\right)^{7/3}}{7 b^3 d}\right)","\frac{a^{7/3} \log \left(a-b x^3\right)}{3\ 2^{2/3} b^3 d}-\frac{a^{7/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2^{2/3} b^3 d}+\frac{\sqrt[3]{2} a^{7/3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^3 d}-\frac{a^2 \sqrt[3]{a+b x^3}}{b^3 d}-\frac{\left(a+b x^3\right)^{7/3}}{7 b^3 d}",1,"((-3*a^2*(a + b*x^3)^(1/3))/(b^3*d) - (3*(a + b*x^3)^(7/3))/(7*b^3*d) - ((2*2^(1/3)*a^(7/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)])/(b*d) - (2^(1/3)*a^(7/3)*(2*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + b*x^3)^(1/3))/(Sqrt[3]*a^(1/3))] + Log[2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]))/(b*d))/(2*b^2))/3","A",1
569,1,186,172,0.0962768,"\int \frac{x^5 \sqrt[3]{a+b x^3}}{a d-b d x^3} \, dx","Integrate[(x^5*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]","-\frac{4 \sqrt[3]{2} a^{4/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)-2 \sqrt[3]{2} a^{4/3} \log \left(2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-4 \sqrt[3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)+3 b x^3 \sqrt[3]{a+b x^3}+15 a \sqrt[3]{a+b x^3}}{12 b^2 d}","\frac{a^{4/3} \log \left(a-b x^3\right)}{3\ 2^{2/3} b^2 d}-\frac{a^{4/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2^{2/3} b^2 d}+\frac{\sqrt[3]{2} a^{4/3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^2 d}-\frac{a \sqrt[3]{a+b x^3}}{b^2 d}-\frac{\left(a+b x^3\right)^{4/3}}{4 b^2 d}",1,"-1/12*(15*a*(a + b*x^3)^(1/3) + 3*b*x^3*(a + b*x^3)^(1/3) - 4*2^(1/3)*Sqrt[3]*a^(4/3)*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] + 4*2^(1/3)*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)] - 2*2^(1/3)*a^(4/3)*Log[2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(b^2*d)","A",1
570,1,167,150,0.0527966,"\int \frac{x^2 \sqrt[3]{a+b x^3}}{a d-b d x^3} \, dx","Integrate[(x^2*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]","\frac{\sqrt[3]{2} \sqrt[3]{a} \log \left(2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-6 \sqrt[3]{a+b x^3}-2 \sqrt[3]{2} \sqrt[3]{a} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt[3]{2} \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)}{6 b d}","-\frac{\sqrt[3]{a+b x^3}}{b d}+\frac{\sqrt[3]{a} \log \left(a-b x^3\right)}{3\ 2^{2/3} b d}-\frac{\sqrt[3]{a} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2^{2/3} b d}+\frac{\sqrt[3]{2} \sqrt[3]{a} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b d}",1,"(-6*(a + b*x^3)^(1/3) + 2*2^(1/3)*Sqrt[3]*a^(1/3)*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*2^(1/3)*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)] + 2^(1/3)*a^(1/3)*Log[2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(6*b*d)","A",1
571,1,233,214,0.1072811,"\int \frac{\sqrt[3]{a+b x^3}}{x \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x*(a*d - b*d*x^3)),x]","-\frac{\log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-\sqrt[3]{2} \log \left(2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt[3]{2} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-2 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)}{6 a^{2/3} d}","\frac{\log \left(a-b x^3\right)}{3\ 2^{2/3} a^{2/3} d}+\frac{\log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2 a^{2/3} d}-\frac{\log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2^{2/3} a^{2/3} d}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} d}+\frac{\sqrt[3]{2} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} d}-\frac{\log (x)}{2 a^{2/3} d}",1,"-1/6*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) - (a + b*x^3)^(1/3)] + 2*2^(1/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)] - 2^(1/3)*Log[2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(a^(2/3)*d)","A",1
572,1,280,268,0.1167296,"\int \frac{\sqrt[3]{a+b x^3}}{x^4 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^4*(a*d - b*d*x^3)),x]","-\frac{6 a^{2/3} \sqrt[3]{a+b x^3}+4 b x^3 \log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-3 \sqrt[3]{2} b x^3 \log \left(2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-8 b x^3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+6 \sqrt[3]{2} b x^3 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+8 \sqrt{3} b x^3 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-6 \sqrt[3]{2} \sqrt{3} b x^3 \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)}{18 a^{5/3} d x^3}","\frac{b \log \left(a-b x^3\right)}{3\ 2^{2/3} a^{5/3} d}+\frac{2 b \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{3 a^{5/3} d}-\frac{b \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2^{2/3} a^{5/3} d}-\frac{4 b \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} d}+\frac{\sqrt[3]{2} b \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{5/3} d}-\frac{2 b \log (x)}{3 a^{5/3} d}-\frac{\left(a+b x^3\right)^{4/3}}{3 a^2 d x^3}+\frac{b \sqrt[3]{a+b x^3}}{3 a^2 d}",1,"-1/18*(6*a^(2/3)*(a + b*x^3)^(1/3) + 8*Sqrt[3]*b*x^3*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 6*2^(1/3)*Sqrt[3]*b*x^3*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 8*b*x^3*Log[a^(1/3) - (a + b*x^3)^(1/3)] + 6*2^(1/3)*b*x^3*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)] + 4*b*x^3*Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)] - 3*2^(1/3)*b*x^3*Log[2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(a^(5/3)*d*x^3)","A",1
573,1,314,283,0.1375217,"\int \frac{\sqrt[3]{a+b x^3}}{x^7 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^7*(a*d - b*d*x^3)),x]","-\frac{11 b^2 x^6 \log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-9 \sqrt[3]{2} b^2 x^6 \log \left(2^{2/3} a^{2/3}+\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)+21 a^{2/3} b x^3 \sqrt[3]{a+b x^3}+9 a^{5/3} \sqrt[3]{a+b x^3}-22 b^2 x^6 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+18 \sqrt[3]{2} b^2 x^6 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+22 \sqrt{3} b^2 x^6 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-18 \sqrt[3]{2} \sqrt{3} b^2 x^6 \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)}{54 a^{8/3} d x^6}","\frac{b^2 \log \left(a-b x^3\right)}{3\ 2^{2/3} a^{8/3} d}+\frac{11 b^2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{18 a^{8/3} d}-\frac{b^2 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2^{2/3} a^{8/3} d}-\frac{11 b^2 \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{8/3} d}+\frac{\sqrt[3]{2} b^2 \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{8/3} d}-\frac{11 b^2 \log (x)}{18 a^{8/3} d}-\frac{2 b \sqrt[3]{a+b x^3}}{9 a^2 d x^3}-\frac{\left(a+b x^3\right)^{4/3}}{6 a^2 d x^6}",1,"-1/54*(9*a^(5/3)*(a + b*x^3)^(1/3) + 21*a^(2/3)*b*x^3*(a + b*x^3)^(1/3) + 22*Sqrt[3]*b^2*x^6*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 18*2^(1/3)*Sqrt[3]*b^2*x^6*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 22*b^2*x^6*Log[a^(1/3) - (a + b*x^3)^(1/3)] + 18*2^(1/3)*b^2*x^6*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)] + 11*b^2*x^6*Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)] - 9*2^(1/3)*b^2*x^6*Log[2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(a^(8/3)*d*x^6)","A",1
574,1,177,268,0.2096135,"\int \frac{x^7 \sqrt[3]{a+b x^3}}{a d-b d x^3} \, dx","Integrate[(x^7*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]","\frac{22 a b x^5 \left(1-\frac{b^2 x^6}{a^2}\right)^{2/3} F_1\left(\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-5 x^2 \left(\left(1-\frac{b x^3}{a}\right)^{2/3} \left(7 a^2+10 a b x^3+3 b^2 x^6\right)-7 a^2 \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{2 b x^3}{a-b x^3}\right)\right)}{90 b^2 d \left(a+b x^3\right)^{2/3} \left(1-\frac{b x^3}{a}\right)^{2/3}}","\frac{a^2 \log \left(a d-b d x^3\right)}{3\ 2^{2/3} b^{8/3} d}+\frac{11 a^2 \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{18 b^{8/3} d}-\frac{a^2 \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2^{2/3} b^{8/3} d}+\frac{11 a^2 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{9 \sqrt{3} b^{8/3} d}-\frac{\sqrt[3]{2} a^2 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{8/3} d}-\frac{7 a x^2 \sqrt[3]{a+b x^3}}{18 b^2 d}-\frac{x^5 \sqrt[3]{a+b x^3}}{6 b d}",1,"(22*a*b*x^5*(1 - (b^2*x^6)/a^2)^(2/3)*AppellF1[5/3, 2/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a] - 5*x^2*((1 - (b*x^3)/a)^(2/3)*(7*a^2 + 10*a*b*x^3 + 3*b^2*x^6) - 7*a^2*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (-2*b*x^3)/(a - b*x^3)]))/(90*b^2*d*(a + b*x^3)^(2/3)*(1 - (b*x^3)/a)^(2/3))","C",0
575,1,160,233,0.1664951,"\int \frac{x^4 \sqrt[3]{a+b x^3}}{a d-b d x^3} \, dx","Integrate[(x^4*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]","\frac{4 b x^5 \left(1-\frac{b^2 x^6}{a^2}\right)^{2/3} F_1\left(\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-5 x^2 \left(\left(a+b x^3\right) \left(1-\frac{b x^3}{a}\right)^{2/3}-a \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{2 b x^3}{a-b x^3}\right)\right)}{15 b d \left(a+b x^3\right)^{2/3} \left(1-\frac{b x^3}{a}\right)^{2/3}}","\frac{a \log \left(a d-b d x^3\right)}{3\ 2^{2/3} b^{5/3} d}+\frac{2 a \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{3 b^{5/3} d}-\frac{a \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2^{2/3} b^{5/3} d}+\frac{4 a \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{5/3} d}-\frac{\sqrt[3]{2} a \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{5/3} d}-\frac{x^2 \sqrt[3]{a+b x^3}}{3 b d}",1,"(4*b*x^5*(1 - (b^2*x^6)/a^2)^(2/3)*AppellF1[5/3, 2/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a] - 5*x^2*((a + b*x^3)*(1 - (b*x^3)/a)^(2/3) - a*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (-2*b*x^3)/(a - b*x^3)]))/(15*b*d*(a + b*x^3)^(2/3)*(1 - (b*x^3)/a)^(2/3))","C",0
576,1,63,201,0.0412171,"\int \frac{x \sqrt[3]{a+b x^3}}{a d-b d x^3} \, dx","Integrate[(x*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]","\frac{x^2 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{2}{3};-\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{2 d \left(a+b x^3\right)^{2/3}}","\frac{\log \left(a d-b d x^3\right)}{3\ 2^{2/3} b^{2/3} d}+\frac{\log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2 b^{2/3} d}-\frac{\log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2^{2/3} b^{2/3} d}+\frac{\tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{2/3} d}-\frac{\sqrt[3]{2} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{2/3} d}",1,"(x^2*(1 + (b*x^3)/a)^(2/3)*AppellF1[2/3, -1/3, 1, 5/3, -((b*x^3)/a), (b*x^3)/a])/(2*d*(a + b*x^3)^(2/3))","C",0
577,1,45,156,0.0175886,"\int \frac{\sqrt[3]{a+b x^3}}{x^2 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^2*(a*d - b*d*x^3)),x]","-\frac{\sqrt[3]{a+b x^3} \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{2 b x^3}{b x^3+a}\right)}{a d x}","-\frac{\sqrt[3]{a+b x^3}}{a d x}+\frac{\sqrt[3]{b} \log \left(a d-b d x^3\right)}{3\ 2^{2/3} a d}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2^{2/3} a d}-\frac{\sqrt[3]{2} \sqrt[3]{b} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} a d}",1,"-(((a + b*x^3)^(1/3)*Hypergeometric2F1[-1/3, 1, 2/3, (2*b*x^3)/(a + b*x^3)])/(a*d*x))","C",1
578,1,125,183,5.1039991,"\int \frac{\sqrt[3]{a+b x^3}}{x^5 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^5*(a*d - b*d*x^3)),x]","\frac{b^2 x^2 \left(\frac{a+b x^3}{a}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{2 b x^3}{a \left(1-\frac{b x^3}{a}\right)}\right)}{a^2 d \left(a+b x^3\right)^{2/3} \left(1-\frac{b x^3}{a}\right)^{2/3}}-\frac{\left(\frac{5 b}{4 a^2 x}+\frac{1}{4 a x^4}\right) \sqrt[3]{a+b x^3}}{d}","\frac{b^{4/3} \log \left(a d-b d x^3\right)}{3\ 2^{2/3} a^2 d}-\frac{b^{4/3} \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2^{2/3} a^2 d}-\frac{\sqrt[3]{2} b^{4/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} a^2 d}-\frac{5 b \sqrt[3]{a+b x^3}}{4 a^2 d x}-\frac{\sqrt[3]{a+b x^3}}{4 a d x^4}",1,"-(((1/(4*a*x^4) + (5*b)/(4*a^2*x))*(a + b*x^3)^(1/3))/d) + (b^2*x^2*((a + b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (-2*b*x^3)/(a*(1 - (b*x^3)/a))])/(a^2*d*(a + b*x^3)^(2/3)*(1 - (b*x^3)/a)^(2/3))","C",0
579,1,135,210,5.121834,"\int \frac{\sqrt[3]{a+b x^3}}{x^8 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^8*(a*d - b*d*x^3)),x]","\frac{7 b^3 x^9 \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{2 b x^3}{a-b x^3}\right)-\left(1-\frac{b x^3}{a}\right)^{2/3} \left(a^3+3 a^2 b x^3+10 a b^2 x^6+8 b^3 x^9\right)}{7 a^3 d x^7 \left(a+b x^3\right)^{2/3} \left(1-\frac{b x^3}{a}\right)^{2/3}}","\frac{b^{7/3} \log \left(a d-b d x^3\right)}{3\ 2^{2/3} a^3 d}-\frac{b^{7/3} \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2^{2/3} a^3 d}-\frac{\sqrt[3]{2} b^{7/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} a^3 d}-\frac{8 b^2 \sqrt[3]{a+b x^3}}{7 a^3 d x}-\frac{2 b \sqrt[3]{a+b x^3}}{7 a^2 d x^4}-\frac{\sqrt[3]{a+b x^3}}{7 a d x^7}",1,"(-((1 - (b*x^3)/a)^(2/3)*(a^3 + 3*a^2*b*x^3 + 10*a*b^2*x^6 + 8*b^3*x^9)) + 7*b^3*x^9*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (-2*b*x^3)/(a - b*x^3)])/(7*a^3*d*x^7*(a + b*x^3)^(2/3)*(1 - (b*x^3)/a)^(2/3))","C",0
580,1,130,237,5.2013181,"\int \frac{\sqrt[3]{a+b x^3}}{x^{11} \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^11*(a*d - b*d*x^3)),x]","\frac{-14 a^4-36 a^3 b x^3-59 a^2 b^2 x^6+\frac{140 b^4 x^{12} \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{2 b x^3}{a-b x^3}\right)}{\left(1-\frac{b x^3}{a}\right)^{2/3}}-206 a b^3 x^9-169 b^4 x^{12}}{140 a^4 d x^{10} \left(a+b x^3\right)^{2/3}}","\frac{b^{10/3} \log \left(a d-b d x^3\right)}{3\ 2^{2/3} a^4 d}-\frac{b^{10/3} \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2^{2/3} a^4 d}-\frac{\sqrt[3]{2} b^{10/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} a^4 d}-\frac{169 b^3 \sqrt[3]{a+b x^3}}{140 a^4 d x}-\frac{37 b^2 \sqrt[3]{a+b x^3}}{140 a^3 d x^4}-\frac{11 b \sqrt[3]{a+b x^3}}{70 a^2 d x^7}-\frac{\sqrt[3]{a+b x^3}}{10 a d x^{10}}",1,"(-14*a^4 - 36*a^3*b*x^3 - 59*a^2*b^2*x^6 - 206*a*b^3*x^9 - 169*b^4*x^12 + (140*b^4*x^12*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (-2*b*x^3)/(a - b*x^3)])/(1 - (b*x^3)/a)^(2/3))/(140*a^4*d*x^10*(a + b*x^3)^(2/3))","C",0
581,1,234,521,0.3773794,"\int \frac{x^6 \sqrt[3]{a+b x^3}}{a d-b d x^3} \, dx","Integrate[(x^6*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]","\frac{\frac{48 a^4 x F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{\left(a-b x^3\right) \left(b x^3 \left(3 F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-2 F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)+4 a F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)}+7 a b x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-4 \left(a+b x^3\right) \left(3 a x+b x^4\right)}{20 b^2 d \left(a+b x^3\right)^{2/3}}","-\frac{a^{5/3} \log \left(2^{2/3}-\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)}{3\ 2^{2/3} b^{7/3} d}+\frac{a^{5/3} \log \left(\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}-\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3\ 2^{2/3} b^{7/3} d}-\frac{\sqrt[3]{2} a^{5/3} \log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 b^{7/3} d}+\frac{a^{5/3} \log \left(\frac{\left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}+\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+2 \sqrt[3]{2}\right)}{6\ 2^{2/3} b^{7/3} d}-\frac{\sqrt[3]{2} a^{5/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}}{\sqrt{3}}\right)}{\sqrt{3} b^{7/3} d}-\frac{a^{5/3} \tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} b^{7/3} d}-\frac{2 a^2 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{5 b^2 d \left(a+b x^3\right)^{2/3}}-\frac{3 a x \sqrt[3]{a+b x^3}}{5 b^2 d}-\frac{x^4 \sqrt[3]{a+b x^3}}{5 b d}",1,"(-4*(a + b*x^3)*(3*a*x + b*x^4) + 7*a*b*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a] + (48*a^4*x*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/((a - b*x^3)*(4*a*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a] + b*x^3*(3*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), (b*x^3)/a] - 2*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a]))))/(20*b^2*d*(a + b*x^3)^(2/3))","C",0
582,1,225,494,0.2122043,"\int \frac{x^3 \sqrt[3]{a+b x^3}}{a d-b d x^3} \, dx","Integrate[(x^3*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]","\frac{x \left(\frac{4 \left(\frac{4 a^3 F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{\left(a-b x^3\right) \left(b x^3 \left(3 F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-2 F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)+4 a F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)}-a-b x^3\right)}{b}+3 x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)}{8 d \left(a+b x^3\right)^{2/3}}","-\frac{a^{2/3} \log \left(2^{2/3}-\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)}{3\ 2^{2/3} b^{4/3} d}+\frac{a^{2/3} \log \left(\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}-\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3\ 2^{2/3} b^{4/3} d}-\frac{\sqrt[3]{2} a^{2/3} \log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 b^{4/3} d}+\frac{a^{2/3} \log \left(\frac{\left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}+\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+2 \sqrt[3]{2}\right)}{6\ 2^{2/3} b^{4/3} d}-\frac{\sqrt[3]{2} a^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}}{\sqrt{3}}\right)}{\sqrt{3} b^{4/3} d}-\frac{a^{2/3} \tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} b^{4/3} d}-\frac{a x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 b d \left(a+b x^3\right)^{2/3}}-\frac{x \sqrt[3]{a+b x^3}}{2 b d}",1,"(x*(3*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a] + (4*(-a - b*x^3 + (4*a^3*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/((a - b*x^3)*(4*a*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a] + b*x^3*(3*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), (b*x^3)/a] - 2*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])))))/b))/(8*d*(a + b*x^3)^(2/3))","C",0
583,1,154,416,0.1537797,"\int \frac{\sqrt[3]{a+b x^3}}{a d-b d x^3} \, dx","Integrate[(a + b*x^3)^(1/3)/(a*d - b*d*x^3),x]","\frac{4 a x \sqrt[3]{a+b x^3} F_1\left(\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{d \left(a-b x^3\right) \left(b x^3 \left(3 F_1\left(\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)+F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)+4 a F_1\left(\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)}","-\frac{\log \left(2^{2/3}-\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)}{3\ 2^{2/3} \sqrt[3]{a} \sqrt[3]{b} d}+\frac{\log \left(\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}-\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3\ 2^{2/3} \sqrt[3]{a} \sqrt[3]{b} d}-\frac{\sqrt[3]{2} \log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d}+\frac{\log \left(\frac{\left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}+\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+2 \sqrt[3]{2}\right)}{6\ 2^{2/3} \sqrt[3]{a} \sqrt[3]{b} d}-\frac{\sqrt[3]{2} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{a} \sqrt[3]{b} d}-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} \sqrt[3]{a} \sqrt[3]{b} d}",1,"(4*a*x*(a + b*x^3)^(1/3)*AppellF1[1/3, -1/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/(d*(a - b*x^3)*(4*a*AppellF1[1/3, -1/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a] + b*x^3*(3*AppellF1[4/3, -1/3, 2, 7/3, -((b*x^3)/a), (b*x^3)/a] + AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])))","C",0
584,1,231,496,0.1488164,"\int \frac{\sqrt[3]{a+b x^3}}{x^3 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^3*(a*d - b*d*x^3)),x]","\frac{\frac{48 a^3 b x^3 F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{\left(a-b x^3\right) \left(b x^3 \left(3 F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-2 F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)+4 a F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)}+b^2 x^6 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-4 a \left(a+b x^3\right)}{8 a^2 d x^2 \left(a+b x^3\right)^{2/3}}","-\frac{b^{2/3} \log \left(2^{2/3}-\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)}{3\ 2^{2/3} a^{4/3} d}+\frac{b^{2/3} \log \left(\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}-\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3\ 2^{2/3} a^{4/3} d}-\frac{\sqrt[3]{2} b^{2/3} \log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 a^{4/3} d}+\frac{b^{2/3} \log \left(\frac{\left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}+\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+2 \sqrt[3]{2}\right)}{6\ 2^{2/3} a^{4/3} d}-\frac{\sqrt[3]{2} b^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}}{\sqrt{3}}\right)}{\sqrt{3} a^{4/3} d}-\frac{b^{2/3} \tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} a^{4/3} d}+\frac{b x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 a d \left(a+b x^3\right)^{2/3}}-\frac{\sqrt[3]{a+b x^3}}{2 a d x^2}",1,"(-4*a*(a + b*x^3) + b^2*x^6*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a] + (48*a^3*b*x^3*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/((a - b*x^3)*(4*a*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a] + b*x^3*(3*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), (b*x^3)/a] - 2*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a]))))/(8*a^2*d*x^2*(a + b*x^3)^(2/3))","C",0
585,1,243,523,0.1664593,"\int \frac{\sqrt[3]{a+b x^3}}{x^6 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^6*(a*d - b*d*x^3)),x]","\frac{\frac{3 b^3 x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{a^3}-\frac{4 \left(a^2+4 a b x^3+3 b^2 x^6\right)}{a^2 x^5}+\frac{112 b^2 x F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{\left(a-b x^3\right) \left(b x^3 \left(3 F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-2 F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)+4 a F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)}}{20 d \left(a+b x^3\right)^{2/3}}","-\frac{b^{5/3} \log \left(2^{2/3}-\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)}{3\ 2^{2/3} a^{7/3} d}+\frac{b^{5/3} \log \left(\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}-\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3\ 2^{2/3} a^{7/3} d}-\frac{\sqrt[3]{2} b^{5/3} \log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 a^{7/3} d}+\frac{b^{5/3} \log \left(\frac{\left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}+\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+2 \sqrt[3]{2}\right)}{6\ 2^{2/3} a^{7/3} d}-\frac{\sqrt[3]{2} b^{5/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}}{\sqrt{3}}\right)}{\sqrt{3} a^{7/3} d}-\frac{b^{5/3} \tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} a^{7/3} d}+\frac{2 b^2 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{5 a^2 d \left(a+b x^3\right)^{2/3}}-\frac{3 b \sqrt[3]{a+b x^3}}{5 a^2 d x^2}-\frac{\sqrt[3]{a+b x^3}}{5 a d x^5}",1,"((-4*(a^2 + 4*a*b*x^3 + 3*b^2*x^6))/(a^2*x^5) + (3*b^3*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])/a^3 + (112*b^2*x*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/((a - b*x^3)*(4*a*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a] + b*x^3*(3*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), (b*x^3)/a] - 2*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a]))))/(20*d*(a + b*x^3)^(2/3))","C",0
586,1,163,223,0.2756951,"\int \frac{x^{11} \left(a+b x^3\right)^{2/3}}{a d-b d x^3} \, dx","Integrate[(x^11*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]","-\frac{-220\ 2^{2/3} a^{11/3} \log \left(a-b x^3\right)+660\ 2^{2/3} a^{11/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+440\ 2^{2/3} \sqrt{3} a^{11/3} \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)+3 \left(a+b x^3\right)^{2/3} \left(293 a^3+98 a^2 b x^3+65 a b^2 x^6+40 b^3 x^9\right)}{1320 b^4 d}","\frac{a^{11/3} \log \left(a-b x^3\right)}{3 \sqrt[3]{2} b^4 d}-\frac{a^{11/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} b^4 d}-\frac{2^{2/3} a^{11/3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^4 d}-\frac{a^3 \left(a+b x^3\right)^{2/3}}{2 b^4 d}-\frac{a^2 \left(a+b x^3\right)^{5/3}}{5 b^4 d}+\frac{a \left(a+b x^3\right)^{8/3}}{8 b^4 d}-\frac{\left(a+b x^3\right)^{11/3}}{11 b^4 d}",1,"-1/1320*(3*(a + b*x^3)^(2/3)*(293*a^3 + 98*a^2*b*x^3 + 65*a*b^2*x^6 + 40*b^3*x^9) + 440*2^(2/3)*Sqrt[3]*a^(11/3)*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 220*2^(2/3)*a^(11/3)*Log[a - b*x^3] + 660*2^(2/3)*a^(11/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)])/(b^4*d)","A",1
587,1,153,177,0.1719888,"\int \frac{x^8 \left(a+b x^3\right)^{2/3}}{a d-b d x^3} \, dx","Integrate[(x^8*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]","\frac{4\ 2^{2/3} a^{8/3} \log \left(a-b x^3\right)-8\ 2^{2/3} \sqrt{3} a^{8/3} \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-3 \left(4\ 2^{2/3} a^{8/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+\left(a+b x^3\right)^{2/3} \left(5 a^2+2 a b x^3+b^2 x^6\right)\right)}{24 b^3 d}","\frac{a^{8/3} \log \left(a-b x^3\right)}{3 \sqrt[3]{2} b^3 d}-\frac{a^{8/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} b^3 d}-\frac{2^{2/3} a^{8/3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^3 d}-\frac{a^2 \left(a+b x^3\right)^{2/3}}{2 b^3 d}-\frac{\left(a+b x^3\right)^{8/3}}{8 b^3 d}",1,"(-8*2^(2/3)*Sqrt[3]*a^(8/3)*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] + 4*2^(2/3)*a^(8/3)*Log[a - b*x^3] - 3*((a + b*x^3)^(2/3)*(5*a^2 + 2*a*b*x^3 + b^2*x^6) + 4*2^(2/3)*a^(8/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)]))/(24*b^3*d)","A",1
588,1,143,175,0.1473301,"\int \frac{x^5 \left(a+b x^3\right)^{2/3}}{a d-b d x^3} \, dx","Integrate[(x^5*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]","\frac{5\ 2^{2/3} a^{5/3} \log \left(a-b x^3\right)-3 \left(5\ 2^{2/3} a^{5/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+\left(a+b x^3\right)^{2/3} \left(7 a+2 b x^3\right)\right)-10\ 2^{2/3} \sqrt{3} a^{5/3} \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)}{30 b^2 d}","\frac{a^{5/3} \log \left(a-b x^3\right)}{3 \sqrt[3]{2} b^2 d}-\frac{a^{5/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} b^2 d}-\frac{2^{2/3} a^{5/3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^2 d}-\frac{a \left(a+b x^3\right)^{2/3}}{2 b^2 d}-\frac{\left(a+b x^3\right)^{5/3}}{5 b^2 d}",1,"(-10*2^(2/3)*Sqrt[3]*a^(5/3)*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] + 5*2^(2/3)*a^(5/3)*Log[a - b*x^3] - 3*((a + b*x^3)^(2/3)*(7*a + 2*b*x^3) + 5*2^(2/3)*a^(5/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)]))/(30*b^2*d)","A",1
589,1,130,153,0.0915116,"\int \frac{x^2 \left(a+b x^3\right)^{2/3}}{a d-b d x^3} \, dx","Integrate[(x^2*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]","\frac{2^{2/3} a^{2/3} \log \left(a-b x^3\right)-3 \left(2^{2/3} a^{2/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+\left(a+b x^3\right)^{2/3}\right)-2\ 2^{2/3} \sqrt{3} a^{2/3} \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)}{6 b d}","\frac{a^{2/3} \log \left(a-b x^3\right)}{3 \sqrt[3]{2} b d}-\frac{a^{2/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} b d}-\frac{2^{2/3} a^{2/3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b d}-\frac{\left(a+b x^3\right)^{2/3}}{2 b d}",1,"(-2*2^(2/3)*Sqrt[3]*a^(2/3)*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] + 2^(2/3)*a^(2/3)*Log[a - b*x^3] - 3*((a + b*x^3)^(2/3) + 2^(2/3)*a^(2/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)]))/(6*b*d)","A",1
590,1,164,214,0.0799807,"\int \frac{\left(a+b x^3\right)^{2/3}}{x \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x*(a*d - b*d*x^3)),x]","\frac{2^{2/3} \log \left(a-b x^3\right)+3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)-3\ 2^{2/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-2\ 2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-3 \log (x)}{6 \sqrt[3]{a} d}","\frac{\log \left(a-b x^3\right)}{3 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{\log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2 \sqrt[3]{a} d}-\frac{\log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} \sqrt[3]{a} d}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} d}-\frac{2^{2/3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} d}-\frac{\log (x)}{2 \sqrt[3]{a} d}",1,"(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*2^(2/3)*Sqrt[3]*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 3*Log[x] + 2^(2/3)*Log[a - b*x^3] + 3*Log[a^(1/3) - (a + b*x^3)^(1/3)] - 3*2^(2/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)])/(6*a^(1/3)*d)","A",1
591,1,213,269,0.0933259,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^4 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^4*(a*d - b*d*x^3)),x]","\frac{10 \sqrt{3} b x^3 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-3 \left(2 \sqrt[3]{a} \left(a+b x^3\right)^{2/3}-2^{2/3} b x^3 \log \left(a-b x^3\right)-5 b x^3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+3\ 2^{2/3} b x^3 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2\ 2^{2/3} \sqrt{3} b x^3 \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)+5 b x^3 \log (x)\right)}{18 a^{4/3} d x^3}","\frac{b \log \left(a-b x^3\right)}{3 \sqrt[3]{2} a^{4/3} d}+\frac{5 b \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{6 a^{4/3} d}-\frac{b \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} a^{4/3} d}+\frac{5 b \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{4/3} d}-\frac{2^{2/3} b \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{4/3} d}-\frac{5 b \log (x)}{6 a^{4/3} d}-\frac{\left(a+b x^3\right)^{5/3}}{3 a^2 d x^3}+\frac{b \left(a+b x^3\right)^{2/3}}{3 a^2 d}",1,"(10*Sqrt[3]*b*x^3*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 3*(2*a^(1/3)*(a + b*x^3)^(2/3) + 2*2^(2/3)*Sqrt[3]*b*x^3*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] + 5*b*x^3*Log[x] - 2^(2/3)*b*x^3*Log[a - b*x^3] - 5*b*x^3*Log[a^(1/3) - (a + b*x^3)^(1/3)] + 3*2^(2/3)*b*x^3*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)]))/(18*a^(4/3)*d*x^3)","A",1
592,1,247,284,0.1556318,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^7 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^7*(a*d - b*d*x^3)),x]","\frac{28 \sqrt{3} b^2 x^6 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-3 \left(3 a^{4/3} \left(a+b x^3\right)^{2/3}-3\ 2^{2/3} b^2 x^6 \log \left(a-b x^3\right)-14 b^2 x^6 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+9\ 2^{2/3} b^2 x^6 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+6\ 2^{2/3} \sqrt{3} b^2 x^6 \tan ^{-1}\left(\frac{\frac{2^{2/3} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)+8 \sqrt[3]{a} b x^3 \left(a+b x^3\right)^{2/3}+14 b^2 x^6 \log (x)\right)}{54 a^{7/3} d x^6}","\frac{b^2 \log \left(a-b x^3\right)}{3 \sqrt[3]{2} a^{7/3} d}+\frac{7 b^2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{9 a^{7/3} d}-\frac{b^2 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} a^{7/3} d}+\frac{14 b^2 \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{7/3} d}-\frac{2^{2/3} b^2 \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{7/3} d}-\frac{7 b^2 \log (x)}{9 a^{7/3} d}-\frac{5 b \left(a+b x^3\right)^{2/3}}{18 a^2 d x^3}-\frac{\left(a+b x^3\right)^{5/3}}{6 a^2 d x^6}",1,"(28*Sqrt[3]*b^2*x^6*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 3*(3*a^(4/3)*(a + b*x^3)^(2/3) + 8*a^(1/3)*b*x^3*(a + b*x^3)^(2/3) + 6*2^(2/3)*Sqrt[3]*b^2*x^6*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] + 14*b^2*x^6*Log[x] - 3*2^(2/3)*b^2*x^6*Log[a - b*x^3] - 14*b^2*x^6*Log[a^(1/3) - (a + b*x^3)^(1/3)] + 9*2^(2/3)*b^2*x^6*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)]))/(54*a^(7/3)*d*x^6)","A",1
593,1,244,264,0.3049163,"\int \frac{x^6 \left(a+b x^3\right)^{2/3}}{a d-b d x^3} \, dx","Integrate[(x^6*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]","\frac{2\ 2^{2/3} a^2 \sqrt[3]{a+b x^3} \left(\log \left(\frac{2^{2/3} b^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)\right)+21 a b^{4/3} x^4 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-3 \sqrt[3]{b} \left(a+b x^3\right) \left(8 a x+3 b x^4\right)}{54 b^{7/3} d \sqrt[3]{a+b x^3}}","\frac{a^2 \log \left(a d-b d x^3\right)}{3 \sqrt[3]{2} b^{7/3} d}-\frac{a^2 \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} b^{7/3} d}+\frac{7 a^2 \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{9 b^{7/3} d}-\frac{14 a^2 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{9 \sqrt{3} b^{7/3} d}+\frac{2^{2/3} a^2 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{7/3} d}-\frac{4 a x \left(a+b x^3\right)^{2/3}}{9 b^2 d}-\frac{x^4 \left(a+b x^3\right)^{2/3}}{6 b d}",1,"(-3*b^(1/3)*(a + b*x^3)*(8*a*x + 3*b*x^4) + 21*a*b^(4/3)*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a] + 2*2^(2/3)*a^2*(a + b*x^3)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)] + Log[1 + (2^(2/3)*b^(2/3)*x^2)/(b + a*x^3)^(2/3) + (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)]))/(54*b^(7/3)*d*(a + b*x^3)^(1/3))","C",0
594,1,216,229,0.2169365,"\int \frac{x^3 \left(a+b x^3\right)^{2/3}}{a d-b d x^3} \, dx","Integrate[(x^3*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]","\frac{\frac{15 x^4 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{\sqrt[3]{a+b x^3}}+\frac{2^{2/3} a \left(\log \left(\frac{2^{2/3} b^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)\right)}{b^{4/3}}-\frac{12 x \left(a+b x^3\right)^{2/3}}{b}}{36 d}","\frac{a \log \left(a d-b d x^3\right)}{3 \sqrt[3]{2} b^{4/3} d}-\frac{a \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} b^{4/3} d}+\frac{5 a \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{6 b^{4/3} d}-\frac{5 a \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{4/3} d}+\frac{2^{2/3} a \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{4/3} d}-\frac{x \left(a+b x^3\right)^{2/3}}{3 b d}",1,"((-12*x*(a + b*x^3)^(2/3))/b + (15*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])/(a + b*x^3)^(1/3) + (2^(2/3)*a*(2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)] + Log[1 + (2^(2/3)*b^(2/3)*x^2)/(b + a*x^3)^(2/3) + (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)]))/b^(4/3))/(36*d)","C",0
595,1,156,200,0.1628803,"\int \frac{\left(a+b x^3\right)^{2/3}}{a d-b d x^3} \, dx","Integrate[(a + b*x^3)^(2/3)/(a*d - b*d*x^3),x]","\frac{4 a x \left(a+b x^3\right)^{2/3} F_1\left(\frac{1}{3};-\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{d \left(a-b x^3\right) \left(b x^3 \left(3 F_1\left(\frac{4}{3};-\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)+2 F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)+4 a F_1\left(\frac{1}{3};-\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)\right)}","\frac{\log \left(a d-b d x^3\right)}{3 \sqrt[3]{2} \sqrt[3]{b} d}-\frac{\log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} \sqrt[3]{b} d}+\frac{\log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 \sqrt[3]{b} d}-\frac{\tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b} d}+\frac{2^{2/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b} d}",1,"(4*a*x*(a + b*x^3)^(2/3)*AppellF1[1/3, -2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/(d*(a - b*x^3)*(4*a*AppellF1[1/3, -2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a] + b*x^3*(3*AppellF1[4/3, -2/3, 2, 7/3, -((b*x^3)/a), (b*x^3)/a] + 2*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])))","C",0
596,1,47,157,0.039884,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^3 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^3*(a*d - b*d*x^3)),x]","-\frac{\left(a+b x^3\right)^{2/3} \, _2F_1\left(-\frac{2}{3},1;\frac{1}{3};\frac{2 b x^3}{b x^3+a}\right)}{2 a d x^2}","\frac{b^{2/3} \log \left(a d-b d x^3\right)}{3 \sqrt[3]{2} a d}-\frac{b^{2/3} \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} a d}+\frac{2^{2/3} b^{2/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} a d}-\frac{\left(a+b x^3\right)^{2/3}}{2 a d x^2}",1,"-1/2*((a + b*x^3)^(2/3)*Hypergeometric2F1[-2/3, 1, 1/3, (2*b*x^3)/(a + b*x^3)])/(a*d*x^2)","C",1
597,1,170,182,5.2030235,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^6 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^6*(a*d - b*d*x^3)),x]","\frac{5\ 2^{2/3} b^{5/3} \left(\log \left(\frac{2^{2/3} b^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)\right)-\frac{3 \left(a+b x^3\right)^{2/3} \left(2 a+7 b x^3\right)}{x^5}}{30 a^2 d}","\frac{b^{5/3} \log \left(a d-b d x^3\right)}{3 \sqrt[3]{2} a^2 d}-\frac{b^{5/3} \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} a^2 d}+\frac{2^{2/3} b^{5/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} a^2 d}-\frac{7 b \left(a+b x^3\right)^{2/3}}{10 a^2 d x^2}-\frac{\left(a+b x^3\right)^{2/3}}{5 a d x^5}",1,"((-3*(a + b*x^3)^(2/3)*(2*a + 7*b*x^3))/x^5 + 5*2^(2/3)*b^(5/3)*(2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)] + Log[1 + (2^(2/3)*b^(2/3)*x^2)/(b + a*x^3)^(2/3) + (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)]))/(30*a^2*d)","A",0
598,1,179,209,5.2683968,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^9 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^9*(a*d - b*d*x^3)),x]","\frac{4\ 2^{2/3} b^{8/3} \left(\log \left(\frac{2^{2/3} b^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)\right)-\frac{3 \left(a+b x^3\right)^{2/3} \left(a^2+2 a b x^3+5 b^2 x^6\right)}{x^8}}{24 a^3 d}","\frac{b^{8/3} \log \left(a d-b d x^3\right)}{3 \sqrt[3]{2} a^3 d}-\frac{b^{8/3} \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} a^3 d}+\frac{2^{2/3} b^{8/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} a^3 d}-\frac{5 b^2 \left(a+b x^3\right)^{2/3}}{8 a^3 d x^2}-\frac{b \left(a+b x^3\right)^{2/3}}{4 a^2 d x^5}-\frac{\left(a+b x^3\right)^{2/3}}{8 a d x^8}",1,"((-3*(a + b*x^3)^(2/3)*(a^2 + 2*a*b*x^3 + 5*b^2*x^6))/x^8 + 4*2^(2/3)*b^(8/3)*(2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)] + Log[1 + (2^(2/3)*b^(2/3)*x^2)/(b + a*x^3)^(2/3) + (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)]))/(24*a^3*d)","A",0
599,1,196,236,5.3502262,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^{12} \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^12*(a*d - b*d*x^3)),x]","\frac{\frac{2^{2/3} b^{11/3} \left(\log \left(\frac{2^{2/3} b^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)\right)}{a^4}-\frac{3 \left(a+b x^3\right)^{2/3} \left(40 a^3+65 a^2 b x^3+98 a b^2 x^6+293 b^3 x^9\right)}{220 a^4 x^{11}}}{6 d}","\frac{b^{11/3} \log \left(a d-b d x^3\right)}{3 \sqrt[3]{2} a^4 d}-\frac{b^{11/3} \log \left(\sqrt[3]{2} \sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{\sqrt[3]{2} a^4 d}+\frac{2^{2/3} b^{11/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} a^4 d}-\frac{293 b^3 \left(a+b x^3\right)^{2/3}}{440 a^4 d x^2}-\frac{49 b^2 \left(a+b x^3\right)^{2/3}}{220 a^3 d x^5}-\frac{13 b \left(a+b x^3\right)^{2/3}}{88 a^2 d x^8}-\frac{\left(a+b x^3\right)^{2/3}}{11 a d x^{11}}",1,"((-3*(a + b*x^3)^(2/3)*(40*a^3 + 65*a^2*b*x^3 + 98*a*b^2*x^6 + 293*b^3*x^9))/(220*a^4*x^11) + (2^(2/3)*b^(11/3)*(2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)] + Log[1 + (2^(2/3)*b^(2/3)*x^2)/(b + a*x^3)^(2/3) + (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)]))/a^4)/(6*d)","A",0
600,1,147,512,0.1141968,"\int \frac{x^7 \left(a+b x^3\right)^{2/3}}{a d-b d x^3} \, dx","Integrate[(x^7*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]","\frac{45 a^2 x^2 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-5 \left(9 a^2 x^2+13 a b x^5+4 b^2 x^8\right)+38 a b x^5 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{140 b^2 d \sqrt[3]{a+b x^3}}","\frac{a^{7/3} \log \left(\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}-\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 \sqrt[3]{2} b^{8/3} d}-\frac{2^{2/3} a^{7/3} \log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 b^{8/3} d}-\frac{a^{7/3} \log \left(\frac{\sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a}}-\frac{2^{2/3} \sqrt[3]{b} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}\right)}{2 \sqrt[3]{2} b^{8/3} d}+\frac{2^{2/3} a^{7/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}}{\sqrt{3}}\right)}{\sqrt{3} b^{8/3} d}+\frac{a^{7/3} \tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} b^{8/3} d}+\frac{a^{7/3} \log \left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{b} x\right)^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a}\right)}{6 \sqrt[3]{2} b^{8/3} d}-\frac{19 a^2 x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{28 b^2 d \sqrt[3]{a+b x^3}}-\frac{9 a x^2 \left(a+b x^3\right)^{2/3}}{28 b^2 d}-\frac{x^5 \left(a+b x^3\right)^{2/3}}{7 b d}",1,"(-5*(9*a^2*x^2 + 13*a*b*x^5 + 4*b^2*x^8) + 45*a^2*x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), (b*x^3)/a] + 38*a*b*x^5*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a])/(140*b^2*d*(a + b*x^3)^(1/3))","C",0
601,1,127,485,0.1738794,"\int \frac{x^4 \left(a+b x^3\right)^{2/3}}{a d-b d x^3} \, dx","Integrate[(x^4*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]","\frac{x^2 \left(6 b x^3 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)+5 a \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-5 \left(a+b x^3\right)\right)}{20 b d \sqrt[3]{a+b x^3}}","\frac{a^{4/3} \log \left(\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}-\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 \sqrt[3]{2} b^{5/3} d}-\frac{2^{2/3} a^{4/3} \log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 b^{5/3} d}-\frac{a^{4/3} \log \left(\frac{\sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a}}-\frac{2^{2/3} \sqrt[3]{b} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}\right)}{2 \sqrt[3]{2} b^{5/3} d}+\frac{2^{2/3} a^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}}{\sqrt{3}}\right)}{\sqrt{3} b^{5/3} d}+\frac{a^{4/3} \tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} b^{5/3} d}+\frac{a^{4/3} \log \left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{b} x\right)^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a}\right)}{6 \sqrt[3]{2} b^{5/3} d}-\frac{3 a x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{4 b d \sqrt[3]{a+b x^3}}-\frac{x^2 \left(a+b x^3\right)^{2/3}}{4 b d}",1,"(x^2*(-5*(a + b*x^3) + 5*a*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), (b*x^3)/a] + 6*b*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a]))/(20*b*d*(a + b*x^3)^(1/3))","C",0
602,1,63,457,0.042633,"\int \frac{x \left(a+b x^3\right)^{2/3}}{a d-b d x^3} \, dx","Integrate[(x*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]","\frac{x^2 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};-\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{2 d \sqrt[3]{a+b x^3}}","\frac{\sqrt[3]{a} \log \left(\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}-\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 \sqrt[3]{2} b^{2/3} d}-\frac{2^{2/3} \sqrt[3]{a} \log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 b^{2/3} d}-\frac{\sqrt[3]{a} \log \left(\frac{\sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a}}-\frac{2^{2/3} \sqrt[3]{b} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}\right)}{2 \sqrt[3]{2} b^{2/3} d}+\frac{2^{2/3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}}{\sqrt{3}}\right)}{\sqrt{3} b^{2/3} d}+\frac{\sqrt[3]{a} \tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} b^{2/3} d}+\frac{\sqrt[3]{a} \log \left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{b} x\right)^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a}\right)}{6 \sqrt[3]{2} b^{2/3} d}-\frac{x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 d \sqrt[3]{a+b x^3}}",1,"(x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, -2/3, 1, 5/3, -((b*x^3)/a), (b*x^3)/a])/(2*d*(a + b*x^3)^(1/3))","C",0
603,1,136,483,0.1124027,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^2 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^2*(a*d - b*d*x^3)),x]","\frac{15 a b x^3 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)-2 \left(b^2 x^6 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)+5 a \left(a+b x^3\right)\right)}{10 a^2 d x \sqrt[3]{a+b x^3}}","\frac{\sqrt[3]{b} \log \left(\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}-\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 \sqrt[3]{2} a^{2/3} d}-\frac{2^{2/3} \sqrt[3]{b} \log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 a^{2/3} d}-\frac{\sqrt[3]{b} \log \left(\frac{\sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a}}-\frac{2^{2/3} \sqrt[3]{b} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}\right)}{2 \sqrt[3]{2} a^{2/3} d}+\frac{2^{2/3} \sqrt[3]{b} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}}{\sqrt{3}}\right)}{\sqrt{3} a^{2/3} d}+\frac{\sqrt[3]{b} \tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} a^{2/3} d}+\frac{\sqrt[3]{b} \log \left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{b} x\right)^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a}\right)}{6 \sqrt[3]{2} a^{2/3} d}+\frac{b x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 a d \sqrt[3]{a+b x^3}}-\frac{\left(a+b x^3\right)^{2/3}}{a d x}",1,"(15*a*b*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), (b*x^3)/a] - 2*(5*a*(a + b*x^3) + b^2*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a]))/(10*a^2*d*x*(a + b*x^3)^(1/3))","C",0
604,1,148,512,0.0899244,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^5 \left(a d-b d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^5*(a*d - b*d*x^3)),x]","\frac{-5 a \left(a^2+7 a b x^3+6 b^2 x^6\right)-6 b^3 x^9 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)+35 a b^2 x^6 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right)}{20 a^3 d x^4 \sqrt[3]{a+b x^3}}","\frac{b^{4/3} \log \left(\frac{2^{2/3} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{\left(a+b x^3\right)^{2/3}}-\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 \sqrt[3]{2} a^{5/3} d}-\frac{2^{2/3} b^{4/3} \log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{3 a^{5/3} d}-\frac{b^{4/3} \log \left(\frac{\sqrt[3]{b} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a}}-\frac{2^{2/3} \sqrt[3]{b} \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}\right)}{2 \sqrt[3]{2} a^{5/3} d}+\frac{2^{2/3} b^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}}{\sqrt{3}}\right)}{\sqrt{3} a^{5/3} d}+\frac{b^{4/3} \tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} a^{5/3} d}+\frac{b^{4/3} \log \left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{b} x\right)^2 \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a}\right)}{6 \sqrt[3]{2} a^{5/3} d}+\frac{3 b^2 x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{4 a^2 d \sqrt[3]{a+b x^3}}-\frac{3 b \left(a+b x^3\right)^{2/3}}{2 a^2 d x}-\frac{\left(a+b x^3\right)^{2/3}}{4 a d x^4}",1,"(-5*a*(a^2 + 7*a*b*x^3 + 6*b^2*x^6) + 35*a*b^2*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), (b*x^3)/a] - 6*b^3*x^9*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a])/(20*a^3*d*x^4*(a + b*x^3)^(1/3))","C",0
605,1,113,127,0.104114,"\int \frac{x^{14}}{\sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[x^14/((1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{1}{660} \left(-55\ 2^{2/3} \log \left(x^3+1\right)+165\ 2^{2/3} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)+110\ 2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)+3 \left(1-x^3\right)^{2/3} \left(-20 x^9+5 x^6-38 x^3+53\right)\right)","\frac{1}{11} \left(1-x^3\right)^{11/3}-\frac{1}{4} \left(1-x^3\right)^{8/3}+\frac{2}{5} \left(1-x^3\right)^{5/3}-\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}+\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}",1,"(3*(1 - x^3)^(2/3)*(53 - 38*x^3 + 5*x^6 - 20*x^9) + 110*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - 55*2^(2/3)*Log[1 + x^3] + 165*2^(2/3)*Log[2^(1/3) - (1 - x^3)^(1/3)])/660","A",1
606,1,128,128,0.0571131,"\int \frac{x^{11}}{\sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[x^11/((1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{1}{120} \left(6 \left(1-x^3\right)^{2/3} x^3-51 \left(1-x^3\right)^{2/3}+10\ 2^{2/3} \log \left(x^3+1\right)-30\ 2^{2/3} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)-20\ 2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)-15 \left(1-x^3\right)^{2/3} x^6\right)","-\frac{1}{8} \left(1-x^3\right)^{8/3}+\frac{1}{5} \left(1-x^3\right)^{5/3}-\frac{1}{2} \left(1-x^3\right)^{2/3}+\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}-\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}",1,"(-51*(1 - x^3)^(2/3) + 6*x^3*(1 - x^3)^(2/3) - 15*x^6*(1 - x^3)^(2/3) - 20*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] + 10*2^(2/3)*Log[1 + x^3] - 30*2^(2/3)*Log[2^(1/3) - (1 - x^3)^(1/3)])/120","A",1
607,1,96,97,0.0658712,"\int \frac{x^8}{\sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[x^8/((1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{1}{60} \left(12 \left(1-x^3\right)^{5/3}-5\ 2^{2/3} \log \left(x^3+1\right)+15\ 2^{2/3} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)+10\ 2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)\right)","\frac{1}{5} \left(1-x^3\right)^{5/3}-\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}+\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}",1,"(12*(1 - x^3)^(5/3) + 10*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - 5*2^(2/3)*Log[1 + x^3] + 15*2^(2/3)*Log[2^(1/3) - (1 - x^3)^(1/3)])/60","A",1
608,1,95,98,0.0331059,"\int \frac{x^5}{\sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[x^5/((1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{1}{12} \left(-6 \left(1-x^3\right)^{2/3}+2^{2/3} \log \left(x^3+1\right)-3\ 2^{2/3} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)-2\ 2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)\right)","-\frac{1}{2} \left(1-x^3\right)^{2/3}+\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}-\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}",1,"(-6*(1 - x^3)^(2/3) - 2*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] + 2^(2/3)*Log[1 + x^3] - 3*2^(2/3)*Log[2^(1/3) - (1 - x^3)^(1/3)])/12","A",1
609,1,73,82,0.0248057,"\int \frac{x^2}{\sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[x^2/((1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{-\log \left(x^3+1\right)+3 \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{6 \sqrt[3]{2}}","-\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}+\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}",1,"(2*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - Log[1 + x^3] + 3*Log[2^(1/3) - (1 - x^3)^(1/3)])/(6*2^(1/3))","A",1
610,1,133,137,0.044512,"\int \frac{1}{x \sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[1/(x*(1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{1}{12} \left(2^{2/3} \log \left(x^3+1\right)+6 \log \left(1-\sqrt[3]{1-x^3}\right)-3\ 2^{2/3} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)+4 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)-2\ 2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)-6 \log (x)\right)","\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}+\frac{1}{2} \log \left(1-\sqrt[3]{1-x^3}\right)-\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\log (x)}{2}",1,"(4*Sqrt[3]*ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt[3]] - 2*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - 6*Log[x] + 2^(2/3)*Log[1 + x^3] + 6*Log[1 - (1 - x^3)^(1/3)] - 3*2^(2/3)*Log[2^(1/3) - (1 - x^3)^(1/3)])/12","A",1
611,1,153,157,0.2179019,"\int \frac{1}{x^4 \sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[1/(x^4*(1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{1}{36} \left(3 \left(-\frac{4 \left(1-x^3\right)^{2/3}}{x^3}-2^{2/3} \log \left(x^3+1\right)-4 \log \left(1-\sqrt[3]{1-x^3}\right)+3\ 2^{2/3} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)+2\ 2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)+4 \log (x)\right)-8 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)\right)","-\frac{\left(1-x^3\right)^{2/3}}{3 x^3}-\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}-\frac{1}{3} \log \left(1-\sqrt[3]{1-x^3}\right)+\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2 \sqrt[3]{2}}-\frac{2 \tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log (x)}{3}",1,"(-8*Sqrt[3]*ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt[3]] + 3*((-4*(1 - x^3)^(2/3))/x^3 + 2*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] + 4*Log[x] - 2^(2/3)*Log[1 + x^3] - 4*Log[1 - (1 - x^3)^(1/3)] + 3*2^(2/3)*Log[2^(1/3) - (1 - x^3)^(1/3)]))/36","A",1
612,1,144,154,0.1581304,"\int \frac{x^6}{\sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[x^6/((1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{1}{36} \left(-6 x^4 F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right)-12 \left(1-x^3\right)^{2/3} x+2^{2/3} \left(2 \log \left(\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+1\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} x}{\sqrt[3]{x^3-1}}-1}{\sqrt{3}}\right)-\log \left(-\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+\frac{2^{2/3} x^2}{\left(x^3-1\right)^{2/3}}+1\right)\right)\right)","-\frac{1}{3} \left(1-x^3\right)^{2/3} x-\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}+\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{2 \sqrt[3]{2}}-\frac{1}{3} \log \left(\sqrt[3]{1-x^3}+x\right)+\frac{2 \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}",1,"(-12*x*(1 - x^3)^(2/3) - 6*x^4*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3] + 2^(2/3)*(2*Sqrt[3]*ArcTan[(-1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] - Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1 + x^3)^(1/3)] + 2*Log[1 + (2^(1/3)*x)/(-1 + x^3)^(1/3)]))/36","C",0
613,1,26,135,0.0195782,"\int \frac{x^3}{\sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[x^3/((1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{1}{4} x^4 F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right)","\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}-\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{2 \sqrt[3]{2}}+\frac{1}{2} \log \left(\sqrt[3]{1-x^3}+x\right)-\frac{\tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}",1,"(x^4*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3])/4","C",0
614,1,112,88,0.0795056,"\int \frac{1}{\sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[1/((1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{2 \log \left(\frac{\sqrt[3]{2} x}{\sqrt[3]{1-x^3}}+1\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}-1}{\sqrt{3}}\right)-\log \left(-\frac{\sqrt[3]{2} x}{\sqrt[3]{1-x^3}}+\frac{2^{2/3} x^2}{\left(1-x^3\right)^{2/3}}+1\right)}{6 \sqrt[3]{2}}","-\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}+\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{2 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}",1,"(2*Sqrt[3]*ArcTan[(-1 + (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]] - Log[1 + (2^(2/3)*x^2)/(1 - x^3)^(2/3) - (2^(1/3)*x)/(1 - x^3)^(1/3)] + 2*Log[1 + (2^(1/3)*x)/(1 - x^3)^(1/3)])/(6*2^(1/3))","A",1
615,1,82,105,0.5965787,"\int \frac{1}{x^3 \sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[1/(x^3*(1 - x^3)^(1/3)*(1 + x^3)),x]","-\frac{\left(-6 x^6+4 x^3+2\right) \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right)-3 x^3 \left(x^3+1\right) \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{2 x^3}{x^3-1}\right)}{4 x^2 \left(1-x^3\right)^{4/3}}","\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}-\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{2 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\left(1-x^3\right)^{2/3}}{2 x^2}",1,"-1/4*((2 + 4*x^3 - 6*x^6)*Hypergeometric2F1[1/3, 1, 4/3, (2*x^3)/(-1 + x^3)] - 3*x^3*(1 + x^3)*Hypergeometric2F1[4/3, 2, 7/3, (2*x^3)/(-1 + x^3)])/(x^2*(1 - x^3)^(4/3))","C",0
616,1,123,124,5.2455258,"\int \frac{1}{x^6 \sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[1/(x^6*(1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{2 \log \left(\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+1\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} x}{\sqrt[3]{x^3-1}}-1}{\sqrt{3}}\right)-\log \left(-\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+\frac{2^{2/3} x^2}{\left(x^3-1\right)^{2/3}}+1\right)}{6 \sqrt[3]{2}}-\frac{\left(1-x^3\right)^{5/3}}{5 x^5}","-\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}+\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{2 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\left(1-x^3\right)^{2/3}}{5 x^5}+\frac{\left(1-x^3\right)^{2/3}}{5 x^2}",1,"-1/5*(1 - x^3)^(5/3)/x^5 + (2*Sqrt[3]*ArcTan[(-1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] - Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1 + x^3)^(1/3)] + 2*Log[1 + (2^(1/3)*x)/(-1 + x^3)^(1/3)])/(6*2^(1/3))","A",0
617,1,133,141,5.1145019,"\int \frac{1}{x^9 \sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[1/(x^9*(1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{-2 \log \left(\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+1\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} x}{\sqrt[3]{x^3-1}}-1}{\sqrt{3}}\right)+\log \left(-\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+\frac{2^{2/3} x^2}{\left(x^3-1\right)^{2/3}}+1\right)}{6 \sqrt[3]{2}}-\frac{\left(1-x^3\right)^{2/3} \left(17 x^6-2 x^3+5\right)}{40 x^8}","\frac{\log \left(x^3+1\right)}{6 \sqrt[3]{2}}-\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{2 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\left(1-x^3\right)^{2/3}}{8 x^8}+\frac{\left(1-x^3\right)^{2/3}}{20 x^5}-\frac{17 \left(1-x^3\right)^{2/3}}{40 x^2}",1,"-1/40*((1 - x^3)^(2/3)*(5 - 2*x^3 + 17*x^6))/x^8 + (-2*Sqrt[3]*ArcTan[(-1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] + Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1 + x^3)^(1/3)] - 2*Log[1 + (2^(1/3)*x)/(-1 + x^3)^(1/3)])/(6*2^(1/3))","A",0
618,1,40,271,0.0396124,"\int \frac{x^7}{\sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[x^7/((1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{1}{4} x^2 \left(F_1\left(\frac{2}{3};-\frac{2}{3},1;\frac{5}{3};x^3,-x^3\right)-\left(1-x^3\right)^{2/3}\right)","-\frac{1}{4} x^2 \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right)+\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6 \sqrt[3]{2}}-\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2^{2/3} \sqrt[3]{1-x^3}+x-1\right)}{4 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{1}{4} \left(1-x^3\right)^{2/3} x^2+\frac{\log \left((1-x) (x+1)^2\right)}{12 \sqrt[3]{2}}",1,"(x^2*(-(1 - x^3)^(2/3) + AppellF1[2/3, -2/3, 1, 5/3, x^3, -x^3]))/4","C",0
619,1,26,254,0.0189262,"\int \frac{x^4}{\sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[x^4/((1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{1}{5} x^5 F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right)","\frac{1}{2} x^2 \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right)-\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6 \sqrt[3]{2}}+\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{3 \sqrt[3]{2}}+\frac{\log \left(2^{2/3} \sqrt[3]{1-x^3}+x-1\right)}{4 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\log \left((1-x) (x+1)^2\right)}{12 \sqrt[3]{2}}",1,"(x^5*AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3])/5","C",0
620,1,26,233,0.017307,"\int \frac{x}{\sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[x/((1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{1}{2} x^2 F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right)","\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6 \sqrt[3]{2}}-\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2^{2/3} \sqrt[3]{1-x^3}+x-1\right)}{4 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}+\frac{\log \left((1-x) (x+1)^2\right)}{12 \sqrt[3]{2}}",1,"(x^2*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3])/2","C",0
621,1,67,270,0.0509548,"\int \frac{1}{x^2 \sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[1/(x^2*(1 - x^3)^(1/3)*(1 + x^3)),x]","-\frac{1}{5} x^5 F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right)-x^2 F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right)-\frac{\left(1-x^3\right)^{2/3}}{x}","-\frac{1}{2} x^2 \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right)-\frac{\left(1-x^3\right)^{2/3}}{x}-\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6 \sqrt[3]{2}}+\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{3 \sqrt[3]{2}}+\frac{\log \left(2^{2/3} \sqrt[3]{1-x^3}+x-1\right)}{4 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\log \left((1-x) (x+1)^2\right)}{12 \sqrt[3]{2}}",1,"-((1 - x^3)^(2/3)/x) - x^2*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3] - (x^5*AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3])/5","C",0
622,1,76,289,0.0572309,"\int \frac{1}{x^5 \sqrt[3]{1-x^3} \left(1+x^3\right)} \, dx","Integrate[1/(x^5*(1 - x^3)^(1/3)*(1 + x^3)),x]","\frac{2 x^9 F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right)+15 x^6 F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right)+5 \left(1-x^3\right)^{2/3} \left(2 x^3-1\right)}{20 x^4}","\frac{1}{4} x^2 \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right)+\frac{\left(1-x^3\right)^{2/3}}{2 x}+\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6 \sqrt[3]{2}}-\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2^{2/3} \sqrt[3]{1-x^3}+x-1\right)}{4 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\left(1-x^3\right)^{2/3}}{4 x^4}+\frac{\log \left((1-x) (x+1)^2\right)}{12 \sqrt[3]{2}}",1,"(5*(1 - x^3)^(2/3)*(-1 + 2*x^3) + 15*x^6*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3] + 2*x^9*AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3])/(20*x^4)","C",0
623,1,151,125,0.1067004,"\int \frac{x^{11}}{\left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[x^11/((1 - x^3)^(2/3)*(1 + x^3)),x]","\frac{1}{84} \left(3 \sqrt[3]{1-x^3} x^3-75 \sqrt[3]{1-x^3}-14 \sqrt[3]{2} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)+7 \sqrt[3]{2} \log \left(\left(1-x^3\right)^{2/3}+\sqrt[3]{2-2 x^3}+2^{2/3}\right)+14 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)-12 \sqrt[3]{1-x^3} x^6\right)","-\frac{1}{7} \left(1-x^3\right)^{7/3}+\frac{1}{4} \left(1-x^3\right)^{4/3}-\sqrt[3]{1-x^3}+\frac{\log \left(x^3+1\right)}{6\ 2^{2/3}}-\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}",1,"(-75*(1 - x^3)^(1/3) + 3*x^3*(1 - x^3)^(1/3) - 12*x^6*(1 - x^3)^(1/3) + 14*2^(1/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - 14*2^(1/3)*Log[2^(1/3) - (1 - x^3)^(1/3)] + 7*2^(1/3)*Log[2^(2/3) + (2 - 2*x^3)^(1/3) + (1 - x^3)^(2/3)])/84","A",1
624,1,135,98,0.0888926,"\int \frac{x^8}{\left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[x^8/((1 - x^3)^(2/3)*(1 + x^3)),x]","\frac{1}{12} \left(-3 \sqrt[3]{1-x^3} x^3+3 \sqrt[3]{1-x^3}+2 \sqrt[3]{2} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)-\sqrt[3]{2} \log \left(\left(1-x^3\right)^{2/3}+\sqrt[3]{2-2 x^3}+2^{2/3}\right)-2 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)\right)","\frac{1}{4} \left(1-x^3\right)^{4/3}-\frac{\log \left(x^3+1\right)}{6\ 2^{2/3}}+\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}",1,"(3*(1 - x^3)^(1/3) - 3*x^3*(1 - x^3)^(1/3) - 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] + 2*2^(1/3)*Log[2^(1/3) - (1 - x^3)^(1/3)] - 2^(1/3)*Log[2^(2/3) + (2 - 2*x^3)^(1/3) + (1 - x^3)^(2/3)])/12","A",1
625,1,118,95,0.0500494,"\int \frac{x^5}{\left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[x^5/((1 - x^3)^(2/3)*(1 + x^3)),x]","\frac{1}{12} \left(-12 \sqrt[3]{1-x^3}-2 \sqrt[3]{2} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)+\sqrt[3]{2} \log \left(\left(1-x^3\right)^{2/3}+\sqrt[3]{2-2 x^3}+2^{2/3}\right)+2 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)\right)","-\sqrt[3]{1-x^3}+\frac{\log \left(x^3+1\right)}{6\ 2^{2/3}}-\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}",1,"(-12*(1 - x^3)^(1/3) + 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - 2*2^(1/3)*Log[2^(1/3) - (1 - x^3)^(1/3)] + 2^(1/3)*Log[2^(2/3) + (2 - 2*x^3)^(1/3) + (1 - x^3)^(2/3)])/12","A",1
626,1,94,83,0.0349095,"\int \frac{x^2}{\left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[x^2/((1 - x^3)^(2/3)*(1 + x^3)),x]","-\frac{-2 \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)+\log \left(\left(1-x^3\right)^{2/3}+\sqrt[3]{2-2 x^3}+2^{2/3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{6\ 2^{2/3}}","-\frac{\log \left(x^3+1\right)}{6\ 2^{2/3}}+\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}",1,"-1/6*(2*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - 2*Log[2^(1/3) - (1 - x^3)^(1/3)] + Log[2^(2/3) + (2 - 2*x^3)^(1/3) + (1 - x^3)^(2/3)])/2^(2/3)","A",1
627,1,179,137,0.0523023,"\int \frac{1}{x \left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[1/(x*(1 - x^3)^(2/3)*(1 + x^3)),x]","\frac{1}{12} \left(4 \log \left(1-\sqrt[3]{1-x^3}\right)-2 \sqrt[3]{2} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)+\sqrt[3]{2} \log \left(\left(1-x^3\right)^{2/3}+\sqrt[3]{2-2 x^3}+2^{2/3}\right)-2 \log \left(\left(1-x^3\right)^{2/3}+\sqrt[3]{1-x^3}+1\right)-4 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)+2 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)\right)","\frac{\log \left(x^3+1\right)}{6\ 2^{2/3}}+\frac{1}{2} \log \left(1-\sqrt[3]{1-x^3}\right)-\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}-\frac{\log (x)}{2}",1,"(-4*Sqrt[3]*ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt[3]] + 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] + 4*Log[1 - (1 - x^3)^(1/3)] - 2*2^(1/3)*Log[2^(1/3) - (1 - x^3)^(1/3)] + 2^(1/3)*Log[2^(2/3) + (2 - 2*x^3)^(1/3) + (1 - x^3)^(2/3)] - 2*Log[1 + (1 - x^3)^(1/3) + (1 - x^3)^(2/3)])/12","A",1
628,1,196,158,0.0995922,"\int \frac{1}{x^4 \left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[1/(x^4*(1 - x^3)^(2/3)*(1 + x^3)),x]","\frac{1}{36} \left(-\frac{12 \sqrt[3]{1-x^3}}{x^3}-4 \log \left(1-\sqrt[3]{1-x^3}\right)+6 \sqrt[3]{2} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)-3 \sqrt[3]{2} \log \left(\left(1-x^3\right)^{2/3}+\sqrt[3]{2-2 x^3}+2^{2/3}\right)+2 \log \left(\left(1-x^3\right)^{2/3}+\sqrt[3]{1-x^3}+1\right)+4 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)-6 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)\right)","-\frac{\sqrt[3]{1-x^3}}{3 x^3}-\frac{\log \left(x^3+1\right)}{6\ 2^{2/3}}-\frac{1}{6} \log \left(1-\sqrt[3]{1-x^3}\right)+\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}+\frac{\log (x)}{6}",1,"((-12*(1 - x^3)^(1/3))/x^3 + 4*Sqrt[3]*ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt[3]] - 6*2^(1/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - 4*Log[1 - (1 - x^3)^(1/3)] + 6*2^(1/3)*Log[2^(1/3) - (1 - x^3)^(1/3)] - 3*2^(1/3)*Log[2^(2/3) + (2 - 2*x^3)^(1/3) + (1 - x^3)^(2/3)] + 2*Log[1 + (1 - x^3)^(1/3) + (1 - x^3)^(2/3)])/36","A",1
629,1,78,160,0.0861414,"\int \frac{x^7}{\left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[x^7/((1 - x^3)^(2/3)*(1 + x^3)),x]","\frac{1}{15} x^2 \left(x^3 \left(-F_1\left(\frac{5}{3};\frac{2}{3},1;\frac{8}{3};x^3,-x^3\right)\right)+\frac{5 \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{2 x^3}{x^3+1}\right)}{\left(x^3+1\right)^{2/3}}-5 \sqrt[3]{1-x^3}\right)","\frac{\log \left(x^3+1\right)}{6\ 2^{2/3}}+\frac{1}{6} \log \left(-\sqrt[3]{1-x^3}-x\right)-\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}-\frac{1}{3} \sqrt[3]{1-x^3} x^2",1,"(x^2*(-5*(1 - x^3)^(1/3) - x^3*AppellF1[5/3, 2/3, 1, 8/3, x^3, -x^3] + (5*Hypergeometric2F1[2/3, 2/3, 5/3, (2*x^3)/(1 + x^3)])/(1 + x^3)^(2/3)))/15","C",0
630,1,26,139,0.0512278,"\int \frac{x^4}{\left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[x^4/((1 - x^3)^(2/3)*(1 + x^3)),x]","\frac{1}{5} x^5 F_1\left(\frac{5}{3};\frac{2}{3},1;\frac{8}{3};x^3,-x^3\right)","-\frac{\log \left(x^3+1\right)}{6\ 2^{2/3}}-\frac{1}{2} \log \left(-\sqrt[3]{1-x^3}-x\right)+\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}",1,"(x^5*AppellF1[5/3, 2/3, 1, 8/3, x^3, -x^3])/5","C",0
631,1,38,88,0.0066334,"\int \frac{x}{\left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[x/((1 - x^3)^(2/3)*(1 + x^3)),x]","\frac{x^2 \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{2 x^3}{x^3+1}\right)}{2 \left(x^3+1\right)^{2/3}}","\frac{\log \left(x^3+1\right)}{6\ 2^{2/3}}-\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}",1,"(x^2*Hypergeometric2F1[2/3, 2/3, 5/3, (2*x^3)/(1 + x^3)])/(2*(1 + x^3)^(2/3))","C",0
632,1,81,103,0.5547226,"\int \frac{1}{x^2 \left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[1/(x^2*(1 - x^3)^(2/3)*(1 + x^3)),x]","-\frac{5 \left(-3 x^6+x^3+2\right) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{2 x^3}{x^3-1}\right)-12 x^3 \left(x^3+1\right) \, _2F_1\left(\frac{5}{3},2;\frac{8}{3};\frac{2 x^3}{x^3-1}\right)}{10 x \left(1-x^3\right)^{5/3}}","-\frac{\sqrt[3]{1-x^3}}{x}-\frac{\log \left(x^3+1\right)}{6\ 2^{2/3}}+\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}",1,"-1/10*(5*(2 + x^3 - 3*x^6)*Hypergeometric2F1[2/3, 1, 5/3, (2*x^3)/(-1 + x^3)] - 12*x^3*(1 + x^3)*Hypergeometric2F1[5/3, 2, 8/3, (2*x^3)/(-1 + x^3)])/(x*(1 - x^3)^(5/3))","C",0
633,1,145,124,13.8994868,"\int \frac{1}{x^5 \left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[1/(x^5*(1 - x^3)^(2/3)*(1 + x^3)),x]","-\frac{81 \left(x^3+1\right)^2 x^3 \, _4F_3\left(\frac{2}{3},2,2,2;1,1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right)+216 \left(x^9+x^6\right) \, _3F_2\left(\frac{2}{3},2,2;1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right)+5 \left(\left(9 x^9-20 x^6-13 x^3+4\right) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{2 x^3}{x^3-1}\right)+9 x^9+x^6-9 x^3-1\right)}{60 x^4 \left(1-x^3\right)^{5/3}}","\frac{\sqrt[3]{1-x^3}}{4 x}+\frac{\log \left(x^3+1\right)}{6\ 2^{2/3}}-\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}-\frac{\sqrt[3]{1-x^3}}{4 x^4}",1,"-1/60*(5*(-1 - 9*x^3 + x^6 + 9*x^9 + (4 - 13*x^3 - 20*x^6 + 9*x^9)*Hypergeometric2F1[2/3, 1, 5/3, (2*x^3)/(-1 + x^3)]) + 216*(x^6 + x^9)*HypergeometricPFQ[{2/3, 2, 2}, {1, 8/3}, (2*x^3)/(-1 + x^3)] + 81*x^3*(1 + x^3)^2*HypergeometricPFQ[{2/3, 2, 2, 2}, {1, 1, 8/3}, (2*x^3)/(-1 + x^3)])/(x^4*(1 - x^3)^(5/3))","C",0
634,1,115,291,0.1629953,"\int \frac{x^6}{\left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[x^6/((1 - x^3)^(2/3)*(1 + x^3)),x]","\frac{1}{2} x \sqrt[3]{1-x^3} \left(-\frac{4 F_1\left(\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right)}{\left(x^3+1\right) \left(x^3 \left(3 F_1\left(\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};x^3,-x^3\right)+F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right)\right)-4 F_1\left(\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right)\right)}-1\right)","-\frac{1}{2} \sqrt[3]{1-x^3} x+\frac{\log \left(2^{2/3}-\frac{1-x}{\sqrt[3]{1-x^3}}\right)}{6\ 2^{2/3}}-\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6\ 2^{2/3}}+\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{3\ 2^{2/3}}-\frac{\log \left(\frac{(1-x)^2}{\left(1-x^3\right)^{2/3}}+\frac{2^{2/3} (1-x)}{\sqrt[3]{1-x^3}}+2 \sqrt[3]{2}\right)}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{2\ 2^{2/3} \sqrt{3}}",1,"(x*(1 - x^3)^(1/3)*(-1 - (4*AppellF1[1/3, -1/3, 1, 4/3, x^3, -x^3])/((1 + x^3)*(-4*AppellF1[1/3, -1/3, 1, 4/3, x^3, -x^3] + x^3*(3*AppellF1[4/3, -1/3, 2, 7/3, x^3, -x^3] + AppellF1[4/3, 2/3, 1, 7/3, x^3, -x^3])))))/2","C",0
635,1,26,294,0.0371078,"\int \frac{x^3}{\left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[x^3/((1 - x^3)^(2/3)*(1 + x^3)),x]","\frac{1}{4} x^4 F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right)","\frac{1}{2} x \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};x^3\right)-\frac{\log \left(2^{2/3}-\frac{1-x}{\sqrt[3]{1-x^3}}\right)}{6\ 2^{2/3}}+\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6\ 2^{2/3}}-\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{3\ 2^{2/3}}+\frac{\log \left(\frac{(1-x)^2}{\left(1-x^3\right)^{2/3}}+\frac{2^{2/3} (1-x)}{\sqrt[3]{1-x^3}}+2 \sqrt[3]{2}\right)}{12\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{2\ 2^{2/3} \sqrt{3}}",1,"(x^4*AppellF1[4/3, 2/3, 1, 7/3, x^3, -x^3])/4","C",0
636,1,111,293,0.2055265,"\int \frac{1}{\left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[1/((1 - x^3)^(2/3)*(1 + x^3)),x]","-\frac{4 x F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};x^3,-x^3\right)}{\left(1-x^3\right)^{2/3} \left(x^3+1\right) \left(x^3 \left(3 F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};x^3,-x^3\right)-2 F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};x^3,-x^3\right)\right)-4 F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};x^3,-x^3\right)\right)}","\frac{1}{2} x \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};x^3\right)+\frac{\log \left(2^{2/3}-\frac{1-x}{\sqrt[3]{1-x^3}}\right)}{6\ 2^{2/3}}-\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6\ 2^{2/3}}+\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{3\ 2^{2/3}}-\frac{\log \left(\frac{(1-x)^2}{\left(1-x^3\right)^{2/3}}+\frac{2^{2/3} (1-x)}{\sqrt[3]{1-x^3}}+2 \sqrt[3]{2}\right)}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{2\ 2^{2/3} \sqrt{3}}",1,"(-4*x*AppellF1[1/3, 2/3, 1, 4/3, x^3, -x^3])/((1 - x^3)^(2/3)*(1 + x^3)*(-4*AppellF1[1/3, 2/3, 1, 4/3, x^3, -x^3] + x^3*(3*AppellF1[4/3, 2/3, 2, 7/3, x^3, -x^3] - 2*AppellF1[4/3, 5/3, 1, 7/3, x^3, -x^3])))","C",0
637,1,120,294,0.1004033,"\int \frac{1}{x^3 \left(1-x^3\right)^{2/3} \left(1+x^3\right)} \, dx","Integrate[1/(x^3*(1 - x^3)^(2/3)*(1 + x^3)),x]","\frac{\sqrt[3]{1-x^3} \left(\frac{4 x^3 F_1\left(\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right)}{\left(x^3+1\right) \left(x^3 \left(3 F_1\left(\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};x^3,-x^3\right)+F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right)\right)-4 F_1\left(\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right)\right)}-1\right)}{2 x^2}","-\frac{\log \left(2^{2/3}-\frac{1-x}{\sqrt[3]{1-x^3}}\right)}{6\ 2^{2/3}}+\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6\ 2^{2/3}}-\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{3\ 2^{2/3}}+\frac{\log \left(\frac{(1-x)^2}{\left(1-x^3\right)^{2/3}}+\frac{2^{2/3} (1-x)}{\sqrt[3]{1-x^3}}+2 \sqrt[3]{2}\right)}{12\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{2\ 2^{2/3} \sqrt{3}}-\frac{\sqrt[3]{1-x^3}}{2 x^2}",1,"((1 - x^3)^(1/3)*(-1 + (4*x^3*AppellF1[1/3, -1/3, 1, 4/3, x^3, -x^3])/((1 + x^3)*(-4*AppellF1[1/3, -1/3, 1, 4/3, x^3, -x^3] + x^3*(3*AppellF1[4/3, -1/3, 2, 7/3, x^3, -x^3] + AppellF1[4/3, 2/3, 1, 7/3, x^3, -x^3])))))/(2*x^2)","C",0
638,1,53,141,0.0368917,"\int \frac{x^{14}}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x^14/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{20 \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{1}{2} \left(1-x^3\right)\right)-5 x^9-x^6-23 x^3+29}{40 \sqrt[3]{1-x^3}}","\frac{1}{8} \left(1-x^3\right)^{8/3}-\frac{2}{5} \left(1-x^3\right)^{5/3}+\left(1-x^3\right)^{2/3}+\frac{1}{2 \sqrt[3]{1-x^3}}-\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}+\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{4 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}",1,"(29 - 23*x^3 - x^6 - 5*x^9 + 20*Hypergeometric2F1[-1/3, 1, 2/3, (1 - x^3)/2])/(40*(1 - x^3)^(1/3))","C",1
639,1,48,130,0.0188364,"\int \frac{x^{11}}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x^11/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{-5 \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{1}{2} \left(1-x^3\right)\right)-2 x^6-x^3+13}{10 \sqrt[3]{1-x^3}}","-\frac{1}{5} \left(1-x^3\right)^{5/3}+\frac{1}{2} \left(1-x^3\right)^{2/3}+\frac{1}{2 \sqrt[3]{1-x^3}}+\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}-\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{4 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}",1,"(13 - x^3 - 2*x^6 - 5*Hypergeometric2F1[-1/3, 1, 2/3, (1 - x^3)/2])/(10*(1 - x^3)^(1/3))","C",1
640,1,41,115,0.0139348,"\int \frac{x^8}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x^8/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{\, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{1}{2} \left(1-x^3\right)\right)-x^3+1}{2 \sqrt[3]{1-x^3}}","\frac{1}{2} \left(1-x^3\right)^{2/3}+\frac{1}{2 \sqrt[3]{1-x^3}}-\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}+\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{4 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}",1,"(1 - x^3 + Hypergeometric2F1[-1/3, 1, 2/3, (1 - x^3)/2])/(2*(1 - x^3)^(1/3))","C",1
641,1,95,100,0.0930097,"\int \frac{x^5}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x^5/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{1}{24} \left(\frac{12}{\sqrt[3]{1-x^3}}+2^{2/3} \log \left(x^3+1\right)-3\ 2^{2/3} \log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)-2\ 2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)\right)","\frac{1}{2 \sqrt[3]{1-x^3}}+\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}-\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{4 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}",1,"(12/(1 - x^3)^(1/3) - 2*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] + 2^(2/3)*Log[1 + x^3] - 3*2^(2/3)*Log[2^(1/3) - (1 - x^3)^(1/3)])/24","A",1
642,1,34,100,0.0102739,"\int \frac{x^2}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x^2/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{\, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{1}{2} \left(1-x^3\right)\right)}{2 \sqrt[3]{1-x^3}}","\frac{1}{2 \sqrt[3]{1-x^3}}-\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}+\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{4 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}",1,"Hypergeometric2F1[-1/3, 1, 2/3, (1 - x^3)/2]/(2*(1 - x^3)^(1/3))","C",1
643,1,54,154,0.0233507,"\int \frac{1}{x \left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[1/(x*(1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{2 \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};1-x^3\right)-\, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{1}{2} \left(1-x^3\right)\right)}{2 \sqrt[3]{1-x^3}}","\frac{1}{2 \sqrt[3]{1-x^3}}+\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}+\frac{1}{2} \log \left(1-\sqrt[3]{1-x^3}\right)-\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{4 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\log (x)}{2}",1,"(-Hypergeometric2F1[-1/3, 1, 2/3, (1 - x^3)/2] + 2*Hypergeometric2F1[-1/3, 1, 2/3, 1 - x^3])/(2*(1 - x^3)^(1/3))","C",1
644,1,64,175,0.0324347,"\int \frac{1}{x^4 \left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[1/(x^4*(1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{3 x^3 \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{1}{2} \left(1-x^3\right)\right)+2 x^3 \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};1-x^3\right)-2}{6 x^3 \sqrt[3]{1-x^3}}","-\frac{1}{3 x^3 \sqrt[3]{1-x^3}}+\frac{5}{6 \sqrt[3]{1-x^3}}-\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}+\frac{1}{6} \log \left(1-\sqrt[3]{1-x^3}\right)+\frac{\log \left(\sqrt[3]{2}-\sqrt[3]{1-x^3}\right)}{4 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2^{2/3} \sqrt[3]{1-x^3}+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\log (x)}{6}",1,"(-2 + 3*x^3*Hypergeometric2F1[-1/3, 1, 2/3, (1 - x^3)/2] + 2*x^3*Hypergeometric2F1[-1/3, 1, 2/3, 1 - x^3])/(6*x^3*(1 - x^3)^(1/3))","C",1
645,1,152,174,0.2482087,"\int \frac{x^9}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x^9/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{1}{72} \left(-6 x^4 F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right)-\frac{12 \left(2 x^3-5\right) x}{\sqrt[3]{1-x^3}}-5\ 2^{2/3} \left(2 \log \left(\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+1\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} x}{\sqrt[3]{x^3-1}}-1}{\sqrt{3}}\right)-\log \left(-\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+\frac{2^{2/3} x^2}{\left(x^3-1\right)^{2/3}}+1\right)\right)\right)","\frac{5}{6} \left(1-x^3\right)^{2/3} x+\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}-\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{4 \sqrt[3]{2}}-\frac{1}{6} \log \left(\sqrt[3]{1-x^3}+x\right)+\frac{\tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}+\frac{x^4}{2 \sqrt[3]{1-x^3}}",1,"((-12*x*(-5 + 2*x^3))/(1 - x^3)^(1/3) - 6*x^4*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3] - 5*2^(2/3)*(2*Sqrt[3]*ArcTan[(-1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] - Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1 + x^3)^(1/3)] + 2*Log[1 + (2^(1/3)*x)/(-1 + x^3)^(1/3)]))/72","C",0
646,1,142,153,0.1644812,"\int \frac{x^6}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x^6/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{1}{24} \left(-6 x^4 F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right)+\frac{12 x}{\sqrt[3]{1-x^3}}+2^{2/3} \left(-2 \log \left(\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+1\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} x}{\sqrt[3]{x^3-1}}-1}{\sqrt{3}}\right)+\log \left(-\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+\frac{2^{2/3} x^2}{\left(x^3-1\right)^{2/3}}+1\right)\right)\right)","\frac{x}{2 \sqrt[3]{1-x^3}}-\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}+\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{4 \sqrt[3]{2}}-\frac{1}{2} \log \left(\sqrt[3]{1-x^3}+x\right)+\frac{\tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}",1,"((12*x)/(1 - x^3)^(1/3) - 6*x^4*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3] + 2^(2/3)*(-2*Sqrt[3]*ArcTan[(-1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] + Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1 + x^3)^(1/3)] - 2*Log[1 + (2^(1/3)*x)/(-1 + x^3)^(1/3)]))/24","C",0
647,1,38,106,0.0089635,"\int \frac{x^3}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x^3/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{x^4 \, _2F_1\left(\frac{4}{3},\frac{4}{3};\frac{7}{3};\frac{2 x^3}{x^3+1}\right)}{4 \left(x^3+1\right)^{4/3}}","\frac{x}{2 \sqrt[3]{1-x^3}}+\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}-\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{4 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}",1,"(x^4*Hypergeometric2F1[4/3, 4/3, 7/3, (2*x^3)/(1 + x^3)])/(4*(1 + x^3)^(4/3))","C",0
648,1,100,106,0.0477985,"\int \frac{1}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[1/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{-7 \left(3 x^3+4\right) \left(x^3-1\right)^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right)-12 \left(x^9+x^6\right) \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right)+7 \left(3 x^3+4\right) \left(x^3-1\right)^2}{14 x^2 \left(1-x^3\right)^{7/3}}","\frac{x}{2 \sqrt[3]{1-x^3}}-\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}+\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{4 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}",1,"(7*(-1 + x^3)^2*(4 + 3*x^3) - 7*(-1 + x^3)^2*(4 + 3*x^3)*Hypergeometric2F1[1/3, 1, 4/3, (2*x^3)/(-1 + x^3)] - 12*(x^6 + x^9)*Hypergeometric2F1[2, 7/3, 10/3, (2*x^3)/(-1 + x^3)])/(14*x^2*(1 - x^3)^(7/3))","C",0
649,1,192,124,3.1083485,"\int \frac{1}{x^3 \left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[1/(x^3*(1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{18 \left(x^3+1\right)^2 x^6 \, _3F_2\left(2,2,\frac{7}{3};1,\frac{10}{3};\frac{2 x^3}{x^3-1}\right)+54 x^{12} \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right)+84 x^9 \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right)+30 x^6 \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right)+7 \left(x^3-1\right)^2 \left(9 x^6+12 x^3+2\right) \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right)-63 x^{12}+42 x^9+91 x^6-56 x^3-14}{14 x^5 \left(1-x^3\right)^{7/3}}","\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}-\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{4 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\left(1-x^3\right)^{2/3}}{x^2}+\frac{1}{2 x^2 \sqrt[3]{1-x^3}}",1,"(-14 - 56*x^3 + 91*x^6 + 42*x^9 - 63*x^12 + 7*(-1 + x^3)^2*(2 + 12*x^3 + 9*x^6)*Hypergeometric2F1[1/3, 1, 4/3, (2*x^3)/(-1 + x^3)] + 30*x^6*Hypergeometric2F1[2, 7/3, 10/3, (2*x^3)/(-1 + x^3)] + 84*x^9*Hypergeometric2F1[2, 7/3, 10/3, (2*x^3)/(-1 + x^3)] + 54*x^12*Hypergeometric2F1[2, 7/3, 10/3, (2*x^3)/(-1 + x^3)] + 18*x^6*(1 + x^3)^2*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, (2*x^3)/(-1 + x^3)])/(14*x^5*(1 - x^3)^(7/3))","C",0
650,1,133,144,5.1583958,"\int \frac{1}{x^6 \left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[1/(x^6*(1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{2 \log \left(\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+1\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} x}{\sqrt[3]{x^3-1}}-1}{\sqrt{3}}\right)-\log \left(-\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+\frac{2^{2/3} x^2}{\left(x^3-1\right)^{2/3}}+1\right)}{12 \sqrt[3]{2}}-\frac{-8 x^6+x^3+2}{10 x^5 \sqrt[3]{1-x^3}}","-\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}+\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{4 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{7 \left(1-x^3\right)^{2/3}}{10 x^5}+\frac{1}{2 x^5 \sqrt[3]{1-x^3}}-\frac{4 \left(1-x^3\right)^{2/3}}{5 x^2}",1,"-1/10*(2 + x^3 - 8*x^6)/(x^5*(1 - x^3)^(1/3)) + (2*Sqrt[3]*ArcTan[(-1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] - Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1 + x^3)^(1/3)] + 2*Log[1 + (2^(1/3)*x)/(-1 + x^3)^(1/3)])/(12*2^(1/3))","A",0
651,1,136,162,5.2053686,"\int \frac{1}{x^9 \left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[1/(x^9*(1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{1}{120} \left(5\ 2^{2/3} \left(-2 \log \left(\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+1\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{2} x}{\sqrt[3]{x^3-1}}-1}{\sqrt{3}}\right)+\log \left(-\frac{\sqrt[3]{2} x}{\sqrt[3]{x^3-1}}+\frac{2^{2/3} x^2}{\left(x^3-1\right)^{2/3}}+1\right)\right)-\frac{3 \left(-49 x^9+23 x^6+x^3+5\right)}{x^8 \sqrt[3]{1-x^3}}\right)","\frac{\log \left(x^3+1\right)}{12 \sqrt[3]{2}}-\frac{\log \left(-\sqrt[3]{1-x^3}-\sqrt[3]{2} x\right)}{4 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{5 \left(1-x^3\right)^{2/3}}{8 x^8}+\frac{1}{2 x^8 \sqrt[3]{1-x^3}}-\frac{13 \left(1-x^3\right)^{2/3}}{20 x^5}-\frac{49 \left(1-x^3\right)^{2/3}}{40 x^2}",1,"((-3*(5 + x^3 + 23*x^6 - 49*x^9))/(x^8*(1 - x^3)^(1/3)) + 5*2^(2/3)*(-2*Sqrt[3]*ArcTan[(-1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] + Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1 + x^3)^(1/3)] - 2*Log[1 + (2^(1/3)*x)/(-1 + x^3)^(1/3)]))/120","A",0
652,1,71,292,0.1387244,"\int \frac{x^{10}}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x^10/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{1}{20} x^2 \left(-4 x^3 F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right)-15 F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right)-\frac{5 \left(x^3-3\right)}{\sqrt[3]{1-x^3}}\right)","-\frac{1}{2} x^2 \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right)-\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{12 \sqrt[3]{2}}+\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6 \sqrt[3]{2}}+\frac{\log \left(2^{2/3} \sqrt[3]{1-x^3}+x-1\right)}{8 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{4 \sqrt[3]{2} \sqrt{3}}+\frac{x^5}{2 \sqrt[3]{1-x^3}}+\frac{3}{4} \left(1-x^3\right)^{2/3} x^2-\frac{\log \left((1-x) (x+1)^2\right)}{24 \sqrt[3]{2}}",1,"(x^2*((-5*(-3 + x^3))/(1 - x^3)^(1/3) - 15*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3] - 4*x^3*AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3]))/20","C",0
653,1,66,274,0.1224712,"\int \frac{x^7}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x^7/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{1}{10} x^2 \left(-3 x^3 F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right)-5 F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right)+\frac{5}{\sqrt[3]{1-x^3}}\right)","-\frac{3}{4} x^2 \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right)+\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{12 \sqrt[3]{2}}-\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6 \sqrt[3]{2}}-\frac{\log \left(2^{2/3} \sqrt[3]{1-x^3}+x-1\right)}{8 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{4 \sqrt[3]{2} \sqrt{3}}+\frac{x^2}{2 \sqrt[3]{1-x^3}}+\frac{\log \left((1-x) (x+1)^2\right)}{24 \sqrt[3]{2}}",1,"(x^2*(5/(1 - x^3)^(1/3) - 5*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3] - 3*x^3*AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3]))/10","C",0
654,1,66,274,0.0877495,"\int \frac{x^4}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x^4/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{1}{10} x^2 \left(x^3 \left(-F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right)\right)-5 F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right)+\frac{5}{\sqrt[3]{1-x^3}}\right)","-\frac{1}{4} x^2 \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right)-\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{12 \sqrt[3]{2}}+\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6 \sqrt[3]{2}}+\frac{\log \left(2^{2/3} \sqrt[3]{1-x^3}+x-1\right)}{8 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{4 \sqrt[3]{2} \sqrt{3}}+\frac{x^2}{2 \sqrt[3]{1-x^3}}-\frac{\log \left((1-x) (x+1)^2\right)}{24 \sqrt[3]{2}}",1,"(x^2*(5/(1 - x^3)^(1/3) - 5*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3] - x^3*AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3]))/10","C",0
655,1,45,274,0.0569024,"\int \frac{x}{\left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[x/((1 - x^3)^(4/3)*(1 + x^3)),x]","\frac{x^2}{2 \sqrt[3]{1-x^3}}-\frac{1}{10} x^5 F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right)","-\frac{1}{4} x^2 \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right)+\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{12 \sqrt[3]{2}}-\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6 \sqrt[3]{2}}-\frac{\log \left(2^{2/3} \sqrt[3]{1-x^3}+x-1\right)}{8 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{4 \sqrt[3]{2} \sqrt{3}}+\frac{x^2}{2 \sqrt[3]{1-x^3}}+\frac{\log \left((1-x) (x+1)^2\right)}{24 \sqrt[3]{2}}",1,"x^2/(2*(1 - x^3)^(1/3)) - (x^5*AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3])/10","C",0
656,1,76,292,0.0641409,"\int \frac{1}{x^2 \left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[1/(x^2*(1 - x^3)^(4/3)*(1 + x^3)),x]","-\frac{3}{10} x^5 F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right)-x^2 F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right)+\frac{3 x^3-2}{2 \sqrt[3]{1-x^3} x}","-\frac{3}{4} x^2 \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right)-\frac{3 \left(1-x^3\right)^{2/3}}{2 x}+\frac{1}{2 \sqrt[3]{1-x^3} x}-\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{12 \sqrt[3]{2}}+\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6 \sqrt[3]{2}}+\frac{\log \left(2^{2/3} \sqrt[3]{1-x^3}+x-1\right)}{8 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{4 \sqrt[3]{2} \sqrt{3}}-\frac{\log \left((1-x) (x+1)^2\right)}{24 \sqrt[3]{2}}",1,"(-2 + 3*x^3)/(2*x*(1 - x^3)^(1/3)) - x^2*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3] - (3*x^5*AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3])/10","C",0
657,1,79,308,0.1021306,"\int \frac{1}{x^5 \left(1-x^3\right)^{4/3} \left(1+x^3\right)} \, dx","Integrate[1/(x^5*(1 - x^3)^(4/3)*(1 + x^3)),x]","-\frac{4 x^9 F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right)+5 x^6 F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right)+\frac{5 \left(-4 x^6+x^3+1\right)}{\sqrt[3]{1-x^3}}}{20 x^4}","-\frac{1}{2} x^2 \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right)-\frac{\left(1-x^3\right)^{2/3}}{x}+\frac{\log \left(\frac{2^{2/3} (1-x)^2}{\left(1-x^3\right)^{2/3}}-\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{12 \sqrt[3]{2}}-\frac{\log \left(\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1\right)}{6 \sqrt[3]{2}}-\frac{\log \left(2^{2/3} \sqrt[3]{1-x^3}+x-1\right)}{8 \sqrt[3]{2}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{4 \sqrt[3]{2} \sqrt{3}}-\frac{3 \left(1-x^3\right)^{2/3}}{4 x^4}+\frac{1}{2 \sqrt[3]{1-x^3} x^4}+\frac{\log \left((1-x) (x+1)^2\right)}{24 \sqrt[3]{2}}",1,"-1/20*((5*(1 + x^3 - 4*x^6))/(1 - x^3)^(1/3) + 5*x^6*AppellF1[2/3, 1/3, 1, 5/3, x^3, -x^3] + 4*x^9*AppellF1[5/3, 1/3, 1, 8/3, x^3, -x^3])/x^4","C",0
658,1,270,264,0.6832549,"\int \frac{x^{11} \sqrt[3]{a+b x^3}}{c+d x^3} \, dx","Integrate[(x^11*(a + b*x^3)^(1/3))/(c + d*x^3),x]","\frac{\frac{105 d \left(a+b x^3\right)^{4/3} \left(a^2 d^2+a b c d+b^2 c^2\right)}{b^3}-\frac{60 d^2 \left(a+b x^3\right)^{7/3} (2 a d+b c)}{b^3}+\frac{42 d^3 \left(a+b x^3\right)^{10/3}}{b^3}-\frac{70 c^3 \sqrt[3]{b c-a d} \left(\log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)\right)}{\sqrt[3]{d}}-420 c^3 \sqrt[3]{a+b x^3}}{420 d^4}","\frac{\left(a+b x^3\right)^{4/3} \left(a^2 d^2+a b c d+b^2 c^2\right)}{4 b^3 d^3}-\frac{\left(a+b x^3\right)^{7/3} (2 a d+b c)}{7 b^3 d^2}+\frac{\left(a+b x^3\right)^{10/3}}{10 b^3 d}-\frac{c^3 \sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 d^{13/3}}+\frac{c^3 \sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{13/3}}-\frac{c^3 \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{13/3}}-\frac{c^3 \sqrt[3]{a+b x^3}}{d^4}",1,"(-420*c^3*(a + b*x^3)^(1/3) + (105*d*(b^2*c^2 + a*b*c*d + a^2*d^2)*(a + b*x^3)^(4/3))/b^3 - (60*d^2*(b*c + 2*a*d)*(a + b*x^3)^(7/3))/b^3 + (42*d^3*(a + b*x^3)^(10/3))/b^3 - (70*c^3*(b*c - a*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/d^(1/3))/(420*d^4)","A",1
659,1,230,220,0.4076407,"\int \frac{x^8 \sqrt[3]{a+b x^3}}{c+d x^3} \, dx","Integrate[(x^8*(a + b*x^3)^(1/3))/(c + d*x^3),x]","\frac{-\frac{21 d \left(a+b x^3\right)^{4/3} (a d+b c)}{b^2}+\frac{12 d^2 \left(a+b x^3\right)^{7/3}}{b^2}+\frac{14 c^2 \sqrt[3]{b c-a d} \left(\log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)\right)}{\sqrt[3]{d}}+84 c^2 \sqrt[3]{a+b x^3}}{84 d^3}","-\frac{\left(a+b x^3\right)^{4/3} (a d+b c)}{4 b^2 d^2}+\frac{\left(a+b x^3\right)^{7/3}}{7 b^2 d}+\frac{c^2 \sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 d^{10/3}}-\frac{c^2 \sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{10/3}}+\frac{c^2 \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{10/3}}+\frac{c^2 \sqrt[3]{a+b x^3}}{d^3}",1,"(84*c^2*(a + b*x^3)^(1/3) - (21*d*(b*c + a*d)*(a + b*x^3)^(4/3))/b^2 + (12*d^2*(a + b*x^3)^(7/3))/b^2 + (14*c^2*(b*c - a*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/d^(1/3))/(84*d^3)","A",1
660,1,204,186,0.4663853,"\int \frac{x^5 \sqrt[3]{a+b x^3}}{c+d x^3} \, dx","Integrate[(x^5*(a + b*x^3)^(1/3))/(c + d*x^3),x]","\frac{c \sqrt[3]{b c-a d} \left(-\log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)+2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)\right)}{6 d^{7/3}}-\frac{c \sqrt[3]{a+b x^3}}{d^2}+\frac{\left(a+b x^3\right)^{4/3}}{4 b d}","-\frac{c \sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 d^{7/3}}+\frac{c \sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{7/3}}-\frac{c \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{7/3}}-\frac{c \sqrt[3]{a+b x^3}}{d^2}+\frac{\left(a+b x^3\right)^{4/3}}{4 b d}",1,"-((c*(a + b*x^3)^(1/3))/d^2) + (a + b*x^3)^(4/3)/(4*b*d) + (c*(b*c - a*d)^(1/3)*(2*Sqrt[3]*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] + 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] - Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/(6*d^(7/3))","A",1
661,1,205,159,0.2872356,"\int \frac{x^2 \sqrt[3]{a+b x^3}}{c+d x^3} \, dx","Integrate[(x^2*(a + b*x^3)^(1/3))/(c + d*x^3),x]","\frac{\sqrt[3]{b c-a d} \log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)-2 \sqrt{3} \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)+6 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{6 d^{4/3}}","\frac{\sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 d^{4/3}}-\frac{\sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{4/3}}+\frac{\sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{4/3}}+\frac{\sqrt[3]{a+b x^3}}{d}",1,"(6*d^(1/3)*(a + b*x^3)^(1/3) - 2*Sqrt[3]*(b*c - a*d)^(1/3)*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*(b*c - a*d)^(1/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + (b*c - a*d)^(1/3)*Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)])/(6*d^(4/3))","A",1
662,1,268,246,0.4652633,"\int \frac{\sqrt[3]{a+b x^3}}{x \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x*(c + d*x^3)),x]","\frac{\frac{\sqrt[3]{b c-a d} \left(-\log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)+2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)\right)}{\sqrt[3]{d}}-\sqrt[3]{a} \left(\log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)}{6 c}","-\frac{\sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 c \sqrt[3]{d}}+\frac{\sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c \sqrt[3]{d}}-\frac{\sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c \sqrt[3]{d}}+\frac{\sqrt[3]{a} \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2 c}-\frac{\sqrt[3]{a} \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} c}-\frac{\sqrt[3]{a} \log (x)}{2 c}",1,"(-(a^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) - (a + b*x^3)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])) + ((b*c - a*d)^(1/3)*(2*Sqrt[3]*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] + 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] - Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/d^(1/3))/(6*c)","A",1
663,1,366,340,1.2421427,"\int \frac{\sqrt[3]{a+b x^3}}{x^4 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^4*(c + d*x^3)),x]","\frac{\frac{(b c-3 a d) \left(3 \sqrt[3]{a+b x^3}-\frac{1}{2} \sqrt[3]{a} \left(\log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)\right)}{3 c}+\frac{a d^{2/3} \left(\sqrt[3]{b c-a d} \log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)-2 \sqrt{3} \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)+6 \sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c}-\frac{\left(a+b x^3\right)^{4/3}}{x^3}}{3 a c}","\frac{(b c-3 a d) \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{6 a^{2/3} c^2}-\frac{(b c-3 a d) \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{2/3} c^2}-\frac{\log (x) (b c-3 a d)}{6 a^{2/3} c^2}+\frac{d^{2/3} \sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 c^2}-\frac{d^{2/3} \sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c^2}+\frac{d^{2/3} \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c^2}+\frac{d \sqrt[3]{a+b x^3}}{c^2}+\frac{\sqrt[3]{a+b x^3} (b c-3 a d)}{3 a c^2}-\frac{\left(a+b x^3\right)^{4/3}}{3 a c x^3}",1,"(-((a + b*x^3)^(4/3)/x^3) + ((b*c - 3*a*d)*(3*(a + b*x^3)^(1/3) - (a^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) - (a + b*x^3)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]))/2))/(3*c) + (a*d^(2/3)*(6*d^(1/3)*(a + b*x^3)^(1/3) - 2*Sqrt[3]*(b*c - a*d)^(1/3)*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*(b*c - a*d)^(1/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + (b*c - a*d)^(1/3)*Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/(2*c))/(3*a*c)","A",1
664,1,411,370,1.8372819,"\int \frac{\sqrt[3]{a+b x^3}}{x^7 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^7*(c + d*x^3)),x]","-\frac{\frac{2 \left(-9 a^2 d^2+3 a b c d+b^2 c^2\right) \left(3 \sqrt[3]{a+b x^3}-\frac{1}{2} \sqrt[3]{a} \left(\log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)\right)}{9 a c^2}+\frac{a d^{5/3} \left(\sqrt[3]{b c-a d} \log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)-2 \sqrt{3} \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)+6 \sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{c^2}-\frac{2 \left(a+b x^3\right)^{4/3} (3 a d+b c)}{3 a c x^3}+\frac{\left(a+b x^3\right)^{4/3}}{x^6}}{6 a c}","-\frac{\left(-9 a^2 d^2+3 a b c d+b^2 c^2\right) \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{18 a^{5/3} c^3}+\frac{\left(-9 a^2 d^2+3 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{5/3} c^3}+\frac{\log (x) \left(-9 a^2 d^2+3 a b c d+b^2 c^2\right)}{18 a^{5/3} c^3}-\frac{d^{5/3} \sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 c^3}+\frac{d^{5/3} \sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c^3}-\frac{d^{5/3} \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c^3}+\frac{\sqrt[3]{a+b x^3} (3 a d+b c)}{9 a c^2 x^3}-\frac{\left(a+b x^3\right)^{4/3}}{6 a c x^6}",1,"-1/6*((a + b*x^3)^(4/3)/x^6 - (2*(b*c + 3*a*d)*(a + b*x^3)^(4/3))/(3*a*c*x^3) + (2*(b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*(3*(a + b*x^3)^(1/3) - (a^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) - (a + b*x^3)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]))/2))/(9*a*c^2) + (a*d^(5/3)*(6*d^(1/3)*(a + b*x^3)^(1/3) - 2*Sqrt[3]*(b*c - a*d)^(1/3)*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*(b*c - a*d)^(1/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + (b*c - a*d)^(1/3)*Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/c^2)/(a*c)","A",1
665,1,226,336,0.3725304,"\int \frac{x^7 \sqrt[3]{a+b x^3}}{c+d x^3} \, dx","Integrate[(x^7*(a + b*x^3)^(1/3))/(c + d*x^3),x]","\frac{5 c x^2 \left(a \left(\frac{b x^3}{a}+1\right)^{2/3} (6 b c-a d) \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)+\left(a+b x^3\right) \left(\frac{d x^3}{c}+1\right)^{2/3} \left(a d-6 b c+3 b d x^3\right)\right)-2 x^5 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3} \left(a^2 d^2+3 a b c d-9 b^2 c^2\right) F_1\left(\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{90 b c d^2 \left(a+b x^3\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3}}","-\frac{\left(-a^2 d^2-3 a b c d+9 b^2 c^2\right) \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{18 b^{5/3} d^3}-\frac{\left(-a^2 d^2-3 a b c d+9 b^2 c^2\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{9 \sqrt{3} b^{5/3} d^3}-\frac{c^{5/3} \sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 d^3}+\frac{c^{5/3} \sqrt[3]{b c-a d} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 d^3}+\frac{c^{5/3} \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^3}-\frac{x^2 \sqrt[3]{a+b x^3} (6 b c-a d)}{18 b d^2}+\frac{x^5 \sqrt[3]{a+b x^3}}{6 d}",1,"(-2*(-9*b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^5*(1 + (b*x^3)/a)^(2/3)*(1 + (d*x^3)/c)^(2/3)*AppellF1[5/3, 2/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)] + 5*c*x^2*((a + b*x^3)*(-6*b*c + a*d + 3*b*d*x^3)*(1 + (d*x^3)/c)^(2/3) + a*(6*b*c - a*d)*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))]))/(90*b*c*d^2*(a + b*x^3)^(2/3)*(1 + (d*x^3)/c)^(2/3))","C",0
666,1,185,276,0.1935603,"\int \frac{x^4 \sqrt[3]{a+b x^3}}{c+d x^3} \, dx","Integrate[(x^4*(a + b*x^3)^(1/3))/(c + d*x^3),x]","\frac{x^5 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3} (a d-3 b c) F_1\left(\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 c x^2 \left(\left(a+b x^3\right) \left(\frac{d x^3}{c}+1\right)^{2/3}-a \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)\right)}{15 c d \left(a+b x^3\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3}}","\frac{(3 b c-a d) \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{6 b^{2/3} d^2}+\frac{(3 b c-a d) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{2/3} d^2}+\frac{c^{2/3} \sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 d^2}-\frac{c^{2/3} \sqrt[3]{b c-a d} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 d^2}-\frac{c^{2/3} \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^2}+\frac{x^2 \sqrt[3]{a+b x^3}}{3 d}",1,"((-3*b*c + a*d)*x^5*(1 + (b*x^3)/a)^(2/3)*(1 + (d*x^3)/c)^(2/3)*AppellF1[5/3, 2/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)] + 5*c*x^2*((a + b*x^3)*(1 + (d*x^3)/c)^(2/3) - a*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))]))/(15*c*d*(a + b*x^3)^(2/3)*(1 + (d*x^3)/c)^(2/3))","C",0
667,1,65,234,0.0354432,"\int \frac{x \sqrt[3]{a+b x^3}}{c+d x^3} \, dx","Integrate[(x*(a + b*x^3)^(1/3))/(c + d*x^3),x]","\frac{x^2 \sqrt[3]{a+b x^3} F_1\left(\frac{2}{3};-\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 c \sqrt[3]{\frac{a+b x^3}{a}}}","-\frac{\sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 \sqrt[3]{c} d}+\frac{\sqrt[3]{b c-a d} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 \sqrt[3]{c} d}+\frac{\sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{c} d}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2 d}-\frac{\sqrt[3]{b} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d}",1,"(x^2*(a + b*x^3)^(1/3)*AppellF1[2/3, -1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)])/(2*c*((a + b*x^3)/a)^(1/3))","C",0
668,1,81,168,0.0425793,"\int \frac{\sqrt[3]{a+b x^3}}{x^2 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^2*(c + d*x^3)),x]","-\frac{\sqrt[3]{a+b x^3} \sqrt[3]{\frac{d x^3}{c}+1} \, _2F_1\left(-\frac{1}{3},-\frac{1}{3};\frac{2}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{c x \sqrt[3]{\frac{b x^3}{a}+1}}","\frac{\sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 c^{4/3}}-\frac{\sqrt[3]{b c-a d} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{4/3}}-\frac{\sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{4/3}}-\frac{\sqrt[3]{a+b x^3}}{c x}",1,"-(((a + b*x^3)^(1/3)*(1 + (d*x^3)/c)^(1/3)*Hypergeometric2F1[-1/3, -1/3, 2/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c*x*(1 + (b*x^3)/a)^(1/3)))","C",0
669,1,146,204,0.775768,"\int \frac{\sqrt[3]{a+b x^3}}{x^5 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^5*(c + d*x^3)),x]","-\frac{x^3 \left(3 d x^3-c\right) (b c-a d) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+3 x^3 \left(c+d x^3\right) (b c-a d) \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+2 c \left(a+b x^3\right) \left(c-3 d x^3\right)}{8 c^3 x^4 \left(a+b x^3\right)^{2/3}}","-\frac{d \sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 c^{7/3}}+\frac{d \sqrt[3]{b c-a d} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{7/3}}+\frac{d \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{7/3}}-\frac{\sqrt[3]{a+b x^3} (b c-4 a d)}{4 a c^2 x}-\frac{\sqrt[3]{a+b x^3}}{4 c x^4}",1,"-1/8*(2*c*(a + b*x^3)*(c - 3*d*x^3) + (b*c - a*d)*x^3*(-c + 3*d*x^3)*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 3*(b*c - a*d)*x^3*(c + d*x^3)*Hypergeometric2F1[2/3, 2, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^3*x^4*(a + b*x^3)^(2/3))","C",0
670,1,451,258,2.7530292,"\int \frac{\sqrt[3]{a+b x^3}}{x^8 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^8*(c + d*x^3)),x]","-\frac{-9 x^3 \left(c+d x^3\right)^2 (b c-a d) \, _3F_2\left(\frac{2}{3},2,2;1,\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+15 b c^3 x^3 \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-2 x^3 \left(2 c^2-3 c d x^3+9 d^2 x^6\right) (b c-a d) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-15 a c^2 d x^3 \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-12 b c^2 d x^6 \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+27 a d^3 x^9 \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-27 b c d^2 x^9 \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+12 a c d^2 x^6 \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+8 a c^3-12 a c^2 d x^3+36 a c d^2 x^6+8 b c^3 x^3-12 b c^2 d x^6+36 b c d^2 x^9}{56 c^4 x^7 \left(a+b x^3\right)^{2/3}}","\frac{\sqrt[3]{a+b x^3} \left(-28 a^2 d^2+7 a b c d+3 b^2 c^2\right)}{28 a^2 c^3 x}+\frac{d^2 \sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 c^{10/3}}-\frac{d^2 \sqrt[3]{b c-a d} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{10/3}}-\frac{d^2 \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{10/3}}-\frac{\sqrt[3]{a+b x^3} (b c-7 a d)}{28 a c^2 x^4}-\frac{\sqrt[3]{a+b x^3}}{7 c x^7}",1,"-1/56*(8*a*c^3 + 8*b*c^3*x^3 - 12*a*c^2*d*x^3 - 12*b*c^2*d*x^6 + 36*a*c*d^2*x^6 + 36*b*c*d^2*x^9 - 2*(b*c - a*d)*x^3*(2*c^2 - 3*c*d*x^3 + 9*d^2*x^6)*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 15*b*c^3*x^3*Hypergeometric2F1[2/3, 2, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 15*a*c^2*d*x^3*Hypergeometric2F1[2/3, 2, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 12*b*c^2*d*x^6*Hypergeometric2F1[2/3, 2, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 12*a*c*d^2*x^6*Hypergeometric2F1[2/3, 2, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 27*b*c*d^2*x^9*Hypergeometric2F1[2/3, 2, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*a*d^3*x^9*Hypergeometric2F1[2/3, 2, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 9*(b*c - a*d)*x^3*(c + d*x^3)^2*HypergeometricPFQ[{2/3, 2, 2}, {1, 5/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^4*x^7*(a + b*x^3)^(2/3))","C",0
671,1,214,318,5.2366704,"\int \frac{\sqrt[3]{a+b x^3}}{x^{11} \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^11*(c + d*x^3)),x]","\frac{\frac{\left(a+b x^3\right) \left(a^3 \left(-14 c^3+20 c^2 d x^3-35 c d^2 x^6+140 d^3 x^9\right)+a^2 b c x^3 \left(-2 c^2+5 c d x^3-35 d^2 x^6\right)+3 a b^2 c^2 x^6 \left(c-5 d x^3\right)-9 b^3 c^3 x^9\right)}{70 a^3 c^4 x^{10}}+\frac{d^3 x^2 \left(\frac{b x^3}{a}+1\right)^{2/3} (a d-b c) \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{c^5 \left(\frac{d x^3}{c}+1\right)^{2/3}}}{2 \left(a+b x^3\right)^{2/3}}","\frac{\sqrt[3]{a+b x^3} \left(-35 a^2 d^2+5 a b c d+3 b^2 c^2\right)}{140 a^2 c^3 x^4}-\frac{\sqrt[3]{a+b x^3} \left(-140 a^3 d^3+35 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right)}{140 a^3 c^4 x}-\frac{d^3 \sqrt[3]{b c-a d} \log \left(c+d x^3\right)}{6 c^{13/3}}+\frac{d^3 \sqrt[3]{b c-a d} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{13/3}}+\frac{d^3 \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{13/3}}-\frac{\sqrt[3]{a+b x^3} (b c-10 a d)}{70 a c^2 x^7}-\frac{\sqrt[3]{a+b x^3}}{10 c x^{10}}",1,"(((a + b*x^3)*(-9*b^3*c^3*x^9 + 3*a*b^2*c^2*x^6*(c - 5*d*x^3) + a^2*b*c*x^3*(-2*c^2 + 5*c*d*x^3 - 35*d^2*x^6) + a^3*(-14*c^3 + 20*c^2*d*x^3 - 35*c*d^2*x^6 + 140*d^3*x^9)))/(70*a^3*c^4*x^10) + (d^3*(-(b*c) + a*d)*x^2*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c^5*(1 + (d*x^3)/c)^(2/3)))/(2*(a + b*x^3)^(2/3))","C",0
672,1,281,64,0.6367414,"\int \frac{x^6 \sqrt[3]{a+b x^3}}{c+d x^3} \, dx","Integrate[(x^6*(a + b*x^3)^(1/3))/(c + d*x^3),x]","\frac{x \left(-\frac{x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(a^2 d^2+5 a b c d-10 b^2 c^2\right) F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c}+\frac{16 a^2 c^2 (a d-5 b c) F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\left(c+d x^3\right) \left(x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}+4 \left(a+b x^3\right) \left(a d-5 b c+2 b d x^3\right)\right)}{40 b d^2 \left(a+b x^3\right)^{2/3}}","\frac{x^7 \sqrt[3]{a+b x^3} F_1\left(\frac{7}{3};-\frac{1}{3},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{7 c \sqrt[3]{\frac{b x^3}{a}+1}}",1,"(x*(4*(a + b*x^3)*(-5*b*c + a*d + 2*b*d*x^3) - ((-10*b^2*c^2 + 5*a*b*c*d + a^2*d^2)*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/c + (16*a^2*c^2*(-5*b*c + a*d)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))))/(40*b*d^2*(a + b*x^3)^(2/3))","B",0
673,1,240,64,0.3717172,"\int \frac{x^3 \sqrt[3]{a+b x^3}}{c+d x^3} \, dx","Integrate[(x^3*(a + b*x^3)^(1/3))/(c + d*x^3),x]","\frac{x \left(4 \left(\frac{4 a^2 c^2 F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\left(c+d x^3\right) \left(x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}+a+b x^3\right)+\frac{x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} (a d-2 b c) F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c}\right)}{8 d \left(a+b x^3\right)^{2/3}}","\frac{x^4 \sqrt[3]{a+b x^3} F_1\left(\frac{4}{3};-\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 c \sqrt[3]{\frac{b x^3}{a}+1}}",1,"(x*(((-2*b*c + a*d)*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/c + 4*(a + b*x^3 + (4*a^2*c^2*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))))))/(8*d*(a + b*x^3)^(2/3))","B",0
674,1,160,59,0.044223,"\int \frac{\sqrt[3]{a+b x^3}}{c+d x^3} \, dx","Integrate[(a + b*x^3)^(1/3)/(c + d*x^3),x]","\frac{4 a c x \sqrt[3]{a+b x^3} F_1\left(\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\left(c+d x^3\right) \left(x^3 \left(b c F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-3 a d F_1\left(\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)+4 a c F_1\left(\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}","\frac{x \sqrt[3]{a+b x^3} F_1\left(\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c \sqrt[3]{\frac{b x^3}{a}+1}}",1,"(4*a*c*x*(a + b*x^3)^(1/3)*AppellF1[1/3, -1/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((c + d*x^3)*(4*a*c*AppellF1[1/3, -1/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(-3*a*d*AppellF1[4/3, -1/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + b*c*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))","B",0
675,1,327,64,0.2643324,"\int \frac{\sqrt[3]{a+b x^3}}{x^3 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^3*(c + d*x^3)),x]","\frac{\frac{c \left(16 a c \left(a \left(c+3 d x^3\right)+b d x^6\right) F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-4 x^3 \left(a+b x^3\right) \left(c+d x^3\right) \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)\right)}{\left(c+d x^3\right) \left(x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}-b d x^6 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{8 c^2 x^2 \left(a+b x^3\right)^{2/3}}","-\frac{\sqrt[3]{a+b x^3} F_1\left(-\frac{2}{3};-\frac{1}{3},1;\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 c x^2 \sqrt[3]{\frac{b x^3}{a}+1}}",1,"(-(b*d*x^6*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]) + (c*(16*a*c*(b*d*x^6 + a*(c + 3*d*x^3))*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] - 4*x^3*(a + b*x^3)*(c + d*x^3)*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))/((c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))))/(8*c^2*x^2*(a + b*x^3)^(2/3))","B",0
676,1,289,64,0.4634912,"\int \frac{\sqrt[3]{a+b x^3}}{x^6 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(1/3)/(x^6*(c + d*x^3)),x]","\frac{\frac{16 x \left(-10 a^2 d^2+5 a b c d+b^2 c^2\right) F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c \left(c+d x^3\right) \left(x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}+\frac{b d x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} (5 a d-b c) F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a c^3}-\frac{4 \left(a+b x^3\right) \left(2 a c-5 a d x^3+b c x^3\right)}{a c^2 x^5}}{40 \left(a+b x^3\right)^{2/3}}","-\frac{\sqrt[3]{a+b x^3} F_1\left(-\frac{5}{3};-\frac{1}{3},1;-\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{5 c x^5 \sqrt[3]{\frac{b x^3}{a}+1}}",1,"((-4*(a + b*x^3)*(2*a*c + b*c*x^3 - 5*a*d*x^3))/(a*c^2*x^5) + (b*d*(-(b*c) + 5*a*d)*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/(a*c^3) + (16*(b^2*c^2 + 5*a*b*c*d - 10*a^2*d^2)*x*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*(c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))))/(40*(a + b*x^3)^(2/3))","B",0
677,1,148,266,0.1243143,"\int \frac{x^{11} \left(a+b x^3\right)^{2/3}}{c+d x^3} \, dx","Integrate[(x^11*(a + b*x^3)^(2/3))/(c + d*x^3),x]","\frac{\left(a+b x^3\right)^{2/3} \left(18 a^3 d^3+3 a^2 b d^2 \left(11 c-4 d x^3\right)+220 b^3 c^3 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+2 a b^2 d \left(44 c^2-11 c d x^3+5 d^2 x^6\right)+b^3 \left(-220 c^3+88 c^2 d x^3-55 c d^2 x^6+40 d^3 x^9\right)\right)}{440 b^3 d^4}","\frac{\left(a+b x^3\right)^{5/3} \left(a^2 d^2+a b c d+b^2 c^2\right)}{5 b^3 d^3}-\frac{\left(a+b x^3\right)^{8/3} (2 a d+b c)}{8 b^3 d^2}+\frac{\left(a+b x^3\right)^{11/3}}{11 b^3 d}+\frac{c^3 (b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 d^{14/3}}-\frac{c^3 (b c-a d)^{2/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{14/3}}-\frac{c^3 (b c-a d)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{14/3}}-\frac{c^3 \left(a+b x^3\right)^{2/3}}{2 d^4}",1,"((a + b*x^3)^(2/3)*(18*a^3*d^3 + 3*a^2*b*d^2*(11*c - 4*d*x^3) + 2*a*b^2*d*(44*c^2 - 11*c*d*x^3 + 5*d^2*x^6) + b^3*(-220*c^3 + 88*c^2*d*x^3 - 55*c*d^2*x^6 + 40*d^3*x^9) + 220*b^3*c^3*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)]))/(440*b^3*d^4)","C",1
678,1,104,223,0.0981534,"\int \frac{x^8 \left(a+b x^3\right)^{2/3}}{c+d x^3} \, dx","Integrate[(x^8*(a + b*x^3)^(2/3))/(c + d*x^3),x]","\frac{\left(a+b x^3\right)^{2/3} \left(-3 a^2 d^2-20 b^2 c^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+2 a b d \left(d x^3-4 c\right)+b^2 \left(20 c^2-8 c d x^3+5 d^2 x^6\right)\right)}{40 b^2 d^3}","-\frac{\left(a+b x^3\right)^{5/3} (a d+b c)}{5 b^2 d^2}+\frac{\left(a+b x^3\right)^{8/3}}{8 b^2 d}-\frac{c^2 (b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 d^{11/3}}+\frac{c^2 (b c-a d)^{2/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{11/3}}+\frac{c^2 (b c-a d)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{11/3}}+\frac{c^2 \left(a+b x^3\right)^{2/3}}{2 d^3}",1,"((a + b*x^3)^(2/3)*(-3*a^2*d^2 + 2*a*b*d*(-4*c + d*x^3) + b^2*(20*c^2 - 8*c*d*x^3 + 5*d^2*x^6) - 20*b^2*c^2*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)]))/(40*b^2*d^3)","C",1
679,1,68,188,0.0462733,"\int \frac{x^5 \left(a+b x^3\right)^{2/3}}{c+d x^3} \, dx","Integrate[(x^5*(a + b*x^3)^(2/3))/(c + d*x^3),x]","\frac{\left(a+b x^3\right)^{2/3} \left(5 b c \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+2 a d-5 b c+2 b d x^3\right)}{10 b d^2}","\frac{c (b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 d^{8/3}}-\frac{c (b c-a d)^{2/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{8/3}}-\frac{c (b c-a d)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{8/3}}-\frac{c \left(a+b x^3\right)^{2/3}}{2 d^2}+\frac{\left(a+b x^3\right)^{5/3}}{5 b d}",1,"((a + b*x^3)^(2/3)*(-5*b*c + 2*a*d + 2*b*d*x^3 + 5*b*c*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)]))/(10*b*d^2)","C",1
680,1,47,162,0.0260029,"\int \frac{x^2 \left(a+b x^3\right)^{2/3}}{c+d x^3} \, dx","Integrate[(x^2*(a + b*x^3)^(2/3))/(c + d*x^3),x]","-\frac{\left(a+b x^3\right)^{2/3} \left(\, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)-1\right)}{2 d}","-\frac{(b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 d^{5/3}}+\frac{(b c-a d)^{2/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{5/3}}+\frac{(b c-a d)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{5/3}}+\frac{\left(a+b x^3\right)^{2/3}}{2 d}",1,"-1/2*((a + b*x^3)^(2/3)*(-1 + Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)]))/d","C",1
681,1,115,245,0.1211295,"\int \frac{\left(a+b x^3\right)^{2/3}}{x \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x*(c + d*x^3)),x]","\frac{a^{2/3} \left(3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-3 \log (x)\right)+3 \left(a+b x^3\right)^{2/3} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)}{6 c}","\frac{a^{2/3} \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2 c}+\frac{a^{2/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} c}-\frac{a^{2/3} \log (x)}{2 c}+\frac{(b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 c d^{2/3}}-\frac{(b c-a d)^{2/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c d^{2/3}}-\frac{(b c-a d)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c d^{2/3}}",1,"(3*(a + b*x^3)^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)] + a^(2/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 3*Log[x] + 3*Log[a^(1/3) - (a + b*x^3)^(1/3)]))/(6*c)","C",1
682,1,202,347,0.1581073,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^4 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^4*(c + d*x^3)),x]","\frac{-9 \sqrt[3]{a} d x^3 \left(a+b x^3\right)^{2/3} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+2 \sqrt{3} x^3 (2 b c-3 a d) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-6 \sqrt[3]{a} c \left(a+b x^3\right)^{2/3}+6 b c x^3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)-9 a d x^3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+9 a d x^3 \log (x)-6 b c x^3 \log (x)}{18 \sqrt[3]{a} c^2 x^3}","\frac{d \left(a+b x^3\right)^{2/3}}{2 c^2}+\frac{\left(a+b x^3\right)^{2/3} (2 b c-3 a d)}{6 a c^2}-\frac{\sqrt[3]{d} (b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 c^2}+\frac{(2 b c-3 a d) \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{6 \sqrt[3]{a} c^2}+\frac{\sqrt[3]{d} (b c-a d)^{2/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c^2}+\frac{(2 b c-3 a d) \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} \sqrt[3]{a} c^2}+\frac{\sqrt[3]{d} (b c-a d)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c^2}-\frac{\log (x) (2 b c-3 a d)}{6 \sqrt[3]{a} c^2}-\frac{\left(a+b x^3\right)^{5/3}}{3 a c x^3}",1,"(-6*a^(1/3)*c*(a + b*x^3)^(2/3) + 2*Sqrt[3]*(2*b*c - 3*a*d)*x^3*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 9*a^(1/3)*d*x^3*(a + b*x^3)^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)] - 6*b*c*x^3*Log[x] + 9*a*d*x^3*Log[x] + 6*b*c*x^3*Log[a^(1/3) - (a + b*x^3)^(1/3)] - 9*a*d*x^3*Log[a^(1/3) - (a + b*x^3)^(1/3)])/(18*a^(1/3)*c^2*x^3)","C",1
683,1,240,370,0.376827,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^7 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^7*(c + d*x^3)),x]","\frac{27 a^{4/3} d^2 x^6 \left(a+b x^3\right)^{2/3} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+3 x^6 \log (x) \left(-9 a^2 d^2+6 a b c d+b^2 c^2\right)-3 x^6 \left(-9 a^2 d^2+6 a b c d+b^2 c^2\right) \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)-2 \sqrt{3} x^6 \left(-9 a^2 d^2+6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)+3 \sqrt[3]{a} c \left(a+b x^3\right)^{2/3} \left(-3 a c+6 a d x^3-2 b c x^3\right)}{54 a^{4/3} c^3 x^6}","-\frac{\left(-9 a^2 d^2+6 a b c d+b^2 c^2\right) \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{18 a^{4/3} c^3}-\frac{\left(-9 a^2 d^2+6 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{4/3} c^3}+\frac{\log (x) \left(-9 a^2 d^2+6 a b c d+b^2 c^2\right)}{18 a^{4/3} c^3}+\frac{d^{4/3} (b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 c^3}-\frac{d^{4/3} (b c-a d)^{2/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c^3}-\frac{d^{4/3} (b c-a d)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c^3}+\frac{\left(a+b x^3\right)^{2/3} (6 a d+b c)}{18 a c^2 x^3}-\frac{\left(a+b x^3\right)^{5/3}}{6 a c x^6}",1,"(3*a^(1/3)*c*(a + b*x^3)^(2/3)*(-3*a*c - 2*b*c*x^3 + 6*a*d*x^3) - 2*Sqrt[3]*(b^2*c^2 + 6*a*b*c*d - 9*a^2*d^2)*x^6*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] + 27*a^(4/3)*d^2*x^6*(a + b*x^3)^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)] + 3*(b^2*c^2 + 6*a*b*c*d - 9*a^2*d^2)*x^6*Log[x] - 3*(b^2*c^2 + 6*a*b*c*d - 9*a^2*d^2)*x^6*Log[a^(1/3) - (a + b*x^3)^(1/3)])/(54*a^(4/3)*c^3*x^6)","C",1
684,1,525,334,0.7567452,"\int \frac{x^6 \left(a+b x^3\right)^{2/3}}{c+d x^3} \, dx","Integrate[(x^6*(a + b*x^3)^(2/3))/(c + d*x^3),x]","\frac{3 x^4 \sqrt[3]{\frac{b x^3}{a}+1} \sqrt[3]{b c-a d} \left(-a^2 d^2-6 a b c d+9 b^2 c^2\right) F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 c \left(-a^2 \sqrt[3]{c} d \sqrt[3]{a+b x^3} \log \left(\frac{\sqrt[3]{c} x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}+\frac{x^2 (b c-a d)^{2/3}}{\left(a x^3+b\right)^{2/3}}+c^{2/3}\right)+6 a^2 d x \sqrt[3]{b c-a d}+9 b^2 d x^7 \sqrt[3]{b c-a d}-18 b^2 c x^4 \sqrt[3]{b c-a d}+3 a b c^{4/3} \sqrt[3]{a+b x^3} \log \left(\frac{\sqrt[3]{c} x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}+\frac{x^2 (b c-a d)^{2/3}}{\left(a x^3+b\right)^{2/3}}+c^{2/3}\right)+15 a b d x^4 \sqrt[3]{b c-a d}+2 a \sqrt[3]{c} \sqrt[3]{a+b x^3} (a d-3 b c) \log \left(\sqrt[3]{c}-\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}\right)-2 \sqrt{3} a \sqrt[3]{c} \sqrt[3]{a+b x^3} (a d-3 b c) \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-18 a b c x \sqrt[3]{b c-a d}\right)}{108 b c d^2 \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}}","-\frac{\left(-a^2 d^2-6 a b c d+9 b^2 c^2\right) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{18 b^{4/3} d^3}+\frac{\left(-a^2 d^2-6 a b c d+9 b^2 c^2\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{9 \sqrt{3} b^{4/3} d^3}-\frac{c^{4/3} (b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 d^3}+\frac{c^{4/3} (b c-a d)^{2/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 d^3}-\frac{c^{4/3} (b c-a d)^{2/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^3}-\frac{x \left(a+b x^3\right)^{2/3} (3 b c-a d)}{9 b d^2}+\frac{x^4 \left(a+b x^3\right)^{2/3}}{6 d}",1,"(3*(b*c - a*d)^(1/3)*(9*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*c*(-18*a*b*c*(b*c - a*d)^(1/3)*x + 6*a^2*d*(b*c - a*d)^(1/3)*x - 18*b^2*c*(b*c - a*d)^(1/3)*x^4 + 15*a*b*d*(b*c - a*d)^(1/3)*x^4 + 9*b^2*d*(b*c - a*d)^(1/3)*x^7 - 2*Sqrt[3]*a*c^(1/3)*(-3*b*c + a*d)*(a + b*x^3)^(1/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] + 2*a*c^(1/3)*(-3*b*c + a*d)*(a + b*x^3)^(1/3)*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)] + 3*a*b*c^(4/3)*(a + b*x^3)^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)] - a^2*c^(1/3)*d*(a + b*x^3)^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(108*b*c*d^2*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3))","C",0
685,1,286,272,0.5939461,"\int \frac{x^3 \left(a+b x^3\right)^{2/3}}{c+d x^3} \, dx","Integrate[(x^3*(a + b*x^3)^(2/3))/(c + d*x^3),x]","\frac{\frac{3 x^4 \sqrt[3]{\frac{b x^3}{a}+1} (2 a d-3 b c) F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c \sqrt[3]{a+b x^3}}+\frac{2 \left(-a \sqrt[3]{c} \log \left(\frac{\sqrt[3]{c} x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}+\frac{x^2 (b c-a d)^{2/3}}{\left(a x^3+b\right)^{2/3}}+c^{2/3}\right)+6 x \left(a+b x^3\right)^{2/3} \sqrt[3]{b c-a d}+2 a \sqrt[3]{c} \log \left(\sqrt[3]{c}-\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}\right)-2 \sqrt{3} a \sqrt[3]{c} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)\right)}{\sqrt[3]{b c-a d}}}{36 d}","\frac{\sqrt[3]{c} (b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 d^2}-\frac{\sqrt[3]{c} (b c-a d)^{2/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 d^2}+\frac{(3 b c-2 a d) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{6 \sqrt[3]{b} d^2}-\frac{(3 b c-2 a d) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} \sqrt[3]{b} d^2}+\frac{\sqrt[3]{c} (b c-a d)^{2/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^2}+\frac{x \left(a+b x^3\right)^{2/3}}{3 d}",1,"((3*(-3*b*c + 2*a*d)*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*(a + b*x^3)^(1/3)) + (2*(6*(b*c - a*d)^(1/3)*x*(a + b*x^3)^(2/3) - 2*Sqrt[3]*a*c^(1/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] + 2*a*c^(1/3)*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)] - a*c^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(b*c - a*d)^(1/3))/(36*d)","C",0
686,1,161,233,0.0588865,"\int \frac{\left(a+b x^3\right)^{2/3}}{c+d x^3} \, dx","Integrate[(a + b*x^3)^(2/3)/(c + d*x^3),x]","\frac{4 a c x \left(a+b x^3\right)^{2/3} F_1\left(\frac{1}{3};-\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\left(c+d x^3\right) \left(x^3 \left(2 b c F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-3 a d F_1\left(\frac{4}{3};-\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)+4 a c F_1\left(\frac{1}{3};-\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}","-\frac{b^{2/3} \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 d}+\frac{b^{2/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d}-\frac{(b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 c^{2/3} d}+\frac{(b c-a d)^{2/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{2/3} d}-\frac{(b c-a d)^{2/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{2/3} d}",1,"(4*a*c*x*(a + b*x^3)^(2/3)*AppellF1[1/3, -2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((c + d*x^3)*(4*a*c*AppellF1[1/3, -2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(-3*a*d*AppellF1[4/3, -2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))","C",0
687,1,83,169,0.0452247,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^3 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^3*(c + d*x^3)),x]","-\frac{\left(a+b x^3\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3} \, _2F_1\left(-\frac{2}{3},-\frac{2}{3};\frac{1}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{2 c x^2 \left(\frac{b x^3}{a}+1\right)^{2/3}}","\frac{(b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 c^{5/3}}-\frac{(b c-a d)^{2/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{5/3}}+\frac{(b c-a d)^{2/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{5/3}}-\frac{\left(a+b x^3\right)^{2/3}}{2 c x^2}",1,"-1/2*((a + b*x^3)^(2/3)*(1 + (d*x^3)/c)^(2/3)*Hypergeometric2F1[-2/3, -2/3, 1/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c*x^2*(1 + (b*x^3)/a)^(2/3))","C",0
688,1,148,206,0.6424016,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^6 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^6*(c + d*x^3)),x]","-\frac{2 x^3 \left(3 d x^3-2 c\right) (b c-a d) \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+6 x^3 \left(c+d x^3\right) (b c-a d) \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+c \left(a+b x^3\right) \left(2 c-3 d x^3\right)}{10 c^3 x^5 \sqrt[3]{a+b x^3}}","-\frac{d (b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 c^{8/3}}+\frac{d (b c-a d)^{2/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{8/3}}-\frac{d (b c-a d)^{2/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{8/3}}-\frac{\left(a+b x^3\right)^{2/3} (2 b c-5 a d)}{10 a c^2 x^2}-\frac{\left(a+b x^3\right)^{2/3}}{5 c x^5}",1,"-1/10*(c*(a + b*x^3)*(2*c - 3*d*x^3) + 2*(b*c - a*d)*x^3*(-2*c + 3*d*x^3)*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 6*(b*c - a*d)*x^3*(c + d*x^3)*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^3*x^5*(a + b*x^3)^(1/3))","C",0
689,1,451,257,2.1957316,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^9 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^9*(c + d*x^3)),x]","-\frac{-9 x^3 \left(c+d x^3\right)^2 (b c-a d) \, _3F_2\left(\frac{1}{3},2,2;1,\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+21 b c^3 x^3 \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-2 x^3 \left(5 c^2-6 c d x^3+9 d^2 x^6\right) (b c-a d) \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-21 a c^2 d x^3 \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-6 b c^2 d x^6 \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+27 a d^3 x^9 \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-27 b c d^2 x^9 \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+6 a c d^2 x^6 \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+5 a c^3-6 a c^2 d x^3+9 a c d^2 x^6+5 b c^3 x^3-6 b c^2 d x^6+9 b c d^2 x^9}{40 c^4 x^8 \sqrt[3]{a+b x^3}}","\frac{\left(a+b x^3\right)^{2/3} \left(-20 a^2 d^2+8 a b c d+3 b^2 c^2\right)}{40 a^2 c^3 x^2}+\frac{d^2 (b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 c^{11/3}}-\frac{d^2 (b c-a d)^{2/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{11/3}}+\frac{d^2 (b c-a d)^{2/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{11/3}}-\frac{\left(a+b x^3\right)^{2/3} (b c-4 a d)}{20 a c^2 x^5}-\frac{\left(a+b x^3\right)^{2/3}}{8 c x^8}",1,"-1/40*(5*a*c^3 + 5*b*c^3*x^3 - 6*a*c^2*d*x^3 - 6*b*c^2*d*x^6 + 9*a*c*d^2*x^6 + 9*b*c*d^2*x^9 - 2*(b*c - a*d)*x^3*(5*c^2 - 6*c*d*x^3 + 9*d^2*x^6)*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 21*b*c^3*x^3*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 21*a*c^2*d*x^3*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 6*b*c^2*d*x^6*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 6*a*c*d^2*x^6*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 27*b*c*d^2*x^9*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*a*d^3*x^9*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 9*(b*c - a*d)*x^3*(c + d*x^3)^2*HypergeometricPFQ[{1/3, 2, 2}, {1, 4/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^4*x^8*(a + b*x^3)^(1/3))","C",0
690,1,819,320,4.9947721,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^{12} \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^12*(c + d*x^3)),x]","-\frac{-81 b c d^3 x^{12}-162 a d^4 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}+162 b c d^3 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}-297 a d^4 \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}+297 b c d^3 \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}-81 a c d^3 x^9+54 b c^2 d^2 x^9+108 a c d^3 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9-108 b c^2 d^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9+54 a c^2 d^2 x^6-45 b c^3 d x^6-90 a c^2 d^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6+90 b c^3 d \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6+99 a c^2 d^2 \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6-99 b c^3 d \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6+40 b c^4 x^3-45 a c^3 d x^3-80 b c^4 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3+80 a c^3 d \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3+198 b c^4 \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3-198 a c^3 d \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3+27 (b c-a d) \left(d x^3+c\right)^2 \left(6 d x^3-5 c\right) \, _3F_2\left(\frac{1}{3},2,2;1,\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3+27 (b c-a d) \left(d x^3+c\right)^3 \, _4F_3\left(\frac{1}{3},2,2,2;1,1,\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3+40 a c^4}{440 c^5 x^{11} \sqrt[3]{b x^3+a}}","\frac{\left(a+b x^3\right)^{2/3} \left(-44 a^2 d^2+11 a b c d+6 b^2 c^2\right)}{220 a^2 c^3 x^5}-\frac{\left(a+b x^3\right)^{2/3} \left(-220 a^3 d^3+88 a^2 b c d^2+33 a b^2 c^2 d+18 b^3 c^3\right)}{440 a^3 c^4 x^2}-\frac{d^3 (b c-a d)^{2/3} \log \left(c+d x^3\right)}{6 c^{14/3}}+\frac{d^3 (b c-a d)^{2/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{14/3}}-\frac{d^3 (b c-a d)^{2/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{14/3}}-\frac{\left(a+b x^3\right)^{2/3} (2 b c-11 a d)}{88 a c^2 x^8}-\frac{\left(a+b x^3\right)^{2/3}}{11 c x^{11}}",1,"-1/440*(40*a*c^4 + 40*b*c^4*x^3 - 45*a*c^3*d*x^3 - 45*b*c^3*d*x^6 + 54*a*c^2*d^2*x^6 + 54*b*c^2*d^2*x^9 - 81*a*c*d^3*x^9 - 81*b*c*d^3*x^12 - 80*b*c^4*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 80*a*c^3*d*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 90*b*c^3*d*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 90*a*c^2*d^2*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 108*b*c^2*d^2*x^9*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 108*a*c*d^3*x^9*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 162*b*c*d^3*x^12*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 162*a*d^4*x^12*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 198*b*c^4*x^3*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 198*a*c^3*d*x^3*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 99*b*c^3*d*x^6*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 99*a*c^2*d^2*x^6*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 297*b*c*d^3*x^12*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 297*a*d^4*x^12*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*(b*c - a*d)*x^3*(c + d*x^3)^2*(-5*c + 6*d*x^3)*HypergeometricPFQ[{1/3, 2, 2}, {1, 4/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*(b*c - a*d)*x^3*(c + d*x^3)^3*HypergeometricPFQ[{1/3, 2, 2, 2}, {1, 1, 4/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^5*x^11*(a + b*x^3)^(1/3))","C",0
691,1,181,64,0.2376827,"\int \frac{x^7 \left(a+b x^3\right)^{2/3}}{c+d x^3} \, dx","Integrate[(x^7*(a + b*x^3)^(2/3))/(c + d*x^3),x]","\frac{x^2 \left(-2 x^3 \sqrt[3]{\frac{b x^3}{a}+1} \left(2 a^2 d^2+7 a b c d-14 b^2 c^2\right) F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 a c \sqrt[3]{\frac{b x^3}{a}+1} (7 b c-2 a d) F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 c \left(a+b x^3\right) \left(2 a d-7 b c+4 b d x^3\right)\right)}{140 b c d^2 \sqrt[3]{a+b x^3}}","\frac{x^8 \left(a+b x^3\right)^{2/3} F_1\left(\frac{8}{3};-\frac{2}{3},1;\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{8 c \left(\frac{b x^3}{a}+1\right)^{2/3}}",1,"(x^2*(5*c*(a + b*x^3)*(-7*b*c + 2*a*d + 4*b*d*x^3) + 5*a*c*(7*b*c - 2*a*d)*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] - 2*(-14*b^2*c^2 + 7*a*b*c*d + 2*a^2*d^2)*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)]))/(140*b*c*d^2*(a + b*x^3)^(1/3))","B",0
692,1,141,64,0.115383,"\int \frac{x^4 \left(a+b x^3\right)^{2/3}}{c+d x^3} \, dx","Integrate[(x^4*(a + b*x^3)^(2/3))/(c + d*x^3),x]","\frac{2 x^5 \sqrt[3]{\frac{b x^3}{a}+1} (a d-2 b c) F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-5 a c x^2 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 c x^2 \left(a+b x^3\right)}{20 c d \sqrt[3]{a+b x^3}}","\frac{x^5 \left(a+b x^3\right)^{2/3} F_1\left(\frac{5}{3};-\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{5 c \left(\frac{b x^3}{a}+1\right)^{2/3}}",1,"(5*c*x^2*(a + b*x^3) - 5*a*c*x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*(-2*b*c + a*d)*x^5*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(20*c*d*(a + b*x^3)^(1/3))","B",0
693,1,65,64,0.0433935,"\int \frac{x \left(a+b x^3\right)^{2/3}}{c+d x^3} \, dx","Integrate[(x*(a + b*x^3)^(2/3))/(c + d*x^3),x]","\frac{x^2 \left(a+b x^3\right)^{2/3} F_1\left(\frac{2}{3};-\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 c \left(\frac{a+b x^3}{a}\right)^{2/3}}","\frac{x^2 \left(a+b x^3\right)^{2/3} F_1\left(\frac{2}{3};-\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 c \left(\frac{b x^3}{a}+1\right)^{2/3}}",1,"(x^2*(a + b*x^3)^(2/3)*AppellF1[2/3, -2/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)])/(2*c*((a + b*x^3)/a)^(2/3))","A",1
694,1,138,62,0.1419679,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^2 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^2*(c + d*x^3)),x]","\frac{-5 x^3 \sqrt[3]{\frac{b x^3}{a}+1} (a d-2 b c) F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b d x^6 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-10 c \left(a+b x^3\right)}{10 c^2 x \sqrt[3]{a+b x^3}}","-\frac{\left(a+b x^3\right)^{2/3} F_1\left(-\frac{1}{3};-\frac{2}{3},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c x \left(\frac{b x^3}{a}+1\right)^{2/3}}",1,"(-10*c*(a + b*x^3) - 5*(-2*b*c + a*d)*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*d*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(10*c^2*x*(a + b*x^3)^(1/3))","B",0
695,1,181,64,0.1779917,"\int \frac{\left(a+b x^3\right)^{2/3}}{x^5 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(2/3)/(x^5*(c + d*x^3)),x]","\frac{5 x^6 \sqrt[3]{\frac{b x^3}{a}+1} \left(2 a^2 d^2-4 a b c d+b^2 c^2\right) F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b d x^9 \sqrt[3]{\frac{b x^3}{a}+1} (b c-2 a d) F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-5 c \left(a+b x^3\right) \left(a \left(c-4 d x^3\right)+2 b c x^3\right)}{20 a c^3 x^4 \sqrt[3]{a+b x^3}}","-\frac{\left(a+b x^3\right)^{2/3} F_1\left(-\frac{4}{3};-\frac{2}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 c x^4 \left(\frac{b x^3}{a}+1\right)^{2/3}}",1,"(-5*c*(a + b*x^3)*(2*b*c*x^3 + a*(c - 4*d*x^3)) + 5*(b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2)*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*d*(b*c - 2*a*d)*x^9*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(20*a*c^3*x^4*(a + b*x^3)^(1/3))","B",0
696,1,258,251,0.5379871,"\int \frac{x^8 \left(a+b x^3\right)^{4/3}}{c+d x^3} \, dx","Integrate[(x^8*(a + b*x^3)^(4/3))/(c + d*x^3),x]","\frac{-\frac{60 d \left(a+b x^3\right)^{7/3} (a d+b c)}{b^2}+\frac{42 d^2 \left(a+b x^3\right)^{10/3}}{b^2}-\frac{70 c^2 (b c-a d) \left(\sqrt[3]{b c-a d} \left(\log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)\right)+6 \sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{d^{4/3}}+105 c^2 \left(a+b x^3\right)^{4/3}}{420 d^3}","-\frac{\left(a+b x^3\right)^{7/3} (a d+b c)}{7 b^2 d^2}+\frac{\left(a+b x^3\right)^{10/3}}{10 b^2 d}-\frac{c^2 (b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 d^{13/3}}+\frac{c^2 (b c-a d)^{4/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{13/3}}-\frac{c^2 (b c-a d)^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{13/3}}-\frac{c^2 \sqrt[3]{a+b x^3} (b c-a d)}{d^4}+\frac{c^2 \left(a+b x^3\right)^{4/3}}{4 d^3}",1,"(105*c^2*(a + b*x^3)^(4/3) - (60*d*(b*c + a*d)*(a + b*x^3)^(7/3))/b^2 + (42*d^2*(a + b*x^3)^(10/3))/b^2 - (70*c^2*(b*c - a*d)*(6*d^(1/3)*(a + b*x^3)^(1/3) + (b*c - a*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)])))/d^(4/3))/(420*d^3)","A",1
697,1,255,211,0.336637,"\int \frac{x^5 \left(a+b x^3\right)^{4/3}}{c+d x^3} \, dx","Integrate[(x^5*(a + b*x^3)^(4/3))/(c + d*x^3),x]","\frac{c (b c-a d) \left(\sqrt[3]{b c-a d} \log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)-2 \sqrt{3} \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)+6 \sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{6 d^{10/3}}-\frac{c \left(a+b x^3\right)^{4/3}}{4 d^2}+\frac{\left(a+b x^3\right)^{7/3}}{7 b d}","\frac{c (b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 d^{10/3}}-\frac{c (b c-a d)^{4/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{10/3}}+\frac{c (b c-a d)^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{10/3}}+\frac{c \sqrt[3]{a+b x^3} (b c-a d)}{d^3}-\frac{c \left(a+b x^3\right)^{4/3}}{4 d^2}+\frac{\left(a+b x^3\right)^{7/3}}{7 b d}",1,"-1/4*(c*(a + b*x^3)^(4/3))/d^2 + (a + b*x^3)^(7/3)/(7*b*d) + (c*(b*c - a*d)*(6*d^(1/3)*(a + b*x^3)^(1/3) - 2*Sqrt[3]*(b*c - a*d)^(1/3)*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*(b*c - a*d)^(1/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + (b*c - a*d)^(1/3)*Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/(6*d^(10/3))","A",1
698,1,232,187,0.3248432,"\int \frac{x^2 \left(a+b x^3\right)^{4/3}}{c+d x^3} \, dx","Integrate[(x^2*(a + b*x^3)^(4/3))/(c + d*x^3),x]","\frac{(a d-b c) \left(\sqrt[3]{b c-a d} \log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)-2 \sqrt{3} \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)+6 \sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{6 d^{7/3}}+\frac{\left(a+b x^3\right)^{4/3}}{4 d}","-\frac{(b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 d^{7/3}}+\frac{(b c-a d)^{4/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{7/3}}-\frac{(b c-a d)^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{7/3}}-\frac{\sqrt[3]{a+b x^3} (b c-a d)}{d^2}+\frac{\left(a+b x^3\right)^{4/3}}{4 d}",1,"(a + b*x^3)^(4/3)/(4*d) + ((-(b*c) + a*d)*(6*d^(1/3)*(a + b*x^3)^(1/3) - 2*Sqrt[3]*(b*c - a*d)^(1/3)*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*(b*c - a*d)^(1/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + (b*c - a*d)^(1/3)*Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/(6*d^(7/3))","A",1
699,1,331,261,0.6687706,"\int \frac{\left(a+b x^3\right)^{4/3}}{x \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(4/3)/(x*(c + d*x^3)),x]","\frac{-\left(a^{4/3} \left(\log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)\right)+\frac{(b c-a d) \left(\sqrt[3]{b c-a d} \log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \sqrt[3]{b c-a d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)-2 \sqrt{3} \sqrt[3]{b c-a d} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)+6 \sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{d^{4/3}}+6 a \sqrt[3]{a+b x^3}}{6 c}","\frac{a^{4/3} \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2 c}-\frac{a^{4/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} c}-\frac{a^{4/3} \log (x)}{2 c}+\frac{(b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 c d^{4/3}}-\frac{(b c-a d)^{4/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c d^{4/3}}+\frac{(b c-a d)^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c d^{4/3}}+\frac{b \sqrt[3]{a+b x^3}}{d}",1,"(6*a*(a + b*x^3)^(1/3) - a^(4/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) - (a + b*x^3)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]) + ((b*c - a*d)*(6*d^(1/3)*(a + b*x^3)^(1/3) - 2*Sqrt[3]*(b*c - a*d)^(1/3)*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*(b*c - a*d)^(1/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + (b*c - a*d)^(1/3)*Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/d^(4/3))/(6*c)","A",1
700,1,389,399,1.5484714,"\int \frac{\left(a+b x^3\right)^{4/3}}{x^4 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(4/3)/(x^4*(c + d*x^3)),x]","\frac{\frac{(4 b c-3 a d) \left(-\frac{1}{2} a^{4/3} \left(\log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)+3 a \sqrt[3]{a+b x^3}+\frac{3}{4} \left(a+b x^3\right)^{4/3}\right)}{3 c}+\frac{a \left(3 d^{4/3} \left(a+b x^3\right)^{4/3}-2 (b c-a d) \left(\sqrt[3]{b c-a d} \left(\log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)\right)+6 \sqrt[3]{d} \sqrt[3]{a+b x^3}\right)\right)}{4 c \sqrt[3]{d}}-\frac{\left(a+b x^3\right)^{7/3}}{x^3}}{3 a c}","\frac{d \left(a+b x^3\right)^{4/3}}{4 c^2}+\frac{\left(a+b x^3\right)^{4/3} (4 b c-3 a d)}{12 a c^2}+\frac{\sqrt[3]{a+b x^3} (4 b c-3 a d)}{3 c^2}-\frac{\sqrt[3]{a+b x^3} (b c-a d)}{c^2}-\frac{(b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 c^2 \sqrt[3]{d}}+\frac{\sqrt[3]{a} (4 b c-3 a d) \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{6 c^2}+\frac{(b c-a d)^{4/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c^2 \sqrt[3]{d}}-\frac{\sqrt[3]{a} (4 b c-3 a d) \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} c^2}-\frac{(b c-a d)^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c^2 \sqrt[3]{d}}-\frac{\sqrt[3]{a} \log (x) (4 b c-3 a d)}{6 c^2}-\frac{\left(a+b x^3\right)^{7/3}}{3 a c x^3}",1,"(-((a + b*x^3)^(7/3)/x^3) + ((4*b*c - 3*a*d)*(3*a*(a + b*x^3)^(1/3) + (3*(a + b*x^3)^(4/3))/4 - (a^(4/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) - (a + b*x^3)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]))/2))/(3*c) + (a*(3*d^(4/3)*(a + b*x^3)^(4/3) - 2*(b*c - a*d)*(6*d^(1/3)*(a + b*x^3)^(1/3) + (b*c - a*d)^(1/3)*(-2*Sqrt[3]*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))))/(4*c*d^(1/3)))/(3*a*c)","A",1
701,1,429,440,1.2543377,"\int \frac{\left(a+b x^3\right)^{4/3}}{x^7 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(4/3)/(x^7*(c + d*x^3)),x]","\frac{x^6 \left(4 \left(9 a^2 d^2-12 a b c d+2 b^2 c^2\right) \left(-\frac{1}{2} a^{4/3} \left(\log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)+3 a \sqrt[3]{a+b x^3}+\frac{3}{4} \left(a+b x^3\right)^{4/3}\right)-9 a^2 d^{2/3} \left(3 d^{4/3} \left(a+b x^3\right)^{4/3}-2 (b c-a d) \left(\sqrt[3]{b c-a d} \left(\log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)\right)+6 \sqrt[3]{d} \sqrt[3]{a+b x^3}\right)\right)\right)-18 a c^2 \left(a+b x^3\right)^{7/3}-6 c x^3 \left(a+b x^3\right)^{7/3} (b c-6 a d)}{108 a^2 c^3 x^6}","\frac{\sqrt[3]{a+b x^3} \left(9 a^2 d^2-12 a b c d+2 b^2 c^2\right)}{9 a c^3}+\frac{\left(9 a^2 d^2-12 a b c d+2 b^2 c^2\right) \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{18 a^{2/3} c^3}-\frac{\left(9 a^2 d^2-12 a b c d+2 b^2 c^2\right) \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{2/3} c^3}-\frac{\log (x) \left(9 a^2 d^2-12 a b c d+2 b^2 c^2\right)}{18 a^{2/3} c^3}+\frac{d^{2/3} (b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 c^3}-\frac{d^{2/3} (b c-a d)^{4/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c^3}+\frac{d^{2/3} (b c-a d)^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c^3}+\frac{d \sqrt[3]{a+b x^3} (b c-a d)}{c^3}-\frac{\left(a+b x^3\right)^{4/3} (b c-6 a d)}{18 a c^2 x^3}-\frac{\left(a+b x^3\right)^{7/3}}{6 a c x^6}",1,"(-18*a*c^2*(a + b*x^3)^(7/3) - 6*c*(b*c - 6*a*d)*x^3*(a + b*x^3)^(7/3) + x^6*(4*(2*b^2*c^2 - 12*a*b*c*d + 9*a^2*d^2)*(3*a*(a + b*x^3)^(1/3) + (3*(a + b*x^3)^(4/3))/4 - (a^(4/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) - (a + b*x^3)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]))/2) - 9*a^2*d^(2/3)*(3*d^(4/3)*(a + b*x^3)^(4/3) - 2*(b*c - a*d)*(6*d^(1/3)*(a + b*x^3)^(1/3) + (b*c - a*d)^(1/3)*(-2*Sqrt[3]*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)])))))/(108*a^2*c^3*x^6)","A",1
702,1,225,334,0.3117319,"\int \frac{x^4 \left(a+b x^3\right)^{4/3}}{c+d x^3} \, dx","Integrate[(x^4*(a + b*x^3)^(4/3))/(c + d*x^3),x]","\frac{2 x^5 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3} \left(2 a^2 d^2-12 a b c d+9 b^2 c^2\right) F_1\left(\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 c x^2 \left(a \left(\frac{b x^3}{a}+1\right)^{2/3} (6 b c-7 a d) \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)+\left(a+b x^3\right) \left(\frac{d x^3}{c}+1\right)^{2/3} \left(7 a d-6 b c+3 b d x^3\right)\right)}{90 c d^2 \left(a+b x^3\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3}}","-\frac{\left(2 a^2 d^2-12 a b c d+9 b^2 c^2\right) \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{18 b^{2/3} d^3}-\frac{\left(2 a^2 d^2-12 a b c d+9 b^2 c^2\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{9 \sqrt{3} b^{2/3} d^3}-\frac{c^{2/3} (b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 d^3}+\frac{c^{2/3} (b c-a d)^{4/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 d^3}+\frac{c^{2/3} (b c-a d)^{4/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^3}-\frac{x^2 \sqrt[3]{a+b x^3} (6 b c-7 a d)}{18 d^2}+\frac{b x^5 \sqrt[3]{a+b x^3}}{6 d}",1,"(2*(9*b^2*c^2 - 12*a*b*c*d + 2*a^2*d^2)*x^5*(1 + (b*x^3)/a)^(2/3)*(1 + (d*x^3)/c)^(2/3)*AppellF1[5/3, 2/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)] + 5*c*x^2*((a + b*x^3)*(-6*b*c + 7*a*d + 3*b*d*x^3)*(1 + (d*x^3)/c)^(2/3) + a*(6*b*c - 7*a*d)*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))]))/(90*c*d^2*(a + b*x^3)^(2/3)*(1 + (d*x^3)/c)^(2/3))","C",0
703,1,198,277,0.3158285,"\int \frac{x \left(a+b x^3\right)^{4/3}}{c+d x^3} \, dx","Integrate[(x*(a + b*x^3)^(4/3))/(c + d*x^3),x]","\frac{2 b x^5 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3} (4 a d-3 b c) F_1\left(\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 x^2 \left(a \left(\frac{b x^3}{a}+1\right)^{2/3} (3 a d-2 b c) \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)+2 b c \left(a+b x^3\right) \left(\frac{d x^3}{c}+1\right)^{2/3}\right)}{30 c d \left(a+b x^3\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3}}","\frac{(b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 \sqrt[3]{c} d^2}+\frac{\sqrt[3]{b} (3 b c-4 a d) \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{6 d^2}-\frac{(b c-a d)^{4/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 \sqrt[3]{c} d^2}+\frac{\sqrt[3]{b} (3 b c-4 a d) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} d^2}-\frac{(b c-a d)^{4/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{c} d^2}+\frac{b x^2 \sqrt[3]{a+b x^3}}{3 d}",1,"(2*b*(-3*b*c + 4*a*d)*x^5*(1 + (b*x^3)/a)^(2/3)*(1 + (d*x^3)/c)^(2/3)*AppellF1[5/3, 2/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)] + 5*x^2*(2*b*c*(a + b*x^3)*(1 + (d*x^3)/c)^(2/3) + a*(-2*b*c + 3*a*d)*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))]))/(30*c*d*(a + b*x^3)^(2/3)*(1 + (d*x^3)/c)^(2/3))","C",0
704,1,161,254,0.5465942,"\int \frac{\left(a+b x^3\right)^{4/3}}{x^2 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(4/3)/(x^2*(c + d*x^3)),x]","\frac{2 b^2 c x^6 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-\frac{5 a x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} (a d-2 b c) \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{\left(\frac{d x^3}{c}+1\right)^{2/3}}-10 a c \left(a+b x^3\right)}{10 c^2 x \left(a+b x^3\right)^{2/3}}","-\frac{b^{4/3} \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2 d}-\frac{b^{4/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d}-\frac{(b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 c^{4/3} d}+\frac{(b c-a d)^{4/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{4/3} d}+\frac{(b c-a d)^{4/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{4/3} d}-\frac{a \sqrt[3]{a+b x^3}}{c x}",1,"(-10*a*c*(a + b*x^3) + 2*b^2*c*x^6*(1 + (b*x^3)/a)^(2/3)*AppellF1[5/3, 2/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)] - (5*a*(-2*b*c + a*d)*x^3*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(1 + (d*x^3)/c)^(2/3))/(10*c^2*x*(a + b*x^3)^(2/3))","C",0
705,1,84,201,0.0399874,"\int \frac{\left(a+b x^3\right)^{4/3}}{x^5 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(4/3)/(x^5*(c + d*x^3)),x]","-\frac{a \sqrt[3]{a+b x^3} \left(\frac{d x^3}{c}+1\right)^{4/3} \, _2F_1\left(-\frac{4}{3},-\frac{4}{3};-\frac{1}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{4 c x^4 \sqrt[3]{\frac{b x^3}{a}+1}}","\frac{(b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 c^{7/3}}-\frac{(b c-a d)^{4/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{7/3}}-\frac{(b c-a d)^{4/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{7/3}}-\frac{\sqrt[3]{a+b x^3} (5 b c-4 a d)}{4 c^2 x}-\frac{a \sqrt[3]{a+b x^3}}{4 c x^4}",1,"-1/4*(a*(a + b*x^3)^(1/3)*(1 + (d*x^3)/c)^(4/3)*Hypergeometric2F1[-4/3, -4/3, -1/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c*x^4*(1 + (b*x^3)/a)^(1/3))","C",0
706,1,179,250,0.602795,"\int \frac{\left(a+b x^3\right)^{4/3}}{x^8 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(4/3)/(x^8*(c + d*x^3)),x]","-\frac{a \left(\frac{b x^3}{a}+1\right) \left(12 c x^3 \left(a+b x^3\right) \left(c+d x^3\right) (a d-b c) \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+\left(4 c-3 d x^3\right) \left(c \left(a+b x^3\right) \left(a \left(c-4 d x^3\right)+5 b c x^3\right)-2 x^6 (b c-a d)^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)\right)\right)}{28 c^4 x^7 \left(a+b x^3\right)^{5/3}}","-\frac{\sqrt[3]{a+b x^3} \left(28 a^2 d^2-35 a b c d+4 b^2 c^2\right)}{28 a c^3 x}-\frac{d (b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 c^{10/3}}+\frac{d (b c-a d)^{4/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{10/3}}+\frac{d (b c-a d)^{4/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{10/3}}-\frac{\sqrt[3]{a+b x^3} (8 b c-7 a d)}{28 c^2 x^4}-\frac{a \sqrt[3]{a+b x^3}}{7 c x^7}",1,"-1/28*(a*(1 + (b*x^3)/a)*(12*c*(-(b*c) + a*d)*x^3*(a + b*x^3)*(c + d*x^3)*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + (4*c - 3*d*x^3)*(c*(a + b*x^3)*(5*b*c*x^3 + a*(c - 4*d*x^3)) - 2*(b*c - a*d)^2*x^6*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])))/(c^4*x^7*(a + b*x^3)^(5/3))","C",0
707,1,270,318,2.7008906,"\int \frac{\left(a+b x^3\right)^{4/3}}{x^{11} \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(4/3)/(x^11*(c + d*x^3)),x]","-\frac{a \left(\frac{b x^3}{a}+1\right) \left(18 c x^3 \left(a+b x^3\right) \left(c+d x^3\right)^2 (b c-a d) \, _3F_2\left(-\frac{1}{3},2,2;\frac{2}{3},1;\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+6 c x^3 \left(a+b x^3\right) \left(-11 c^2-2 c d x^3+9 d^2 x^6\right) (b c-a d) \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+\left(14 c^2-12 c d x^3+9 d^2 x^6\right) \left(c \left(a+b x^3\right) \left(a \left(c-4 d x^3\right)+5 b c x^3\right)-2 x^6 (b c-a d)^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)\right)\right)}{140 c^5 x^{10} \left(a+b x^3\right)^{5/3}}","-\frac{\sqrt[3]{a+b x^3} \left(35 a^2 d^2-40 a b c d+2 b^2 c^2\right)}{140 a c^3 x^4}+\frac{\sqrt[3]{a+b x^3} \left(140 a^3 d^3-175 a^2 b c d^2+20 a b^2 c^2 d+6 b^3 c^3\right)}{140 a^2 c^4 x}+\frac{d^2 (b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 c^{13/3}}-\frac{d^2 (b c-a d)^{4/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{13/3}}-\frac{d^2 (b c-a d)^{4/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{13/3}}-\frac{\sqrt[3]{a+b x^3} (11 b c-10 a d)}{70 c^2 x^7}-\frac{a \sqrt[3]{a+b x^3}}{10 c x^{10}}",1,"-1/140*(a*(1 + (b*x^3)/a)*(6*c*(b*c - a*d)*x^3*(a + b*x^3)*(-11*c^2 - 2*c*d*x^3 + 9*d^2*x^6)*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + (14*c^2 - 12*c*d*x^3 + 9*d^2*x^6)*(c*(a + b*x^3)*(5*b*c*x^3 + a*(c - 4*d*x^3)) - 2*(b*c - a*d)^2*x^6*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]) + 18*c*(b*c - a*d)*x^3*(a + b*x^3)*(c + d*x^3)^2*HypergeometricPFQ[{-1/3, 2, 2}, {2/3, 1}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]))/(c^5*x^10*(a + b*x^3)^(5/3))","C",0
708,1,1446,392,5.4420129,"\int \frac{\left(a+b x^3\right)^{4/3}}{x^{14} \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(4/3)/(x^14*(c + d*x^3)),x]","\frac{-324 a b c d^4 x^{15}+405 b^2 c^2 d^3 x^{15}-594 a b c d^4 \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{15}+594 b^2 c^2 d^3 \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{15}-162 a^2 d^5 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{15}+324 a b c d^4 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{15}-162 b^2 c^2 d^3 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{15}-324 a^2 c d^4 x^{12}+918 a b c^2 d^3 x^{12}-540 b^2 c^3 d^2 x^{12}-594 a^2 c d^4 \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}+918 a b c^2 d^3 \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}-324 b^2 c^3 d^2 \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}+216 a^2 c d^4 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}-432 a b c^2 d^3 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}+216 b^2 c^3 d^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}+513 a^2 c^2 d^3 x^9-1152 a b c^3 d^2 x^9+630 b^2 c^4 d x^9+324 a^2 c^2 d^3 \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9-234 a b c^3 d^2 \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9-90 b^2 c^4 d \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9-252 a^2 c^2 d^3 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9+504 a b c^3 d^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9-252 b^2 c^4 d \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9-700 b^2 c^5 x^6-612 a^2 c^3 d^2 x^6+1316 a b c^4 d x^6+828 b^2 c^5 \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6+90 a^2 c^3 d^2 \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6-918 a b c^4 d \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6+280 b^2 c^5 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6+280 a^2 c^3 d^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6-560 a b c^4 d \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6-840 a b c^5 x^3+686 a^2 c^4 d x^3+828 a b c^5 \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3-828 a^2 c^4 d \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3+54 c (b c-a d) \left(b x^3+a\right) \left(d x^3+c\right)^2 \left(6 d x^3-7 c\right) \, _3F_2\left(-\frac{1}{3},2,2;\frac{2}{3},1;\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3+54 c (b c-a d) \left(b x^3+a\right) \left(d x^3+c\right)^3 \, _4F_3\left(-\frac{1}{3},2,2,2;\frac{2}{3},1,1;\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3-140 a^2 c^5}{1820 c^6 x^{13} \left(b x^3+a\right)^{2/3}}","-\frac{\sqrt[3]{a+b x^3} \left(130 a^2 d^2-143 a b c d+4 b^2 c^2\right)}{910 a c^3 x^7}+\frac{\sqrt[3]{a+b x^3} \left(455 a^3 d^3-520 a^2 b c d^2+26 a b^2 c^2 d+12 b^3 c^3\right)}{1820 a^2 c^4 x^4}-\frac{\sqrt[3]{a+b x^3} \left(1820 a^4 d^4-2275 a^3 b c d^3+260 a^2 b^2 c^2 d^2+78 a b^3 c^3 d+36 b^4 c^4\right)}{1820 a^3 c^5 x}-\frac{d^3 (b c-a d)^{4/3} \log \left(c+d x^3\right)}{6 c^{16/3}}+\frac{d^3 (b c-a d)^{4/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{16/3}}+\frac{d^3 (b c-a d)^{4/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{16/3}}-\frac{\sqrt[3]{a+b x^3} (14 b c-13 a d)}{130 c^2 x^{10}}-\frac{a \sqrt[3]{a+b x^3}}{13 c x^{13}}",1,"(-140*a^2*c^5 - 840*a*b*c^5*x^3 + 686*a^2*c^4*d*x^3 - 700*b^2*c^5*x^6 + 1316*a*b*c^4*d*x^6 - 612*a^2*c^3*d^2*x^6 + 630*b^2*c^4*d*x^9 - 1152*a*b*c^3*d^2*x^9 + 513*a^2*c^2*d^3*x^9 - 540*b^2*c^3*d^2*x^12 + 918*a*b*c^2*d^3*x^12 - 324*a^2*c*d^4*x^12 + 405*b^2*c^2*d^3*x^15 - 324*a*b*c*d^4*x^15 + 828*a*b*c^5*x^3*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 828*a^2*c^4*d*x^3*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 828*b^2*c^5*x^6*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 918*a*b*c^4*d*x^6*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 90*a^2*c^3*d^2*x^6*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 90*b^2*c^4*d*x^9*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 234*a*b*c^3*d^2*x^9*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 324*a^2*c^2*d^3*x^9*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 324*b^2*c^3*d^2*x^12*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 918*a*b*c^2*d^3*x^12*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 594*a^2*c*d^4*x^12*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 594*b^2*c^2*d^3*x^15*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 594*a*b*c*d^4*x^15*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 280*b^2*c^5*x^6*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 560*a*b*c^4*d*x^6*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 280*a^2*c^3*d^2*x^6*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 252*b^2*c^4*d*x^9*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 504*a*b*c^3*d^2*x^9*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 252*a^2*c^2*d^3*x^9*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 216*b^2*c^3*d^2*x^12*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 432*a*b*c^2*d^3*x^12*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 216*a^2*c*d^4*x^12*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 162*b^2*c^2*d^3*x^15*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 324*a*b*c*d^4*x^15*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 162*a^2*d^5*x^15*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 54*c*(b*c - a*d)*x^3*(a + b*x^3)*(c + d*x^3)^2*(-7*c + 6*d*x^3)*HypergeometricPFQ[{-1/3, 2, 2}, {2/3, 1}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 54*c*(b*c - a*d)*x^3*(a + b*x^3)*(c + d*x^3)^3*HypergeometricPFQ[{-1/3, 2, 2, 2}, {2/3, 1, 1}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(1820*c^6*x^13*(a + b*x^3)^(2/3))","C",0
709,1,343,65,0.744733,"\int \frac{x^6 \left(a+b x^3\right)^{4/3}}{c+d x^3} \, dx","Integrate[(x^6*(a + b*x^3)^(4/3))/(c + d*x^3),x]","\frac{x \left(\frac{16 a^2 c^2 \left(a^2 d^2-12 a b c d+10 b^2 c^2\right) F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\left(c+d x^3\right) \left(x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}+2 \left(a+b x^3\right) \left(2 a^2 d^2+3 a b d \left(3 d x^3-8 c\right)+b^2 \left(20 c^2-8 c d x^3+5 d^2 x^6\right)\right)-\frac{x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(a^3 d^3+8 a^2 b c d^2-30 a b^2 c^2 d+20 b^3 c^3\right) F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c}\right)}{80 b d^3 \left(a+b x^3\right)^{2/3}}","\frac{a x^7 \sqrt[3]{a+b x^3} F_1\left(\frac{7}{3};-\frac{4}{3},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{7 c \sqrt[3]{\frac{b x^3}{a}+1}}",1,"(x*(2*(a + b*x^3)*(2*a^2*d^2 + 3*a*b*d*(-8*c + 3*d*x^3) + b^2*(20*c^2 - 8*c*d*x^3 + 5*d^2*x^6)) - ((20*b^3*c^3 - 30*a*b^2*c^2*d + 8*a^2*b*c*d^2 + a^3*d^3)*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/c + (16*a^2*c^2*(10*b^2*c^2 - 12*a*b*c*d + a^2*d^2)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))))/(80*b*d^3*(a + b*x^3)^(2/3))","B",0
710,1,280,65,0.6250267,"\int \frac{x^3 \left(a+b x^3\right)^{4/3}}{c+d x^3} \, dx","Integrate[(x^3*(a + b*x^3)^(4/3))/(c + d*x^3),x]","\frac{x \left(\frac{x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(4 a^2 d^2-15 a b c d+10 b^2 c^2\right) F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c}+\frac{16 a^2 c^2 (6 a d-5 b c) F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\left(c+d x^3\right) \left(x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}+4 \left(a+b x^3\right) \left(6 a d-5 b c+2 b d x^3\right)\right)}{40 d^2 \left(a+b x^3\right)^{2/3}}","\frac{a x^4 \sqrt[3]{a+b x^3} F_1\left(\frac{4}{3};-\frac{4}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 c \sqrt[3]{\frac{b x^3}{a}+1}}",1,"(x*(4*(a + b*x^3)*(-5*b*c + 6*a*d + 2*b*d*x^3) + ((10*b^2*c^2 - 15*a*b*c*d + 4*a^2*d^2)*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/c + (16*a^2*c^2*(-5*b*c + 6*a*d)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))))/(40*d^2*(a + b*x^3)^(2/3))","B",0
711,1,346,60,0.412219,"\int \frac{\left(a+b x^3\right)^{4/3}}{c+d x^3} \, dx","Integrate[(a + b*x^3)^(4/3)/(c + d*x^3),x]","\frac{x \left(\frac{4 \left(b x^3 \left(a+b x^3\right) \left(c+d x^3\right) \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c \left(2 a^2 d+a b d x^3+b^2 x^3 \left(c+d x^3\right)\right) F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}{\left(c+d x^3\right) \left(x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}+\frac{b x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} (3 a d-2 b c) F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c}\right)}{8 d \left(a+b x^3\right)^{2/3}}","\frac{a x \sqrt[3]{a+b x^3} F_1\left(\frac{1}{3};-\frac{4}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c \sqrt[3]{\frac{b x^3}{a}+1}}",1,"(x*((b*(-2*b*c + 3*a*d)*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/c + (4*(-4*a*c*(2*a^2*d + a*b*d*x^3 + b^2*x^3*(c + d*x^3))*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + b*x^3*(a + b*x^3)*(c + d*x^3)*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))/((c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))))/(8*d*(a + b*x^3)^(2/3))","B",0
712,1,341,65,0.3785089,"\int \frac{\left(a+b x^3\right)^{4/3}}{x^3 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(4/3)/(x^3*(c + d*x^3)),x]","-\frac{b x^6 \left(\frac{b x^3}{a}+1\right)^{2/3} (a d-2 b c) F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+\frac{4 a c \left(x^3 \left(a+b x^3\right) \left(c+d x^3\right) \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c \left(a c+3 a d x^3-2 b c x^3+b d x^6\right) F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}{\left(c+d x^3\right) \left(x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}}{8 c^2 x^2 \left(a+b x^3\right)^{2/3}}","-\frac{a \sqrt[3]{a+b x^3} F_1\left(-\frac{2}{3};-\frac{4}{3},1;\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 c x^2 \sqrt[3]{\frac{b x^3}{a}+1}}",1,"-1/8*(b*(-2*b*c + a*d)*x^6*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + (4*a*c*(-4*a*c*(a*c - 2*b*c*x^3 + 3*a*d*x^3 + b*d*x^6)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(a + b*x^3)*(c + d*x^3)*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))/((c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))))/(c^2*x^2*(a + b*x^3)^(2/3))","B",0
713,1,286,65,0.4392708,"\int \frac{\left(a+b x^3\right)^{4/3}}{x^6 \left(c+d x^3\right)} \, dx","Integrate[(a + b*x^3)^(4/3)/(x^6*(c + d*x^3)),x]","\frac{-\frac{16 a x \left(10 a^2 d^2-15 a b c d+4 b^2 c^2\right) F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c \left(c+d x^3\right) \left(x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}+\frac{b d x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} (5 a d-6 b c) F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c^3}-\frac{4 \left(a+b x^3\right) \left(2 a c-5 a d x^3+6 b c x^3\right)}{c^2 x^5}}{40 \left(a+b x^3\right)^{2/3}}","-\frac{a \sqrt[3]{a+b x^3} F_1\left(-\frac{5}{3};-\frac{4}{3},1;-\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{5 c x^5 \sqrt[3]{\frac{b x^3}{a}+1}}",1,"((-4*(a + b*x^3)*(2*a*c + 6*b*c*x^3 - 5*a*d*x^3))/(c^2*x^5) + (b*d*(-6*b*c + 5*a*d)*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/c^3 - (16*a*(4*b^2*c^2 - 15*a*b*c*d + 10*a^2*d^2)*x*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*(c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))))/(40*(a + b*x^3)^(2/3))","B",0
714,1,157,290,0.2577427,"\int \frac{x^{14}}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[x^14/((a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{\left(a+b x^3\right)^{2/3} \left(\frac{-81 a^3 d^3+9 a^2 b d^2 \left(6 d x^3-11 c\right)-3 a b^2 d \left(44 c^2-22 c d x^3+15 d^2 x^6\right)+b^3 \left(-220 c^3+88 c^2 d x^3-55 c d^2 x^6+40 d^3 x^9\right)}{b^4}+\frac{220 c^4 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)}{b c-a d}\right)}{440 d^4}","-\frac{\left(a+b x^3\right)^{2/3} (a d+b c) \left(a^2 d^2+b^2 c^2\right)}{2 b^4 d^4}+\frac{\left(a+b x^3\right)^{5/3} \left(3 a^2 d^2+2 a b c d+b^2 c^2\right)}{5 b^4 d^3}-\frac{\left(a+b x^3\right)^{8/3} (3 a d+b c)}{8 b^4 d^2}+\frac{\left(a+b x^3\right)^{11/3}}{11 b^4 d}+\frac{c^4 \log \left(c+d x^3\right)}{6 d^{14/3} \sqrt[3]{b c-a d}}-\frac{c^4 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{14/3} \sqrt[3]{b c-a d}}-\frac{c^4 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{14/3} \sqrt[3]{b c-a d}}",1,"((a + b*x^3)^(2/3)*((-81*a^3*d^3 + 9*a^2*b*d^2*(-11*c + 6*d*x^3) - 3*a*b^2*d*(44*c^2 - 22*c*d*x^3 + 15*d^2*x^6) + b^3*(-220*c^3 + 88*c^2*d*x^3 - 55*c*d^2*x^6 + 40*d^3*x^9))/b^4 + (220*c^4*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)])/(b*c - a*d)))/(440*d^4)","C",1
715,1,145,244,0.1383658,"\int \frac{x^{11}}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[x^11/((a + b*x^3)^(1/3)*(c + d*x^3)),x]","-\frac{\left(a+b x^3\right)^{2/3} \left(9 a^3 d^3+3 a^2 b d^2 \left(c-2 d x^3\right)+20 b^3 c^3 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+a b^2 d \left(8 c^2-2 c d x^3+5 d^2 x^6\right)+b^3 c \left(-20 c^2+8 c d x^3-5 d^2 x^6\right)\right)}{40 b^3 d^3 (b c-a d)}","\frac{\left(a+b x^3\right)^{2/3} \left(a^2 d^2+a b c d+b^2 c^2\right)}{2 b^3 d^3}-\frac{\left(a+b x^3\right)^{5/3} (2 a d+b c)}{5 b^3 d^2}+\frac{\left(a+b x^3\right)^{8/3}}{8 b^3 d}-\frac{c^3 \log \left(c+d x^3\right)}{6 d^{11/3} \sqrt[3]{b c-a d}}+\frac{c^3 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{11/3} \sqrt[3]{b c-a d}}+\frac{c^3 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{11/3} \sqrt[3]{b c-a d}}",1,"-1/40*((a + b*x^3)^(2/3)*(9*a^3*d^3 + 3*a^2*b*d^2*(c - 2*d*x^3) + b^3*c*(-20*c^2 + 8*c*d*x^3 - 5*d^2*x^6) + a*b^2*d*(8*c^2 - 2*c*d*x^3 + 5*d^2*x^6) + 20*b^3*c^3*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)]))/(b^3*d^3*(b*c - a*d))","C",1
716,1,103,203,0.0949641,"\int \frac{x^8}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[x^8/((a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{\left(a+b x^3\right)^{2/3} \left(3 a^2 d^2+5 b^2 c^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+2 a b d \left(c-d x^3\right)+b^2 c \left(2 d x^3-5 c\right)\right)}{10 b^2 d^2 (b c-a d)}","-\frac{\left(a+b x^3\right)^{2/3} (a d+b c)}{2 b^2 d^2}+\frac{\left(a+b x^3\right)^{5/3}}{5 b^2 d}+\frac{c^2 \log \left(c+d x^3\right)}{6 d^{8/3} \sqrt[3]{b c-a d}}-\frac{c^2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{8/3} \sqrt[3]{b c-a d}}-\frac{c^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{8/3} \sqrt[3]{b c-a d}}",1,"((a + b*x^3)^(2/3)*(3*a^2*d^2 + 2*a*b*d*(c - d*x^3) + b^2*c*(-5*c + 2*d*x^3) + 5*b^2*c^2*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)]))/(10*b^2*d^2*(b*c - a*d))","C",1
717,1,69,168,0.0306552,"\int \frac{x^5}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[x^5/((a + b*x^3)^(1/3)*(c + d*x^3)),x]","-\frac{\left(a+b x^3\right)^{2/3} \left(b c \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+a d-b c\right)}{2 b d (b c-a d)}","-\frac{c \log \left(c+d x^3\right)}{6 d^{5/3} \sqrt[3]{b c-a d}}+\frac{c \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{5/3} \sqrt[3]{b c-a d}}+\frac{c \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{5/3} \sqrt[3]{b c-a d}}+\frac{\left(a+b x^3\right)^{2/3}}{2 b d}",1,"-1/2*((a + b*x^3)^(2/3)*(-(b*c) + a*d + b*c*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)]))/(b*d*(b*c - a*d))","C",1
718,1,50,145,0.0187442,"\int \frac{x^2}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[x^2/((a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{\left(a+b x^3\right)^{2/3} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)}{2 b c-2 a d}","\frac{\log \left(c+d x^3\right)}{6 d^{2/3} \sqrt[3]{b c-a d}}-\frac{\log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{2/3} \sqrt[3]{b c-a d}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{2/3} \sqrt[3]{b c-a d}}",1,"((a + b*x^3)^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)])/(2*b*c - 2*a*d)","C",1
719,1,140,244,0.1781659,"\int \frac{1}{x \sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[1/(x*(a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{3 \sqrt[3]{a} d \left(a+b x^3\right)^{2/3} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)-(b c-a d) \left(3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-3 \log (x)\right)}{6 \sqrt[3]{a} c (a d-b c)}","-\frac{\sqrt[3]{d} \log \left(c+d x^3\right)}{6 c \sqrt[3]{b c-a d}}+\frac{\sqrt[3]{d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c \sqrt[3]{b c-a d}}+\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c \sqrt[3]{b c-a d}}+\frac{\log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2 \sqrt[3]{a} c}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} c}-\frac{\log (x)}{2 \sqrt[3]{a} c}",1,"(3*a^(1/3)*d*(a + b*x^3)^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)] - (b*c - a*d)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 3*Log[x] + 3*Log[a^(1/3) - (a + b*x^3)^(1/3)]))/(6*a^(1/3)*c*(-(b*c) + a*d))","C",1
720,1,156,296,0.611853,"\int \frac{1}{x^4 \sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^4*(a + b*x^3)^(1/3)*(c + d*x^3)),x]","-\frac{\frac{(3 a d+b c) \left(3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-3 \log (x)\right)}{a^{4/3}}-\frac{9 d^2 \left(a+b x^3\right)^{2/3} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)}{b c-a d}+\frac{6 c \left(a+b x^3\right)^{2/3}}{a x^3}}{18 c^2}","-\frac{(3 a d+b c) \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{6 a^{4/3} c^2}-\frac{(3 a d+b c) \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{4/3} c^2}+\frac{\log (x) (3 a d+b c)}{6 a^{4/3} c^2}+\frac{d^{4/3} \log \left(c+d x^3\right)}{6 c^2 \sqrt[3]{b c-a d}}-\frac{d^{4/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c^2 \sqrt[3]{b c-a d}}-\frac{d^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c^2 \sqrt[3]{b c-a d}}-\frac{\left(a+b x^3\right)^{2/3}}{3 a c x^3}",1,"-1/18*((6*c*(a + b*x^3)^(2/3))/(a*x^3) - (9*d^2*(a + b*x^3)^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)])/(b*c - a*d) + ((b*c + 3*a*d)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 3*Log[x] + 3*Log[a^(1/3) - (a + b*x^3)^(1/3)]))/a^(4/3))/c^2","C",1
721,1,288,273,0.8184959,"\int \frac{x^6}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[x^6/((a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{\frac{2 \left(-a \sqrt[3]{c} \log \left(\frac{\sqrt[3]{c} x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}+\frac{x^2 (b c-a d)^{2/3}}{\left(a x^3+b\right)^{2/3}}+c^{2/3}\right)+6 x \left(a+b x^3\right)^{2/3} \sqrt[3]{b c-a d}+2 a \sqrt[3]{c} \log \left(\sqrt[3]{c}-\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}\right)-2 \sqrt{3} a \sqrt[3]{c} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)\right)}{\sqrt[3]{b c-a d}}-\frac{3 x^4 \sqrt[3]{\frac{b x^3}{a}+1} (a d+3 b c) F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c \sqrt[3]{a+b x^3}}}{36 b d}","\frac{(a d+3 b c) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{6 b^{4/3} d^2}-\frac{(a d+3 b c) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{4/3} d^2}+\frac{c^{4/3} \log \left(c+d x^3\right)}{6 d^2 \sqrt[3]{b c-a d}}-\frac{c^{4/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 d^2 \sqrt[3]{b c-a d}}+\frac{c^{4/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^2 \sqrt[3]{b c-a d}}+\frac{x \left(a+b x^3\right)^{2/3}}{3 b d}",1,"((-3*(3*b*c + a*d)*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*(a + b*x^3)^(1/3)) + (2*(6*(b*c - a*d)^(1/3)*x*(a + b*x^3)^(2/3) - 2*Sqrt[3]*a*c^(1/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] + 2*a*c^(1/3)*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)] - a*c^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(b*c - a*d)^(1/3))/(36*b*d)","C",0
722,1,65,233,0.0406399,"\int \frac{x^3}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[x^3/((a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{x^4 \sqrt[3]{\frac{a+b x^3}{a}} F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 c \sqrt[3]{a+b x^3}}","-\frac{\sqrt[3]{c} \log \left(c+d x^3\right)}{6 d \sqrt[3]{b c-a d}}+\frac{\sqrt[3]{c} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 d \sqrt[3]{b c-a d}}-\frac{\sqrt[3]{c} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d \sqrt[3]{b c-a d}}-\frac{\log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 \sqrt[3]{b} d}+\frac{\tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b} d}",1,"(x^4*((a + b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/(4*c*(a + b*x^3)^(1/3))","C",0
723,1,168,148,0.2403449,"\int \frac{1}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[1/((a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{\log \left(\frac{\sqrt[3]{c} x \sqrt[3]{b c-a d}}{\sqrt[3]{a+b x^3}}+\frac{x^2 (b c-a d)^{2/3}}{\left(a+b x^3\right)^{2/3}}+c^{2/3}\right)-2 \log \left(\sqrt[3]{c}-\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{6 c^{2/3} \sqrt[3]{b c-a d}}","\frac{\log \left(c+d x^3\right)}{6 c^{2/3} \sqrt[3]{b c-a d}}-\frac{\log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{2/3} \sqrt[3]{b c-a d}}+\frac{\tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{2/3} \sqrt[3]{b c-a d}}",1,"(2*Sqrt[3]*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]] - 2*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(a + b*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(a + b*x^3)^(1/3)])/(6*c^(2/3)*(b*c - a*d)^(1/3))","A",1
724,1,124,176,0.0661329,"\int \frac{1}{x^3 \sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^3*(a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{-3 x^3 \left(c+d x^3\right) (b c-a d) \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-4 c \left(a+b x^3\right) \left(c+3 d x^3\right) \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)}{8 c^3 x^2 \left(a+b x^3\right)^{4/3}}","-\frac{d \log \left(c+d x^3\right)}{6 c^{5/3} \sqrt[3]{b c-a d}}+\frac{d \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{5/3} \sqrt[3]{b c-a d}}-\frac{d \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{5/3} \sqrt[3]{b c-a d}}-\frac{\left(a+b x^3\right)^{2/3}}{2 a c x^2}",1,"(-4*c*(a + b*x^3)*(c + 3*d*x^3)*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 3*(b*c - a*d)*x^3*(c + d*x^3)*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(8*c^3*x^2*(a + b*x^3)^(4/3))","C",0
725,1,207,214,1.0254364,"\int \frac{1}{x^6 \sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^6*(a + b*x^3)^(1/3)*(c + d*x^3)),x]","-\frac{-9 x^3 \left(c+d x^3\right)^2 (b c-a d) \, _3F_2\left(\frac{4}{3},2,2;1,\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-3 x^3 \left(-c^2+8 c d x^3+9 d^2 x^6\right) (b c-a d) \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+8 c \left(a+b x^3\right) \left(c^2-3 c d x^3-9 d^2 x^6\right) \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)}{40 c^4 x^5 \left(a+b x^3\right)^{4/3}}","\frac{\left(a+b x^3\right)^{2/3} (5 a d+3 b c)}{10 a^2 c^2 x^2}+\frac{d^2 \log \left(c+d x^3\right)}{6 c^{8/3} \sqrt[3]{b c-a d}}-\frac{d^2 \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{8/3} \sqrt[3]{b c-a d}}+\frac{d^2 \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{8/3} \sqrt[3]{b c-a d}}-\frac{\left(a+b x^3\right)^{2/3}}{5 a c x^5}",1,"-1/40*(8*c*(a + b*x^3)*(c^2 - 3*c*d*x^3 - 9*d^2*x^6)*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 3*(b*c - a*d)*x^3*(-c^2 + 8*c*d*x^3 + 9*d^2*x^6)*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 9*(b*c - a*d)*x^3*(c + d*x^3)^2*HypergeometricPFQ[{4/3, 2, 2}, {1, 7/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^4*x^5*(a + b*x^3)^(4/3))","C",0
726,1,821,262,2.545744,"\int \frac{1}{x^9 \sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^9*(a + b*x^3)^(1/3)*(c + d*x^3)),x]","-\frac{648 b c d^3 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}-297 a d^4 \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}+297 b c d^3 \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}+648 a c d^3 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9+216 b c^2 d^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9-243 a c d^3 \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9+243 b c^2 d^2 \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9+216 a c^2 d^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6-72 b c^3 d \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6+45 a c^2 d^2 \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6-45 b c^3 d \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6+40 b c^4 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3-72 a c^3 d \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3+9 b c^4 \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3-9 a c^3 d \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3-54 (b c-a d) \left(c-3 d x^3\right) \left(d x^3+c\right)^2 \, _3F_2\left(\frac{4}{3},2,2;1,\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3+27 (b c-a d) \left(d x^3+c\right)^3 \, _4F_3\left(\frac{4}{3},2,2,2;1,1,\frac{7}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3+40 a c^4 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)}{320 c^5 x^8 \left(b x^3+a\right)^{4/3}}","\frac{\left(a+b x^3\right)^{2/3} (4 a d+3 b c)}{20 a^2 c^2 x^5}-\frac{\left(a+b x^3\right)^{2/3} \left(20 a^2 d^2+12 a b c d+9 b^2 c^2\right)}{40 a^3 c^3 x^2}-\frac{d^3 \log \left(c+d x^3\right)}{6 c^{11/3} \sqrt[3]{b c-a d}}+\frac{d^3 \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{11/3} \sqrt[3]{b c-a d}}-\frac{d^3 \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{11/3} \sqrt[3]{b c-a d}}-\frac{\left(a+b x^3\right)^{2/3}}{8 a c x^8}",1,"-1/320*(40*a*c^4*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 40*b*c^4*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 72*a*c^3*d*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 72*b*c^3*d*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 216*a*c^2*d^2*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 216*b*c^2*d^2*x^9*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 648*a*c*d^3*x^9*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 648*b*c*d^3*x^12*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 9*b*c^4*x^3*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 9*a*c^3*d*x^3*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 45*b*c^3*d*x^6*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 45*a*c^2*d^2*x^6*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 243*b*c^2*d^2*x^9*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 243*a*c*d^3*x^9*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 297*b*c*d^3*x^12*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 297*a*d^4*x^12*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 54*(b*c - a*d)*x^3*(c - 3*d*x^3)*(c + d*x^3)^2*HypergeometricPFQ[{4/3, 2, 2}, {1, 7/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*(b*c - a*d)*x^3*(c + d*x^3)^3*HypergeometricPFQ[{4/3, 2, 2, 2}, {1, 1, 7/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^5*x^8*(a + b*x^3)^(4/3))","C",0
727,1,144,64,0.128926,"\int \frac{x^7}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[x^7/((a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{-2 x^5 \sqrt[3]{\frac{b x^3}{a}+1} (a d+2 b c) F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-5 a c x^2 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 c x^2 \left(a+b x^3\right)}{20 b c d \sqrt[3]{a+b x^3}}","\frac{x^8 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{8}{3};\frac{1}{3},1;\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{8 c \sqrt[3]{a+b x^3}}",1,"(5*c*x^2*(a + b*x^3) - 5*a*c*x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] - 2*(2*b*c + a*d)*x^5*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(20*b*c*d*(a + b*x^3)^(1/3))","B",0
728,1,65,64,0.0329017,"\int \frac{x^4}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[x^4/((a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{x^5 \sqrt[3]{\frac{a+b x^3}{a}} F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{5 c \sqrt[3]{a+b x^3}}","\frac{x^5 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{5 c \sqrt[3]{a+b x^3}}",1,"(x^5*((a + b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(5*c*(a + b*x^3)^(1/3))","A",1
729,1,65,64,0.0532096,"\int \frac{x}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[x/((a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{x^2 \sqrt[3]{\frac{a+b x^3}{a}} F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 c \sqrt[3]{a+b x^3}}","\frac{x^2 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 c \sqrt[3]{a+b x^3}}",1,"(x^2*((a + b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)])/(2*c*(a + b*x^3)^(1/3))","A",1
730,1,141,62,0.1272089,"\int \frac{1}{x^2 \sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^2*(a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{5 x^3 \sqrt[3]{\frac{b x^3}{a}+1} (b c-a d) F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b d x^6 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-10 c \left(a+b x^3\right)}{10 a c^2 x \sqrt[3]{a+b x^3}}","-\frac{\sqrt[3]{\frac{b x^3}{a}+1} F_1\left(-\frac{1}{3};\frac{1}{3},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c x \sqrt[3]{a+b x^3}}",1,"(-10*c*(a + b*x^3) + 5*(b*c - a*d)*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*d*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(10*a*c^2*x*(a + b*x^3)^(1/3))","B",0
731,1,183,64,0.2249075,"\int \frac{1}{x^5 \sqrt[3]{a+b x^3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^5*(a + b*x^3)^(1/3)*(c + d*x^3)),x]","\frac{5 x^6 \sqrt[3]{\frac{b x^3}{a}+1} \left(2 a^2 d^2-2 a b c d-b^2 c^2\right) F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-2 b d x^9 \sqrt[3]{\frac{b x^3}{a}+1} (2 a d+b c) F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 c \left(a+b x^3\right) \left(-a c+4 a d x^3+2 b c x^3\right)}{20 a^2 c^3 x^4 \sqrt[3]{a+b x^3}}","-\frac{\sqrt[3]{\frac{b x^3}{a}+1} F_1\left(-\frac{4}{3};\frac{1}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 c x^4 \sqrt[3]{a+b x^3}}",1,"(5*c*(a + b*x^3)*(-(a*c) + 2*b*c*x^3 + 4*a*d*x^3) + 5*(-(b^2*c^2) - 2*a*b*c*d + 2*a^2*d^2)*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] - 2*b*d*(b*c + 2*a*d)*x^9*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(20*a^2*c^3*x^4*(a + b*x^3)^(1/3))","B",0
732,1,251,241,0.6636906,"\int \frac{x^{11}}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[x^11/((a + b*x^3)^(2/3)*(c + d*x^3)),x]","\frac{\frac{84 \sqrt[3]{a+b x^3} \left(a^2 d^2+a b c d+b^2 c^2\right)}{b^3}-\frac{21 d \left(a+b x^3\right)^{4/3} (2 a d+b c)}{b^3}+\frac{12 d^2 \left(a+b x^3\right)^{7/3}}{b^3}+\frac{14 c^3 \left(\log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)\right)}{\sqrt[3]{d} (b c-a d)^{2/3}}}{84 d^3}","\frac{\sqrt[3]{a+b x^3} \left(a^2 d^2+a b c d+b^2 c^2\right)}{b^3 d^3}-\frac{\left(a+b x^3\right)^{4/3} (2 a d+b c)}{4 b^3 d^2}+\frac{\left(a+b x^3\right)^{7/3}}{7 b^3 d}+\frac{c^3 \log \left(c+d x^3\right)}{6 d^{10/3} (b c-a d)^{2/3}}-\frac{c^3 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{10/3} (b c-a d)^{2/3}}+\frac{c^3 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{10/3} (b c-a d)^{2/3}}",1,"((84*(b^2*c^2 + a*b*c*d + a^2*d^2)*(a + b*x^3)^(1/3))/b^3 - (21*d*(b*c + 2*a*d)*(a + b*x^3)^(4/3))/b^3 + (12*d^2*(a + b*x^3)^(7/3))/b^3 + (14*c^3*(2*Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/(d^(1/3)*(b*c - a*d)^(2/3)))/(84*d^3)","A",1
733,1,211,201,0.4253113,"\int \frac{x^8}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[x^8/((a + b*x^3)^(2/3)*(c + d*x^3)),x]","\frac{-\frac{12 \sqrt[3]{a+b x^3} (a d+b c)}{b^2}+\frac{3 d \left(a+b x^3\right)^{4/3}}{b^2}-\frac{2 c^2 \left(\log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)\right)}{\sqrt[3]{d} (b c-a d)^{2/3}}}{12 d^2}","-\frac{\sqrt[3]{a+b x^3} (a d+b c)}{b^2 d^2}+\frac{\left(a+b x^3\right)^{4/3}}{4 b^2 d}-\frac{c^2 \log \left(c+d x^3\right)}{6 d^{7/3} (b c-a d)^{2/3}}+\frac{c^2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{7/3} (b c-a d)^{2/3}}-\frac{c^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{7/3} (b c-a d)^{2/3}}",1,"((-12*(b*c + a*d)*(a + b*x^3)^(1/3))/b^2 + (3*d*(a + b*x^3)^(4/3))/b^2 - (2*c^2*(2*Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/(d^(1/3)*(b*c - a*d)^(2/3)))/(12*d^2)","A",1
734,1,202,165,0.18011,"\int \frac{x^5}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[x^5/((a + b*x^3)^(2/3)*(c + d*x^3)),x]","\frac{b c \log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)+6 \sqrt[3]{d} \sqrt[3]{a+b x^3} (b c-a d)^{2/3}-2 b c \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)-2 \sqrt{3} b c \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)}{6 b d^{4/3} (b c-a d)^{2/3}}","\frac{c \log \left(c+d x^3\right)}{6 d^{4/3} (b c-a d)^{2/3}}-\frac{c \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{4/3} (b c-a d)^{2/3}}+\frac{c \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{4/3} (b c-a d)^{2/3}}+\frac{\sqrt[3]{a+b x^3}}{b d}",1,"(6*d^(1/3)*(b*c - a*d)^(2/3)*(a + b*x^3)^(1/3) - 2*Sqrt[3]*b*c*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - 2*b*c*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + b*c*Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)])/(6*b*d^(4/3)*(b*c - a*d)^(2/3))","A",1
735,1,164,145,0.0622744,"\int \frac{x^2}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[x^2/((a + b*x^3)^(2/3)*(c + d*x^3)),x]","\frac{-\log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)+2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)}{6 \sqrt[3]{d} (b c-a d)^{2/3}}","-\frac{\log \left(c+d x^3\right)}{6 \sqrt[3]{d} (b c-a d)^{2/3}}+\frac{\log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 \sqrt[3]{d} (b c-a d)^{2/3}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{d} (b c-a d)^{2/3}}",1,"(2*Sqrt[3]*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] + 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] - Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)])/(6*d^(1/3)*(b*c - a*d)^(2/3))","A",1
736,1,308,245,0.4455218,"\int \frac{1}{x \left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x*(a + b*x^3)^(2/3)*(c + d*x^3)),x]","-\frac{-\frac{2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{a^{2/3}}+\frac{\log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)}{a^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)}{a^{2/3}}+\frac{2 d^{2/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{(b c-a d)^{2/3}}-\frac{d^{2/3} \log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)}{(b c-a d)^{2/3}}+\frac{2 \sqrt{3} d^{2/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)}{(b c-a d)^{2/3}}}{6 c}","\frac{\log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2 a^{2/3} c}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} c}-\frac{\log (x)}{2 a^{2/3} c}+\frac{d^{2/3} \log \left(c+d x^3\right)}{6 c (b c-a d)^{2/3}}-\frac{d^{2/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c (b c-a d)^{2/3}}+\frac{d^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c (b c-a d)^{2/3}}",1,"-1/6*((2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*Sqrt[3]*d^(2/3)*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]])/(b*c - a*d)^(2/3) - (2*Log[a^(1/3) - (a + b*x^3)^(1/3)])/a^(2/3) + (2*d^(2/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(b*c - a*d)^(2/3) + Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]/a^(2/3) - (d^(2/3)*Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)])/(b*c - a*d)^(2/3))/c","A",1
737,1,303,299,0.59079,"\int \frac{1}{x^4 \left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^4*(a + b*x^3)^(2/3)*(c + d*x^3)),x]","\frac{\frac{(3 a d+2 b c) \left(\log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^3}+\left(a+b x^3\right)^{2/3}\right)-2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^3}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)}{a^{2/3} c}+\frac{3 a d^{5/3} \left(-\log \left(-\sqrt[3]{d} \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left(a+b x^3\right)^{2/3}\right)+2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}-1}{\sqrt{3}}\right)\right)}{c (b c-a d)^{2/3}}-\frac{6 \sqrt[3]{a+b x^3}}{x^3}}{18 a c}","-\frac{(3 a d+2 b c) \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{6 a^{5/3} c^2}+\frac{(3 a d+2 b c) \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} c^2}+\frac{\log (x) (3 a d+2 b c)}{6 a^{5/3} c^2}-\frac{d^{5/3} \log \left(c+d x^3\right)}{6 c^2 (b c-a d)^{2/3}}+\frac{d^{5/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c^2 (b c-a d)^{2/3}}-\frac{d^{5/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c^2 (b c-a d)^{2/3}}-\frac{\sqrt[3]{a+b x^3}}{3 a c x^3}",1,"((-6*(a + b*x^3)^(1/3))/x^3 + ((2*b*c + 3*a*d)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) - (a + b*x^3)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]))/(a^(2/3)*c) + (3*a*d^(5/3)*(2*Sqrt[3]*ArcTan[(-1 + (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] + 2*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] - Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)]))/(c*(b*c - a*d)^(2/3)))/(18*a*c)","A",1
738,1,190,279,0.2914612,"\int \frac{x^7}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[x^7/((a + b*x^3)^(2/3)*(c + d*x^3)),x]","\frac{5 c x^2 \left(\left(a+b x^3\right) \left(\frac{d x^3}{c}+1\right)^{2/3}-a \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)\right)-x^5 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3} (2 a d+3 b c) F_1\left(\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{15 b c d \left(a+b x^3\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3}}","\frac{(2 a d+3 b c) \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{6 b^{5/3} d^2}+\frac{(2 a d+3 b c) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{5/3} d^2}+\frac{c^{5/3} \log \left(c+d x^3\right)}{6 d^2 (b c-a d)^{2/3}}-\frac{c^{5/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 d^2 (b c-a d)^{2/3}}-\frac{c^{5/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^2 (b c-a d)^{2/3}}+\frac{x^2 \sqrt[3]{a+b x^3}}{3 b d}",1,"(-((3*b*c + 2*a*d)*x^5*(1 + (b*x^3)/a)^(2/3)*(1 + (d*x^3)/c)^(2/3)*AppellF1[5/3, 2/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)]) + 5*c*x^2*((a + b*x^3)*(1 + (d*x^3)/c)^(2/3) - a*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))]))/(15*b*c*d*(a + b*x^3)^(2/3)*(1 + (d*x^3)/c)^(2/3))","C",0
739,1,65,234,0.0354684,"\int \frac{x^4}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[x^4/((a + b*x^3)^(2/3)*(c + d*x^3)),x]","\frac{x^5 \left(\frac{a+b x^3}{a}\right)^{2/3} F_1\left(\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{5 c \left(a+b x^3\right)^{2/3}}","-\frac{\log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2 b^{2/3} d}-\frac{\tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{2/3} d}-\frac{c^{2/3} \log \left(c+d x^3\right)}{6 d (b c-a d)^{2/3}}+\frac{c^{2/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 d (b c-a d)^{2/3}}+\frac{c^{2/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d (b c-a d)^{2/3}}",1,"(x^5*((a + b*x^3)/a)^(2/3)*AppellF1[5/3, 2/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(5*c*(a + b*x^3)^(2/3))","C",0
740,1,83,149,0.0443337,"\int \frac{x}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[x/((a + b*x^3)^(2/3)*(c + d*x^3)),x]","\frac{x^2 \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{2 c \left(a+b x^3\right)^{2/3} \left(\frac{d x^3}{c}+1\right)^{2/3}}","\frac{\log \left(c+d x^3\right)}{6 \sqrt[3]{c} (b c-a d)^{2/3}}-\frac{\log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 \sqrt[3]{c} (b c-a d)^{2/3}}-\frac{\tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{c} (b c-a d)^{2/3}}",1,"(x^2*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(2*c*(a + b*x^3)^(2/3)*(1 + (d*x^3)/c)^(2/3))","C",0
741,1,128,173,0.0648211,"\int \frac{1}{x^2 \left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^2*(a + b*x^3)^(2/3)*(c + d*x^3)),x]","-\frac{\frac{6 x^3 \left(c+d x^3\right) (b c-a d) \, _2F_1\left(\frac{5}{3},2;\frac{8}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)}{a+b x^3}+5 c \left(2 c+3 d x^3\right) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)}{10 c^3 x \left(a+b x^3\right)^{2/3}}","-\frac{d \log \left(c+d x^3\right)}{6 c^{4/3} (b c-a d)^{2/3}}+\frac{d \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{4/3} (b c-a d)^{2/3}}+\frac{d \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{4/3} (b c-a d)^{2/3}}-\frac{\sqrt[3]{a+b x^3}}{a c x}",1,"-1/10*(5*c*(2*c + 3*d*x^3)*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + (6*(b*c - a*d)*x^3*(c + d*x^3)*Hypergeometric2F1[5/3, 2, 8/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(a + b*x^3))/(c^3*x*(a + b*x^3)^(2/3))","C",0
742,1,267,215,1.8882671,"\int \frac{1}{x^5 \left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^5*(a + b*x^3)^(2/3)*(c + d*x^3)),x]","-\frac{-81 x^3 \left(c+d x^3\right)^2 (b c-a d) \, _4F_3\left(\frac{2}{3},2,2,2;1,1,\frac{8}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+216 d x^6 \left(c+d x^3\right) (a d-b c) \, _3F_2\left(\frac{2}{3},2,2;1,\frac{8}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-5 \left(2 c \left(a+b x^3\right) \left(c^2+10 c d x^3+9 d^2 x^6\right)+\left(a \left(-8 c^3+17 c^2 d x^3+46 c d^2 x^6+9 d^3 x^9\right)+3 b c x^3 \left(-3 c^2+2 c d x^3+9 d^2 x^6\right)\right) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)\right)}{120 c^4 x^4 \left(a+b x^3\right)^{5/3}}","\frac{\sqrt[3]{a+b x^3} (4 a d+3 b c)}{4 a^2 c^2 x}+\frac{d^2 \log \left(c+d x^3\right)}{6 c^{7/3} (b c-a d)^{2/3}}-\frac{d^2 \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{7/3} (b c-a d)^{2/3}}-\frac{d^2 \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{7/3} (b c-a d)^{2/3}}-\frac{\sqrt[3]{a+b x^3}}{4 a c x^4}",1,"-1/120*(-5*(2*c*(a + b*x^3)*(c^2 + 10*c*d*x^3 + 9*d^2*x^6) + (3*b*c*x^3*(-3*c^2 + 2*c*d*x^3 + 9*d^2*x^6) + a*(-8*c^3 + 17*c^2*d*x^3 + 46*c*d^2*x^6 + 9*d^3*x^9))*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]) + 216*d*(-(b*c) + a*d)*x^6*(c + d*x^3)*HypergeometricPFQ[{2/3, 2, 2}, {1, 8/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 81*(b*c - a*d)*x^3*(c + d*x^3)^2*HypergeometricPFQ[{2/3, 2, 2, 2}, {1, 1, 8/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^4*x^4*(a + b*x^3)^(5/3))","C",0
743,1,249,64,0.3781392,"\int \frac{x^6}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[x^6/((a + b*x^3)^(2/3)*(c + d*x^3)),x]","\frac{x \left(4 \left(\frac{4 a^2 c^2 F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{b \left(c+d x^3\right) \left(x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}+\frac{a}{b}+x^3\right)-\frac{x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} (a d+2 b c) F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{b c}\right)}{8 d \left(a+b x^3\right)^{2/3}}","\frac{x^7 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{7}{3};\frac{2}{3},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{7 c \left(a+b x^3\right)^{2/3}}",1,"(x*(-(((2*b*c + a*d)*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/(b*c)) + 4*(a/b + x^3 + (4*a^2*c^2*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(b*(c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))))))/(8*d*(a + b*x^3)^(2/3))","B",0
744,1,65,64,0.035016,"\int \frac{x^3}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[x^3/((a + b*x^3)^(2/3)*(c + d*x^3)),x]","\frac{x^4 \left(\frac{a+b x^3}{a}\right)^{2/3} F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 c \left(a+b x^3\right)^{2/3}}","\frac{x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 c \left(a+b x^3\right)^{2/3}}",1,"(x^4*((a + b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/(4*c*(a + b*x^3)^(2/3))","A",1
745,1,161,59,0.0376524,"\int \frac{1}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[1/((a + b*x^3)^(2/3)*(c + d*x^3)),x]","-\frac{4 a c x F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right) \left(x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c \left(a+b x^3\right)^{2/3}}",1,"(-4*a*c*x*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((a + b*x^3)^(2/3)*(c + d*x^3)*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))","B",0
746,1,338,64,0.3283937,"\int \frac{1}{x^3 \left(a+b x^3\right)^{2/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^3*(a + b*x^3)^(2/3)*(c + d*x^3)),x]","\frac{\frac{4 c \left(x^3 \left(a+b x^3\right) \left(c+d x^3\right) \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c \left(a c+3 a d x^3+2 b c x^3+b d x^6\right) F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}{\left(c+d x^3\right) \left(4 a c F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-x^3 \left(3 a d F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 b c F_1\left(\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)\right)}-b d x^6 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{8 a c^2 x^2 \left(a+b x^3\right)^{2/3}}","-\frac{\left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(-\frac{2}{3};\frac{2}{3},1;\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 c x^2 \left(a+b x^3\right)^{2/3}}",1,"(-(b*d*x^6*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]) + (4*c*(-4*a*c*(a*c + 2*b*c*x^3 + 3*a*d*x^3 + b*d*x^6)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(a + b*x^3)*(c + d*x^3)*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))/((c + d*x^3)*(4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] - x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))))/(8*a*c^2*x^2*(a + b*x^3)^(2/3))","B",0
747,1,157,347,0.2652581,"\int \frac{x^{14}}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x^14/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{\frac{81 a^3 d^3+9 a^2 b d^2 \left(8 c+3 d x^3\right)+3 a b^2 d \left(20 c^2+8 c d x^3-3 d^2 x^6\right)+b^3 \left(40 c^3+20 c^2 d x^3-8 c d^2 x^6+5 d^3 x^9\right)}{b^4}-\frac{40 c^4 \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)}{b c-a d}}{40 d^4 \sqrt[3]{a+b x^3}}","-\frac{a^4}{b^4 \sqrt[3]{a+b x^3} (b c-a d)}+\frac{a^2 \left(a+b x^3\right)^{2/3}}{2 b^4 d}+\frac{\left(a+b x^3\right)^{2/3} \left(a^2 d^2+a b c d+b^2 c^2\right)}{2 b^4 d^3}+\frac{a \left(a+b x^3\right)^{2/3} (a d+b c)}{2 b^4 d^2}-\frac{\left(a+b x^3\right)^{5/3} (a d+b c)}{5 b^4 d^2}-\frac{2 a \left(a+b x^3\right)^{5/3}}{5 b^4 d}+\frac{\left(a+b x^3\right)^{8/3}}{8 b^4 d}-\frac{c^4 \log \left(c+d x^3\right)}{6 d^{11/3} (b c-a d)^{4/3}}+\frac{c^4 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{11/3} (b c-a d)^{4/3}}+\frac{c^4 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{11/3} (b c-a d)^{4/3}}",1,"((81*a^3*d^3 + 9*a^2*b*d^2*(8*c + 3*d*x^3) + 3*a*b^2*d*(20*c^2 + 8*c*d*x^3 - 3*d^2*x^6) + b^3*(40*c^3 + 20*c^2*d*x^3 - 8*c*d^2*x^6 + 5*d^3*x^9))/b^4 - (40*c^4*Hypergeometric2F1[-1/3, 1, 2/3, (d*(a + b*x^3))/(-(b*c) + a*d)])/(b*c - a*d))/(40*d^4*(a + b*x^3)^(1/3))","C",1
748,1,147,253,0.1105866,"\int \frac{x^{11}}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x^11/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{18 a^3 d^3+3 a^2 b d^2 \left(2 d x^3-c\right)+10 b^3 c^3 \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)-a b^2 d \left(5 c^2+c d x^3+2 d^2 x^6\right)+b^3 c \left(-10 c^2-5 c d x^3+2 d^2 x^6\right)}{10 b^3 d^3 \sqrt[3]{a+b x^3} (b c-a d)}","\frac{a^3}{b^3 \sqrt[3]{a+b x^3} (b c-a d)}-\frac{\left(a+b x^3\right)^{2/3} (a d+b c)}{2 b^3 d^2}-\frac{a \left(a+b x^3\right)^{2/3}}{2 b^3 d}+\frac{\left(a+b x^3\right)^{5/3}}{5 b^3 d}+\frac{c^3 \log \left(c+d x^3\right)}{6 d^{8/3} (b c-a d)^{4/3}}-\frac{c^3 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{8/3} (b c-a d)^{4/3}}-\frac{c^3 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{8/3} (b c-a d)^{4/3}}",1,"(18*a^3*d^3 + 3*a^2*b*d^2*(-c + 2*d*x^3) + b^3*c*(-10*c^2 - 5*c*d*x^3 + 2*d^2*x^6) - a*b^2*d*(5*c^2 + c*d*x^3 + 2*d^2*x^6) + 10*b^3*c^3*Hypergeometric2F1[-1/3, 1, 2/3, (d*(a + b*x^3))/(-(b*c) + a*d)])/(10*b^3*d^3*(b*c - a*d)*(a + b*x^3)^(1/3))","C",1
749,1,101,203,0.067697,"\int \frac{x^8}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x^8/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{-3 a^2 d^2-2 b^2 c^2 \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+a b d \left(c-d x^3\right)+b^2 c \left(2 c+d x^3\right)}{2 b^2 d^2 \sqrt[3]{a+b x^3} (b c-a d)}","-\frac{a^2}{b^2 \sqrt[3]{a+b x^3} (b c-a d)}+\frac{\left(a+b x^3\right)^{2/3}}{2 b^2 d}-\frac{c^2 \log \left(c+d x^3\right)}{6 d^{5/3} (b c-a d)^{4/3}}+\frac{c^2 \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{5/3} (b c-a d)^{4/3}}+\frac{c^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{5/3} (b c-a d)^{4/3}}",1,"(-3*a^2*d^2 + a*b*d*(c - d*x^3) + b^2*c*(2*c + d*x^3) - 2*b^2*c^2*Hypergeometric2F1[-1/3, 1, 2/3, (d*(a + b*x^3))/(-(b*c) + a*d)])/(2*b^2*d^2*(b*c - a*d)*(a + b*x^3)^(1/3))","C",1
750,1,77,174,0.0344496,"\int \frac{x^5}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x^5/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{b c \left(a+b x^3\right) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+2 a (b c-a d)}{2 b \sqrt[3]{a+b x^3} (b c-a d)^2}","\frac{c \log \left(c+d x^3\right)}{6 d^{2/3} (b c-a d)^{4/3}}-\frac{c \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 d^{2/3} (b c-a d)^{4/3}}-\frac{c \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} d^{2/3} (b c-a d)^{4/3}}+\frac{a}{b \sqrt[3]{a+b x^3} (b c-a d)}",1,"(2*a*(b*c - a*d) + b*c*(a + b*x^3)*Hypergeometric2F1[2/3, 1, 5/3, (d*(a + b*x^3))/(-(b*c) + a*d)])/(2*b*(b*c - a*d)^2*(a + b*x^3)^(1/3))","C",1
751,1,50,167,0.0213673,"\int \frac{x^2}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x^2/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","-\frac{\, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)}{\sqrt[3]{a+b x^3} (b c-a d)}","-\frac{1}{\sqrt[3]{a+b x^3} (b c-a d)}-\frac{\sqrt[3]{d} \log \left(c+d x^3\right)}{6 (b c-a d)^{4/3}}+\frac{\sqrt[3]{d} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 (b c-a d)^{4/3}}+\frac{\sqrt[3]{d} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} (b c-a d)^{4/3}}",1,"-(Hypergeometric2F1[-1/3, 1, 2/3, (d*(a + b*x^3))/(-(b*c) + a*d)]/((b*c - a*d)*(a + b*x^3)^(1/3)))","C",1
752,1,86,271,0.0451248,"\int \frac{1}{x \left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x*(a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{a d \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+(b c-a d) \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{b x^3}{a}+1\right)}{a c \sqrt[3]{a+b x^3} (b c-a d)}","\frac{\log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{2 a^{4/3} c}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{4/3} c}-\frac{\log (x)}{2 a^{4/3} c}+\frac{d^{4/3} \log \left(c+d x^3\right)}{6 c (b c-a d)^{4/3}}-\frac{d^{4/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c (b c-a d)^{4/3}}-\frac{d^{4/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c (b c-a d)^{4/3}}+\frac{b}{a \sqrt[3]{a+b x^3} (b c-a d)}",1,"(a*d*Hypergeometric2F1[-1/3, 1, 2/3, (d*(a + b*x^3))/(-(b*c) + a*d)] + (b*c - a*d)*Hypergeometric2F1[-1/3, 1, 2/3, 1 + (b*x^3)/a])/(a*c*(b*c - a*d)*(a + b*x^3)^(1/3))","C",1
753,1,117,357,0.0540456,"\int \frac{1}{x^4 \left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^4*(a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{3 a^2 d^2 x^3 \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{d \left(b x^3+a\right)}{a d-b c}\right)+(b c-a d) \left(x^3 (3 a d+4 b c) \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{b x^3}{a}+1\right)+a c\right)}{3 a^2 c^2 x^3 \sqrt[3]{a+b x^3} (a d-b c)}","-\frac{(3 a d+4 b c) \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^3}\right)}{6 a^{7/3} c^2}-\frac{(3 a d+4 b c) \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^3}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{7/3} c^2}+\frac{\log (x) (3 a d+4 b c)}{6 a^{7/3} c^2}-\frac{3 a d+4 b c}{3 a^2 c^2 \sqrt[3]{a+b x^3}}-\frac{d^{7/3} \log \left(c+d x^3\right)}{6 c^2 (b c-a d)^{4/3}}+\frac{d^{7/3} \log \left(\sqrt[3]{b c-a d}+\sqrt[3]{d} \sqrt[3]{a+b x^3}\right)}{2 c^2 (b c-a d)^{4/3}}+\frac{d^{7/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} \sqrt[3]{a+b x^3}}{\sqrt[3]{b c-a d}}}{\sqrt{3}}\right)}{\sqrt{3} c^2 (b c-a d)^{4/3}}-\frac{d^2}{c^2 \sqrt[3]{a+b x^3} (b c-a d)}-\frac{1}{3 a c x^3 \sqrt[3]{a+b x^3}}",1,"(3*a^2*d^2*x^3*Hypergeometric2F1[-1/3, 1, 2/3, (d*(a + b*x^3))/(-(b*c) + a*d)] + (b*c - a*d)*(a*c + (4*b*c + 3*a*d)*x^3*Hypergeometric2F1[-1/3, 1, 2/3, 1 + (b*x^3)/a]))/(3*a^2*c^2*(-(b*c) + a*d)*x^3*(a + b*x^3)^(1/3))","C",1
754,1,504,322,1.0786307,"\int \frac{x^9}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x^9/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","-\frac{3 x^4 \sqrt[3]{\frac{b x^3}{a}+1} \sqrt[3]{b c-a d} \left(-4 a^2 d^2+a b c d+3 b^2 c^2\right) F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+2 c \left(-4 a^2 \sqrt[3]{c} d \sqrt[3]{a+b x^3} \log \left(\frac{\sqrt[3]{c} x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}+\frac{x^2 (b c-a d)^{2/3}}{\left(a x^3+b\right)^{2/3}}+c^{2/3}\right)+24 a^2 d x \sqrt[3]{b c-a d}-6 b^2 c x^4 \sqrt[3]{b c-a d}+a b c^{4/3} \sqrt[3]{a+b x^3} \log \left(\frac{\sqrt[3]{c} x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}+\frac{x^2 (b c-a d)^{2/3}}{\left(a x^3+b\right)^{2/3}}+c^{2/3}\right)+6 a b d x^4 \sqrt[3]{b c-a d}+2 a \sqrt[3]{c} \sqrt[3]{a+b x^3} (4 a d-b c) \log \left(\sqrt[3]{c}-\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}\right)-2 \sqrt{3} a \sqrt[3]{c} \sqrt[3]{a+b x^3} (4 a d-b c) \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-6 a b c x \sqrt[3]{b c-a d}\right)}{36 b^2 c d \sqrt[3]{a+b x^3} (b c-a d)^{4/3}}","\frac{(4 a d+3 b c) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{6 b^{7/3} d^2}-\frac{(4 a d+3 b c) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{7/3} d^2}+\frac{x \left(a+b x^3\right)^{2/3} (b c-4 a d)}{3 b^2 d (b c-a d)}+\frac{c^{7/3} \log \left(c+d x^3\right)}{6 d^2 (b c-a d)^{4/3}}-\frac{c^{7/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 d^2 (b c-a d)^{4/3}}+\frac{c^{7/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^2 (b c-a d)^{4/3}}+\frac{a x^4}{b \sqrt[3]{a+b x^3} (b c-a d)}",1,"-1/36*(3*(b*c - a*d)^(1/3)*(3*b^2*c^2 + a*b*c*d - 4*a^2*d^2)*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*c*(-6*a*b*c*(b*c - a*d)^(1/3)*x + 24*a^2*d*(b*c - a*d)^(1/3)*x - 6*b^2*c*(b*c - a*d)^(1/3)*x^4 + 6*a*b*d*(b*c - a*d)^(1/3)*x^4 - 2*Sqrt[3]*a*c^(1/3)*(-(b*c) + 4*a*d)*(a + b*x^3)^(1/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] + 2*a*c^(1/3)*(-(b*c) + 4*a*d)*(a + b*x^3)^(1/3)*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)] + a*b*c^(4/3)*(a + b*x^3)^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)] - 4*a^2*c^(1/3)*d*(a + b*x^3)^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(b^2*c*d*(b*c - a*d)^(4/3)*(a + b*x^3)^(1/3))","C",0
755,1,309,260,0.4455156,"\int \frac{x^6}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x^6/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{3 x^4 \sqrt[3]{\frac{b x^3}{a}+1} (b c-a d)^{4/3} F_1\left(\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-2 a c \left(\sqrt[3]{c} \sqrt[3]{a+b x^3} \log \left(\frac{\sqrt[3]{c} x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}+\frac{x^2 (b c-a d)^{2/3}}{\left(a x^3+b\right)^{2/3}}+c^{2/3}\right)-2 \sqrt[3]{c} \sqrt[3]{a+b x^3} \log \left(\sqrt[3]{c}-\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}\right)+2 \sqrt{3} \sqrt[3]{c} \sqrt[3]{a+b x^3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-6 x \sqrt[3]{b c-a d}\right)}{12 b c \sqrt[3]{a+b x^3} (b c-a d)^{4/3}}","-\frac{\log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 b^{4/3} d}+\frac{\tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{4/3} d}-\frac{c^{4/3} \log \left(c+d x^3\right)}{6 d (b c-a d)^{4/3}}+\frac{c^{4/3} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 d (b c-a d)^{4/3}}-\frac{c^{4/3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d (b c-a d)^{4/3}}+\frac{a x}{b \sqrt[3]{a+b x^3} (b c-a d)}",1,"(3*(b*c - a*d)^(4/3)*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)] - 2*a*c*(-6*(b*c - a*d)^(1/3)*x + 2*Sqrt[3]*c^(1/3)*(a + b*x^3)^(1/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] - 2*c^(1/3)*(a + b*x^3)^(1/3)*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)] + c^(1/3)*(a + b*x^3)^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(12*b*c*(b*c - a*d)^(4/3)*(a + b*x^3)^(1/3))","C",0
756,1,86,172,0.0610536,"\int \frac{x^3}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x^3/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{x^4 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{4}{3},\frac{4}{3};\frac{7}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{4 a c \sqrt[3]{a+b x^3} \left(\frac{d x^3}{c}+1\right)^{4/3}}","-\frac{x}{\sqrt[3]{a+b x^3} (b c-a d)}+\frac{\sqrt[3]{c} \log \left(c+d x^3\right)}{6 (b c-a d)^{4/3}}-\frac{\sqrt[3]{c} \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 (b c-a d)^{4/3}}+\frac{\sqrt[3]{c} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} (b c-a d)^{4/3}}",1,"(x^4*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[4/3, 4/3, 7/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(4*a*c*(a + b*x^3)^(1/3)*(1 + (d*x^3)/c)^(4/3))","C",0
757,1,256,179,0.1179954,"\int \frac{1}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[1/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","-\frac{28 c^3 \left(a+b x^3\right)^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-28 c^3 \left(a+b x^3\right)^2+21 c^2 d x^3 \left(a+b x^3\right)^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-21 c^2 d x^3 \left(a+b x^3\right)^2+3 d x^9 (b c-a d)^2 \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+3 c x^6 (b c-a d)^2 \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)}{7 c^3 x^2 \left(a+b x^3\right)^{7/3} (a d-b c)}","-\frac{d \log \left(c+d x^3\right)}{6 c^{2/3} (b c-a d)^{4/3}}+\frac{d \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{2/3} (b c-a d)^{4/3}}-\frac{d \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{2/3} (b c-a d)^{4/3}}+\frac{b x}{a \sqrt[3]{a+b x^3} (b c-a d)}",1,"-1/7*(-28*c^3*(a + b*x^3)^2 - 21*c^2*d*x^3*(a + b*x^3)^2 + 28*c^3*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 21*c^2*d*x^3*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 3*c*(b*c - a*d)^2*x^6*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 3*d*(b*c - a*d)^2*x^9*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^3*(-(b*c) + a*d)*x^2*(a + b*x^3)^(7/3))","C",0
758,1,542,229,0.832384,"\int \frac{1}{x^3 \left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^3*(a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{9 c^2 x^6 (b c-a d)^2 \, _3F_2\left(2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+9 d^2 x^{12} (b c-a d)^2 \, _3F_2\left(2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+18 c d x^9 (b c-a d)^2 \, _3F_2\left(2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+28 c^4 \left(a+b x^3\right)^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-28 c^4 \left(a+b x^3\right)^2+168 c^3 d x^3 \left(a+b x^3\right)^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-168 c^3 d x^3 \left(a+b x^3\right)^2+126 c^2 d^2 x^6 \left(a+b x^3\right)^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)-126 c^2 d^2 x^6 \left(a+b x^3\right)^2+15 c^2 x^6 (b c-a d)^2 \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+27 d^2 x^{12} (b c-a d)^2 \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+42 c d x^9 (b c-a d)^2 \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)}{14 c^4 x^5 \left(a+b x^3\right)^{7/3} (a d-b c)}","-\frac{\left(a+b x^3\right)^{2/3} (3 b c-a d)}{2 a^2 c x^2 (b c-a d)}+\frac{d^2 \log \left(c+d x^3\right)}{6 c^{5/3} (b c-a d)^{4/3}}-\frac{d^2 \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{5/3} (b c-a d)^{4/3}}+\frac{d^2 \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{5/3} (b c-a d)^{4/3}}+\frac{b}{a x^2 \sqrt[3]{a+b x^3} (b c-a d)}",1,"(-28*c^4*(a + b*x^3)^2 - 168*c^3*d*x^3*(a + b*x^3)^2 - 126*c^2*d^2*x^6*(a + b*x^3)^2 + 28*c^4*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 168*c^3*d*x^3*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 126*c^2*d^2*x^6*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 15*c^2*(b*c - a*d)^2*x^6*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 42*c*d*(b*c - a*d)^2*x^9*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*d^2*(b*c - a*d)^2*x^12*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 9*c^2*(b*c - a*d)^2*x^6*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 18*c*d*(b*c - a*d)^2*x^9*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 9*d^2*(b*c - a*d)^2*x^12*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(14*c^4*(-(b*c) + a*d)*x^5*(a + b*x^3)^(7/3))","C",0
759,1,950,287,2.311505,"\int \frac{1}{x^6 \left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^6*(a + b*x^3)^(4/3)*(c + d*x^3)),x]","-\frac{297 d^3 (b c-a d)^2 \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{15}+162 d^3 (b c-a d)^2 \, _3F_2\left(2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{15}+27 d^3 (b c-a d)^2 \, _4F_3\left(2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{15}+486 c d^2 (b c-a d)^2 \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}+351 c d^2 (b c-a d)^2 \, _3F_2\left(2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}+81 c d^2 (b c-a d)^2 \, _4F_3\left(2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^{12}-1134 c^2 d^3 \left(b x^3+a\right)^2 x^9+1134 c^2 d^3 \left(b x^3+a\right)^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9+171 c^2 d (b c-a d)^2 \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9+216 c^2 d (b c-a d)^2 \, _3F_2\left(2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9+81 c^2 d (b c-a d)^2 \, _4F_3\left(2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^9-1512 c^3 d^2 \left(b x^3+a\right)^2 x^6+1512 c^3 d^2 \left(b x^3+a\right)^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6-18 c^3 (b c-a d)^2 \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6+27 c^3 (b c-a d)^2 \, _3F_2\left(2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6+27 c^3 (b c-a d)^2 \, _4F_3\left(2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^6-252 c^4 d \left(b x^3+a\right)^2 x^3+252 c^4 d \left(b x^3+a\right)^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right) x^3+56 c^5 \left(b x^3+a\right)^2-56 c^5 \left(b x^3+a\right)^2 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)}{70 c^5 (a d-b c) x^8 \left(b x^3+a\right)^{7/3}}","-\frac{\left(a+b x^3\right)^{2/3} (6 b c-a d)}{5 a^2 c x^5 (b c-a d)}+\frac{\left(a+b x^3\right)^{2/3} \left(-5 a^2 d^2-3 a b c d+18 b^2 c^2\right)}{10 a^3 c^2 x^2 (b c-a d)}-\frac{d^3 \log \left(c+d x^3\right)}{6 c^{8/3} (b c-a d)^{4/3}}+\frac{d^3 \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{8/3} (b c-a d)^{4/3}}-\frac{d^3 \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{8/3} (b c-a d)^{4/3}}+\frac{b}{a x^5 \sqrt[3]{a+b x^3} (b c-a d)}",1,"-1/70*(56*c^5*(a + b*x^3)^2 - 252*c^4*d*x^3*(a + b*x^3)^2 - 1512*c^3*d^2*x^6*(a + b*x^3)^2 - 1134*c^2*d^3*x^9*(a + b*x^3)^2 - 56*c^5*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 252*c^4*d*x^3*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 1512*c^3*d^2*x^6*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 1134*c^2*d^3*x^9*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 18*c^3*(b*c - a*d)^2*x^6*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 171*c^2*d*(b*c - a*d)^2*x^9*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 486*c*d^2*(b*c - a*d)^2*x^12*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 297*d^3*(b*c - a*d)^2*x^15*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*c^3*(b*c - a*d)^2*x^6*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 216*c^2*d*(b*c - a*d)^2*x^9*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 351*c*d^2*(b*c - a*d)^2*x^12*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 162*d^3*(b*c - a*d)^2*x^15*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*c^3*(b*c - a*d)^2*x^6*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 81*c^2*d*(b*c - a*d)^2*x^9*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 81*c*d^2*(b*c - a*d)^2*x^12*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*d^3*(b*c - a*d)^2*x^15*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^5*(-(b*c) + a*d)*x^8*(a + b*x^3)^(7/3))","C",0
760,1,278,351,5.8564095,"\int \frac{1}{x^9 \left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^9*(a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{\left(a+b x^3\right)^{2/3} \left(-\frac{x^6 \left(20 a^2 d^2+32 a b c d+41 b^2 c^2\right)}{c^3}-\frac{5 a^2}{c}+\frac{40 b^4 x^9}{\left(a+b x^3\right) (a d-b c)}+\frac{2 a x^3 (4 a d+7 b c)}{c^2}\right)}{40 a^4 x^8}+\frac{d^4 \left(\log \left(\frac{\sqrt[3]{c} x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}+\frac{x^2 (b c-a d)^{2/3}}{\left(a x^3+b\right)^{2/3}}+c^{2/3}\right)-2 \log \left(\sqrt[3]{c}-\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{a x^3+b}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)\right)}{6 c^{11/3} (b c-a d)^{4/3}}","\frac{\left(a+b x^3\right)^{2/3} (9 b c-4 a d) (a d+3 b c)}{20 a^3 c^2 x^5 (b c-a d)}-\frac{\left(a+b x^3\right)^{2/3} (9 b c-a d)}{8 a^2 c x^8 (b c-a d)}-\frac{\left(a+b x^3\right)^{2/3} \left(-20 a^3 d^3-12 a^2 b c d^2-9 a b^2 c^2 d+81 b^3 c^3\right)}{40 a^4 c^3 x^2 (b c-a d)}+\frac{d^4 \log \left(c+d x^3\right)}{6 c^{11/3} (b c-a d)^{4/3}}-\frac{d^4 \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{2 c^{11/3} (b c-a d)^{4/3}}+\frac{d^4 \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{11/3} (b c-a d)^{4/3}}+\frac{b}{a x^8 \sqrt[3]{a+b x^3} (b c-a d)}",1,"((a + b*x^3)^(2/3)*((-5*a^2)/c + (2*a*(7*b*c + 4*a*d)*x^3)/c^2 - ((41*b^2*c^2 + 32*a*b*c*d + 20*a^2*d^2)*x^6)/c^3 + (40*b^4*x^9)/((-(b*c) + a*d)*(a + b*x^3))))/(40*a^4*x^8) + (d^4*(2*Sqrt[3]*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] - 2*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)] + Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(6*c^(11/3)*(b*c - a*d)^(4/3))","A",0
761,1,194,67,0.2903171,"\int \frac{x^{10}}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x^10/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{x^2 \left(2 x^3 \sqrt[3]{\frac{b x^3}{a}+1} \left(5 a^2 d^2-a b c d-2 b^2 c^2\right) F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 c \left(-5 a^2 d+a b \left(c-d x^3\right)+b^2 c x^3\right)+5 a c \sqrt[3]{\frac{b x^3}{a}+1} (5 a d-b c) F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}{20 b^2 c d \sqrt[3]{a+b x^3} (b c-a d)}","\frac{x^{11} \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{11}{3};\frac{4}{3},1;\frac{14}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{11 a c \sqrt[3]{a+b x^3}}",1,"(x^2*(5*c*(-5*a^2*d + b^2*c*x^3 + a*b*(c - d*x^3)) + 5*a*c*(-(b*c) + 5*a*d)*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*(-2*b^2*c^2 - a*b*c*d + 5*a^2*d^2)*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)]))/(20*b^2*c*d*(b*c - a*d)*(a + b*x^3)^(1/3))","B",0
762,1,144,67,0.1350968,"\int \frac{x^7}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x^7/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{x^5 \sqrt[3]{\frac{b x^3}{a}+1} (b c-2 a d) F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-5 a c x^2 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 a c x^2}{5 b c \sqrt[3]{a+b x^3} (b c-a d)}","\frac{x^8 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{8}{3};\frac{4}{3},1;\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{8 a c \sqrt[3]{a+b x^3}}",1,"(5*a*c*x^2 - 5*a*c*x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] + (b*c - 2*a*d)*x^5*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(5*b*c*(b*c - a*d)*(a + b*x^3)^(1/3))","B",0
763,1,129,67,0.1167186,"\int \frac{x^4}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x^4/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{x^2 \left(d x^3 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 c \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-5 c\right)}{5 c \sqrt[3]{a+b x^3} (b c-a d)}","\frac{x^5 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{4}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{5 a c \sqrt[3]{a+b x^3}}",1,"(x^2*(-5*c + 5*c*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] + d*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)]))/(5*c*(b*c - a*d)*(a + b*x^3)^(1/3))","A",0
764,1,141,67,0.1262297,"\int \frac{x}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[x/((a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{x^2 \left(2 b d x^3 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 \sqrt[3]{\frac{b x^3}{a}+1} (a d+b c) F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-10 b c\right)}{10 a c \sqrt[3]{a+b x^3} (a d-b c)}","\frac{x^2 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{4}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{2 a c \sqrt[3]{a+b x^3}}",1,"(x^2*(-10*b*c + 5*(b*c + a*d)*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*d*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)]))/(10*a*c*(-(b*c) + a*d)*(a + b*x^3)^(1/3))","B",0
765,1,193,65,0.238116,"\int \frac{1}{x^2 \left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^2*(a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{-5 x^3 \sqrt[3]{\frac{b x^3}{a}+1} \left(a^2 d^2-a b c d+2 b^2 c^2\right) F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+10 c \left(-a^2 d+a b \left(c-d x^3\right)+2 b^2 c x^3\right)+2 b d x^6 \sqrt[3]{\frac{b x^3}{a}+1} (a d-2 b c) F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{10 a^2 c^2 x \sqrt[3]{a+b x^3} (a d-b c)}","-\frac{\sqrt[3]{\frac{b x^3}{a}+1} F_1\left(-\frac{1}{3};\frac{4}{3},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a c x \sqrt[3]{a+b x^3}}",1,"(10*c*(-(a^2*d) + 2*b^2*c*x^3 + a*b*(c - d*x^3)) - 5*(2*b^2*c^2 - a*b*c*d + a^2*d^2)*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*d*(-2*b*c + a*d)*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(10*a^2*c^2*(-(b*c) + a*d)*x*(a + b*x^3)^(1/3))","B",0
766,1,264,67,0.3714238,"\int \frac{1}{x^5 \left(a+b x^3\right)^{4/3} \left(c+d x^3\right)} \, dx","Integrate[1/(x^5*(a + b*x^3)^(4/3)*(c + d*x^3)),x]","\frac{-2 b d x^9 \sqrt[3]{\frac{b x^3}{a}+1} \left(2 a^2 d^2+a b c d-5 b^2 c^2\right) F_1\left(\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 x^6 \sqrt[3]{\frac{b x^3}{a}+1} \left(2 a^3 d^3-2 a^2 b c d^2-a b^2 c^2 d+5 b^3 c^3\right) F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+5 c \left(a^3 d \left(4 d x^3-c\right)+a^2 b \left(c^2+c d x^3+4 d^2 x^6\right)+a b^2 c x^3 \left(2 d x^3-5 c\right)-10 b^3 c^2 x^6\right)}{20 a^3 c^3 x^4 \sqrt[3]{a+b x^3} (a d-b c)}","-\frac{\sqrt[3]{\frac{b x^3}{a}+1} F_1\left(-\frac{4}{3};\frac{4}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{4 a c x^4 \sqrt[3]{a+b x^3}}",1,"(5*c*(-10*b^3*c^2*x^6 + a*b^2*c*x^3*(-5*c + 2*d*x^3) + a^3*d*(-c + 4*d*x^3) + a^2*b*(c^2 + c*d*x^3 + 4*d^2*x^6)) + 5*(5*b^3*c^3 - a*b^2*c^2*d - 2*a^2*b*c*d^2 + 2*a^3*d^3)*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d*x^3)/c)] - 2*b*d*(-5*b^2*c^2 + a*b*c*d + 2*a^2*d^2)*x^9*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(20*a^3*c^3*(-(b*c) + a*d)*x^4*(a + b*x^3)^(1/3))","B",0
767,1,92,90,0.0660549,"\int \frac{x^{15}}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x^15/((a + b*x^4)*(c + d*x^4)),x]","-\frac{a^3 \log \left(a+b x^4\right)}{4 b^3 (b c-a d)}+\frac{x^4 (-a d-b c)}{4 b^2 d^2}+\frac{c^3 \log \left(c+d x^4\right)}{4 d^3 (b c-a d)}+\frac{x^8}{8 b d}","-\frac{a^3 \log \left(a+b x^4\right)}{4 b^3 (b c-a d)}-\frac{x^4 (a d+b c)}{4 b^2 d^2}+\frac{c^3 \log \left(c+d x^4\right)}{4 d^3 (b c-a d)}+\frac{x^8}{8 b d}",1,"((-(b*c) - a*d)*x^4)/(4*b^2*d^2) + x^8/(8*b*d) - (a^3*Log[a + b*x^4])/(4*b^3*(b*c - a*d)) + (c^3*Log[c + d*x^4])/(4*d^3*(b*c - a*d))","A",1
768,1,66,70,0.0489884,"\int \frac{x^{11}}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x^11/((a + b*x^4)*(c + d*x^4)),x]","\frac{a^2 d^2 \log \left(a+b x^4\right)-b \left(d x^4 (a d-b c)+b c^2 \log \left(c+d x^4\right)\right)}{4 b^2 d^2 (b c-a d)}","\frac{a^2 \log \left(a+b x^4\right)}{4 b^2 (b c-a d)}-\frac{c^2 \log \left(c+d x^4\right)}{4 d^2 (b c-a d)}+\frac{x^4}{4 b d}",1,"(a^2*d^2*Log[a + b*x^4] - b*(d*(-(b*c) + a*d)*x^4 + b*c^2*Log[c + d*x^4]))/(4*b^2*d^2*(b*c - a*d))","A",1
769,1,43,53,0.0255346,"\int \frac{x^7}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x^7/((a + b*x^4)*(c + d*x^4)),x]","-\frac{a d \log \left(a+b x^4\right)-b c \log \left(c+d x^4\right)}{4 b^2 c d-4 a b d^2}","\frac{c \log \left(c+d x^4\right)}{4 d (b c-a d)}-\frac{a \log \left(a+b x^4\right)}{4 b (b c-a d)}",1,"-((a*d*Log[a + b*x^4] - b*c*Log[c + d*x^4])/(4*b^2*c*d - 4*a*b*d^2))","A",1
770,1,31,45,0.0204328,"\int \frac{x^3}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x^3/((a + b*x^4)*(c + d*x^4)),x]","\frac{\log \left(a+b x^4\right)-\log \left(c+d x^4\right)}{4 b c-4 a d}","\frac{\log \left(a+b x^4\right)}{4 (b c-a d)}-\frac{\log \left(c+d x^4\right)}{4 (b c-a d)}",1,"(Log[a + b*x^4] - Log[c + d*x^4])/(4*b*c - 4*a*d)","A",1
771,1,54,62,0.0334611,"\int \frac{1}{x \left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[1/(x*(a + b*x^4)*(c + d*x^4)),x]","\frac{-b c \log \left(a+b x^4\right)+a d \log \left(c+d x^4\right)-4 a d \log (x)+4 b c \log (x)}{4 a b c^2-4 a^2 c d}","-\frac{b \log \left(a+b x^4\right)}{4 a (b c-a d)}+\frac{d \log \left(c+d x^4\right)}{4 c (b c-a d)}+\frac{\log (x)}{a c}",1,"(4*b*c*Log[x] - 4*a*d*Log[x] - b*c*Log[a + b*x^4] + a*d*Log[c + d*x^4])/(4*a*b*c^2 - 4*a^2*c*d)","A",1
772,1,88,87,0.0446103,"\int \frac{1}{x^5 \left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[1/(x^5*(a + b*x^4)*(c + d*x^4)),x]","-\frac{b^2 \log \left(a+b x^4\right)}{4 a^2 (a d-b c)}+\frac{\log (x) (-a d-b c)}{a^2 c^2}-\frac{d^2 \log \left(c+d x^4\right)}{4 c^2 (b c-a d)}-\frac{1}{4 a c x^4}","\frac{b^2 \log \left(a+b x^4\right)}{4 a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log \left(c+d x^4\right)}{4 c^2 (b c-a d)}-\frac{1}{4 a c x^4}",1,"-1/4*1/(a*c*x^4) + ((-(b*c) - a*d)*Log[x])/(a^2*c^2) - (b^2*Log[a + b*x^4])/(4*a^2*(-(b*c) + a*d)) - (d^2*Log[c + d*x^4])/(4*c^2*(b*c - a*d))","A",1
773,1,104,112,0.1699811,"\int \frac{x^{13}}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x^13/((a + b*x^4)*(c + d*x^4)),x]","\frac{1}{6} \left(\frac{3 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{b^{5/2} (a d-b c)}+\frac{x^2 \left(-3 a d-3 b c+b d x^4\right)}{b^2 d^2}+\frac{3 c^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c}}\right)}{d^{5/2} (b c-a d)}\right)","-\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 b^{5/2} (b c-a d)}-\frac{x^2 (a d+b c)}{2 b^2 d^2}+\frac{c^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c}}\right)}{2 d^{5/2} (b c-a d)}+\frac{x^6}{6 b d}",1,"((x^2*(-3*b*c - 3*a*d + b*d*x^4))/(b^2*d^2) + (3*a^(5/2)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(b^(5/2)*(-(b*c) + a*d)) + (3*c^(5/2)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(d^(5/2)*(b*c - a*d)))/6","A",1
774,1,82,92,0.1413623,"\int \frac{x^9}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x^9/((a + b*x^4)*(c + d*x^4)),x]","\frac{\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{b^{3/2}}+x^2 \left(\frac{c}{d}-\frac{a}{b}\right)-\frac{c^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c}}\right)}{d^{3/2}}}{2 b c-2 a d}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 b^{3/2} (b c-a d)}-\frac{c^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c}}\right)}{2 d^{3/2} (b c-a d)}+\frac{x^2}{2 b d}",1,"((-(a/b) + c/d)*x^2 + (a^(3/2)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/b^(3/2) - (c^(3/2)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/d^(3/2))/(2*b*c - 2*a*d)","A",1
775,1,66,79,0.0406545,"\int \frac{x^5}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x^5/((a + b*x^4)*(c + d*x^4)),x]","\frac{\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c}}\right)}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{\sqrt{b}}}{2 b c-2 a d}","\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c}}\right)}{2 \sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{b} (b c-a d)}",1,"(-((Sqrt[a]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[b]) + (Sqrt[c]*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/Sqrt[d])/(2*b*c - 2*a*d)","A",1
776,1,66,79,0.0499129,"\int \frac{x}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x/((a + b*x^4)*(c + d*x^4)),x]","\frac{\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c}}\right)}{\sqrt{c}}}{2 b c-2 a d}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 \sqrt{a} (b c-a d)}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c}}\right)}{2 \sqrt{c} (b c-a d)}",1,"((Sqrt[b]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[a] - (Sqrt[d]*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/Sqrt[c])/(2*b*c - 2*a*d)","A",1
777,1,169,92,0.2296205,"\int \frac{1}{x^3 \left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[1/(x^3*(a + b*x^4)*(c + d*x^4)),x]","\frac{-\frac{b^{3/2} x^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{a^{3/2}}-\frac{b^{3/2} x^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{a^{3/2}}+\frac{b}{a}+\frac{d^{3/2} x^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{c^{3/2}}+\frac{d^{3/2} x^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{c^{3/2}}-\frac{d}{c}}{2 x^2 (a d-b c)}","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 a^{3/2} (b c-a d)}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c}}\right)}{2 c^{3/2} (b c-a d)}-\frac{1}{2 a c x^2}",1,"(b/a - d/c - (b^(3/2)*x^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/2) - (b^(3/2)*x^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/2) + (d^(3/2)*x^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(3/2) + (d^(3/2)*x^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(3/2))/(2*(-(b*c) + a*d)*x^2)","A",1
778,1,193,112,0.2618952,"\int \frac{1}{x^7 \left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[1/(x^7*(a + b*x^4)*(c + d*x^4)),x]","\frac{\frac{3 b^{5/2} x^6 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{a^{5/2}}+\frac{3 b^{5/2} x^6 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{a^{5/2}}-\frac{3 b^2 x^4}{a^2}+\frac{b}{a}-\frac{3 d^{5/2} x^6 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{c^{5/2}}-\frac{3 d^{5/2} x^6 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{c^{5/2}}+\frac{3 d^2 x^4}{c^2}-\frac{d}{c}}{6 x^6 (a d-b c)}","\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)}{2 a^{5/2} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c}}\right)}{2 c^{5/2} (b c-a d)}-\frac{1}{6 a c x^6}",1,"(b/a - d/c - (3*b^2*x^4)/a^2 + (3*d^2*x^4)/c^2 + (3*b^(5/2)*x^6*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(5/2) + (3*b^(5/2)*x^6*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(5/2) - (3*d^(5/2)*x^6*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(5/2) - (3*d^(5/2)*x^6*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(5/2))/(6*(-(b*c) + a*d)*x^6)","A",1
779,1,377,457,0.2450947,"\int \frac{x^8}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x^8/((a + b*x^4)*(c + d*x^4)),x]","\frac{-\frac{\sqrt{2} a^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{5/4}}+\frac{\sqrt{2} a^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{b^{5/4}}-\frac{2 \sqrt{2} a^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{b^{5/4}}+\frac{2 \sqrt{2} a^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{b^{5/4}}-\frac{8 a x}{b}+\frac{\sqrt{2} c^{5/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{d^{5/4}}-\frac{\sqrt{2} c^{5/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{d^{5/4}}+\frac{2 \sqrt{2} c^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{d^{5/4}}-\frac{2 \sqrt{2} c^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{d^{5/4}}+\frac{8 c x}{d}}{8 b c-8 a d}","-\frac{a^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} b^{5/4} (b c-a d)}+\frac{a^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} b^{5/4} (b c-a d)}-\frac{a^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} b^{5/4} (b c-a d)}+\frac{a^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} b^{5/4} (b c-a d)}+\frac{c^{5/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} d^{5/4} (b c-a d)}-\frac{c^{5/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} d^{5/4} (b c-a d)}+\frac{c^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{2 \sqrt{2} d^{5/4} (b c-a d)}-\frac{c^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} d^{5/4} (b c-a d)}+\frac{x}{b d}",1,"((-8*a*x)/b + (8*c*x)/d - (2*Sqrt[2]*a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(5/4) + (2*Sqrt[2]*a^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(5/4) + (2*Sqrt[2]*c^(5/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/d^(5/4) - (2*Sqrt[2]*c^(5/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/d^(5/4) - (Sqrt[2]*a^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(5/4) + (Sqrt[2]*a^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(5/4) + (Sqrt[2]*c^(5/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/d^(5/4) - (Sqrt[2]*c^(5/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/d^(5/4))/(8*b*c - 8*a*d)","A",1
780,1,340,449,0.1214535,"\int \frac{x^6}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x^6/((a + b*x^4)*(c + d*x^4)),x]","\frac{-a^{3/4} d^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+a^{3/4} d^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+2 a^{3/4} d^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)-2 a^{3/4} d^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)+b^{3/4} c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)-b^{3/4} c^{3/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)-2 b^{3/4} c^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)+2 b^{3/4} c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} b^{3/4} d^{3/4} (b c-a d)}","-\frac{a^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} b^{3/4} (b c-a d)}+\frac{a^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} b^{3/4} (b c-a d)}+\frac{a^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} b^{3/4} (b c-a d)}-\frac{a^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} b^{3/4} (b c-a d)}+\frac{c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} d^{3/4} (b c-a d)}-\frac{c^{3/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} d^{3/4} (b c-a d)}-\frac{c^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{2 \sqrt{2} d^{3/4} (b c-a d)}+\frac{c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} d^{3/4} (b c-a d)}",1,"(2*a^(3/4)*d^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 2*a^(3/4)*d^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 2*b^(3/4)*c^(3/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 2*b^(3/4)*c^(3/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - a^(3/4)*d^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + a^(3/4)*d^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + b^(3/4)*c^(3/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] - b^(3/4)*c^(3/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*b^(3/4)*d^(3/4)*(b*c - a*d))","A",1
781,1,340,449,0.1127903,"\int \frac{x^4}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x^4/((a + b*x^4)*(c + d*x^4)),x]","\frac{\sqrt[4]{a} \sqrt[4]{d} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-\sqrt[4]{a} \sqrt[4]{d} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+2 \sqrt[4]{a} \sqrt[4]{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)-2 \sqrt[4]{a} \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)-\sqrt[4]{b} \sqrt[4]{c} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)+\sqrt[4]{b} \sqrt[4]{c} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)-2 \sqrt[4]{b} \sqrt[4]{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)+2 \sqrt[4]{b} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} \sqrt[4]{b} \sqrt[4]{d} (b c-a d)}","\frac{\sqrt[4]{a} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} \sqrt[4]{b} (b c-a d)}-\frac{\sqrt[4]{a} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} \sqrt[4]{b} (b c-a d)}-\frac{\sqrt[4]{c} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} \sqrt[4]{d} (b c-a d)}+\frac{\sqrt[4]{c} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} \sqrt[4]{d} (b c-a d)}+\frac{\sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} \sqrt[4]{b} (b c-a d)}-\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} \sqrt[4]{b} (b c-a d)}-\frac{\sqrt[4]{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{2 \sqrt{2} \sqrt[4]{d} (b c-a d)}+\frac{\sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} \sqrt[4]{d} (b c-a d)}",1,"(2*a^(1/4)*d^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 2*a^(1/4)*d^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 2*b^(1/4)*c^(1/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 2*b^(1/4)*c^(1/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + a^(1/4)*d^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - a^(1/4)*d^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - b^(1/4)*c^(1/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + b^(1/4)*c^(1/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*b^(1/4)*d^(1/4)*(b*c - a*d))","A",1
782,1,340,449,0.1391208,"\int \frac{x^2}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[x^2/((a + b*x^4)*(c + d*x^4)),x]","\frac{\sqrt[4]{b} \sqrt[4]{c} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-\sqrt[4]{b} \sqrt[4]{c} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-2 \sqrt[4]{b} \sqrt[4]{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+2 \sqrt[4]{b} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)-\sqrt[4]{a} \sqrt[4]{d} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)+\sqrt[4]{a} \sqrt[4]{d} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)+2 \sqrt[4]{a} \sqrt[4]{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)-2 \sqrt[4]{a} \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{4 \sqrt{2} \sqrt[4]{a} \sqrt[4]{c} (b c-a d)}","\frac{\sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)}-\frac{\sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} (b c-a d)}-\frac{\sqrt[4]{d} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)}+\frac{\sqrt[4]{d} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} \sqrt[4]{c} (b c-a d)}-\frac{\sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} \sqrt[4]{a} (b c-a d)}+\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} \sqrt[4]{a} (b c-a d)}+\frac{\sqrt[4]{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{2 \sqrt{2} \sqrt[4]{c} (b c-a d)}-\frac{\sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} \sqrt[4]{c} (b c-a d)}",1,"(-2*b^(1/4)*c^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*b^(1/4)*c^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*a^(1/4)*d^(1/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - 2*a^(1/4)*d^(1/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + b^(1/4)*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - b^(1/4)*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - a^(1/4)*d^(1/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + a^(1/4)*d^(1/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*a^(1/4)*c^(1/4)*(b*c - a*d))","A",1
783,1,340,449,0.196047,"\int \frac{1}{\left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[1/((a + b*x^4)*(c + d*x^4)),x]","\frac{a^{3/4} d^{3/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)-a^{3/4} d^{3/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)+2 a^{3/4} d^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)-2 a^{3/4} d^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)-b^{3/4} c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+b^{3/4} c^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-2 b^{3/4} c^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+2 b^{3/4} c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{3/4} c^{3/4} (b c-a d)}","-\frac{b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{3/4} (b c-a d)}-\frac{b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} (b c-a d)}+\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} (b c-a d)}+\frac{d^{3/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{3/4} (b c-a d)}+\frac{d^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{2 \sqrt{2} c^{3/4} (b c-a d)}-\frac{d^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} c^{3/4} (b c-a d)}",1,"(-2*b^(3/4)*c^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*b^(3/4)*c^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*a^(3/4)*d^(3/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - 2*a^(3/4)*d^(3/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - b^(3/4)*c^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + b^(3/4)*c^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + a^(3/4)*d^(3/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] - a^(3/4)*d^(3/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(4*Sqrt[2]*a^(3/4)*c^(3/4)*(b*c - a*d))","A",1
784,1,385,460,0.2199977,"\int \frac{1}{x^2 \left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[1/(x^2*(a + b*x^4)*(c + d*x^4)),x]","\frac{\frac{\sqrt{2} b^{5/4} x \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{5/4}}-\frac{\sqrt{2} b^{5/4} x \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{5/4}}-\frac{2 \sqrt{2} b^{5/4} x \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{a^{5/4}}+\frac{2 \sqrt{2} b^{5/4} x \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{a^{5/4}}+\frac{8 b}{a}-\frac{\sqrt{2} d^{5/4} x \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{c^{5/4}}+\frac{\sqrt{2} d^{5/4} x \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{c^{5/4}}+\frac{2 \sqrt{2} d^{5/4} x \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{c^{5/4}}-\frac{2 \sqrt{2} d^{5/4} x \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{c^{5/4}}-\frac{8 d}{c}}{8 a d x-8 b c x}","-\frac{b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{5/4} (b c-a d)}-\frac{b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{2 \sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{1}{a c x}",1,"((8*b)/a - (8*d)/c - (2*Sqrt[2]*b^(5/4)*x*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(5/4) + (2*Sqrt[2]*b^(5/4)*x*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(5/4) + (2*Sqrt[2]*d^(5/4)*x*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(5/4) - (2*Sqrt[2]*d^(5/4)*x*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(5/4) + (Sqrt[2]*b^(5/4)*x*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(5/4) - (Sqrt[2]*b^(5/4)*x*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(5/4) - (Sqrt[2]*d^(5/4)*x*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c^(5/4) + (Sqrt[2]*d^(5/4)*x*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c^(5/4))/(-8*b*c*x + 8*a*d*x)","A",1
785,1,406,462,0.2625057,"\int \frac{1}{x^4 \left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[1/(x^4*(a + b*x^4)*(c + d*x^4)),x]","\frac{-\frac{6 \sqrt{2} b^{7/4} x^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{a^{7/4}}+\frac{6 \sqrt{2} b^{7/4} x^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{a^{7/4}}-\frac{3 \sqrt{2} b^{7/4} x^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}+\frac{3 \sqrt{2} b^{7/4} x^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}+\frac{8 b}{a}+\frac{6 \sqrt{2} d^{7/4} x^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{c^{7/4}}-\frac{6 \sqrt{2} d^{7/4} x^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{c^{7/4}}+\frac{3 \sqrt{2} d^{7/4} x^3 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{c^{7/4}}-\frac{3 \sqrt{2} d^{7/4} x^3 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{c^{7/4}}-\frac{8 d}{c}}{24 x^3 (a d-b c)}","\frac{b^{7/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{7/4} (b c-a d)}-\frac{b^{7/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{7/4} (b c-a d)}+\frac{b^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{7/4} (b c-a d)}-\frac{b^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{7/4} (b c-a d)}-\frac{d^{7/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{7/4} (b c-a d)}-\frac{d^{7/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{2 \sqrt{2} c^{7/4} (b c-a d)}+\frac{d^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} c^{7/4} (b c-a d)}-\frac{1}{3 a c x^3}",1,"((8*b)/a - (8*d)/c - (6*Sqrt[2]*b^(7/4)*x^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (6*Sqrt[2]*b^(7/4)*x^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (6*Sqrt[2]*d^(7/4)*x^3*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(7/4) - (6*Sqrt[2]*d^(7/4)*x^3*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(7/4) - (3*Sqrt[2]*b^(7/4)*x^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (3*Sqrt[2]*b^(7/4)*x^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (3*Sqrt[2]*d^(7/4)*x^3*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c^(7/4) - (3*Sqrt[2]*d^(7/4)*x^3*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c^(7/4))/(24*(-(b*c) + a*d)*x^3)","A",1
786,1,428,479,0.3784759,"\int \frac{1}{x^6 \left(a+b x^4\right) \left(c+d x^4\right)} \, dx","Integrate[1/(x^6*(a + b*x^4)*(c + d*x^4)),x]","\frac{\frac{10 \sqrt{2} b^{9/4} x^5 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{a^{9/4}}-\frac{10 \sqrt{2} b^{9/4} x^5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{a^{9/4}}-\frac{5 \sqrt{2} b^{9/4} x^5 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{9/4}}+\frac{5 \sqrt{2} b^{9/4} x^5 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{9/4}}-\frac{40 b^2 x^4}{a^2}+\frac{8 b}{a}-\frac{10 \sqrt{2} d^{9/4} x^5 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{c^{9/4}}+\frac{10 \sqrt{2} d^{9/4} x^5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{c^{9/4}}+\frac{5 \sqrt{2} d^{9/4} x^5 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{c^{9/4}}-\frac{5 \sqrt{2} d^{9/4} x^5 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{c^{9/4}}+\frac{40 d^2 x^4}{c^2}-\frac{8 d}{c}}{40 x^5 (a d-b c)}","\frac{b^{9/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{9/4} (b c-a d)}-\frac{b^{9/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{4 \sqrt{2} a^{9/4} (b c-a d)}-\frac{b^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{9/4} (b c-a d)}+\frac{b^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{9/4} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{9/4} \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{9/4} (b c-a d)}+\frac{d^{9/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{2 \sqrt{2} c^{9/4} (b c-a d)}-\frac{d^{9/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} c^{9/4} (b c-a d)}-\frac{1}{5 a c x^5}",1,"((8*b)/a - (8*d)/c - (40*b^2*x^4)/a^2 + (40*d^2*x^4)/c^2 + (10*Sqrt[2]*b^(9/4)*x^5*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(9/4) - (10*Sqrt[2]*b^(9/4)*x^5*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(9/4) - (10*Sqrt[2]*d^(9/4)*x^5*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(9/4) + (10*Sqrt[2]*d^(9/4)*x^5*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(9/4) - (5*Sqrt[2]*b^(9/4)*x^5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(9/4) + (5*Sqrt[2]*b^(9/4)*x^5*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(9/4) + (5*Sqrt[2]*d^(9/4)*x^5*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c^(9/4) - (5*Sqrt[2]*d^(9/4)*x^5*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c^(9/4))/(40*(-(b*c) + a*d)*x^5)","A",1
787,1,88,93,0.1094202,"\int \frac{x^7 \sqrt{c+d x^4}}{a+b x^4} \, dx","Integrate[(x^7*Sqrt[c + d*x^4])/(a + b*x^4),x]","\frac{a \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 b^{5/2}}+\frac{\sqrt{c+d x^4} \left(b \left(c+d x^4\right)-3 a d\right)}{6 b^2 d}","\frac{a \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 b^{5/2}}-\frac{a \sqrt{c+d x^4}}{2 b^2}+\frac{\left(c+d x^4\right)^{3/2}}{6 b d}",1,"(Sqrt[c + d*x^4]*(-3*a*d + b*(c + d*x^4)))/(6*b^2*d) + (a*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(2*b^(5/2))","A",1
788,1,114,120,0.1915495,"\int \frac{x^5 \sqrt{c+d x^4}}{a+b x^4} \, dx","Integrate[(x^5*Sqrt[c + d*x^4])/(a + b*x^4),x]","\frac{\frac{(b c-2 a d) \log \left(\sqrt{d} \sqrt{c+d x^4}+d x^2\right)}{\sqrt{d}}-2 \sqrt{a} \sqrt{b c-a d} \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)+b x^2 \sqrt{c+d x^4}}{4 b^2}","-\frac{\sqrt{a} \sqrt{b c-a d} \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{2 b^2}+\frac{(b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right)}{4 b^2 \sqrt{d}}+\frac{x^2 \sqrt{c+d x^4}}{4 b}",1,"(b*x^2*Sqrt[c + d*x^4] - 2*Sqrt[a]*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])] + ((b*c - 2*a*d)*Log[d*x^2 + Sqrt[d]*Sqrt[c + d*x^4]])/Sqrt[d])/(4*b^2)","A",1
789,1,69,70,0.0511613,"\int \frac{x^3 \sqrt{c+d x^4}}{a+b x^4} \, dx","Integrate[(x^3*Sqrt[c + d*x^4])/(a + b*x^4),x]","\frac{1}{2} \left(\frac{\sqrt{c+d x^4}}{b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{b^{3/2}}\right)","\frac{\sqrt{c+d x^4}}{2 b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 b^{3/2}}",1,"(Sqrt[c + d*x^4]/b - (Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/b^(3/2))/2","A",1
790,1,89,91,0.0650126,"\int \frac{x \sqrt{c+d x^4}}{a+b x^4} \, dx","Integrate[(x*Sqrt[c + d*x^4])/(a + b*x^4),x]","\frac{\frac{\sqrt{b c-a d} \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{\sqrt{a}}+\sqrt{d} \log \left(\sqrt{d} \sqrt{c+d x^4}+d x^2\right)}{2 b}","\frac{\sqrt{b c-a d} \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{2 \sqrt{a} b}+\frac{\sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right)}{2 b}",1,"((Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/Sqrt[a] + Sqrt[d]*Log[d*x^2 + Sqrt[d]*Sqrt[c + d*x^4]])/(2*b)","A",1
791,1,81,85,0.0939898,"\int \frac{\sqrt{c+d x^4}}{x \left(a+b x^4\right)} \, dx","Integrate[Sqrt[c + d*x^4]/(x*(a + b*x^4)),x]","\frac{\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{\sqrt{b}}-\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{2 a}","\frac{\sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 a \sqrt{b}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{2 a}",1,"(-(Sqrt[c]*ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]]) + (Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/Sqrt[b])/(2*a)","A",1
792,1,53,76,0.0637107,"\int \frac{\sqrt{c+d x^4}}{x^3 \left(a+b x^4\right)} \, dx","Integrate[Sqrt[c + d*x^4]/(x^3*(a + b*x^4)),x]","-\frac{\sqrt{c+d x^4} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{(a d-b c) x^4}{a \left(d x^4+c\right)}\right)}{2 a x^2}","-\frac{\sqrt{b c-a d} \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{2 a^{3/2}}-\frac{\sqrt{c+d x^4}}{2 a x^2}",1,"-1/2*(Sqrt[c + d*x^4]*Hypergeometric2F1[-1/2, 1, 1/2, ((-(b*c) + a*d)*x^4)/(a*(c + d*x^4))])/(a*x^2)","C",1
793,1,107,115,0.1341756,"\int \frac{\sqrt{c+d x^4}}{x^5 \left(a+b x^4\right)} \, dx","Integrate[Sqrt[c + d*x^4]/(x^5*(a + b*x^4)),x]","\frac{\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{\sqrt{c}}-2 \sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)-\frac{a \sqrt{c+d x^4}}{x^4}}{4 a^2}","\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{4 a^2 \sqrt{c}}-\frac{\sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 a^2}-\frac{\sqrt{c+d x^4}}{4 a x^4}",1,"(-((a*Sqrt[c + d*x^4])/x^4) + ((2*b*c - a*d)*ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]])/Sqrt[c] - 2*Sqrt[b]*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(4*a^2)","A",1
794,1,130,110,5.3303397,"\int \frac{\sqrt{c+d x^4}}{x^7 \left(a+b x^4\right)} \, dx","Integrate[Sqrt[c + d*x^4]/(x^7*(a + b*x^4)),x]","\frac{3 b c^2 x^4 \sqrt{\frac{d x^4}{c}+1} \sqrt{x^4 \left(\frac{b}{a}-\frac{d}{c}\right)} \sin ^{-1}\left(\frac{\sqrt{x^4 \left(\frac{b}{a}-\frac{d}{c}\right)}}{\sqrt{\frac{b x^4}{a}+1}}\right)+\left(c+d x^4\right) \left(3 b c x^4-a \left(c+d x^4\right)\right)}{6 a^2 c x^6 \sqrt{c+d x^4}}","\frac{b \sqrt{b c-a d} \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{2 a^{5/2}}+\frac{\sqrt{c+d x^4} (3 b c-a d)}{6 a^2 c x^2}-\frac{\sqrt{c+d x^4}}{6 a x^6}",1,"((c + d*x^4)*(3*b*c*x^4 - a*(c + d*x^4)) + 3*b*c^2*x^4*Sqrt[(b/a - d/c)*x^4]*Sqrt[1 + (d*x^4)/c]*ArcSin[Sqrt[(b/a - d/c)*x^4]/Sqrt[1 + (b*x^4)/a]])/(6*a^2*c*x^6*Sqrt[c + d*x^4])","A",0
795,1,141,857,0.1154556,"\int \frac{x^6 \sqrt{c+d x^4}}{a+b x^4} \, dx","Integrate[(x^6*Sqrt[c + d*x^4])/(a + b*x^4),x]","\frac{x^7 \sqrt{\frac{d x^4}{c}+1} (2 b c-5 a d) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)-7 a c x^3 \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+7 a x^3 \left(c+d x^4\right)}{35 a b \sqrt{c+d x^4}}","\frac{\sqrt{d x^4+c} x^3}{5 b}+\frac{(2 b c-5 a d) \sqrt{d x^4+c} x}{5 b^2 \sqrt{d} \left(\sqrt{d} x^2+\sqrt{c}\right)}-\frac{a \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right)}{4 b^2}-\frac{a \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right)}{4 b^2}-\frac{\sqrt[4]{c} (2 b c-5 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{5 b^2 d^{3/4} \sqrt{d x^4+c}}+\frac{\sqrt[4]{c} \left(b^2 c^2+a b d c-5 a^2 d^2\right) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{5 b^2 d^{3/4} (b c+a d) \sqrt{d x^4+c}}+\frac{a \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right) (b c-a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b^{5/2} \sqrt[4]{c} \left(\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right) \sqrt[4]{d} \sqrt{d x^4+c}}-\frac{a \left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right) (b c-a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\sqrt{c} \left(\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b^{5/2} \sqrt[4]{c} \left(\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right) \sqrt[4]{d} \sqrt{d x^4+c}}",1,"(7*a*x^3*(c + d*x^4) - 7*a*c*x^3*Sqrt[1 + (d*x^4)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)] + (2*b*c - 5*a*d)*x^7*Sqrt[1 + (d*x^4)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(35*a*b*Sqrt[c + d*x^4])","C",0
796,1,241,700,0.4758048,"\int \frac{x^4 \sqrt{c+d x^4}}{a+b x^4} \, dx","Integrate[(x^4*Sqrt[c + d*x^4])/(a + b*x^4),x]","\frac{x \left(5 \left(\frac{5 a^2 c^2 F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{\left(a+b x^4\right) \left(2 x^4 \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)}+c+d x^4\right)+\frac{x^4 \sqrt{\frac{d x^4}{c}+1} (2 b c-3 a d) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{a}\right)}{15 b \sqrt{c+d x^4}}","-\frac{(b c-a d) \tan ^{-1}\left(\frac{x \sqrt{\frac{\sqrt{-a} \left(\frac{b c}{a}-d\right)}{\sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 b^2 \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{(b c-a d) \tan ^{-1}\left(\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 b^2 \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right) (b c-a d) \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b^2 \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)}-\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right) (b c-a d) \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b^2 \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right)}+\frac{c^{3/4} \left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} (b c-2 a d) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{3 b \sqrt[4]{d} \sqrt{c+d x^4} (a d+b c)}+\frac{x \sqrt{c+d x^4}}{3 b}",1,"(x*(((2*b*c - 3*a*d)*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])/a + 5*(c + d*x^4 + (5*a^2*c^2*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)])/((a + b*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]))))))/(15*b*Sqrt[c + d*x^4])","C",0
797,1,65,786,0.0393909,"\int \frac{x^2 \sqrt{c+d x^4}}{a+b x^4} \, dx","Integrate[(x^2*Sqrt[c + d*x^4])/(a + b*x^4),x]","\frac{x^3 \sqrt{c+d x^4} F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{3 a \sqrt{\frac{c+d x^4}{c}}}","-\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right) (b c-a d) \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b^{3/2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right)}+\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right) (b c-a d) \Pi \left(-\frac{\sqrt{c} \left(\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b^{3/2} \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{-a} \sqrt{b} \sqrt{c}+a \sqrt{d}\right)}+\frac{a \sqrt[4]{c} d^{5/4} \left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{b \sqrt{c+d x^4} (a d+b c)}+\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{x \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 b}+\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 b}+\frac{\sqrt{d} x \sqrt{c+d x^4}}{b \left(\sqrt{c}+\sqrt{d} x^2\right)}-\frac{\sqrt[4]{c} \sqrt[4]{d} \left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{b \sqrt{c+d x^4}}",1,"(x^3*Sqrt[c + d*x^4]*AppellF1[3/4, -1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)])/(3*a*Sqrt[(c + d*x^4)/c])","C",0
798,1,161,679,0.2061488,"\int \frac{\sqrt{c+d x^4}}{a+b x^4} \, dx","Integrate[Sqrt[c + d*x^4]/(a + b*x^4),x]","\frac{5 a c x \sqrt{c+d x^4} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{\left(a+b x^4\right) \left(2 x^4 \left(a d F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)-2 b c F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)+5 a c F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)}","\frac{c^{3/4} d^{3/4} \left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{\sqrt{c+d x^4} (a d+b c)}+\frac{(b c-a d) \tan ^{-1}\left(\frac{x \sqrt{\frac{\sqrt{-a} \left(\frac{b c}{a}-d\right)}{\sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 a b \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{(b c-a d) \tan ^{-1}\left(\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 a b \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right) (b c-a d) \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a b \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)}+\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right) (b c-a d) \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a b \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right)}",1,"(5*a*c*x*Sqrt[c + d*x^4]*AppellF1[1/4, -1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)])/((a + b*x^4)*(5*a*c*AppellF1[1/4, -1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(-2*b*c*AppellF1[5/4, -1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])))","C",0
799,1,138,809,0.1323811,"\int \frac{\sqrt{c+d x^4}}{x^2 \left(a+b x^4\right)} \, dx","Integrate[Sqrt[c + d*x^4]/(x^2*(a + b*x^4)),x]","\frac{-7 x^4 \sqrt{\frac{d x^4}{c}+1} (b c-2 a d) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+3 b d x^8 \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)-21 a \left(c+d x^4\right)}{21 a^2 x \sqrt{c+d x^4}}","\frac{b \sqrt[4]{d} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) c^{5/4}}{a (b c+a d) \sqrt{d x^4+c}}-\frac{\sqrt[4]{d} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \sqrt[4]{c}}{a \sqrt{d x^4+c}}-\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right)}{4 a}-\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right)}{4 a}-\frac{\sqrt{d x^4+c}}{a x}+\frac{\sqrt{d} x \sqrt{d x^4+c}}{a \left(\sqrt{d} x^2+\sqrt{c}\right)}-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right) (b c-a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 \sqrt{b} \left(\sqrt{d} a^2+(-a)^{3/2} \sqrt{b} \sqrt{c}\right) \sqrt[4]{d} \sqrt{d x^4+c} \sqrt[4]{c}}-\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right) (b c-a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\sqrt{c} \left(\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a \sqrt{b} \left(\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right) \sqrt[4]{d} \sqrt{d x^4+c} \sqrt[4]{c}}",1,"(-21*a*(c + d*x^4) - 7*(b*c - 2*a*d)*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)] + 3*b*d*x^8*Sqrt[1 + (d*x^4)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(21*a^2*x*Sqrt[c + d*x^4])","C",0
800,1,333,703,0.3031026,"\int \frac{\sqrt{c+d x^4}}{x^4 \left(a+b x^4\right)} \, dx","Integrate[Sqrt[c + d*x^4]/(x^4*(a + b*x^4)),x]","\frac{\frac{a \left(25 a c \left(a c-a d x^4+4 b c x^4+b d x^8\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)-10 x^4 \left(a+b x^4\right) \left(c+d x^4\right) \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)\right)}{\left(a+b x^4\right) \left(2 x^4 \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)}-b d x^8 \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{15 a^2 x^3 \sqrt{c+d x^4}}","-\frac{(b c-a d) \tan ^{-1}\left(\frac{x \sqrt{\frac{\sqrt{-a} \left(\frac{b c}{a}-d\right)}{\sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 a^2 \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{(b c-a d) \tan ^{-1}\left(\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 a^2 \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right) (b c-a d) \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a^2 \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)}-\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right) (b c-a d) \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a^2 \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right)}-\frac{d^{3/4} \left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} (2 b c-a d) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{3 a \sqrt[4]{c} \sqrt{c+d x^4} (a d+b c)}-\frac{\sqrt{c+d x^4}}{3 a x^3}",1,"(-(b*d*x^8*Sqrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]) + (a*(25*a*c*(a*c + 4*b*c*x^4 - a*d*x^4 + b*d*x^8)*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] - 10*x^4*(a + b*x^4)*(c + d*x^4)*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])))/((a + b*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]))))/(15*a^2*x^3*Sqrt[c + d*x^4])","C",0
801,1,70,71,0.0523362,"\int \frac{(e x)^{3/2} \sqrt{c+d x^4}}{a+b x^4} \, dx","Integrate[((e*x)^(3/2)*Sqrt[c + d*x^4])/(a + b*x^4),x]","\frac{2 x (e x)^{3/2} \sqrt{c+d x^4} F_1\left(\frac{5}{8};-\frac{1}{2},1;\frac{13}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{5 a \sqrt{\frac{c+d x^4}{c}}}","\frac{2 (e x)^{5/2} \sqrt{c+d x^4} F_1\left(\frac{5}{8};1,-\frac{1}{2};\frac{13}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{5 a e \sqrt{\frac{d x^4}{c}+1}}",1,"(2*x*(e*x)^(3/2)*Sqrt[c + d*x^4]*AppellF1[5/8, -1/2, 1, 13/8, -((d*x^4)/c), -((b*x^4)/a)])/(5*a*Sqrt[(c + d*x^4)/c])","A",0
802,1,70,71,0.0356187,"\int \frac{\sqrt{e x} \sqrt{c+d x^4}}{a+b x^4} \, dx","Integrate[(Sqrt[e*x]*Sqrt[c + d*x^4])/(a + b*x^4),x]","\frac{2 x \sqrt{e x} \sqrt{c+d x^4} F_1\left(\frac{3}{8};-\frac{1}{2},1;\frac{11}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{3 a \sqrt{\frac{c+d x^4}{c}}}","\frac{2 (e x)^{3/2} \sqrt{c+d x^4} F_1\left(\frac{3}{8};1,-\frac{1}{2};\frac{11}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{3 a e \sqrt{\frac{d x^4}{c}+1}}",1,"(2*x*Sqrt[e*x]*Sqrt[c + d*x^4]*AppellF1[3/8, -1/2, 1, 11/8, -((d*x^4)/c), -((b*x^4)/a)])/(3*a*Sqrt[(c + d*x^4)/c])","A",0
803,1,68,69,0.0498657,"\int \frac{\sqrt{c+d x^4}}{\sqrt{e x} \left(a+b x^4\right)} \, dx","Integrate[Sqrt[c + d*x^4]/(Sqrt[e*x]*(a + b*x^4)),x]","\frac{2 x \sqrt{c+d x^4} F_1\left(\frac{1}{8};-\frac{1}{2},1;\frac{9}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{a \sqrt{e x} \sqrt{\frac{c+d x^4}{c}}}","\frac{2 \sqrt{e x} \sqrt{c+d x^4} F_1\left(\frac{1}{8};1,-\frac{1}{2};\frac{9}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a e \sqrt{\frac{d x^4}{c}+1}}",1,"(2*x*Sqrt[c + d*x^4]*AppellF1[1/8, -1/2, 1, 9/8, -((d*x^4)/c), -((b*x^4)/a)])/(a*Sqrt[e*x]*Sqrt[(c + d*x^4)/c])","A",0
804,1,143,69,0.1781876,"\int \frac{\sqrt{c+d x^4}}{(e x)^{3/2} \left(a+b x^4\right)} \, dx","Integrate[Sqrt[c + d*x^4]/((e*x)^(3/2)*(a + b*x^4)),x]","\frac{x \left(-10 x^4 \sqrt{\frac{d x^4}{c}+1} (b c-4 a d) F_1\left(\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+14 b d x^8 \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)-70 a \left(c+d x^4\right)\right)}{35 a^2 (e x)^{3/2} \sqrt{c+d x^4}}","-\frac{2 \sqrt{c+d x^4} F_1\left(-\frac{1}{8};1,-\frac{1}{2};\frac{7}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a e \sqrt{e x} \sqrt{\frac{d x^4}{c}+1}}",1,"(x*(-70*a*(c + d*x^4) - 10*(b*c - 4*a*d)*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[7/8, 1/2, 1, 15/8, -((d*x^4)/c), -((b*x^4)/a)] + 14*b*d*x^8*Sqrt[1 + (d*x^4)/c]*AppellF1[15/8, 1/2, 1, 23/8, -((d*x^4)/c), -((b*x^4)/a)]))/(35*a^2*(e*x)^(3/2)*Sqrt[c + d*x^4])","B",0
805,1,91,104,0.2573857,"\int \frac{x^{11}}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[x^11/((a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{\sqrt{c+d x^4} \left(-3 a d-2 b c+b d x^4\right)}{6 b^2 d^2}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 b^{5/2} \sqrt{b c-a d}}","-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^4} (a d+b c)}{2 b^2 d^2}+\frac{\left(c+d x^4\right)^{3/2}}{6 b d^2}",1,"(Sqrt[c + d*x^4]*(-2*b*c - 3*a*d + b*d*x^4))/(6*b^2*d^2) - (a^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(2*b^(5/2)*Sqrt[b*c - a*d])","A",1
806,1,72,74,0.0793303,"\int \frac{x^7}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[x^7/((a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{1}{2} \left(\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^4}}{b d}\right)","\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^4}}{2 b d}",1,"(Sqrt[c + d*x^4]/(b*d) + (a*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(b^(3/2)*Sqrt[b*c - a*d]))/2","A",1
807,1,51,51,0.0177045,"\int \frac{x^3}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[x^3/((a + b*x^4)*Sqrt[c + d*x^4]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 \sqrt{b} \sqrt{b c-a d}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 \sqrt{b} \sqrt{b c-a d}}",1,"-1/2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]]/(Sqrt[b]*Sqrt[b*c - a*d])","A",1
808,1,81,85,0.0858661,"\int \frac{1}{x \left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[1/(x*(a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{\sqrt{c}}}{2 a}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 a \sqrt{b c-a d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{2 a \sqrt{c}}",1,"(-(ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]]/Sqrt[c]) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d])/(2*a)","A",1
809,1,151,117,0.1698775,"\int \frac{1}{x^5 \left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[1/(x^5*(a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{b^{3/2} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 a^2 (a d-b c)}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{2 a^2 \sqrt{c}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{4 a c^{3/2}}-\frac{\sqrt{c+d x^4}}{4 a c x^4}","-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{2 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{4 a^2 c^{3/2}}-\frac{\sqrt{c+d x^4}}{4 a c x^4}",1,"-1/4*Sqrt[c + d*x^4]/(a*c*x^4) + (b*ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]])/(2*a^2*Sqrt[c]) + (d*ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]])/(4*a*c^(3/2)) + (b^(3/2)*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(2*a^2*(-(b*c) + a*d))","A",1
810,1,118,123,0.2103235,"\int \frac{x^9}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[x^9/((a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{\frac{2 a^{3/2} \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{\sqrt{b c-a d}}-\frac{(2 a d+b c) \log \left(\sqrt{d} \sqrt{c+d x^4}+d x^2\right)}{d^{3/2}}+\frac{b x^2 \sqrt{c+d x^4}}{d}}{4 b^2}","\frac{a^{3/2} \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{2 b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right)}{4 b^2 d^{3/2}}+\frac{x^2 \sqrt{c+d x^4}}{4 b d}",1,"((b*x^2*Sqrt[c + d*x^4])/d + (2*a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/Sqrt[b*c - a*d] - ((b*c + 2*a*d)*Log[d*x^2 + Sqrt[d]*Sqrt[c + d*x^4]])/d^(3/2))/(4*b^2)","A",1
811,1,90,91,0.0692318,"\int \frac{x^5}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[x^5/((a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{\frac{\log \left(\sqrt{d} \sqrt{c+d x^4}+d x^2\right)}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{\sqrt{b c-a d}}}{2 b}","\frac{\tanh ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right)}{2 b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{2 b \sqrt{b c-a d}}",1,"(-((Sqrt[a]*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/Sqrt[b*c - a*d]) + Log[d*x^2 + Sqrt[d]*Sqrt[c + d*x^4]]/Sqrt[d])/(2*b)","A",1
812,1,95,54,0.0634386,"\int \frac{x}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[x/((a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{x^2 \sqrt{\frac{d x^4}{c}+1} \tanh ^{-1}\left(\frac{\sqrt{\frac{d x^4}{c}-\frac{b x^4}{a}}}{\sqrt{\frac{d x^4}{c}+1}}\right)}{2 a \sqrt{c+d x^4} \sqrt{\frac{d x^4}{c}-\frac{b x^4}{a}}}","\frac{\tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{2 \sqrt{a} \sqrt{b c-a d}}",1,"(x^2*Sqrt[1 + (d*x^4)/c]*ArcTanh[Sqrt[-((b*x^4)/a) + (d*x^4)/c]/Sqrt[1 + (d*x^4)/c]])/(2*a*Sqrt[c + d*x^4]*Sqrt[-((b*x^4)/a) + (d*x^4)/c])","A",1
813,1,179,80,4.8299697,"\int \frac{1}{x^3 \left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[1/(x^3*(a + b*x^4)*Sqrt[c + d*x^4]),x]","-\frac{\left(\frac{d x^4}{c}+1\right) \left(\frac{4 x^4 \left(c+d x^4\right) (b c-a d) \, _2F_1\left(2,2;\frac{5}{2};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{3 c^2 \left(a+b x^4\right)}+\frac{\left(c+2 d x^4\right) \sin ^{-1}\left(\sqrt{\frac{x^4 (b c-a d)}{c \left(a+b x^4\right)}}\right)}{c \sqrt{\frac{a x^4 \left(c+d x^4\right) (b c-a d)}{c^2 \left(a+b x^4\right)^2}}}\right)}{2 x^2 \left(a+b x^4\right) \sqrt{c+d x^4}}","-\frac{b \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{2 a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^4}}{2 a c x^2}",1,"-1/2*((1 + (d*x^4)/c)*(((c + 2*d*x^4)*ArcSin[Sqrt[((b*c - a*d)*x^4)/(c*(a + b*x^4))]])/(c*Sqrt[(a*(b*c - a*d)*x^4*(c + d*x^4))/(c^2*(a + b*x^4)^2)]) + (4*(b*c - a*d)*x^4*(c + d*x^4)*Hypergeometric2F1[2, 2, 5/2, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(3*c^2*(a + b*x^4))))/(x^2*(a + b*x^4)*Sqrt[c + d*x^4])","C",0
814,1,137,115,5.5487587,"\int \frac{1}{x^7 \left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[1/(x^7*(a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{\frac{b^2 x^2 \sqrt{\frac{d x^4}{c}+1} \sin ^{-1}\left(\frac{\sqrt{x^4 \left(\frac{b}{a}-\frac{d}{c}\right)}}{\sqrt{\frac{b x^4}{a}+1}}\right)}{a^3 \sqrt{\frac{x^4 (b c-a d)}{a c}}}+\frac{\left(c+d x^4\right) \left(-a c+2 a d x^4+3 b c x^4\right)}{3 a^2 c^2 x^6}}{2 \sqrt{c+d x^4}}","\frac{b^2 \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{2 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^4} (2 a d+3 b c)}{6 a^2 c^2 x^2}-\frac{\sqrt{c+d x^4}}{6 a c x^6}",1,"(((c + d*x^4)*(-(a*c) + 3*b*c*x^4 + 2*a*d*x^4))/(3*a^2*c^2*x^6) + (b^2*x^2*Sqrt[1 + (d*x^4)/c]*ArcSin[Sqrt[(b/a - d/c)*x^4]/Sqrt[1 + (b*x^4)/a]])/(a^3*Sqrt[((b*c - a*d)*x^4)/(a*c)]))/(2*Sqrt[c + d*x^4])","A",0
815,1,249,872,0.3472549,"\int \frac{x^8}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[x^8/((a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{x \left(5 \left(\frac{5 a^2 c^2 F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{d \left(a+b x^4\right) \left(2 x^4 \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)}+\frac{c}{d}+x^4\right)-\frac{x^4 \sqrt{\frac{d x^4}{c}+1} (3 a d+b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{a d}\right)}{15 b \sqrt{c+d x^4}}","\frac{\left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right) \sqrt[4]{d} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) a^2}{4 b^2 \sqrt[4]{c} (b c+a d) \sqrt{d x^4+c}}+\frac{\left(\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right) \sqrt[4]{d} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) a}{4 b^2 \sqrt[4]{c} (b c+a d) \sqrt{d x^4+c}}+\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) a}{8 b^2 \sqrt[4]{c} \sqrt[4]{d} (b c+a d) \sqrt{d x^4+c}}+\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2 \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) a}{8 b^2 \sqrt[4]{c} \sqrt[4]{d} (b c+a d) \sqrt{d x^4+c}}-\frac{(-a)^{5/4} \tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{4 b^{7/4} \sqrt{b c-a d}}-\frac{(-a)^{5/4} \tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{4 b^{7/4} \sqrt{a d-b c}}-\frac{(b c+3 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{6 b^2 \sqrt[4]{c} d^{5/4} \sqrt{d x^4+c}}+\frac{x \sqrt{d x^4+c}}{3 b d}",1,"(x*(-(((b*c + 3*a*d)*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])/(a*d)) + 5*(c/d + x^4 + (5*a^2*c^2*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)])/(d*(a + b*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]))))))/(15*b*Sqrt[c + d*x^4])","C",0
816,1,65,638,0.053746,"\int \frac{x^4}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[x^4/((a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{x^5 \sqrt{\frac{c+d x^4}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{5 a \sqrt{c+d x^4}}","\frac{c^{3/4} \left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{d} \sqrt{c+d x^4} (a d+b c)}-\frac{\tan ^{-1}\left(\frac{x \sqrt{\frac{\sqrt{-a} \left(\frac{b c}{a}-d\right)}{\sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 b \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\tan ^{-1}\left(\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 b \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right) \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)}-\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right) \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right)}",1,"(x^5*Sqrt[(c + d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])/(5*a*Sqrt[c + d*x^4])","C",0
817,1,161,638,0.0495241,"\int \frac{1}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[1/((a + b*x^4)*Sqrt[c + d*x^4]),x]","-\frac{5 a c x F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{\left(a+b x^4\right) \sqrt{c+d x^4} \left(2 x^4 \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)}","\frac{d^{3/4} \left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{c} \sqrt{c+d x^4} (a d+b c)}+\frac{\tan ^{-1}\left(\frac{x \sqrt{\frac{\sqrt{-a} \left(\frac{b c}{a}-d\right)}{\sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 a \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{\tan ^{-1}\left(\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 a \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}+\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right) \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)}+\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right) \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right)}",1,"(-5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)])/((a + b*x^4)*Sqrt[c + d*x^4]*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])))","C",0
818,1,337,677,0.3757114,"\int \frac{1}{x^4 \left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[1/(x^4*(a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{\frac{5 a \left(2 x^4 \left(a+b x^4\right) \left(c+d x^4\right) \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)-5 a c \left(a c+2 a d x^4+4 b c x^4+b d x^8\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)}{\left(a+b x^4\right) \left(5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)-2 x^4 \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)\right)}-b d x^8 \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{15 a^2 c x^3 \sqrt{c+d x^4}}","-\frac{b \tan ^{-1}\left(\frac{x \sqrt{\frac{\sqrt{-a} \left(\frac{b c}{a}-d\right)}{\sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 a^2 \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{b \tan ^{-1}\left(\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 a^2 \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{b \left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right) \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a^2 \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)}-\frac{b \left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right) \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a^2 \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right)}-\frac{d^{3/4} \left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} (a d+4 b c) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{6 a c^{5/4} \sqrt{c+d x^4} (a d+b c)}-\frac{\sqrt{c+d x^4}}{3 a c x^3}",1,"(-(b*d*x^8*Sqrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]) + (5*a*(-5*a*c*(a*c + 4*b*c*x^4 + 2*a*d*x^4 + b*d*x^8)*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(a + b*x^4)*(c + d*x^4)*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])))/((a + b*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] - 2*x^4*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]))))/(15*a^2*c*x^3*Sqrt[c + d*x^4])","C",0
819,1,65,804,0.0439509,"\int \frac{x^6}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[x^6/((a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{x^7 \sqrt{\frac{c+d x^4}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{7 a \sqrt{c+d x^4}}","\frac{\sqrt{d x^4+c} x}{b \sqrt{d} \left(\sqrt{d} x^2+\sqrt{c}\right)}-\frac{a \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right)}{4 b (b c-a d)}-\frac{a \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right)}{4 b (b c-a d)}-\frac{\sqrt[4]{c} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{b d^{3/4} \sqrt{d x^4+c}}+\frac{\sqrt[4]{c} (b c+2 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{2 b d^{3/4} (b c+a d) \sqrt{d x^4+c}}+\frac{a \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b^{3/2} \sqrt[4]{c} \left(\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right) \sqrt[4]{d} \sqrt{d x^4+c}}-\frac{a \left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\sqrt{c} \left(\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b^{3/2} \sqrt[4]{c} \left(\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right) \sqrt[4]{d} \sqrt{d x^4+c}}",1,"(x^7*Sqrt[(c + d*x^4)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(7*a*Sqrt[c + d*x^4])","C",0
820,1,65,656,0.033307,"\int \frac{x^2}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[x^2/((a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{x^3 \sqrt{\frac{c+d x^4}{c}} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{3 a \sqrt{c+d x^4}}","\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{x \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 (b c-a d)}+\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right)}{4 (b c-a d)}-\frac{\sqrt[4]{c} \sqrt[4]{d} \left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{2 \sqrt{c+d x^4} (a d+b c)}-\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right) \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 \sqrt{b} \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right)}+\frac{\left(\sqrt{c}+\sqrt{d} x^2\right) \sqrt{\frac{c+d x^4}{\left(\sqrt{c}+\sqrt{d} x^2\right)^2}} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right) \Pi \left(-\frac{\sqrt{c} \left(\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 \sqrt{b} \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^4} \left(\sqrt{-a} \sqrt{b} \sqrt{c}+a \sqrt{d}\right)}",1,"(x^3*Sqrt[(c + d*x^4)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)])/(3*a*Sqrt[c + d*x^4])","C",0
821,1,141,833,0.1386617,"\int \frac{1}{x^2 \left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[1/(x^2*(a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{7 x^4 \sqrt{\frac{d x^4}{c}+1} (a d-b c) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+3 b d x^8 \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)-21 a \left(c+d x^4\right)}{21 a^2 c x \sqrt{c+d x^4}}","\frac{\sqrt{d} \sqrt{d x^4+c} x}{a c \left(\sqrt{d} x^2+\sqrt{c}\right)}-\frac{b \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right)}{4 a (b c-a d)}-\frac{b \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left(\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right)}{4 a (b c-a d)}-\frac{\sqrt[4]{d} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{a c^{3/4} \sqrt{d x^4+c}}+\frac{\sqrt[4]{d} (2 b c+a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{2 a c^{3/4} (b c+a d) \sqrt{d x^4+c}}+\frac{\sqrt{b} \left(\frac{\sqrt{b} \sqrt[4]{c}}{\sqrt[4]{d}}-\frac{\sqrt{-a} \sqrt[4]{d}}{\sqrt[4]{c}}\right) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a \left(\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right) \sqrt{d x^4+c}}-\frac{\sqrt{b} \left(\frac{\sqrt[4]{d} \sqrt{-a}}{\sqrt[4]{c}}+\frac{\sqrt{b} \sqrt[4]{c}}{\sqrt[4]{d}}\right) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\sqrt{c} \left(\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a \left(\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right) \sqrt{d x^4+c}}-\frac{\sqrt{d x^4+c}}{a c x}",1,"(-21*a*(c + d*x^4) + 7*(-(b*c) + a*d)*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)] + 3*b*d*x^8*Sqrt[1 + (d*x^4)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(21*a^2*c*x*Sqrt[c + d*x^4])","C",0
822,1,175,175,0.3058917,"\int \frac{x^{15}}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x^15/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{a^2 (5 a d-6 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{4 b^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} \left(-15 a^3 d^2+2 a^2 b d \left(4 c-5 d x^4\right)+2 a b^2 \left(2 c^2+3 c d x^4+d^2 x^8\right)+2 b^3 c x^4 \left(2 c-d x^4\right)\right)}{12 b^3 d^2 \left(a+b x^4\right) (b c-a d)}","-\frac{a^2 (6 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{4 b^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} \left(-15 a^2 d^2-b d x^4 (2 b c-5 a d)+8 a b c d+4 b^2 c^2\right)}{12 b^3 d^2 (b c-a d)}+\frac{a x^8 \sqrt{c+d x^4}}{4 b \left(a+b x^4\right) (b c-a d)}",1,"-1/12*(Sqrt[c + d*x^4]*(-15*a^3*d^2 + 2*a^2*b*d*(4*c - 5*d*x^4) + 2*b^3*c*x^4*(2*c - d*x^4) + 2*a*b^2*(2*c^2 + 3*c*d*x^4 + d^2*x^8)))/(b^3*d^2*(b*c - a*d)*(a + b*x^4)) + (a^2*(-6*b*c + 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(4*b^(7/2)*(b*c - a*d)^(3/2))","A",1
823,1,107,123,0.355176,"\int \frac{x^{11}}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x^11/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{1}{4} \left(\frac{\sqrt{c+d x^4} \left(\frac{a^2}{\left(a+b x^4\right) (a d-b c)}+\frac{2}{d}\right)}{b^2}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{b^{5/2} (b c-a d)^{3/2}}\right)","-\frac{a^2 \sqrt{c+d x^4}}{4 b^2 \left(a+b x^4\right) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{4 b^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^4}}{2 b^2 d}",1,"((Sqrt[c + d*x^4]*(2/d + a^2/((-(b*c) + a*d)*(a + b*x^4))))/b^2 + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(3/2)))/4","A",1
824,1,98,99,0.1133755,"\int \frac{x^7}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x^7/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{\frac{a \sqrt{b} \sqrt{c+d x^4}}{\left(a+b x^4\right) (b c-a d)}+\frac{(a d-2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}}{4 b^{3/2}}","\frac{a \sqrt{c+d x^4}}{4 b \left(a+b x^4\right) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{4 b^{3/2} (b c-a d)^{3/2}}",1,"((a*Sqrt[b]*Sqrt[c + d*x^4])/((b*c - a*d)*(a + b*x^4)) + ((-2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(4*b^(3/2))","A",1
825,1,85,87,0.1023064,"\int \frac{x^3}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x^3/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{1}{4} \left(\frac{\sqrt{c+d x^4}}{\left(a+b x^4\right) (a d-b c)}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{a d-b c}}\right)}{\sqrt{b} (a d-b c)^{3/2}}\right)","\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{4 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4}}{4 \left(a+b x^4\right) (b c-a d)}",1,"(Sqrt[c + d*x^4]/((-(b*c) + a*d)*(a + b*x^4)) + (d*ArcTan[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[-(b*c) + a*d]])/(Sqrt[b]*(-(b*c) + a*d)^(3/2)))/4","A",1
826,1,123,132,0.3336908,"\int \frac{1}{x \left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[1/(x*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{\frac{a b \sqrt{c+d x^4}}{\left(a+b x^4\right) (b c-a d)}+\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{\sqrt{c}}}{4 a^2}","\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{4 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{2 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^4}}{4 a \left(a+b x^4\right) (b c-a d)}",1,"((a*b*Sqrt[c + d*x^4])/((b*c - a*d)*(a + b*x^4)) - (2*ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]])/Sqrt[c] + (Sqrt[b]*(2*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(4*a^2)","A",1
827,1,163,185,0.6953826,"\int \frac{1}{x^5 \left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[1/(x^5*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{\frac{a \sqrt{c+d x^4} \left(a^2 d+a b \left(d x^4-c\right)-2 b^2 c x^4\right)}{x^4 \left(a+b x^4\right) (b c-a d)}+\frac{b^{3/2} c (5 a d-4 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{\sqrt{c}}}{4 a^3 c}","-\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right)}{4 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right)}{4 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^4} (2 b c-a d)}{4 a^2 c \left(a+b x^4\right) (b c-a d)}-\frac{\sqrt{c+d x^4}}{4 a c x^4 \left(a+b x^4\right)}",1,"((a*Sqrt[c + d*x^4]*(a^2*d - 2*b^2*c*x^4 + a*b*(-c + d*x^4)))/((b*c - a*d)*x^4*(a + b*x^4)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]])/Sqrt[c] + (b^(3/2)*c*(-4*b*c + 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(4*a^3*c)","A",1
828,1,150,191,0.5660915,"\int \frac{x^{13}}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x^13/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{\frac{a^{3/2} (5 b c-4 a d) \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{(b c-a d)^{3/2}}+b x^2 \sqrt{c+d x^4} \left(\frac{a^2}{\left(a+b x^4\right) (a d-b c)}+\frac{1}{d}\right)-\frac{(4 a d+b c) \log \left(\sqrt{d} \sqrt{c+d x^4}+d x^2\right)}{d^{3/2}}}{4 b^3}","\frac{a^{3/2} (5 b c-4 a d) \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{4 b^3 (b c-a d)^{3/2}}-\frac{(4 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right)}{4 b^3 d^{3/2}}+\frac{x^2 \sqrt{c+d x^4} (b c-2 a d)}{4 b^2 d (b c-a d)}+\frac{a x^6 \sqrt{c+d x^4}}{4 b \left(a+b x^4\right) (b c-a d)}",1,"(b*x^2*Sqrt[c + d*x^4]*(d^(-1) + a^2/((-(b*c) + a*d)*(a + b*x^4))) + (a^(3/2)*(5*b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(b*c - a*d)^(3/2) - ((b*c + 4*a*d)*Log[d*x^2 + Sqrt[d]*Sqrt[c + d*x^4]])/d^(3/2))/(4*b^3)","A",1
829,1,135,141,0.2240148,"\int \frac{x^9}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x^9/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{\frac{a b x^2 \sqrt{c+d x^4}}{\left(a+b x^4\right) (b c-a d)}+\frac{\sqrt{a} (2 a d-3 b c) \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{(b c-a d)^{3/2}}+\frac{2 \log \left(\sqrt{d} \sqrt{c+d x^4}+d x^2\right)}{\sqrt{d}}}{4 b^2}","-\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{4 b^2 (b c-a d)^{3/2}}+\frac{a x^2 \sqrt{c+d x^4}}{4 b \left(a+b x^4\right) (b c-a d)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right)}{2 b^2 \sqrt{d}}",1,"((a*b*x^2*Sqrt[c + d*x^4])/((b*c - a*d)*(a + b*x^4)) + (Sqrt[a]*(-3*b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(b*c - a*d)^(3/2) + (2*Log[d*x^2 + Sqrt[d]*Sqrt[c + d*x^4]])/Sqrt[d])/(4*b^2)","A",1
830,1,124,93,0.5834617,"\int \frac{x^5}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x^5/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{\sqrt{c+d x^4} \left(-\frac{x^4 (b c-a d)}{a+b x^4}-\frac{c \sqrt{x^4 \left(\frac{d}{c}-\frac{b}{a}\right)} \tanh ^{-1}\left(\frac{\sqrt{x^4 \left(\frac{d}{c}-\frac{b}{a}\right)}}{\sqrt{\frac{d x^4}{c}+1}}\right)}{\sqrt{\frac{d x^4}{c}+1}}\right)}{4 x^2 (b c-a d)^2}","\frac{c \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{4 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^2 \sqrt{c+d x^4}}{4 \left(a+b x^4\right) (b c-a d)}",1,"(Sqrt[c + d*x^4]*(-(((b*c - a*d)*x^4)/(a + b*x^4)) - (c*Sqrt[(-(b/a) + d/c)*x^4]*ArcTanh[Sqrt[(-(b/a) + d/c)*x^4]/Sqrt[1 + (d*x^4)/c]])/Sqrt[1 + (d*x^4)/c]))/(4*(b*c - a*d)^2*x^2)","A",1
831,1,407,104,6.0126439,"\int \frac{x}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{x^2 \sqrt{c+d x^4} \left(-30 d x^4 \sqrt{\frac{a x^4 \left(c+d x^4\right) (b c-a d)}{c^2 \left(a+b x^4\right)^2}}-45 c \sqrt{\frac{a x^4 \left(c+d x^4\right) (b c-a d)}{c^2 \left(a+b x^4\right)^2}}+16 d x^4 \left(\frac{x^4 (b c-a d)}{c \left(a+b x^4\right)}\right)^{5/2} \sqrt{\frac{a \left(c+d x^4\right)}{c \left(a+b x^4\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)+16 c \left(\frac{x^4 (b c-a d)}{c \left(a+b x^4\right)}\right)^{5/2} \sqrt{\frac{a \left(c+d x^4\right)}{c \left(a+b x^4\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)+30 d x^4 \sin ^{-1}\left(\sqrt{\frac{x^4 (b c-a d)}{c \left(a+b x^4\right)}}\right)+45 c \sin ^{-1}\left(\sqrt{\frac{x^4 (b c-a d)}{c \left(a+b x^4\right)}}\right)\right)}{60 c^2 \left(a+b x^4\right)^2 \left(\frac{x^4 (b c-a d)}{c \left(a+b x^4\right)}\right)^{3/2} \sqrt{\frac{a \left(c+d x^4\right)}{c \left(a+b x^4\right)}}}","\frac{(b c-2 a d) \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{4 a^{3/2} (b c-a d)^{3/2}}+\frac{b x^2 \sqrt{c+d x^4}}{4 a \left(a+b x^4\right) (b c-a d)}",1,"(x^2*Sqrt[c + d*x^4]*(-45*c*Sqrt[(a*(b*c - a*d)*x^4*(c + d*x^4))/(c^2*(a + b*x^4)^2)] - 30*d*x^4*Sqrt[(a*(b*c - a*d)*x^4*(c + d*x^4))/(c^2*(a + b*x^4)^2)] + 45*c*ArcSin[Sqrt[((b*c - a*d)*x^4)/(c*(a + b*x^4))]] + 30*d*x^4*ArcSin[Sqrt[((b*c - a*d)*x^4)/(c*(a + b*x^4))]] + 16*c*(((b*c - a*d)*x^4)/(c*(a + b*x^4)))^(5/2)*Sqrt[(a*(c + d*x^4))/(c*(a + b*x^4))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 16*d*x^4*(((b*c - a*d)*x^4)/(c*(a + b*x^4)))^(5/2)*Sqrt[(a*(c + d*x^4))/(c*(a + b*x^4))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^4)/(c*(a + b*x^4))]))/(60*c^2*(((b*c - a*d)*x^4)/(c*(a + b*x^4)))^(3/2)*(a + b*x^4)^2*Sqrt[(a*(c + d*x^4))/(c*(a + b*x^4))])","C",0
832,1,155,149,5.6328064,"\int \frac{1}{x^3 \left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[1/(x^3*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{a^2 \left(c+d x^4\right) \left(\frac{b^2 x^4}{\left(a+b x^4\right) (a d-b c)}-\frac{2}{c}\right)-\frac{b x^8 \sqrt{\frac{d x^4}{c}+1} (3 b c-4 a d) \sin ^{-1}\left(\frac{\sqrt{x^4 \left(\frac{b}{a}-\frac{d}{c}\right)}}{\sqrt{\frac{b x^4}{a}+1}}\right)}{c \left(\frac{x^4 (b c-a d)}{a c}\right)^{3/2}}}{4 a^4 x^2 \sqrt{c+d x^4}}","-\frac{b (3 b c-4 a d) \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{4 a^{5/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} (3 b c-2 a d)}{4 a^2 c x^2 (b c-a d)}+\frac{b \sqrt{c+d x^4}}{4 a x^2 \left(a+b x^4\right) (b c-a d)}",1,"(a^2*(c + d*x^4)*(-2/c + (b^2*x^4)/((-(b*c) + a*d)*(a + b*x^4))) - (b*(3*b*c - 4*a*d)*x^8*Sqrt[1 + (d*x^4)/c]*ArcSin[Sqrt[(b/a - d/c)*x^4]/Sqrt[1 + (b*x^4)/a]])/(c*(((b*c - a*d)*x^4)/(a*c))^(3/2)))/(4*a^4*x^2*Sqrt[c + d*x^4])","A",0
833,1,175,208,5.8711494,"\int \frac{1}{x^7 \left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[1/(x^7*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{a^2 \left(c+d x^4\right) \left(\frac{3 b^3 x^8}{\left(a+b x^4\right) (b c-a d)}+\frac{4 x^4 (a d+3 b c)}{c^2}-\frac{2 a}{c}\right)+\frac{3 b^2 x^{12} \sqrt{\frac{d x^4}{c}+1} (5 b c-6 a d) \sin ^{-1}\left(\frac{\sqrt{x^4 \left(\frac{b}{a}-\frac{d}{c}\right)}}{\sqrt{\frac{b x^4}{a}+1}}\right)}{c \left(\frac{x^4 (b c-a d)}{a c}\right)^{3/2}}}{12 a^5 x^6 \sqrt{c+d x^4}}","\frac{b^2 (5 b c-6 a d) \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right)}{4 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} (5 b c-2 a d)}{12 a^2 c x^6 (b c-a d)}+\frac{\sqrt{c+d x^4} \left(-4 a^2 d^2-8 a b c d+15 b^2 c^2\right)}{12 a^3 c^2 x^2 (b c-a d)}+\frac{b \sqrt{c+d x^4}}{4 a x^6 \left(a+b x^4\right) (b c-a d)}",1,"(a^2*(c + d*x^4)*((-2*a)/c + (4*(3*b*c + a*d)*x^4)/c^2 + (3*b^3*x^8)/((b*c - a*d)*(a + b*x^4))) + (3*b^2*(5*b*c - 6*a*d)*x^12*Sqrt[1 + (d*x^4)/c]*ArcSin[Sqrt[(b/a - d/c)*x^4]/Sqrt[1 + (b*x^4)/a]])/(c*(((b*c - a*d)*x^4)/(a*c))^(3/2)))/(12*a^5*x^6*Sqrt[c + d*x^4])","A",0
834,1,253,996,0.3515937,"\int \frac{x^8}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x^8/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{x \left(\frac{5 a \left(\frac{5 a c^2 F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{2 x^4 \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}+c+d x^4\right)}{b \left(a+b x^4\right)}+\frac{x^4 \sqrt{\frac{d x^4}{c}+1} (4 b c-3 a d) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{a b}\right)}{20 \sqrt{c+d x^4} (b c-a d)}","-\frac{(5 b c-3 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{32 b^2 \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{\sqrt{-a} \sqrt[4]{d} (5 b c-3 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)}{16 b^2 \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{\sqrt[4]{-a} (5 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 b^{7/4} (b c-a d)^{3/2}}+\frac{\sqrt[4]{-a} (5 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 b^{7/4} (a d-b c)^{3/2}}+\frac{(4 b c-3 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b^2 \sqrt[4]{c} \sqrt[4]{d} (b c-a d) \sqrt{d x^4+c}}-\frac{a \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right) \sqrt[4]{d} (5 b c-3 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 b^2 \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 (5 b c-3 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 b^2 \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{a x \sqrt{d x^4+c}}{4 b (b c-a d) \left(b x^4+a\right)}",1,"(x*(((4*b*c - 3*a*d)*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])/(a*b) + (5*a*(c + d*x^4 + (5*a*c^2*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)])/(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]))))/(b*(a + b*x^4))))/(20*(b*c - a*d)*Sqrt[c + d*x^4])","C",0
835,1,238,908,0.1927199,"\int \frac{x^4}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x^4/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{x \left(\frac{5 \left(\frac{5 a c^2 F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{2 x^4 \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}+c+d x^4\right)}{a+b x^4}+\frac{d x^4 \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{a}\right)}{20 \sqrt{c+d x^4} (a d-b c)}","\frac{\left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{32 a b \sqrt[4]{c} \sqrt[4]{d} (b c-a d) \sqrt{d x^4+c}}-\frac{(b c+a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 (-a)^{3/4} b^{3/4} (b c-a d)^{3/2}}+\frac{(b c+a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 (-a)^{3/4} b^{3/4} (a d-b c)^{3/2}}-\frac{d^{3/4} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b \sqrt[4]{c} (b c-a d) \sqrt{d x^4+c}}+\frac{\left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right) \sqrt[4]{d} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 b \sqrt[4]{c} (b c-a d) \sqrt{d x^4+c}}+\frac{\left(\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right) \sqrt[4]{d} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a b \sqrt[4]{c} (b c-a d) \sqrt{d x^4+c}}+\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 a b \sqrt[4]{c} \sqrt[4]{d} (b c-a d) \sqrt{d x^4+c}}-\frac{x \sqrt{d x^4+c}}{4 (b c-a d) \left(b x^4+a\right)}",1,"(x*((d*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])/a + (5*(c + d*x^4 + (5*a*c^2*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)])/(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]))))/(a + b*x^4)))/(20*(-(b*c) + a*d)*Sqrt[c + d*x^4])","C",0
836,1,392,983,0.3942523,"\int \frac{1}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[1/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{2 b x^5 \left(d x^4 \left(a+b x^4\right) \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+5 a \left(c+d x^4\right)\right) \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)-5 a c x F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right) \left(b d x^4 \left(a+b x^4\right) \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+5 a \left(-4 a d+4 b c+b d x^4\right)\right)}{20 a^2 \left(a+b x^4\right) \sqrt{c+d x^4} (b c-a d) \left(2 x^4 \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)}","\frac{(3 b c-5 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{32 a^2 \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{\sqrt[4]{d} (3 b c-5 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)}{16 (-a)^{3/2} \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{\sqrt[4]{b} (3 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 (-a)^{7/4} (b c-a d)^{3/2}}-\frac{\sqrt[4]{b} (3 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 (-a)^{7/4} (a d-b c)^{3/2}}+\frac{d^{3/4} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a \sqrt[4]{c} (b c-a d) \sqrt{d x^4+c}}+\frac{\left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right) \sqrt[4]{d} (3 b c-5 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 (3 b c-5 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 a^2 \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{b x \sqrt{d x^4+c}}{4 a (b c-a d) \left(b x^4+a\right)}",1,"(-5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)]*(5*a*(4*b*c - 4*a*d + b*d*x^4) + b*d*x^4*(a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]) + 2*b*x^5*(5*a*(c + d*x^4) + d*x^4*(a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]))/(20*a^2*(b*c - a*d)*(a + b*x^4)*Sqrt[c + d*x^4]*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])))","C",0
837,1,408,1046,0.7032072,"\int \frac{1}{x^4 \left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[1/(x^4*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{\frac{a \left(25 a c \left(4 a^2 d \left(c+2 d x^4\right)+4 a b \left(-c^2+5 c d x^4+d^2 x^8\right)-7 b^2 c x^4 \left(4 c+d x^4\right)\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+10 x^4 \left(c+d x^4\right) \left(-4 a^2 d+4 a b \left(c-d x^4\right)+7 b^2 c x^4\right) \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)\right)}{\left(a+b x^4\right) \left(2 x^4 \left(2 b c F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)}+b d x^8 \sqrt{\frac{d x^4}{c}+1} (7 b c-4 a d) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{60 a^3 c x^3 \sqrt{c+d x^4} (a d-b c)}","-\frac{b (7 b c-9 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{32 a^3 \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{b \sqrt[4]{d} (7 b c-9 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)}{16 (-a)^{5/2} \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{b^{5/4} (7 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 (-a)^{11/4} (b c-a d)^{3/2}}-\frac{b^{5/4} (7 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 (-a)^{11/4} (a d-b c)^{3/2}}-\frac{d^{3/4} (7 b c-4 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{24 a^2 c^{5/4} (b c-a d) \sqrt{d x^4+c}}-\frac{b \left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right) \sqrt[4]{d} (7 b c-9 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 (-a)^{5/2} \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{b \left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 (7 b c-9 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 a^3 \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{b \sqrt{d x^4+c}}{4 a (b c-a d) x^3 \left(b x^4+a\right)}-\frac{(7 b c-4 a d) \sqrt{d x^4+c}}{12 a^2 c (b c-a d) x^3}",1,"(b*d*(7*b*c - 4*a*d)*x^8*Sqrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + (a*(25*a*c*(-7*b^2*c*x^4*(4*c + d*x^4) + 4*a^2*d*(c + 2*d*x^4) + 4*a*b*(-c^2 + 5*c*d*x^4 + d^2*x^8))*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 10*x^4*(c + d*x^4)*(-4*a^2*d + 7*b^2*c*x^4 + 4*a*b*(c - d*x^4))*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])))/((a + b*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]))))/(60*a^3*c*(-(b*c) + a*d)*x^3*Sqrt[c + d*x^4])","C",0
838,1,162,1146,0.170389,"\int \frac{x^6}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x^6/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{d x^7 \left(a+b x^4\right) \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+7 c x^3 \left(a+b x^4\right) \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)-7 a x^3 \left(c+d x^4\right)}{28 a \left(a+b x^4\right) \sqrt{c+d x^4} (b c-a d)}","-\frac{\sqrt{d x^4+c} x^3}{4 (b c-a d) \left(b x^4+a\right)}+\frac{\sqrt{d} \sqrt{d x^4+c} x}{4 b (b c-a d) \left(\sqrt{d} x^2+\sqrt{c}\right)}+\frac{(3 b c-a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 \sqrt[4]{-a} b^{5/4} (b c-a d)^{3/2}}+\frac{(3 b c-a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 \sqrt[4]{-a} b^{5/4} (a d-b c)^{3/2}}-\frac{\sqrt[4]{c} \sqrt[4]{d} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{4 b (b c-a d) \sqrt{d x^4+c}}+\frac{\sqrt[4]{c} \sqrt[4]{d} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b (b c-a d) \sqrt{d x^4+c}}-\frac{\left(\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (3 b c-a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 b \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{\left(\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (3 b c-a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 b \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 (3 b c-a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 \sqrt{-a} b^{3/2} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2 (3 b c-a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 \sqrt{-a} b^{3/2} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}",1,"(-7*a*x^3*(c + d*x^4) + 7*c*x^3*(a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)] + d*x^7*(a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(28*a*(b*c - a*d)*(a + b*x^4)*Sqrt[c + d*x^4])","C",0
839,1,172,1144,0.1548689,"\int \frac{x^2}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[x^2/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{-3 b d x^7 \left(a+b x^4\right) \sqrt{\frac{d x^4}{c}+1} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+7 x^3 \left(a+b x^4\right) \sqrt{\frac{d x^4}{c}+1} (b c-4 a d) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+21 a b x^3 \left(c+d x^4\right)}{84 a^2 \left(a+b x^4\right) \sqrt{c+d x^4} (b c-a d)}","\frac{b \sqrt{d x^4+c} x^3}{4 a (b c-a d) \left(b x^4+a\right)}-\frac{\sqrt{d} \sqrt{d x^4+c} x}{4 a (b c-a d) \left(\sqrt{d} x^2+\sqrt{c}\right)}-\frac{(b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 (-a)^{5/4} \sqrt[4]{b} (b c-a d)^{3/2}}-\frac{(b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 (-a)^{5/4} \sqrt[4]{b} (a d-b c)^{3/2}}+\frac{\sqrt[4]{c} \sqrt[4]{d} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{4 a (b c-a d) \sqrt{d x^4+c}}-\frac{\sqrt[4]{c} \sqrt[4]{d} \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a (b c-a d) \sqrt{d x^4+c}}-\frac{\left(\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (b c-3 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{\left(\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (b c-3 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 (b c-3 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 (-a)^{3/2} \sqrt{b} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2 (b c-3 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 (-a)^{3/2} \sqrt{b} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}",1,"(21*a*b*x^3*(c + d*x^4) + 7*(b*c - 4*a*d)*x^3*(a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)] - 3*b*d*x^7*(a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(84*a^2*(b*c - a*d)*(a + b*x^4)*Sqrt[c + d*x^4])","C",0
840,1,226,1225,0.3310515,"\int \frac{1}{x^2 \left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[1/(x^2*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{-7 x^4 \left(a+b x^4\right) \sqrt{\frac{d x^4}{c}+1} \left(4 a^2 d^2-12 a b c d+5 b^2 c^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+21 a \left(c+d x^4\right) \left(4 a^2 d-4 a b \left(c-d x^4\right)-5 b^2 c x^4\right)+3 b d x^8 \left(a+b x^4\right) \sqrt{\frac{d x^4}{c}+1} (5 b c-4 a d) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{84 a^3 c x \left(a+b x^4\right) \sqrt{c+d x^4} (b c-a d)}","\frac{\sqrt{b} (5 b c-7 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{32 (-a)^{5/2} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{b^{3/4} (5 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 (-a)^{9/4} (b c-a d)^{3/2}}-\frac{b^{3/4} (5 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^4+c}}\right)}{16 (-a)^{9/4} (a d-b c)^{3/2}}-\frac{\sqrt[4]{d} (5 b c-4 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{4 a^2 c^{3/4} (b c-a d) \sqrt{d x^4+c}}+\frac{\sqrt[4]{d} (5 b c-4 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a^2 c^{3/4} (b c-a d) \sqrt{d x^4+c}}+\frac{b \left(\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (5 b c-7 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a^2 \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{b \left(\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (5 b c-7 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a^2 \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{\sqrt{b} \left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 (5 b c-7 a d) \left(\sqrt{d} x^2+\sqrt{c}\right) \sqrt{\frac{d x^4+c}{\left(\sqrt{d} x^2+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 (-a)^{5/2} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{(5 b c-4 a d) \sqrt{d x^4+c}}{4 a^2 c (b c-a d) x}+\frac{\sqrt{d} (5 b c-4 a d) x \sqrt{d x^4+c}}{4 a^2 c (b c-a d) \left(\sqrt{d} x^2+\sqrt{c}\right)}+\frac{b \sqrt{d x^4+c}}{4 a (b c-a d) x \left(b x^4+a\right)}",1,"(21*a*(c + d*x^4)*(4*a^2*d - 5*b^2*c*x^4 - 4*a*b*(c - d*x^4)) - 7*(5*b^2*c^2 - 12*a*b*c*d + 4*a^2*d^2)*x^4*(a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)] + 3*b*d*(5*b*c - 4*a*d)*x^8*(a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(84*a^3*c*(b*c - a*d)*x*(a + b*x^4)*Sqrt[c + d*x^4])","C",0
841,1,164,200,0.177322,"\int \frac{(e x)^m \left(a+b x^4\right)^2}{\sqrt{c+d x^4}} \, dx","Integrate[((e*x)^m*(a + b*x^4)^2)/Sqrt[c + d*x^4],x]","\frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \left(a^2 \left(m^2+14 m+45\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)+b (m+1) x^4 \left(2 a (m+9) \, _2F_1\left(\frac{1}{2},\frac{m+5}{4};\frac{m+9}{4};-\frac{d x^4}{c}\right)+b (m+5) x^4 \, _2F_1\left(\frac{1}{2},\frac{m+9}{4};\frac{m+13}{4};-\frac{d x^4}{c}\right)\right)\right)}{(m+1) (m+5) (m+9) \sqrt{c+d x^4}}","\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \left(a^2 d^2 (m+3) (m+7)+b c (m+1) (b c (m+5)-2 a d (m+7))\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)}{d^2 e (m+1) (m+3) (m+7) \sqrt{c+d x^4}}-\frac{b \sqrt{c+d x^4} (e x)^{m+1} (b c (m+5)-2 a d (m+7))}{d^2 e (m+3) (m+7)}+\frac{b^2 \sqrt{c+d x^4} (e x)^{m+5}}{d e^5 (m+7)}",1,"(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*(a^2*(45 + 14*m + m^2)*Hypergeometric2F1[1/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)] + b*(1 + m)*x^4*(2*a*(9 + m)*Hypergeometric2F1[1/2, (5 + m)/4, (9 + m)/4, -((d*x^4)/c)] + b*(5 + m)*x^4*Hypergeometric2F1[1/2, (9 + m)/4, (13 + m)/4, -((d*x^4)/c)])))/((1 + m)*(5 + m)*(9 + m)*Sqrt[c + d*x^4])","A",1
842,1,110,123,0.102973,"\int \frac{(e x)^m \left(a+b x^4\right)}{\sqrt{c+d x^4}} \, dx","Integrate[((e*x)^m*(a + b*x^4))/Sqrt[c + d*x^4],x]","\frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \left(a (m+5) \, _2F_1\left(\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)+b (m+1) x^4 \, _2F_1\left(\frac{1}{2},\frac{m+5}{4};\frac{m+9}{4};-\frac{d x^4}{c}\right)\right)}{(m+1) (m+5) \sqrt{c+d x^4}}","\frac{b \sqrt{c+d x^4} (e x)^{m+1}}{d e (m+3)}-\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} (b c (m+1)-a d (m+3)) \, _2F_1\left(\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)}{d e (m+1) (m+3) \sqrt{c+d x^4}}",1,"(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*(a*(5 + m)*Hypergeometric2F1[1/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)] + b*(1 + m)*x^4*Hypergeometric2F1[1/2, (5 + m)/4, (9 + m)/4, -((d*x^4)/c)]))/((1 + m)*(5 + m)*Sqrt[c + d*x^4])","A",1
843,1,66,68,0.0160685,"\int \frac{(e x)^m}{\sqrt{c+d x^4}} \, dx","Integrate[(e*x)^m/Sqrt[c + d*x^4],x]","\frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \, _2F_1\left(\frac{1}{2},\frac{m+1}{4};\frac{m+1}{4}+1;-\frac{d x^4}{c}\right)}{(m+1) \sqrt{c+d x^4}}","\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)}{e (m+1) \sqrt{c+d x^4}}",1,"(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*Hypergeometric2F1[1/2, (1 + m)/4, 1 + (1 + m)/4, -((d*x^4)/c)])/((1 + m)*Sqrt[c + d*x^4])","A",1
844,1,125,81,0.13525,"\int \frac{(e x)^m}{\left(a+b x^4\right) \sqrt{c+d x^4}} \, dx","Integrate[(e*x)^m/((a + b*x^4)*Sqrt[c + d*x^4]),x]","\frac{x \sqrt{c+d x^4} (e x)^m \left(b c F_1\left(\frac{m+1}{4};-\frac{1}{2},1;\frac{m+5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)-a d \, _2F_1\left(\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)\right)}{a c (m+1) \sqrt{\frac{d x^4}{c}+1} (b c-a d)}","\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left(\frac{m+1}{4};1,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a e (m+1) \sqrt{c+d x^4}}",1,"(x*(e*x)^m*Sqrt[c + d*x^4]*(b*c*AppellF1[(1 + m)/4, -1/2, 1, (5 + m)/4, -((d*x^4)/c), -((b*x^4)/a)] - a*d*Hypergeometric2F1[1/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)]))/(a*c*(b*c - a*d)*(1 + m)*Sqrt[1 + (d*x^4)/c])","A",0
845,1,179,81,0.2302534,"\int \frac{(e x)^m}{\left(a+b x^4\right)^2 \sqrt{c+d x^4}} \, dx","Integrate[(e*x)^m/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]","\frac{x \sqrt{c+d x^4} (e x)^m \left(a^2 d^2 \, _2F_1\left(\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)-a b c d F_1\left(\frac{m+1}{4};-\frac{1}{2},1;\frac{m+5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+b c (b c-a d) F_1\left(\frac{m+1}{4};2,-\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}{a^2 c (m+1) \sqrt{\frac{d x^4}{c}+1} (b c-a d)^2}","\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left(\frac{m+1}{4};2,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a^2 e (m+1) \sqrt{c+d x^4}}",1,"(x*(e*x)^m*Sqrt[c + d*x^4]*(-(a*b*c*d*AppellF1[(1 + m)/4, -1/2, 1, (5 + m)/4, -((d*x^4)/c), -((b*x^4)/a)]) + b*c*(b*c - a*d)*AppellF1[(1 + m)/4, 2, -1/2, (5 + m)/4, -((b*x^4)/a), -((d*x^4)/c)] + a^2*d^2*Hypergeometric2F1[1/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)]))/(a^2*c*(b*c - a*d)^2*(1 + m)*Sqrt[1 + (d*x^4)/c])","B",0
846,1,77,81,0.1134287,"\int \frac{(e x)^m}{\left(a+b x^4\right)^3 \sqrt{c+d x^4}} \, dx","Integrate[(e*x)^m/((a + b*x^4)^3*Sqrt[c + d*x^4]),x]","\frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m F_1\left(\frac{m+1}{4};3,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a^3 (m+1) \sqrt{c+d x^4}}","\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left(\frac{m+1}{4};3,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a^3 e (m+1) \sqrt{c+d x^4}}",1,"(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*AppellF1[(1 + m)/4, 3, 1/2, (5 + m)/4, -((b*x^4)/a), -((d*x^4)/c)])/(a^3*(1 + m)*Sqrt[c + d*x^4])","A",1
847,1,167,198,0.1587496,"\int \frac{(e x)^m \left(a+b x^4\right)^2}{\left(c+d x^4\right)^{3/2}} \, dx","Integrate[((e*x)^m*(a + b*x^4)^2)/(c + d*x^4)^(3/2),x]","\frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \left(a^2 \left(m^2+14 m+45\right) \, _2F_1\left(\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)+b (m+1) x^4 \left(2 a (m+9) \, _2F_1\left(\frac{3}{2},\frac{m+5}{4};\frac{m+9}{4};-\frac{d x^4}{c}\right)+b (m+5) x^4 \, _2F_1\left(\frac{3}{2},\frac{m+9}{4};\frac{m+13}{4};-\frac{d x^4}{c}\right)\right)\right)}{c (m+1) (m+5) (m+9) \sqrt{c+d x^4}}","-\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \left(2 b^2 c^2 (m+1)-(m+3) \left(2 a^2 d^2-(m+1) (b c-a d)^2\right)\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)}{2 c d^2 e (m+1) (m+3) \sqrt{c+d x^4}}+\frac{(e x)^{m+1} (b c-a d)^2}{2 c d^2 e \sqrt{c+d x^4}}+\frac{b^2 \sqrt{c+d x^4} (e x)^{m+1}}{d^2 e (m+3)}",1,"(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*(a^2*(45 + 14*m + m^2)*Hypergeometric2F1[3/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)] + b*(1 + m)*x^4*(2*a*(9 + m)*Hypergeometric2F1[3/2, (5 + m)/4, (9 + m)/4, -((d*x^4)/c)] + b*(5 + m)*x^4*Hypergeometric2F1[3/2, (9 + m)/4, (13 + m)/4, -((d*x^4)/c)])))/(c*(1 + m)*(5 + m)*(9 + m)*Sqrt[c + d*x^4])","A",1
848,1,113,132,0.1009308,"\int \frac{(e x)^m \left(a+b x^4\right)}{\left(c+d x^4\right)^{3/2}} \, dx","Integrate[((e*x)^m*(a + b*x^4))/(c + d*x^4)^(3/2),x]","\frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \left(a (m+5) \, _2F_1\left(\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)+b (m+1) x^4 \, _2F_1\left(\frac{3}{2},\frac{m+5}{4};\frac{m+9}{4};-\frac{d x^4}{c}\right)\right)}{c (m+1) (m+5) \sqrt{c+d x^4}}","\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} (a d (1-m)+b c (m+1)) \, _2F_1\left(\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)}{2 c d e (m+1) \sqrt{c+d x^4}}-\frac{(e x)^{m+1} (b c-a d)}{2 c d e \sqrt{c+d x^4}}",1,"(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*(a*(5 + m)*Hypergeometric2F1[3/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)] + b*(1 + m)*x^4*Hypergeometric2F1[3/2, (5 + m)/4, (9 + m)/4, -((d*x^4)/c)]))/(c*(1 + m)*(5 + m)*Sqrt[c + d*x^4])","A",1
849,1,69,71,0.0173565,"\int \frac{(e x)^m}{\left(c+d x^4\right)^{3/2}} \, dx","Integrate[(e*x)^m/(c + d*x^4)^(3/2),x]","\frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \, _2F_1\left(\frac{3}{2},\frac{m+1}{4};\frac{m+1}{4}+1;-\frac{d x^4}{c}\right)}{c (m+1) \sqrt{c+d x^4}}","\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)}{c e (m+1) \sqrt{c+d x^4}}",1,"(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*Hypergeometric2F1[3/2, (1 + m)/4, 1 + (1 + m)/4, -((d*x^4)/c)])/(c*(1 + m)*Sqrt[c + d*x^4])","A",1
850,1,169,84,0.2066425,"\int \frac{(e x)^m}{\left(a+b x^4\right) \left(c+d x^4\right)^{3/2}} \, dx","Integrate[(e*x)^m/((a + b*x^4)*(c + d*x^4)^(3/2)),x]","\frac{x \sqrt{c+d x^4} (e x)^m \left(b^2 c^2 F_1\left(\frac{m+1}{4};-\frac{1}{2},1;\frac{m+5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)+a d \left((a d-b c) \, _2F_1\left(\frac{3}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)-b c \, _2F_1\left(\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right)\right)\right)}{a c^2 (m+1) \sqrt{\frac{d x^4}{c}+1} (b c-a d)^2}","\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left(\frac{m+1}{4};1,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a c e (m+1) \sqrt{c+d x^4}}",1,"(x*(e*x)^m*Sqrt[c + d*x^4]*(b^2*c^2*AppellF1[(1 + m)/4, -1/2, 1, (5 + m)/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*(-(b*c*Hypergeometric2F1[1/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)]) + (-(b*c) + a*d)*Hypergeometric2F1[3/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)])))/(a*c^2*(b*c - a*d)^2*(1 + m)*Sqrt[1 + (d*x^4)/c])","B",0
851,1,77,84,0.1063944,"\int \frac{(e x)^m}{\left(a+b x^4\right)^2 \left(c+d x^4\right)^{3/2}} \, dx","Integrate[(e*x)^m/((a + b*x^4)^2*(c + d*x^4)^(3/2)),x]","\frac{x \left(\frac{d x^4}{c}+1\right)^{3/2} (e x)^m F_1\left(\frac{m+1}{4};2,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a^2 (m+1) \left(c+d x^4\right)^{3/2}}","\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left(\frac{m+1}{4};2,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a^2 c e (m+1) \sqrt{c+d x^4}}",1,"(x*(e*x)^m*(1 + (d*x^4)/c)^(3/2)*AppellF1[(1 + m)/4, 2, 3/2, (5 + m)/4, -((b*x^4)/a), -((d*x^4)/c)])/(a^2*(1 + m)*(c + d*x^4)^(3/2))","A",1
852,1,77,84,0.0948042,"\int \frac{(e x)^m}{\left(a+b x^4\right)^3 \left(c+d x^4\right)^{3/2}} \, dx","Integrate[(e*x)^m/((a + b*x^4)^3*(c + d*x^4)^(3/2)),x]","\frac{x \left(\frac{d x^4}{c}+1\right)^{3/2} (e x)^m F_1\left(\frac{m+1}{4};3,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a^3 (m+1) \left(c+d x^4\right)^{3/2}}","\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left(\frac{m+1}{4};3,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a^3 c e (m+1) \sqrt{c+d x^4}}",1,"(x*(e*x)^m*(1 + (d*x^4)/c)^(3/2)*AppellF1[(1 + m)/4, 3, 3/2, (5 + m)/4, -((b*x^4)/a), -((d*x^4)/c)])/(a^3*(1 + m)*(c + d*x^4)^(3/2))","A",1
853,1,91,104,0.2768992,"\int \frac{x^{17}}{\left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[x^17/((a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{\sqrt{c+d x^6} \left(-3 a d-2 b c+b d x^6\right)}{9 b^2 d^2}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{3 b^{5/2} \sqrt{b c-a d}}","-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{3 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^6} (a d+b c)}{3 b^2 d^2}+\frac{\left(c+d x^6\right)^{3/2}}{9 b d^2}",1,"(Sqrt[c + d*x^6]*(-2*b*c - 3*a*d + b*d*x^6))/(9*b^2*d^2) - (a^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(3*b^(5/2)*Sqrt[b*c - a*d])","A",1
854,1,72,74,0.1298526,"\int \frac{x^{11}}{\left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[x^11/((a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{1}{3} \left(\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^6}}{b d}\right)","\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{3 b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^6}}{3 b d}",1,"(Sqrt[c + d*x^6]/(b*d) + (a*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(b^(3/2)*Sqrt[b*c - a*d]))/3","A",1
855,1,51,51,0.0204761,"\int \frac{x^5}{\left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[x^5/((a + b*x^6)*Sqrt[c + d*x^6]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{3 \sqrt{b} \sqrt{b c-a d}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{3 \sqrt{b} \sqrt{b c-a d}}",1,"-1/3*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]]/(Sqrt[b]*Sqrt[b*c - a*d])","A",1
856,1,81,85,0.1264231,"\int \frac{1}{x \left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[1/(x*(a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right)}{\sqrt{c}}}{3 a}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{3 a \sqrt{b c-a d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right)}{3 a \sqrt{c}}",1,"(-(ArcTanh[Sqrt[c + d*x^6]/Sqrt[c]]/Sqrt[c]) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d])/(3*a)","A",1
857,1,151,117,0.1814437,"\int \frac{1}{x^7 \left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[1/(x^7*(a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{b^{3/2} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{3 a^2 (a d-b c)}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right)}{3 a^2 \sqrt{c}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right)}{6 a c^{3/2}}-\frac{\sqrt{c+d x^6}}{6 a c x^6}","-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{3 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right)}{6 a^2 c^{3/2}}-\frac{\sqrt{c+d x^6}}{6 a c x^6}",1,"-1/6*Sqrt[c + d*x^6]/(a*c*x^6) + (b*ArcTanh[Sqrt[c + d*x^6]/Sqrt[c]])/(3*a^2*Sqrt[c]) + (d*ArcTanh[Sqrt[c + d*x^6]/Sqrt[c]])/(6*a*c^(3/2)) + (b^(3/2)*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(3*a^2*(-(b*c) + a*d))","A",1
858,1,118,123,0.2356623,"\int \frac{x^{14}}{\left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[x^14/((a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{\frac{2 a^{3/2} \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{\sqrt{b c-a d}}-\frac{(2 a d+b c) \log \left(\sqrt{d} \sqrt{c+d x^6}+d x^3\right)}{d^{3/2}}+\frac{b x^3 \sqrt{c+d x^6}}{d}}{6 b^2}","\frac{a^{3/2} \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{3 b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right)}{6 b^2 d^{3/2}}+\frac{x^3 \sqrt{c+d x^6}}{6 b d}",1,"((b*x^3*Sqrt[c + d*x^6])/d + (2*a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/Sqrt[b*c - a*d] - ((b*c + 2*a*d)*Log[d*x^3 + Sqrt[d]*Sqrt[c + d*x^6]])/d^(3/2))/(6*b^2)","A",1
859,1,90,91,0.0914308,"\int \frac{x^8}{\left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[x^8/((a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{\frac{\log \left(\sqrt{d} \sqrt{c+d x^6}+d x^3\right)}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{\sqrt{b c-a d}}}{3 b}","\frac{\tanh ^{-1}\left(\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right)}{3 b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{3 b \sqrt{b c-a d}}",1,"(-((Sqrt[a]*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/Sqrt[b*c - a*d]) + Log[d*x^3 + Sqrt[d]*Sqrt[c + d*x^6]]/Sqrt[d])/(3*b)","A",1
860,1,95,54,0.0901721,"\int \frac{x^2}{\left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[x^2/((a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{x^3 \sqrt{\frac{d x^6}{c}+1} \tanh ^{-1}\left(\frac{\sqrt{\frac{d x^6}{c}-\frac{b x^6}{a}}}{\sqrt{\frac{d x^6}{c}+1}}\right)}{3 a \sqrt{c+d x^6} \sqrt{\frac{d x^6}{c}-\frac{b x^6}{a}}}","\frac{\tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{3 \sqrt{a} \sqrt{b c-a d}}",1,"(x^3*Sqrt[1 + (d*x^6)/c]*ArcTanh[Sqrt[-((b*x^6)/a) + (d*x^6)/c]/Sqrt[1 + (d*x^6)/c]])/(3*a*Sqrt[c + d*x^6]*Sqrt[-((b*x^6)/a) + (d*x^6)/c])","A",1
861,1,179,80,1.169039,"\int \frac{1}{x^4 \left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[1/(x^4*(a + b*x^6)*Sqrt[c + d*x^6]),x]","-\frac{\left(\frac{d x^6}{c}+1\right) \left(\frac{4 x^6 \left(c+d x^6\right) (b c-a d) \, _2F_1\left(2,2;\frac{5}{2};\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)}{3 c^2 \left(a+b x^6\right)}+\frac{\left(c+2 d x^6\right) \sin ^{-1}\left(\sqrt{\frac{x^6 (b c-a d)}{c \left(a+b x^6\right)}}\right)}{c \sqrt{\frac{a x^6 \left(c+d x^6\right) (b c-a d)}{c^2 \left(a+b x^6\right)^2}}}\right)}{3 x^3 \left(a+b x^6\right) \sqrt{c+d x^6}}","-\frac{b \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{3 a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^6}}{3 a c x^3}",1,"-1/3*((1 + (d*x^6)/c)*(((c + 2*d*x^6)*ArcSin[Sqrt[((b*c - a*d)*x^6)/(c*(a + b*x^6))]])/(c*Sqrt[(a*(b*c - a*d)*x^6*(c + d*x^6))/(c^2*(a + b*x^6)^2)]) + (4*(b*c - a*d)*x^6*(c + d*x^6)*Hypergeometric2F1[2, 2, 5/2, ((b*c - a*d)*x^6)/(c*(a + b*x^6))])/(3*c^2*(a + b*x^6))))/(x^3*(a + b*x^6)*Sqrt[c + d*x^6])","C",0
862,1,253,115,4.6499344,"\int \frac{1}{x^{10} \left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[1/(x^10*(a + b*x^6)*Sqrt[c + d*x^6]),x]","-\frac{\left(\frac{d x^6}{c}+1\right) \left(-\frac{8 x^6 \left(c+d x^6\right)^2 (b c-a d) \, _3F_2\left(2,2,2;1,\frac{5}{2};\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)}{a+b x^6}+\frac{3 c \left(c^2-4 c d x^6-8 d^2 x^{12}\right) \sin ^{-1}\left(\sqrt{\frac{x^6 (b c-a d)}{c \left(a+b x^6\right)}}\right)}{\sqrt{\frac{a x^6 \left(c+d x^6\right) (b c-a d)}{c^2 \left(a+b x^6\right)^2}}}+\frac{24 d x^{12} \left(c+d x^6\right) (a d-b c) \, _2F_1\left(2,2;\frac{5}{2};\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)}{a+b x^6}\right)}{27 c^3 x^9 \left(a+b x^6\right) \sqrt{c+d x^6}}","\frac{b^2 \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{3 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^6} (2 a d+3 b c)}{9 a^2 c^2 x^3}-\frac{\sqrt{c+d x^6}}{9 a c x^9}",1,"-1/27*((1 + (d*x^6)/c)*((3*c*(c^2 - 4*c*d*x^6 - 8*d^2*x^12)*ArcSin[Sqrt[((b*c - a*d)*x^6)/(c*(a + b*x^6))]])/Sqrt[(a*(b*c - a*d)*x^6*(c + d*x^6))/(c^2*(a + b*x^6)^2)] + (24*d*(-(b*c) + a*d)*x^12*(c + d*x^6)*Hypergeometric2F1[2, 2, 5/2, ((b*c - a*d)*x^6)/(c*(a + b*x^6))])/(a + b*x^6) - (8*(b*c - a*d)*x^6*(c + d*x^6)^2*HypergeometricPFQ[{2, 2, 2}, {1, 5/2}, ((b*c - a*d)*x^6)/(c*(a + b*x^6))])/(a + b*x^6)))/(c^3*x^9*(a + b*x^6)*Sqrt[c + d*x^6])","C",0
863,1,65,64,0.0482473,"\int \frac{x^4}{\left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[x^4/((a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{x^5 \sqrt{\frac{c+d x^6}{c}} F_1\left(\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)}{5 a \sqrt{c+d x^6}}","\frac{x^5 \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{5}{6};1,\frac{1}{2};\frac{11}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{5 a \sqrt{c+d x^6}}",1,"(x^5*Sqrt[(c + d*x^6)/c]*AppellF1[5/6, 1/2, 1, 11/6, -((d*x^6)/c), -((b*x^6)/a)])/(5*a*Sqrt[c + d*x^6])","A",0
864,1,65,64,0.0458447,"\int \frac{x^3}{\left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[x^3/((a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{x^4 \sqrt{\frac{c+d x^6}{c}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)}{4 a \sqrt{c+d x^6}}","\frac{x^4 \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{2}{3};1,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{4 a \sqrt{c+d x^6}}",1,"(x^4*Sqrt[(c + d*x^6)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^6)/c), -((b*x^6)/a)])/(4*a*Sqrt[c + d*x^6])","A",0
865,1,65,64,0.0391728,"\int \frac{x}{\left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[x/((a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{x^2 \sqrt{\frac{c+d x^6}{c}} F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)}{2 a \sqrt{c+d x^6}}","\frac{x^2 \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{1}{3};1,\frac{1}{2};\frac{4}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{2 a \sqrt{c+d x^6}}",1,"(x^2*Sqrt[(c + d*x^6)/c]*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^6)/c), -((b*x^6)/a)])/(2*a*Sqrt[c + d*x^6])","A",0
866,1,161,59,0.2012823,"\int \frac{1}{\left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[1/((a + b*x^6)*Sqrt[c + d*x^6]),x]","-\frac{7 a c x F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)}{\left(a+b x^6\right) \sqrt{c+d x^6} \left(3 x^6 \left(2 b c F_1\left(\frac{7}{6};\frac{1}{2},2;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+a d F_1\left(\frac{7}{6};\frac{3}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)\right)-7 a c F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)\right)}","\frac{x \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{1}{6};1,\frac{1}{2};\frac{7}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{a \sqrt{c+d x^6}}",1,"(-7*a*c*x*AppellF1[1/6, 1/2, 1, 7/6, -((d*x^6)/c), -((b*x^6)/a)])/((a + b*x^6)*Sqrt[c + d*x^6]*(-7*a*c*AppellF1[1/6, 1/2, 1, 7/6, -((d*x^6)/c), -((b*x^6)/a)] + 3*x^6*(2*b*c*AppellF1[7/6, 1/2, 2, 13/6, -((d*x^6)/c), -((b*x^6)/a)] + a*d*AppellF1[7/6, 3/2, 1, 13/6, -((d*x^6)/c), -((b*x^6)/a)])))","B",0
867,1,141,62,0.1281266,"\int \frac{1}{x^2 \left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[1/(x^2*(a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{-11 x^6 \sqrt{\frac{d x^6}{c}+1} (b c-2 a d) F_1\left(\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+10 b d x^{12} \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)-55 a \left(c+d x^6\right)}{55 a^2 c x \sqrt{c+d x^6}}","-\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left(-\frac{1}{6};1,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{a x \sqrt{c+d x^6}}",1,"(-55*a*(c + d*x^6) - 11*(b*c - 2*a*d)*x^6*Sqrt[1 + (d*x^6)/c]*AppellF1[5/6, 1/2, 1, 11/6, -((d*x^6)/c), -((b*x^6)/a)] + 10*b*d*x^12*Sqrt[1 + (d*x^6)/c]*AppellF1[11/6, 1/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)])/(55*a^2*c*x*Sqrt[c + d*x^6])","B",0
868,1,141,64,0.1195137,"\int \frac{1}{x^3 \left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[1/(x^3*(a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{5 x^6 \sqrt{\frac{d x^6}{c}+1} (a d-2 b c) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+2 b d x^{12} \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)-20 a \left(c+d x^6\right)}{40 a^2 c x^2 \sqrt{c+d x^6}}","-\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left(-\frac{1}{3};1,\frac{1}{2};\frac{2}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{2 a x^2 \sqrt{c+d x^6}}",1,"(-20*a*(c + d*x^6) + 5*(-2*b*c + a*d)*x^6*Sqrt[1 + (d*x^6)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^6)/c), -((b*x^6)/a)] + 2*b*d*x^12*Sqrt[1 + (d*x^6)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^6)/c), -((b*x^6)/a)])/(40*a^2*c*x^2*Sqrt[c + d*x^6])","B",0
869,1,141,64,0.164766,"\int \frac{1}{x^5 \left(a+b x^6\right) \sqrt{c+d x^6}} \, dx","Integrate[1/(x^5*(a + b*x^6)*Sqrt[c + d*x^6]),x]","\frac{-4 x^6 \sqrt{\frac{d x^6}{c}+1} (a d+4 b c) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)-b d x^{12} \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)-8 a \left(c+d x^6\right)}{32 a^2 c x^4 \sqrt{c+d x^6}}","-\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left(-\frac{2}{3};1,\frac{1}{2};\frac{1}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{4 a x^4 \sqrt{c+d x^6}}",1,"(-8*a*(c + d*x^6) - 4*(4*b*c + a*d)*x^6*Sqrt[1 + (d*x^6)/c]*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^6)/c), -((b*x^6)/a)] - b*d*x^12*Sqrt[1 + (d*x^6)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^6)/c), -((b*x^6)/a)])/(32*a^2*c*x^4*Sqrt[c + d*x^6])","B",0
870,1,107,123,0.3573684,"\int \frac{x^{17}}{\left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[x^17/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{1}{6} \left(\frac{\sqrt{c+d x^6} \left(\frac{a^2}{\left(a+b x^6\right) (a d-b c)}+\frac{2}{d}\right)}{b^2}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{b^{5/2} (b c-a d)^{3/2}}\right)","-\frac{a^2 \sqrt{c+d x^6}}{6 b^2 \left(a+b x^6\right) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{6 b^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^6}}{3 b^2 d}",1,"((Sqrt[c + d*x^6]*(2/d + a^2/((-(b*c) + a*d)*(a + b*x^6))))/b^2 + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(3/2)))/6","A",1
871,1,98,99,0.0964778,"\int \frac{x^{11}}{\left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[x^11/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{\frac{a \sqrt{b} \sqrt{c+d x^6}}{\left(a+b x^6\right) (b c-a d)}+\frac{(a d-2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}}{6 b^{3/2}}","\frac{a \sqrt{c+d x^6}}{6 b \left(a+b x^6\right) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{6 b^{3/2} (b c-a d)^{3/2}}",1,"((a*Sqrt[b]*Sqrt[c + d*x^6])/((b*c - a*d)*(a + b*x^6)) + ((-2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(6*b^(3/2))","A",1
872,1,85,87,0.1010301,"\int \frac{x^5}{\left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[x^5/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{1}{6} \left(\frac{\sqrt{c+d x^6}}{\left(a+b x^6\right) (a d-b c)}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{a d-b c}}\right)}{\sqrt{b} (a d-b c)^{3/2}}\right)","\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{6 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^6}}{6 \left(a+b x^6\right) (b c-a d)}",1,"(Sqrt[c + d*x^6]/((-(b*c) + a*d)*(a + b*x^6)) + (d*ArcTan[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[-(b*c) + a*d]])/(Sqrt[b]*(-(b*c) + a*d)^(3/2)))/6","A",1
873,1,123,132,0.3544513,"\int \frac{1}{x \left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[1/(x*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{\frac{a b \sqrt{c+d x^6}}{\left(a+b x^6\right) (b c-a d)}+\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right)}{\sqrt{c}}}{6 a^2}","\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{6 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right)}{3 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^6}}{6 a \left(a+b x^6\right) (b c-a d)}",1,"((a*b*Sqrt[c + d*x^6])/((b*c - a*d)*(a + b*x^6)) - (2*ArcTanh[Sqrt[c + d*x^6]/Sqrt[c]])/Sqrt[c] + (Sqrt[b]*(2*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(6*a^2)","A",1
874,1,163,185,0.6831571,"\int \frac{1}{x^7 \left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[1/(x^7*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{\frac{a \sqrt{c+d x^6} \left(a^2 d+a b \left(d x^6-c\right)-2 b^2 c x^6\right)}{x^6 \left(a+b x^6\right) (b c-a d)}+\frac{b^{3/2} c (5 a d-4 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right)}{\sqrt{c}}}{6 a^3 c}","-\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right)}{6 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right)}{6 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^6} (2 b c-a d)}{6 a^2 c \left(a+b x^6\right) (b c-a d)}-\frac{\sqrt{c+d x^6}}{6 a c x^6 \left(a+b x^6\right)}",1,"((a*Sqrt[c + d*x^6]*(a^2*d - 2*b^2*c*x^6 + a*b*(-c + d*x^6)))/((b*c - a*d)*x^6*(a + b*x^6)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^6]/Sqrt[c]])/Sqrt[c] + (b^(3/2)*c*(-4*b*c + 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(6*a^3*c)","A",1
875,1,135,141,0.2756104,"\int \frac{x^{14}}{\left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[x^14/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{\frac{a b x^3 \sqrt{c+d x^6}}{\left(a+b x^6\right) (b c-a d)}+\frac{\sqrt{a} (2 a d-3 b c) \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{(b c-a d)^{3/2}}+\frac{2 \log \left(\sqrt{d} \sqrt{c+d x^6}+d x^3\right)}{\sqrt{d}}}{6 b^2}","-\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{6 b^2 (b c-a d)^{3/2}}+\frac{a x^3 \sqrt{c+d x^6}}{6 b \left(a+b x^6\right) (b c-a d)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right)}{3 b^2 \sqrt{d}}",1,"((a*b*x^3*Sqrt[c + d*x^6])/((b*c - a*d)*(a + b*x^6)) + (Sqrt[a]*(-3*b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(b*c - a*d)^(3/2) + (2*Log[d*x^3 + Sqrt[d]*Sqrt[c + d*x^6]])/Sqrt[d])/(6*b^2)","A",1
876,1,124,93,0.5813956,"\int \frac{x^8}{\left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[x^8/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{\sqrt{c+d x^6} \left(-\frac{x^6 (b c-a d)}{a+b x^6}-\frac{c \sqrt{x^6 \left(\frac{d}{c}-\frac{b}{a}\right)} \tanh ^{-1}\left(\frac{\sqrt{x^6 \left(\frac{d}{c}-\frac{b}{a}\right)}}{\sqrt{\frac{d x^6}{c}+1}}\right)}{\sqrt{\frac{d x^6}{c}+1}}\right)}{6 x^3 (b c-a d)^2}","\frac{c \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{6 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^3 \sqrt{c+d x^6}}{6 \left(a+b x^6\right) (b c-a d)}",1,"(Sqrt[c + d*x^6]*(-(((b*c - a*d)*x^6)/(a + b*x^6)) - (c*Sqrt[(-(b/a) + d/c)*x^6]*ArcTanh[Sqrt[(-(b/a) + d/c)*x^6]/Sqrt[1 + (d*x^6)/c]])/Sqrt[1 + (d*x^6)/c]))/(6*(b*c - a*d)^2*x^3)","A",1
877,1,407,104,0.9980723,"\int \frac{x^2}{\left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[x^2/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{x^3 \sqrt{c+d x^6} \left(-30 d x^6 \sqrt{\frac{a x^6 \left(c+d x^6\right) (b c-a d)}{c^2 \left(a+b x^6\right)^2}}-45 c \sqrt{\frac{a x^6 \left(c+d x^6\right) (b c-a d)}{c^2 \left(a+b x^6\right)^2}}+16 d x^6 \left(\frac{x^6 (b c-a d)}{c \left(a+b x^6\right)}\right)^{5/2} \sqrt{\frac{a \left(c+d x^6\right)}{c \left(a+b x^6\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)+16 c \left(\frac{x^6 (b c-a d)}{c \left(a+b x^6\right)}\right)^{5/2} \sqrt{\frac{a \left(c+d x^6\right)}{c \left(a+b x^6\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)+30 d x^6 \sin ^{-1}\left(\sqrt{\frac{x^6 (b c-a d)}{c \left(a+b x^6\right)}}\right)+45 c \sin ^{-1}\left(\sqrt{\frac{x^6 (b c-a d)}{c \left(a+b x^6\right)}}\right)\right)}{90 c^2 \left(a+b x^6\right)^2 \left(\frac{x^6 (b c-a d)}{c \left(a+b x^6\right)}\right)^{3/2} \sqrt{\frac{a \left(c+d x^6\right)}{c \left(a+b x^6\right)}}}","\frac{(b c-2 a d) \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{6 a^{3/2} (b c-a d)^{3/2}}+\frac{b x^3 \sqrt{c+d x^6}}{6 a \left(a+b x^6\right) (b c-a d)}",1,"(x^3*Sqrt[c + d*x^6]*(-45*c*Sqrt[(a*(b*c - a*d)*x^6*(c + d*x^6))/(c^2*(a + b*x^6)^2)] - 30*d*x^6*Sqrt[(a*(b*c - a*d)*x^6*(c + d*x^6))/(c^2*(a + b*x^6)^2)] + 45*c*ArcSin[Sqrt[((b*c - a*d)*x^6)/(c*(a + b*x^6))]] + 30*d*x^6*ArcSin[Sqrt[((b*c - a*d)*x^6)/(c*(a + b*x^6))]] + 16*c*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^6)/(c*(a + b*x^6))] + 16*d*x^6*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^6)/(c*(a + b*x^6))]))/(90*c^2*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(3/2)*(a + b*x^6)^2*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))])","C",0
878,1,869,149,5.4009257,"\int \frac{1}{x^4 \left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[1/(x^4*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]","-\frac{\sqrt{d x^6+c} \left(120 d^2 \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}}\right) x^{12}+96 d^2 \left(\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^6+c\right)}{c \left(b x^6+a\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right) x^{12}+32 d^2 \left(\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^6+c\right)}{c \left(b x^6+a\right)}} \, _3F_2\left(2,2,3;1,\frac{7}{2};\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right) x^{12}-120 d^2 \sqrt{\frac{a (b c-a d) x^6 \left(d x^6+c\right)}{c^2 \left(b x^6+a\right)^2}} x^{12}+180 c d \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}}\right) x^6+160 c d \left(\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^6+c\right)}{c \left(b x^6+a\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right) x^6+64 c d \left(\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^6+c\right)}{c \left(b x^6+a\right)}} \, _3F_2\left(2,2,3;1,\frac{7}{2};\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right) x^6-180 c d \sqrt{\frac{a (b c-a d) x^6 \left(d x^6+c\right)}{c^2 \left(b x^6+a\right)^2}} x^6+45 c^2 \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}}\right)+64 c^2 \left(\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^6+c\right)}{c \left(b x^6+a\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)+32 c^2 \left(\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^6+c\right)}{c \left(b x^6+a\right)}} \, _3F_2\left(2,2,3;1,\frac{7}{2};\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)-45 c^2 \sqrt{\frac{a (b c-a d) x^6 \left(d x^6+c\right)}{c^2 \left(b x^6+a\right)^2}}\right)}{90 c^3 x^3 \left(\frac{(b c-a d) x^6}{c \left(b x^6+a\right)}\right)^{3/2} \left(b x^6+a\right)^2 \sqrt{\frac{a \left(d x^6+c\right)}{c \left(b x^6+a\right)}}}","-\frac{b (3 b c-4 a d) \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{6 a^{5/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^6} (3 b c-2 a d)}{6 a^2 c x^3 (b c-a d)}+\frac{b \sqrt{c+d x^6}}{6 a x^3 \left(a+b x^6\right) (b c-a d)}",1,"-1/90*(Sqrt[c + d*x^6]*(-45*c^2*Sqrt[(a*(b*c - a*d)*x^6*(c + d*x^6))/(c^2*(a + b*x^6)^2)] - 180*c*d*x^6*Sqrt[(a*(b*c - a*d)*x^6*(c + d*x^6))/(c^2*(a + b*x^6)^2)] - 120*d^2*x^12*Sqrt[(a*(b*c - a*d)*x^6*(c + d*x^6))/(c^2*(a + b*x^6)^2)] + 45*c^2*ArcSin[Sqrt[((b*c - a*d)*x^6)/(c*(a + b*x^6))]] + 180*c*d*x^6*ArcSin[Sqrt[((b*c - a*d)*x^6)/(c*(a + b*x^6))]] + 120*d^2*x^12*ArcSin[Sqrt[((b*c - a*d)*x^6)/(c*(a + b*x^6))]] + 64*c^2*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^6)/(c*(a + b*x^6))] + 160*c*d*x^6*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^6)/(c*(a + b*x^6))] + 96*d^2*x^12*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^6)/(c*(a + b*x^6))] + 32*c^2*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*HypergeometricPFQ[{2, 2, 3}, {1, 7/2}, ((b*c - a*d)*x^6)/(c*(a + b*x^6))] + 64*c*d*x^6*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*HypergeometricPFQ[{2, 2, 3}, {1, 7/2}, ((b*c - a*d)*x^6)/(c*(a + b*x^6))] + 32*d^2*x^12*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*HypergeometricPFQ[{2, 2, 3}, {1, 7/2}, ((b*c - a*d)*x^6)/(c*(a + b*x^6))]))/(c^3*x^3*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(3/2)*(a + b*x^6)^2*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))])","C",0
879,1,175,208,5.8580985,"\int \frac{1}{x^{10} \left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[1/(x^10*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{a^2 \left(c+d x^6\right) \left(\frac{3 b^3 x^{12}}{\left(a+b x^6\right) (b c-a d)}+\frac{4 x^6 (a d+3 b c)}{c^2}-\frac{2 a}{c}\right)+\frac{3 b^2 x^{18} \sqrt{\frac{d x^6}{c}+1} (5 b c-6 a d) \sin ^{-1}\left(\frac{\sqrt{x^6 \left(\frac{b}{a}-\frac{d}{c}\right)}}{\sqrt{\frac{b x^6}{a}+1}}\right)}{c \left(\frac{x^6 (b c-a d)}{a c}\right)^{3/2}}}{18 a^5 x^9 \sqrt{c+d x^6}}","\frac{b^2 (5 b c-6 a d) \tan ^{-1}\left(\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right)}{6 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^6} (5 b c-2 a d)}{18 a^2 c x^9 (b c-a d)}+\frac{\sqrt{c+d x^6} \left(-4 a^2 d^2-8 a b c d+15 b^2 c^2\right)}{18 a^3 c^2 x^3 (b c-a d)}+\frac{b \sqrt{c+d x^6}}{6 a x^9 \left(a+b x^6\right) (b c-a d)}",1,"(a^2*(c + d*x^6)*((-2*a)/c + (4*(3*b*c + a*d)*x^6)/c^2 + (3*b^3*x^12)/((b*c - a*d)*(a + b*x^6))) + (3*b^2*(5*b*c - 6*a*d)*x^18*Sqrt[1 + (d*x^6)/c]*ArcSin[Sqrt[(b/a - d/c)*x^6]/Sqrt[1 + (b*x^6)/a]])/(c*(((b*c - a*d)*x^6)/(a*c))^(3/2)))/(18*a^5*x^9*Sqrt[c + d*x^6])","A",0
880,1,169,64,0.2181477,"\int \frac{x^4}{\left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[x^4/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{x^5 \left(-10 b d x^6 \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+11 \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} (b c-6 a d) F_1\left(\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+55 a b \left(c+d x^6\right)\right)}{330 a^2 \left(a+b x^6\right) \sqrt{c+d x^6} (b c-a d)}","\frac{x^5 \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{5}{6};2,\frac{1}{2};\frac{11}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{5 a^2 \sqrt{c+d x^6}}",1,"(x^5*(55*a*b*(c + d*x^6) + 11*(b*c - 6*a*d)*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[5/6, 1/2, 1, 11/6, -((d*x^6)/c), -((b*x^6)/a)] - 10*b*d*x^6*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[11/6, 1/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)]))/(330*a^2*(b*c - a*d)*(a + b*x^6)*Sqrt[c + d*x^6])","B",0
881,1,168,64,0.2437321,"\int \frac{x^3}{\left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[x^3/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]","-\frac{x^4 \left(b d x^6 \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)-5 \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} (b c-3 a d) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)-10 a b \left(c+d x^6\right)\right)}{60 a^2 \left(a+b x^6\right) \sqrt{c+d x^6} (b c-a d)}","\frac{x^4 \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{4 a^2 \sqrt{c+d x^6}}",1,"-1/60*(x^4*(-10*a*b*(c + d*x^6) - 5*(b*c - 3*a*d)*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^6)/c), -((b*x^6)/a)] + b*d*x^6*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^6)/c), -((b*x^6)/a)]))/(a^2*(b*c - a*d)*(a + b*x^6)*Sqrt[c + d*x^6])","B",0
882,1,172,64,0.1921641,"\int \frac{x}{\left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[x/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{b d x^8 \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+8 x^2 \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} (2 b c-3 a d) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+8 a b x^2 \left(c+d x^6\right)}{48 a^2 \left(a+b x^6\right) \sqrt{c+d x^6} (b c-a d)}","\frac{x^2 \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{1}{3};2,\frac{1}{2};\frac{4}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{2 a^2 \sqrt{c+d x^6}}",1,"(8*a*b*x^2*(c + d*x^6) + 8*(2*b*c - 3*a*d)*x^2*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^6)/c), -((b*x^6)/a)] + b*d*x^8*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^6)/c), -((b*x^6)/a)])/(48*a^2*(b*c - a*d)*(a + b*x^6)*Sqrt[c + d*x^6])","B",0
883,1,329,59,0.3949371,"\int \frac{1}{\left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[1/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{x \left(-2 b d x^6 \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{7}{6};\frac{1}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)-\frac{7 a \left(3 b x^6 \left(c+d x^6\right) \left(2 b c F_1\left(\frac{7}{6};\frac{1}{2},2;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+a d F_1\left(\frac{7}{6};\frac{3}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)\right)+7 a c \left(6 a d-b \left(6 c+d x^6\right)\right) F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)\right)}{\left(a+b x^6\right) \left(3 x^6 \left(2 b c F_1\left(\frac{7}{6};\frac{1}{2},2;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+a d F_1\left(\frac{7}{6};\frac{3}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)\right)-7 a c F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)\right)}\right)}{42 a^2 \sqrt{c+d x^6} (a d-b c)}","\frac{x \sqrt{\frac{d x^6}{c}+1} F_1\left(\frac{1}{6};2,\frac{1}{2};\frac{7}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{a^2 \sqrt{c+d x^6}}",1,"(x*(-2*b*d*x^6*Sqrt[1 + (d*x^6)/c]*AppellF1[7/6, 1/2, 1, 13/6, -((d*x^6)/c), -((b*x^6)/a)] - (7*a*(7*a*c*(6*a*d - b*(6*c + d*x^6))*AppellF1[1/6, 1/2, 1, 7/6, -((d*x^6)/c), -((b*x^6)/a)] + 3*b*x^6*(c + d*x^6)*(2*b*c*AppellF1[7/6, 1/2, 2, 13/6, -((d*x^6)/c), -((b*x^6)/a)] + a*d*AppellF1[7/6, 3/2, 1, 13/6, -((d*x^6)/c), -((b*x^6)/a)])))/((a + b*x^6)*(-7*a*c*AppellF1[1/6, 1/2, 1, 7/6, -((d*x^6)/c), -((b*x^6)/a)] + 3*x^6*(2*b*c*AppellF1[7/6, 1/2, 2, 13/6, -((d*x^6)/c), -((b*x^6)/a)] + a*d*AppellF1[7/6, 3/2, 1, 13/6, -((d*x^6)/c), -((b*x^6)/a)])))))/(42*a^2*(-(b*c) + a*d)*Sqrt[c + d*x^6])","B",0
884,1,226,62,0.2885146,"\int \frac{1}{x^2 \left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[1/(x^2*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{-11 x^6 \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} \left(12 a^2 d^2-24 a b c d+7 b^2 c^2\right) F_1\left(\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+55 a \left(c+d x^6\right) \left(6 a^2 d-6 a b \left(c-d x^6\right)-7 b^2 c x^6\right)+10 b d x^{12} \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} (7 b c-6 a d) F_1\left(\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)}{330 a^3 c x \left(a+b x^6\right) \sqrt{c+d x^6} (b c-a d)}","-\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left(-\frac{1}{6};2,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{a^2 x \sqrt{c+d x^6}}",1,"(55*a*(c + d*x^6)*(6*a^2*d - 7*b^2*c*x^6 - 6*a*b*(c - d*x^6)) - 11*(7*b^2*c^2 - 24*a*b*c*d + 12*a^2*d^2)*x^6*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[5/6, 1/2, 1, 11/6, -((d*x^6)/c), -((b*x^6)/a)] + 10*b*d*(7*b*c - 6*a*d)*x^12*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[11/6, 1/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)])/(330*a^3*c*(b*c - a*d)*x*(a + b*x^6)*Sqrt[c + d*x^6])","B",0
885,1,226,64,0.3749192,"\int \frac{1}{x^3 \left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[1/(x^3*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{-5 x^6 \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} \left(3 a^2 d^2-15 a b c d+8 b^2 c^2\right) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+20 a \left(c+d x^6\right) \left(3 a^2 d-3 a b \left(c-d x^6\right)-4 b^2 c x^6\right)+2 b d x^{12} \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} (4 b c-3 a d) F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)}{120 a^3 c x^2 \left(a+b x^6\right) \sqrt{c+d x^6} (b c-a d)}","-\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left(-\frac{1}{3};2,\frac{1}{2};\frac{2}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{2 a^2 x^2 \sqrt{c+d x^6}}",1,"(20*a*(c + d*x^6)*(3*a^2*d - 4*b^2*c*x^6 - 3*a*b*(c - d*x^6)) - 5*(8*b^2*c^2 - 15*a*b*c*d + 3*a^2*d^2)*x^6*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^6)/c), -((b*x^6)/a)] + 2*b*d*(4*b*c - 3*a*d)*x^12*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^6)/c), -((b*x^6)/a)])/(120*a^3*c*(b*c - a*d)*x^2*(a + b*x^6)*Sqrt[c + d*x^6])","B",0
886,1,225,64,0.2848656,"\int \frac{1}{x^5 \left(a+b x^6\right)^2 \sqrt{c+d x^6}} \, dx","Integrate[1/(x^5*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]","\frac{4 x^6 \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} \left(3 a^2 d^2+21 a b c d-20 b^2 c^2\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)+8 a \left(c+d x^6\right) \left(3 a^2 d-3 a b \left(c-d x^6\right)-5 b^2 c x^6\right)+b d x^{12} \left(a+b x^6\right) \sqrt{\frac{d x^6}{c}+1} (3 a d-5 b c) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right)}{96 a^3 c x^4 \left(a+b x^6\right) \sqrt{c+d x^6} (b c-a d)}","-\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left(-\frac{2}{3};2,\frac{1}{2};\frac{1}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right)}{4 a^2 x^4 \sqrt{c+d x^6}}",1,"(8*a*(c + d*x^6)*(3*a^2*d - 5*b^2*c*x^6 - 3*a*b*(c - d*x^6)) + 4*(-20*b^2*c^2 + 21*a*b*c*d + 3*a^2*d^2)*x^6*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^6)/c), -((b*x^6)/a)] + b*d*(-5*b*c + 3*a*d)*x^12*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^6)/c), -((b*x^6)/a)])/(96*a^3*c*(b*c - a*d)*x^4*(a + b*x^6)*Sqrt[c + d*x^6])","B",0
887,1,91,104,0.2156447,"\int \frac{x^{23}}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x^23/((a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{\sqrt{c+d x^8} \left(-3 a d-2 b c+b d x^8\right)}{12 b^2 d^2}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{4 b^{5/2} \sqrt{b c-a d}}","-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{4 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^8} (a d+b c)}{4 b^2 d^2}+\frac{\left(c+d x^8\right)^{3/2}}{12 b d^2}",1,"(Sqrt[c + d*x^8]*(-2*b*c - 3*a*d + b*d*x^8))/(12*b^2*d^2) - (a^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(4*b^(5/2)*Sqrt[b*c - a*d])","A",1
888,1,72,74,0.0776874,"\int \frac{x^{15}}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x^15/((a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{1}{4} \left(\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^8}}{b d}\right)","\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{4 b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^8}}{4 b d}",1,"(Sqrt[c + d*x^8]/(b*d) + (a*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(b^(3/2)*Sqrt[b*c - a*d]))/4","A",1
889,1,51,51,0.0166766,"\int \frac{x^7}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x^7/((a + b*x^8)*Sqrt[c + d*x^8]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{4 \sqrt{b} \sqrt{b c-a d}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{4 \sqrt{b} \sqrt{b c-a d}}",1,"-1/4*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]]/(Sqrt[b]*Sqrt[b*c - a*d])","A",1
890,1,81,85,0.0988318,"\int \frac{1}{x \left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[1/(x*(a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right)}{\sqrt{c}}}{4 a}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{4 a \sqrt{b c-a d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right)}{4 a \sqrt{c}}",1,"(-(ArcTanh[Sqrt[c + d*x^8]/Sqrt[c]]/Sqrt[c]) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d])/(4*a)","A",1
891,1,151,117,0.1390815,"\int \frac{1}{x^9 \left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[1/(x^9*(a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{b^{3/2} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{4 a^2 (a d-b c)}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right)}{4 a^2 \sqrt{c}}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right)}{8 a c^{3/2}}-\frac{\sqrt{c+d x^8}}{8 a c x^8}","-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{4 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right)}{8 a^2 c^{3/2}}-\frac{\sqrt{c+d x^8}}{8 a c x^8}",1,"-1/8*Sqrt[c + d*x^8]/(a*c*x^8) + (b*ArcTanh[Sqrt[c + d*x^8]/Sqrt[c]])/(4*a^2*Sqrt[c]) + (d*ArcTanh[Sqrt[c + d*x^8]/Sqrt[c]])/(8*a*c^(3/2)) + (b^(3/2)*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(4*a^2*(-(b*c) + a*d))","A",1
892,1,118,123,0.2399523,"\int \frac{x^{19}}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x^19/((a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{\frac{2 a^{3/2} \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{\sqrt{b c-a d}}-\frac{(2 a d+b c) \log \left(\sqrt{d} \sqrt{c+d x^8}+d x^4\right)}{d^{3/2}}+\frac{b x^4 \sqrt{c+d x^8}}{d}}{8 b^2}","\frac{a^{3/2} \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{4 b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right)}{8 b^2 d^{3/2}}+\frac{x^4 \sqrt{c+d x^8}}{8 b d}",1,"((b*x^4*Sqrt[c + d*x^8])/d + (2*a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/Sqrt[b*c - a*d] - ((b*c + 2*a*d)*Log[d*x^4 + Sqrt[d]*Sqrt[c + d*x^8]])/d^(3/2))/(8*b^2)","A",1
893,1,90,91,0.067744,"\int \frac{x^{11}}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x^11/((a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{\frac{\log \left(\sqrt{d} \sqrt{c+d x^8}+d x^4\right)}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{\sqrt{b c-a d}}}{4 b}","\frac{\tanh ^{-1}\left(\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right)}{4 b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{4 b \sqrt{b c-a d}}",1,"(-((Sqrt[a]*ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/Sqrt[b*c - a*d]) + Log[d*x^4 + Sqrt[d]*Sqrt[c + d*x^8]]/Sqrt[d])/(4*b)","A",1
894,1,95,54,0.0676257,"\int \frac{x^3}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x^3/((a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{x^4 \sqrt{\frac{d x^8}{c}+1} \tanh ^{-1}\left(\frac{\sqrt{\frac{d x^8}{c}-\frac{b x^8}{a}}}{\sqrt{\frac{d x^8}{c}+1}}\right)}{4 a \sqrt{c+d x^8} \sqrt{\frac{d x^8}{c}-\frac{b x^8}{a}}}","\frac{\tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{4 \sqrt{a} \sqrt{b c-a d}}",1,"(x^4*Sqrt[1 + (d*x^8)/c]*ArcTanh[Sqrt[-((b*x^8)/a) + (d*x^8)/c]/Sqrt[1 + (d*x^8)/c]])/(4*a*Sqrt[c + d*x^8]*Sqrt[-((b*x^8)/a) + (d*x^8)/c])","A",1
895,1,179,80,0.8322057,"\int \frac{1}{x^5 \left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[1/(x^5*(a + b*x^8)*Sqrt[c + d*x^8]),x]","-\frac{\left(\frac{d x^8}{c}+1\right) \left(\frac{4 x^8 \left(c+d x^8\right) (b c-a d) \, _2F_1\left(2,2;\frac{5}{2};\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)}{3 c^2 \left(a+b x^8\right)}+\frac{\left(c+2 d x^8\right) \sin ^{-1}\left(\sqrt{\frac{x^8 (b c-a d)}{c \left(a+b x^8\right)}}\right)}{c \sqrt{\frac{a x^8 \left(c+d x^8\right) (b c-a d)}{c^2 \left(a+b x^8\right)^2}}}\right)}{4 x^4 \left(a+b x^8\right) \sqrt{c+d x^8}}","-\frac{b \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{4 a^{3/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x^8}}{4 a c x^4}",1,"-1/4*((1 + (d*x^8)/c)*(((c + 2*d*x^8)*ArcSin[Sqrt[((b*c - a*d)*x^8)/(c*(a + b*x^8))]])/(c*Sqrt[(a*(b*c - a*d)*x^8*(c + d*x^8))/(c^2*(a + b*x^8)^2)]) + (4*(b*c - a*d)*x^8*(c + d*x^8)*Hypergeometric2F1[2, 2, 5/2, ((b*c - a*d)*x^8)/(c*(a + b*x^8))])/(3*c^2*(a + b*x^8))))/(x^4*(a + b*x^8)*Sqrt[c + d*x^8])","C",0
896,1,253,115,2.2725681,"\int \frac{1}{x^{13} \left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[1/(x^13*(a + b*x^8)*Sqrt[c + d*x^8]),x]","-\frac{\left(\frac{d x^8}{c}+1\right) \left(-\frac{8 x^8 \left(c+d x^8\right)^2 (b c-a d) \, _3F_2\left(2,2,2;1,\frac{5}{2};\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)}{a+b x^8}+\frac{3 c \left(c^2-4 c d x^8-8 d^2 x^{16}\right) \sin ^{-1}\left(\sqrt{\frac{x^8 (b c-a d)}{c \left(a+b x^8\right)}}\right)}{\sqrt{\frac{a x^8 \left(c+d x^8\right) (b c-a d)}{c^2 \left(a+b x^8\right)^2}}}+\frac{24 d x^{16} \left(c+d x^8\right) (a d-b c) \, _2F_1\left(2,2;\frac{5}{2};\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)}{a+b x^8}\right)}{36 c^3 x^{12} \left(a+b x^8\right) \sqrt{c+d x^8}}","\frac{b^2 \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{4 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^8} (2 a d+3 b c)}{12 a^2 c^2 x^4}-\frac{\sqrt{c+d x^8}}{12 a c x^{12}}",1,"-1/36*((1 + (d*x^8)/c)*((3*c*(c^2 - 4*c*d*x^8 - 8*d^2*x^16)*ArcSin[Sqrt[((b*c - a*d)*x^8)/(c*(a + b*x^8))]])/Sqrt[(a*(b*c - a*d)*x^8*(c + d*x^8))/(c^2*(a + b*x^8)^2)] + (24*d*(-(b*c) + a*d)*x^16*(c + d*x^8)*Hypergeometric2F1[2, 2, 5/2, ((b*c - a*d)*x^8)/(c*(a + b*x^8))])/(a + b*x^8) - (8*(b*c - a*d)*x^8*(c + d*x^8)^2*HypergeometricPFQ[{2, 2, 2}, {1, 5/2}, ((b*c - a*d)*x^8)/(c*(a + b*x^8))])/(a + b*x^8)))/(c^3*x^12*(a + b*x^8)*Sqrt[c + d*x^8])","C",0
897,1,65,851,0.0432159,"\int \frac{x^9}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x^9/((a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{x^{10} \sqrt{\frac{c+d x^8}{c}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)}{10 a \sqrt{c+d x^8}}","-\frac{\left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{16 b \sqrt[4]{c} \sqrt[4]{d} (b c+a d) \sqrt{d x^8+c}}-\frac{\sqrt[4]{-a} \tan ^{-1}\left(\frac{\sqrt{b c-a d} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{8 b^{3/4} \sqrt{b c-a d}}-\frac{\sqrt[4]{-a} \tan ^{-1}\left(\frac{\sqrt{a d-b c} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{8 b^{3/4} \sqrt{a d-b c}}+\frac{\left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{4 b \sqrt[4]{c} \sqrt[4]{d} \sqrt{d x^8+c}}-\frac{a \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right) \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b \sqrt[4]{c} (b c+a d) \sqrt{d x^8+c}}-\frac{\left(\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right) \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b \sqrt[4]{c} (b c+a d) \sqrt{d x^8+c}}-\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 b \sqrt[4]{c} \sqrt[4]{d} (b c+a d) \sqrt{d x^8+c}}",1,"(x^10*Sqrt[(c + d*x^8)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^8)/c), -((b*x^8)/a)])/(10*a*Sqrt[c + d*x^8])","C",0
898,1,65,754,0.0452684,"\int \frac{x}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x/((a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{x^2 \sqrt{\frac{c+d x^8}{c}} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)}{2 a \sqrt{c+d x^8}}","-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{c+d x^8}}\right)}{8 (-a)^{3/4} \sqrt{b c-a d}}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{x^2 \sqrt{a d-b c}}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{c+d x^8}}\right)}{8 (-a)^{3/4} \sqrt{a d-b c}}+\frac{\sqrt[4]{d} \left(\sqrt{c}+\sqrt{d} x^4\right) \sqrt{\frac{c+d x^8}{\left(\sqrt{c}+\sqrt{d} x^4\right)^2}} \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 \sqrt[4]{c} \sqrt{c+d x^8} (a d+b c)}+\frac{\sqrt[4]{d} \left(\sqrt{c}+\sqrt{d} x^4\right) \sqrt{\frac{c+d x^8}{\left(\sqrt{c}+\sqrt{d} x^4\right)^2}} \left(\sqrt{-a} \sqrt{b} \sqrt{c}+a \sqrt{d}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a \sqrt[4]{c} \sqrt{c+d x^8} (a d+b c)}+\frac{\left(\sqrt{c}+\sqrt{d} x^4\right) \sqrt{\frac{c+d x^8}{\left(\sqrt{c}+\sqrt{d} x^4\right)^2}} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2 \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^8} (a d+b c)}+\frac{\left(\sqrt{c}+\sqrt{d} x^4\right) \sqrt{\frac{c+d x^8}{\left(\sqrt{c}+\sqrt{d} x^4\right)^2}} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right)^2 \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^8} (a d+b c)}",1,"(x^2*Sqrt[(c + d*x^8)/c]*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b*x^8)/a)])/(2*a*Sqrt[c + d*x^8])","C",0
899,1,141,878,0.1632997,"\int \frac{1}{x^7 \left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[1/(x^7*(a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{-5 x^8 \sqrt{\frac{d x^8}{c}+1} (a d+3 b c) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)-b d x^{16} \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)-5 a \left(c+d x^8\right)}{30 a^2 c x^6 \sqrt{c+d x^8}}","-\frac{b \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{16 a^2 \sqrt[4]{c} \sqrt[4]{d} (b c+a d) \sqrt{d x^8+c}}-\frac{b^{5/4} \tan ^{-1}\left(\frac{\sqrt{b c-a d} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{8 (-a)^{7/4} \sqrt{b c-a d}}-\frac{b^{5/4} \tan ^{-1}\left(\frac{\sqrt{a d-b c} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{8 (-a)^{7/4} \sqrt{a d-b c}}-\frac{d^{3/4} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{12 a c^{5/4} \sqrt{d x^8+c}}-\frac{b \left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right) \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a \sqrt[4]{c} (b c+a d) \sqrt{d x^8+c}}-\frac{b \left(\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right) \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a^2 \sqrt[4]{c} (b c+a d) \sqrt{d x^8+c}}-\frac{b \left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a^2 \sqrt[4]{c} \sqrt[4]{d} (b c+a d) \sqrt{d x^8+c}}-\frac{\sqrt{d x^8+c}}{6 a c x^6}",1,"(-5*a*(c + d*x^8) - 5*(3*b*c + a*d)*x^8*Sqrt[1 + (d*x^8)/c]*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b*x^8)/a)] - b*d*x^16*Sqrt[1 + (d*x^8)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^8)/c), -((b*x^8)/a)])/(30*a^2*c*x^6*Sqrt[c + d*x^8])","C",0
900,1,65,1005,0.0677318,"\int \frac{x^{13}}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x^13/((a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{x^{14} \sqrt{\frac{c+d x^8}{c}} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)}{14 a \sqrt{c+d x^8}}","-\frac{\sqrt{-a} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{16 b^{3/2} \sqrt[4]{c} \sqrt[4]{d} (b c+a d) \sqrt{d x^8+c}}+\frac{(-a)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b c-a d} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{8 b^{5/4} \sqrt{b c-a d}}-\frac{(-a)^{3/4} \tan ^{-1}\left(\frac{\sqrt{a d-b c} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{8 b^{5/4} \sqrt{a d-b c}}-\frac{\sqrt[4]{c} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{2 b d^{3/4} \sqrt{d x^8+c}}+\frac{\sqrt[4]{c} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{4 b d^{3/4} \sqrt{d x^8+c}}+\frac{a \left(\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b \sqrt[4]{c} (b c+a d) \sqrt{d x^8+c}}+\frac{a \left(\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b \sqrt[4]{c} (b c+a d) \sqrt{d x^8+c}}+\frac{\sqrt{-a} \left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 b^{3/2} \sqrt[4]{c} \sqrt[4]{d} (b c+a d) \sqrt{d x^8+c}}+\frac{x^2 \sqrt{d x^8+c}}{2 b \sqrt{d} \left(\sqrt{d} x^4+\sqrt{c}\right)}",1,"(x^14*Sqrt[(c + d*x^8)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)])/(14*a*Sqrt[c + d*x^8])","C",0
901,1,65,768,0.0433156,"\int \frac{x^5}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x^5/((a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{x^6 \sqrt{\frac{c+d x^8}{c}} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)}{6 a \sqrt{c+d x^8}}","\frac{\tan ^{-1}\left(\frac{x^2 \sqrt{b c-a d}}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{c+d x^8}}\right)}{8 \sqrt[4]{-a} \sqrt[4]{b} \sqrt{b c-a d}}-\frac{\tan ^{-1}\left(\frac{x^2 \sqrt{a d-b c}}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{c+d x^8}}\right)}{8 \sqrt[4]{-a} \sqrt[4]{b} \sqrt{a d-b c}}-\frac{\sqrt[4]{d} \left(\sqrt{c}+\sqrt{d} x^4\right) \sqrt{\frac{c+d x^8}{\left(\sqrt{c}+\sqrt{d} x^4\right)^2}} \left(\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 \sqrt[4]{c} \sqrt{c+d x^8} (a d+b c)}-\frac{\sqrt[4]{d} \left(\sqrt{c}+\sqrt{d} x^4\right) \sqrt{\frac{c+d x^8}{\left(\sqrt{c}+\sqrt{d} x^4\right)^2}} \left(\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}+\sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 \sqrt[4]{c} \sqrt{c+d x^8} (a d+b c)}-\frac{\left(\sqrt{c}+\sqrt{d} x^4\right) \sqrt{\frac{c+d x^8}{\left(\sqrt{c}+\sqrt{d} x^4\right)^2}} \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2 \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 \sqrt{-a} \sqrt{b} \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^8} (a d+b c)}+\frac{\left(\sqrt{c}+\sqrt{d} x^4\right) \sqrt{\frac{c+d x^8}{\left(\sqrt{c}+\sqrt{d} x^4\right)^2}} \left(\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right)^2 \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 \sqrt{-a} \sqrt{b} \sqrt[4]{c} \sqrt[4]{d} \sqrt{c+d x^8} (a d+b c)}",1,"(x^6*Sqrt[(c + d*x^8)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^8)/c), -((b*x^8)/a)])/(6*a*Sqrt[c + d*x^8])","C",0
902,1,141,1032,0.1281242,"\int \frac{1}{x^3 \left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[1/(x^3*(a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{7 x^8 \sqrt{\frac{d x^8}{c}+1} (a d-b c) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+3 b d x^{16} \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)-21 a \left(c+d x^8\right)}{42 a^2 c x^2 \sqrt{c+d x^8}}","-\frac{\sqrt{b} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{16 (-a)^{3/2} \sqrt[4]{c} \sqrt[4]{d} (b c+a d) \sqrt{d x^8+c}}+\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt{b c-a d} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{8 (-a)^{5/4} \sqrt{b c-a d}}-\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt{a d-b c} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{8 (-a)^{5/4} \sqrt{a d-b c}}-\frac{\sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{2 a c^{3/4} \sqrt{d x^8+c}}+\frac{\sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{4 a c^{3/4} \sqrt{d x^8+c}}+\frac{b \left(\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a \sqrt[4]{c} (b c+a d) \sqrt{d x^8+c}}+\frac{b \left(\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a \sqrt[4]{c} (b c+a d) \sqrt{d x^8+c}}+\frac{\sqrt{b} \left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 (-a)^{3/2} \sqrt[4]{c} \sqrt[4]{d} (b c+a d) \sqrt{d x^8+c}}+\frac{\sqrt{d} x^2 \sqrt{d x^8+c}}{2 a c \left(\sqrt{d} x^4+\sqrt{c}\right)}-\frac{\sqrt{d x^8+c}}{2 a c x^2}",1,"(-21*a*(c + d*x^8) + 7*(-(b*c) + a*d)*x^8*Sqrt[1 + (d*x^8)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^8)/c), -((b*x^8)/a)] + 3*b*d*x^16*Sqrt[1 + (d*x^8)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)])/(42*a^2*c*x^2*Sqrt[c + d*x^8])","C",0
903,1,65,64,0.0535921,"\int \frac{x^4}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x^4/((a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{x^5 \sqrt{\frac{c+d x^8}{c}} F_1\left(\frac{5}{8};\frac{1}{2},1;\frac{13}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)}{5 a \sqrt{c+d x^8}}","\frac{x^5 \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{5}{8};1,\frac{1}{2};\frac{13}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right)}{5 a \sqrt{c+d x^8}}",1,"(x^5*Sqrt[(c + d*x^8)/c]*AppellF1[5/8, 1/2, 1, 13/8, -((d*x^8)/c), -((b*x^8)/a)])/(5*a*Sqrt[c + d*x^8])","A",0
904,1,65,64,0.0675827,"\int \frac{x^2}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[x^2/((a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{x^3 \sqrt{\frac{c+d x^8}{c}} F_1\left(\frac{3}{8};\frac{1}{2},1;\frac{11}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)}{3 a \sqrt{c+d x^8}}","\frac{x^3 \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{3}{8};1,\frac{1}{2};\frac{11}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right)}{3 a \sqrt{c+d x^8}}",1,"(x^3*Sqrt[(c + d*x^8)/c]*AppellF1[3/8, 1/2, 1, 11/8, -((d*x^8)/c), -((b*x^8)/a)])/(3*a*Sqrt[c + d*x^8])","A",0
905,1,161,59,0.2977561,"\int \frac{1}{\left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[1/((a + b*x^8)*Sqrt[c + d*x^8]),x]","-\frac{9 a c x F_1\left(\frac{1}{8};\frac{1}{2},1;\frac{9}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)}{\left(a+b x^8\right) \sqrt{c+d x^8} \left(4 x^8 \left(2 b c F_1\left(\frac{9}{8};\frac{1}{2},2;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+a d F_1\left(\frac{9}{8};\frac{3}{2},1;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)\right)-9 a c F_1\left(\frac{1}{8};\frac{1}{2},1;\frac{9}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)\right)}","\frac{x \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{1}{8};1,\frac{1}{2};\frac{9}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right)}{a \sqrt{c+d x^8}}",1,"(-9*a*c*x*AppellF1[1/8, 1/2, 1, 9/8, -((d*x^8)/c), -((b*x^8)/a)])/((a + b*x^8)*Sqrt[c + d*x^8]*(-9*a*c*AppellF1[1/8, 1/2, 1, 9/8, -((d*x^8)/c), -((b*x^8)/a)] + 4*x^8*(2*b*c*AppellF1[9/8, 1/2, 2, 17/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[9/8, 3/2, 1, 17/8, -((d*x^8)/c), -((b*x^8)/a)])))","B",0
906,1,141,62,0.1472894,"\int \frac{1}{x^2 \left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[1/(x^2*(a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{-5 x^8 \sqrt{\frac{d x^8}{c}+1} (b c-3 a d) F_1\left(\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+7 b d x^{16} \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)-35 a \left(c+d x^8\right)}{35 a^2 c x \sqrt{c+d x^8}}","-\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left(-\frac{1}{8};1,\frac{1}{2};\frac{7}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right)}{a x \sqrt{c+d x^8}}",1,"(-35*a*(c + d*x^8) - 5*(b*c - 3*a*d)*x^8*Sqrt[1 + (d*x^8)/c]*AppellF1[7/8, 1/2, 1, 15/8, -((d*x^8)/c), -((b*x^8)/a)] + 7*b*d*x^16*Sqrt[1 + (d*x^8)/c]*AppellF1[15/8, 1/2, 1, 23/8, -((d*x^8)/c), -((b*x^8)/a)])/(35*a^2*c*x*Sqrt[c + d*x^8])","B",0
907,1,141,64,0.1382487,"\int \frac{1}{x^4 \left(a+b x^8\right) \sqrt{c+d x^8}} \, dx","Integrate[1/(x^4*(a + b*x^8)*Sqrt[c + d*x^8]),x]","\frac{13 x^8 \sqrt{\frac{d x^8}{c}+1} (a d-3 b c) F_1\left(\frac{5}{8};\frac{1}{2},1;\frac{13}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+5 b d x^{16} \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{13}{8};\frac{1}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)-65 a \left(c+d x^8\right)}{195 a^2 c x^3 \sqrt{c+d x^8}}","-\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left(-\frac{3}{8};1,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right)}{3 a x^3 \sqrt{c+d x^8}}",1,"(-65*a*(c + d*x^8) + 13*(-3*b*c + a*d)*x^8*Sqrt[1 + (d*x^8)/c]*AppellF1[5/8, 1/2, 1, 13/8, -((d*x^8)/c), -((b*x^8)/a)] + 5*b*d*x^16*Sqrt[1 + (d*x^8)/c]*AppellF1[13/8, 1/2, 1, 21/8, -((d*x^8)/c), -((b*x^8)/a)])/(195*a^2*c*x^3*Sqrt[c + d*x^8])","B",0
908,1,107,123,0.2570172,"\int \frac{x^{23}}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x^23/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{1}{8} \left(\frac{\sqrt{c+d x^8} \left(\frac{a^2}{\left(a+b x^8\right) (a d-b c)}+\frac{2}{d}\right)}{b^2}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{b^{5/2} (b c-a d)^{3/2}}\right)","-\frac{a^2 \sqrt{c+d x^8}}{8 b^2 \left(a+b x^8\right) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{8 b^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^8}}{4 b^2 d}",1,"((Sqrt[c + d*x^8]*(2/d + a^2/((-(b*c) + a*d)*(a + b*x^8))))/b^2 + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(3/2)))/8","A",1
909,1,98,99,0.0922192,"\int \frac{x^{15}}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x^15/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{\frac{a \sqrt{b} \sqrt{c+d x^8}}{\left(a+b x^8\right) (b c-a d)}+\frac{(a d-2 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}}{8 b^{3/2}}","\frac{a \sqrt{c+d x^8}}{8 b \left(a+b x^8\right) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{8 b^{3/2} (b c-a d)^{3/2}}",1,"((a*Sqrt[b]*Sqrt[c + d*x^8])/((b*c - a*d)*(a + b*x^8)) + ((-2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(8*b^(3/2))","A",1
910,1,85,87,0.0958341,"\int \frac{x^7}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x^7/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{1}{8} \left(\frac{\sqrt{c+d x^8}}{\left(a+b x^8\right) (a d-b c)}+\frac{d \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{a d-b c}}\right)}{\sqrt{b} (a d-b c)^{3/2}}\right)","\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{8 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^8}}{8 \left(a+b x^8\right) (b c-a d)}",1,"(Sqrt[c + d*x^8]/((-(b*c) + a*d)*(a + b*x^8)) + (d*ArcTan[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[-(b*c) + a*d]])/(Sqrt[b]*(-(b*c) + a*d)^(3/2)))/8","A",1
911,1,123,132,0.3554829,"\int \frac{1}{x \left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[1/(x*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{\frac{a b \sqrt{c+d x^8}}{\left(a+b x^8\right) (b c-a d)}+\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right)}{\sqrt{c}}}{8 a^2}","\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{8 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right)}{4 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^8}}{8 a \left(a+b x^8\right) (b c-a d)}",1,"((a*b*Sqrt[c + d*x^8])/((b*c - a*d)*(a + b*x^8)) - (2*ArcTanh[Sqrt[c + d*x^8]/Sqrt[c]])/Sqrt[c] + (Sqrt[b]*(2*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(8*a^2)","A",1
912,1,163,185,0.6018639,"\int \frac{1}{x^9 \left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[1/(x^9*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{\frac{a \sqrt{c+d x^8} \left(a^2 d+a b \left(d x^8-c\right)-2 b^2 c x^8\right)}{x^8 \left(a+b x^8\right) (b c-a d)}+\frac{b^{3/2} c (5 a d-4 b c) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right)}{\sqrt{c}}}{8 a^3 c}","-\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right)}{8 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+d x^8}}{\sqrt{c}}\right)}{8 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^8} (2 b c-a d)}{8 a^2 c \left(a+b x^8\right) (b c-a d)}-\frac{\sqrt{c+d x^8}}{8 a c x^8 \left(a+b x^8\right)}",1,"((a*Sqrt[c + d*x^8]*(a^2*d - 2*b^2*c*x^8 + a*b*(-c + d*x^8)))/((b*c - a*d)*x^8*(a + b*x^8)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^8]/Sqrt[c]])/Sqrt[c] + (b^(3/2)*c*(-4*b*c + 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(8*a^3*c)","A",1
913,1,135,141,0.2644904,"\int \frac{x^{19}}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x^19/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{\frac{a b x^4 \sqrt{c+d x^8}}{\left(a+b x^8\right) (b c-a d)}+\frac{\sqrt{a} (2 a d-3 b c) \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{(b c-a d)^{3/2}}+\frac{2 \log \left(\sqrt{d} \sqrt{c+d x^8}+d x^4\right)}{\sqrt{d}}}{8 b^2}","-\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{8 b^2 (b c-a d)^{3/2}}+\frac{a x^4 \sqrt{c+d x^8}}{8 b \left(a+b x^8\right) (b c-a d)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{d} x^4}{\sqrt{c+d x^8}}\right)}{4 b^2 \sqrt{d}}",1,"((a*b*x^4*Sqrt[c + d*x^8])/((b*c - a*d)*(a + b*x^8)) + (Sqrt[a]*(-3*b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/(b*c - a*d)^(3/2) + (2*Log[d*x^4 + Sqrt[d]*Sqrt[c + d*x^8]])/Sqrt[d])/(8*b^2)","A",1
914,1,124,93,0.5485106,"\int \frac{x^{11}}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x^11/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{\sqrt{c+d x^8} \left(-\frac{x^8 (b c-a d)}{a+b x^8}-\frac{c \sqrt{x^8 \left(\frac{d}{c}-\frac{b}{a}\right)} \tanh ^{-1}\left(\frac{\sqrt{x^8 \left(\frac{d}{c}-\frac{b}{a}\right)}}{\sqrt{\frac{d x^8}{c}+1}}\right)}{\sqrt{\frac{d x^8}{c}+1}}\right)}{8 x^4 (b c-a d)^2}","\frac{c \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{8 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^4 \sqrt{c+d x^8}}{8 \left(a+b x^8\right) (b c-a d)}",1,"(Sqrt[c + d*x^8]*(-(((b*c - a*d)*x^8)/(a + b*x^8)) - (c*Sqrt[(-(b/a) + d/c)*x^8]*ArcTanh[Sqrt[(-(b/a) + d/c)*x^8]/Sqrt[1 + (d*x^8)/c]])/Sqrt[1 + (d*x^8)/c]))/(8*(b*c - a*d)^2*x^4)","A",1
915,1,407,104,0.9413284,"\int \frac{x^3}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x^3/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{x^4 \sqrt{c+d x^8} \left(-30 d x^8 \sqrt{\frac{a x^8 \left(c+d x^8\right) (b c-a d)}{c^2 \left(a+b x^8\right)^2}}-45 c \sqrt{\frac{a x^8 \left(c+d x^8\right) (b c-a d)}{c^2 \left(a+b x^8\right)^2}}+16 d x^8 \left(\frac{x^8 (b c-a d)}{c \left(a+b x^8\right)}\right)^{5/2} \sqrt{\frac{a \left(c+d x^8\right)}{c \left(a+b x^8\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)+16 c \left(\frac{x^8 (b c-a d)}{c \left(a+b x^8\right)}\right)^{5/2} \sqrt{\frac{a \left(c+d x^8\right)}{c \left(a+b x^8\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)+30 d x^8 \sin ^{-1}\left(\sqrt{\frac{x^8 (b c-a d)}{c \left(a+b x^8\right)}}\right)+45 c \sin ^{-1}\left(\sqrt{\frac{x^8 (b c-a d)}{c \left(a+b x^8\right)}}\right)\right)}{120 c^2 \left(a+b x^8\right)^2 \left(\frac{x^8 (b c-a d)}{c \left(a+b x^8\right)}\right)^{3/2} \sqrt{\frac{a \left(c+d x^8\right)}{c \left(a+b x^8\right)}}}","\frac{(b c-2 a d) \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{8 a^{3/2} (b c-a d)^{3/2}}+\frac{b x^4 \sqrt{c+d x^8}}{8 a \left(a+b x^8\right) (b c-a d)}",1,"(x^4*Sqrt[c + d*x^8]*(-45*c*Sqrt[(a*(b*c - a*d)*x^8*(c + d*x^8))/(c^2*(a + b*x^8)^2)] - 30*d*x^8*Sqrt[(a*(b*c - a*d)*x^8*(c + d*x^8))/(c^2*(a + b*x^8)^2)] + 45*c*ArcSin[Sqrt[((b*c - a*d)*x^8)/(c*(a + b*x^8))]] + 30*d*x^8*ArcSin[Sqrt[((b*c - a*d)*x^8)/(c*(a + b*x^8))]] + 16*c*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 16*d*x^8*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^8)/(c*(a + b*x^8))]))/(120*c^2*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(3/2)*(a + b*x^8)^2*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))])","C",0
916,1,869,149,2.034015,"\int \frac{1}{x^5 \left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[1/(x^5*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]","-\frac{\sqrt{d x^8+c} \left(120 d^2 \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}}\right) x^{16}+96 d^2 \left(\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^8+c\right)}{c \left(b x^8+a\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right) x^{16}+32 d^2 \left(\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^8+c\right)}{c \left(b x^8+a\right)}} \, _3F_2\left(2,2,3;1,\frac{7}{2};\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right) x^{16}-120 d^2 \sqrt{\frac{a (b c-a d) x^8 \left(d x^8+c\right)}{c^2 \left(b x^8+a\right)^2}} x^{16}+180 c d \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}}\right) x^8+160 c d \left(\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^8+c\right)}{c \left(b x^8+a\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right) x^8+64 c d \left(\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^8+c\right)}{c \left(b x^8+a\right)}} \, _3F_2\left(2,2,3;1,\frac{7}{2};\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right) x^8-180 c d \sqrt{\frac{a (b c-a d) x^8 \left(d x^8+c\right)}{c^2 \left(b x^8+a\right)^2}} x^8+45 c^2 \sin ^{-1}\left(\sqrt{\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}}\right)+64 c^2 \left(\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^8+c\right)}{c \left(b x^8+a\right)}} \, _2F_1\left(2,3;\frac{7}{2};\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)+32 c^2 \left(\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)^{5/2} \sqrt{\frac{a \left(d x^8+c\right)}{c \left(b x^8+a\right)}} \, _3F_2\left(2,2,3;1,\frac{7}{2};\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)-45 c^2 \sqrt{\frac{a (b c-a d) x^8 \left(d x^8+c\right)}{c^2 \left(b x^8+a\right)^2}}\right)}{120 c^3 x^4 \left(\frac{(b c-a d) x^8}{c \left(b x^8+a\right)}\right)^{3/2} \left(b x^8+a\right)^2 \sqrt{\frac{a \left(d x^8+c\right)}{c \left(b x^8+a\right)}}}","-\frac{b (3 b c-4 a d) \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{8 a^{5/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^8} (3 b c-2 a d)}{8 a^2 c x^4 (b c-a d)}+\frac{b \sqrt{c+d x^8}}{8 a x^4 \left(a+b x^8\right) (b c-a d)}",1,"-1/120*(Sqrt[c + d*x^8]*(-45*c^2*Sqrt[(a*(b*c - a*d)*x^8*(c + d*x^8))/(c^2*(a + b*x^8)^2)] - 180*c*d*x^8*Sqrt[(a*(b*c - a*d)*x^8*(c + d*x^8))/(c^2*(a + b*x^8)^2)] - 120*d^2*x^16*Sqrt[(a*(b*c - a*d)*x^8*(c + d*x^8))/(c^2*(a + b*x^8)^2)] + 45*c^2*ArcSin[Sqrt[((b*c - a*d)*x^8)/(c*(a + b*x^8))]] + 180*c*d*x^8*ArcSin[Sqrt[((b*c - a*d)*x^8)/(c*(a + b*x^8))]] + 120*d^2*x^16*ArcSin[Sqrt[((b*c - a*d)*x^8)/(c*(a + b*x^8))]] + 64*c^2*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 160*c*d*x^8*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 96*d^2*x^16*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 32*c^2*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*HypergeometricPFQ[{2, 2, 3}, {1, 7/2}, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 64*c*d*x^8*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*HypergeometricPFQ[{2, 2, 3}, {1, 7/2}, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 32*d^2*x^16*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*HypergeometricPFQ[{2, 2, 3}, {1, 7/2}, ((b*c - a*d)*x^8)/(c*(a + b*x^8))]))/(c^3*x^4*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(3/2)*(a + b*x^8)^2*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))])","C",0
917,1,175,208,5.846678,"\int \frac{1}{x^{13} \left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[1/(x^13*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{a^2 \left(c+d x^8\right) \left(\frac{3 b^3 x^{16}}{\left(a+b x^8\right) (b c-a d)}+\frac{4 x^8 (a d+3 b c)}{c^2}-\frac{2 a}{c}\right)+\frac{3 b^2 x^{24} \sqrt{\frac{d x^8}{c}+1} (5 b c-6 a d) \sin ^{-1}\left(\frac{\sqrt{x^8 \left(\frac{b}{a}-\frac{d}{c}\right)}}{\sqrt{\frac{b x^8}{a}+1}}\right)}{c \left(\frac{x^8 (b c-a d)}{a c}\right)^{3/2}}}{24 a^5 x^{12} \sqrt{c+d x^8}}","\frac{b^2 (5 b c-6 a d) \tan ^{-1}\left(\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right)}{8 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^8} (5 b c-2 a d)}{24 a^2 c x^{12} (b c-a d)}+\frac{\sqrt{c+d x^8} \left(-4 a^2 d^2-8 a b c d+15 b^2 c^2\right)}{24 a^3 c^2 x^4 (b c-a d)}+\frac{b \sqrt{c+d x^8}}{8 a x^{12} \left(a+b x^8\right) (b c-a d)}",1,"(a^2*(c + d*x^8)*((-2*a)/c + (4*(3*b*c + a*d)*x^8)/c^2 + (3*b^3*x^16)/((b*c - a*d)*(a + b*x^8))) + (3*b^2*(5*b*c - 6*a*d)*x^24*Sqrt[1 + (d*x^8)/c]*ArcSin[Sqrt[(b/a - d/c)*x^8]/Sqrt[1 + (b*x^8)/a]])/(c*(((b*c - a*d)*x^8)/(a*c))^(3/2)))/(24*a^5*x^12*Sqrt[c + d*x^8])","A",0
918,1,159,924,0.1674446,"\int \frac{x^9}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x^9/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","-\frac{x^2 \left(d x^8 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)-5 c \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+5 a \left(c+d x^8\right)\right)}{40 a \left(a+b x^8\right) \sqrt{c+d x^8} (b c-a d)}","\frac{\left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{64 a b \sqrt[4]{c} \sqrt[4]{d} (b c-a d) \sqrt{d x^8+c}}-\frac{(b c+a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 (-a)^{3/4} b^{3/4} (b c-a d)^{3/2}}+\frac{(b c+a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 (-a)^{3/4} b^{3/4} (a d-b c)^{3/2}}-\frac{d^{3/4} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 b \sqrt[4]{c} (b c-a d) \sqrt{d x^8+c}}+\frac{\left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right) \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 b \sqrt[4]{c} (b c-a d) \sqrt{d x^8+c}}+\frac{\left(\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right) \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 a b \sqrt[4]{c} (b c-a d) \sqrt{d x^8+c}}+\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{64 a b \sqrt[4]{c} \sqrt[4]{d} (b c-a d) \sqrt{d x^8+c}}-\frac{x^2 \sqrt{d x^8+c}}{8 (b c-a d) \left(b x^8+a\right)}",1,"-1/40*(x^2*(5*a*(c + d*x^8) - 5*c*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b*x^8)/a)] + d*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^8)/c), -((b*x^8)/a)]))/(a*(b*c - a*d)*(a + b*x^8)*Sqrt[c + d*x^8])","C",0
919,1,169,999,0.2191607,"\int \frac{x}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{x^2 \left(b d x^8 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+5 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} (3 b c-4 a d) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+5 a b \left(c+d x^8\right)\right)}{40 a^2 \left(a+b x^8\right) \sqrt{c+d x^8} (b c-a d)}","\frac{(3 b c-5 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{64 a^2 \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{\sqrt[4]{d} (3 b c-5 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)}{32 (-a)^{3/2} \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{\sqrt[4]{b} (3 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 (-a)^{7/4} (b c-a d)^{3/2}}-\frac{\sqrt[4]{b} (3 b c-5 a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 (-a)^{7/4} (a d-b c)^{3/2}}+\frac{d^{3/4} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a \sqrt[4]{c} (b c-a d) \sqrt{d x^8+c}}+\frac{\left(\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right) \sqrt[4]{d} (3 b c-5 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 a \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 (3 b c-5 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{64 a^2 \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{b x^2 \sqrt{d x^8+c}}{8 a (b c-a d) \left(b x^8+a\right)}",1,"(x^2*(5*a*b*(c + d*x^8) + 5*(3*b*c - 4*a*d)*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b*x^8)/a)] + b*d*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^8)/c), -((b*x^8)/a)]))/(40*a^2*(b*c - a*d)*(a + b*x^8)*Sqrt[c + d*x^8])","C",0
920,1,225,1060,0.3876148,"\int \frac{1}{x^7 \left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[1/(x^7*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{5 x^8 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} \left(4 a^2 d^2+20 a b c d-21 b^2 c^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+5 a \left(c+d x^8\right) \left(4 a^2 d-4 a b \left(c-d x^8\right)-7 b^2 c x^8\right)+b d x^{16} \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} (4 a d-7 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)}{120 a^3 c x^6 \left(a+b x^8\right) \sqrt{c+d x^8} (b c-a d)}","-\frac{b (7 b c-9 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{64 a^3 \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{b \sqrt[4]{d} (7 b c-9 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)}{32 (-a)^{5/2} \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{b^{5/4} (7 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 (-a)^{11/4} (b c-a d)^{3/2}}-\frac{b^{5/4} (7 b c-9 a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 (-a)^{11/4} (a d-b c)^{3/2}}-\frac{d^{3/4} (7 b c-4 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{48 a^2 c^{5/4} (b c-a d) \sqrt{d x^8+c}}-\frac{b \left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right) \sqrt[4]{d} (7 b c-9 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 (-a)^{5/2} \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}-\frac{b \left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 (7 b c-9 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{64 a^3 \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{b \sqrt{d x^8+c}}{8 a (b c-a d) x^6 \left(b x^8+a\right)}-\frac{(7 b c-4 a d) \sqrt{d x^8+c}}{24 a^2 c (b c-a d) x^6}",1,"(5*a*(c + d*x^8)*(4*a^2*d - 7*b^2*c*x^8 - 4*a*b*(c - d*x^8)) + 5*(-21*b^2*c^2 + 20*a*b*c*d + 4*a^2*d^2)*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^8)/c), -((b*x^8)/a)] + b*d*(-7*b*c + 4*a*d)*x^16*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^8)/c), -((b*x^8)/a)])/(120*a^3*c*(b*c - a*d)*x^6*(a + b*x^8)*Sqrt[c + d*x^8])","C",0
921,1,159,1164,0.1651972,"\int \frac{x^{13}}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x^13/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{x^6 \left(d x^8 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+7 c \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)-7 a \left(c+d x^8\right)\right)}{56 a \left(a+b x^8\right) \sqrt{c+d x^8} (b c-a d)}","-\frac{\sqrt{d x^8+c} x^6}{8 (b c-a d) \left(b x^8+a\right)}+\frac{\sqrt{d} \sqrt{d x^8+c} x^2}{8 b (b c-a d) \left(\sqrt{d} x^4+\sqrt{c}\right)}+\frac{(3 b c-a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 \sqrt[4]{-a} b^{5/4} (b c-a d)^{3/2}}+\frac{(3 b c-a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 \sqrt[4]{-a} b^{5/4} (a d-b c)^{3/2}}-\frac{\sqrt[4]{c} \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 b (b c-a d) \sqrt{d x^8+c}}+\frac{\sqrt[4]{c} \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 b (b c-a d) \sqrt{d x^8+c}}-\frac{\left(\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (3 b c-a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 b \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}-\frac{\left(\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (3 b c-a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 b \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 (3 b c-a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{64 \sqrt{-a} b^{3/2} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2 (3 b c-a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{64 \sqrt{-a} b^{3/2} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}",1,"(x^6*(-7*a*(c + d*x^8) + 7*c*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^8)/c), -((b*x^8)/a)] + d*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)]))/(56*a*(b*c - a*d)*(a + b*x^8)*Sqrt[c + d*x^8])","C",0
922,1,169,1162,0.1917917,"\int \frac{x^5}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x^5/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{x^6 \left(-3 b d x^8 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+7 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} (b c-4 a d) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+21 a b \left(c+d x^8\right)\right)}{168 a^2 \left(a+b x^8\right) \sqrt{c+d x^8} (b c-a d)}","\frac{b \sqrt{d x^8+c} x^6}{8 a (b c-a d) \left(b x^8+a\right)}-\frac{\sqrt{d} \sqrt{d x^8+c} x^2}{8 a (b c-a d) \left(\sqrt{d} x^4+\sqrt{c}\right)}-\frac{(b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 (-a)^{5/4} \sqrt[4]{b} (b c-a d)^{3/2}}-\frac{(b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 (-a)^{5/4} \sqrt[4]{b} (a d-b c)^{3/2}}+\frac{\sqrt[4]{c} \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a (b c-a d) \sqrt{d x^8+c}}-\frac{\sqrt[4]{c} \sqrt[4]{d} \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a (b c-a d) \sqrt{d x^8+c}}-\frac{\left(\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (b c-3 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 a \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}-\frac{\left(\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (b c-3 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 a \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}-\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 (b c-3 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{64 (-a)^{3/2} \sqrt{b} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2 (b c-3 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{64 (-a)^{3/2} \sqrt{b} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}",1,"(x^6*(21*a*b*(c + d*x^8) + 7*(b*c - 4*a*d)*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^8)/c), -((b*x^8)/a)] - 3*b*d*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)]))/(168*a^2*(b*c - a*d)*(a + b*x^8)*Sqrt[c + d*x^8])","C",0
923,1,226,1243,0.3375035,"\int \frac{1}{x^3 \left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[1/(x^3*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{-7 x^8 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} \left(4 a^2 d^2-12 a b c d+5 b^2 c^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+21 a \left(c+d x^8\right) \left(4 a^2 d-4 a b \left(c-d x^8\right)-5 b^2 c x^8\right)+3 b d x^{16} \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} (5 b c-4 a d) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)}{168 a^3 c x^2 \left(a+b x^8\right) \sqrt{c+d x^8} (b c-a d)}","\frac{\sqrt{b} (5 b c-7 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(\frac{\left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{64 (-a)^{5/2} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}-\frac{b^{3/4} (5 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{b c-a d} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 (-a)^{9/4} (b c-a d)^{3/2}}-\frac{b^{3/4} (5 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{a d-b c} x^2}{\sqrt[4]{-a} \sqrt[4]{b} \sqrt{d x^8+c}}\right)}{32 (-a)^{9/4} (a d-b c)^{3/2}}-\frac{\sqrt[4]{d} (5 b c-4 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{8 a^2 c^{3/4} (b c-a d) \sqrt{d x^8+c}}+\frac{\sqrt[4]{d} (5 b c-4 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{16 a^2 c^{3/4} (b c-a d) \sqrt{d x^8+c}}+\frac{b \left(\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (5 b c-7 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 a^2 \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{b \left(\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right) \sqrt[4]{d} (5 b c-7 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{32 a^2 \sqrt[4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}-\frac{\sqrt{b} \left(\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right)^2 (5 b c-7 a d) \left(\sqrt{d} x^4+\sqrt{c}\right) \sqrt{\frac{d x^8+c}{\left(\sqrt{d} x^4+\sqrt{c}\right)^2}} \Pi \left(-\frac{\left(\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right)^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x^2}{\sqrt[4]{c}}\right)|\frac{1}{2}\right)}{64 (-a)^{5/2} \sqrt[4]{c} \sqrt[4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{\sqrt{d} (5 b c-4 a d) x^2 \sqrt{d x^8+c}}{8 a^2 c (b c-a d) \left(\sqrt{d} x^4+\sqrt{c}\right)}+\frac{b \sqrt{d x^8+c}}{8 a (b c-a d) x^2 \left(b x^8+a\right)}-\frac{(5 b c-4 a d) \sqrt{d x^8+c}}{8 a^2 c (b c-a d) x^2}",1,"(21*a*(c + d*x^8)*(4*a^2*d - 5*b^2*c*x^8 - 4*a*b*(c - d*x^8)) - 7*(5*b^2*c^2 - 12*a*b*c*d + 4*a^2*d^2)*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^8)/c), -((b*x^8)/a)] + 3*b*d*(5*b*c - 4*a*d)*x^16*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^8)/c), -((b*x^8)/a)])/(168*a^3*c*(b*c - a*d)*x^2*(a + b*x^8)*Sqrt[c + d*x^8])","C",0
924,1,170,64,0.2041096,"\int \frac{x^4}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x^4/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{x^5 \left(-5 b d x^8 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{13}{8};\frac{1}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+13 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} (3 b c-8 a d) F_1\left(\frac{5}{8};\frac{1}{2},1;\frac{13}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+65 a b \left(c+d x^8\right)\right)}{520 a^2 \left(a+b x^8\right) \sqrt{c+d x^8} (b c-a d)}","\frac{x^5 \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{5}{8};2,\frac{1}{2};\frac{13}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right)}{5 a^2 \sqrt{c+d x^8}}",1,"(x^5*(65*a*b*(c + d*x^8) + 13*(3*b*c - 8*a*d)*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[5/8, 1/2, 1, 13/8, -((d*x^8)/c), -((b*x^8)/a)] - 5*b*d*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[13/8, 1/2, 1, 21/8, -((d*x^8)/c), -((b*x^8)/a)]))/(520*a^2*(b*c - a*d)*(a + b*x^8)*Sqrt[c + d*x^8])","B",0
925,1,170,64,0.2199148,"\int \frac{x^2}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[x^2/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{x^3 \left(3 b d x^8 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{11}{8};\frac{1}{2},1;\frac{19}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+11 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} (5 b c-8 a d) F_1\left(\frac{3}{8};\frac{1}{2},1;\frac{11}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+33 a b \left(c+d x^8\right)\right)}{264 a^2 \left(a+b x^8\right) \sqrt{c+d x^8} (b c-a d)}","\frac{x^3 \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{3}{8};2,\frac{1}{2};\frac{11}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right)}{3 a^2 \sqrt{c+d x^8}}",1,"(x^3*(33*a*b*(c + d*x^8) + 11*(5*b*c - 8*a*d)*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[3/8, 1/2, 1, 11/8, -((d*x^8)/c), -((b*x^8)/a)] + 3*b*d*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[11/8, 1/2, 1, 19/8, -((d*x^8)/c), -((b*x^8)/a)]))/(264*a^2*(b*c - a*d)*(a + b*x^8)*Sqrt[c + d*x^8])","B",0
926,1,328,59,0.3627889,"\int \frac{1}{\left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[1/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]","-\frac{x \left(b d x^8 \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{9}{8};\frac{1}{2},1;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+\frac{3 a \left(4 b x^8 \left(c+d x^8\right) \left(2 b c F_1\left(\frac{9}{8};\frac{1}{2},2;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+a d F_1\left(\frac{9}{8};\frac{3}{2},1;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)\right)+9 a c \left(8 a d-b \left(8 c+d x^8\right)\right) F_1\left(\frac{1}{8};\frac{1}{2},1;\frac{9}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)\right)}{\left(a+b x^8\right) \left(4 x^8 \left(2 b c F_1\left(\frac{9}{8};\frac{1}{2},2;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+a d F_1\left(\frac{9}{8};\frac{3}{2},1;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)\right)-9 a c F_1\left(\frac{1}{8};\frac{1}{2},1;\frac{9}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)\right)}\right)}{24 a^2 \sqrt{c+d x^8} (a d-b c)}","\frac{x \sqrt{\frac{d x^8}{c}+1} F_1\left(\frac{1}{8};2,\frac{1}{2};\frac{9}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right)}{a^2 \sqrt{c+d x^8}}",1,"-1/24*(x*(b*d*x^8*Sqrt[1 + (d*x^8)/c]*AppellF1[9/8, 1/2, 1, 17/8, -((d*x^8)/c), -((b*x^8)/a)] + (3*a*(9*a*c*(8*a*d - b*(8*c + d*x^8))*AppellF1[1/8, 1/2, 1, 9/8, -((d*x^8)/c), -((b*x^8)/a)] + 4*b*x^8*(c + d*x^8)*(2*b*c*AppellF1[9/8, 1/2, 2, 17/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[9/8, 3/2, 1, 17/8, -((d*x^8)/c), -((b*x^8)/a)])))/((a + b*x^8)*(-9*a*c*AppellF1[1/8, 1/2, 1, 9/8, -((d*x^8)/c), -((b*x^8)/a)] + 4*x^8*(2*b*c*AppellF1[9/8, 1/2, 2, 17/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[9/8, 3/2, 1, 17/8, -((d*x^8)/c), -((b*x^8)/a)])))))/(a^2*(-(b*c) + a*d)*Sqrt[c + d*x^8])","B",0
927,1,226,62,0.3223743,"\int \frac{1}{x^2 \left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[1/(x^2*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{-5 x^8 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} \left(24 a^2 d^2-40 a b c d+9 b^2 c^2\right) F_1\left(\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+35 a \left(c+d x^8\right) \left(8 a^2 d-8 a b \left(c-d x^8\right)-9 b^2 c x^8\right)+7 b d x^{16} \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} (9 b c-8 a d) F_1\left(\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)}{280 a^3 c x \left(a+b x^8\right) \sqrt{c+d x^8} (b c-a d)}","-\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left(-\frac{1}{8};2,\frac{1}{2};\frac{7}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right)}{a^2 x \sqrt{c+d x^8}}",1,"(35*a*(c + d*x^8)*(8*a^2*d - 9*b^2*c*x^8 - 8*a*b*(c - d*x^8)) - 5*(9*b^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[7/8, 1/2, 1, 15/8, -((d*x^8)/c), -((b*x^8)/a)] + 7*b*d*(9*b*c - 8*a*d)*x^16*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[15/8, 1/2, 1, 23/8, -((d*x^8)/c), -((b*x^8)/a)])/(280*a^3*c*(b*c - a*d)*x*(a + b*x^8)*Sqrt[c + d*x^8])","B",0
928,1,226,64,0.3222562,"\int \frac{1}{x^4 \left(a+b x^8\right)^2 \sqrt{c+d x^8}} \, dx","Integrate[1/(x^4*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]","\frac{-13 x^8 \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} \left(8 a^2 d^2-56 a b c d+33 b^2 c^2\right) F_1\left(\frac{5}{8};\frac{1}{2},1;\frac{13}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)+65 a \left(c+d x^8\right) \left(8 a^2 d-8 a b \left(c-d x^8\right)-11 b^2 c x^8\right)+5 b d x^{16} \left(a+b x^8\right) \sqrt{\frac{d x^8}{c}+1} (11 b c-8 a d) F_1\left(\frac{13}{8};\frac{1}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right)}{1560 a^3 c x^3 \left(a+b x^8\right) \sqrt{c+d x^8} (b c-a d)}","-\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left(-\frac{3}{8};2,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right)}{3 a^2 x^3 \sqrt{c+d x^8}}",1,"(65*a*(c + d*x^8)*(8*a^2*d - 11*b^2*c*x^8 - 8*a*b*(c - d*x^8)) - 13*(33*b^2*c^2 - 56*a*b*c*d + 8*a^2*d^2)*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[5/8, 1/2, 1, 13/8, -((d*x^8)/c), -((b*x^8)/a)] + 5*b*d*(11*b*c - 8*a*d)*x^16*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[13/8, 1/2, 1, 21/8, -((d*x^8)/c), -((b*x^8)/a)])/(1560*a^3*c*(b*c - a*d)*x^3*(a + b*x^8)*Sqrt[c + d*x^8])","B",0
929,1,121,123,0.256806,"\int \left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}} x^5 \, dx","Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^5,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(\sqrt{c} x \sqrt{\frac{c x^2}{d}+1} \left(a \left(8 c^2 x^4+2 c d x^2-3 d^2\right)+6 b c \left(2 c x^2+d\right)\right)+3 d^{3/2} (a d-2 b c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}{48 c^{5/2} \sqrt{\frac{c x^2}{d}+1}}","-\frac{d^2 (2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)}{16 c^{5/2}}+\frac{d x^2 \sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{16 c^2}+\frac{x^4 \sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{8 c}+\frac{a x^6 \left(c+\frac{d}{x^2}\right)^{3/2}}{6 c}",1,"(Sqrt[c + d/x^2]*x*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/d]*(6*b*c*(d + 2*c*x^2) + a*(-3*d^2 + 2*c*d*x^2 + 8*c^2*x^4)) + 3*d^(3/2)*(-2*b*c + a*d)*ArcSinh[(Sqrt[c]*x)/Sqrt[d]]))/(48*c^(5/2)*Sqrt[1 + (c*x^2)/d])","A",1
930,1,100,90,0.1360065,"\int \left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}} x^3 \, dx","Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^3,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(\sqrt{c} x \sqrt{\frac{c x^2}{d}+1} \left(a \left(2 c x^2+d\right)+4 b c\right)+\sqrt{d} (4 b c-a d) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}{8 c^{3/2} \sqrt{\frac{c x^2}{d}+1}}","\frac{d (4 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)}{8 c^{3/2}}+\frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-a d)}{8 c}+\frac{a x^4 \left(c+\frac{d}{x^2}\right)^{3/2}}{4 c}",1,"(Sqrt[c + d/x^2]*x*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/d]*(4*b*c + a*(d + 2*c*x^2)) + Sqrt[d]*(4*b*c - a*d)*ArcSinh[(Sqrt[c]*x)/Sqrt[d]]))/(8*c^(3/2)*Sqrt[1 + (c*x^2)/d])","A",1
931,1,71,84,0.1884511,"\int \left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}} x \, dx","Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x,x]","\frac{1}{2} \sqrt{c+\frac{d}{x^2}} \left(\frac{x (a d+2 b c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{\sqrt{c} \sqrt{d} \sqrt{\frac{c x^2}{d}+1}}+a x^2-2 b\right)","-\frac{\sqrt{c+\frac{d}{x^2}} (a d+2 b c)}{2 c}+\frac{(a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)}{2 \sqrt{c}}+\frac{a x^2 \left(c+\frac{d}{x^2}\right)^{3/2}}{2 c}",1,"(Sqrt[c + d/x^2]*(-2*b + a*x^2 + ((2*b*c + a*d)*x*ArcSinh[(Sqrt[c]*x)/Sqrt[d]])/(Sqrt[c]*Sqrt[d]*Sqrt[1 + (c*x^2)/d])))/2","A",1
932,1,82,59,0.1549497,"\int \frac{\left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}}}{x} \, dx","Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x,x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(\frac{3 a \sqrt{c} \sqrt{d} x^3 \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{\sqrt{\frac{c x^2}{d}+1}}-3 a d x^2-b \left(c x^2+d\right)\right)}{3 d x^2}","-a \sqrt{c+\frac{d}{x^2}}+a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)-\frac{b \left(c+\frac{d}{x^2}\right)^{3/2}}{3 d}",1,"(Sqrt[c + d/x^2]*(-3*a*d*x^2 - b*(d + c*x^2) + (3*a*Sqrt[c]*Sqrt[d]*x^3*ArcSinh[(Sqrt[c]*x)/Sqrt[d]])/Sqrt[1 + (c*x^2)/d]))/(3*d*x^2)","A",1
933,1,47,46,0.0174847,"\int \frac{\left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}}}{x^3} \, dx","Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^3,x]","-\frac{\sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right) \left(5 a d x^2-2 b c x^2+3 b d\right)}{15 d^2 x^4}","\frac{\left(c+\frac{d}{x^2}\right)^{3/2} (b c-a d)}{3 d^2}-\frac{b \left(c+\frac{d}{x^2}\right)^{5/2}}{5 d^2}",1,"-1/15*(Sqrt[c + d/x^2]*(d + c*x^2)*(3*b*d - 2*b*c*x^2 + 5*a*d*x^2))/(d^2*x^4)","A",1
934,1,69,74,0.0245125,"\int \frac{\left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}}}{x^5} \, dx","Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^5,x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right) \left(7 a d x^2 \left(2 c x^2-3 d\right)+b \left(-8 c^2 x^4+12 c d x^2-15 d^2\right)\right)}{105 d^3 x^6}","\frac{\left(c+\frac{d}{x^2}\right)^{5/2} (2 b c-a d)}{5 d^3}-\frac{c \left(c+\frac{d}{x^2}\right)^{3/2} (b c-a d)}{3 d^3}-\frac{b \left(c+\frac{d}{x^2}\right)^{7/2}}{7 d^3}",1,"(Sqrt[c + d/x^2]*(d + c*x^2)*(7*a*d*x^2*(-3*d + 2*c*x^2) + b*(-15*d^2 + 12*c*d*x^2 - 8*c^2*x^4)))/(105*d^3*x^6)","A",1
935,1,79,104,0.0644966,"\int \frac{\left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}}}{x^7} \, dx","Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^7,x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(x^2 \left(\frac{c x^2}{d}+1\right) \left(8 c^2 x^4-12 c d x^2+15 d^2\right) (6 b c-9 a d)-105 b d^2 \left(c x^2+d\right)\right)}{945 d^3 x^8}","\frac{c^2 \left(c+\frac{d}{x^2}\right)^{3/2} (b c-a d)}{3 d^4}+\frac{\left(c+\frac{d}{x^2}\right)^{7/2} (3 b c-a d)}{7 d^4}-\frac{c \left(c+\frac{d}{x^2}\right)^{5/2} (3 b c-2 a d)}{5 d^4}-\frac{b \left(c+\frac{d}{x^2}\right)^{9/2}}{9 d^4}",1,"(Sqrt[c + d/x^2]*(-105*b*d^2*(d + c*x^2) + (6*b*c - 9*a*d)*x^2*(1 + (c*x^2)/d)*(15*d^2 - 12*c*d*x^2 + 8*c^2*x^4)))/(945*d^3*x^8)","A",1
936,1,90,134,0.078183,"\int \frac{\left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}}}{x^9} \, dx","Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^9,x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(x^2 \left(\frac{c x^2}{d}+1\right) \left(-16 c^3 x^6+24 c^2 d x^4-30 c d^2 x^2+35 d^3\right) (8 b c-11 a d)-315 b d^3 \left(c x^2+d\right)\right)}{3465 d^4 x^{10}}","-\frac{c^3 \left(c+\frac{d}{x^2}\right)^{3/2} (b c-a d)}{3 d^5}+\frac{c^2 \left(c+\frac{d}{x^2}\right)^{5/2} (4 b c-3 a d)}{5 d^5}+\frac{\left(c+\frac{d}{x^2}\right)^{9/2} (4 b c-a d)}{9 d^5}-\frac{3 c \left(c+\frac{d}{x^2}\right)^{7/2} (2 b c-a d)}{7 d^5}-\frac{b \left(c+\frac{d}{x^2}\right)^{11/2}}{11 d^5}",1,"(Sqrt[c + d/x^2]*(-315*b*d^3*(d + c*x^2) + (8*b*c - 11*a*d)*x^2*(1 + (c*x^2)/d)*(35*d^3 - 30*c*d^2*x^2 + 24*c^2*d*x^4 - 16*c^3*x^6)))/(3465*d^4*x^10)","A",1
937,1,108,150,0.0843926,"\int \left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}} x^{10} \, dx","Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^10,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right) \left(a \left(315 c^4 x^8-280 c^3 d x^6+240 c^2 d^2 x^4-192 c d^3 x^2+128 d^4\right)+11 b c \left(35 c^3 x^6-30 c^2 d x^4+24 c d^2 x^2-16 d^3\right)\right)}{3465 c^5}","-\frac{16 d^3 x^3 \left(c+\frac{d}{x^2}\right)^{3/2} (11 b c-8 a d)}{3465 c^5}+\frac{8 d^2 x^5 \left(c+\frac{d}{x^2}\right)^{3/2} (11 b c-8 a d)}{1155 c^4}-\frac{2 d x^7 \left(c+\frac{d}{x^2}\right)^{3/2} (11 b c-8 a d)}{231 c^3}+\frac{x^9 \left(c+\frac{d}{x^2}\right)^{3/2} (11 b c-8 a d)}{99 c^2}+\frac{a x^{11} \left(c+\frac{d}{x^2}\right)^{3/2}}{11 c}",1,"(Sqrt[c + d/x^2]*x*(d + c*x^2)*(11*b*c*(-16*d^3 + 24*c*d^2*x^2 - 30*c^2*d*x^4 + 35*c^3*x^6) + a*(128*d^4 - 192*c*d^3*x^2 + 240*c^2*d^2*x^4 - 280*c^3*d*x^6 + 315*c^4*x^8)))/(3465*c^5)","A",1
938,1,86,117,0.0652401,"\int \left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}} x^8 \, dx","Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^8,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right) \left(a \left(35 c^3 x^6-30 c^2 d x^4+24 c d^2 x^2-16 d^3\right)+3 b c \left(15 c^2 x^4-12 c d x^2+8 d^2\right)\right)}{315 c^4}","\frac{8 d^2 x^3 \left(c+\frac{d}{x^2}\right)^{3/2} (3 b c-2 a d)}{315 c^4}-\frac{4 d x^5 \left(c+\frac{d}{x^2}\right)^{3/2} (3 b c-2 a d)}{105 c^3}+\frac{x^7 \left(c+\frac{d}{x^2}\right)^{3/2} (3 b c-2 a d)}{21 c^2}+\frac{a x^9 \left(c+\frac{d}{x^2}\right)^{3/2}}{9 c}",1,"(Sqrt[c + d/x^2]*x*(d + c*x^2)*(3*b*c*(8*d^2 - 12*c*d*x^2 + 15*c^2*x^4) + a*(-16*d^3 + 24*c*d^2*x^2 - 30*c^2*d*x^4 + 35*c^3*x^6)))/(315*c^4)","A",1
939,1,64,84,0.0457149,"\int \left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}} x^6 \, dx","Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^6,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right) \left(a \left(15 c^2 x^4-12 c d x^2+8 d^2\right)+7 b c \left(3 c x^2-2 d\right)\right)}{105 c^3}","-\frac{2 d x^3 \left(c+\frac{d}{x^2}\right)^{3/2} (7 b c-4 a d)}{105 c^3}+\frac{x^5 \left(c+\frac{d}{x^2}\right)^{3/2} (7 b c-4 a d)}{35 c^2}+\frac{a x^7 \left(c+\frac{d}{x^2}\right)^{3/2}}{7 c}",1,"(Sqrt[c + d/x^2]*x*(d + c*x^2)*(7*b*c*(-2*d + 3*c*x^2) + a*(8*d^2 - 12*c*d*x^2 + 15*c^2*x^4)))/(105*c^3)","A",1
940,1,42,53,0.0273419,"\int \left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}} x^4 \, dx","Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^4,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right) \left(3 a c x^2-2 a d+5 b c\right)}{15 c^2}","\frac{x^3 \left(c+\frac{d}{x^2}\right)^{3/2} (5 b c-2 a d)}{15 c^2}+\frac{a x^5 \left(c+\frac{d}{x^2}\right)^{3/2}}{5 c}",1,"(Sqrt[c + d/x^2]*x*(d + c*x^2)*(5*b*c - 2*a*d + 3*a*c*x^2))/(15*c^2)","A",1
941,1,84,66,0.0570444,"\int \left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}} x^2 \, dx","Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^2,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(\sqrt{c x^2+d} \left(a \left(c x^2+d\right)+3 b c\right)-3 b c \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{c x^2+d}}{\sqrt{d}}\right)\right)}{3 c \sqrt{c x^2+d}}","\frac{a x^3 \left(c+\frac{d}{x^2}\right)^{3/2}}{3 c}+b x \sqrt{c+\frac{d}{x^2}}-b \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)",1,"(Sqrt[c + d/x^2]*x*(Sqrt[d + c*x^2]*(3*b*c + a*(d + c*x^2)) - 3*b*c*Sqrt[d]*ArcTanh[Sqrt[d + c*x^2]/Sqrt[d]]))/(3*c*Sqrt[d + c*x^2])","A",1
942,1,75,85,0.0904697,"\int \left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}} \, dx","Integrate[(a + b/x^2)*Sqrt[c + d/x^2],x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(-\frac{x^2 (2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{c x^2+d}}{\sqrt{d}}\right)}{\sqrt{d} \sqrt{c x^2+d}}+2 a x^2-b\right)}{2 x}","-\frac{\sqrt{c+\frac{d}{x^2}} (2 a d+b c)}{2 c x}-\frac{(2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{2 \sqrt{d}}+\frac{a x \left(c+\frac{d}{x^2}\right)^{3/2}}{c}",1,"(Sqrt[c + d/x^2]*(-b + 2*a*x^2 - ((b*c + 2*a*d)*x^2*ArcTanh[Sqrt[d + c*x^2]/Sqrt[d]])/(Sqrt[d]*Sqrt[d + c*x^2])))/(2*x)","A",1
943,1,100,91,0.1853579,"\int \frac{\left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}}}{x^2} \, dx","Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^2,x]","-\frac{\sqrt{c+\frac{d}{x^2}} \left(\left(c x^2+d\right) \left(4 a d x^2+b c x^2+2 b d\right)+c x^4 \sqrt{\frac{c x^2}{d}+1} (4 a d-b c) \tanh ^{-1}\left(\sqrt{\frac{c x^2}{d}+1}\right)\right)}{8 d x^3 \left(c x^2+d\right)}","\frac{c (b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{8 d^{3/2}}+\frac{\sqrt{c+\frac{d}{x^2}} (b c-4 a d)}{8 d x}-\frac{b \left(c+\frac{d}{x^2}\right)^{3/2}}{4 d x}",1,"-1/8*(Sqrt[c + d/x^2]*((d + c*x^2)*(2*b*d + b*c*x^2 + 4*a*d*x^2) + c*(-(b*c) + 4*a*d)*x^4*Sqrt[1 + (c*x^2)/d]*ArcTanh[Sqrt[1 + (c*x^2)/d]]))/(d*x^3*(d + c*x^2))","A",1
944,1,68,123,0.0298792,"\int \frac{\left(a+\frac{b}{x^2}\right) \sqrt{c+\frac{d}{x^2}}}{x^4} \, dx","Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^4,x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right) \left(c^2 x^6 (b c-2 a d) \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{c x^2}{d}+1\right)-b d^3\right)}{6 d^4 x^5}","-\frac{c^2 (b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{16 d^{5/2}}+\frac{c \sqrt{c+\frac{d}{x^2}} (b c-2 a d)}{16 d^2 x}+\frac{\sqrt{c+\frac{d}{x^2}} (b c-2 a d)}{8 d x^3}-\frac{b \left(c+\frac{d}{x^2}\right)^{3/2}}{6 d x^3}",1,"(Sqrt[c + d/x^2]*(d + c*x^2)*(-(b*d^3) + c^2*(b*c - 2*a*d)*x^6*Hypergeometric2F1[3/2, 3, 5/2, 1 + (c*x^2)/d]))/(6*d^4*x^5)","C",1
945,1,123,123,0.1419369,"\int \left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2} x^5 \, dx","Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^5,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(\sqrt{c} x \sqrt{\frac{c x^2}{d}+1} \left(a \left(8 c^2 x^4+14 c d x^2+3 d^2\right)+6 b c \left(2 c x^2+5 d\right)\right)-3 d^{3/2} (a d-6 b c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)\right)}{48 c^{3/2} \sqrt{\frac{c x^2}{d}+1}}","\frac{d^2 (6 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)}{16 c^{3/2}}+\frac{d x^2 \sqrt{c+\frac{d}{x^2}} (6 b c-a d)}{16 c}+\frac{x^4 \left(c+\frac{d}{x^2}\right)^{3/2} (6 b c-a d)}{24 c}+\frac{a x^6 \left(c+\frac{d}{x^2}\right)^{5/2}}{6 c}",1,"(Sqrt[c + d/x^2]*x*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/d]*(6*b*c*(5*d + 2*c*x^2) + a*(3*d^2 + 14*c*d*x^2 + 8*c^2*x^4)) - 3*d^(3/2)*(-6*b*c + a*d)*ArcSinh[(Sqrt[c]*x)/Sqrt[d]]))/(48*c^(3/2)*Sqrt[1 + (c*x^2)/d])","A",1
946,1,89,115,0.2450447,"\int \left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2} x^3 \, dx","Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^3,x]","\frac{1}{8} \sqrt{c+\frac{d}{x^2}} \left(\frac{3 \sqrt{d} x (a d+4 b c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{\sqrt{c} \sqrt{\frac{c x^2}{d}+1}}+2 a c x^4+5 a d x^2+4 b c x^2-8 b d\right)","\frac{x^2 \left(c+\frac{d}{x^2}\right)^{3/2} (a d+4 b c)}{8 c}-\frac{3 d \sqrt{c+\frac{d}{x^2}} (a d+4 b c)}{8 c}+\frac{3 d (a d+4 b c) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)}{8 \sqrt{c}}+\frac{a x^4 \left(c+\frac{d}{x^2}\right)^{5/2}}{4 c}",1,"(Sqrt[c + d/x^2]*(-8*b*d + 4*b*c*x^2 + 5*a*d*x^2 + 2*a*c*x^4 + (3*Sqrt[d]*(4*b*c + a*d)*x*ArcSinh[(Sqrt[c]*x)/Sqrt[d]])/(Sqrt[c]*Sqrt[1 + (c*x^2)/d])))/8","A",1
947,1,78,110,0.0951019,"\int \left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2} x \, dx","Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x,x]","\frac{1}{3} \sqrt{c+\frac{d}{x^2}} \left(-\frac{(3 a d+2 b c) \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x^2}{d}\right)}{\sqrt{\frac{c x^2}{d}+1}}-\frac{b \left(c x^2+d\right)^2}{d x^2}\right)","-\frac{\left(c+\frac{d}{x^2}\right)^{3/2} (3 a d+2 b c)}{6 c}-\frac{1}{2} \sqrt{c+\frac{d}{x^2}} (3 a d+2 b c)+\frac{1}{2} \sqrt{c} (3 a d+2 b c) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)+\frac{a x^2 \left(c+\frac{d}{x^2}\right)^{5/2}}{2 c}",1,"(Sqrt[c + d/x^2]*(-((b*(d + c*x^2)^2)/(d*x^2)) - ((2*b*c + 3*a*d)*Hypergeometric2F1[-3/2, -1/2, 1/2, -((c*x^2)/d)])/Sqrt[1 + (c*x^2)/d]))/3","C",1
948,1,90,76,0.0714102,"\int \frac{\left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2}}{x} \, dx","Integrate[((a + b/x^2)*(c + d/x^2)^(3/2))/x,x]","-\frac{\sqrt{c+\frac{d}{x^2}} \left(5 a d^2 x^2 \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{c x^2}{d}\right)+3 b \left(c x^2+d\right)^2 \sqrt{\frac{c x^2}{d}+1}\right)}{15 d x^4 \sqrt{\frac{c x^2}{d}+1}}","a c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)-\frac{1}{3} a \left(c+\frac{d}{x^2}\right)^{3/2}-a c \sqrt{c+\frac{d}{x^2}}-\frac{b \left(c+\frac{d}{x^2}\right)^{5/2}}{5 d}",1,"-1/15*(Sqrt[c + d/x^2]*(3*b*(d + c*x^2)^2*Sqrt[1 + (c*x^2)/d] + 5*a*d^2*x^2*Hypergeometric2F1[-3/2, -3/2, -1/2, -((c*x^2)/d)]))/(d*x^4*Sqrt[1 + (c*x^2)/d])","C",1
949,1,49,46,0.0218402,"\int \frac{\left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2}}{x^3} \, dx","Integrate[((a + b/x^2)*(c + d/x^2)^(3/2))/x^3,x]","-\frac{\sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right)^2 \left(7 a d x^2-2 b c x^2+5 b d\right)}{35 d^2 x^6}","\frac{\left(c+\frac{d}{x^2}\right)^{5/2} (b c-a d)}{5 d^2}-\frac{b \left(c+\frac{d}{x^2}\right)^{7/2}}{7 d^2}",1,"-1/35*(Sqrt[c + d/x^2]*(d + c*x^2)^2*(5*b*d - 2*b*c*x^2 + 7*a*d*x^2))/(d^2*x^6)","A",1
950,1,71,74,0.0327141,"\int \frac{\left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2}}{x^5} \, dx","Integrate[((a + b/x^2)*(c + d/x^2)^(3/2))/x^5,x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right)^2 \left(9 a d x^2 \left(2 c x^2-5 d\right)+b \left(-8 c^2 x^4+20 c d x^2-35 d^2\right)\right)}{315 d^3 x^8}","\frac{\left(c+\frac{d}{x^2}\right)^{7/2} (2 b c-a d)}{7 d^3}-\frac{c \left(c+\frac{d}{x^2}\right)^{5/2} (b c-a d)}{5 d^3}-\frac{b \left(c+\frac{d}{x^2}\right)^{9/2}}{9 d^3}",1,"(Sqrt[c + d/x^2]*(d + c*x^2)^2*(9*a*d*x^2*(-5*d + 2*c*x^2) + b*(-35*d^2 + 20*c*d*x^2 - 8*c^2*x^4)))/(315*d^3*x^8)","A",1
951,1,94,104,0.0395135,"\int \frac{\left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2}}{x^7} \, dx","Integrate[((a + b/x^2)*(c + d/x^2)^(3/2))/x^7,x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right)^2 \left(-11 a d x^2 \left(8 c^2 x^4-20 c d x^2+35 d^2\right)-3 b \left(-16 c^3 x^6+40 c^2 d x^4-70 c d^2 x^2+105 d^3\right)\right)}{3465 d^4 x^{10}}","\frac{c^2 \left(c+\frac{d}{x^2}\right)^{5/2} (b c-a d)}{5 d^4}+\frac{\left(c+\frac{d}{x^2}\right)^{9/2} (3 b c-a d)}{9 d^4}-\frac{c \left(c+\frac{d}{x^2}\right)^{7/2} (3 b c-2 a d)}{7 d^4}-\frac{b \left(c+\frac{d}{x^2}\right)^{11/2}}{11 d^4}",1,"(Sqrt[c + d/x^2]*(d + c*x^2)^2*(-11*a*d*x^2*(35*d^2 - 20*c*d*x^2 + 8*c^2*x^4) - 3*b*(105*d^3 - 70*c*d^2*x^2 + 40*c^2*d*x^4 - 16*c^3*x^6)))/(3465*d^4*x^10)","A",1
952,1,115,134,0.0606545,"\int \frac{\left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2}}{x^9} \, dx","Integrate[((a + b/x^2)*(c + d/x^2)^(3/2))/x^9,x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right)^2 \left(13 a d x^2 \left(16 c^3 x^6-40 c^2 d x^4+70 c d^2 x^2-105 d^3\right)+b \left(-128 c^4 x^8+320 c^3 d x^6-560 c^2 d^2 x^4+840 c d^3 x^2-1155 d^4\right)\right)}{15015 d^5 x^{12}}","-\frac{c^3 \left(c+\frac{d}{x^2}\right)^{5/2} (b c-a d)}{5 d^5}+\frac{c^2 \left(c+\frac{d}{x^2}\right)^{7/2} (4 b c-3 a d)}{7 d^5}+\frac{\left(c+\frac{d}{x^2}\right)^{11/2} (4 b c-a d)}{11 d^5}-\frac{c \left(c+\frac{d}{x^2}\right)^{9/2} (2 b c-a d)}{3 d^5}-\frac{b \left(c+\frac{d}{x^2}\right)^{13/2}}{13 d^5}",1,"(Sqrt[c + d/x^2]*(d + c*x^2)^2*(13*a*d*x^2*(-105*d^3 + 70*c*d^2*x^2 - 40*c^2*d*x^4 + 16*c^3*x^6) + b*(-1155*d^4 + 840*c*d^3*x^2 - 560*c^2*d^2*x^4 + 320*c^3*d*x^6 - 128*c^4*x^8)))/(15015*d^5*x^12)","A",1
953,1,110,150,0.110001,"\int \left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2} x^{12} \, dx","Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^12,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right)^2 \left(a \left(1155 c^4 x^8-840 c^3 d x^6+560 c^2 d^2 x^4-320 c d^3 x^2+128 d^4\right)+13 b c \left(105 c^3 x^6-70 c^2 d x^4+40 c d^2 x^2-16 d^3\right)\right)}{15015 c^5}","-\frac{16 d^3 x^5 \left(c+\frac{d}{x^2}\right)^{5/2} (13 b c-8 a d)}{15015 c^5}+\frac{8 d^2 x^7 \left(c+\frac{d}{x^2}\right)^{5/2} (13 b c-8 a d)}{3003 c^4}-\frac{2 d x^9 \left(c+\frac{d}{x^2}\right)^{5/2} (13 b c-8 a d)}{429 c^3}+\frac{x^{11} \left(c+\frac{d}{x^2}\right)^{5/2} (13 b c-8 a d)}{143 c^2}+\frac{a x^{13} \left(c+\frac{d}{x^2}\right)^{5/2}}{13 c}",1,"(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(13*b*c*(-16*d^3 + 40*c*d^2*x^2 - 70*c^2*d*x^4 + 105*c^3*x^6) + a*(128*d^4 - 320*c*d^3*x^2 + 560*c^2*d^2*x^4 - 840*c^3*d*x^6 + 1155*c^4*x^8)))/(15015*c^5)","A",1
954,1,89,117,0.0898242,"\int \left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2} x^{10} \, dx","Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^10,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right)^2 \left(3 a \left(105 c^3 x^6-70 c^2 d x^4+40 c d^2 x^2-16 d^3\right)+11 b c \left(35 c^2 x^4-20 c d x^2+8 d^2\right)\right)}{3465 c^4}","\frac{8 d^2 x^5 \left(c+\frac{d}{x^2}\right)^{5/2} (11 b c-6 a d)}{3465 c^4}-\frac{4 d x^7 \left(c+\frac{d}{x^2}\right)^{5/2} (11 b c-6 a d)}{693 c^3}+\frac{x^9 \left(c+\frac{d}{x^2}\right)^{5/2} (11 b c-6 a d)}{99 c^2}+\frac{a x^{11} \left(c+\frac{d}{x^2}\right)^{5/2}}{11 c}",1,"(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(11*b*c*(8*d^2 - 20*c*d*x^2 + 35*c^2*x^4) + 3*a*(-16*d^3 + 40*c*d^2*x^2 - 70*c^2*d*x^4 + 105*c^3*x^6)))/(3465*c^4)","A",1
955,1,66,84,0.0552017,"\int \left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2} x^8 \, dx","Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^8,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right)^2 \left(a \left(35 c^2 x^4-20 c d x^2+8 d^2\right)+9 b c \left(5 c x^2-2 d\right)\right)}{315 c^3}","-\frac{2 d x^5 \left(c+\frac{d}{x^2}\right)^{5/2} (9 b c-4 a d)}{315 c^3}+\frac{x^7 \left(c+\frac{d}{x^2}\right)^{5/2} (9 b c-4 a d)}{63 c^2}+\frac{a x^9 \left(c+\frac{d}{x^2}\right)^{5/2}}{9 c}",1,"(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(9*b*c*(-2*d + 5*c*x^2) + a*(8*d^2 - 20*c*d*x^2 + 35*c^2*x^4)))/(315*c^3)","A",1
956,1,44,53,0.0364012,"\int \left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2} x^6 \, dx","Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^6,x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right)^2 \left(5 a c x^2-2 a d+7 b c\right)}{35 c^2}","\frac{x^5 \left(c+\frac{d}{x^2}\right)^{5/2} (7 b c-2 a d)}{35 c^2}+\frac{a x^7 \left(c+\frac{d}{x^2}\right)^{5/2}}{7 c}",1,"(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(7*b*c - 2*a*d + 5*a*c*x^2))/(35*c^2)","A",1
957,1,81,86,0.1474083,"\int \left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2} x^4 \, dx","Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^4,x]","\frac{1}{15} x \sqrt{c+\frac{d}{x^2}} \left(\frac{3 a \left(c x^2+d\right)^2}{c}-\frac{15 b d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c x^2+d}}{\sqrt{d}}\right)}{\sqrt{c x^2+d}}+5 b \left(c x^2+4 d\right)\right)","\frac{a x^5 \left(c+\frac{d}{x^2}\right)^{5/2}}{5 c}-b d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)+b d x \sqrt{c+\frac{d}{x^2}}+\frac{1}{3} b x^3 \left(c+\frac{d}{x^2}\right)^{3/2}",1,"(Sqrt[c + d/x^2]*x*((3*a*(d + c*x^2)^2)/c + 5*b*(4*d + c*x^2) - (15*b*d^(3/2)*ArcTanh[Sqrt[d + c*x^2]/Sqrt[d]])/Sqrt[d + c*x^2]))/15","A",1
958,1,105,121,0.0876882,"\int \left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2} x^2 \, dx","Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^2,x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(\sqrt{c x^2+d} \left(2 a c x^4+8 a d x^2+6 b c x^2-3 b d\right)-3 \sqrt{d} x^2 (2 a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{c x^2+d}}{\sqrt{d}}\right)\right)}{6 x \sqrt{c x^2+d}}","\frac{x \left(c+\frac{d}{x^2}\right)^{3/2} (2 a d+3 b c)}{3 c}-\frac{d \sqrt{c+\frac{d}{x^2}} (2 a d+3 b c)}{2 c x}-\frac{1}{2} \sqrt{d} (2 a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)+\frac{a x^3 \left(c+\frac{d}{x^2}\right)^{5/2}}{3 c}",1,"(Sqrt[c + d/x^2]*(Sqrt[d + c*x^2]*(-3*b*d + 6*b*c*x^2 + 8*a*d*x^2 + 2*a*c*x^4) - 3*Sqrt[d]*(3*b*c + 2*a*d)*x^2*ArcTanh[Sqrt[d + c*x^2]/Sqrt[d]]))/(6*x*Sqrt[d + c*x^2])","A",1
959,1,68,112,0.0339432,"\int \left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2} \, dx","Integrate[(a + b/x^2)*(c + d/x^2)^(3/2),x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right)^2 \left(c x^4 (4 a d+b c) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{c x^2}{d}+1\right)-5 b d^2\right)}{20 d^3 x^3}","-\frac{\left(c+\frac{d}{x^2}\right)^{3/2} (4 a d+b c)}{4 c x}-\frac{3 \sqrt{c+\frac{d}{x^2}} (4 a d+b c)}{8 x}-\frac{3 c (4 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{8 \sqrt{d}}+\frac{a x \left(c+\frac{d}{x^2}\right)^{5/2}}{c}",1,"(Sqrt[c + d/x^2]*(d + c*x^2)^2*(-5*b*d^2 + c*(b*c + 4*a*d)*x^4*Hypergeometric2F1[2, 5/2, 7/2, 1 + (c*x^2)/d]))/(20*d^3*x^3)","C",1
960,1,126,123,0.2443832,"\int \frac{\left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2}}{x^2} \, dx","Integrate[((a + b/x^2)*(c + d/x^2)^(3/2))/x^2,x]","-\frac{\sqrt{c+\frac{d}{x^2}} \left(\left(c x^2+d\right) \left(6 a d x^2 \left(5 c x^2+2 d\right)+b \left(3 c^2 x^4+14 c d x^2+8 d^2\right)\right)+3 c^2 x^6 \sqrt{\frac{c x^2}{d}+1} (6 a d-b c) \tanh ^{-1}\left(\sqrt{\frac{c x^2}{d}+1}\right)\right)}{48 d x^5 \left(c x^2+d\right)}","\frac{c^2 (b c-6 a d) \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{16 d^{3/2}}+\frac{\left(c+\frac{d}{x^2}\right)^{3/2} (b c-6 a d)}{24 d x}+\frac{c \sqrt{c+\frac{d}{x^2}} (b c-6 a d)}{16 d x}-\frac{b \left(c+\frac{d}{x^2}\right)^{5/2}}{6 d x}",1,"-1/48*(Sqrt[c + d/x^2]*((d + c*x^2)*(6*a*d*x^2*(2*d + 5*c*x^2) + b*(8*d^2 + 14*c*d*x^2 + 3*c^2*x^4)) + 3*c^2*(-(b*c) + 6*a*d)*x^6*Sqrt[1 + (c*x^2)/d]*ArcTanh[Sqrt[1 + (c*x^2)/d]]))/(d*x^5*(d + c*x^2))","A",1
961,1,71,159,0.0486515,"\int \frac{\left(a+\frac{b}{x^2}\right) \left(c+\frac{d}{x^2}\right)^{3/2}}{x^4} \, dx","Integrate[((a + b/x^2)*(c + d/x^2)^(3/2))/x^4,x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(c x^2+d\right)^2 \left(c^3 x^8 (8 a d-3 b c) \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{c x^2}{d}+1\right)-5 b d^4\right)}{40 d^5 x^7}","-\frac{c^3 (3 b c-8 a d) \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{128 d^{5/2}}+\frac{c^2 \sqrt{c+\frac{d}{x^2}} (3 b c-8 a d)}{128 d^2 x}+\frac{c \sqrt{c+\frac{d}{x^2}} (3 b c-8 a d)}{64 d x^3}+\frac{\left(c+\frac{d}{x^2}\right)^{3/2} (3 b c-8 a d)}{48 d x^3}-\frac{b \left(c+\frac{d}{x^2}\right)^{5/2}}{8 d x^3}",1,"(Sqrt[c + d/x^2]*(d + c*x^2)^2*(-5*b*d^4 + c^3*(-3*b*c + 8*a*d)*x^8*Hypergeometric2F1[5/2, 4, 7/2, 1 + (c*x^2)/d]))/(40*d^5*x^7)","C",1
962,1,95,90,0.1122635,"\int \frac{\left(a+\frac{b}{x^2}\right) x^3}{\sqrt{c+\frac{d}{x^2}}} \, dx","Integrate[((a + b/x^2)*x^3)/Sqrt[c + d/x^2],x]","\frac{\sqrt{c} x \left(c x^2+d\right) \left(2 a c x^2-3 a d+4 b c\right)+d \sqrt{c x^2+d} (3 a d-4 b c) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{c x^2+d}}\right)}{8 c^{5/2} x \sqrt{c+\frac{d}{x^2}}}","-\frac{d (4 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)}{8 c^{5/2}}+\frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-3 a d)}{8 c^2}+\frac{a x^4 \sqrt{c+\frac{d}{x^2}}}{4 c}",1,"(Sqrt[c]*x*(d + c*x^2)*(4*b*c - 3*a*d + 2*a*c*x^2) + d*(-4*b*c + 3*a*d)*Sqrt[d + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[d + c*x^2]])/(8*c^(5/2)*Sqrt[c + d/x^2]*x)","A",1
963,1,79,59,0.0551769,"\int \frac{\left(a+\frac{b}{x^2}\right) x}{\sqrt{c+\frac{d}{x^2}}} \, dx","Integrate[((a + b/x^2)*x)/Sqrt[c + d/x^2],x]","\frac{\sqrt{c x^2+d} (2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{c x^2+d}}\right)+a \sqrt{c} x \left(c x^2+d\right)}{2 c^{3/2} x \sqrt{c+\frac{d}{x^2}}}","\frac{(2 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)}{2 c^{3/2}}+\frac{a x^2 \sqrt{c+\frac{d}{x^2}}}{2 c}",1,"(a*Sqrt[c]*x*(d + c*x^2) + (2*b*c - a*d)*Sqrt[d + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[d + c*x^2]])/(2*c^(3/2)*Sqrt[c + d/x^2]*x)","A",1
964,1,73,43,0.0485509,"\int \frac{a+\frac{b}{x^2}}{\sqrt{c+\frac{d}{x^2}} x} \, dx","Integrate[(a + b/x^2)/(Sqrt[c + d/x^2]*x),x]","\frac{a d x \sqrt{c x^2+d} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{c x^2+d}}\right)-b \sqrt{c} \left(c x^2+d\right)}{\sqrt{c} d x^2 \sqrt{c+\frac{d}{x^2}}}","\frac{a \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)}{\sqrt{c}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d}",1,"(-(b*Sqrt[c]*(d + c*x^2)) + a*d*x*Sqrt[d + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[d + c*x^2]])/(Sqrt[c]*d*Sqrt[c + d/x^2]*x^2)","A",1
965,1,39,43,0.0481202,"\int \frac{a+\frac{b}{x^2}}{\sqrt{c+\frac{d}{x^2}} x^3} \, dx","Integrate[(a + b/x^2)/(Sqrt[c + d/x^2]*x^3),x]","-\frac{\sqrt{c+\frac{d}{x^2}} \left(3 a d x^2+b \left(d-2 c x^2\right)\right)}{3 d^2 x^2}","\frac{\sqrt{c+\frac{d}{x^2}} (b c-a d)}{d^2}-\frac{b \left(c+\frac{d}{x^2}\right)^{3/2}}{3 d^2}",1,"-1/3*(Sqrt[c + d/x^2]*(3*a*d*x^2 + b*(d - 2*c*x^2)))/(d^2*x^2)","A",1
966,1,60,72,0.0385372,"\int \frac{a+\frac{b}{x^2}}{\sqrt{c+\frac{d}{x^2}} x^5} \, dx","Integrate[(a + b/x^2)/(Sqrt[c + d/x^2]*x^5),x]","\frac{\sqrt{c+\frac{d}{x^2}} \left(b \left(-8 c^2 x^4+4 c d x^2-3 d^2\right)-5 a d x^2 \left(d-2 c x^2\right)\right)}{15 d^3 x^4}","\frac{\left(c+\frac{d}{x^2}\right)^{3/2} (2 b c-a d)}{3 d^3}-\frac{c \sqrt{c+\frac{d}{x^2}} (b c-a d)}{d^3}-\frac{b \left(c+\frac{d}{x^2}\right)^{5/2}}{5 d^3}",1,"(Sqrt[c + d/x^2]*(-5*a*d*x^2*(d - 2*c*x^2) + b*(-3*d^2 + 4*c*d*x^2 - 8*c^2*x^4)))/(15*d^3*x^4)","A",1
967,1,91,101,0.0482807,"\int \frac{a+\frac{b}{x^2}}{\sqrt{c+\frac{d}{x^2}} x^7} \, dx","Integrate[(a + b/x^2)/(Sqrt[c + d/x^2]*x^7),x]","\frac{\left(\frac{c x^2}{d}+1\right) \left(8 c^2 x^4-4 c d x^2+3 d^2\right) (6 b c-7 a d)}{105 d^3 x^6 \sqrt{c+\frac{d}{x^2}}}-\frac{b \left(c x^2+d\right)}{7 d x^8 \sqrt{c+\frac{d}{x^2}}}","\frac{c^2 \sqrt{c+\frac{d}{x^2}} (b c-a d)}{d^4}+\frac{\left(c+\frac{d}{x^2}\right)^{5/2} (3 b c-a d)}{5 d^4}-\frac{c \left(c+\frac{d}{x^2}\right)^{3/2} (3 b c-2 a d)}{3 d^4}-\frac{b \left(c+\frac{d}{x^2}\right)^{7/2}}{7 d^4}",1,"-1/7*(b*(d + c*x^2))/(d*Sqrt[c + d/x^2]*x^8) + ((6*b*c - 7*a*d)*(1 + (c*x^2)/d)*(3*d^2 - 4*c*d*x^2 + 8*c^2*x^4))/(105*d^3*Sqrt[c + d/x^2]*x^6)","A",1
968,1,56,82,0.0746507,"\int \frac{\left(a+\frac{b}{x^2}\right) x^4}{\sqrt{c+\frac{d}{x^2}}} \, dx","Integrate[((a + b/x^2)*x^4)/Sqrt[c + d/x^2],x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(a \left(3 c^2 x^4-4 c d x^2+8 d^2\right)+5 b c \left(c x^2-2 d\right)\right)}{15 c^3}","-\frac{2 d x \sqrt{c+\frac{d}{x^2}} (5 b c-4 a d)}{15 c^3}+\frac{x^3 \sqrt{c+\frac{d}{x^2}} (5 b c-4 a d)}{15 c^2}+\frac{a x^5 \sqrt{c+\frac{d}{x^2}}}{5 c}",1,"(Sqrt[c + d/x^2]*x*(5*b*c*(-2*d + c*x^2) + a*(8*d^2 - 4*c*d*x^2 + 3*c^2*x^4)))/(15*c^3)","A",1
969,1,34,51,0.0561063,"\int \frac{\left(a+\frac{b}{x^2}\right) x^2}{\sqrt{c+\frac{d}{x^2}}} \, dx","Integrate[((a + b/x^2)*x^2)/Sqrt[c + d/x^2],x]","\frac{x \sqrt{c+\frac{d}{x^2}} \left(a c x^2-2 a d+3 b c\right)}{3 c^2}","\frac{x \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{3 c^2}+\frac{a x^3 \sqrt{c+\frac{d}{x^2}}}{3 c}",1,"(Sqrt[c + d/x^2]*x*(3*b*c - 2*a*d + a*c*x^2))/(3*c^2)","A",1
970,1,71,47,0.041131,"\int \frac{a+\frac{b}{x^2}}{\sqrt{c+\frac{d}{x^2}}} \, dx","Integrate[(a + b/x^2)/Sqrt[c + d/x^2],x]","\frac{a \sqrt{d} \left(c x^2+d\right)-b c \sqrt{c x^2+d} \tanh ^{-1}\left(\frac{\sqrt{c x^2+d}}{\sqrt{d}}\right)}{c \sqrt{d} x \sqrt{c+\frac{d}{x^2}}}","\frac{a x \sqrt{c+\frac{d}{x^2}}}{c}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{\sqrt{d}}",1,"(a*Sqrt[d]*(d + c*x^2) - b*c*Sqrt[d + c*x^2]*ArcTanh[Sqrt[d + c*x^2]/Sqrt[d]])/(c*Sqrt[d]*Sqrt[c + d/x^2]*x)","A",1
971,1,80,61,0.0637627,"\int \frac{a+\frac{b}{x^2}}{\sqrt{c+\frac{d}{x^2}} x^2} \, dx","Integrate[(a + b/x^2)/(Sqrt[c + d/x^2]*x^2),x]","\frac{x^2 \sqrt{c x^2+d} (b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{c x^2+d}}{\sqrt{d}}\right)-b \sqrt{d} \left(c x^2+d\right)}{2 d^{3/2} x^3 \sqrt{c+\frac{d}{x^2}}}","\frac{(b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{2 d^{3/2}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{2 d x}",1,"(-(b*Sqrt[d]*(d + c*x^2)) + (b*c - 2*a*d)*x^2*Sqrt[d + c*x^2]*ArcTanh[Sqrt[d + c*x^2]/Sqrt[d]])/(2*d^(3/2)*Sqrt[c + d/x^2]*x^3)","A",1
972,1,107,93,0.1793778,"\int \frac{a+\frac{b}{x^2}}{\sqrt{c+\frac{d}{x^2}} x^4} \, dx","Integrate[(a + b/x^2)/(Sqrt[c + d/x^2]*x^4),x]","-\frac{\left(c x^2+d\right) \left(d \sqrt{\frac{c x^2}{d}+1} \left(4 a d x^2-3 b c x^2+2 b d\right)+c x^4 (3 b c-4 a d) \tanh ^{-1}\left(\sqrt{\frac{c x^2}{d}+1}\right)\right)}{8 d^3 x^5 \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c x^2}{d}+1}}","-\frac{c (3 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{8 d^{5/2}}+\frac{\sqrt{c+\frac{d}{x^2}} (3 b c-4 a d)}{8 d^2 x}-\frac{b \sqrt{c+\frac{d}{x^2}}}{4 d x^3}",1,"-1/8*((d + c*x^2)*(d*Sqrt[1 + (c*x^2)/d]*(2*b*d - 3*b*c*x^2 + 4*a*d*x^2) + c*(3*b*c - 4*a*d)*x^4*ArcTanh[Sqrt[1 + (c*x^2)/d]]))/(d^3*Sqrt[c + d/x^2]*x^5*Sqrt[1 + (c*x^2)/d])","A",1
973,1,111,118,0.192235,"\int \frac{\left(a+\frac{b}{x^2}\right) x^3}{\left(c+\frac{d}{x^2}\right)^{3/2}} \, dx","Integrate[((a + b/x^2)*x^3)/(c + d/x^2)^(3/2),x]","\frac{\sqrt{c} x \left(a \left(2 c^2 x^4-5 c d x^2-15 d^2\right)+4 b c \left(c x^2+3 d\right)\right)+3 d^{3/2} \sqrt{\frac{c x^2}{d}+1} (5 a d-4 b c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{8 c^{7/2} x \sqrt{c+\frac{d}{x^2}}}","-\frac{3 d (4 b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)}{8 c^{7/2}}+\frac{3 d (4 b c-5 a d)}{8 c^3 \sqrt{c+\frac{d}{x^2}}}+\frac{x^2 (4 b c-5 a d)}{8 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^4}{4 c \sqrt{c+\frac{d}{x^2}}}",1,"(Sqrt[c]*x*(4*b*c*(3*d + c*x^2) + a*(-15*d^2 - 5*c*d*x^2 + 2*c^2*x^4)) + 3*d^(3/2)*(-4*b*c + 5*a*d)*Sqrt[1 + (c*x^2)/d]*ArcSinh[(Sqrt[c]*x)/Sqrt[d]])/(8*c^(7/2)*Sqrt[c + d/x^2]*x)","A",1
974,1,89,86,0.2168757,"\int \frac{\left(a+\frac{b}{x^2}\right) x}{\left(c+\frac{d}{x^2}\right)^{3/2}} \, dx","Integrate[((a + b/x^2)*x)/(c + d/x^2)^(3/2),x]","\frac{\sqrt{c} x \left(a c x^2+3 a d-2 b c\right)-\sqrt{d} \sqrt{\frac{c x^2}{d}+1} (3 a d-2 b c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{2 c^{5/2} x \sqrt{c+\frac{d}{x^2}}}","\frac{(2 b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)}{2 c^{5/2}}-\frac{2 b c-3 a d}{2 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^2}{2 c \sqrt{c+\frac{d}{x^2}}}",1,"(Sqrt[c]*x*(-2*b*c + 3*a*d + a*c*x^2) - Sqrt[d]*(-2*b*c + 3*a*d)*Sqrt[1 + (c*x^2)/d]*ArcSinh[(Sqrt[c]*x)/Sqrt[d]])/(2*c^(5/2)*Sqrt[c + d/x^2]*x)","A",1
975,1,73,52,0.0936077,"\int \frac{a+\frac{b}{x^2}}{\left(c+\frac{d}{x^2}\right)^{3/2} x} \, dx","Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x),x]","\frac{\sqrt{c} x (b c-a d)+a d^{3/2} \sqrt{\frac{c x^2}{d}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{c^{3/2} d x \sqrt{c+\frac{d}{x^2}}}","\frac{b c-a d}{c d \sqrt{c+\frac{d}{x^2}}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right)}{c^{3/2}}",1,"(Sqrt[c]*(b*c - a*d)*x + a*d^(3/2)*Sqrt[1 + (c*x^2)/d]*ArcSinh[(Sqrt[c]*x)/Sqrt[d]])/(c^(3/2)*d*Sqrt[c + d/x^2]*x)","A",1
976,1,36,42,0.0210024,"\int \frac{a+\frac{b}{x^2}}{\left(c+\frac{d}{x^2}\right)^{3/2} x^3} \, dx","Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^3),x]","\frac{a d x^2-b \left(2 c x^2+d\right)}{d^2 x^2 \sqrt{c+\frac{d}{x^2}}}","-\frac{b c-a d}{d^2 \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d^2}",1,"(a*d*x^2 - b*(d + 2*c*x^2))/(d^2*Sqrt[c + d/x^2]*x^2)","A",1
977,1,60,68,0.0273359,"\int \frac{a+\frac{b}{x^2}}{\left(c+\frac{d}{x^2}\right)^{3/2} x^5} \, dx","Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^5),x]","\frac{b \left(8 c^2 x^4+4 c d x^2-d^2\right)-3 a d x^2 \left(2 c x^2+d\right)}{3 d^3 x^4 \sqrt{c+\frac{d}{x^2}}}","\frac{\sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{d^3}+\frac{c (b c-a d)}{d^3 \sqrt{c+\frac{d}{x^2}}}-\frac{b \left(c+\frac{d}{x^2}\right)^{3/2}}{3 d^3}",1,"(-3*a*d*x^2*(d + 2*c*x^2) + b*(-d^2 + 4*c*d*x^2 + 8*c^2*x^4))/(3*d^3*Sqrt[c + d/x^2]*x^4)","A",1
978,1,81,100,0.0299655,"\int \frac{a+\frac{b}{x^2}}{\left(c+\frac{d}{x^2}\right)^{3/2} x^7} \, dx","Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^7),x]","\frac{-5 a d x^2 \left(-8 c^2 x^4-4 c d x^2+d^2\right)-3 b \left(16 c^3 x^6+8 c^2 d x^4-2 c d^2 x^2+d^3\right)}{15 d^4 x^6 \sqrt{c+\frac{d}{x^2}}}","-\frac{c^2 (b c-a d)}{d^4 \sqrt{c+\frac{d}{x^2}}}+\frac{\left(c+\frac{d}{x^2}\right)^{3/2} (3 b c-a d)}{3 d^4}-\frac{c \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{d^4}-\frac{b \left(c+\frac{d}{x^2}\right)^{5/2}}{5 d^4}",1,"(-5*a*d*x^2*(d^2 - 4*c*d*x^2 - 8*c^2*x^4) - 3*b*(d^3 - 2*c*d^2*x^2 + 8*c^2*d*x^4 + 16*c^3*x^6))/(15*d^4*Sqrt[c + d/x^2]*x^6)","A",1
979,1,104,126,0.0354702,"\int \frac{a+\frac{b}{x^2}}{\left(c+\frac{d}{x^2}\right)^{3/2} x^9} \, dx","Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^9),x]","\frac{b \left(128 c^4 x^8+64 c^3 d x^6-16 c^2 d^2 x^4+8 c d^3 x^2-5 d^4\right)-7 a d x^2 \left(16 c^3 x^6+8 c^2 d x^4-2 c d^2 x^2+d^3\right)}{35 d^5 x^8 \sqrt{c+\frac{d}{x^2}}}","\frac{c^3 (b c-a d)}{d^5 \sqrt{c+\frac{d}{x^2}}}+\frac{c^2 \sqrt{c+\frac{d}{x^2}} (4 b c-3 a d)}{d^5}+\frac{\left(c+\frac{d}{x^2}\right)^{5/2} (4 b c-a d)}{5 d^5}-\frac{c \left(c+\frac{d}{x^2}\right)^{3/2} (2 b c-a d)}{d^5}-\frac{b \left(c+\frac{d}{x^2}\right)^{7/2}}{7 d^5}",1,"(-7*a*d*x^2*(d^3 - 2*c*d^2*x^2 + 8*c^2*d*x^4 + 16*c^3*x^6) + b*(-5*d^4 + 8*c*d^3*x^2 - 16*c^2*d^2*x^4 + 64*c^3*d*x^6 + 128*c^4*x^8))/(35*d^5*Sqrt[c + d/x^2]*x^8)","A",1
980,1,80,111,0.0757377,"\int \frac{\left(a+\frac{b}{x^2}\right) x^4}{\left(c+\frac{d}{x^2}\right)^{3/2}} \, dx","Integrate[((a + b/x^2)*x^4)/(c + d/x^2)^(3/2),x]","\frac{3 a \left(c^3 x^6-2 c^2 d x^4+8 c d^2 x^2+16 d^3\right)+5 b c \left(c^2 x^4-4 c d x^2-8 d^2\right)}{15 c^4 x \sqrt{c+\frac{d}{x^2}}}","-\frac{8 d x \sqrt{c+\frac{d}{x^2}} (5 b c-6 a d)}{15 c^4}+\frac{4 d x (5 b c-6 a d)}{15 c^3 \sqrt{c+\frac{d}{x^2}}}+\frac{x^3 (5 b c-6 a d)}{15 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^5}{5 c \sqrt{c+\frac{d}{x^2}}}",1,"(5*b*c*(-8*d^2 - 4*c*d*x^2 + c^2*x^4) + 3*a*(16*d^3 + 8*c*d^2*x^2 - 2*c^2*d*x^4 + c^3*x^6))/(15*c^4*Sqrt[c + d/x^2]*x)","A",1
981,1,57,79,0.0500692,"\int \frac{\left(a+\frac{b}{x^2}\right) x^2}{\left(c+\frac{d}{x^2}\right)^{3/2}} \, dx","Integrate[((a + b/x^2)*x^2)/(c + d/x^2)^(3/2),x]","\frac{a \left(c^2 x^4-4 c d x^2-8 d^2\right)+3 b c \left(c x^2+2 d\right)}{3 c^3 x \sqrt{c+\frac{d}{x^2}}}","\frac{2 x \sqrt{c+\frac{d}{x^2}} (3 b c-4 a d)}{3 c^3}-\frac{x (3 b c-4 a d)}{3 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^3}{3 c \sqrt{c+\frac{d}{x^2}}}",1,"(3*b*c*(2*d + c*x^2) + a*(-8*d^2 - 4*c*d*x^2 + c^2*x^4))/(3*c^3*Sqrt[c + d/x^2]*x)","A",1
982,1,33,45,0.0262292,"\int \frac{a+\frac{b}{x^2}}{\left(c+\frac{d}{x^2}\right)^{3/2}} \, dx","Integrate[(a + b/x^2)/(c + d/x^2)^(3/2),x]","\frac{a c x^2+2 a d-b c}{c^2 x \sqrt{c+\frac{d}{x^2}}}","\frac{a x}{c \sqrt{c+\frac{d}{x^2}}}-\frac{b c-2 a d}{c^2 x \sqrt{c+\frac{d}{x^2}}}",1,"(-(b*c) + 2*a*d + a*c*x^2)/(c^2*Sqrt[c + d/x^2]*x)","A",1
983,1,71,59,0.0385268,"\int \frac{a+\frac{b}{x^2}}{\left(c+\frac{d}{x^2}\right)^{3/2} x^2} \, dx","Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^2),x]","\frac{\sqrt{d} (b c-a d)-b c \sqrt{c x^2+d} \tanh ^{-1}\left(\frac{\sqrt{c x^2+d}}{\sqrt{d}}\right)}{c d^{3/2} x \sqrt{c+\frac{d}{x^2}}}","\frac{b c-a d}{c d x \sqrt{c+\frac{d}{x^2}}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{d^{3/2}}",1,"(Sqrt[d]*(b*c - a*d) - b*c*Sqrt[d + c*x^2]*ArcTanh[Sqrt[d + c*x^2]/Sqrt[d]])/(c*d^(3/2)*Sqrt[c + d/x^2]*x)","A",1
984,1,57,92,0.0290416,"\int \frac{a+\frac{b}{x^2}}{\left(c+\frac{d}{x^2}\right)^{3/2} x^4} \, dx","Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^4),x]","\frac{x^2 (2 a d-3 b c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c x^2}{d}+1\right)-b d}{2 d^2 x^3 \sqrt{c+\frac{d}{x^2}}}","\frac{(3 b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{2 d^{5/2}}-\frac{3 b c-2 a d}{2 d^2 x \sqrt{c+\frac{d}{x^2}}}-\frac{b}{2 d x^3 \sqrt{c+\frac{d}{x^2}}}",1,"(-(b*d) + (-3*b*c + 2*a*d)*x^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (c*x^2)/d])/(2*d^2*Sqrt[c + d/x^2]*x^3)","C",1
985,1,60,123,0.0297425,"\int \frac{a+\frac{b}{x^2}}{\left(c+\frac{d}{x^2}\right)^{3/2} x^6} \, dx","Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^6),x]","\frac{c x^4 (5 b c-4 a d) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{c x^2}{d}+1\right)-b d^2}{4 d^3 x^5 \sqrt{c+\frac{d}{x^2}}}","-\frac{3 c (5 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right)}{8 d^{7/2}}+\frac{3 \sqrt{c+\frac{d}{x^2}} (5 b c-4 a d)}{8 d^3 x}-\frac{5 b c-4 a d}{4 d^2 x^3 \sqrt{c+\frac{d}{x^2}}}-\frac{b}{4 d x^5 \sqrt{c+\frac{d}{x^2}}}",1,"(-(b*d^2) + c*(5*b*c - 4*a*d)*x^4*Hypergeometric2F1[-1/2, 2, 1/2, 1 + (c*x^2)/d])/(4*d^3*Sqrt[c + d/x^2]*x^5)","C",1
986,1,115,105,0.1547694,"\int \left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q (e x)^m \, dx","Integrate[(a + b/x^2)^p*(c + d/x^2)^q*(e*x)^m,x]","\frac{x (e x)^m \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2} (m-2 p-2 q+1);-p,-q;\frac{1}{2} (m-2 p-2 q+3);-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{m-2 p-2 q+1}","\frac{(e x)^{m+1} \left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(\frac{1}{2} (-m-1);-p,-q;\frac{1-m}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)}{e (m+1)}",1,"((a + b/x^2)^p*(c + d/x^2)^q*x*(e*x)^m*AppellF1[(1 + m - 2*p - 2*q)/2, -p, -q, (3 + m - 2*p - 2*q)/2, -((a*x^2)/b), -((c*x^2)/d)])/((1 + m - 2*p - 2*q)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q)","A",0
987,1,106,84,0.2042861,"\int \left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q x^4 \, dx","Integrate[(a + b/x^2)^p*(c + d/x^2)^q*x^4,x]","-\frac{x^5 \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q+\frac{5}{2};-p,-q;-p-q+\frac{7}{2};-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{2 p+2 q-5}","\frac{1}{5} x^5 \left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(-\frac{5}{2};-p,-q;-\frac{3}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)",1,"-(((a + b/x^2)^p*(c + d/x^2)^q*x^5*AppellF1[5/2 - p - q, -p, -q, 7/2 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-5 + 2*p + 2*q)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q))","A",0
988,1,100,100,0.1073711,"\int \left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q x^3 \, dx","Integrate[(a + b/x^2)^p*(c + d/x^2)^q*x^3,x]","-\frac{x^4 \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q+2;-p,-q;-p-q+3;-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{2 (p+q-2)}","\frac{b^2 \left(a+\frac{b}{x^2}\right)^{p+1} \left(c+\frac{d}{x^2}\right)^q \left(\frac{b \left(c+\frac{d}{x^2}\right)}{b c-a d}\right)^{-q} F_1\left(p+1;-q,3;p+2;-\frac{d \left(a+\frac{b}{x^2}\right)}{b c-a d},\frac{a+\frac{b}{x^2}}{a}\right)}{2 a^3 (p+1)}",1,"-1/2*((a + b/x^2)^p*(c + d/x^2)^q*x^4*AppellF1[2 - p - q, -p, -q, 3 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-2 + p + q)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q)","A",0
989,1,106,84,0.1062969,"\int \left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q x^2 \, dx","Integrate[(a + b/x^2)^p*(c + d/x^2)^q*x^2,x]","-\frac{x^3 \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q+\frac{3}{2};-p,-q;-p-q+\frac{5}{2};-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{2 p+2 q-3}","\frac{1}{3} x^3 \left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(-\frac{3}{2};-p,-q;-\frac{1}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)",1,"-(((a + b/x^2)^p*(c + d/x^2)^q*x^3*AppellF1[3/2 - p - q, -p, -q, 5/2 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-3 + 2*p + 2*q)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q))","A",0
990,1,100,98,0.1117114,"\int \left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q x \, dx","Integrate[(a + b/x^2)^p*(c + d/x^2)^q*x,x]","-\frac{x^2 \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q+1;-p,-q;-p-q+2;-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{2 (p+q-1)}","-\frac{b \left(a+\frac{b}{x^2}\right)^{p+1} \left(c+\frac{d}{x^2}\right)^q \left(\frac{b \left(c+\frac{d}{x^2}\right)}{b c-a d}\right)^{-q} F_1\left(p+1;-q,2;p+2;-\frac{d \left(a+\frac{b}{x^2}\right)}{b c-a d},\frac{a+\frac{b}{x^2}}{a}\right)}{2 a^2 (p+1)}",1,"-1/2*((a + b/x^2)^p*(c + d/x^2)^q*x^2*AppellF1[1 - p - q, -p, -q, 2 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-1 + p + q)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q)","A",0
991,1,104,79,0.0950383,"\int \left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q \, dx","Integrate[(a + b/x^2)^p*(c + d/x^2)^q,x]","-\frac{x \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q+\frac{1}{2};-p,-q;-p-q+\frac{3}{2};-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{2 p+2 q-1}","x \left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(-\frac{1}{2};-p,-q;\frac{1}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)",1,"-(((a + b/x^2)^p*(c + d/x^2)^q*x*AppellF1[1/2 - p - q, -p, -q, 3/2 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-1 + 2*p + 2*q)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q))","A",0
992,1,95,97,0.1521887,"\int \frac{\left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q}{x} \, dx","Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/x,x]","-\frac{\left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q;-p,-q;-p-q+1;-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{2 (p+q)}","\frac{\left(a+\frac{b}{x^2}\right)^{p+1} \left(c+\frac{d}{x^2}\right)^q \left(\frac{b \left(c+\frac{d}{x^2}\right)}{b c-a d}\right)^{-q} F_1\left(p+1;-q,1;p+2;-\frac{d \left(a+\frac{b}{x^2}\right)}{b c-a d},\frac{a+\frac{b}{x^2}}{a}\right)}{2 a (p+1)}",1,"-1/2*((a + b/x^2)^p*(c + d/x^2)^q*AppellF1[-p - q, -p, -q, 1 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((p + q)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q)","A",0
993,1,106,82,0.1551611,"\int \frac{\left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q}{x^2} \, dx","Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/x^2,x]","-\frac{\left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q-\frac{1}{2};-p,-q;-p-q+\frac{1}{2};-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{x (2 p+2 q+1)}","-\frac{\left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)}{x}",1,"-(((a + b/x^2)^p*(c + d/x^2)^q*AppellF1[-1/2 - p - q, -p, -q, 1/2 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((1 + 2*p + 2*q)*x*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q))","A",0
994,1,110,85,0.0832575,"\int \frac{\left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q}{x^3} \, dx","Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/x^3,x]","-\frac{\left(c x^2+d\right) \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(\frac{c x^2}{d}+1\right)^p \left(c+\frac{d}{x^2}\right)^q \, _2F_1\left(-p,-p-q-1;-p-q;\frac{(b c-a d) x^2}{b \left(c x^2+d\right)}\right)}{2 d x^2 (p+q+1)}","-\frac{\left(a+\frac{b}{x^2}\right)^{p+1} \left(c+\frac{d}{x^2}\right)^q \left(\frac{b \left(c+\frac{d}{x^2}\right)}{b c-a d}\right)^{-q} \, _2F_1\left(p+1,-q;p+2;-\frac{d \left(a+\frac{b}{x^2}\right)}{b c-a d}\right)}{2 b (p+1)}",1,"-1/2*((a + b/x^2)^p*(c + d/x^2)^q*(d + c*x^2)*(1 + (c*x^2)/d)^p*Hypergeometric2F1[-p, -1 - p - q, -p - q, ((b*c - a*d)*x^2)/(b*(d + c*x^2))])/(d*(1 + p + q)*x^2*(1 + (a*x^2)/b)^p)","A",0
995,1,106,84,0.3670563,"\int \frac{\left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q}{x^4} \, dx","Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/x^4,x]","-\frac{\left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q-\frac{3}{2};-p,-q;-p-q-\frac{1}{2};-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{x^3 (2 p+2 q+3)}","-\frac{\left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(\frac{3}{2};-p,-q;\frac{5}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)}{3 x^3}",1,"-(((a + b/x^2)^p*(c + d/x^2)^q*AppellF1[-3/2 - p - q, -p, -q, -1/2 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((3 + 2*p + 2*q)*x^3*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q))","A",0
996,1,111,91,0.1968744,"\int \left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q (e x)^{5/2} \, dx","Integrate[(a + b/x^2)^p*(c + d/x^2)^q*(e*x)^(5/2),x]","-\frac{2 x (e x)^{5/2} \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q+\frac{7}{4};-p,-q;-p-q+\frac{11}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{4 p+4 q-7}","\frac{2 (e x)^{7/2} \left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(-\frac{7}{4};-p,-q;-\frac{3}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)}{7 e}",1,"(-2*(a + b/x^2)^p*(c + d/x^2)^q*x*(e*x)^(5/2)*AppellF1[7/4 - p - q, -p, -q, 11/4 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-7 + 4*p + 4*q)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q)","A",0
997,1,111,91,0.1836171,"\int \left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q (e x)^{3/2} \, dx","Integrate[(a + b/x^2)^p*(c + d/x^2)^q*(e*x)^(3/2),x]","-\frac{2 x (e x)^{3/2} \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q+\frac{5}{4};-p,-q;-p-q+\frac{9}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{4 p+4 q-5}","\frac{2 (e x)^{5/2} \left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(-\frac{5}{4};-p,-q;-\frac{1}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)}{5 e}",1,"(-2*(a + b/x^2)^p*(c + d/x^2)^q*x*(e*x)^(3/2)*AppellF1[5/4 - p - q, -p, -q, 9/4 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-5 + 4*p + 4*q)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q)","A",0
998,1,111,91,0.1460991,"\int \left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q \sqrt{e x} \, dx","Integrate[(a + b/x^2)^p*(c + d/x^2)^q*Sqrt[e*x],x]","-\frac{2 x \sqrt{e x} \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q+\frac{3}{4};-p,-q;-p-q+\frac{7}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{4 p+4 q-3}","\frac{2 (e x)^{3/2} \left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(-\frac{3}{4};-p,-q;\frac{1}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)}{3 e}",1,"(-2*(a + b/x^2)^p*(c + d/x^2)^q*x*Sqrt[e*x]*AppellF1[3/4 - p - q, -p, -q, 7/4 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-3 + 4*p + 4*q)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q)","A",0
999,1,111,89,0.1581109,"\int \frac{\left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q}{\sqrt{e x}} \, dx","Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/Sqrt[e*x],x]","-\frac{2 x \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q+\frac{1}{4};-p,-q;-p-q+\frac{5}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{\sqrt{e x} (4 p+4 q-1)}","\frac{2 \sqrt{e x} \left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(-\frac{1}{4};-p,-q;\frac{3}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)}{e}",1,"(-2*(a + b/x^2)^p*(c + d/x^2)^q*x*AppellF1[1/4 - p - q, -p, -q, 5/4 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-1 + 4*p + 4*q)*Sqrt[e*x]*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q)","A",0
1000,1,111,89,0.1680631,"\int \frac{\left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q}{(e x)^{3/2}} \, dx","Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/(e*x)^(3/2),x]","-\frac{2 x \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q-\frac{1}{4};-p,-q;-p-q+\frac{3}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{(e x)^{3/2} (4 p+4 q+1)}","-\frac{2 \left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)}{e \sqrt{e x}}",1,"(-2*(a + b/x^2)^p*(c + d/x^2)^q*x*AppellF1[-1/4 - p - q, -p, -q, 3/4 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((1 + 4*p + 4*q)*(e*x)^(3/2)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q)","A",0
1001,1,111,91,0.1764,"\int \frac{\left(a+\frac{b}{x^2}\right)^p \left(c+\frac{d}{x^2}\right)^q}{(e x)^{5/2}} \, dx","Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/(e*x)^(5/2),x]","-\frac{2 x \left(a+\frac{b}{x^2}\right)^p \left(\frac{a x^2}{b}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{c x^2}{d}+1\right)^{-q} F_1\left(-p-q-\frac{3}{4};-p,-q;-p-q+\frac{1}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right)}{(e x)^{5/2} (4 p+4 q+3)}","-\frac{2 \left(a+\frac{b}{x^2}\right)^p \left(\frac{b}{a x^2}+1\right)^{-p} \left(c+\frac{d}{x^2}\right)^q \left(\frac{d}{c x^2}+1\right)^{-q} F_1\left(\frac{3}{4};-p,-q;\frac{7}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right)}{3 e (e x)^{3/2}}",1,"(-2*(a + b/x^2)^p*(c + d/x^2)^q*x*AppellF1[-3/4 - p - q, -p, -q, 1/4 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((3 + 4*p + 4*q)*(e*x)^(5/2)*(1 + (a*x^2)/b)^p*(1 + (c*x^2)/d)^q)","A",0
1002,1,111,135,0.0686335,"\int \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2} \, dx","Integrate[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2),x]","\frac{\sqrt{\sqrt{x}+1} \sqrt{x} \left(48 x^{7/2}-8 x^{5/2}-10 x^{3/2}-48 x^3+8 x^2+10 x-15 \sqrt{x}+15\right)+30 \sqrt{1-\sqrt{x}} \sin ^{-1}\left(\frac{\sqrt{1-\sqrt{x}}}{\sqrt{2}}\right)}{192 \sqrt{\sqrt{x}-1}}","\frac{1}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{7/2}-\frac{1}{24} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{5}{96} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{5}{64} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{5}{64} \cosh ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[1 + Sqrt[x]]*Sqrt[x]*(15 - 15*Sqrt[x] + 10*x - 10*x^(3/2) + 8*x^2 - 8*x^(5/2) - 48*x^3 + 48*x^(7/2)) + 30*Sqrt[1 - Sqrt[x]]*ArcSin[Sqrt[1 - Sqrt[x]]/Sqrt[2]])/(192*Sqrt[-1 + Sqrt[x]])","A",1
1003,1,99,104,0.0562274,"\int \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2} \, dx","Integrate[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2),x]","\frac{\sqrt{\sqrt{x}+1} \sqrt{x} \left(8 x^{5/2}-2 x^{3/2}-8 x^2+2 x-3 \sqrt{x}+3\right)+6 \sqrt{1-\sqrt{x}} \sin ^{-1}\left(\frac{\sqrt{1-\sqrt{x}}}{\sqrt{2}}\right)}{24 \sqrt{\sqrt{x}-1}}","\frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{1}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{1}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{1}{8} \cosh ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[1 + Sqrt[x]]*Sqrt[x]*(3 - 3*Sqrt[x] + 2*x - 2*x^(3/2) - 8*x^2 + 8*x^(5/2)) + 6*Sqrt[1 - Sqrt[x]]*ArcSin[Sqrt[1 - Sqrt[x]]/Sqrt[2]])/(24*Sqrt[-1 + Sqrt[x]])","A",1
1004,1,87,73,0.0751439,"\int \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x} \, dx","Integrate[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x],x]","\frac{\sqrt{\sqrt{x}+1} \sqrt{x} \left(2 x^{3/2}-2 x-\sqrt{x}+1\right)+2 \sqrt{1-\sqrt{x}} \sin ^{-1}\left(\frac{\sqrt{1-\sqrt{x}}}{\sqrt{2}}\right)}{4 \sqrt{\sqrt{x}-1}}","\frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{1}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{1}{4} \cosh ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[1 + Sqrt[x]]*Sqrt[x]*(1 - Sqrt[x] - 2*x + 2*x^(3/2)) + 2*Sqrt[1 - Sqrt[x]]*ArcSin[Sqrt[1 - Sqrt[x]]/Sqrt[2]])/(4*Sqrt[-1 + Sqrt[x]])","A",1
1005,1,72,37,0.0347693,"\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{\sqrt{x}} \, dx","Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/Sqrt[x],x]","\frac{\sqrt{\sqrt{x}+1} \sqrt{x} \left(\sqrt{x}-1\right)+2 \sqrt{1-\sqrt{x}} \sin ^{-1}\left(\frac{\sqrt{1-\sqrt{x}}}{\sqrt{2}}\right)}{\sqrt{\sqrt{x}-1}}","\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\cosh ^{-1}\left(\sqrt{x}\right)",1,"((-1 + Sqrt[x])*Sqrt[1 + Sqrt[x]]*Sqrt[x] + 2*Sqrt[1 - Sqrt[x]]*ArcSin[Sqrt[1 - Sqrt[x]]/Sqrt[2]])/Sqrt[-1 + Sqrt[x]]","A",1
1006,1,74,67,0.078629,"\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{3/2}} \, dx","Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(3/2),x]","\frac{2 \left(-\frac{\sqrt{\sqrt{x}+1} \left(\sqrt{x}-1\right)}{\sqrt{x}}-2 \sqrt{1-\sqrt{x}} \sin ^{-1}\left(\frac{\sqrt{1-\sqrt{x}}}{\sqrt{2}}\right)\right)}{\sqrt{\sqrt{x}-1}}","\frac{2 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{\sqrt{x}}-2 \sqrt{\sqrt{x}-1} \sqrt{x} \sqrt{\sqrt{x}+1}+2 \cosh ^{-1}\left(\sqrt{x}\right)",1,"(2*(-(((-1 + Sqrt[x])*Sqrt[1 + Sqrt[x]])/Sqrt[x]) - 2*Sqrt[1 - Sqrt[x]]*ArcSin[Sqrt[1 - Sqrt[x]]/Sqrt[2]]))/Sqrt[-1 + Sqrt[x]]","A",1
1007,1,31,31,0.0160509,"\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{5/2}} \, dx","Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(5/2),x]","\frac{2 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{3 x^{3/2}}","\frac{2 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{3 x^{3/2}}",1,"(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(3*x^(3/2))","A",1
1008,1,36,63,0.0206463,"\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{7/2}} \, dx","Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(7/2),x]","\frac{2 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2} (2 x+3)}{15 x^{5/2}}","\frac{4 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{15 x^{3/2}}+\frac{2 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{5 x^{5/2}}",1,"(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2)*(3 + 2*x))/(15*x^(5/2))","A",1
1009,1,41,94,0.0309268,"\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{9/2}} \, dx","Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(9/2),x]","\frac{2 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2} \left(8 x^2+12 x+15\right)}{105 x^{7/2}}","\frac{16 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{105 x^{3/2}}+\frac{8 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{35 x^{5/2}}+\frac{2 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{7 x^{7/2}}",1,"(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2)*(15 + 12*x + 8*x^2))/(105*x^(7/2))","A",1
1010,1,46,125,0.0331635,"\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{11/2}} \, dx","Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(11/2),x]","\frac{2 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2} \left(16 x^3+24 x^2+30 x+35\right)}{315 x^{9/2}}","\frac{32 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{315 x^{3/2}}+\frac{16 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{105 x^{5/2}}+\frac{4 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{21 x^{7/2}}+\frac{2 \left(\sqrt{x}-1\right)^{3/2} \left(\sqrt{x}+1\right)^{3/2}}{9 x^{9/2}}",1,"(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2)*(35 + 30*x + 24*x^2 + 16*x^3))/(315*x^(9/2))","A",1
1011,1,67,104,0.054114,"\int \frac{x^{5/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx","Integrate[x^(5/2)/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]),x]","\frac{1}{24} \left(\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} \left(8 x^2+10 x+15\right)+30 \tanh ^{-1}\left(\sqrt{\frac{\sqrt{x}-1}{\sqrt{x}+1}}\right)\right)","\frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}+\frac{5}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{5}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{5}{8} \cosh ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x]*(15 + 10*x + 8*x^2) + 30*ArcTanh[Sqrt[(-1 + Sqrt[x])/(1 + Sqrt[x])]])/24","A",0
1012,1,62,73,0.0362106,"\int \frac{x^{3/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx","Integrate[x^(3/2)/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]),x]","\frac{1}{4} \left(\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} (2 x+3)+6 \tanh ^{-1}\left(\sqrt{\frac{\sqrt{x}-1}{\sqrt{x}+1}}\right)\right)","\frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{3}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{3}{4} \cosh ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x]*(3 + 2*x) + 6*ArcTanh[Sqrt[(-1 + Sqrt[x])/(1 + Sqrt[x])]])/4","A",0
1013,1,55,35,0.0116248,"\int \frac{\sqrt{x}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx","Integrate[Sqrt[x]/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]),x]","\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+2 \tanh ^{-1}\left(\frac{\sqrt{\sqrt{x}-1}}{\sqrt{\sqrt{x}+1}}\right)","\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\cosh ^{-1}\left(\sqrt{x}\right)",1,"Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] + 2*ArcTanh[Sqrt[-1 + Sqrt[x]]/Sqrt[1 + Sqrt[x]]]","A",1
1014,1,26,8,0.0068611,"\int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}} \, dx","Integrate[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x]),x]","4 \tanh ^{-1}\left(\frac{\sqrt{\sqrt{x}-1}}{\sqrt{\sqrt{x}+1}}\right)","2 \cosh ^{-1}\left(\sqrt{x}\right)",1,"4*ArcTanh[Sqrt[-1 + Sqrt[x]]/Sqrt[1 + Sqrt[x]]]","B",1
1015,1,29,29,0.0104781,"\int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}} \, dx","Integrate[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2)),x]","\frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}}","\frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}}",1,"(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/Sqrt[x]","A",1
1016,1,36,63,0.0143062,"\int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}} \, dx","Integrate[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2)),x]","\frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} (2 x+1)}{3 x^{3/2}}","\frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 x^{3/2}}+\frac{4 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 \sqrt{x}}",1,"(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*(1 + 2*x))/(3*x^(3/2))","A",1
1017,1,41,94,0.0294004,"\int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{7/2}} \, dx","Integrate[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(7/2)),x]","\frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \left(8 x^2+4 x+3\right)}{15 x^{5/2}}","\frac{8 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{15 x^{3/2}}+\frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{5 x^{5/2}}+\frac{16 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{15 \sqrt{x}}",1,"(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*(3 + 4*x + 8*x^2))/(15*x^(5/2))","A",1
1018,1,80,78,0.0430811,"\int x^2 \left(-a+b x^n\right)^p \left(a+b x^n\right)^p \, dx","Integrate[x^2*(-a + b*x^n)^p*(a + b*x^n)^p,x]","\frac{1}{3} x^3 \left(b x^n-a\right)^p \left(a+b x^n\right)^p \left(1-\frac{b^2 x^{2 n}}{a^2}\right)^{-p} \, _2F_1\left(\frac{3}{2 n},-p;1+\frac{3}{2 n};\frac{b^2 x^{2 n}}{a^2}\right)","\frac{1}{3} x^3 \left(b x^n-a\right)^p \left(a+b x^n\right)^p \left(1-\frac{b^2 x^{2 n}}{a^2}\right)^{-p} \, _2F_1\left(\frac{3}{2 n},-p;1+\frac{3}{2 n};\frac{b^2 x^{2 n}}{a^2}\right)",1,"(x^3*(-a + b*x^n)^p*(a + b*x^n)^p*HypergeometricPFQ[{3/(2*n), -p}, {1 + 3/(2*n)}, (b^2*x^(2*n))/a^2])/(3*(1 - (b^2*x^(2*n))/a^2)^p)","A",1
1019,1,72,70,0.0176031,"\int x \left(-a+b x^n\right)^p \left(a+b x^n\right)^p \, dx","Integrate[x*(-a + b*x^n)^p*(a + b*x^n)^p,x]","\frac{1}{2} x^2 \left(b x^n-a\right)^p \left(a+b x^n\right)^p \left(1-\frac{b^2 x^{2 n}}{a^2}\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};\frac{b^2 x^{2 n}}{a^2}\right)","\frac{1}{2} x^2 \left(b x^n-a\right)^p \left(a+b x^n\right)^p \left(1-\frac{b^2 x^{2 n}}{a^2}\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};\frac{b^2 x^{2 n}}{a^2}\right)",1,"(x^2*(-a + b*x^n)^p*(a + b*x^n)^p*HypergeometricPFQ[{n^(-1), -p}, {1 + n^(-1)}, (b^2*x^(2*n))/a^2])/(2*(1 - (b^2*x^(2*n))/a^2)^p)","A",1
1020,1,73,73,0.0262891,"\int \left(-a+b x^n\right)^p \left(a+b x^n\right)^p \, dx","Integrate[(-a + b*x^n)^p*(a + b*x^n)^p,x]","x \left(b x^n-a\right)^p \left(a+b x^n\right)^p \left(1-\frac{b^2 x^{2 n}}{a^2}\right)^{-p} \, _2F_1\left(\frac{1}{2 n},-p;1+\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right)","x \left(b x^n-a\right)^p \left(a+b x^n\right)^p \left(1-\frac{b^2 x^{2 n}}{a^2}\right)^{-p} \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);\frac{b^2 x^{2 n}}{a^2}\right)",1,"(x*(-a + b*x^n)^p*(a + b*x^n)^p*Hypergeometric2F1[1/(2*n), -p, 1 + 1/(2*n), (b^2*x^(2*n))/a^2])/(1 - (b^2*x^(2*n))/a^2)^p","A",1
1021,1,73,72,0.0698116,"\int \frac{\left(-a+b x^n\right)^p \left(a+b x^n\right)^p}{x} \, dx","Integrate[((-a + b*x^n)^p*(a + b*x^n)^p)/x,x]","\frac{\left(b^2 x^{2 n}-a^2\right) \left(b x^n-a\right)^p \left(a+b x^n\right)^p \, _2F_1\left(1,p+1;p+2;1-\frac{b^2 x^{2 n}}{a^2}\right)}{2 a^2 n (p+1)}","-\frac{\left(a^2-b^2 x^{2 n}\right) \left(b x^n-a\right)^p \left(a+b x^n\right)^p \, _2F_1\left(1,p+1;p+2;1-\frac{b^2 x^{2 n}}{a^2}\right)}{2 a^2 n (p+1)}",1,"((-a + b*x^n)^p*(a + b*x^n)^p*(-a^2 + b^2*x^(2*n))*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (b^2*x^(2*n))/a^2])/(2*a^2*n*(1 + p))","A",1
1022,1,78,76,0.0290227,"\int \frac{\left(-a+b x^n\right)^p \left(a+b x^n\right)^p}{x^2} \, dx","Integrate[((-a + b*x^n)^p*(a + b*x^n)^p)/x^2,x]","-\frac{\left(b x^n-a\right)^p \left(a+b x^n\right)^p \left(1-\frac{b^2 x^{2 n}}{a^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2 n},-p;1-\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right)}{x}","-\frac{\left(b x^n-a\right)^p \left(a+b x^n\right)^p \left(1-\frac{b^2 x^{2 n}}{a^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2 n},-p;1-\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right)}{x}",1,"-(((-a + b*x^n)^p*(a + b*x^n)^p*HypergeometricPFQ[{-1/2*1/n, -p}, {1 - 1/(2*n)}, (b^2*x^(2*n))/a^2])/(x*(1 - (b^2*x^(2*n))/a^2)^p))","A",1
1023,1,15,15,0.0058013,"\int \frac{1+x^6}{x \left(1-x^6\right)} \, dx","Integrate[(1 + x^6)/(x*(1 - x^6)),x]","\log (x)-\frac{1}{3} \log \left(1-x^6\right)","\log (x)-\frac{1}{3} \log \left(1-x^6\right)",1,"Log[x] - Log[1 - x^6]/3","A",1
1024,1,110,22,0.1660218,"\int (e x)^m \left(a+b x^n\right)^p \left(a (1+m)+b (1+m+n+n p) x^n\right) \, dx","Integrate[(e*x)^m*(a + b*x^n)^p*(a*(1 + m) + b*(1 + m + n + n*p)*x^n),x]","x (e x)^m \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(\frac{b x^n (m+n p+n+1) \, _2F_1\left(\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right)}{m+n+1}+a \, _2F_1\left(\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right)\right)","\frac{(e x)^{m+1} \left(a+b x^n\right)^{p+1}}{e}",1,"(x*(e*x)^m*(a + b*x^n)^p*(a*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)] + (b*(1 + m + n + n*p)*x^n*Hypergeometric2F1[(1 + m + n)/n, -p, (1 + m + 2*n)/n, -((b*x^n)/a)])/(1 + m + n)))/(1 + (b*x^n)/a)^p","C",1
1025,1,88,114,0.1185108,"\int \frac{(e x)^m}{\left(a+b x^n\right) \left(c+d x^n\right)} \, dx","Integrate[(e*x)^m/((a + b*x^n)*(c + d*x^n)),x]","\frac{x (e x)^m \left(a d \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)-b c \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)\right)}{a c (m+1) (a d-b c)}","\frac{b (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a e (m+1) (b c-a d)}-\frac{d (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c e (m+1) (b c-a d)}",1,"(x*(e*x)^m*(-(b*c*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]) + a*d*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(a*c*(-(b*c) + a*d)*(1 + m))","A",1
1026,1,78,89,0.113428,"\int \frac{x^2}{\left(a+b x^n\right) \left(c+d x^n\right)} \, dx","Integrate[x^2/((a + b*x^n)*(c + d*x^n)),x]","\frac{b c x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right)-a d x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{d x^n}{c}\right)}{3 a b c^2-3 a^2 c d}","\frac{b x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right)}{3 a (b c-a d)}-\frac{d x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{d x^n}{c}\right)}{3 c (b c-a d)}",1,"(b*c*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, -((b*x^n)/a)] - a*d*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, -((d*x^n)/c)])/(3*a*b*c^2 - 3*a^2*c*d)","A",1
1027,1,78,89,0.0662068,"\int \frac{x}{\left(a+b x^n\right) \left(c+d x^n\right)} \, dx","Integrate[x/((a + b*x^n)*(c + d*x^n)),x]","\frac{b c x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right)-a d x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right)}{2 a b c^2-2 a^2 c d}","\frac{b x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right)}{2 a (b c-a d)}-\frac{d x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right)}{2 c (b c-a d)}",1,"(b*c*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, -((b*x^n)/a)] - a*d*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, -((d*x^n)/c)])/(2*a*b*c^2 - 2*a^2*c*d)","A",1
1028,1,64,72,0.0443913,"\int \frac{1}{\left(a+b x^n\right) \left(c+d x^n\right)} \, dx","Integrate[1/((a + b*x^n)*(c + d*x^n)),x]","\frac{x \left(a d \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)-b c \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)\right)}{a c (a d-b c)}","\frac{b x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a (b c-a d)}-\frac{d x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c (b c-a d)}",1,"(x*(-(b*c*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)]) + a*d*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)]))/(a*c*(-(b*c) + a*d))","A",1
1029,1,56,63,0.0647182,"\int \frac{1}{x \left(a+b x^n\right) \left(c+d x^n\right)} \, dx","Integrate[1/(x*(a + b*x^n)*(c + d*x^n)),x]","\frac{-b c \log \left(a+b x^n\right)+a d \log \left(c+d x^n\right)-a d n \log (x)+b c n \log (x)}{a b c^2 n-a^2 c d n}","-\frac{b \log \left(a+b x^n\right)}{a n (b c-a d)}+\frac{d \log \left(c+d x^n\right)}{c n (b c-a d)}+\frac{\log (x)}{a c}",1,"(b*c*n*Log[x] - a*d*n*Log[x] - b*c*Log[a + b*x^n] + a*d*Log[c + d*x^n])/(a*b*c^2*n - a^2*c*d*n)","A",1
1030,1,74,90,0.0791715,"\int \frac{1}{x^2 \left(a+b x^n\right) \left(c+d x^n\right)} \, dx","Integrate[1/(x^2*(a + b*x^n)*(c + d*x^n)),x]","\frac{b c \, _2F_1\left(1,-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right)-a d \, _2F_1\left(1,-\frac{1}{n};\frac{n-1}{n};-\frac{d x^n}{c}\right)}{a c x (a d-b c)}","\frac{d \, _2F_1\left(1,-\frac{1}{n};-\frac{1-n}{n};-\frac{d x^n}{c}\right)}{c x (b c-a d)}-\frac{b \, _2F_1\left(1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right)}{a x (b c-a d)}",1,"(b*c*Hypergeometric2F1[1, -n^(-1), (-1 + n)/n, -((b*x^n)/a)] - a*d*Hypergeometric2F1[1, -n^(-1), (-1 + n)/n, -((d*x^n)/c)])/(a*c*(-(b*c) + a*d)*x)","A",1
1031,1,77,95,0.0972429,"\int \frac{1}{x^3 \left(a+b x^n\right) \left(c+d x^n\right)} \, dx","Integrate[1/(x^3*(a + b*x^n)*(c + d*x^n)),x]","\frac{b c \, _2F_1\left(1,-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right)-a d \, _2F_1\left(1,-\frac{2}{n};\frac{n-2}{n};-\frac{d x^n}{c}\right)}{2 a c x^2 (a d-b c)}","\frac{d \, _2F_1\left(1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right)}{2 c x^2 (b c-a d)}-\frac{b \, _2F_1\left(1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right)}{2 a x^2 (b c-a d)}",1,"(b*c*Hypergeometric2F1[1, -2/n, (-2 + n)/n, -((b*x^n)/a)] - a*d*Hypergeometric2F1[1, -2/n, (-2 + n)/n, -((d*x^n)/c)])/(2*a*c*(-(b*c) + a*d)*x^2)","A",1
1032,1,141,175,0.2851489,"\int \frac{(e x)^m}{\left(a+b x^n\right)^2 \left(c+d x^n\right)} \, dx","Integrate[(e*x)^m/((a + b*x^n)^2*(c + d*x^n)),x]","\frac{x (e x)^m \left(\frac{b^2 c-a b d}{a^2 n+a b n x^n}+\frac{b (a d (m-2 n+1)-b c (m-n+1)) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^2 (m+1) n}+\frac{d^2 \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c m+c}\right)}{(b c-a d)^2}","\frac{b (e x)^{m+1} (a d (m-2 n+1)-b c (m-n+1)) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^2 e (m+1) n (b c-a d)^2}+\frac{d^2 (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c e (m+1) (b c-a d)^2}+\frac{b (e x)^{m+1}}{a e n (b c-a d) \left(a+b x^n\right)}",1,"(x*(e*x)^m*((b^2*c - a*b*d)/(a^2*n + a*b*n*x^n) + (b*(a*d*(1 + m - 2*n) - b*c*(1 + m - n))*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a^2*(1 + m)*n) + (d^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c + c*m)))/(b*c - a*d)^2","A",1
1033,1,135,142,0.235727,"\int \frac{x^2}{\left(a+b x^n\right)^2 \left(c+d x^n\right)} \, dx","Integrate[x^2/((a + b*x^n)^2*(c + d*x^n)),x]","\frac{x^3 \left(a \left(a d^2 n \left(a+b x^n\right) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{d x^n}{c}\right)+3 b c (b c-a d)\right)+b c \left(a+b x^n\right) (a d (3-2 n)+b c (n-3)) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right)\right)}{3 a^2 c n (b c-a d)^2 \left(a+b x^n\right)}","\frac{b x^3 (a d (3-2 n)-b c (3-n)) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right)}{3 a^2 n (b c-a d)^2}+\frac{d^2 x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{d x^n}{c}\right)}{3 c (b c-a d)^2}+\frac{b x^3}{a n (b c-a d) \left(a+b x^n\right)}",1,"(x^3*(b*c*(a*d*(3 - 2*n) + b*c*(-3 + n))*(a + b*x^n)*Hypergeometric2F1[1, 3/n, (3 + n)/n, -((b*x^n)/a)] + a*(3*b*c*(b*c - a*d) + a*d^2*n*(a + b*x^n)*Hypergeometric2F1[1, 3/n, (3 + n)/n, -((d*x^n)/c)])))/(3*a^2*c*(b*c - a*d)^2*n*(a + b*x^n))","A",1
1034,1,134,143,0.2326307,"\int \frac{x}{\left(a+b x^n\right)^2 \left(c+d x^n\right)} \, dx","Integrate[x/((a + b*x^n)^2*(c + d*x^n)),x]","\frac{x^2 \left(a \left(a d^2 n \left(a+b x^n\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right)+2 b c (b c-a d)\right)+b c \left(a+b x^n\right) (b c (n-2)-2 a d (n-1)) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right)\right)}{2 a^2 c n (b c-a d)^2 \left(a+b x^n\right)}","\frac{b x^2 (2 a d (1-n)-b c (2-n)) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right)}{2 a^2 n (b c-a d)^2}+\frac{d^2 x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right)}{2 c (b c-a d)^2}+\frac{b x^2}{a n (b c-a d) \left(a+b x^n\right)}",1,"(x^2*(b*c*(b*c*(-2 + n) - 2*a*d*(-1 + n))*(a + b*x^n)*Hypergeometric2F1[1, 2/n, (2 + n)/n, -((b*x^n)/a)] + a*(2*b*c*(b*c - a*d) + a*d^2*n*(a + b*x^n)*Hypergeometric2F1[1, 2/n, (2 + n)/n, -((d*x^n)/c)])))/(2*a^2*c*(b*c - a*d)^2*n*(a + b*x^n))","A",1
1035,1,108,122,0.1891278,"\int \frac{1}{\left(a+b x^n\right)^2 \left(c+d x^n\right)} \, dx","Integrate[1/((a + b*x^n)^2*(c + d*x^n)),x]","\frac{x \left(\frac{b^2 c-a b d}{a^2 n+a b n x^n}+\frac{b (a d (1-2 n)+b c (n-1)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2 n}+\frac{d^2 \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c}\right)}{(b c-a d)^2}","\frac{b x (a d (1-2 n)-b c (1-n)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2 n (b c-a d)^2}+\frac{d^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c (b c-a d)^2}+\frac{b x}{a n (b c-a d) \left(a+b x^n\right)}",1,"(x*((b^2*c - a*b*d)/(a^2*n + a*b*n*x^n) + (b*(a*d*(1 - 2*n) + b*c*(-1 + n))*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a^2*n) + (d^2*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/c))/(b*c - a*d)^2","A",1
1036,1,97,101,0.3285374,"\int \frac{1}{x \left(a+b x^n\right)^2 \left(c+d x^n\right)} \, dx","Integrate[1/(x*(a + b*x^n)^2*(c + d*x^n)),x]","\frac{-\frac{b (b c-2 a d) \log \left(a+b x^n\right)}{a^2 (b c-a d)^2}+\frac{n \log (x)}{a^2 c}-\frac{d^2 \log \left(c+d x^n\right)}{c (b c-a d)^2}+\frac{b}{a (b c-a d) \left(a+b x^n\right)}}{n}","-\frac{b (b c-2 a d) \log \left(a+b x^n\right)}{a^2 n (b c-a d)^2}+\frac{\log (x)}{a^2 c}-\frac{d^2 \log \left(c+d x^n\right)}{c n (b c-a d)^2}+\frac{b}{a n (b c-a d) \left(a+b x^n\right)}",1,"(b/(a*(b*c - a*d)*(a + b*x^n)) + (n*Log[x])/(a^2*c) - (b*(b*c - 2*a*d)*Log[a + b*x^n])/(a^2*(b*c - a*d)^2) - (d^2*Log[c + d*x^n])/(c*(b*c - a*d)^2))/n","A",1
1037,1,133,142,0.2697185,"\int \frac{1}{x^2 \left(a+b x^n\right)^2 \left(c+d x^n\right)} \, dx","Integrate[1/(x^2*(a + b*x^n)^2*(c + d*x^n)),x]","\frac{b c \left(a+b x^n\right) (a d (2 n+1)-b c (n+1)) \, _2F_1\left(1,-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right)-a \left(a d^2 n \left(a+b x^n\right) \, _2F_1\left(1,-\frac{1}{n};\frac{n-1}{n};-\frac{d x^n}{c}\right)+b c (a d-b c)\right)}{a^2 c n x (b c-a d)^2 \left(a+b x^n\right)}","-\frac{b (b c (n+1)-a d (2 n+1)) \, _2F_1\left(1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right)}{a^2 n x (b c-a d)^2}-\frac{d^2 \, _2F_1\left(1,-\frac{1}{n};-\frac{1-n}{n};-\frac{d x^n}{c}\right)}{c x (b c-a d)^2}+\frac{b}{a n x (b c-a d) \left(a+b x^n\right)}",1,"(b*c*(-(b*c*(1 + n)) + a*d*(1 + 2*n))*(a + b*x^n)*Hypergeometric2F1[1, -n^(-1), (-1 + n)/n, -((b*x^n)/a)] - a*(b*c*(-(b*c) + a*d) + a*d^2*n*(a + b*x^n)*Hypergeometric2F1[1, -n^(-1), (-1 + n)/n, -((d*x^n)/c)]))/(a^2*c*(b*c - a*d)^2*n*x*(a + b*x^n))","A",1
1038,1,136,145,0.2314281,"\int \frac{1}{x^3 \left(a+b x^n\right)^2 \left(c+d x^n\right)} \, dx","Integrate[1/(x^3*(a + b*x^n)^2*(c + d*x^n)),x]","\frac{b c \left(a+b x^n\right) (2 a d (n+1)-b c (n+2)) \, _2F_1\left(1,-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right)-a \left(a d^2 n \left(a+b x^n\right) \, _2F_1\left(1,-\frac{2}{n};\frac{n-2}{n};-\frac{d x^n}{c}\right)+2 b c (a d-b c)\right)}{2 a^2 c n x^2 (b c-a d)^2 \left(a+b x^n\right)}","\frac{b (2 a d (n+1)-b c (n+2)) \, _2F_1\left(1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right)}{2 a^2 n x^2 (b c-a d)^2}-\frac{d^2 \, _2F_1\left(1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right)}{2 c x^2 (b c-a d)^2}+\frac{b}{a n x^2 (b c-a d) \left(a+b x^n\right)}",1,"(b*c*(2*a*d*(1 + n) - b*c*(2 + n))*(a + b*x^n)*Hypergeometric2F1[1, -2/n, (-2 + n)/n, -((b*x^n)/a)] - a*(2*b*c*(-(b*c) + a*d) + a*d^2*n*(a + b*x^n)*Hypergeometric2F1[1, -2/n, (-2 + n)/n, -((d*x^n)/c)]))/(2*a^2*c*(b*c - a*d)^2*n*x^2*(a + b*x^n))","A",1
1039,1,115,130,0.353826,"\int \frac{x^{-1+2 n} \left(a+b x^n\right)^3}{c+d x^n} \, dx","Integrate[(x^(-1 + 2*n)*(a + b*x^n)^3)/(c + d*x^n),x]","\frac{6 b d^2 x^{2 n} \left(3 a^2 d^2-3 a b c d+b^2 c^2\right)-4 b^2 d^3 x^{3 n} (b c-3 a d)+12 d x^n (a d-b c)^3+12 c (b c-a d)^3 \log \left(c+d x^n\right)+3 b^3 d^4 x^{4 n}}{12 d^5 n}","\frac{b x^{2 n} \left(3 a^2 d^2-3 a b c d+b^2 c^2\right)}{2 d^3 n}-\frac{b^2 x^{3 n} (b c-3 a d)}{3 d^2 n}+\frac{c (b c-a d)^3 \log \left(c+d x^n\right)}{d^5 n}-\frac{x^n (b c-a d)^3}{d^4 n}+\frac{b^3 x^{4 n}}{4 d n}",1,"(12*d*(-(b*c) + a*d)^3*x^n + 6*b*d^2*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*x^(2*n) - 4*b^2*d^3*(b*c - 3*a*d)*x^(3*n) + 3*b^3*d^4*x^(4*n) + 12*c*(b*c - a*d)^3*Log[c + d*x^n])/(12*d^5*n)","A",1
1040,1,82,90,0.1377824,"\int \frac{x^{-1+2 n} \left(a+b x^n\right)^2}{c+d x^n} \, dx","Integrate[(x^(-1 + 2*n)*(a + b*x^n)^2)/(c + d*x^n),x]","\frac{-\frac{c (b c-a d)^2 \log \left(c+d x^n\right)}{d^4}+\frac{x^n (b c-a d)^2}{d^3}-\frac{b x^{2 n} (b c-2 a d)}{2 d^2}+\frac{b^2 x^{3 n}}{3 d}}{n}","-\frac{c (b c-a d)^2 \log \left(c+d x^n\right)}{d^4 n}+\frac{x^n (b c-a d)^2}{d^3 n}-\frac{b x^{2 n} (b c-2 a d)}{2 d^2 n}+\frac{b^2 x^{3 n}}{3 d n}",1,"(((b*c - a*d)^2*x^n)/d^3 - (b*(b*c - 2*a*d)*x^(2*n))/(2*d^2) + (b^2*x^(3*n))/(3*d) - (c*(b*c - a*d)^2*Log[c + d*x^n])/d^4)/n","A",1
1041,1,50,60,0.0582686,"\int \frac{x^{-1+2 n} \left(a+b x^n\right)}{c+d x^n} \, dx","Integrate[(x^(-1 + 2*n)*(a + b*x^n))/(c + d*x^n),x]","\frac{d x^n \left(2 a d-2 b c+b d x^n\right)+2 c (b c-a d) \log \left(c+d x^n\right)}{2 d^3 n}","\frac{c (b c-a d) \log \left(c+d x^n\right)}{d^3 n}-\frac{x^n (b c-a d)}{d^2 n}+\frac{b x^{2 n}}{2 d n}",1,"(d*x^n*(-2*b*c + 2*a*d + b*d*x^n) + 2*c*(b*c - a*d)*Log[c + d*x^n])/(2*d^3*n)","A",1
1042,1,44,54,0.0451558,"\int \frac{x^{-1+2 n}}{\left(a+b x^n\right) \left(c+d x^n\right)} \, dx","Integrate[x^(-1 + 2*n)/((a + b*x^n)*(c + d*x^n)),x]","-\frac{a d \log \left(a+b x^n\right)-b c \log \left(c+d x^n\right)}{b^2 c d n-a b d^2 n}","\frac{c \log \left(c+d x^n\right)}{d n (b c-a d)}-\frac{a \log \left(a+b x^n\right)}{b n (b c-a d)}",1,"-((a*d*Log[a + b*x^n] - b*c*Log[c + d*x^n])/(b^2*c*d*n - a*b*d^2*n))","A",1
1043,1,58,75,0.1354245,"\int \frac{x^{-1+2 n}}{\left(a+b x^n\right)^2 \left(c+d x^n\right)} \, dx","Integrate[x^(-1 + 2*n)/((a + b*x^n)^2*(c + d*x^n)),x]","\frac{\frac{a (b c-a d)}{b \left(a+b x^n\right)}+c \log \left(a+b x^n\right)-c \log \left(c+d x^n\right)}{n (b c-a d)^2}","\frac{a}{b n (b c-a d) \left(a+b x^n\right)}+\frac{c \log \left(a+b x^n\right)}{n (b c-a d)^2}-\frac{c \log \left(c+d x^n\right)}{n (b c-a d)^2}",1,"((a*(b*c - a*d))/(b*(a + b*x^n)) + c*Log[a + b*x^n] - c*Log[c + d*x^n])/((b*c - a*d)^2*n)","A",1
1044,1,97,105,0.1676452,"\int \frac{x^{-1+2 n}}{\left(a+b x^n\right)^3 \left(c+d x^n\right)} \, dx","Integrate[x^(-1 + 2*n)/((a + b*x^n)^3*(c + d*x^n)),x]","\frac{\frac{a}{2 b (b c-a d) \left(a+b x^n\right)^2}-\frac{c}{(b c-a d)^2 \left(a+b x^n\right)}-\frac{c d \log \left(a+b x^n\right)}{(b c-a d)^3}+\frac{c d \log \left(c+d x^n\right)}{(b c-a d)^3}}{n}","\frac{a}{2 b n (b c-a d) \left(a+b x^n\right)^2}-\frac{c}{n (b c-a d)^2 \left(a+b x^n\right)}-\frac{c d \log \left(a+b x^n\right)}{n (b c-a d)^3}+\frac{c d \log \left(c+d x^n\right)}{n (b c-a d)^3}",1,"(a/(2*b*(b*c - a*d)*(a + b*x^n)^2) - c/((b*c - a*d)^2*(a + b*x^n)) - (c*d*Log[a + b*x^n])/(b*c - a*d)^3 + (c*d*Log[c + d*x^n])/(b*c - a*d)^3)/n","A",1
1045,1,138,158,0.3136197,"\int \frac{x^{-1+3 n} \left(a+b x^n\right)^3}{c+d x^n} \, dx","Integrate[(x^(-1 + 3*n)*(a + b*x^n)^3)/(c + d*x^n),x]","\frac{20 b d^3 x^{3 n} \left(3 a^2 d^2-3 a b c d+b^2 c^2\right)-15 b^2 d^4 x^{4 n} (b c-3 a d)-60 c^2 (b c-a d)^3 \log \left(c+d x^n\right)+30 d^2 x^{2 n} (a d-b c)^3+60 c d x^n (b c-a d)^3+12 b^3 d^5 x^{5 n}}{60 d^6 n}","\frac{b x^{3 n} \left(3 a^2 d^2-3 a b c d+b^2 c^2\right)}{3 d^3 n}-\frac{b^2 x^{4 n} (b c-3 a d)}{4 d^2 n}-\frac{c^2 (b c-a d)^3 \log \left(c+d x^n\right)}{d^6 n}+\frac{c x^n (b c-a d)^3}{d^5 n}-\frac{x^{2 n} (b c-a d)^3}{2 d^4 n}+\frac{b^3 x^{5 n}}{5 d n}",1,"(60*c*d*(b*c - a*d)^3*x^n + 30*d^2*(-(b*c) + a*d)^3*x^(2*n) + 20*b*d^3*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*x^(3*n) - 15*b^2*d^4*(b*c - 3*a*d)*x^(4*n) + 12*b^3*d^5*x^(5*n) - 60*c^2*(b*c - a*d)^3*Log[c + d*x^n])/(60*d^6*n)","A",1
1046,1,103,118,0.2691684,"\int \frac{x^{-1+3 n} \left(a+b x^n\right)^2}{c+d x^n} \, dx","Integrate[(x^(-1 + 3*n)*(a + b*x^n)^2)/(c + d*x^n),x]","\frac{12 c^2 (b c-a d)^2 \log \left(c+d x^n\right)-4 b d^3 x^{3 n} (b c-2 a d)+6 d^2 x^{2 n} (b c-a d)^2-12 c d x^n (b c-a d)^2+3 b^2 d^4 x^{4 n}}{12 d^5 n}","\frac{c^2 (b c-a d)^2 \log \left(c+d x^n\right)}{d^5 n}-\frac{c x^n (b c-a d)^2}{d^4 n}+\frac{x^{2 n} (b c-a d)^2}{2 d^3 n}-\frac{b x^{3 n} (b c-2 a d)}{3 d^2 n}+\frac{b^2 x^{4 n}}{4 d n}",1,"(-12*c*d*(b*c - a*d)^2*x^n + 6*d^2*(b*c - a*d)^2*x^(2*n) - 4*b*d^3*(b*c - 2*a*d)*x^(3*n) + 3*b^2*d^4*x^(4*n) + 12*c^2*(b*c - a*d)^2*Log[c + d*x^n])/(12*d^5*n)","A",1
1047,1,78,86,0.0986453,"\int \frac{x^{-1+3 n} \left(a+b x^n\right)}{c+d x^n} \, dx","Integrate[(x^(-1 + 3*n)*(a + b*x^n))/(c + d*x^n),x]","\frac{-\frac{c^2 (b c-a d) \log \left(c+d x^n\right)}{d^4}+\frac{c x^n (b c-a d)}{d^3}-\frac{x^{2 n} (b c-a d)}{2 d^2}+\frac{b x^{3 n}}{3 d}}{n}","-\frac{c^2 (b c-a d) \log \left(c+d x^n\right)}{d^4 n}+\frac{c x^n (b c-a d)}{d^3 n}-\frac{x^{2 n} (b c-a d)}{2 d^2 n}+\frac{b x^{3 n}}{3 d n}",1,"((c*(b*c - a*d)*x^n)/d^3 - ((b*c - a*d)*x^(2*n))/(2*d^2) + (b*x^(3*n))/(3*d) - (c^2*(b*c - a*d)*Log[c + d*x^n])/d^4)/n","A",1
1048,1,66,71,0.1054334,"\int \frac{x^{-1+3 n}}{\left(a+b x^n\right) \left(c+d x^n\right)} \, dx","Integrate[x^(-1 + 3*n)/((a + b*x^n)*(c + d*x^n)),x]","\frac{\frac{a^2 \log \left(a+b x^n\right)}{b^2 (b c-a d)}-\frac{c^2 \log \left(c+d x^n\right)}{d^2 (b c-a d)}+\frac{x^n}{b d}}{n}","\frac{a^2 \log \left(a+b x^n\right)}{b^2 n (b c-a d)}-\frac{c^2 \log \left(c+d x^n\right)}{d^2 n (b c-a d)}+\frac{x^n}{b d n}",1,"(x^n/(b*d) + (a^2*Log[a + b*x^n])/(b^2*(b*c - a*d)) - (c^2*Log[c + d*x^n])/(d^2*(b*c - a*d)))/n","A",1
1049,1,90,95,0.1096858,"\int \frac{x^{-1+3 n}}{\left(a+b x^n\right)^2 \left(c+d x^n\right)} \, dx","Integrate[x^(-1 + 3*n)/((a + b*x^n)^2*(c + d*x^n)),x]","\frac{-\frac{a^2}{b^2 (b c-a d) \left(a+b x^n\right)}-\frac{a (2 b c-a d) \log \left(a+b x^n\right)}{b^2 (b c-a d)^2}+\frac{c^2 \log \left(c+d x^n\right)}{d (b c-a d)^2}}{n}","-\frac{a^2}{b^2 n (b c-a d) \left(a+b x^n\right)}-\frac{a (2 b c-a d) \log \left(a+b x^n\right)}{b^2 n (b c-a d)^2}+\frac{c^2 \log \left(c+d x^n\right)}{d n (b c-a d)^2}",1,"(-(a^2/(b^2*(b*c - a*d)*(a + b*x^n))) - (a*(2*b*c - a*d)*Log[a + b*x^n])/(b^2*(b*c - a*d)^2) + (c^2*Log[c + d*x^n])/(d*(b*c - a*d)^2))/n","A",1
1050,1,112,120,0.2748193,"\int \frac{x^{-1+3 n}}{\left(a+b x^n\right)^3 \left(c+d x^n\right)} \, dx","Integrate[x^(-1 + 3*n)/((a + b*x^n)^3*(c + d*x^n)),x]","\frac{-\frac{a^2}{2 b^2 (b c-a d) \left(a+b x^n\right)^2}+\frac{a (2 b c-a d)}{b^2 (b c-a d)^2 \left(a+b x^n\right)}+\frac{c^2 \log \left(a+b x^n\right)}{(b c-a d)^3}-\frac{c^2 \log \left(c+d x^n\right)}{(b c-a d)^3}}{n}","-\frac{a^2}{2 b^2 n (b c-a d) \left(a+b x^n\right)^2}+\frac{a (2 b c-a d)}{b^2 n (b c-a d)^2 \left(a+b x^n\right)}+\frac{c^2 \log \left(a+b x^n\right)}{n (b c-a d)^3}-\frac{c^2 \log \left(c+d x^n\right)}{n (b c-a d)^3}",1,"(-1/2*a^2/(b^2*(b*c - a*d)*(a + b*x^n)^2) + (a*(2*b*c - a*d))/(b^2*(b*c - a*d)^2*(a + b*x^n)) + (c^2*Log[a + b*x^n])/(b*c - a*d)^3 - (c^2*Log[c + d*x^n])/(b*c - a*d)^3)/n","A",1
1051,1,172,14,0.0062008,"\int x^{13} (b+c x)^{13} (b+2 c x) \, dx","Integrate[x^13*(b + c*x)^13*(b + 2*c*x),x]","\frac{b^{14} x^{14}}{14}+b^{13} c x^{15}+\frac{13}{2} b^{12} c^2 x^{16}+26 b^{11} c^3 x^{17}+\frac{143}{2} b^{10} c^4 x^{18}+143 b^9 c^5 x^{19}+\frac{429}{2} b^8 c^6 x^{20}+\frac{1716}{7} b^7 c^7 x^{21}+\frac{429}{2} b^6 c^8 x^{22}+143 b^5 c^9 x^{23}+\frac{143}{2} b^4 c^{10} x^{24}+26 b^3 c^{11} x^{25}+\frac{13}{2} b^2 c^{12} x^{26}+b c^{13} x^{27}+\frac{c^{14} x^{28}}{14}","\frac{1}{14} x^{14} (b+c x)^{14}",1,"(b^14*x^14)/14 + b^13*c*x^15 + (13*b^12*c^2*x^16)/2 + 26*b^11*c^3*x^17 + (143*b^10*c^4*x^18)/2 + 143*b^9*c^5*x^19 + (429*b^8*c^6*x^20)/2 + (1716*b^7*c^7*x^21)/7 + (429*b^6*c^8*x^22)/2 + 143*b^5*c^9*x^23 + (143*b^4*c^10*x^24)/2 + 26*b^3*c^11*x^25 + (13*b^2*c^12*x^26)/2 + b*c^13*x^27 + (c^14*x^28)/14","B",1
1052,1,182,16,0.0062635,"\int x^{27} \left(b+c x^2\right)^{13} \left(b+2 c x^2\right) \, dx","Integrate[x^27*(b + c*x^2)^13*(b + 2*c*x^2),x]","\frac{b^{14} x^{28}}{28}+\frac{1}{2} b^{13} c x^{30}+\frac{13}{4} b^{12} c^2 x^{32}+13 b^{11} c^3 x^{34}+\frac{143}{4} b^{10} c^4 x^{36}+\frac{143}{2} b^9 c^5 x^{38}+\frac{429}{4} b^8 c^6 x^{40}+\frac{858}{7} b^7 c^7 x^{42}+\frac{429}{4} b^6 c^8 x^{44}+\frac{143}{2} b^5 c^9 x^{46}+\frac{143}{4} b^4 c^{10} x^{48}+13 b^3 c^{11} x^{50}+\frac{13}{4} b^2 c^{12} x^{52}+\frac{1}{2} b c^{13} x^{54}+\frac{c^{14} x^{56}}{28}","\frac{1}{28} x^{28} \left(b+c x^2\right)^{14}",1,"(b^14*x^28)/28 + (b^13*c*x^30)/2 + (13*b^12*c^2*x^32)/4 + 13*b^11*c^3*x^34 + (143*b^10*c^4*x^36)/4 + (143*b^9*c^5*x^38)/2 + (429*b^8*c^6*x^40)/4 + (858*b^7*c^7*x^42)/7 + (429*b^6*c^8*x^44)/4 + (143*b^5*c^9*x^46)/2 + (143*b^4*c^10*x^48)/4 + 13*b^3*c^11*x^50 + (13*b^2*c^12*x^52)/4 + (b*c^13*x^54)/2 + (c^14*x^56)/28","B",1
1053,1,186,16,0.0205947,"\int x^{41} \left(b+c x^3\right)^{13} \left(b+2 c x^3\right) \, dx","Integrate[x^41*(b + c*x^3)^13*(b + 2*c*x^3),x]","\frac{b^{14} x^{42}}{42}+\frac{1}{3} b^{13} c x^{45}+\frac{13}{6} b^{12} c^2 x^{48}+\frac{26}{3} b^{11} c^3 x^{51}+\frac{143}{6} b^{10} c^4 x^{54}+\frac{143}{3} b^9 c^5 x^{57}+\frac{143}{2} b^8 c^6 x^{60}+\frac{572}{7} b^7 c^7 x^{63}+\frac{143}{2} b^6 c^8 x^{66}+\frac{143}{3} b^5 c^9 x^{69}+\frac{143}{6} b^4 c^{10} x^{72}+\frac{26}{3} b^3 c^{11} x^{75}+\frac{13}{6} b^2 c^{12} x^{78}+\frac{1}{3} b c^{13} x^{81}+\frac{c^{14} x^{84}}{42}","\frac{1}{42} x^{42} \left(b+c x^3\right)^{14}",1,"(b^14*x^42)/42 + (b^13*c*x^45)/3 + (13*b^12*c^2*x^48)/6 + (26*b^11*c^3*x^51)/3 + (143*b^10*c^4*x^54)/6 + (143*b^9*c^5*x^57)/3 + (143*b^8*c^6*x^60)/2 + (572*b^7*c^7*x^63)/7 + (143*b^6*c^8*x^66)/2 + (143*b^5*c^9*x^69)/3 + (143*b^4*c^10*x^72)/6 + (26*b^3*c^11*x^75)/3 + (13*b^2*c^12*x^78)/6 + (b*c^13*x^81)/3 + (c^14*x^84)/42","B",1
1054,1,21,21,0.1899397,"\int x^{-1+14 n} \left(b+c x^n\right)^{13} \left(b+2 c x^n\right) \, dx","Integrate[x^(-1 + 14*n)*(b + c*x^n)^13*(b + 2*c*x^n),x]","\frac{x^{14 n} \left(b+c x^n\right)^{14}}{14 n}","\frac{x^{14 n} \left(b+c x^n\right)^{14}}{14 n}",1,"(x^(14*n)*(b + c*x^n)^14)/(14*n)","A",1
1055,1,107,13,0.1839356,"\int x^{-1+m} \left(a+b x^n\right)^{-1+p} \left(a m+b (m+n p) x^n\right) \, dx","Integrate[x^(-1 + m)*(a + b*x^n)^(-1 + p)*(a*m + b*(m + n*p)*x^n),x]","\frac{x^m \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(b x^n (m+n p) \, _2F_1\left(\frac{m+n}{n},1-p;\frac{m}{n}+2;-\frac{b x^n}{a}\right)+a (m+n) \, _2F_1\left(\frac{m}{n},1-p;\frac{m+n}{n};-\frac{b x^n}{a}\right)\right)}{a (m+n)}","x^m \left(a+b x^n\right)^p",1,"(x^m*(a + b*x^n)^p*(a*(m + n)*Hypergeometric2F1[m/n, 1 - p, (m + n)/n, -((b*x^n)/a)] + b*(m + n*p)*x^n*Hypergeometric2F1[(m + n)/n, 1 - p, 2 + m/n, -((b*x^n)/a)]))/(a*(m + n)*(1 + (b*x^n)/a)^p)","C",1
1056,1,9,8,0.0181014,"\int \frac{b+2 c x}{x (b+c x)} \, dx","Integrate[(b + 2*c*x)/(x*(b + c*x)),x]","\log (b+c x)+\log (x)","\log (x (b+c x))",1,"Log[x] + Log[b + c*x]","A",1
1057,1,15,15,0.00928,"\int \frac{b+2 c x^2}{x \left(b+c x^2\right)} \, dx","Integrate[(b + 2*c*x^2)/(x*(b + c*x^2)),x]","\frac{1}{2} \log \left(b+c x^2\right)+\log (x)","\frac{1}{2} \log \left(b+c x^2\right)+\log (x)",1,"Log[x] + Log[b + c*x^2]/2","A",1
1058,1,15,15,0.0199735,"\int \frac{b+2 c x^3}{x \left(b+c x^3\right)} \, dx","Integrate[(b + 2*c*x^3)/(x*(b + c*x^3)),x]","\frac{1}{3} \log \left(b+c x^3\right)+\log (x)","\frac{1}{3} \log \left(b+c x^3\right)+\log (x)",1,"Log[x] + Log[b + c*x^3]/3","A",1
1059,1,15,15,0.0156889,"\int \frac{b+2 c x^n}{x \left(b+c x^n\right)} \, dx","Integrate[(b + 2*c*x^n)/(x*(b + c*x^n)),x]","\frac{\log \left(b+c x^n\right)}{n}+\log (x)","\frac{\log \left(b+c x^n\right)}{n}+\log (x)",1,"Log[x] + Log[b + c*x^n]/n","A",1
1060,1,14,14,0.0242832,"\int \frac{b+2 c x}{x^8 (b+c x)^8} \, dx","Integrate[(b + 2*c*x)/(x^8*(b + c*x)^8),x]","-\frac{1}{7 x^7 (b+c x)^7}","-\frac{1}{7 x^7 (b+c x)^7}",1,"-1/7*1/(x^7*(b + c*x)^7)","A",1
1061,1,16,16,0.0470226,"\int \frac{b+2 c x^2}{x^{15} \left(b+c x^2\right)^8} \, dx","Integrate[(b + 2*c*x^2)/(x^15*(b + c*x^2)^8),x]","-\frac{1}{14 x^{14} \left(b+c x^2\right)^7}","-\frac{1}{14 x^{14} \left(b+c x^2\right)^7}",1,"-1/14*1/(x^14*(b + c*x^2)^7)","A",1
1062,1,16,16,0.0542479,"\int \frac{b+2 c x^3}{x^{22} \left(b+c x^3\right)^8} \, dx","Integrate[(b + 2*c*x^3)/(x^22*(b + c*x^3)^8),x]","-\frac{1}{21 x^{21} \left(b+c x^3\right)^7}","-\frac{1}{21 x^{21} \left(b+c x^3\right)^7}",1,"-1/21*1/(x^21*(b + c*x^3)^7)","A",1
1063,1,21,21,0.2511835,"\int \frac{x^{-1-7 n} \left(b+2 c x^n\right)}{\left(b+c x^n\right)^8} \, dx","Integrate[(x^(-1 - 7*n)*(b + 2*c*x^n))/(b + c*x^n)^8,x]","-\frac{x^{-7 n}}{7 n \left(b+c x^n\right)^7}","-\frac{x^{-7 n}}{7 n \left(b+c x^n\right)^7}",1,"-1/7*1/(n*x^(7*n)*(b + c*x^n)^7)","A",1
1064,1,44,52,0.04327,"\int \frac{x^{31} \sqrt{1+x^{16}}}{1-x^{16}} \, dx","Integrate[(x^31*Sqrt[1 + x^16])/(1 - x^16),x]","\frac{1}{24} \left(3 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{x^{16}+1}}{\sqrt{2}}\right)-\sqrt{x^{16}+1} \left(x^{16}+4\right)\right)","-\frac{1}{24} \left(x^{16}+1\right)^{3/2}-\frac{\sqrt{x^{16}+1}}{8}+\frac{\tanh ^{-1}\left(\frac{\sqrt{x^{16}+1}}{\sqrt{2}}\right)}{4 \sqrt{2}}",1,"(-(Sqrt[1 + x^16]*(4 + x^16)) + 3*Sqrt[2]*ArcTanh[Sqrt[1 + x^16]/Sqrt[2]])/24","A",1
1065,1,142,93,0.681525,"\int \frac{\sqrt{c+\frac{d}{x}}}{\sqrt{a+\frac{b}{x}} x} \, dx","Integrate[Sqrt[c + d/x]/(Sqrt[a + b/x]*x),x]","\frac{2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right)}{\sqrt{a}}-\frac{2 \sqrt{d} x \sqrt{c+\frac{d}{x}} \sqrt{b c-a d} \sqrt{\frac{b (c x+d)}{x (b c-a d)}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{b c x+b d}","\frac{2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right)}{\sqrt{a}}-\frac{2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b} \sqrt{c+\frac{d}{x}}}\right)}{\sqrt{b}}",1,"(-2*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[c + d/x]*x*Sqrt[(b*(d + c*x))/((b*c - a*d)*x)]*ArcSinh[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(b*d + b*c*x) + (2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])/(Sqrt[a]*Sqrt[c + d/x])])/Sqrt[a]","A",1
1066,1,223,252,1.0931548,"\int \frac{x^{-1+2 n} \left(a+b x^n\right)^{5/2}}{\sqrt{c+d x^n}} \, dx","Integrate[(x^(-1 + 2*n)*(a + b*x^n)^(5/2))/Sqrt[c + d*x^n],x]","\frac{b \sqrt{d} \sqrt{a+b x^n} \left(c+d x^n\right) \left(15 a^3 d^3+a^2 b d^2 \left(118 d x^n-191 c\right)+a b^2 d \left(265 c^2-172 c d x^n+136 d^2 x^{2 n}\right)+b^3 \left(-105 c^3+70 c^2 d x^n-56 c d^2 x^{2 n}+48 d^3 x^{3 n}\right)\right)+15 (a d+7 b c) (b c-a d)^{7/2} \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right)}{192 b^2 d^{9/2} n \sqrt{c+d x^n}}","\frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right)}{64 b^{3/2} d^{9/2} n}-\frac{5 (b c-a d)^2 (a d+7 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b d^4 n}+\frac{5 (b c-a d) (a d+7 b c) \left(a+b x^n\right)^{3/2} \sqrt{c+d x^n}}{96 b d^3 n}-\frac{(a d+7 b c) \left(a+b x^n\right)^{5/2} \sqrt{c+d x^n}}{24 b d^2 n}+\frac{\left(a+b x^n\right)^{7/2} \sqrt{c+d x^n}}{4 b d n}",1,"(b*Sqrt[d]*Sqrt[a + b*x^n]*(c + d*x^n)*(15*a^3*d^3 + a^2*b*d^2*(-191*c + 118*d*x^n) + a*b^2*d*(265*c^2 - 172*c*d*x^n + 136*d^2*x^(2*n)) + b^3*(-105*c^3 + 70*c^2*d*x^n - 56*c*d^2*x^(2*n) + 48*d^3*x^(3*n))) + 15*(b*c - a*d)^(7/2)*(7*b*c + a*d)*Sqrt[(b*(c + d*x^n))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(192*b^2*d^(9/2)*n*Sqrt[c + d*x^n])","A",1
1067,1,178,199,0.6720431,"\int \frac{x^{-1+2 n} \left(a+b x^n\right)^{3/2}}{\sqrt{c+d x^n}} \, dx","Integrate[(x^(-1 + 2*n)*(a + b*x^n)^(3/2))/Sqrt[c + d*x^n],x]","\frac{b \sqrt{d} \sqrt{a+b x^n} \left(c+d x^n\right) \left(3 a^2 d^2+2 a b d \left(7 d x^n-11 c\right)+b^2 \left(15 c^2-10 c d x^n+8 d^2 x^{2 n}\right)\right)-3 (b c-a d)^{5/2} (a d+5 b c) \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right)}{24 b^2 d^{7/2} n \sqrt{c+d x^n}}","-\frac{(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right)}{8 b^{3/2} d^{7/2} n}+\frac{(b c-a d) (a d+5 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b d^3 n}-\frac{(a d+5 b c) \left(a+b x^n\right)^{3/2} \sqrt{c+d x^n}}{12 b d^2 n}+\frac{\left(a+b x^n\right)^{5/2} \sqrt{c+d x^n}}{3 b d n}",1,"(b*Sqrt[d]*Sqrt[a + b*x^n]*(c + d*x^n)*(3*a^2*d^2 + 2*a*b*d*(-11*c + 7*d*x^n) + b^2*(15*c^2 - 10*c*d*x^n + 8*d^2*x^(2*n))) - 3*(b*c - a*d)^(5/2)*(5*b*c + a*d)*Sqrt[(b*(c + d*x^n))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(24*b^2*d^(7/2)*n*Sqrt[c + d*x^n])","A",1
1068,1,141,146,0.5132726,"\int \frac{x^{-1+2 n} \sqrt{a+b x^n}}{\sqrt{c+d x^n}} \, dx","Integrate[(x^(-1 + 2*n)*Sqrt[a + b*x^n])/Sqrt[c + d*x^n],x]","\frac{b \sqrt{d} \sqrt{a+b x^n} \left(c+d x^n\right) \left(a d-3 b c+2 b d x^n\right)+(a d+3 b c) (b c-a d)^{3/2} \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right)}{4 b^2 d^{5/2} n \sqrt{c+d x^n}}","\frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right)}{4 b^{3/2} d^{5/2} n}-\frac{(a d+3 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b d^2 n}+\frac{\left(a+b x^n\right)^{3/2} \sqrt{c+d x^n}}{2 b d n}",1,"(b*Sqrt[d]*Sqrt[a + b*x^n]*(c + d*x^n)*(-3*b*c + a*d + 2*b*d*x^n) + (b*c - a*d)^(3/2)*(3*b*c + a*d)*Sqrt[(b*(c + d*x^n))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(4*b^2*d^(5/2)*n*Sqrt[c + d*x^n])","A",1
1069,1,123,89,0.321195,"\int \frac{x^{-1+2 n}}{\sqrt{a+b x^n} \sqrt{c+d x^n}} \, dx","Integrate[x^(-1 + 2*n)/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),x]","\frac{b \sqrt{d} \sqrt{a+b x^n} \left(c+d x^n\right)-\sqrt{b c-a d} (a d+b c) \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right)}{b^2 d^{3/2} n \sqrt{c+d x^n}}","\frac{\sqrt{a+b x^n} \sqrt{c+d x^n}}{b d n}-\frac{(a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right)}{b^{3/2} d^{3/2} n}",1,"(b*Sqrt[d]*Sqrt[a + b*x^n]*(c + d*x^n) - Sqrt[b*c - a*d]*(b*c + a*d)*Sqrt[(b*(c + d*x^n))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(b^2*d^(3/2)*n*Sqrt[c + d*x^n])","A",1
1070,1,122,91,0.9009822,"\int \frac{x^{-1+2 n}}{\left(a+b x^n\right)^{3/2} \sqrt{c+d x^n}} \, dx","Integrate[x^(-1 + 2*n)/((a + b*x^n)^(3/2)*Sqrt[c + d*x^n]),x]","\frac{2 \left(\frac{a b \left(c+d x^n\right)}{(b c-a d) \sqrt{a+b x^n}}+\frac{\sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right)}{\sqrt{d}}\right)}{b^2 n \sqrt{c+d x^n}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right)}{b^{3/2} \sqrt{d} n}+\frac{2 a \sqrt{c+d x^n}}{b n (b c-a d) \sqrt{a+b x^n}}",1,"(2*((a*b*(c + d*x^n))/((b*c - a*d)*Sqrt[a + b*x^n]) + (Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^n))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/Sqrt[d]))/(b^2*n*Sqrt[c + d*x^n])","A",1
1071,1,57,95,0.0749728,"\int \frac{x^{-1+2 n}}{\left(a+b x^n\right)^{5/2} \sqrt{c+d x^n}} \, dx","Integrate[x^(-1 + 2*n)/((a + b*x^n)^(5/2)*Sqrt[c + d*x^n]),x]","\frac{2 \sqrt{c+d x^n} \left(-2 a c+a d x^n-3 b c x^n\right)}{3 n (b c-a d)^2 \left(a+b x^n\right)^{3/2}}","\frac{2 a \sqrt{c+d x^n}}{3 b n (b c-a d) \left(a+b x^n\right)^{3/2}}-\frac{2 (3 b c-a d) \sqrt{c+d x^n}}{3 b n (b c-a d)^2 \sqrt{a+b x^n}}",1,"(2*Sqrt[c + d*x^n]*(-2*a*c - 3*b*c*x^n + a*d*x^n))/(3*(b*c - a*d)^2*n*(a + b*x^n)^(3/2))","A",1
1072,1,274,358,2.024221,"\int \frac{x^{-1+3 n} \left(a+b x^n\right)^{5/2}}{\sqrt{c+d x^n}} \, dx","Integrate[(x^(-1 + 3*n)*(a + b*x^n)^(5/2))/Sqrt[c + d*x^n],x]","\frac{\sqrt{c+d x^n} \left(\frac{5 (b c-a d)^3 \left(3 a^2 d^2+14 a b c d+63 b^2 c^2\right) \left(-\frac{16 d^3 \left(a+b x^n\right)^3}{15 (a d-b c)^3}-\frac{4 d^2 \left(a+b x^n\right)^2}{3 (b c-a d)^2}-\frac{2 d \left(a+b x^n\right)}{a d-b c}-\frac{2 \sqrt{d} \sqrt{a+b x^n} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}}}\right)}{4 b d^5}-\frac{24 (a d+3 b c) \left(a+b x^n\right)^4}{b d}+64 x^n \left(a+b x^n\right)^4\right)}{320 b d n \sqrt{a+b x^n}}","\frac{(b c-a d)^2 \left(3 a^2 d^2+14 a b c d+63 b^2 c^2\right) \sqrt{a+b x^n} \sqrt{c+d x^n}}{128 b^2 d^5 n}-\frac{(b c-a d) \left(3 a^2 d^2+14 a b c d+63 b^2 c^2\right) \left(a+b x^n\right)^{3/2} \sqrt{c+d x^n}}{192 b^2 d^4 n}+\frac{\left(3 a^2 d^2+14 a b c d+63 b^2 c^2\right) \left(a+b x^n\right)^{5/2} \sqrt{c+d x^n}}{240 b^2 d^3 n}-\frac{(b c-a d)^3 \left(3 a^2 d^2+14 a b c d+63 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right)}{128 b^{5/2} d^{11/2} n}-\frac{3 (a d+3 b c) \left(a+b x^n\right)^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left(a+b x^n\right)^{7/2} \sqrt{c+d x^n}}{5 b d n}",1,"(Sqrt[c + d*x^n]*((-24*(3*b*c + a*d)*(a + b*x^n)^4)/(b*d) + 64*x^n*(a + b*x^n)^4 + (5*(b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*((-2*d*(a + b*x^n))/(-(b*c) + a*d) - (4*d^2*(a + b*x^n)^2)/(3*(b*c - a*d)^2) - (16*d^3*(a + b*x^n)^3)/(15*(-(b*c) + a*d)^3) - (2*Sqrt[d]*Sqrt[a + b*x^n]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^n))/(b*c - a*d)])))/(4*b*d^5)))/(320*b*d*n*Sqrt[a + b*x^n])","A",1
1073,1,241,291,1.0067508,"\int \frac{x^{-1+3 n} \left(a+b x^n\right)^{3/2}}{\sqrt{c+d x^n}} \, dx","Integrate[(x^(-1 + 3*n)*(a + b*x^n)^(3/2))/Sqrt[c + d*x^n],x]","\frac{3 (b c-a d)^{5/2} \left(3 a^2 d^2+10 a b c d+35 b^2 c^2\right) \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right)-b \sqrt{d} \sqrt{a+b x^n} \left(c+d x^n\right) \left(9 a^3 d^3+3 a^2 b d^2 \left(5 c-2 d x^n\right)-a b^2 d \left(145 c^2-92 c d x^n+72 d^2 x^{2 n}\right)+b^3 \left(105 c^3-70 c^2 d x^n+56 c d^2 x^{2 n}-48 d^3 x^{3 n}\right)\right)}{192 b^3 d^{9/2} n \sqrt{c+d x^n}}","-\frac{(b c-a d) \left(3 a^2 d^2+10 a b c d+35 b^2 c^2\right) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b^2 d^4 n}+\frac{\left(3 a^2 d^2+10 a b c d+35 b^2 c^2\right) \left(a+b x^n\right)^{3/2} \sqrt{c+d x^n}}{96 b^2 d^3 n}+\frac{(b c-a d)^2 \left(3 a^2 d^2+10 a b c d+35 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right)}{64 b^{5/2} d^{9/2} n}-\frac{(3 a d+7 b c) \left(a+b x^n\right)^{5/2} \sqrt{c+d x^n}}{24 b^2 d^2 n}+\frac{x^n \left(a+b x^n\right)^{5/2} \sqrt{c+d x^n}}{4 b d n}",1,"(-(b*Sqrt[d]*Sqrt[a + b*x^n]*(c + d*x^n)*(9*a^3*d^3 + 3*a^2*b*d^2*(5*c - 2*d*x^n) - a*b^2*d*(145*c^2 - 92*c*d*x^n + 72*d^2*x^(2*n)) + b^3*(105*c^3 - 70*c^2*d*x^n + 56*c*d^2*x^(2*n) - 48*d^3*x^(3*n)))) + 3*(b*c - a*d)^(5/2)*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[(b*(c + d*x^n))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(192*b^3*d^(9/2)*n*Sqrt[c + d*x^n])","A",1
1074,1,191,221,0.6241045,"\int \frac{x^{-1+3 n} \sqrt{a+b x^n}}{\sqrt{c+d x^n}} \, dx","Integrate[(x^(-1 + 3*n)*Sqrt[a + b*x^n])/Sqrt[c + d*x^n],x]","\frac{b \sqrt{d} \sqrt{a+b x^n} \left(c+d x^n\right) \left(-3 a^2 d^2+2 a b d \left(d x^n-2 c\right)+b^2 \left(15 c^2-10 c d x^n+8 d^2 x^{2 n}\right)\right)-3 (b c-a d)^{3/2} \left(a^2 d^2+2 a b c d+5 b^2 c^2\right) \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right)}{24 b^3 d^{7/2} n \sqrt{c+d x^n}}","\frac{\left(a^2 d^2+2 a b c d+5 b^2 c^2\right) \sqrt{a+b x^n} \sqrt{c+d x^n}}{8 b^2 d^3 n}-\frac{(b c-a d) \left(a^2 d^2+2 a b c d+5 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right)}{8 b^{5/2} d^{7/2} n}-\frac{(3 a d+5 b c) \left(a+b x^n\right)^{3/2} \sqrt{c+d x^n}}{12 b^2 d^2 n}+\frac{x^n \left(a+b x^n\right)^{3/2} \sqrt{c+d x^n}}{3 b d n}",1,"(b*Sqrt[d]*Sqrt[a + b*x^n]*(c + d*x^n)*(-3*a^2*d^2 + 2*a*b*d*(-2*c + d*x^n) + b^2*(15*c^2 - 10*c*d*x^n + 8*d^2*x^(2*n))) - 3*(b*c - a*d)^(3/2)*(5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Sqrt[(b*(c + d*x^n))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(24*b^3*d^(7/2)*n*Sqrt[c + d*x^n])","A",1
1075,1,157,150,0.4639019,"\int \frac{x^{-1+3 n}}{\sqrt{a+b x^n} \sqrt{c+d x^n}} \, dx","Integrate[x^(-1 + 3*n)/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),x]","\frac{\sqrt{b c-a d} \left(3 a^2 d^2+2 a b c d+3 b^2 c^2\right) \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right)+b \sqrt{d} \sqrt{a+b x^n} \left(c+d x^n\right) \left(-3 a d-3 b c+2 b d x^n\right)}{4 b^3 d^{5/2} n \sqrt{c+d x^n}}","-\frac{\left(4 a b c d-3 (a d+b c)^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right)}{4 b^{5/2} d^{5/2} n}-\frac{3 (a d+b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b^2 d^2 n}+\frac{x^n \sqrt{a+b x^n} \sqrt{c+d x^n}}{2 b d n}",1,"(b*Sqrt[d]*Sqrt[a + b*x^n]*(c + d*x^n)*(-3*b*c - 3*a*d + 2*b*d*x^n) + Sqrt[b*c - a*d]*(3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*Sqrt[(b*(c + d*x^n))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(4*b^3*d^(5/2)*n*Sqrt[c + d*x^n])","A",1
1076,1,185,133,0.5223862,"\int \frac{x^{-1+3 n}}{\left(a+b x^n\right)^{3/2} \sqrt{c+d x^n}} \, dx","Integrate[x^(-1 + 3*n)/((a + b*x^n)^(3/2)*Sqrt[c + d*x^n]),x]","\frac{\sqrt{b c-a d} \left(-3 a^2 d^2+2 a b c d+b^2 c^2\right) \sqrt{a+b x^n} \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right)-b \sqrt{d} \left(c+d x^n\right) \left(-3 a^2 d+a b \left(c-d x^n\right)+b^2 c x^n\right)}{b^3 d^{3/2} n (a d-b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}","-\frac{2 a^2 \sqrt{c+d x^n}}{b^2 n (b c-a d) \sqrt{a+b x^n}}-\frac{(3 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right)}{b^{5/2} d^{3/2} n}+\frac{\sqrt{a+b x^n} \sqrt{c+d x^n}}{b^2 d n}",1,"(-(b*Sqrt[d]*(c + d*x^n)*(-3*a^2*d + b^2*c*x^n + a*b*(c - d*x^n))) + Sqrt[b*c - a*d]*(b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x^n]*Sqrt[(b*(c + d*x^n))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(b^3*d^(3/2)*(-(b*c) + a*d)*n*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])","A",1
1077,1,217,147,0.8115146,"\int \frac{x^{-1+3 n}}{\left(a+b x^n\right)^{5/2} \sqrt{c+d x^n}} \, dx","Integrate[x^(-1 + 3*n)/((a + b*x^n)^(5/2)*Sqrt[c + d*x^n]),x]","\frac{2 \sqrt{c+d x^n} \left(\frac{\left(3 b^2 c^2-a^2 d^2\right) \left(a+b x^n\right)}{d (b c-a d)^2}+\frac{a^2}{a d-b c}-\frac{3 \left(a+b x^n\right) \left(\sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}}-\sqrt{d} \sqrt{a+b x^n} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right)\right)}{d \sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^n\right)}{b c-a d}}}\right)}{3 b^2 n \left(a+b x^n\right)^{3/2}}","-\frac{2 a^2 \sqrt{c+d x^n}}{3 b^2 n (b c-a d) \left(a+b x^n\right)^{3/2}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right)}{b^{5/2} \sqrt{d} n}+\frac{4 a (3 b c-2 a d) \sqrt{c+d x^n}}{3 b^2 n (b c-a d)^2 \sqrt{a+b x^n}}",1,"(2*Sqrt[c + d*x^n]*(a^2/(-(b*c) + a*d) + ((3*b^2*c^2 - a^2*d^2)*(a + b*x^n))/(d*(b*c - a*d)^2) - (3*(a + b*x^n)*(Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^n))/(b*c - a*d)] - Sqrt[d]*Sqrt[a + b*x^n]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]]))/(d*Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^n))/(b*c - a*d)])))/(3*b^2*n*(a + b*x^n)^(3/2))","A",1
1078,1,20,20,0.0176604,"\int x^p (b+c x)^p (b+2 c x) \, dx","Integrate[x^p*(b + c*x)^p*(b + 2*c*x),x]","\frac{x^{p+1} (b+c x)^{p+1}}{p+1}","\frac{x^{p+1} (b+c x)^{p+1}}{p+1}",1,"(x^(1 + p)*(b + c*x)^(1 + p))/(1 + p)","A",1
1079,1,97,27,0.1316338,"\int x^{-1+2 (1+p)} \left(b+c x^2\right)^p \left(b+2 c x^2\right) \, dx","Integrate[x^(-1 + 2*(1 + p))*(b + c*x^2)^p*(b + 2*c*x^2),x]","\frac{x^{2 p+2} \left(b+c x^2\right)^p \left(\frac{c x^2}{b}+1\right)^{-p} \left(2 c (p+1) x^2 \, _2F_1\left(-p,p+2;p+3;-\frac{c x^2}{b}\right)+b (p+2) \, _2F_1\left(-p,p+1;p+2;-\frac{c x^2}{b}\right)\right)}{2 (p+1) (p+2)}","\frac{x^{2 (p+1)} \left(b+c x^2\right)^{p+1}}{2 (p+1)}",1,"(x^(2 + 2*p)*(b + c*x^2)^p*(b*(2 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, -((c*x^2)/b)] + 2*c*(1 + p)*x^2*Hypergeometric2F1[-p, 2 + p, 3 + p, -((c*x^2)/b)]))/(2*(1 + p)*(2 + p)*(1 + (c*x^2)/b)^p)","C",1
1080,1,97,27,0.095674,"\int x^{-1+3 (1+p)} \left(b+c x^3\right)^p \left(b+2 c x^3\right) \, dx","Integrate[x^(-1 + 3*(1 + p))*(b + c*x^3)^p*(b + 2*c*x^3),x]","\frac{x^{3 p+3} \left(b+c x^3\right)^p \left(\frac{c x^3}{b}+1\right)^{-p} \left(2 c (p+1) x^3 \, _2F_1\left(-p,p+2;p+3;-\frac{c x^3}{b}\right)+b (p+2) \, _2F_1\left(-p,p+1;p+2;-\frac{c x^3}{b}\right)\right)}{3 (p+1) (p+2)}","\frac{x^{3 (p+1)} \left(b+c x^3\right)^{p+1}}{3 (p+1)}",1,"(x^(3 + 3*p)*(b + c*x^3)^p*(b*(2 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, -((c*x^3)/b)] + 2*c*(1 + p)*x^3*Hypergeometric2F1[-p, 2 + p, 3 + p, -((c*x^3)/b)]))/(3*(1 + p)*(2 + p)*(1 + (c*x^3)/b)^p)","C",1
1081,1,101,27,0.1620555,"\int x^{-1+n (1+p)} \left(b+c x^n\right)^p \left(b+2 c x^n\right) \, dx","Integrate[x^(-1 + n*(1 + p))*(b + c*x^n)^p*(b + 2*c*x^n),x]","\frac{\left(b+c x^n\right)^p \left(\frac{c x^n}{b}+1\right)^{-p} \left(b (p+2) x^{n (p+1)} \, _2F_1\left(-p,p+1;p+2;-\frac{c x^n}{b}\right)+2 c (p+1) x^{n (p+2)} \, _2F_1\left(-p,p+2;p+3;-\frac{c x^n}{b}\right)\right)}{n (p+1) (p+2)}","\frac{x^{n (p+1)} \left(b+c x^n\right)^{p+1}}{n (p+1)}",1,"((b + c*x^n)^p*(b*(2 + p)*x^(n*(1 + p))*Hypergeometric2F1[-p, 1 + p, 2 + p, -((c*x^n)/b)] + 2*c*(1 + p)*x^(n*(2 + p))*Hypergeometric2F1[-p, 2 + p, 3 + p, -((c*x^n)/b)]))/(n*(1 + p)*(2 + p)*(1 + (c*x^n)/b)^p)","C",1