1,1,94,94,0.0490244,"\int \left(a+b x^3\right) \left(c+d x^3\right)^4 \, dx","Integrate[(a + b*x^3)*(c + d*x^3)^4,x]","\frac{1}{4} c^3 x^4 (4 a d+b c)+\frac{2}{7} c^2 d x^7 (3 a d+2 b c)+\frac{1}{13} d^3 x^{13} (a d+4 b c)+\frac{1}{5} c d^2 x^{10} (2 a d+3 b c)+a c^4 x+\frac{1}{16} b d^4 x^{16}","\frac{1}{4} c^3 x^4 (4 a d+b c)+\frac{2}{7} c^2 d x^7 (3 a d+2 b c)+\frac{1}{13} d^3 x^{13} (a d+4 b c)+\frac{1}{5} c d^2 x^{10} (2 a d+3 b c)+a c^4 x+\frac{1}{16} b d^4 x^{16}",1,"a*c^4*x + (c^3*(b*c + 4*a*d)*x^4)/4 + (2*c^2*d*(2*b*c + 3*a*d)*x^7)/7 + (c*d^2*(3*b*c + 2*a*d)*x^10)/5 + (d^3*(4*b*c + a*d)*x^13)/13 + (b*d^4*x^16)/16","A",1
2,1,70,70,0.0146742,"\int \left(a+b x^3\right) \left(c+d x^3\right)^3 \, dx","Integrate[(a + b*x^3)*(c + d*x^3)^3,x]","\frac{1}{4} c^2 x^4 (3 a d+b c)+\frac{1}{10} d^2 x^{10} (a d+3 b c)+\frac{3}{7} c d x^7 (a d+b c)+a c^3 x+\frac{1}{13} b d^3 x^{13}","\frac{1}{4} c^2 x^4 (3 a d+b c)+\frac{1}{10} d^2 x^{10} (a d+3 b c)+\frac{3}{7} c d x^7 (a d+b c)+a c^3 x+\frac{1}{13} b d^3 x^{13}",1,"a*c^3*x + (c^2*(b*c + 3*a*d)*x^4)/4 + (3*c*d*(b*c + a*d)*x^7)/7 + (d^2*(3*b*c + a*d)*x^10)/10 + (b*d^3*x^13)/13","A",1
3,1,50,50,0.0082163,"\int \left(a+b x^3\right) \left(c+d x^3\right)^2 \, dx","Integrate[(a + b*x^3)*(c + d*x^3)^2,x]","\frac{1}{7} d x^7 (a d+2 b c)+\frac{1}{4} c x^4 (2 a d+b c)+a c^2 x+\frac{1}{10} b d^2 x^{10}","\frac{1}{7} d x^7 (a d+2 b c)+\frac{1}{4} c x^4 (2 a d+b c)+a c^2 x+\frac{1}{10} b d^2 x^{10}",1,"a*c^2*x + (c*(b*c + 2*a*d)*x^4)/4 + (d*(2*b*c + a*d)*x^7)/7 + (b*d^2*x^10)/10","A",1
4,1,28,28,0.0052755,"\int \left(a+b x^3\right) \left(c+d x^3\right) \, dx","Integrate[(a + b*x^3)*(c + d*x^3),x]","\frac{1}{4} x^4 (a d+b c)+a c x+\frac{1}{7} b d x^7","\frac{1}{4} x^4 (a d+b c)+a c x+\frac{1}{7} b d x^7",1,"a*c*x + ((b*c + a*d)*x^4)/4 + (b*d*x^7)/7","A",1
5,1,128,144,0.1230405,"\int \frac{a+b x^3}{c+d x^3} \, dx","Integrate[(a + b*x^3)/(c + d*x^3),x]","\frac{(b c-a d) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)-2 (b c-a d) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)+2 \sqrt{3} (b c-a d) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)+6 b c^{2/3} \sqrt[3]{d} x}{6 c^{2/3} d^{4/3}}","\frac{(b c-a d) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c^{2/3} d^{4/3}}-\frac{(b c-a d) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{2/3} d^{4/3}}+\frac{(b c-a d) \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{2/3} d^{4/3}}+\frac{b x}{d}",1,"(6*b*c^(2/3)*d^(1/3)*x + 2*Sqrt[3]*(b*c - a*d)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] - 2*(b*c - a*d)*Log[c^(1/3) + d^(1/3)*x] + (b*c - a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(2/3)*d^(4/3))","A",1
6,1,145,169,0.09641,"\int \frac{a+b x^3}{\left(c+d x^3\right)^2} \, dx","Integrate[(a + b*x^3)/(c + d*x^3)^2,x]","\frac{-(2 a d+b c) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)-\frac{6 c^{2/3} \sqrt[3]{d} x (b c-a d)}{c+d x^3}+2 (2 a d+b c) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)-2 \sqrt{3} (2 a d+b c) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{18 c^{5/3} d^{4/3}}","-\frac{(2 a d+b c) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{18 c^{5/3} d^{4/3}}+\frac{(2 a d+b c) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{9 c^{5/3} d^{4/3}}-\frac{(2 a d+b c) \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{3 \sqrt{3} c^{5/3} d^{4/3}}-\frac{x (b c-a d)}{3 c d \left(c+d x^3\right)}",1,"((-6*c^(2/3)*d^(1/3)*(b*c - a*d)*x)/(c + d*x^3) - 2*Sqrt[3]*(b*c + 2*a*d)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] + 2*(b*c + 2*a*d)*Log[c^(1/3) + d^(1/3)*x] - (b*c + 2*a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(18*c^(5/3)*d^(4/3))","A",1
7,1,175,197,0.1305122,"\int \frac{a+b x^3}{\left(c+d x^3\right)^3} \, dx","Integrate[(a + b*x^3)/(c + d*x^3)^3,x]","\frac{-(5 a d+b c) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)-\frac{9 c^{5/3} \sqrt[3]{d} x (b c-a d)}{\left(c+d x^3\right)^2}+\frac{3 c^{2/3} \sqrt[3]{d} x (5 a d+b c)}{c+d x^3}+2 (5 a d+b c) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)-2 \sqrt{3} (5 a d+b c) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{54 c^{8/3} d^{4/3}}","-\frac{(5 a d+b c) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{54 c^{8/3} d^{4/3}}+\frac{(5 a d+b c) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{27 c^{8/3} d^{4/3}}-\frac{(5 a d+b c) \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{9 \sqrt{3} c^{8/3} d^{4/3}}+\frac{x (5 a d+b c)}{18 c^2 d \left(c+d x^3\right)}-\frac{x (b c-a d)}{6 c d \left(c+d x^3\right)^2}",1,"((-9*c^(5/3)*d^(1/3)*(b*c - a*d)*x)/(c + d*x^3)^2 + (3*c^(2/3)*d^(1/3)*(b*c + 5*a*d)*x)/(c + d*x^3) - 2*Sqrt[3]*(b*c + 5*a*d)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] + 2*(b*c + 5*a*d)*Log[c^(1/3) + d^(1/3)*x] - (b*c + 5*a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(4/3))","A",1
8,1,122,122,0.0193501,"\int \left(a+b x^3\right)^2 \left(c+d x^3\right)^3 \, dx","Integrate[(a + b*x^3)^2*(c + d*x^3)^3,x]","\frac{1}{10} d x^{10} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{7} c x^7 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 x+\frac{1}{4} a c^2 x^4 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{16} b^2 d^3 x^{16}","\frac{1}{10} d x^{10} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{7} c x^7 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 x+\frac{1}{4} a c^2 x^4 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{16} b^2 d^3 x^{16}",1,"a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^4)/4 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^7)/7 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^10)/10 + (b*d^2*(3*b*c + 2*a*d)*x^13)/13 + (b^2*d^3*x^16)/16","A",1
9,1,82,82,0.0122966,"\int \left(a+b x^3\right)^2 \left(c+d x^3\right)^2 \, dx","Integrate[(a + b*x^3)^2*(c + d*x^3)^2,x]","\frac{1}{7} x^7 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 x+\frac{1}{5} b d x^{10} (a d+b c)+\frac{1}{2} a c x^4 (a d+b c)+\frac{1}{13} b^2 d^2 x^{13}","\frac{1}{7} x^7 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 x+\frac{1}{5} b d x^{10} (a d+b c)+\frac{1}{2} a c x^4 (a d+b c)+\frac{1}{13} b^2 d^2 x^{13}",1,"a^2*c^2*x + (a*c*(b*c + a*d)*x^4)/2 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^7)/7 + (b*d*(b*c + a*d)*x^10)/5 + (b^2*d^2*x^13)/13","A",1
10,1,50,50,0.0065102,"\int \left(a+b x^3\right)^2 \left(c+d x^3\right) \, dx","Integrate[(a + b*x^3)^2*(c + d*x^3),x]","a^2 c x+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{4} a x^4 (a d+2 b c)+\frac{1}{10} b^2 d x^{10}","a^2 c x+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{4} a x^4 (a d+2 b c)+\frac{1}{10} b^2 d x^{10}",1,"a^2*c*x + (a*(2*b*c + a*d)*x^4)/4 + (b*(b*c + 2*a*d)*x^7)/7 + (b^2*d*x^10)/10","A",1
11,1,167,173,0.1415555,"\int \frac{\left(a+b x^3\right)^2}{c+d x^3} \, dx","Integrate[(a + b*x^3)^2/(c + d*x^3),x]","\frac{-2 (b c-a d)^2 \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)-12 b c^{2/3} \sqrt[3]{d} x (b c-2 a d)+4 (b c-a d)^2 \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)+4 \sqrt{3} (b c-a d)^2 \tan ^{-1}\left(\frac{2 \sqrt[3]{d} x-\sqrt[3]{c}}{\sqrt{3} \sqrt[3]{c}}\right)+3 b^2 c^{2/3} d^{4/3} x^4}{12 c^{2/3} d^{7/3}}","-\frac{(b c-a d)^2 \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{6 c^{2/3} d^{7/3}}+\frac{(b c-a d)^2 \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{2/3} d^{7/3}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^4}{4 d}",1,"(-12*b*c^(2/3)*d^(1/3)*(b*c - 2*a*d)*x + 3*b^2*c^(2/3)*d^(4/3)*x^4 + 4*Sqrt[3]*(b*c - a*d)^2*ArcTan[(-c^(1/3) + 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))] + 4*(b*c - a*d)^2*Log[c^(1/3) + d^(1/3)*x] - 2*(b*c - a*d)^2*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(12*c^(2/3)*d^(7/3))","A",1
12,1,210,203,0.2205286,"\int \frac{\left(a+b x^3\right)^2}{\left(c+d x^3\right)^2} \, dx","Integrate[(a + b*x^3)^2/(c + d*x^3)^2,x]","\frac{-\frac{2 \left(-a^2 d^2-a b c d+2 b^2 c^2\right) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{c^{5/3}}+\frac{2 \sqrt{3} \left(-a^2 d^2-a b c d+2 b^2 c^2\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{c^{5/3}}+\frac{\left(-a^2 d^2-a b c d+2 b^2 c^2\right) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{c^{5/3}}+\frac{3 \sqrt[3]{d} x (b c-a d)^2}{c \left(c+d x^3\right)}+9 b^2 \sqrt[3]{d} x}{9 d^{7/3}}","\frac{(b c-a d) (a d+2 b c) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{9 c^{5/3} d^{7/3}}-\frac{2 (b c-a d) (a d+2 b c) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{9 c^{5/3} d^{7/3}}+\frac{2 (b c-a d) (a d+2 b c) \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{3 \sqrt{3} c^{5/3} d^{7/3}}+\frac{x (b c-a d)^2}{3 c d^2 \left(c+d x^3\right)}+\frac{b^2 x}{d^2}",1,"(9*b^2*d^(1/3)*x + (3*d^(1/3)*(b*c - a*d)^2*x)/(c*(c + d*x^3)) + (2*Sqrt[3]*(2*b^2*c^2 - a*b*c*d - a^2*d^2)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(5/3) - (2*(2*b^2*c^2 - a*b*c*d - a^2*d^2)*Log[c^(1/3) + d^(1/3)*x])/c^(5/3) + ((2*b^2*c^2 - a*b*c*d - a^2*d^2)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(5/3))/(9*d^(7/3))","A",1
13,1,234,258,0.2761553,"\int \frac{\left(a+b x^3\right)^2}{\left(c+d x^3\right)^3} \, dx","Integrate[(a + b*x^3)^2/(c + d*x^3)^3,x]","\frac{2 \left(5 a^2 d^2+2 a b c d+2 b^2 c^2\right) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)-2 \sqrt{3} \left(5 a^2 d^2+2 a b c d+2 b^2 c^2\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)-\frac{3 c^{2/3} \sqrt[3]{d} x \left(-a^2 d^2 \left(8 c+5 d x^3\right)+2 a b c d \left(2 c-d x^3\right)+b^2 c^2 \left(4 c+7 d x^3\right)\right)}{\left(c+d x^3\right)^2}-\left(5 a^2 d^2+2 a b c d+2 b^2 c^2\right) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{54 c^{8/3} d^{7/3}}","-\frac{\left(5 a^2 d^2+2 a b c d+2 b^2 c^2\right) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{54 c^{8/3} d^{7/3}}+\frac{\left(5 a^2 d^2+2 a b c d+2 b^2 c^2\right) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{27 c^{8/3} d^{7/3}}-\frac{\left(5 a^2 d^2+2 a b c d+2 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{9 \sqrt{3} c^{8/3} d^{7/3}}-\frac{x (b c-a d) (5 a d+4 b c)}{18 c^2 d^2 \left(c+d x^3\right)}-\frac{x \left(a+b x^3\right) (b c-a d)}{6 c d \left(c+d x^3\right)^2}",1,"((-3*c^(2/3)*d^(1/3)*x*(2*a*b*c*d*(2*c - d*x^3) - a^2*d^2*(8*c + 5*d*x^3) + b^2*c^2*(4*c + 7*d*x^3)))/(c + d*x^3)^2 - 2*Sqrt[3]*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] + 2*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(1/3) + d^(1/3)*x] - (2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(7/3))","A",1
14,1,253,252,0.1273882,"\int \frac{\left(c+d x^3\right)^4}{a+b x^3} \, dx","Integrate[(c + d*x^3)^4/(a + b*x^3),x]","\frac{-\frac{70 (b c-a d)^4 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{2/3}}+\frac{140 (b c-a d)^4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{2/3}}+\frac{140 \sqrt{3} (b c-a d)^4 \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{a^{2/3}}+105 b^{4/3} d^2 x^4 \left(a^2 d^2-4 a b c d+6 b^2 c^2\right)+420 \sqrt[3]{b} d x \left(-a^3 d^3+4 a^2 b c d^2-6 a b^2 c^2 d+4 b^3 c^3\right)+60 b^{7/3} d^3 x^7 (4 b c-a d)+42 b^{10/3} d^4 x^{10}}{420 b^{13/3}}","-\frac{(b c-a d)^4 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} b^{13/3}}+\frac{(b c-a d)^4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} b^{13/3}}-\frac{(b c-a d)^4 \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} b^{13/3}}+\frac{d x (2 b c-a d) \left(a^2 d^2-2 a b c d+2 b^2 c^2\right)}{b^4}+\frac{d^2 x^4 \left(a^2 d^2-4 a b c d+6 b^2 c^2\right)}{4 b^3}+\frac{d^3 x^7 (4 b c-a d)}{7 b^2}+\frac{d^4 x^{10}}{10 b}",1,"(420*b^(1/3)*d*(4*b^3*c^3 - 6*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*x + 105*b^(4/3)*d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^4 + 60*b^(7/3)*d^3*(4*b*c - a*d)*x^7 + 42*b^(10/3)*d^4*x^10 + (140*Sqrt[3]*(b*c - a*d)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(2/3) + (140*(b*c - a*d)^4*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) - (70*(b*c - a*d)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(420*b^(13/3))","A",1
15,1,203,208,0.106274,"\int \frac{\left(c+d x^3\right)^3}{a+b x^3} \, dx","Integrate[(c + d*x^3)^3/(a + b*x^3),x]","\frac{\frac{14 (a d-b c)^3 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{2/3}}+\frac{28 (b c-a d)^3 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{2/3}}+\frac{28 \sqrt{3} (b c-a d)^3 \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{a^{2/3}}+84 \sqrt[3]{b} d x \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)+21 b^{4/3} d^2 x^4 (3 b c-a d)+12 b^{7/3} d^3 x^7}{84 b^{10/3}}","-\frac{(b c-a d)^3 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} b^{10/3}}+\frac{(b c-a d)^3 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} b^{10/3}}-\frac{(b c-a d)^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^{10/3}}+\frac{d x \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{b^3}+\frac{d^2 x^4 (3 b c-a d)}{4 b^2}+\frac{d^3 x^7}{7 b}",1,"(84*b^(1/3)*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x + 21*b^(4/3)*d^2*(3*b*c - a*d)*x^4 + 12*b^(7/3)*d^3*x^7 + (28*Sqrt[3]*(b*c - a*d)^3*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(2/3) + (28*(b*c - a*d)^3*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + (14*(-(b*c) + a*d)^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(84*b^(10/3))","A",1
16,1,167,173,0.1490621,"\int \frac{\left(c+d x^3\right)^2}{a+b x^3} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3),x]","\frac{-2 (b c-a d)^2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+3 a^{2/3} b^{4/3} d^2 x^4-12 a^{2/3} \sqrt[3]{b} d x (a d-2 b c)+4 (b c-a d)^2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+4 \sqrt{3} (b c-a d)^2 \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{12 a^{2/3} b^{7/3}}","-\frac{(b c-a d)^2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} b^{7/3}}+\frac{(b c-a d)^2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} b^{7/3}}-\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} b^{7/3}}+\frac{d x (2 b c-a d)}{b^2}+\frac{d^2 x^4}{4 b}",1,"(-12*a^(2/3)*b^(1/3)*d*(-2*b*c + a*d)*x + 3*a^(2/3)*b^(4/3)*d^2*x^4 + 4*Sqrt[3]*(b*c - a*d)^2*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] + 4*(b*c - a*d)^2*Log[a^(1/3) + b^(1/3)*x] - 2*(b*c - a*d)^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(12*a^(2/3)*b^(7/3))","A",1
17,1,129,145,0.0752421,"\int \frac{c+d x^3}{a+b x^3} \, dx","Integrate[(c + d*x^3)/(a + b*x^3),x]","\frac{-(b c-a d) \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} d x+2 (b c-a d) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} (b c-a d) \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{(b c-a d) \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{(b c-a d) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} b^{4/3}}-\frac{(b c-a d) \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{d x}{b}",1,"(6*a^(2/3)*b^(1/3)*d*x - 2*Sqrt[3]*(b*c - a*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(b*c - a*d)*Log[a^(1/3) + b^(1/3)*x] - (b*c - a*d)*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
18,1,224,288,0.1070658,"\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
19,1,336,346,0.2354789,"\int \frac{1}{\left(a+b x^3\right) \left(c+d x^3\right)^2} \, dx","Integrate[1/((a + b*x^3)*(c + d*x^3)^2),x]","\frac{-3 b^{5/3} c^{5/3} \left(c+d x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+a^{2/3} d^{2/3} \left(c+d x^3\right) (5 b c-2 a d) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)+6 a^{2/3} c^{2/3} d x (a d-b c)+2 a^{2/3} d^{2/3} \left(c+d x^3\right) (2 a d-5 b c) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)-2 \sqrt{3} a^{2/3} d^{2/3} \left(c+d x^3\right) (2 a d-5 b c) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)+6 b^{5/3} c^{5/3} \left(c+d x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-6 \sqrt{3} b^{5/3} c^{5/3} \left(c+d x^3\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{18 a^{2/3} c^{5/3} \left(c+d x^3\right) (b c-a d)^2}","-\frac{b^{5/3} \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)^2}+\frac{b^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} (b c-a d)^2}-\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^{2/3} (b c-a d)^2}+\frac{d^{2/3} (5 b c-2 a d) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{18 c^{5/3} (b c-a d)^2}-\frac{d^{2/3} (5 b c-2 a d) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{9 c^{5/3} (b c-a d)^2}+\frac{d^{2/3} (5 b c-2 a d) \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{3 \sqrt{3} c^{5/3} (b c-a d)^2}-\frac{d x}{3 c \left(c+d x^3\right) (b c-a d)}",1,"(6*a^(2/3)*c^(2/3)*d*(-(b*c) + a*d)*x - 6*Sqrt[3]*b^(5/3)*c^(5/3)*(c + d*x^3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*Sqrt[3]*a^(2/3)*d^(2/3)*(-5*b*c + 2*a*d)*(c + d*x^3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] + 6*b^(5/3)*c^(5/3)*(c + d*x^3)*Log[a^(1/3) + b^(1/3)*x] + 2*a^(2/3)*d^(2/3)*(-5*b*c + 2*a*d)*(c + d*x^3)*Log[c^(1/3) + d^(1/3)*x] - 3*b^(5/3)*c^(5/3)*(c + d*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + a^(2/3)*d^(2/3)*(5*b*c - 2*a*d)*(c + d*x^3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(18*a^(2/3)*c^(5/3)*(b*c - a*d)^2*(c + d*x^3))","A",1
20,1,313,320,0.3032867,"\int \frac{\left(c+d x^3\right)^5}{\left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3)^5/(a + b*x^3)^2,x]","\frac{-\frac{70 (b c-a d)^4 (13 a d+2 b c) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{5/3}}+\frac{140 (b c-a d)^4 (13 a d+2 b c) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{5/3}}+\frac{140 \sqrt{3} (b c-a d)^4 (13 a d+2 b c) \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{a^{5/3}}+315 b^{4/3} d^3 x^4 \left(3 a^2 d^2-10 a b c d+10 b^2 c^2\right)+1260 \sqrt[3]{b} d^2 x \left(-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right)+180 b^{7/3} d^4 x^7 (5 b c-2 a d)+\frac{420 \sqrt[3]{b} x (b c-a d)^5}{a \left(a+b x^3\right)}+126 b^{10/3} d^5 x^{10}}{1260 b^{16/3}}","-\frac{(b c-a d)^4 (13 a d+2 b c) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{5/3} b^{16/3}}+\frac{(b c-a d)^4 (13 a d+2 b c) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} b^{16/3}}-\frac{(b c-a d)^4 (13 a d+2 b c) \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^{16/3}}+\frac{d^3 x^4 \left(3 a^2 d^2-10 a b c d+10 b^2 c^2\right)}{4 b^4}+\frac{d^2 x \left(-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right)}{b^5}+\frac{x (b c-a d)^5}{3 a b^5 \left(a+b x^3\right)}+\frac{d^4 x^7 (5 b c-2 a d)}{7 b^3}+\frac{d^5 x^{10}}{10 b^2}",1,"(1260*b^(1/3)*d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x + 315*b^(4/3)*d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*x^4 + 180*b^(7/3)*d^4*(5*b*c - 2*a*d)*x^7 + 126*b^(10/3)*d^5*x^10 + (420*b^(1/3)*(b*c - a*d)^5*x)/(a*(a + b*x^3)) + (140*Sqrt[3]*(b*c - a*d)^4*(2*b*c + 13*a*d)*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(5/3) + (140*(b*c - a*d)^4*(2*b*c + 13*a*d)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (70*(b*c - a*d)^4*(2*b*c + 13*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(1260*b^(16/3))","A",1
21,1,260,267,0.2380641,"\int \frac{\left(c+d x^3\right)^4}{\left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3)^4/(a + b*x^3)^2,x]","\frac{\frac{14 (a d-b c)^3 (5 a d+b c) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{5/3}}+\frac{28 (b c-a d)^3 (5 a d+b c) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{5/3}}+\frac{28 \sqrt{3} (b c-a d)^3 (5 a d+b c) \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{a^{5/3}}+126 \sqrt[3]{b} d^2 x \left(3 a^2 d^2-8 a b c d+6 b^2 c^2\right)+63 b^{4/3} d^3 x^4 (2 b c-a d)+\frac{42 \sqrt[3]{b} x (b c-a d)^4}{a \left(a+b x^3\right)}+18 b^{7/3} d^4 x^7}{126 b^{13/3}}","-\frac{(b c-a d)^3 (5 a d+b c) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{9 a^{5/3} b^{13/3}}+\frac{2 (b c-a d)^3 (5 a d+b c) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} b^{13/3}}-\frac{2 (b c-a d)^3 (5 a d+b c) \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^{13/3}}+\frac{d^2 x \left(3 a^2 d^2-8 a b c d+6 b^2 c^2\right)}{b^4}+\frac{x (b c-a d)^4}{3 a b^4 \left(a+b x^3\right)}+\frac{d^3 x^4 (2 b c-a d)}{2 b^3}+\frac{d^4 x^7}{7 b^2}",1,"(126*b^(1/3)*d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x + 63*b^(4/3)*d^3*(2*b*c - a*d)*x^4 + 18*b^(7/3)*d^4*x^7 + (42*b^(1/3)*(b*c - a*d)^4*x)/(a*(a + b*x^3)) + (28*Sqrt[3]*(b*c - a*d)^3*(b*c + 5*a*d)*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(5/3) + (28*(b*c - a*d)^3*(b*c + 5*a*d)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + (14*(-(b*c) + a*d)^3*(b*c + 5*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(126*b^(13/3))","A",1
22,1,227,234,0.155815,"\int \frac{\left(c+d x^3\right)^3}{\left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3)^3/(a + b*x^3)^2,x]","\frac{-\frac{2 (b c-a d)^2 (7 a d+2 b c) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{5/3}}+\frac{4 (b c-a d)^2 (7 a d+2 b c) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{5/3}}+\frac{4 \sqrt{3} (b c-a d)^2 (7 a d+2 b c) \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{a^{5/3}}+36 \sqrt[3]{b} d^2 x (3 b c-2 a d)+\frac{12 \sqrt[3]{b} x (b c-a d)^3}{a \left(a+b x^3\right)}+9 b^{4/3} d^3 x^4}{36 b^{10/3}}","-\frac{(b c-a d)^2 (7 a d+2 b c) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{5/3} b^{10/3}}+\frac{(b c-a d)^2 (7 a d+2 b c) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} b^{10/3}}-\frac{(b c-a d)^2 (7 a d+2 b c) \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^{10/3}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{3 a b^3 \left(a+b x^3\right)}+\frac{d^3 x^4}{4 b^2}",1,"(36*b^(1/3)*d^2*(3*b*c - 2*a*d)*x + 9*b^(4/3)*d^3*x^4 + (12*b^(1/3)*(b*c - a*d)^3*x)/(a*(a + b*x^3)) + (4*Sqrt[3]*(b*c - a*d)^2*(2*b*c + 7*a*d)*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(5/3) + (4*(b*c - a*d)^2*(2*b*c + 7*a*d)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (2*(b*c - a*d)^2*(2*b*c + 7*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(36*b^(10/3))","A",1
23,1,205,203,0.1977915,"\int \frac{\left(c+d x^3\right)^2}{\left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3)^2,x]","\frac{\frac{2 \left(-2 a^2 d^2+a b c d+b^2 c^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{5/3}}-\frac{2 \sqrt{3} \left(-2 a^2 d^2+a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{5/3}}-\frac{\left(-2 a^2 d^2+a b c d+b^2 c^2\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{5/3}}+\frac{3 \sqrt[3]{b} x (b c-a d)^2}{a \left(a+b x^3\right)}+9 \sqrt[3]{b} d^2 x}{9 b^{7/3}}","-\frac{(b c-a d) (2 a d+b c) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{9 a^{5/3} b^{7/3}}+\frac{2 (b c-a d) (2 a d+b c) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} b^{7/3}}-\frac{2 (b c-a d) (2 a d+b c) \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^{7/3}}+\frac{x (b c-a d)^2}{3 a b^2 \left(a+b x^3\right)}+\frac{d^2 x}{b^2}",1,"(9*b^(1/3)*d^2*x + (3*b^(1/3)*(b*c - a*d)^2*x)/(a*(a + b*x^3)) - (2*Sqrt[3]*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - ((b^2*c^2 + a*b*c*d - 2*a^2*d^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(9*b^(7/3))","A",1
24,1,145,169,0.0910638,"\int \frac{c+d x^3}{\left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3)/(a + b*x^3)^2,x]","\frac{-(a d+2 b c) \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 d-b c)}{a+b x^3}+2 (a d+2 b c) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} (a d+2 b c) \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 d+2 b c) \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 d+2 b c) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} b^{4/3}}-\frac{(a d+2 b c) \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 (b c-a d)}{3 a b \left(a+b x^3\right)}",1,"((-6*a^(2/3)*b^(1/3)*(-(b*c) + a*d)*x)/(a + b*x^3) - 2*Sqrt[3]*(2*b*c + a*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(2*b*c + a*d)*Log[a^(1/3) + b^(1/3)*x] - (2*b*c + a*d)*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
25,1,337,346,0.1952548,"\int \frac{1}{\left(a+b x^3\right)^2 \left(c+d x^3\right)} \, dx","Integrate[1/((a + b*x^3)^2*(c + d*x^3)),x]","\frac{-b^{2/3} c^{2/3} \left(a+b x^3\right) (2 b c-5 a d) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-3 a^{5/3} d^{5/3} \left(a+b x^3\right) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)+6 a^{2/3} b c^{2/3} x (b c-a d)+6 a^{5/3} d^{5/3} \left(a+b x^3\right) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)-6 \sqrt{3} a^{5/3} d^{5/3} \left(a+b x^3\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)+2 b^{2/3} c^{2/3} \left(a+b x^3\right) (2 b c-5 a d) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} b^{2/3} c^{2/3} \left(a+b x^3\right) (2 b c-5 a d) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{18 a^{5/3} c^{2/3} \left(a+b x^3\right) (b c-a d)^2}","-\frac{b^{2/3} (2 b c-5 a d) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{18 a^{5/3} (b c-a d)^2}+\frac{b^{2/3} (2 b c-5 a d) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} (b c-a d)^2}-\frac{b^{2/3} (2 b c-5 a d) \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 c-a d)^2}-\frac{d^{5/3} \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)^2}+\frac{d^{5/3} \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{3 c^{2/3} (b c-a d)^2}-\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^{2/3} (b c-a d)^2}+\frac{b x}{3 a \left(a+b x^3\right) (b c-a d)}",1,"(6*a^(2/3)*b*c^(2/3)*(b*c - a*d)*x - 2*Sqrt[3]*b^(2/3)*c^(2/3)*(2*b*c - 5*a*d)*(a + b*x^3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 6*Sqrt[3]*a^(5/3)*d^(5/3)*(a + b*x^3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] + 2*b^(2/3)*c^(2/3)*(2*b*c - 5*a*d)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x] + 6*a^(5/3)*d^(5/3)*(a + b*x^3)*Log[c^(1/3) + d^(1/3)*x] - b^(2/3)*c^(2/3)*(2*b*c - 5*a*d)*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 3*a^(5/3)*d^(5/3)*(a + b*x^3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(18*a^(5/3)*c^(2/3)*(b*c - a*d)^2*(a + b*x^3))","A",1
26,1,381,419,0.630187,"\int \frac{1}{\left(a+b x^3\right)^2 \left(c+d x^3\right)^2} \, dx","Integrate[1/((a + b*x^3)^2*(c + d*x^3)^2),x]","\frac{1}{9} \left(\frac{b^{5/3} (b c-4 a d) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{5/3} (a d-b c)^3}+\frac{2 b^{5/3} (4 a d-b c) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{5/3} (a d-b c)^3}+\frac{2 \sqrt{3} b^{5/3} (b c-4 a d) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{a^{5/3} (a d-b c)^3}+\frac{3 b^2 x}{a \left(a+b x^3\right) (b c-a d)^2}+\frac{d^{5/3} (a d-4 b c) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{c^{5/3} (b c-a d)^3}+\frac{2 d^{5/3} (4 b c-a d) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{c^{5/3} (b c-a d)^3}+\frac{2 \sqrt{3} d^{5/3} (a d-4 b c) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{d} x}{\sqrt[3]{c}}}{\sqrt{3}}\right)}{c^{5/3} (b c-a d)^3}+\frac{3 d^2 x}{c \left(c+d x^3\right) (b c-a d)^2}\right)","-\frac{b^{5/3} (b c-4 a d) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{9 a^{5/3} (b c-a d)^3}+\frac{2 b^{5/3} (b c-4 a d) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{9 a^{5/3} (b c-a d)^3}-\frac{2 b^{5/3} (b c-4 a d) \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 c-a d)^3}-\frac{d^{5/3} (4 b c-a d) \log \left(c^{2/3}-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3} x^2\right)}{9 c^{5/3} (b c-a d)^3}+\frac{2 d^{5/3} (4 b c-a d) \log \left(\sqrt[3]{c}+\sqrt[3]{d} x\right)}{9 c^{5/3} (b c-a d)^3}-\frac{2 d^{5/3} (4 b c-a d) \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{d} x}{\sqrt{3} \sqrt[3]{c}}\right)}{3 \sqrt{3} c^{5/3} (b c-a d)^3}+\frac{b x}{3 a \left(a+b x^3\right) \left(c+d x^3\right) (b c-a d)}+\frac{d x (a d+b c)}{3 a c \left(c+d x^3\right) (b c-a d)^2}",1,"((3*b^2*x)/(a*(b*c - a*d)^2*(a + b*x^3)) + (3*d^2*x)/(c*(b*c - a*d)^2*(c + d*x^3)) + (2*Sqrt[3]*b^(5/3)*(b*c - 4*a*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(a^(5/3)*(-(b*c) + a*d)^3) + (2*Sqrt[3]*d^(5/3)*(-4*b*c + a*d)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/(c^(5/3)*(b*c - a*d)^3) + (2*b^(5/3)*(-(b*c) + 4*a*d)*Log[a^(1/3) + b^(1/3)*x])/(a^(5/3)*(-(b*c) + a*d)^3) + (2*d^(5/3)*(4*b*c - a*d)*Log[c^(1/3) + d^(1/3)*x])/(c^(5/3)*(b*c - a*d)^3) + (b^(5/3)*(b*c - 4*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(a^(5/3)*(-(b*c) + a*d)^3) + (d^(5/3)*(-4*b*c + a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(c^(5/3)*(b*c - a*d)^3))/9","A",1
27,1,62,112,0.1058462,"\int \left(a-b x^3\right) \left(a+b x^3\right)^{2/3} \, dx","Integrate[(a - b*x^3)*(a + b*x^3)^(2/3),x]","\frac{1}{6} x \left(a+b x^3\right)^{2/3} \left(\frac{7 a \, _2F_1\left(-\frac{2}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\left(\frac{b x^3}{a}+1\right)^{2/3}}-a-b x^3\right)","-\frac{7 a^2 \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{18 \sqrt[3]{b}}+\frac{7 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} \sqrt[3]{b}}+\frac{7}{18} a x \left(a+b x^3\right)^{2/3}-\frac{1}{6} x \left(a+b x^3\right)^{5/3}",1,"(x*(a + b*x^3)^(2/3)*(-a - b*x^3 + (7*a*Hypergeometric2F1[-2/3, 1/3, 4/3, -((b*x^3)/a)])/(1 + (b*x^3)/a)^(2/3)))/6","C",1
28,1,134,91,0.0837707,"\int \frac{a-b x^3}{\sqrt[3]{a+b x^3}} \, dx","Integrate[(a - b*x^3)/(a + b*x^3)^(1/3),x]","\frac{2 a \log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-3 \sqrt[3]{b} x \left(a+b x^3\right)^{2/3}-4 a \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+4 \sqrt{3} a \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{9 \sqrt[3]{b}}","-\frac{1}{3} x \left(a+b x^3\right)^{2/3}-\frac{2 a \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{3 \sqrt[3]{b}}+\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} \sqrt[3]{b}}",1,"(-3*b^(1/3)*x*(a + b*x^3)^(2/3) + 4*Sqrt[3]*a*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 4*a*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + 2*a*Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)])/(9*b^(1/3))","A",1
29,1,62,85,0.0366712,"\int \frac{a-b x^3}{\left(a+b x^3\right)^{4/3}} \, dx","Integrate[(a - b*x^3)/(a + b*x^3)^(4/3),x]","\frac{4 a x-b x^4 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{4}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right)}{4 a \sqrt[3]{a+b x^3}}","\frac{2 x}{\sqrt[3]{a+b x^3}}+\frac{\log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 \sqrt[3]{b}}-\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}}",1,"(4*a*x - b*x^4*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[4/3, 4/3, 7/3, -((b*x^3)/a)])/(4*a*(a + b*x^3)^(1/3))","C",1
30,1,28,47,0.0149574,"\int \frac{a-b x^3}{\left(a+b x^3\right)^{7/3}} \, dx","Integrate[(a - b*x^3)/(a + b*x^3)^(7/3),x]","\frac{x \left(2 a+b x^3\right)}{2 a \left(a+b x^3\right)^{4/3}}","\frac{3 x}{4 a \sqrt[3]{a+b x^3}}+\frac{x \left(a-b x^3\right)}{4 a \left(a+b x^3\right)^{4/3}}",1,"(x*(2*a + b*x^3))/(2*a*(a + b*x^3)^(4/3))","A",1
31,1,40,55,0.0212231,"\int \frac{a-b x^3}{\left(a+b x^3\right)^{10/3}} \, dx","Integrate[(a - b*x^3)/(a + b*x^3)^(10/3),x]","\frac{x \left(28 a^2+35 a b x^3+15 b^2 x^6\right)}{28 a^2 \left(a+b x^3\right)^{7/3}}","\frac{15 x}{28 a^2 \sqrt[3]{a+b x^3}}+\frac{5 x}{28 a \left(a+b x^3\right)^{4/3}}+\frac{2 x}{7 \left(a+b x^3\right)^{7/3}}",1,"(x*(28*a^2 + 35*a*b*x^3 + 15*b^2*x^6))/(28*a^2*(a + b*x^3)^(7/3))","A",1
32,1,51,74,0.0224986,"\int \frac{a-b x^3}{\left(a+b x^3\right)^{13/3}} \, dx","Integrate[(a - b*x^3)/(a + b*x^3)^(13/3),x]","\frac{x \left(35 a^3+70 a^2 b x^3+60 a b^2 x^6+18 b^3 x^9\right)}{35 a^3 \left(a+b x^3\right)^{10/3}}","\frac{18 x}{35 a^3 \sqrt[3]{a+b x^3}}+\frac{6 x}{35 a^2 \left(a+b x^3\right)^{4/3}}+\frac{4 x}{35 a \left(a+b x^3\right)^{7/3}}+\frac{x}{5 \left(a+b x^3\right)^{10/3}}",1,"(x*(35*a^3 + 70*a^2*b*x^3 + 60*a*b^2*x^6 + 18*b^3*x^9))/(35*a^3*(a + b*x^3)^(10/3))","A",1
33,1,62,93,0.0255075,"\int \frac{a-b x^3}{\left(a+b x^3\right)^{16/3}} \, dx","Integrate[(a - b*x^3)/(a + b*x^3)^(16/3),x]","\frac{x \left(1820 a^4+5005 a^3 b x^3+6435 a^2 b^2 x^6+3861 a b^3 x^9+891 b^4 x^{12}\right)}{1820 a^4 \left(a+b x^3\right)^{13/3}}","\frac{891 x}{1820 a^4 \sqrt[3]{a+b x^3}}+\frac{297 x}{1820 a^3 \left(a+b x^3\right)^{4/3}}+\frac{99 x}{910 a^2 \left(a+b x^3\right)^{7/3}}+\frac{11 x}{130 a \left(a+b x^3\right)^{10/3}}+\frac{2 x}{13 \left(a+b x^3\right)^{13/3}}",1,"(x*(1820*a^4 + 5005*a^3*b*x^3 + 6435*a^2*b^2*x^6 + 3861*a*b^3*x^9 + 891*b^4*x^12))/(1820*a^4*(a + b*x^3)^(13/3))","A",1
34,1,232,483,0.383524,"\int \frac{\left(a+b x^3\right)^{7/3}}{a-b x^3} \, dx","Integrate[(a + b*x^3)^(7/3)/(a - b*x^3),x]","\frac{4 x \left(\frac{52 a^4 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)}-8 a^2-9 a b x^3-b^2 x^6\right)+27 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)}{20 \left(a+b x^3\right)^{2/3}}","-\frac{2 \sqrt[3]{2} a^{5/3} \log \left(2^{2/3}-\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)}{3 \sqrt[3]{b}}+\frac{2 \sqrt[3]{2} 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 \sqrt[3]{b}}-\frac{4 \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 \sqrt[3]{b}}+\frac{\sqrt[3]{2} 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)}{3 \sqrt[3]{b}}-\frac{4 \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} \sqrt[3]{b}}-\frac{2 \sqrt[3]{2} 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)}{\sqrt{3} \sqrt[3]{b}}-\frac{7 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 \left(a+b x^3\right)^{2/3}}-\frac{7}{5} a x \sqrt[3]{a+b x^3}-\frac{1}{5} x \left(a+b x^3\right)^{4/3}",1,"(27*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] + 4*x*(-8*a^2 - 9*a*b*x^3 - b^2*x^6 + (52*a^4*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*(a + b*x^3)^(2/3))","C",0
35,1,217,464,0.116216,"\int \frac{\left(a+b x^3\right)^{4/3}}{a-b x^3} \, dx","Integrate[(a + b*x^3)^(4/3)/(a - b*x^3),x]","\frac{x \left(\frac{48 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)}+5 b 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)-4 \left(a+b x^3\right)\right)}{8 \left(a+b x^3\right)^{2/3}}","-\frac{\sqrt[3]{2} a^{2/3} \log \left(2^{2/3}-\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)}{3 \sqrt[3]{b}}+\frac{\sqrt[3]{2} 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 \sqrt[3]{b}}-\frac{2 \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 \sqrt[3]{b}}+\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)}{3\ 2^{2/3} \sqrt[3]{b}}-\frac{2 \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} \sqrt[3]{b}}-\frac{\sqrt[3]{2} 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)}{\sqrt{3} \sqrt[3]{b}}-\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 \left(a+b x^3\right)^{2/3}}-\frac{1}{2} x \sqrt[3]{a+b x^3}",1,"(x*(-4*(a + b*x^3) + 5*b*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] + (48*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])))))/(8*(a + b*x^3)^(2/3))","C",0
36,1,151,398,0.1336026,"\int \frac{\sqrt[3]{a+b x^3}}{a-b x^3} \, dx","Integrate[(a + b*x^3)^(1/3)/(a - b*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)}{\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}}+\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}}-\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}}+\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}}-\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}}-\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}}",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])/((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
37,1,153,452,0.1334733,"\int \frac{1}{\left(a-b x^3\right) \left(a+b x^3\right)^{2/3}} \, dx","Integrate[1/((a - b*x^3)*(a + b*x^3)^(2/3)),x]","\frac{4 a 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(a+b x^3\right)^{2/3} \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)}","-\frac{\log \left(2^{2/3}-\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)}{6\ 2^{2/3} a^{4/3} \sqrt[3]{b}}+\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)}{6\ 2^{2/3} a^{4/3} \sqrt[3]{b}}-\frac{\log \left(\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} \sqrt[3]{b}}+\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)}{12\ 2^{2/3} a^{4/3} \sqrt[3]{b}}-\frac{\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)}{2^{2/3} \sqrt{3} a^{4/3} \sqrt[3]{b}}-\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^{2/3} \sqrt{3} a^{4/3} \sqrt[3]{b}}+\frac{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 \left(a+b x^3\right)^{2/3}}",1,"(4*a*x*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/((a - b*x^3)*(a + b*x^3)^(2/3)*(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])))","C",0
38,1,213,473,0.1388427,"\int \frac{1}{\left(a-b x^3\right) \left(a+b x^3\right)^{5/3}} \, dx","Integrate[1/((a - b*x^3)*(a + b*x^3)^(5/3)),x]","\frac{x \left(-\frac{b 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)}{a^3}+\frac{4}{a^2}+\frac{48 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)}\right)}{16 \left(a+b x^3\right)^{2/3}}","-\frac{\log \left(2^{2/3}-\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)}{12\ 2^{2/3} a^{7/3} \sqrt[3]{b}}+\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)}{12\ 2^{2/3} a^{7/3} \sqrt[3]{b}}-\frac{\log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{6\ 2^{2/3} a^{7/3} \sqrt[3]{b}}+\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)}{24\ 2^{2/3} a^{7/3} \sqrt[3]{b}}-\frac{\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)}{2\ 2^{2/3} \sqrt{3} a^{7/3} \sqrt[3]{b}}-\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)}{4\ 2^{2/3} \sqrt{3} a^{7/3} \sqrt[3]{b}}+\frac{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^2 \left(a+b x^3\right)^{2/3}}+\frac{x}{4 a^2 \left(a+b x^3\right)^{2/3}}",1,"(x*(4/a^2 - (b*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])/a^3 + (48*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])))))/(16*(a + b*x^3)^(2/3))","C",0
39,1,240,492,0.1758539,"\int \frac{1}{\left(a-b x^3\right) \left(a+b x^3\right)^{8/3}} \, dx","Integrate[1/((a - b*x^3)*(a + b*x^3)^(8/3)),x]","\frac{x \left(\frac{368 a^3 \left(a+b x^3\right) 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)}+16 a^2-13 b x^3 \left(a+b x^3\right) \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)+52 a \left(a+b x^3\right)\right)}{160 a^4 \left(a+b x^3\right)^{5/3}}","-\frac{\log \left(2^{2/3}-\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)}{24\ 2^{2/3} a^{10/3} \sqrt[3]{b}}+\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)}{24\ 2^{2/3} a^{10/3} \sqrt[3]{b}}-\frac{\log \left(\frac{\sqrt[3]{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{a+b x^3}}+1\right)}{12\ 2^{2/3} a^{10/3} \sqrt[3]{b}}+\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)}{48\ 2^{2/3} a^{10/3} \sqrt[3]{b}}-\frac{\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)}{4\ 2^{2/3} \sqrt{3} a^{10/3} \sqrt[3]{b}}-\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)}{8\ 2^{2/3} \sqrt{3} a^{10/3} \sqrt[3]{b}}+\frac{9 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)}{20 a^3 \left(a+b x^3\right)^{2/3}}+\frac{13 x}{40 a^3 \left(a+b x^3\right)^{2/3}}+\frac{x}{10 a^2 \left(a+b x^3\right)^{5/3}}",1,"(x*(16*a^2 + 52*a*(a + b*x^3) - 13*b*x^3*(a + b*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] + (368*a^3*(a + 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])))))/(160*a^4*(a + b*x^3)^(5/3))","C",0
40,1,151,139,0.137441,"\int \left(a-b x^3\right)^2 \left(a+b x^3\right)^{2/3} \, dx","Integrate[(a - b*x^3)^2*(a + b*x^3)^(2/3),x]","\frac{1}{243} \left(\frac{38 a^3 \left(\log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)\right)}{\sqrt[3]{b}}+3 \left(a+b x^3\right)^{2/3} \left(5 a^2 x-24 a b x^4+9 b^2 x^7\right)\right)","-\frac{38 a^3 \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{81 \sqrt[3]{b}}+\frac{76 a^3 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{81 \sqrt{3} \sqrt[3]{b}}+\frac{38}{81} a^2 x \left(a+b x^3\right)^{2/3}-\frac{8}{27} a x \left(a+b x^3\right)^{5/3}-\frac{1}{9} x \left(a-b x^3\right) \left(a+b x^3\right)^{5/3}",1,"(3*(a + b*x^3)^(2/3)*(5*a^2*x - 24*a*b*x^4 + 9*b^2*x^7) + (38*a^3*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/b^(1/3))/243","A",1
41,1,141,120,0.0783112,"\int \frac{\left(a-b x^3\right)^2}{\sqrt[3]{a+b x^3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(1/3),x]","\frac{17 a^2 \left(\log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)\right)}{54 \sqrt[3]{b}}+\left(a+b x^3\right)^{2/3} \left(\frac{b x^4}{6}-\frac{8 a x}{9}\right)","-\frac{17 a^2 \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{18 \sqrt[3]{b}}+\frac{17 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} \sqrt[3]{b}}-\frac{13}{18} a x \left(a+b x^3\right)^{2/3}-\frac{1}{6} x \left(a-b x^3\right) \left(a+b x^3\right)^{2/3}",1,"(a + b*x^3)^(2/3)*((-8*a*x)/9 + (b*x^4)/6) + (17*a^2*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/(54*b^(1/3))","A",1
42,1,137,113,0.0819813,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{4/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(4/3),x]","\frac{x \left(13 a+b x^3\right)}{3 \sqrt[3]{a+b x^3}}-\frac{5 a \left(\log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)\right)}{9 \sqrt[3]{b}}","\frac{7}{3} x \left(a+b x^3\right)^{2/3}+\frac{2 x \left(a-b x^3\right)}{\sqrt[3]{a+b x^3}}+\frac{5 a \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{3 \sqrt[3]{b}}-\frac{10 a \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}}",1,"(x*(13*a + b*x^3))/(3*(a + b*x^3)^(1/3)) - (5*a*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/(9*b^(1/3))","A",1
43,1,131,110,0.0625069,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{7/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(7/3),x]","\frac{-\frac{6 b^{4/3} x^4}{\left(a+b x^3\right)^{4/3}}+\log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{6 \sqrt[3]{b}}","-\frac{x}{2 \sqrt[3]{a+b x^3}}+\frac{x \left(a-b x^3\right)}{2 \left(a+b x^3\right)^{4/3}}-\frac{\log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 \sqrt[3]{b}}+\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}}",1,"((-6*b^(4/3)*x^4)/(a + b*x^3)^(4/3) + 2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)])/(6*b^(1/3))","A",1
44,1,40,76,0.014813,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{10/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(10/3),x]","\frac{x \left(7 a^2+7 a b x^3+4 b^2 x^6\right)}{7 a \left(a+b x^3\right)^{7/3}}","\frac{x \left(a-b x^3\right)^2}{7 a \left(a+b x^3\right)^{7/3}}+\frac{3 x \left(a-b x^3\right)}{14 a \left(a+b x^3\right)^{4/3}}+\frac{9 x}{14 a \sqrt[3]{a+b x^3}}",1,"(x*(7*a^2 + 7*a*b*x^3 + 4*b^2*x^6))/(7*a*(a + b*x^3)^(7/3))","A",1
45,1,51,105,0.0324681,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{13/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(13/3),x]","\frac{x \left(140 a^3+245 a^2 b x^3+230 a b^2 x^6+69 b^3 x^9\right)}{140 a^2 \left(a+b x^3\right)^{10/3}}","\frac{x \left(a-b x^3\right)^3}{20 a^2 \left(a+b x^3\right)^{10/3}}+\frac{19 x \left(a-b x^3\right)^2}{140 a^2 \left(a+b x^3\right)^{7/3}}+\frac{57 x \left(a-b x^3\right)}{280 a^2 \left(a+b x^3\right)^{4/3}}+\frac{171 x}{280 a^2 \sqrt[3]{a+b x^3}}",1,"(x*(140*a^3 + 245*a^2*b*x^3 + 230*a*b^2*x^6 + 69*b^3*x^9))/(140*a^2*(a + b*x^3)^(10/3))","A",1
46,1,62,98,0.0353584,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{16/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(16/3),x]","\frac{x \left(910 a^4+2275 a^3 b x^3+3055 a^2 b^2 x^6+1833 a b^3 x^9+423 b^4 x^{12}\right)}{910 a^3 \left(a+b x^3\right)^{13/3}}","\frac{423 x}{910 a^3 \sqrt[3]{a+b x^3}}+\frac{141 x}{910 a^2 \left(a+b x^3\right)^{4/3}}+\frac{47 x}{455 a \left(a+b x^3\right)^{7/3}}+\frac{8 x}{65 \left(a+b x^3\right)^{10/3}}+\frac{2 x \left(a-b x^3\right)}{13 \left(a+b x^3\right)^{13/3}}",1,"(x*(910*a^4 + 2275*a^3*b*x^3 + 3055*a^2*b^2*x^6 + 1833*a*b^3*x^9 + 423*b^4*x^12))/(910*a^3*(a + b*x^3)^(13/3))","A",1
47,1,73,117,0.0402142,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{19/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(19/3),x]","\frac{x \left(364 a^5+1183 a^4 b x^3+2080 a^3 b^2 x^6+1872 a^2 b^3 x^9+864 a b^4 x^{12}+162 b^5 x^{15}\right)}{364 a^4 \left(a+b x^3\right)^{16/3}}","\frac{81 x}{182 a^4 \sqrt[3]{a+b x^3}}+\frac{27 x}{182 a^3 \left(a+b x^3\right)^{4/3}}+\frac{9 x}{91 a^2 \left(a+b x^3\right)^{7/3}}+\frac{x}{13 a \left(a+b x^3\right)^{10/3}}+\frac{11 x}{104 \left(a+b x^3\right)^{13/3}}+\frac{x \left(a-b x^3\right)}{8 \left(a+b x^3\right)^{16/3}}",1,"(x*(364*a^5 + 1183*a^4*b*x^3 + 2080*a^3*b^2*x^6 + 1872*a^2*b^3*x^9 + 864*a*b^4*x^12 + 162*b^5*x^15))/(364*a^4*(a + b*x^3)^(16/3))","A",1
48,1,97,94,0.0488196,"\int \left(a-b x^3\right)^2 \left(a+b x^3\right)^{4/3} \, dx","Integrate[(a - b*x^3)^2*(a + b*x^3)^(4/3),x]","\frac{x \left(114 a^4 \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)+106 a^4+53 a^3 b x^3-78 a^2 b^2 x^6-5 a b^3 x^9+20 b^4 x^{12}\right)}{220 \left(a+b x^3\right)^{2/3}}","\frac{57 a^3 x \sqrt[3]{a+b x^3} \, _2F_1\left(-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{44 \sqrt[3]{\frac{b x^3}{a}+1}}-\frac{9}{44} a x \left(a+b x^3\right)^{7/3}-\frac{1}{11} x \left(a-b x^3\right) \left(a+b x^3\right)^{7/3}",1,"(x*(106*a^4 + 53*a^3*b*x^3 - 78*a^2*b^2*x^6 - 5*a*b^3*x^9 + 20*b^4*x^12 + 114*a^4*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)]))/(220*(a + b*x^3)^(2/3))","A",1
49,1,85,94,0.0463797,"\int \left(a-b x^3\right)^2 \sqrt[3]{a+b x^3} \, dx","Integrate[(a - b*x^3)^2*(a + b*x^3)^(1/3),x]","\frac{x \left(6 a^3 \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^3-a^2 b x^3-2 a b^2 x^6+b^3 x^9\right)}{8 \left(a+b x^3\right)^{2/3}}","\frac{3 a^2 x \sqrt[3]{a+b x^3} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 \sqrt[3]{\frac{b x^3}{a}+1}}-\frac{3}{8} a x \left(a+b x^3\right)^{4/3}-\frac{1}{8} x \left(a-b x^3\right) \left(a+b x^3\right)^{4/3}",1,"(x*(2*a^3 - a^2*b*x^3 - 2*a*b^2*x^6 + b^3*x^9 + 6*a^3*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)]))/(8*(a + b*x^3)^(2/3))","A",1
50,1,75,94,0.0473929,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{2/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(2/3),x]","\frac{12 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)-7 a^2 x-6 a b x^4+b^2 x^7}{5 \left(a+b x^3\right)^{2/3}}","\frac{12 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 \left(a+b x^3\right)^{2/3}}-\frac{6}{5} a x \sqrt[3]{a+b x^3}-\frac{1}{5} x \left(a-b x^3\right) \sqrt[3]{a+b x^3}",1,"(-7*a^2*x - 6*a*b*x^4 + b^2*x^7 + 12*a^2*x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)])/(5*(a + b*x^3)^(2/3))","A",1
51,1,62,74,0.0453791,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{5/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(5/3),x]","\frac{-3 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)+5 a x+b x^4}{2 \left(a+b x^3\right)^{2/3}}","\frac{3 b x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right)}{4 \left(a+b x^3\right)^{2/3}}+\frac{x \left(a-b x^3\right)}{\left(a+b x^3\right)^{2/3}}",1,"(5*a*x + b*x^4 - 3*a*x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)])/(2*(a + b*x^3)^(2/3))","A",1
52,1,70,74,0.0693088,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{8/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(8/3),x]","\frac{3 x \left(a+b x^3\right) \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 x \left(a-b x^3\right)}{5 \left(a+b x^3\right)^{5/3}}","\frac{3 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 \left(a+b x^3\right)^{2/3}}+\frac{2 x \left(a-b x^3\right)}{5 \left(a+b x^3\right)^{5/3}}",1,"(2*x*(a - b*x^3) + 3*x*(a + b*x^3)*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)])/(5*(a + b*x^3)^(5/3))","A",1
53,1,85,77,0.1183572,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{11/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(11/3),x]","\frac{7 a^2 x+3 x \left(a+b x^3\right)^2 \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 b x^4+3 b^2 x^7}{10 a \left(a+b x^3\right)^{8/3}}","\frac{3 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{8}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{4 a \left(a+b x^3\right)^{2/3}}+\frac{x \left(a-b x^3\right)}{4 \left(a+b x^3\right)^{8/3}}",1,"(7*a^2*x + 5*a*b*x^4 + 3*b^2*x^7 + 3*x*(a + b*x^3)^2*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)])/(10*a*(a + b*x^3)^(8/3))","A",1
54,1,95,93,0.1261663,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{14/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(14/3),x]","\frac{x \left(16 a^3+23 a^2 b x^3+21 a b^2 x^6+6 \left(a+b x^3\right)^3 \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)+6 b^3 x^9\right)}{22 a^2 \left(a+b x^3\right)^{11/3}}","\frac{15 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{8}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{22 a^2 \left(a+b x^3\right)^{2/3}}+\frac{3 x}{22 \left(a+b x^3\right)^{8/3}}+\frac{2 x \left(a-b x^3\right)}{11 \left(a+b x^3\right)^{11/3}}",1,"(x*(16*a^3 + 23*a^2*b*x^3 + 21*a*b^2*x^6 + 6*b^3*x^9 + 6*(a + b*x^3)^3*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)]))/(22*a^2*(a + b*x^3)^(11/3))","A",1
55,1,106,93,0.1408251,"\int \frac{\left(a-b x^3\right)^2}{\left(a+b x^3\right)^{17/3}} \, dx","Integrate[(a - b*x^3)^2/(a + b*x^3)^(17/3),x]","\frac{x \left(2282 a^4+4879 a^3 b x^3+6270 a^2 b^2 x^6+3591 a b^3 x^9+798 \left(a+b x^3\right)^4 \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)+798 b^4 x^{12}\right)}{3080 a^3 \left(a+b x^3\right)^{14/3}}","\frac{57 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{11}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{77 a^3 \left(a+b x^3\right)^{2/3}}+\frac{9 x}{77 \left(a+b x^3\right)^{11/3}}+\frac{x \left(a-b x^3\right)}{7 \left(a+b x^3\right)^{14/3}}",1,"(x*(2282*a^4 + 4879*a^3*b*x^3 + 6270*a^2*b^2*x^6 + 3591*a*b^3*x^9 + 798*b^4*x^12 + 798*(a + b*x^3)^4*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)]))/(3080*a^3*(a + b*x^3)^(14/3))","A",1
56,1,75,174,0.0677961,"\int \left(a+b x^3\right)^{5/3} \left(c+d x^3\right) \, dx","Integrate[(a + b*x^3)^(5/3)*(c + d*x^3),x]","\frac{x \left(a+b x^3\right)^{2/3} \left(d \left(a+b x^3\right)^2-\frac{a (a d-9 b c) \, _2F_1\left(-\frac{5}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\left(\frac{b x^3}{a}+1\right)^{2/3}}\right)}{9 b}","-\frac{5 a^2 (9 b c-a d) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{162 b^{4/3}}+\frac{5 a^2 (9 b c-a d) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{81 \sqrt{3} b^{4/3}}+\frac{x \left(a+b x^3\right)^{5/3} (9 b c-a d)}{54 b}+\frac{5 a x \left(a+b x^3\right)^{2/3} (9 b c-a d)}{162 b}+\frac{d x \left(a+b x^3\right)^{8/3}}{9 b}",1,"(x*(a + b*x^3)^(2/3)*(d*(a + b*x^3)^2 - (a*(-9*b*c + a*d)*Hypergeometric2F1[-5/3, 1/3, 4/3, -((b*x^3)/a)])/(1 + (b*x^3)/a)^(2/3)))/(9*b)","C",1
57,1,72,141,0.0952896,"\int \left(a+b x^3\right)^{2/3} \left(c+d x^3\right) \, dx","Integrate[(a + b*x^3)^(2/3)*(c + d*x^3),x]","\frac{x \left(a+b x^3\right)^{2/3} \left(\frac{(6 b c-a d) \, _2F_1\left(-\frac{2}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\left(\frac{b x^3}{a}+1\right)^{2/3}}+d \left(a+b x^3\right)\right)}{6 b}","-\frac{a (6 b c-a d) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{18 b^{4/3}}+\frac{a (6 b c-a d) \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}}+\frac{x \left(a+b x^3\right)^{2/3} (6 b c-a d)}{18 b}+\frac{d x \left(a+b x^3\right)^{5/3}}{6 b}",1,"(x*(a + b*x^3)^(2/3)*(d*(a + b*x^3) + ((6*b*c - a*d)*Hypergeometric2F1[-2/3, 1/3, 4/3, -((b*x^3)/a)])/(1 + (b*x^3)/a)^(2/3)))/(6*b)","C",1
58,1,141,111,0.1540893,"\int \frac{c+d x^3}{\sqrt[3]{a+b x^3}} \, dx","Integrate[(c + d*x^3)/(a + b*x^3)^(1/3),x]","\frac{\frac{(3 b c-a d) \left(\log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)\right)}{6 \sqrt[3]{b}}+d x \left(a+b x^3\right)^{2/3}}{3 b}","-\frac{(3 b c-a d) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{6 b^{4/3}}+\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^{4/3}}+\frac{d x \left(a+b x^3\right)^{2/3}}{3 b}",1,"(d*x*(a + b*x^3)^(2/3) + ((3*b*c - a*d)*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/(6*b^(1/3)))/(3*b)","A",1
59,1,61,99,0.0522769,"\int \frac{c+d x^3}{\left(a+b x^3\right)^{4/3}} \, dx","Integrate[(c + d*x^3)/(a + b*x^3)^(4/3),x]","\frac{d x^4 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{4}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right)+4 c x}{4 a \sqrt[3]{a+b x^3}}","-\frac{d \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 b^{4/3}}+\frac{d \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}}+\frac{x (b c-a d)}{a b \sqrt[3]{a+b x^3}}",1,"(4*c*x + d*x^4*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[4/3, 4/3, 7/3, -((b*x^3)/a)])/(4*a*(a + b*x^3)^(1/3))","C",1
60,1,37,47,0.035823,"\int \frac{c+d x^3}{\left(a+b x^3\right)^{7/3}} \, dx","Integrate[(c + d*x^3)/(a + b*x^3)^(7/3),x]","\frac{x \left(4 a c+a d x^3+3 b c x^3\right)}{4 a^2 \left(a+b x^3\right)^{4/3}}","\frac{3 c x}{4 a^2 \sqrt[3]{a+b x^3}}+\frac{x \left(c+d x^3\right)}{4 a \left(a+b x^3\right)^{4/3}}",1,"(x*(4*a*c + 3*b*c*x^3 + a*d*x^3))/(4*a^2*(a + b*x^3)^(4/3))","A",1
61,1,59,91,0.0393141,"\int \frac{c+d x^3}{\left(a+b x^3\right)^{10/3}} \, dx","Integrate[(c + d*x^3)/(a + b*x^3)^(10/3),x]","\frac{7 a^2 \left(4 c x+d x^4\right)+3 a b x^4 \left(14 c+d x^3\right)+18 b^2 c x^7}{28 a^3 \left(a+b x^3\right)^{7/3}}","\frac{3 x (a d+6 b c)}{28 a^3 b \sqrt[3]{a+b x^3}}+\frac{x (a d+6 b c)}{28 a^2 b \left(a+b x^3\right)^{4/3}}+\frac{x (b c-a d)}{7 a b \left(a+b x^3\right)^{7/3}}",1,"(18*b^2*c*x^7 + 3*a*b*x^4*(14*c + d*x^3) + 7*a^2*(4*c*x + d*x^4))/(28*a^3*(a + b*x^3)^(7/3))","A",1
62,1,80,121,0.0343449,"\int \frac{c+d x^3}{\left(a+b x^3\right)^{13/3}} \, dx","Integrate[(c + d*x^3)/(a + b*x^3)^(13/3),x]","\frac{x \left(35 a^3 \left(4 c+d x^3\right)+15 a^2 b x^3 \left(21 c+2 d x^3\right)+9 a b^2 x^6 \left(30 c+d x^3\right)+81 b^3 c x^9\right)}{140 a^4 \left(a+b x^3\right)^{10/3}}","\frac{9 x (a d+9 b c)}{140 a^4 b \sqrt[3]{a+b x^3}}+\frac{3 x (a d+9 b c)}{140 a^3 b \left(a+b x^3\right)^{4/3}}+\frac{x (a d+9 b c)}{70 a^2 b \left(a+b x^3\right)^{7/3}}+\frac{x (b c-a d)}{10 a b \left(a+b x^3\right)^{10/3}}",1,"(x*(81*b^3*c*x^9 + 35*a^3*(4*c + d*x^3) + 9*a*b^2*x^6*(30*c + d*x^3) + 15*a^2*b*x^3*(21*c + 2*d*x^3)))/(140*a^4*(a + b*x^3)^(10/3))","A",1
63,1,100,151,0.0622663,"\int \frac{c+d x^3}{\left(a+b x^3\right)^{16/3}} \, dx","Integrate[(c + d*x^3)/(a + b*x^3)^(16/3),x]","\frac{x \left(455 a^4 \left(4 c+d x^3\right)+195 a^3 b x^3 \left(28 c+3 d x^3\right)+351 a^2 b^2 x^6 \left(20 c+d x^3\right)+81 a b^3 x^9 \left(52 c+d x^3\right)+972 b^4 c x^{12}\right)}{1820 a^5 \left(a+b x^3\right)^{13/3}}","\frac{81 x (a d+12 b c)}{1820 a^5 b \sqrt[3]{a+b x^3}}+\frac{27 x (a d+12 b c)}{1820 a^4 b \left(a+b x^3\right)^{4/3}}+\frac{9 x (a d+12 b c)}{910 a^3 b \left(a+b x^3\right)^{7/3}}+\frac{x (a d+12 b c)}{130 a^2 b \left(a+b x^3\right)^{10/3}}+\frac{x (b c-a d)}{13 a b \left(a+b x^3\right)^{13/3}}",1,"(x*(972*b^4*c*x^12 + 455*a^4*(4*c + d*x^3) + 351*a^2*b^2*x^6*(20*c + d*x^3) + 81*a*b^3*x^9*(52*c + d*x^3) + 195*a^3*b*x^3*(28*c + 3*d*x^3)))/(1820*a^5*(a + b*x^3)^(13/3))","A",1
64,1,77,85,0.0519723,"\int \left(a+b x^3\right)^{7/3} \left(c+d x^3\right) \, dx","Integrate[(a + b*x^3)^(7/3)*(c + d*x^3),x]","\frac{x \sqrt[3]{a+b x^3} \left(d \left(a+b x^3\right)^3-\frac{a^2 (a d-11 b c) \, _2F_1\left(-\frac{7}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt[3]{\frac{b x^3}{a}+1}}\right)}{11 b}","\frac{a^2 x \sqrt[3]{a+b x^3} (11 b c-a d) \, _2F_1\left(-\frac{7}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{11 b \sqrt[3]{\frac{b x^3}{a}+1}}+\frac{d x \left(a+b x^3\right)^{10/3}}{11 b}",1,"(x*(a + b*x^3)^(1/3)*(d*(a + b*x^3)^3 - (a^2*(-11*b*c + a*d)*Hypergeometric2F1[-7/3, 1/3, 4/3, -((b*x^3)/a)])/(1 + (b*x^3)/a)^(1/3)))/(11*b)","A",1
65,1,75,83,0.1433375,"\int \left(a+b x^3\right)^{4/3} \left(c+d x^3\right) \, dx","Integrate[(a + b*x^3)^(4/3)*(c + d*x^3),x]","\frac{x \sqrt[3]{a+b x^3} \left(d \left(a+b x^3\right)^2-\frac{a (a d-8 b c) \, _2F_1\left(-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt[3]{\frac{b x^3}{a}+1}}\right)}{8 b}","\frac{a x \sqrt[3]{a+b x^3} (8 b c-a d) \, _2F_1\left(-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{8 b \sqrt[3]{\frac{b x^3}{a}+1}}+\frac{d x \left(a+b x^3\right)^{7/3}}{8 b}",1,"(x*(a + b*x^3)^(1/3)*(d*(a + b*x^3)^2 - (a*(-8*b*c + a*d)*Hypergeometric2F1[-4/3, 1/3, 4/3, -((b*x^3)/a)])/(1 + (b*x^3)/a)^(1/3)))/(8*b)","A",1
66,1,72,82,0.1381356,"\int \sqrt[3]{a+b x^3} \left(c+d x^3\right) \, dx","Integrate[(a + b*x^3)^(1/3)*(c + d*x^3),x]","\frac{x \sqrt[3]{a+b x^3} \left(\frac{(5 b c-a d) \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\sqrt[3]{\frac{b x^3}{a}+1}}+d \left(a+b x^3\right)\right)}{5 b}","\frac{x \sqrt[3]{a+b x^3} (5 b c-a d) \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{5 b \sqrt[3]{\frac{b x^3}{a}+1}}+\frac{d x \left(a+b x^3\right)^{4/3}}{5 b}",1,"(x*(a + b*x^3)^(1/3)*(d*(a + b*x^3) + ((5*b*c - a*d)*Hypergeometric2F1[-1/3, 1/3, 4/3, -((b*x^3)/a)])/(1 + (b*x^3)/a)^(1/3)))/(5*b)","A",1
67,1,73,82,0.0693044,"\int \frac{c+d x^3}{\left(a+b x^3\right)^{2/3}} \, dx","Integrate[(c + d*x^3)/(a + b*x^3)^(2/3),x]","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} (2 b c-a d) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d x \left(a+b x^3\right)}{2 b \left(a+b x^3\right)^{2/3}}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} (2 b c-a d) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 b \left(a+b x^3\right)^{2/3}}+\frac{d x \sqrt[3]{a+b x^3}}{2 b}",1,"(d*x*(a + b*x^3) + (2*b*c - a*d)*x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)])/(2*b*(a + b*x^3)^(2/3))","A",1
68,1,66,93,0.046971,"\int \frac{c+d x^3}{\left(a+b x^3\right)^{5/3}} \, dx","Integrate[(c + d*x^3)/(a + b*x^3)^(5/3),x]","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} (a d+b c) \, _2F_1\left(\frac{1}{3},\frac{5}{3};\frac{4}{3};-\frac{b x^3}{a}\right)-a d x}{a b \left(a+b x^3\right)^{2/3}}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} (a d+b c) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 a b \left(a+b x^3\right)^{2/3}}+\frac{x (b c-a d)}{2 a b \left(a+b x^3\right)^{2/3}}",1,"(-(a*d*x) + (b*c + a*d)*x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 5/3, 4/3, -((b*x^3)/a)])/(a*b*(a + b*x^3)^(2/3))","A",1
69,1,75,94,0.0440138,"\int \frac{c+d x^3}{\left(a+b x^3\right)^{8/3}} \, dx","Integrate[(c + d*x^3)/(a + b*x^3)^(8/3),x]","\frac{x \left(\frac{\left(a+b x^3\right) \left(\frac{b x^3}{a}+1\right)^{2/3} (a d+4 b c) \, _2F_1\left(\frac{1}{3},\frac{8}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{a^2}-d\right)}{4 b \left(a+b x^3\right)^{5/3}}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} (a d+4 b c) \, _2F_1\left(\frac{1}{3},\frac{5}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{5 a^2 b \left(a+b x^3\right)^{2/3}}+\frac{x (b c-a d)}{5 a b \left(a+b x^3\right)^{5/3}}",1,"(x*(-d + ((4*b*c + a*d)*(a + b*x^3)*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 8/3, 4/3, -((b*x^3)/a)])/a^2))/(4*b*(a + b*x^3)^(5/3))","A",1
70,1,238,262,5.1901283,"\int \left(a+b x^3\right)^{5/3} \left(c+d x^3\right)^2 \, dx","Integrate[(a + b*x^3)^(5/3)*(c + d*x^3)^2,x]","\frac{10 a^2 \left(a^2 d^2-6 a b c d+27 b^2 c^2\right) \left(\log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)\right)+3 \sqrt[3]{b} x \left(a+b x^3\right)^{2/3} \left(-20 a^3 d^2+15 a^2 b d \left(8 c+d x^3\right)+18 a b^2 \left(24 c^2+22 c d x^3+7 d^2 x^6\right)+27 b^3 x^3 \left(6 c^2+8 c d x^3+3 d^2 x^6\right)\right)}{2916 b^{7/3}}","\frac{x \left(a+b x^3\right)^{5/3} \left(a^2 d^2-6 a b c d+27 b^2 c^2\right)}{162 b^2}+\frac{5 a x \left(a+b x^3\right)^{2/3} \left(a^2 d^2-6 a b c d+27 b^2 c^2\right)}{486 b^2}-\frac{5 a^2 \left(a^2 d^2-6 a b c d+27 b^2 c^2\right) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{486 b^{7/3}}+\frac{5 a^2 \left(a^2 d^2-6 a b c d+27 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)}{243 \sqrt{3} b^{7/3}}+\frac{d x \left(a+b x^3\right)^{8/3} (15 b c-4 a d)}{108 b^2}+\frac{d x \left(a+b x^3\right)^{8/3} \left(c+d x^3\right)}{12 b}",1,"(3*b^(1/3)*x*(a + b*x^3)^(2/3)*(-20*a^3*d^2 + 15*a^2*b*d*(8*c + d*x^3) + 27*b^3*x^3*(6*c^2 + 8*c*d*x^3 + 3*d^2*x^6) + 18*a*b^2*(24*c^2 + 22*c*d*x^3 + 7*d^2*x^6)) + 10*a^2*(27*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/(2916*b^(7/3))","A",1
71,1,203,219,5.1708362,"\int \left(a+b x^3\right)^{2/3} \left(c+d x^3\right)^2 \, dx","Integrate[(a + b*x^3)^(2/3)*(c + d*x^3)^2,x]","\frac{3 \sqrt[3]{b} x \left(a+b x^3\right)^{2/3} \left(-4 a^2 d^2+3 a b d \left(6 c+d x^3\right)+9 b^2 \left(3 c^2+3 c d x^3+d^2 x^6\right)\right)+a \left(2 a^2 d^2-9 a b c d+27 b^2 c^2\right) \left(\log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)\right)}{243 b^{7/3}}","\frac{x \left(a+b x^3\right)^{2/3} \left(2 a^2 d^2-9 a b c d+27 b^2 c^2\right)}{81 b^2}-\frac{a \left(2 a^2 d^2-9 a b c d+27 b^2 c^2\right) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{81 b^{7/3}}+\frac{2 a \left(2 a^2 d^2-9 a b c d+27 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)}{81 \sqrt{3} b^{7/3}}+\frac{2 d x \left(a+b x^3\right)^{5/3} (3 b c-a d)}{27 b^2}+\frac{d x \left(a+b x^3\right)^{5/3} \left(c+d x^3\right)}{9 b}",1,"(3*b^(1/3)*x*(a + b*x^3)^(2/3)*(-4*a^2*d^2 + 3*a*b*d*(6*c + d*x^3) + 9*b^2*(3*c^2 + 3*c*d*x^3 + d^2*x^6)) + a*(27*b^2*c^2 - 9*a*b*c*d + 2*a^2*d^2)*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/(243*b^(7/3))","A",1
72,1,172,175,5.1504212,"\int \frac{\left(c+d x^3\right)^2}{\sqrt[3]{a+b x^3}} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3)^(1/3),x]","\frac{\left(2 a^2 d^2-6 a b c d+9 b^2 c^2\right) \left(\log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)\right)+3 \sqrt[3]{b} d x \left(a+b x^3\right)^{2/3} \left(3 b \left(4 c+d x^3\right)-4 a d\right)}{54 b^{7/3}}","-\frac{\left(2 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^{7/3}}+\frac{\left(2 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^{7/3}}+\frac{d x \left(a+b x^3\right)^{2/3} (9 b c-4 a d)}{18 b^2}+\frac{d x \left(a+b x^3\right)^{2/3} \left(c+d x^3\right)}{6 b}",1,"(3*b^(1/3)*d*x*(a + b*x^3)^(2/3)*(-4*a*d + 3*b*(4*c + d*x^3)) + (9*b^2*c^2 - 6*a*b*c*d + 2*a^2*d^2)*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/(54*b^(7/3))","A",1
73,1,168,159,5.163413,"\int \frac{\left(c+d x^3\right)^2}{\left(a+b x^3\right)^{4/3}} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3)^(4/3),x]","\frac{d (3 b c-2 a d) \left(\log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)\right)}{9 b^{7/3}}+\frac{x \left(a+b x^3\right)^{2/3} \left(\frac{3 (b c-a d)^2}{a \left(a+b x^3\right)}+d^2\right)}{3 b^2}","-\frac{d (3 b c-2 a d) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{3 b^{7/3}}+\frac{2 d (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} b^{7/3}}-\frac{d x \left(a+b x^3\right)^{2/3} (3 b c-4 a d)}{3 a b^2}+\frac{x \left(c+d x^3\right) (b c-a d)}{a b \sqrt[3]{a+b x^3}}",1,"(x*(a + b*x^3)^(2/3)*(d^2 + (3*(b*c - a*d)^2)/(a*(a + b*x^3))))/(3*b^2) + (d*(3*b*c - 2*a*d)*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/(9*b^(7/3))","A",1
74,1,180,152,5.2471784,"\int \frac{\left(c+d x^3\right)^2}{\left(a+b x^3\right)^{7/3}} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3)^(7/3),x]","\frac{x \left(\left(a+b x^3\right) \left(-5 a^2 d^2+2 a b c d+3 b^2 c^2\right)+a (b c-a d)^2\right)}{4 a^2 b^2 \left(a+b x^3\right)^{4/3}}+\frac{d^2 \left(\log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)\right)}{6 b^{7/3}}","\frac{x (b c-a d) (4 a d+3 b c)}{4 a^2 b^2 \sqrt[3]{a+b x^3}}-\frac{d^2 \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 b^{7/3}}+\frac{d^2 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{7/3}}+\frac{x \left(c+d x^3\right) (b c-a d)}{4 a b \left(a+b x^3\right)^{4/3}}",1,"(x*(a*(b*c - a*d)^2 + (3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*(a + b*x^3)))/(4*a^2*b^2*(a + b*x^3)^(4/3)) + (d^2*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/(6*b^(7/3))","A",1
75,1,126,78,0.0973305,"\int \frac{\left(c+d x^3\right)^2}{\left(a+b x^3\right)^{10/3}} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3)^(10/3),x]","\frac{x \sqrt[3]{\frac{b x^3}{a}+1} \left(a^2 \left(14 c^2+7 c d x^3+2 d^2 x^6\right)+3 a b c x^3 \left(7 c+d x^3\right)+9 b^2 c^2 x^6\right)}{14 a^3 \left(a+b x^3\right)^{7/3} \sqrt[3]{\frac{d x^3}{c}+1} \sqrt[3]{\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}}}","\frac{9 c^2 x}{14 a^3 \sqrt[3]{a+b x^3}}+\frac{3 c x \left(c+d x^3\right)}{14 a^2 \left(a+b x^3\right)^{4/3}}+\frac{x \left(c+d x^3\right)^2}{7 a \left(a+b x^3\right)^{7/3}}",1,"(x*(1 + (b*x^3)/a)^(1/3)*(9*b^2*c^2*x^6 + 3*a*b*c*x^3*(7*c + d*x^3) + a^2*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6)))/(14*a^3*(a + b*x^3)^(7/3)*((c*(a + b*x^3))/(a*(c + d*x^3)))^(1/3)*(1 + (d*x^3)/c)^(1/3))","A",0
76,1,106,174,5.106124,"\int \frac{\left(c+d x^3\right)^2}{\left(a+b x^3\right)^{13/3}} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3)^(13/3),x]","\frac{x \left(10 a^3 \left(14 c^2+7 c d x^3+2 d^2 x^6\right)+3 a^2 b x^3 \left(105 c^2+20 c d x^3+2 d^2 x^6\right)+18 a b^2 c x^6 \left(15 c+d x^3\right)+81 b^3 c^2 x^9\right)}{140 a^4 \left(a+b x^3\right)^{10/3}}","\frac{9 c^2 x (9 b c-10 a d)}{140 a^4 \sqrt[3]{a+b x^3} (b c-a d)}+\frac{3 c x \left(c+d x^3\right) (9 b c-10 a d)}{140 a^3 \left(a+b x^3\right)^{4/3} (b c-a d)}+\frac{x \left(c+d x^3\right)^2 (9 b c-10 a d)}{70 a^2 \left(a+b x^3\right)^{7/3} (b c-a d)}+\frac{b x \left(c+d x^3\right)^3}{10 a \left(a+b x^3\right)^{10/3} (b c-a d)}",1,"(x*(81*b^3*c^2*x^9 + 18*a*b^2*c*x^6*(15*c + d*x^3) + 10*a^3*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6) + 3*a^2*b*x^3*(105*c^2 + 20*c*d*x^3 + 2*d^2*x^6)))/(140*a^4*(a + b*x^3)^(10/3))","A",1
77,1,138,211,5.175785,"\int \frac{\left(c+d x^3\right)^2}{\left(a+b x^3\right)^{16/3}} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3)^(16/3),x]","\frac{x \left(65 a^4 \left(14 c^2+7 c d x^3+2 d^2 x^6\right)+39 a^3 b x^3 \left(70 c^2+15 c d x^3+2 d^2 x^6\right)+9 a^2 b^2 x^6 \left(390 c^2+39 c d x^3+2 d^2 x^6\right)+81 a b^3 c x^9 \left(26 c+d x^3\right)+486 b^4 c^2 x^{12}\right)}{910 a^5 \left(a+b x^3\right)^{13/3}}","\frac{2 x (b c-a d) (a d+3 b c)}{65 a^2 b^2 \left(a+b x^3\right)^{10/3}}+\frac{9 x \left(2 a^2 d^2+9 a b c d+54 b^2 c^2\right)}{910 a^5 b^2 \sqrt[3]{a+b x^3}}+\frac{3 x \left(2 a^2 d^2+9 a b c d+54 b^2 c^2\right)}{910 a^4 b^2 \left(a+b x^3\right)^{4/3}}+\frac{x \left(2 a^2 d^2+9 a b c d+54 b^2 c^2\right)}{455 a^3 b^2 \left(a+b x^3\right)^{7/3}}+\frac{x \left(c+d x^3\right) (b c-a d)}{13 a b \left(a+b x^3\right)^{13/3}}",1,"(x*(486*b^4*c^2*x^12 + 81*a*b^3*c*x^9*(26*c + d*x^3) + 65*a^4*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6) + 39*a^3*b*x^3*(70*c^2 + 15*c*d*x^3 + 2*d^2*x^6) + 9*a^2*b^2*x^6*(390*c^2 + 39*c*d*x^3 + 2*d^2*x^6)))/(910*a^5*(a + b*x^3)^(13/3))","A",1
78,1,169,253,5.1532971,"\int \frac{\left(c+d x^3\right)^2}{\left(a+b x^3\right)^{19/3}} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3)^(19/3),x]","\frac{x \left(520 a^5 \left(14 c^2+7 c d x^3+2 d^2 x^6\right)+156 a^4 b x^3 \left(175 c^2+40 c d x^3+6 d^2 x^6\right)+144 a^3 b^2 x^6 \left(325 c^2+39 c d x^3+3 d^2 x^6\right)+81 a^2 b^3 x^9 \left(520 c^2+32 c d x^3+d^2 x^6\right)+486 a b^4 c x^{12} \left(40 c+d x^3\right)+3645 b^5 c^2 x^{15}\right)}{7280 a^6 \left(a+b x^3\right)^{16/3}}","\frac{x (b c-a d) (4 a d+15 b c)}{208 a^2 b^2 \left(a+b x^3\right)^{13/3}}+\frac{81 x \left(a^2 d^2+6 a b c d+45 b^2 c^2\right)}{7280 a^6 b^2 \sqrt[3]{a+b x^3}}+\frac{27 x \left(a^2 d^2+6 a b c d+45 b^2 c^2\right)}{7280 a^5 b^2 \left(a+b x^3\right)^{4/3}}+\frac{9 x \left(a^2 d^2+6 a b c d+45 b^2 c^2\right)}{3640 a^4 b^2 \left(a+b x^3\right)^{7/3}}+\frac{x \left(a^2 d^2+6 a b c d+45 b^2 c^2\right)}{520 a^3 b^2 \left(a+b x^3\right)^{10/3}}+\frac{x \left(c+d x^3\right) (b c-a d)}{16 a b \left(a+b x^3\right)^{16/3}}",1,"(x*(3645*b^5*c^2*x^15 + 486*a*b^4*c*x^12*(40*c + d*x^3) + 81*a^2*b^3*x^9*(520*c^2 + 32*c*d*x^3 + d^2*x^6) + 520*a^5*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6) + 144*a^3*b^2*x^6*(325*c^2 + 39*c*d*x^3 + 3*d^2*x^6) + 156*a^4*b*x^3*(175*c^2 + 40*c*d*x^3 + 6*d^2*x^6)))/(7280*a^6*(a + b*x^3)^(16/3))","A",1
79,1,177,135,3.5871535,"\int \left(a+b x^3\right)^{7/3} \left(c+d x^3\right)^2 \, dx","Integrate[(a + b*x^3)^(7/3)*(c + d*x^3)^2,x]","\frac{a x \sqrt[3]{a+b x^3} \left(-9 b x^3 \Gamma \left(-\frac{4}{3}\right) \left(c+d x^3\right)^2 \, _3F_2\left(-\frac{4}{3},\frac{4}{3},2;1,\frac{13}{3};-\frac{b x^3}{a}\right)-3 b x^3 \Gamma \left(-\frac{4}{3}\right) \left(11 c^2+16 c d x^3+5 d^2 x^6\right) \, _2F_1\left(-\frac{4}{3},\frac{4}{3};\frac{13}{3};-\frac{b x^3}{a}\right)+20 a \Gamma \left(-\frac{7}{3}\right) \left(14 c^2+7 c d x^3+2 d^2 x^6\right) \, _2F_1\left(-\frac{7}{3},\frac{1}{3};\frac{10}{3};-\frac{b x^3}{a}\right)\right)}{280 \Gamma \left(-\frac{7}{3}\right) \sqrt[3]{\frac{b x^3}{a}+1}}","\frac{a^2 x \sqrt[3]{a+b x^3} \left(2 a^2 d^2-14 a b c d+77 b^2 c^2\right) \, _2F_1\left(-\frac{7}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{77 b^2 \sqrt[3]{\frac{b x^3}{a}+1}}+\frac{d x \left(a+b x^3\right)^{10/3} (17 b c-4 a d)}{154 b^2}+\frac{d x \left(a+b x^3\right)^{10/3} \left(c+d x^3\right)}{14 b}",1,"(a*x*(a + b*x^3)^(1/3)*(20*a*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6)*Gamma[-7/3]*Hypergeometric2F1[-7/3, 1/3, 10/3, -((b*x^3)/a)] - 3*b*x^3*(11*c^2 + 16*c*d*x^3 + 5*d^2*x^6)*Gamma[-4/3]*Hypergeometric2F1[-4/3, 4/3, 13/3, -((b*x^3)/a)] - 9*b*x^3*(c + d*x^3)^2*Gamma[-4/3]*HypergeometricPFQ[{-4/3, 4/3, 2}, {1, 13/3}, -((b*x^3)/a)]))/(280*(1 + (b*x^3)/a)^(1/3)*Gamma[-7/3])","A",0
80,1,171,133,5.1888067,"\int \left(a+b x^3\right)^{4/3} \left(c+d x^3\right)^2 \, dx","Integrate[(a + b*x^3)^(4/3)*(c + d*x^3)^2,x]","\frac{x \left(2 a^2 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(2 a^2 d^2-11 a b c d+44 b^2 c^2\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)-\left(a+b x^3\right) \left(4 a^3 d^2-2 a^2 b d \left(11 c+d x^3\right)-3 a b^2 \left(44 c^2+33 c d x^3+10 d^2 x^6\right)-b^3 x^3 \left(44 c^2+55 c d x^3+20 d^2 x^6\right)\right)\right)}{220 b^2 \left(a+b x^3\right)^{2/3}}","\frac{a x \sqrt[3]{a+b x^3} \left(2 a^2 d^2-11 a b c d+44 b^2 c^2\right) \, _2F_1\left(-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{44 b^2 \sqrt[3]{\frac{b x^3}{a}+1}}+\frac{d x \left(a+b x^3\right)^{7/3} (7 b c-2 a d)}{44 b^2}+\frac{d x \left(a+b x^3\right)^{7/3} \left(c+d x^3\right)}{11 b}",1,"(x*(-((a + b*x^3)*(4*a^3*d^2 - 2*a^2*b*d*(11*c + d*x^3) - 3*a*b^2*(44*c^2 + 33*c*d*x^3 + 10*d^2*x^6) - b^3*x^3*(44*c^2 + 55*c*d*x^3 + 20*d^2*x^6))) + 2*a^2*(44*b^2*c^2 - 11*a*b*c*d + 2*a^2*d^2)*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)]))/(220*b^2*(a + b*x^3)^(2/3))","A",1
81,1,179,131,3.9969319,"\int \sqrt[3]{a+b x^3} \left(c+d x^3\right)^2 \, dx","Integrate[(a + b*x^3)^(1/3)*(c + d*x^3)^2,x]","\frac{x \sqrt[3]{a+b x^3} \left(-9 b x^3 \Gamma \left(\frac{2}{3}\right) \left(c+d x^3\right)^2 \, _3F_2\left(\frac{2}{3},\frac{4}{3},2;1,\frac{13}{3};-\frac{b x^3}{a}\right)-3 b x^3 \Gamma \left(\frac{2}{3}\right) \left(11 c^2+16 c d x^3+5 d^2 x^6\right) \, _2F_1\left(\frac{2}{3},\frac{4}{3};\frac{13}{3};-\frac{b x^3}{a}\right)+20 a \Gamma \left(-\frac{1}{3}\right) \left(14 c^2+7 c d x^3+2 d^2 x^6\right) \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{10}{3};-\frac{b x^3}{a}\right)\right)}{280 a \Gamma \left(-\frac{1}{3}\right) \sqrt[3]{\frac{b x^3}{a}+1}}","\frac{x \sqrt[3]{a+b x^3} \left(a^2 d^2-4 a b c d+10 b^2 c^2\right) \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{10 b^2 \sqrt[3]{\frac{b x^3}{a}+1}}+\frac{d x \left(a+b x^3\right)^{4/3} (11 b c-4 a d)}{40 b^2}+\frac{d x \left(a+b x^3\right)^{4/3} \left(c+d x^3\right)}{8 b}",1,"(x*(a + b*x^3)^(1/3)*(20*a*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6)*Gamma[-1/3]*Hypergeometric2F1[-1/3, 1/3, 10/3, -((b*x^3)/a)] - 3*b*x^3*(11*c^2 + 16*c*d*x^3 + 5*d^2*x^6)*Gamma[2/3]*Hypergeometric2F1[2/3, 4/3, 13/3, -((b*x^3)/a)] - 9*b*x^3*(c + d*x^3)^2*Gamma[2/3]*HypergeometricPFQ[{2/3, 4/3, 2}, {1, 13/3}, -((b*x^3)/a)]))/(280*a*(1 + (b*x^3)/a)^(1/3)*Gamma[-1/3])","A",0
82,1,104,132,5.1351495,"\int \frac{\left(c+d x^3\right)^2}{\left(a+b x^3\right)^{2/3}} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3)^(2/3),x]","\frac{x \left(\left(\frac{b x^3}{a}+1\right)^{2/3} \left(2 a^2 d^2-5 a b c d+5 b^2 c^2\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)-d \left(a+b x^3\right) \left(2 a d-b \left(5 c+d x^3\right)\right)\right)}{5 b^2 \left(a+b x^3\right)^{2/3}}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} \left(2 a^2 d^2-5 a b c d+5 b^2 c^2\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{5 b^2 \left(a+b x^3\right)^{2/3}}+\frac{2 d x \sqrt[3]{a+b x^3} (2 b c-a d)}{5 b^2}+\frac{d x \sqrt[3]{a+b x^3} \left(c+d x^3\right)}{5 b}",1,"(x*(-(d*(a + b*x^3)*(2*a*d - b*(5*c + d*x^3))) + (5*b^2*c^2 - 5*a*b*c*d + 2*a^2*d^2)*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)]))/(5*b^2*(a + b*x^3)^(2/3))","A",1
83,1,171,146,3.7627266,"\int \frac{\left(c+d x^3\right)^2}{\left(a+b x^3\right)^{5/3}} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3)^(5/3),x]","\frac{x \Gamma \left(\frac{2}{3}\right) \left(\frac{b x^3}{a}+1\right)^{2/3} \left(-3 b x^3 \left(c+d x^3\right)^2 \, _3F_2\left(\frac{4}{3},2,\frac{8}{3};1,\frac{13}{3};-\frac{b x^3}{a}\right)-b x^3 \left(11 c^2+16 c d x^3+5 d^2 x^6\right) \, _2F_1\left(\frac{4}{3},\frac{8}{3};\frac{13}{3};-\frac{b x^3}{a}\right)+4 a \left(14 c^2+7 c d x^3+2 d^2 x^6\right) \, _2F_1\left(\frac{1}{3},\frac{5}{3};\frac{10}{3};-\frac{b x^3}{a}\right)\right)}{84 a^2 \Gamma \left(\frac{5}{3}\right) \left(a+b x^3\right)^{2/3}}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} \left(-2 a^2 d^2+2 a b c d+b^2 c^2\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 a b^2 \left(a+b x^3\right)^{2/3}}-\frac{d x \sqrt[3]{a+b x^3} (b c-2 a d)}{2 a b^2}+\frac{x \left(c+d x^3\right) (b c-a d)}{2 a b \left(a+b x^3\right)^{2/3}}",1,"(x*(1 + (b*x^3)/a)^(2/3)*Gamma[2/3]*(4*a*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6)*Hypergeometric2F1[1/3, 5/3, 10/3, -((b*x^3)/a)] - b*x^3*(11*c^2 + 16*c*d*x^3 + 5*d^2*x^6)*Hypergeometric2F1[4/3, 8/3, 13/3, -((b*x^3)/a)] - 3*b*x^3*(c + d*x^3)^2*HypergeometricPFQ[{4/3, 2, 8/3}, {1, 13/3}, -((b*x^3)/a)]))/(84*a^2*(a + b*x^3)^(2/3)*Gamma[5/3])","A",0
84,1,128,147,5.1623998,"\int \frac{\left(c+d x^3\right)^2}{\left(a+b x^3\right)^{8/3}} \, dx","Integrate[(c + d*x^3)^2/(a + b*x^3)^(8/3),x]","\frac{x \left(\left(a+b x^3\right) \left(\frac{b x^3}{a}+1\right)^{2/3} \left(2 a^2 d^2+a b c d+2 b^2 c^2\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+\left(a+b x^3\right) \left(-3 a^2 d^2+a b c d+2 b^2 c^2\right)+a (b c-a d)^2\right)}{5 a^2 b^2 \left(a+b x^3\right)^{5/3}}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} \left(2 a^2 d^2+a b c d+2 b^2 c^2\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{5 a^2 b^2 \left(a+b x^3\right)^{2/3}}+\frac{2 x \left(\frac{c^2}{a^2}-\frac{d^2}{b^2}\right)}{5 \left(a+b x^3\right)^{2/3}}+\frac{x \left(c+d x^3\right) (b c-a d)}{5 a b \left(a+b x^3\right)^{5/3}}",1,"(x*(a*(b*c - a*d)^2 + (2*b^2*c^2 + a*b*c*d - 3*a^2*d^2)*(a + b*x^3) + (2*b^2*c^2 + a*b*c*d + 2*a^2*d^2)*(a + b*x^3)*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)]))/(5*a^2*b^2*(a + b*x^3)^(5/3))","A",1
85,1,120,109,0.0434592,"\int \frac{\left(a+b x^3\right)^3}{\left(c+d x^3\right)^{13/3}} \, dx","Integrate[(a + b*x^3)^3/(c + d*x^3)^(13/3),x]","\frac{x \left(a^3 \left(140 c^3+315 c^2 d x^3+270 c d^2 x^6+81 d^3 x^9\right)+3 a^2 b c x^3 \left(35 c^2+30 c d x^3+9 d^2 x^6\right)+6 a b^2 c^2 x^6 \left(10 c+3 d x^3\right)+14 b^3 c^3 x^9\right)}{140 c^4 \left(c+d x^3\right)^{10/3}}","\frac{81 a^3 x}{140 c^4 \sqrt[3]{c+d x^3}}+\frac{27 a^2 x \left(a+b x^3\right)}{140 c^3 \left(c+d x^3\right)^{4/3}}+\frac{9 a x \left(a+b x^3\right)^2}{70 c^2 \left(c+d x^3\right)^{7/3}}+\frac{x \left(a+b x^3\right)^3}{10 c \left(c+d x^3\right)^{10/3}}",1,"(x*(14*b^3*c^3*x^9 + 6*a*b^2*c^2*x^6*(10*c + 3*d*x^3) + 3*a^2*b*c*x^3*(35*c^2 + 30*c*d*x^3 + 9*d^2*x^6) + a^3*(140*c^3 + 315*c^2*d*x^3 + 270*c*d^2*x^6 + 81*d^3*x^9)))/(140*c^4*(c + d*x^3)^(10/3))","A",1
86,1,655,331,1.2322042,"\int \frac{\left(a+b x^3\right)^{8/3}}{c+d x^3} \, dx","Integrate[(a + b*x^3)^(8/3)/(c + d*x^3),x]","\frac{3 b x^4 \sqrt[3]{\frac{b x^3}{a}+1} \sqrt[3]{b c-a d} \left(20 a^2 d^2-24 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 \sqrt[3]{c} \left(9 a^3 d^2 \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 a \sqrt[3]{a+b x^3} \left(9 a^2 d^2-7 a b c d+3 b^2 c^2\right) \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]{a+b x^3} \left(9 a^2 d^2-7 a b c d+3 b^2 c^2\right) \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)-7 a^2 b 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)+42 a^2 b c^{2/3} d x \sqrt[3]{b c-a d}+9 b^3 c^{2/3} d x^7 \sqrt[3]{b c-a d}-18 b^3 c^{5/3} x^4 \sqrt[3]{b c-a d}+51 a b^2 c^{2/3} d x^4 \sqrt[3]{b c-a d}-18 a b^2 c^{5/3} x \sqrt[3]{b c-a d}+3 a b^2 c^2 \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)\right)}{108 c d^2 \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}}","-\frac{b^{2/3} \left(20 a^2 d^2-24 a b c d+9 b^2 c^2\right) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{18 d^3}+\frac{b^{2/3} \left(20 a^2 d^2-24 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} d^3}-\frac{(b c-a d)^{8/3} \log \left(c+d x^3\right)}{6 c^{2/3} d^3}+\frac{(b c-a d)^{8/3} \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^3}-\frac{(b c-a d)^{8/3} \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^3}-\frac{b x \left(a+b x^3\right)^{2/3} (6 b c-11 a d)}{18 d^2}+\frac{b x \left(a+b x^3\right)^{5/3}}{6 d}",1,"(3*b*(b*c - a*d)^(1/3)*(9*b^2*c^2 - 24*a*b*c*d + 20*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^(1/3)*(-18*a*b^2*c^(5/3)*(b*c - a*d)^(1/3)*x + 42*a^2*b*c^(2/3)*d*(b*c - a*d)^(1/3)*x - 18*b^3*c^(5/3)*(b*c - a*d)^(1/3)*x^4 + 51*a*b^2*c^(2/3)*d*(b*c - a*d)^(1/3)*x^4 + 9*b^3*c^(2/3)*d*(b*c - a*d)^(1/3)*x^7 + 2*Sqrt[3]*a*(3*b^2*c^2 - 7*a*b*c*d + 9*a^2*d^2)*(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*(3*b^2*c^2 - 7*a*b*c*d + 9*a^2*d^2)*(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^2*c^2*(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)] - 7*a^2*b*c*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)] + 9*a^3*d^2*(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*c*d^2*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3))","C",0
87,1,443,273,0.7107832,"\int \frac{\left(a+b x^3\right)^{5/3}}{c+d x^3} \, dx","Integrate[(a + b*x^3)^(5/3)/(c + d*x^3),x]","\frac{2 \sqrt[3]{c} \left(3 a^2 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 b^2 c^{2/3} x^4 \sqrt[3]{b c-a d}-a b 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)+6 a b c^{2/3} x \sqrt[3]{b c-a d}+2 a \sqrt[3]{a+b x^3} (b c-3 a d) \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]{a+b x^3} (3 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)\right)+3 b x^4 \sqrt[3]{\frac{b x^3}{a}+1} \sqrt[3]{b c-a d} (5 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)}{36 c d \sqrt[3]{a+b x^3} \sqrt[3]{b c-a d}}","\frac{b^{2/3} (3 b c-5 a d) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{6 d^2}-\frac{b^{2/3} (3 b c-5 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)^{5/3} \log \left(c+d x^3\right)}{6 c^{2/3} d^2}-\frac{(b c-a d)^{5/3} \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^2}+\frac{(b c-a d)^{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} c^{2/3} d^2}+\frac{b x \left(a+b x^3\right)^{2/3}}{3 d}",1,"(3*b*(b*c - a*d)^(1/3)*(-3*b*c + 5*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)] + 2*c^(1/3)*(6*a*b*c^(2/3)*(b*c - a*d)^(1/3)*x + 6*b^2*c^(2/3)*(b*c - a*d)^(1/3)*x^4 + 2*Sqrt[3]*a*(-(b*c) + 3*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*(b*c - 3*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*(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)] + 3*a^2*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)]))/(36*c*d*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3))","C",0
88,1,161,233,0.2238404,"\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
89,1,168,148,0.1510599,"\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
90,1,256,179,0.8009513,"\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
91,1,621,226,2.6728503,"\int \frac{1}{\left(a+b x^3\right)^{7/3} \left(c+d x^3\right)} \, dx","Integrate[1/((a + b*x^3)^(7/3)*(c + d*x^3)),x]","\frac{9 c^2 x^9 (b c-a d)^3 \, _3F_2\left(2,2,\frac{10}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+9 d^2 x^{15} (b c-a d)^3 \, _3F_2\left(2,2,\frac{10}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+18 c d x^{12} (b c-a d)^3 \, _3F_2\left(2,2,\frac{10}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+280 c^5 \left(a+b x^3\right)^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)-280 c^5 \left(a+b x^3\right)^3+420 c^4 d x^3 \left(a+b x^3\right)^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)-420 c^4 d x^3 \left(a+b x^3\right)^3-70 c^4 x^3 \left(a+b x^3\right)^2 (b c-a d)+180 c^3 d^2 x^6 \left(a+b x^3\right)^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)-180 c^3 d^2 x^6 \left(a+b x^3\right)^3-105 c^3 d x^6 \left(a+b x^3\right)^2 (b c-a d)-45 c^2 d^2 x^9 \left(a+b x^3\right)^2 (b c-a d)+33 c^2 x^9 (b c-a d)^3 \, _2F_1\left(2,\frac{10}{3};\frac{13}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+27 d^2 x^{15} (b c-a d)^3 \, _2F_1\left(2,\frac{10}{3};\frac{13}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+60 c d x^{12} (b c-a d)^3 \, _2F_1\left(2,\frac{10}{3};\frac{13}{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)^{10/3} (b c-a d)^2}","\frac{b x (3 b c-7 a d)}{4 a^2 \sqrt[3]{a+b x^3} (b c-a d)^2}+\frac{d^2 \log \left(c+d x^3\right)}{6 c^{2/3} (b c-a d)^{7/3}}-\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^{2/3} (b c-a d)^{7/3}}+\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^{2/3} (b c-a d)^{7/3}}+\frac{b x}{4 a \left(a+b x^3\right)^{4/3} (b c-a d)}",1,"(-70*c^4*(b*c - a*d)*x^3*(a + b*x^3)^2 - 105*c^3*d*(b*c - a*d)*x^6*(a + b*x^3)^2 - 45*c^2*d^2*(b*c - a*d)*x^9*(a + b*x^3)^2 - 280*c^5*(a + b*x^3)^3 - 420*c^4*d*x^3*(a + b*x^3)^3 - 180*c^3*d^2*x^6*(a + b*x^3)^3 + 280*c^5*(a + b*x^3)^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 420*c^4*d*x^3*(a + b*x^3)^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 180*c^3*d^2*x^6*(a + b*x^3)^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 33*c^2*(b*c - a*d)^3*x^9*Hypergeometric2F1[2, 10/3, 13/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 60*c*d*(b*c - a*d)^3*x^12*Hypergeometric2F1[2, 10/3, 13/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*d^2*(b*c - a*d)^3*x^15*Hypergeometric2F1[2, 10/3, 13/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 9*c^2*(b*c - a*d)^3*x^9*HypergeometricPFQ[{2, 2, 10/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 18*c*d*(b*c - a*d)^3*x^12*HypergeometricPFQ[{2, 2, 10/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 9*d^2*(b*c - a*d)^3*x^15*HypergeometricPFQ[{2, 2, 10/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(40*c^4*(b*c - a*d)^2*x^5*(a + b*x^3)^(10/3))","C",0
92,1,277,280,5.7462422,"\int \frac{1}{\left(a+b x^3\right)^{10/3} \left(c+d x^3\right)} \, dx","Integrate[1/((a + b*x^3)^(10/3)*(c + d*x^3)),x]","\frac{b x \left(\left(a+b x^3\right)^2 \left(67 a^2 d^2-57 a b c d+18 b^2 c^2\right)+4 a^2 (b c-a d)^2+a \left(a+b x^3\right) (a d-b c) (13 a d-6 b c)\right)}{28 a^3 \left(a+b x^3\right)^{7/3} (b c-a d)^3}-\frac{d^3 \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^{2/3} (b c-a d)^{10/3}}","\frac{b x (6 b c-13 a d)}{28 a^2 \left(a+b x^3\right)^{4/3} (b c-a d)^2}+\frac{b x \left(67 a^2 d^2-57 a b c d+18 b^2 c^2\right)}{28 a^3 \sqrt[3]{a+b x^3} (b c-a d)^3}-\frac{d^3 \log \left(c+d x^3\right)}{6 c^{2/3} (b c-a d)^{10/3}}+\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^{2/3} (b c-a d)^{10/3}}-\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^{2/3} (b c-a d)^{10/3}}+\frac{b x}{7 a \left(a+b x^3\right)^{7/3} (b c-a d)}",1,"(b*x*(4*a^2*(b*c - a*d)^2 + a*(-(b*c) + a*d)*(-6*b*c + 13*a*d)*(a + b*x^3) + (18*b^2*c^2 - 57*a*b*c*d + 67*a^2*d^2)*(a + b*x^3)^2))/(28*a^3*(b*c - a*d)^3*(a + b*x^3)^(7/3)) - (d^3*(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^(2/3)*(b*c - a*d)^(10/3))","A",0
93,1,346,60,0.6112192,"\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
94,1,160,59,0.1664344,"\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
95,1,161,59,0.0393926,"\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
96,1,332,62,0.2787875,"\int \frac{1}{\left(a+b x^3\right)^{5/3} \left(c+d x^3\right)} \, dx","Integrate[1/((a + b*x^3)^(5/3)*(c + d*x^3)),x]","\frac{x \left(\frac{4 \left(b x^3 \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 d-b \left(2 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(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)}-\frac{b d 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{d x^3}{c}\right)}{c}\right)}{8 a \left(a+b x^3\right)^{2/3} (a d-b c)}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{1}{3};\frac{5}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a c \left(a+b x^3\right)^{2/3}}",1,"(x*(-((b*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*d - b*(2*c + d*x^3))*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + 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*(-(b*c) + a*d)*(a + b*x^3)^(2/3))","B",0
97,1,429,62,0.8668958,"\int \frac{1}{\left(a+b x^3\right)^{8/3} \left(c+d x^3\right)} \, dx","Integrate[1/((a + b*x^3)^(8/3)*(c + d*x^3)),x]","-\frac{x \left(\frac{4 \left(b x^3 \left(c+d x^3\right) \left(11 a^2 d+a b \left(9 d x^3-6 c\right)-4 b^2 c 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(10 a^3 d^2-a^2 b d \left(20 c+d x^3\right)+a b^2 \left(10 c^2-12 c d x^3-9 d^2 x^6\right)+4 b^3 c x^3 \left(2 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(a+b x^3\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 d x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} (9 a d-4 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)}{40 a^2 \left(a+b x^3\right)^{2/3} (b c-a d)^2}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{1}{3};\frac{8}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2 c \left(a+b x^3\right)^{2/3}}",1,"-1/40*(x*((b*d*(-4*b*c + 9*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*(10*a^3*d^2 + 4*b^3*c*x^3*(2*c + d*x^3) - a^2*b*d*(20*c + d*x^3) + a*b^2*(10*c^2 - 12*c*d*x^3 - 9*d^2*x^6))*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + b*x^3*(c + d*x^3)*(11*a^2*d - 4*b^2*c*x^3 + a*b*(-6*c + 9*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)])))/((a + b*x^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)])))))/(a^2*(b*c - a*d)^2*(a + b*x^3)^(2/3))","B",0
98,1,698,351,1.0833493,"\int \frac{\left(a+b x^3\right)^{8/3}}{\left(c+d x^3\right)^2} \, dx","Integrate[(a + b*x^3)^(8/3)/(c + d*x^3)^2,x]","\frac{1}{18} \left(\frac{2 a^3 \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)}{c^{5/3} \sqrt[3]{b c-a d}}+\frac{2 a^2 b \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)}{c^{2/3} d \sqrt[3]{b c-a d}}-\frac{9 b^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{d x^3}{c}\right)}{d^2 \sqrt[3]{a+b x^3}}+\frac{12 a b^2 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{d x^3}{c}\right)}{c d \sqrt[3]{a+b x^3}}-\frac{2 a b^2 \sqrt[3]{c} \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)}{d^2 \sqrt[3]{b c-a d}}+\frac{6 x \left(a+b x^3\right)^{2/3} \left(\frac{(b c-a d)^2}{c \left(c+d x^3\right)}+b^2\right)}{d^2}\right)","\frac{b^{5/3} (3 b c-4 a d) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{3 d^3}-\frac{2 b^{5/3} (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^3}+\frac{(b c-a d)^{5/3} (a d+3 b c) \log \left(c+d x^3\right)}{9 c^{5/3} d^3}-\frac{(b c-a d)^{5/3} (a d+3 b c) \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{3 c^{5/3} d^3}+\frac{2 (b c-a d)^{5/3} (a d+3 b c) \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c-a d}}{\sqrt[3]{c} \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} c^{5/3} d^3}+\frac{b x \left(a+b x^3\right)^{2/3} (2 b c-a d)}{3 c d^2}-\frac{x \left(a+b x^3\right)^{5/3} (b c-a d)}{3 c d \left(c+d x^3\right)}",1,"((6*x*(a + b*x^3)^(2/3)*(b^2 + (b*c - a*d)^2/(c*(c + d*x^3))))/d^2 - (9*b^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)])/(d^2*(a + b*x^3)^(1/3)) + (12*a*b^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)])/(c*d*(a + b*x^3)^(1/3)) + (2*a^3*(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)]))/(c^(5/3)*(b*c - a*d)^(1/3)) - (2*a*b^2*c^(1/3)*(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)]))/(d^2*(b*c - a*d)^(1/3)) + (2*a^2*b*(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)]))/(c^(2/3)*d*(b*c - a*d)^(1/3)))/18","C",0
99,1,450,301,0.6433409,"\int \frac{\left(a+b x^3\right)^{5/3}}{\left(c+d x^3\right)^2} \, dx","Integrate[(a + b*x^3)^(5/3)/(c + d*x^3)^2,x]","\frac{\frac{4 a^2 \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)}{\sqrt[3]{b c-a d}}+\frac{9 b^2 c^{2/3} 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{d x^3}{c}\right)}{d \sqrt[3]{a+b x^3}}-\frac{12 c^{2/3} x \left(a+b x^3\right)^{2/3} (b c-a d)}{d \left(c+d x^3\right)}+\frac{2 a b c \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)}{d \sqrt[3]{b c-a d}}}{36 c^{5/3}}","-\frac{b^{5/3} \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 d^2}+\frac{b^{5/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^2}-\frac{(b c-a d)^{2/3} (2 a d+3 b c) \log \left(c+d x^3\right)}{18 c^{5/3} d^2}+\frac{(b c-a d)^{2/3} (2 a d+3 b c) \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{6 c^{5/3} d^2}-\frac{(b c-a d)^{2/3} (2 a d+3 b 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)}{3 \sqrt{3} c^{5/3} d^2}-\frac{x \left(a+b x^3\right)^{2/3} (b c-a d)}{3 c d \left(c+d x^3\right)}",1,"((-12*c^(2/3)*(b*c - a*d)*x*(a + b*x^3)^(2/3))/(d*(c + d*x^3)) + (9*b^2*c^(2/3)*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)])/(d*(a + b*x^3)^(1/3)) + (4*a^2*(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)]))/(b*c - a*d)^(1/3) + (2*a*b*c*(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)]))/(d*(b*c - a*d)^(1/3)))/(36*c^(5/3))","C",0
100,1,78,182,0.040754,"\int \frac{\left(a+b x^3\right)^{2/3}}{\left(c+d x^3\right)^2} \, dx","Integrate[(a + b*x^3)^(2/3)/(c + d*x^3)^2,x]","\frac{x \left(a+b x^3\right)^{2/3} \, _2F_1\left(-\frac{2}{3},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{c^2 \left(\frac{b x^3}{a}+1\right)^{2/3} \sqrt[3]{\frac{d x^3}{c}+1}}","\frac{a \log \left(c+d x^3\right)}{9 c^{5/3} \sqrt[3]{b c-a d}}-\frac{a \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{3 c^{5/3} \sqrt[3]{b c-a d}}+\frac{2 a \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)}{3 \sqrt{3} c^{5/3} \sqrt[3]{b c-a d}}+\frac{x \left(a+b x^3\right)^{2/3}}{3 c \left(c+d x^3\right)}",1,"(x*(a + b*x^3)^(2/3)*Hypergeometric2F1[-2/3, 1/3, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c^2*(1 + (b*x^3)/a)^(2/3)*(1 + (d*x^3)/c)^(1/3))","C",0
101,1,99,217,0.1445347,"\int \frac{1}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)^2} \, dx","Integrate[1/((a + b*x^3)^(1/3)*(c + d*x^3)^2),x]","\frac{x \left(\left(c+d x^3\right) (3 b c-2 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)-c d \left(a+b x^3\right)\right)}{3 c^2 \sqrt[3]{a+b x^3} \left(c+d x^3\right) (b c-a d)}","\frac{(3 b c-2 a d) \log \left(c+d x^3\right)}{18 c^{5/3} (b c-a d)^{4/3}}-\frac{(3 b c-2 a d) \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{6 c^{5/3} (b c-a d)^{4/3}}+\frac{(3 b c-2 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)}{3 \sqrt{3} c^{5/3} (b c-a d)^{4/3}}-\frac{d x \left(a+b x^3\right)^{2/3}}{3 c \left(c+d x^3\right) (b c-a d)}",1,"(x*(-(c*d*(a + b*x^3)) + (3*b*c - 2*a*d)*(c + d*x^3)*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]))/(3*c^2*(b*c - a*d)*(a + b*x^3)^(1/3)*(c + d*x^3))","C",1
102,1,625,261,2.0318622,"\int \frac{1}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)^2} \, dx","Integrate[1/((a + b*x^3)^(4/3)*(c + d*x^3)^2),x]","\frac{c \left(a+b x^3\right)^{2/3} \left(\frac{54 d^2 x^{15} (b c-a d)^3 \, _3F_2\left(2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)}{c^5 \left(a+b x^3\right)^3}+\frac{108 d x^{12} (b c-a d)^3 \, _3F_2\left(2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)}{c^4 \left(a+b x^3\right)^3}+\frac{54 x^9 (b c-a d)^3 \, _3F_2\left(2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)}{c^3 \left(a+b x^3\right)^3}+\frac{2520 d^2 x^9 (a d-b c) \, _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)}{c^3 \left(a+b x^3\right)}+\frac{945 d^2 x^9 (b c-a d)}{c^3 \left(a+b x^3\right)}+\frac{6300 d^2 x^6 \, _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)}{c^2}+\frac{5320 d x^6 (a d-b c) \, _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)}{c^2 \left(a+b x^3\right)}+\frac{1890 d x^6 (b c-a d)}{c^2 \left(a+b x^3\right)}+\frac{13720 d x^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)}{c}-\frac{2240 x^3 (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)}{c \left(a+b x^3\right)}+6860 \, _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)+\frac{525 x^3 (b c-a d)}{c \left(a+b x^3\right)}-\frac{6300 d^2 x^6}{c^2}-\frac{13720 d x^3}{c}-6860\right)}{420 x^5 \left(c+d x^3\right) (b c-a d)^2}","-\frac{d (3 b c-a d) \log \left(c+d x^3\right)}{9 c^{5/3} (b c-a d)^{7/3}}+\frac{d (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)}{3 c^{5/3} (b c-a d)^{7/3}}-\frac{2 d (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)}{3 \sqrt{3} c^{5/3} (b c-a d)^{7/3}}+\frac{b x (a d+3 b c)}{3 a c \sqrt[3]{a+b x^3} (b c-a d)^2}-\frac{d x}{3 c \sqrt[3]{a+b x^3} \left(c+d x^3\right) (b c-a d)}",1,"(c*(a + b*x^3)^(2/3)*(-6860 - (13720*d*x^3)/c - (6300*d^2*x^6)/c^2 + (525*(b*c - a*d)*x^3)/(c*(a + b*x^3)) + (1890*d*(b*c - a*d)*x^6)/(c^2*(a + b*x^3)) + (945*d^2*(b*c - a*d)*x^9)/(c^3*(a + b*x^3)) + 6860*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + (13720*d*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/c + (6300*d^2*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/c^2 - (2240*(b*c - a*d)*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c*(a + b*x^3)) + (5320*d*(-(b*c) + a*d)*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^2*(a + b*x^3)) + (2520*d^2*(-(b*c) + a*d)*x^9*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^3*(a + b*x^3)) + (54*(b*c - a*d)^3*x^9*HypergeometricPFQ[{2, 2, 7/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^3*(a + b*x^3)^3) + (108*d*(b*c - a*d)^3*x^12*HypergeometricPFQ[{2, 2, 7/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^4*(a + b*x^3)^3) + (54*d^2*(b*c - a*d)^3*x^15*HypergeometricPFQ[{2, 2, 7/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^5*(a + b*x^3)^3)))/(420*(b*c - a*d)^2*x^5*(c + d*x^3))","C",0
103,1,288,324,5.9443991,"\int \frac{1}{\left(a+b x^3\right)^{7/3} \left(c+d x^3\right)^2} \, dx","Integrate[1/((a + b*x^3)^(7/3)*(c + d*x^3)^2),x]","\frac{1}{36} \left(3 x \left(a+b x^3\right)^{2/3} \left(\frac{3 b^2 (11 a d-3 b c)}{a^2 \left(a+b x^3\right) (a d-b c)^3}+\frac{3 b^2}{a \left(a+b x^3\right)^2 (b c-a d)^2}-\frac{4 d^3}{c \left(c+d x^3\right) (b c-a d)^3}\right)+\frac{2 d^2 (9 b c-2 a d) \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)}{c^{5/3} (b c-a d)^{10/3}}\right)","\frac{b x \left(-4 a^2 d^2-33 a b c d+9 b^2 c^2\right)}{12 a^2 c \sqrt[3]{a+b x^3} (b c-a d)^3}+\frac{d^2 (9 b c-2 a d) \log \left(c+d x^3\right)}{18 c^{5/3} (b c-a d)^{10/3}}-\frac{d^2 (9 b c-2 a d) \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{6 c^{5/3} (b c-a d)^{10/3}}+\frac{d^2 (9 b c-2 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)}{3 \sqrt{3} c^{5/3} (b c-a d)^{10/3}}-\frac{d x}{3 c \left(a+b x^3\right)^{4/3} \left(c+d x^3\right) (b c-a d)}+\frac{b x (4 a d+3 b c)}{12 a c \left(a+b x^3\right)^{4/3} (b c-a d)^2}",1,"(3*x*(a + b*x^3)^(2/3)*((3*b^2)/(a*(b*c - a*d)^2*(a + b*x^3)^2) + (3*b^2*(-3*b*c + 11*a*d))/(a^2*(-(b*c) + a*d)^3*(a + b*x^3)) - (4*d^3)/(c*(b*c - a*d)^3*(c + d*x^3))) + (2*d^2*(9*b*c - 2*a*d)*(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)]))/(c^(5/3)*(b*c - a*d)^(10/3)))/36","A",0
104,1,341,60,0.3233156,"\int \frac{\left(a+b x^3\right)^{4/3}}{\left(c+d x^3\right)^2} \, dx","Integrate[(a + b*x^3)^(4/3)/(c + d*x^3)^2,x]","\frac{x \left(\frac{4 c \left(x^3 \left(a+b x^3\right) (a d-b c) \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(3 a^2 d+a b d x^3-b^2 c x^3\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)}+b 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)\right)}{12 c^2 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},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c^2 \sqrt[3]{\frac{b x^3}{a}+1}}",1,"(x*(b*(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)] + (4*c*(-4*a*c*(3*a^2*d - b^2*c*x^3 + a*b*d*x^3)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + (-(b*c) + a*d)*x^3*(a + b*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)])))))/(12*c^2*d*(a + b*x^3)^(2/3))","B",0
105,1,232,59,0.2106885,"\int \frac{\sqrt[3]{a+b x^3}}{\left(c+d x^3\right)^2} \, dx","Integrate[(a + b*x^3)^(1/3)/(c + d*x^3)^2,x]","\frac{x \left(\frac{4 \left(\frac{a+b x^3}{c}-\frac{8 a^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)}{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)}{c+d x^3}+\frac{b 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{d x^3}{c}\right)}{c^2}\right)}{12 \left(a+b x^3\right)^{2/3}}","\frac{x \sqrt[3]{a+b x^3} F_1\left(\frac{1}{3};-\frac{1}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c^2 \sqrt[3]{\frac{b x^3}{a}+1}}",1,"(x*((b*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^2 + (4*((a + b*x^3)/c - (8*a^2*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(-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 + d*x^3)))/(12*(a + b*x^3)^(2/3))","B",0
106,1,393,59,0.2675988,"\int \frac{1}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)^2} \, dx","Integrate[1/((a + b*x^3)^(2/3)*(c + d*x^3)^2),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(b d x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(c+d x^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)+4 c \left(3 a d-3 b c+b d x^3\right)\right)-d x^4 \left(b x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(c+d x^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)+4 c \left(a+b x^3\right)\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)}{12 c^2 \left(a+b x^3\right)^{2/3} \left(c+d x^3\right) (b c-a d) \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},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c^2 \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)]*(4*c*(-3*b*c + 3*a*d + b*d*x^3) + b*d*x^3*(1 + (b*x^3)/a)^(2/3)*(c + d*x^3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]) - d*x^4*(4*c*(a + b*x^3) + b*x^3*(1 + (b*x^3)/a)^(2/3)*(c + d*x^3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])*(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)]))/(12*c^2*(b*c - a*d)*(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
107,1,386,62,0.5747689,"\int \frac{1}{\left(a+b x^3\right)^{5/3} \left(c+d x^3\right)^2} \, dx","Integrate[1/((a + b*x^3)^(5/3)*(c + d*x^3)^2),x]","\frac{x \left(\frac{c \left(16 a c \left(6 a^2 d^2+2 a b d \left(d x^3-6 c\right)+3 b^2 c \left(2 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)-4 x^3 \left(2 a^2 d^2+2 a b d^2 x^3+3 b^2 c \left(c+d x^3\right)\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(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^3 \left(\frac{b x^3}{a}+1\right)^{2/3} (2 a d+3 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)\right)}{24 a c^2 \left(a+b x^3\right)^{2/3} (b c-a d)^2}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{1}{3};\frac{5}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a c^2 \left(a+b x^3\right)^{2/3}}",1,"(x*(b*d*(3*b*c + 2*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*(16*a*c*(6*a^2*d^2 + 2*a*b*d*(-6*c + d*x^3) + 3*b^2*c*(2*c + d*x^3))*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] - 4*x^3*(2*a^2*d^2 + 2*a*b*d^2*x^3 + 3*b^2*c*(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)])))))/(24*a*c^2*(b*c - a*d)^2*(a + b*x^3)^(2/3))","B",0
108,1,550,62,0.9864063,"\int \frac{1}{\left(a+b x^3\right)^{8/3} \left(c+d x^3\right)^2} \, dx","Integrate[1/((a + b*x^3)^(8/3)*(c + d*x^3)^2),x]","\frac{\frac{b d x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(5 a^2 d^2+21 a b c d-6 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)}{(a d-b c)^3}+\frac{4 c \left(x^4 \left(5 a^4 d^3+10 a^3 b d^3 x^3+a^2 b^2 d \left(24 c^2+24 c d x^3+5 d^2 x^6\right)+3 a b^3 c \left(-3 c^2+4 c d x^3+7 d^2 x^6\right)-6 b^4 c^2 x^3 \left(c+d x^3\right)\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 x \left(15 a^4 d^3+5 a^3 b d^2 \left(4 d x^3-9 c\right)+a^2 b^2 d \left(45 c^2-21 c d x^3+5 d^2 x^6\right)+3 a b^3 c \left(-5 c^2+11 c d x^3+7 d^2 x^6\right)-6 b^4 c^2 x^3 \left(2 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(a+b x^3\right) \left(c+d x^3\right) (b c-a d)^3 \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)}}{60 a^2 c^2 \left(a+b x^3\right)^{2/3}}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{1}{3};\frac{8}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2 c^2 \left(a+b x^3\right)^{2/3}}",1,"((b*d*(-6*b^2*c^2 + 21*a*b*c*d + 5*a^2*d^2)*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)])/(-(b*c) + a*d)^3 + (4*c*(-4*a*c*x*(15*a^4*d^3 - 6*b^4*c^2*x^3*(2*c + d*x^3) + 5*a^3*b*d^2*(-9*c + 4*d*x^3) + a^2*b^2*d*(45*c^2 - 21*c*d*x^3 + 5*d^2*x^6) + 3*a*b^3*c*(-5*c^2 + 11*c*d*x^3 + 7*d^2*x^6))*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^4*(5*a^4*d^3 + 10*a^3*b*d^3*x^3 - 6*b^4*c^2*x^3*(c + d*x^3) + a^2*b^2*d*(24*c^2 + 24*c*d*x^3 + 5*d^2*x^6) + 3*a*b^3*c*(-3*c^2 + 4*c*d*x^3 + 7*d^2*x^6))*(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*c - a*d)^3*(a + b*x^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)]))))/(60*a^2*c^2*(a + b*x^3)^(2/3))","B",0
109,1,1171,541,2.3406986,"\int \frac{\left(a+b x^3\right)^{14/3}}{\left(c+d x^3\right)^3} \, dx","Integrate[(a + b*x^3)^(14/3)/(c + d*x^3)^3,x]","\frac{1}{108} \left(\frac{10 \left(2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c-a d} x}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-2 \log \left(\sqrt[3]{c}-\frac{\sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}\right)+\log \left(\frac{(b c-a d)^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{c} \sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}+c^{2/3}\right)\right) a^5}{c^{8/3} \sqrt[3]{b c-a d}}+\frac{6 b \left(2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c-a d} x}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-2 \log \left(\sqrt[3]{c}-\frac{\sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}\right)+\log \left(\frac{(b c-a d)^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{c} \sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}+c^{2/3}\right)\right) a^4}{c^{5/3} d \sqrt[3]{b c-a d}}+\frac{30 b^2 \left(2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c-a d} x}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-2 \log \left(\sqrt[3]{c}-\frac{\sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}\right)+\log \left(\frac{(b c-a d)^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{c} \sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}+c^{2/3}\right)\right) a^3}{c^{2/3} d^2 \sqrt[3]{b c-a d}}+\frac{231 b^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{d x^3}{c}\right) a^2}{c d^2 \sqrt[3]{b x^3+a}}-\frac{72 b^3 \sqrt[3]{c} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c-a d} x}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-2 \log \left(\sqrt[3]{c}-\frac{\sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}\right)+\log \left(\frac{(b c-a d)^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{c} \sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}+c^{2/3}\right)\right) a^2}{d^3 \sqrt[3]{b c-a d}}-\frac{378 b^4 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{d x^3}{c}\right) a}{d^3 \sqrt[3]{b x^3+a}}+\frac{36 b^4 c^{4/3} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c-a d} x}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-2 \log \left(\sqrt[3]{c}-\frac{\sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}\right)+\log \left(\frac{(b c-a d)^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{c} \sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}+c^{2/3}\right)\right) a}{d^4 \sqrt[3]{b c-a d}}+\frac{6 x \left(b x^3+a\right)^{2/3} \left(3 d x^3 b^4-2 (9 b c-13 a d) b^3-\frac{(b c-a d)^3 (21 b c+5 a d)}{c^2 \left(d x^3+c\right)}+\frac{3 (b c-a d)^4}{c \left(d x^3+c\right)^2}\right)}{d^4}+\frac{162 b^5 c 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{d x^3}{c}\right)}{d^4 \sqrt[3]{b x^3+a}}\right)","-\frac{b x \left(a+b x^3\right)^{2/3} (2 b c-a d) \left(-5 a^2 d^2-18 a b c d+18 b^2 c^2\right)}{18 c^2 d^4}+\frac{b x \left(a+b x^3\right)^{5/3} \left(-5 a^2 d^2-10 a b c d+18 b^2 c^2\right)}{18 c^2 d^3}-\frac{(b c-a d)^{8/3} \left(5 a^2 d^2+18 a b c d+54 b^2 c^2\right) \log \left(c+d x^3\right)}{54 c^{8/3} d^5}+\frac{(b c-a d)^{8/3} \left(5 a^2 d^2+18 a b c d+54 b^2 c^2\right) \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{18 c^{8/3} d^5}-\frac{(b c-a d)^{8/3} \left(5 a^2 d^2+18 a b c d+54 b^2 c^2\right) \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)}{9 \sqrt{3} c^{8/3} d^5}-\frac{b^{8/3} \left(77 a^2 d^2-126 a b c d+54 b^2 c^2\right) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{18 d^5}+\frac{b^{8/3} \left(77 a^2 d^2-126 a b c d+54 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} d^5}-\frac{x \left(a+b x^3\right)^{8/3} (b c-a d) (5 a d+12 b c)}{18 c^2 d^2 \left(c+d x^3\right)}-\frac{x \left(a+b x^3\right)^{11/3} (b c-a d)}{6 c d \left(c+d x^3\right)^2}",1,"((6*x*(a + b*x^3)^(2/3)*(-2*b^3*(9*b*c - 13*a*d) + 3*b^4*d*x^3 + (3*(b*c - a*d)^4)/(c*(c + d*x^3)^2) - ((b*c - a*d)^3*(21*b*c + 5*a*d))/(c^2*(c + d*x^3))))/d^4 + (162*b^5*c*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)])/(d^4*(a + b*x^3)^(1/3)) - (378*a*b^4*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)])/(d^3*(a + b*x^3)^(1/3)) + (231*a^2*b^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)])/(c*d^2*(a + b*x^3)^(1/3)) + (10*a^5*(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)]))/(c^(8/3)*(b*c - a*d)^(1/3)) + (36*a*b^4*c^(4/3)*(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)]))/(d^4*(b*c - a*d)^(1/3)) - (72*a^2*b^3*c^(1/3)*(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)]))/(d^3*(b*c - a*d)^(1/3)) + (30*a^3*b^2*(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)]))/(c^(2/3)*d^2*(b*c - a*d)^(1/3)) + (6*a^4*b*(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)]))/(c^(5/3)*d*(b*c - a*d)^(1/3)))/108","C",0
110,1,908,458,1.8376885,"\int \frac{\left(a+b x^3\right)^{11/3}}{\left(c+d x^3\right)^3} \, dx","Integrate[(a + b*x^3)^(11/3)/(c + d*x^3)^3,x]","\frac{1}{108} \left(\frac{10 \left(2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c-a d} x}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-2 \log \left(\sqrt[3]{c}-\frac{\sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}\right)+\log \left(\frac{(b c-a d)^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{c} \sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}+c^{2/3}\right)\right) a^4}{c^{8/3} \sqrt[3]{b c-a d}}+\frac{4 b \left(2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c-a d} x}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-2 \log \left(\sqrt[3]{c}-\frac{\sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}\right)+\log \left(\frac{(b c-a d)^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{c} \sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}+c^{2/3}\right)\right) a^3}{c^{5/3} d \sqrt[3]{b c-a d}}+\frac{16 b^2 \left(2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c-a d} x}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-2 \log \left(\sqrt[3]{c}-\frac{\sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}\right)+\log \left(\frac{(b c-a d)^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{c} \sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}+c^{2/3}\right)\right) a^2}{c^{2/3} d^2 \sqrt[3]{b c-a d}}+\frac{99 b^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{d x^3}{c}\right) a}{c d^2 \sqrt[3]{b x^3+a}}-\frac{18 b^3 \sqrt[3]{c} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c-a d} x}{\sqrt[3]{c} \sqrt[3]{a x^3+b}}+1}{\sqrt{3}}\right)-2 \log \left(\sqrt[3]{c}-\frac{\sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}\right)+\log \left(\frac{(b c-a d)^{2/3} x^2}{\left(a x^3+b\right)^{2/3}}+\frac{\sqrt[3]{c} \sqrt[3]{b c-a d} x}{\sqrt[3]{a x^3+b}}+c^{2/3}\right)\right) a}{d^3 \sqrt[3]{b c-a d}}+\frac{6 x \left(b x^3+a\right)^{2/3} \left(6 b^3+\frac{5 (b c-a d)^2 (3 b c+a d)}{c^2 \left(d x^3+c\right)}-\frac{3 (b c-a d)^3}{c \left(d x^3+c\right)^2}\right)}{d^3}-\frac{81 b^4 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{d x^3}{c}\right)}{d^3 \sqrt[3]{b x^3+a}}\right)","\frac{b x \left(a+b x^3\right)^{2/3} \left(-5 a^2 d^2-7 a b c d+18 b^2 c^2\right)}{18 c^2 d^3}+\frac{(b c-a d)^{5/3} \left(5 a^2 d^2+12 a b c d+27 b^2 c^2\right) \log \left(c+d x^3\right)}{54 c^{8/3} d^4}-\frac{(b c-a d)^{5/3} \left(5 a^2 d^2+12 a b c d+27 b^2 c^2\right) \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{18 c^{8/3} d^4}+\frac{(b c-a d)^{5/3} \left(5 a^2 d^2+12 a b c d+27 b^2 c^2\right) \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)}{9 \sqrt{3} c^{8/3} d^4}+\frac{b^{8/3} (9 b c-11 a d) \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{6 d^4}-\frac{b^{8/3} (9 b c-11 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^4}-\frac{x \left(a+b x^3\right)^{5/3} (b c-a d) (5 a d+9 b c)}{18 c^2 d^2 \left(c+d x^3\right)}-\frac{x \left(a+b x^3\right)^{8/3} (b c-a d)}{6 c d \left(c+d x^3\right)^2}",1,"((6*x*(a + b*x^3)^(2/3)*(6*b^3 - (3*(b*c - a*d)^3)/(c*(c + d*x^3)^2) + (5*(b*c - a*d)^2*(3*b*c + a*d))/(c^2*(c + d*x^3))))/d^3 - (81*b^4*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)])/(d^3*(a + b*x^3)^(1/3)) + (99*a*b^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)])/(c*d^2*(a + b*x^3)^(1/3)) + (10*a^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)]))/(c^(8/3)*(b*c - a*d)^(1/3)) - (18*a*b^3*c^(1/3)*(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)]))/(d^3*(b*c - a*d)^(1/3)) + (16*a^2*b^2*(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)]))/(c^(2/3)*d^2*(b*c - a*d)^(1/3)) + (4*a^3*b*(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)]))/(c^(5/3)*d*(b*c - a*d)^(1/3)))/108","C",0
111,1,651,391,1.1636066,"\int \frac{\left(a+b x^3\right)^{8/3}}{\left(c+d x^3\right)^3} \, dx","Integrate[(a + b*x^3)^(8/3)/(c + d*x^3)^3,x]","\frac{\frac{10 a^3 \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)}{\sqrt[3]{b c-a d}}+\frac{2 a^2 b c \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)}{d \sqrt[3]{b c-a d}}+\frac{27 b^3 c^{5/3} 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{d x^3}{c}\right)}{d^2 \sqrt[3]{a+b x^3}}+\frac{6 a b^2 c^2 \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)}{d^2 \sqrt[3]{b c-a d}}+\frac{6 c^{2/3} x \left(a+b x^3\right)^{2/3} (a d-b c) \left(a d \left(8 c+5 d x^3\right)+3 b c \left(2 c+3 d x^3\right)\right)}{d^2 \left(c+d x^3\right)^2}}{108 c^{8/3}}","-\frac{(b c-a d)^{2/3} \left(5 a^2 d^2+6 a b c d+9 b^2 c^2\right) \log \left(c+d x^3\right)}{54 c^{8/3} d^3}+\frac{(b c-a d)^{2/3} \left(5 a^2 d^2+6 a b c d+9 b^2 c^2\right) \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{18 c^{8/3} d^3}-\frac{(b c-a d)^{2/3} \left(5 a^2 d^2+6 a b c d+9 b^2 c^2\right) \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)}{9 \sqrt{3} c^{8/3} d^3}-\frac{b^{8/3} \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 d^3}+\frac{b^{8/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^3}-\frac{x \left(a+b x^3\right)^{2/3} (b c-a d) (5 a d+6 b c)}{18 c^2 d^2 \left(c+d x^3\right)}-\frac{x \left(a+b x^3\right)^{5/3} (b c-a d)}{6 c d \left(c+d x^3\right)^2}",1,"((6*c^(2/3)*(-(b*c) + a*d)*x*(a + b*x^3)^(2/3)*(3*b*c*(2*c + 3*d*x^3) + a*d*(8*c + 5*d*x^3)))/(d^2*(c + d*x^3)^2) + (27*b^3*c^(5/3)*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)])/(d^2*(a + b*x^3)^(1/3)) + (10*a^3*(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)]))/(b*c - a*d)^(1/3) + (6*a*b^2*c^2*(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)]))/(d^2*(b*c - a*d)^(1/3)) + (2*a^2*b*c*(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)]))/(d*(b*c - a*d)^(1/3)))/(108*c^(8/3))","C",0
112,1,79,217,0.032864,"\int \frac{\left(a+b x^3\right)^{5/3}}{\left(c+d x^3\right)^3} \, dx","Integrate[(a + b*x^3)^(5/3)/(c + d*x^3)^3,x]","\frac{a x \left(a+b x^3\right)^{2/3} \, _2F_1\left(-\frac{5}{3},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{c^3 \left(\frac{b x^3}{a}+1\right)^{2/3} \sqrt[3]{\frac{d x^3}{c}+1}}","\frac{5 a^2 \log \left(c+d x^3\right)}{54 c^{8/3} \sqrt[3]{b c-a d}}-\frac{5 a^2 \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{18 c^{8/3} \sqrt[3]{b c-a d}}+\frac{5 a^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)}{9 \sqrt{3} c^{8/3} \sqrt[3]{b c-a d}}+\frac{5 a x \left(a+b x^3\right)^{2/3}}{18 c^2 \left(c+d x^3\right)}+\frac{x \left(a+b x^3\right)^{5/3}}{6 c \left(c+d x^3\right)^2}",1,"(a*x*(a + b*x^3)^(2/3)*Hypergeometric2F1[-5/3, 1/3, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c^3*(1 + (b*x^3)/a)^(2/3)*(1 + (d*x^3)/c)^(1/3))","C",0
113,1,153,267,0.2191553,"\int \frac{\left(a+b x^3\right)^{2/3}}{\left(c+d x^3\right)^3} \, dx","Integrate[(a + b*x^3)^(2/3)/(c + d*x^3)^3,x]","\frac{x \left(c \left(-a^2 d \left(8 c+5 d x^3\right)+a b \left(6 c^2-5 c d x^3-5 d^2 x^6\right)+3 b^2 c x^3 \left(2 c+d x^3\right)\right)-2 a \left(c+d x^3\right)^2 (5 a d-6 b c) \, _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)\right)}{18 c^3 \sqrt[3]{a+b x^3} \left(c+d x^3\right)^2 (b c-a d)}","\frac{a (6 b c-5 a d) \log \left(c+d x^3\right)}{54 c^{8/3} (b c-a d)^{4/3}}-\frac{a (6 b c-5 a d) \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{18 c^{8/3} (b c-a d)^{4/3}}+\frac{a (6 b c-5 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)}{9 \sqrt{3} c^{8/3} (b c-a d)^{4/3}}+\frac{x \left(a+b x^3\right)^{2/3} (6 b c-5 a d)}{18 c^2 \left(c+d x^3\right) (b c-a d)}-\frac{d x \left(a+b x^3\right)^{5/3}}{6 c \left(c+d x^3\right)^2 (b c-a d)}",1,"(x*(c*(3*b^2*c*x^3*(2*c + d*x^3) - a^2*d*(8*c + 5*d*x^3) + a*b*(6*c^2 - 5*c*d*x^3 - 5*d^2*x^6)) - 2*a*(-6*b*c + 5*a*d)*(c + d*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)*(a + b*x^3)^(1/3)*(c + d*x^3)^2)","C",1
114,1,168,307,0.3804652,"\int \frac{1}{\sqrt[3]{a+b x^3} \left(c+d x^3\right)^3} \, dx","Integrate[1/((a + b*x^3)^(1/3)*(c + d*x^3)^3),x]","\frac{x \left(2 \left(c+d x^3\right)^2 \left(5 a^2 d^2-12 a b c d+9 b^2 c^2\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)-c d \left(-a^2 d \left(8 c+5 d x^3\right)+a b \left(12 c^2+c d x^3-5 d^2 x^6\right)+3 b^2 c x^3 \left(4 c+3 d x^3\right)\right)\right)}{18 c^3 \sqrt[3]{a+b x^3} \left(c+d x^3\right)^2 (b c-a d)^2}","\frac{\left(5 a^2 d^2-12 a b c d+9 b^2 c^2\right) \log \left(c+d x^3\right)}{54 c^{8/3} (b c-a d)^{7/3}}-\frac{\left(5 a^2 d^2-12 a b c d+9 b^2 c^2\right) \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{18 c^{8/3} (b c-a d)^{7/3}}+\frac{\left(5 a^2 d^2-12 a b c d+9 b^2 c^2\right) \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)}{9 \sqrt{3} c^{8/3} (b c-a d)^{7/3}}-\frac{d x \left(a+b x^3\right)^{2/3} (9 b c-5 a d)}{18 c^2 \left(c+d x^3\right) (b c-a d)^2}-\frac{d x \left(a+b x^3\right)^{2/3}}{6 c \left(c+d x^3\right)^2 (b c-a d)}",1,"(x*(-(c*d*(3*b^2*c*x^3*(4*c + 3*d*x^3) - a^2*d*(8*c + 5*d*x^3) + a*b*(12*c^2 + c*d*x^3 - 5*d^2*x^6))) + 2*(9*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*(c + d*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*(a + b*x^3)^(1/3)*(c + d*x^3)^2)","C",1
115,1,428,377,3.0844159,"\int \frac{1}{\left(a+b x^3\right)^{4/3} \left(c+d x^3\right)^3} \, dx","Integrate[1/((a + b*x^3)^(4/3)*(c + d*x^3)^3),x]","-\frac{486 x^{12} \left(c+d x^3\right)^3 (b c-a d)^4 \, _4F_3\left(2,2,2,\frac{7}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left(b x^3+a\right)}\right)+65 c^2 \left(a+b x^3\right)^2 \left(28 \left(c+d x^3\right)^2 \left(a^2 \left(500 c^3+843 c^2 d x^3+375 c d^2 x^6+27 d^3 x^9\right)+9 a b c x^3 \left(73 c^2+104 c d x^3+33 d^2 x^6\right)+27 b^2 c^2 x^6 \left(7 c+6 d x^3\right)\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)-14000 a^2 c^5-48104 a^2 c^4 d x^3-60807 a^2 c^3 d^2 x^6-33657 a^2 c^2 d^3 x^9-7155 a^2 c d^4 x^{12}-243 a^2 d^5 x^{15}-21896 a b c^5 x^3-70802 a b c^4 d x^6-81534 a b c^3 d^2 x^9-38652 a b c^2 d^3 x^{12}-5940 a b c d^4 x^{15}-8391 b^2 c^5 x^6-24417 b^2 c^4 d x^9-23409 b^2 c^3 d^2 x^{12}-7425 b^2 c^2 d^3 x^{15}\right)}{16380 c^5 x^8 \left(a+b x^3\right)^{7/3} \left(c+d x^3\right)^2 (a d-b c)^3}","\frac{d x \left(a+b x^3\right)^{2/3} \left(-5 a^2 d^2+15 a b c d+18 b^2 c^2\right)}{18 a c^2 \left(c+d x^3\right) (b c-a d)^3}-\frac{d \left(5 a^2 d^2-18 a b c d+27 b^2 c^2\right) \log \left(c+d x^3\right)}{54 c^{8/3} (b c-a d)^{10/3}}+\frac{d \left(5 a^2 d^2-18 a b c d+27 b^2 c^2\right) \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{18 c^{8/3} (b c-a d)^{10/3}}-\frac{d \left(5 a^2 d^2-18 a b c d+27 b^2 c^2\right) \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)}{9 \sqrt{3} c^{8/3} (b c-a d)^{10/3}}+\frac{b x (a d+6 b c)}{6 a c \sqrt[3]{a+b x^3} \left(c+d x^3\right) (b c-a d)^2}-\frac{d x}{6 c \sqrt[3]{a+b x^3} \left(c+d x^3\right)^2 (b c-a d)}",1,"-1/16380*(65*c^2*(a + b*x^3)^2*(-14000*a^2*c^5 - 21896*a*b*c^5*x^3 - 48104*a^2*c^4*d*x^3 - 8391*b^2*c^5*x^6 - 70802*a*b*c^4*d*x^6 - 60807*a^2*c^3*d^2*x^6 - 24417*b^2*c^4*d*x^9 - 81534*a*b*c^3*d^2*x^9 - 33657*a^2*c^2*d^3*x^9 - 23409*b^2*c^3*d^2*x^12 - 38652*a*b*c^2*d^3*x^12 - 7155*a^2*c*d^4*x^12 - 7425*b^2*c^2*d^3*x^15 - 5940*a*b*c*d^4*x^15 - 243*a^2*d^5*x^15 + 28*(c + d*x^3)^2*(27*b^2*c^2*x^6*(7*c + 6*d*x^3) + 9*a*b*c*x^3*(73*c^2 + 104*c*d*x^3 + 33*d^2*x^6) + a^2*(500*c^3 + 843*c^2*d*x^3 + 375*c*d^2*x^6 + 27*d^3*x^9))*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]) + 486*(b*c - a*d)^4*x^12*(c + d*x^3)^3*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^5*(-(b*c) + a*d)^3*x^8*(a + b*x^3)^(7/3)*(c + d*x^3)^2)","C",0
116,1,337,463,5.8783164,"\int \frac{1}{\left(a+b x^3\right)^{7/3} \left(c+d x^3\right)^3} \, dx","Integrate[1/((a + b*x^3)^(7/3)*(c + d*x^3)^3),x]","\frac{1}{36} x \left(a+b x^3\right)^{2/3} \left(\frac{27 b^3 (b c-5 a d)}{a^2 \left(a+b x^3\right) (b c-a d)^4}-\frac{9 b^3}{a \left(a+b x^3\right)^2 (a d-b c)^3}+\frac{2 d^3 (5 a d-21 b c)}{c^2 \left(c+d x^3\right) (b c-a d)^4}-\frac{6 d^3}{c \left(c+d x^3\right)^2 (b c-a d)^3}\right)+\frac{d^2 \left(5 a^2 d^2-24 a b c d+54 b^2 c^2\right) \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)}{54 c^{8/3} (b c-a d)^{13/3}}","\frac{b x \left(-2 a^2 d^2-42 a b c d+9 b^2 c^2\right)}{12 a^2 c \sqrt[3]{a+b x^3} \left(c+d x^3\right) (b c-a d)^3}+\frac{d^2 \left(5 a^2 d^2-24 a b c d+54 b^2 c^2\right) \log \left(c+d x^3\right)}{54 c^{8/3} (b c-a d)^{13/3}}-\frac{d^2 \left(5 a^2 d^2-24 a b c d+54 b^2 c^2\right) \log \left(\frac{x \sqrt[3]{b c-a d}}{\sqrt[3]{c}}-\sqrt[3]{a+b x^3}\right)}{18 c^{8/3} (b c-a d)^{13/3}}+\frac{d^2 \left(5 a^2 d^2-24 a b c d+54 b^2 c^2\right) \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)}{9 \sqrt{3} c^{8/3} (b c-a d)^{13/3}}+\frac{d x \left(a+b x^3\right)^{2/3} \left(10 a^3 d^3-42 a^2 b c d^2-135 a b^2 c^2 d+27 b^3 c^3\right)}{36 a^2 c^2 \left(c+d x^3\right) (b c-a d)^4}-\frac{d x}{6 c \left(a+b x^3\right)^{4/3} \left(c+d x^3\right)^2 (b c-a d)}+\frac{b x (2 a d+3 b c)}{12 a c \left(a+b x^3\right)^{4/3} \left(c+d x^3\right) (b c-a d)^2}",1,"(x*(a + b*x^3)^(2/3)*((-9*b^3)/(a*(-(b*c) + a*d)^3*(a + b*x^3)^2) + (27*b^3*(b*c - 5*a*d))/(a^2*(b*c - a*d)^4*(a + b*x^3)) - (6*d^3)/(c*(b*c - a*d)^3*(c + d*x^3)^2) + (2*d^3*(-21*b*c + 5*a*d))/(c^2*(b*c - a*d)^4*(c + d*x^3))))/36 + (d^2*(54*b^2*c^2 - 24*a*b*c*d + 5*a^2*d^2)*(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)]))/(54*c^(8/3)*(b*c - a*d)^(13/3))","A",0
117,1,316,60,0.519282,"\int \frac{\left(a+b x^3\right)^{4/3}}{\left(c+d x^3\right)^3} \, dx","Integrate[(a + b*x^3)^(4/3)/(c + d*x^3)^3,x]","\frac{x \left(\frac{4 c \left(\frac{4 a^2 c \left(c+d x^3\right) (10 a d+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)}{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)}+8 a^2 c d+5 a^2 d^2 x^3-a b c^2+10 a b c d x^3+5 a b d^2 x^6-b^2 c^2 x^3+2 b^2 c d x^6\right)}{\left(c+d x^3\right)^2}+b x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} (5 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)\right)}{72 c^3 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},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c^3 \sqrt[3]{\frac{b x^3}{a}+1}}",1,"(x*(b*(2*b*c + 5*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)] + (4*c*(-(a*b*c^2) + 8*a^2*c*d - b^2*c^2*x^3 + 10*a*b*c*d*x^3 + 5*a^2*d^2*x^3 + 2*b^2*c*d*x^6 + 5*a*b*d^2*x^6 + (4*a^2*c*(b*c + 10*a*d)*(c + d*x^3)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(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 + d*x^3)^2))/(72*c^3*d*(a + b*x^3)^(2/3))","B",0
118,1,431,59,0.6169062,"\int \frac{\sqrt[3]{a+b x^3}}{\left(c+d x^3\right)^3} \, dx","Integrate[(a + b*x^3)^(1/3)/(c + d*x^3)^3,x]","\frac{\frac{c \left(16 a c x \left(3 a^2 d \left(6 c+5 d x^3\right)+a b \left(-18 c^2-7 c d x^3+5 d^2 x^6\right)-b^2 c x^3 \left(7 c+4 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)-4 x^4 \left(a^2 d \left(8 c+5 d x^3\right)+a b \left(-7 c^2+4 c d x^3+5 d^2 x^6\right)-b^2 c x^3 \left(7 c+4 d x^3\right)\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)^2 \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 x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} (5 a d-4 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)}{72 c^3 \left(a+b x^3\right)^{2/3} (b c-a d)}","\frac{x \sqrt[3]{a+b x^3} F_1\left(\frac{1}{3};-\frac{1}{3},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c^3 \sqrt[3]{\frac{b x^3}{a}+1}}",1,"(-(b*(-4*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*(16*a*c*x*(-(b^2*c*x^3*(7*c + 4*d*x^3)) + 3*a^2*d*(6*c + 5*d*x^3) + a*b*(-18*c^2 - 7*c*d*x^3 + 5*d^2*x^6))*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] - 4*x^4*(-(b^2*c*x^3*(7*c + 4*d*x^3)) + a^2*d*(8*c + 5*d*x^3) + a*b*(-7*c^2 + 4*c*d*x^3 + 5*d^2*x^6))*(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)^2*(-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)]))))/(72*c^3*(b*c - a*d)*(a + b*x^3)^(2/3))","B",0
119,1,442,59,0.7724549,"\int \frac{1}{\left(a+b x^3\right)^{2/3} \left(c+d x^3\right)^3} \, dx","Integrate[1/((a + b*x^3)^(2/3)*(c + d*x^3)^3),x]","\frac{x \left(\frac{4 c \left(d x^3 \left(a^2 d \left(8 c+5 d x^3\right)+a b \left(-13 c^2-2 c d x^3+5 d^2 x^6\right)-b^2 c x^3 \left(13 c+10 d x^3\right)\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(3 a^2 d^2 \left(6 c+5 d x^3\right)+a b d \left(-36 c^2-25 c d x^3+5 d^2 x^6\right)+b^2 c \left(18 c^2+5 c d x^3-10 d^2 x^6\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)^2 \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)}+5 b d 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)\right)}{72 c^3 \left(a+b x^3\right)^{2/3} (b c-a d)^2}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{1}{3};\frac{2}{3},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c^3 \left(a+b x^3\right)^{2/3}}",1,"(x*(5*b*d*(-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)] + (4*c*(-4*a*c*(3*a^2*d^2*(6*c + 5*d*x^3) + b^2*c*(18*c^2 + 5*c*d*x^3 - 10*d^2*x^6) + a*b*d*(-36*c^2 - 25*c*d*x^3 + 5*d^2*x^6))*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + d*x^3*(a^2*d*(8*c + 5*d*x^3) - b^2*c*x^3*(13*c + 10*d*x^3) + a*b*(-13*c^2 - 2*c*d*x^3 + 5*d^2*x^6))*(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)^2*(-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)])))))/(72*c^3*(b*c - a*d)^2*(a + b*x^3)^(2/3))","B",0
120,1,531,62,1.09519,"\int \frac{1}{\left(a+b x^3\right)^{5/3} \left(c+d x^3\right)^3} \, dx","Integrate[1/((a + b*x^3)^(5/3)*(c + d*x^3)^3),x]","\frac{\frac{b d x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(5 a^2 d^2-16 a b c d-9 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)}{(a d-b c)^3}-\frac{4 c \left(4 a c x \left(3 a^3 d^3 \left(6 c+5 d x^3\right)+a^2 b d^2 \left(-54 c^2-43 c d x^3+5 d^2 x^6\right)+a b^2 c d \left(54 c^2+35 c d x^3-16 d^2 x^6\right)-9 b^3 c^2 \left(2 c^2+3 c d x^3+d^2 x^6\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)+x^4 \left(-a^3 d^3 \left(8 c+5 d x^3\right)+a^2 b d^2 \left(19 c^2+8 c d x^3-5 d^2 x^6\right)+a b^2 c d^2 x^3 \left(19 c+16 d x^3\right)+9 b^3 c^2 \left(c+d x^3\right)^2\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)^2 (b c-a d)^3 \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)}}{72 a c^3 \left(a+b x^3\right)^{2/3}}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{1}{3};\frac{5}{3},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a c^3 \left(a+b x^3\right)^{2/3}}",1,"((b*d*(-9*b^2*c^2 - 16*a*b*c*d + 5*a^2*d^2)*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)])/(-(b*c) + a*d)^3 - (4*c*(4*a*c*x*(3*a^3*d^3*(6*c + 5*d*x^3) + a*b^2*c*d*(54*c^2 + 35*c*d*x^3 - 16*d^2*x^6) - 9*b^3*c^2*(2*c^2 + 3*c*d*x^3 + d^2*x^6) + a^2*b*d^2*(-54*c^2 - 43*c*d*x^3 + 5*d^2*x^6))*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^4*(9*b^3*c^2*(c + d*x^3)^2 - a^3*d^3*(8*c + 5*d*x^3) + a*b^2*c*d^2*x^3*(19*c + 16*d*x^3) + a^2*b*d^2*(19*c^2 + 8*c*d*x^3 - 5*d^2*x^6))*(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*c - a*d)^3*(c + d*x^3)^2*(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)]))))/(72*a*c^3*(a + b*x^3)^(2/3))","B",0
121,1,515,62,1.6159448,"\int \frac{1}{\left(a+b x^3\right)^{8/3} \left(c+d x^3\right)^3} \, dx","Integrate[1/((a + b*x^3)^(8/3)*(c + d*x^3)^3),x]","\frac{x \left(b d x^3 \left(\frac{b x^3}{a}+1\right)^{2/3} \left(25 a^3 d^3-110 a^2 b c d^2-171 a b^2 c^2 d+36 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)+\frac{4 c \left(\frac{4 a c \left(c+d x^3\right) \left(50 a^4 d^4-235 a^3 b c d^3+540 a^2 b^2 c^2 d^2-171 a b^3 c^3 d+36 b^4 c^4\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 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)}+\frac{5 a^5 d^4 \left(8 c+5 d x^3\right)+5 a^4 b d^3 \left(-25 c^2-6 c d x^3+10 d^2 x^6\right)+5 a^3 b^2 d^3 x^3 \left(-50 c^2-36 c d x^3+5 d^2 x^6\right)-a^2 b^3 c d \left(189 c^3+378 c^2 d x^3+314 c d^2 x^6+110 d^3 x^9\right)+9 a b^4 c^2 \left(6 c-19 d x^3\right) \left(c+d x^3\right)^2+36 b^5 c^3 x^3 \left(c+d x^3\right)^2}{a+b x^3}\right)}{\left(c+d x^3\right)^2}\right)}{360 a^2 c^3 \left(a+b x^3\right)^{2/3} (b c-a d)^4}","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{1}{3};\frac{8}{3},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2 c^3 \left(a+b x^3\right)^{2/3}}",1,"(x*(b*d*(36*b^3*c^3 - 171*a*b^2*c^2*d - 110*a^2*b*c*d^2 + 25*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)] + (4*c*((36*b^5*c^3*x^3*(c + d*x^3)^2 + 9*a*b^4*c^2*(6*c - 19*d*x^3)*(c + d*x^3)^2 + 5*a^5*d^4*(8*c + 5*d*x^3) + 5*a^3*b^2*d^3*x^3*(-50*c^2 - 36*c*d*x^3 + 5*d^2*x^6) + 5*a^4*b*d^3*(-25*c^2 - 6*c*d*x^3 + 10*d^2*x^6) - a^2*b^3*c*d*(189*c^3 + 378*c^2*d*x^3 + 314*c*d^2*x^6 + 110*d^3*x^9))/(a + b*x^3) + (4*a*c*(36*b^4*c^4 - 171*a*b^3*c^3*d + 540*a^2*b^2*c^2*d^2 - 235*a^3*b*c*d^3 + 50*a^4*d^4)*(c + d*x^3)*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(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 + d*x^3)^2))/(360*a^2*c^3*(b*c - a*d)^4*(a + b*x^3)^(2/3))","B",0
122,1,90,155,0.0426976,"\int \frac{\left(a+b x^3\right)^{7/4}}{\left(c+d x^3\right)^{37/12}} \, dx","Integrate[(a + b*x^3)^(7/4)/(c + d*x^3)^(37/12),x]","\frac{a x \left(a+b x^3\right)^{3/4} \, _2F_1\left(-\frac{7}{4},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{c^3 \left(\frac{b x^3}{a}+1\right)^{3/4} \sqrt[12]{c+d x^3} \sqrt[4]{\frac{d x^3}{c}+1}}","\frac{189 a^2 x \sqrt[4]{\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}} \, _2F_1\left(\frac{1}{4},\frac{1}{3};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left(d x^3+c\right)}\right)}{325 c^3 \sqrt[4]{a+b x^3} \sqrt[12]{c+d x^3}}+\frac{84 a x \left(a+b x^3\right)^{3/4}}{325 c^2 \left(c+d x^3\right)^{13/12}}+\frac{4 x \left(a+b x^3\right)^{7/4}}{25 c \left(c+d x^3\right)^{25/12}}",1,"(a*x*(a + b*x^3)^(3/4)*Hypergeometric2F1[-7/4, 1/3, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c^3*(1 + (b*x^3)/a)^(3/4)*(c + d*x^3)^(1/12)*(1 + (d*x^3)/c)^(1/4))","A",0
123,1,90,155,0.0314248,"\int \frac{\left(a+b x^3\right)^{5/4}}{\left(c+d x^3\right)^{31/12}} \, dx","Integrate[(a + b*x^3)^(5/4)/(c + d*x^3)^(31/12),x]","\frac{a x \sqrt[4]{a+b x^3} \sqrt[4]{\frac{d x^3}{c}+1} \, _2F_1\left(-\frac{5}{4},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{c^2 \sqrt[4]{\frac{b x^3}{a}+1} \left(c+d x^3\right)^{7/12}}","\frac{45 a^2 x \left(c+d x^3\right)^{5/12} \left(\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}\right)^{3/4} \, _2F_1\left(\frac{1}{3},\frac{3}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left(d x^3+c\right)}\right)}{133 c^3 \left(a+b x^3\right)^{3/4}}+\frac{60 a x \sqrt[4]{a+b x^3}}{133 c^2 \left(c+d x^3\right)^{7/12}}+\frac{4 x \left(a+b x^3\right)^{5/4}}{19 c \left(c+d x^3\right)^{19/12}}",1,"(a*x*(a + b*x^3)^(1/4)*(1 + (d*x^3)/c)^(1/4)*Hypergeometric2F1[-5/4, 1/3, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c^2*(1 + (b*x^3)/a)^(1/4)*(c + d*x^3)^(7/12))","A",0
124,1,89,122,0.0200953,"\int \frac{\left(a+b x^3\right)^{3/4}}{\left(c+d x^3\right)^{25/12}} \, dx","Integrate[(a + b*x^3)^(3/4)/(c + d*x^3)^(25/12),x]","\frac{x \left(a+b x^3\right)^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{c^2 \left(\frac{b x^3}{a}+1\right)^{3/4} \sqrt[12]{c+d x^3} \sqrt[4]{\frac{d x^3}{c}+1}}","\frac{9 a x \sqrt[4]{\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}} \, _2F_1\left(\frac{1}{4},\frac{1}{3};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left(d x^3+c\right)}\right)}{13 c^2 \sqrt[4]{a+b x^3} \sqrt[12]{c+d x^3}}+\frac{4 x \left(a+b x^3\right)^{3/4}}{13 c \left(c+d x^3\right)^{13/12}}",1,"(x*(a + b*x^3)^(3/4)*Hypergeometric2F1[-3/4, 1/3, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c^2*(1 + (b*x^3)/a)^(3/4)*(c + d*x^3)^(1/12)*(1 + (d*x^3)/c)^(1/4))","A",0
125,1,89,122,0.0245576,"\int \frac{\sqrt[4]{a+b x^3}}{\left(c+d x^3\right)^{19/12}} \, dx","Integrate[(a + b*x^3)^(1/4)/(c + d*x^3)^(19/12),x]","\frac{x \sqrt[4]{a+b x^3} \sqrt[4]{\frac{d x^3}{c}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{c \sqrt[4]{\frac{b x^3}{a}+1} \left(c+d x^3\right)^{7/12}}","\frac{3 a x \left(c+d x^3\right)^{5/12} \left(\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}\right)^{3/4} \, _2F_1\left(\frac{1}{3},\frac{3}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left(d x^3+c\right)}\right)}{7 c^2 \left(a+b x^3\right)^{3/4}}+\frac{4 x \sqrt[4]{a+b x^3}}{7 c \left(c+d x^3\right)^{7/12}}",1,"(x*(a + b*x^3)^(1/4)*(1 + (d*x^3)/c)^(1/4)*Hypergeometric2F1[-1/4, 1/3, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c*(1 + (b*x^3)/a)^(1/4)*(c + d*x^3)^(7/12))","A",0
126,1,86,87,0.0623132,"\int \frac{1}{\sqrt[4]{a+b x^3} \left(c+d x^3\right)^{13/12}} \, dx","Integrate[1/((a + b*x^3)^(1/4)*(c + d*x^3)^(13/12)),x]","\frac{x \sqrt[4]{\frac{b x^3}{a}+1} \left(\frac{d x^3}{c}+1\right)^{3/4} \, _2F_1\left(\frac{1}{4},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{\sqrt[4]{a+b x^3} \left(c+d x^3\right)^{13/12}}","\frac{x \sqrt[4]{\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}} \, _2F_1\left(\frac{1}{4},\frac{1}{3};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left(d x^3+c\right)}\right)}{c \sqrt[4]{a+b x^3} \sqrt[12]{c+d x^3}}",1,"(x*(1 + (b*x^3)/a)^(1/4)*(1 + (d*x^3)/c)^(3/4)*Hypergeometric2F1[1/4, 1/3, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/((a + b*x^3)^(1/4)*(c + d*x^3)^(13/12))","A",0
127,1,86,87,0.0215971,"\int \frac{1}{\left(a+b x^3\right)^{3/4} \left(c+d x^3\right)^{7/12}} \, dx","Integrate[1/((a + b*x^3)^(3/4)*(c + d*x^3)^(7/12)),x]","\frac{x \left(\frac{b x^3}{a}+1\right)^{3/4} \sqrt[4]{\frac{d x^3}{c}+1} \, _2F_1\left(\frac{1}{3},\frac{3}{4};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{\left(a+b x^3\right)^{3/4} \left(c+d x^3\right)^{7/12}}","\frac{x \left(c+d x^3\right)^{5/12} \left(\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}\right)^{3/4} \, _2F_1\left(\frac{1}{3},\frac{3}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left(d x^3+c\right)}\right)}{c \left(a+b x^3\right)^{3/4}}",1,"(x*(1 + (b*x^3)/a)^(3/4)*(1 + (d*x^3)/c)^(1/4)*Hypergeometric2F1[1/3, 3/4, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/((a + b*x^3)^(3/4)*(c + d*x^3)^(7/12))","A",0
128,1,89,87,0.0415456,"\int \frac{1}{\left(a+b x^3\right)^{5/4} \sqrt[12]{c+d x^3}} \, dx","Integrate[1/((a + b*x^3)^(5/4)*(c + d*x^3)^(1/12)),x]","\frac{x \sqrt[4]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{5}{4};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{a \sqrt[4]{a+b x^3} \sqrt[12]{c+d x^3} \sqrt[4]{\frac{d x^3}{c}+1}}","\frac{x \left(c+d x^3\right)^{11/12} \left(\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}\right)^{5/4} \, _2F_1\left(\frac{1}{3},\frac{5}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left(d x^3+c\right)}\right)}{c \left(a+b x^3\right)^{5/4}}",1,"(x*(1 + (b*x^3)/a)^(1/4)*Hypergeometric2F1[1/3, 5/4, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(a*(a + b*x^3)^(1/4)*(c + d*x^3)^(1/12)*(1 + (d*x^3)/c)^(1/4))","A",0
129,1,89,121,0.0501304,"\int \frac{\left(c+d x^3\right)^{5/12}}{\left(a+b x^3\right)^{7/4}} \, dx","Integrate[(c + d*x^3)^(5/12)/(a + b*x^3)^(7/4),x]","\frac{x \left(\frac{b x^3}{a}+1\right)^{3/4} \left(c+d x^3\right)^{5/12} \, _2F_1\left(\frac{1}{3},\frac{7}{4};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{a \left(a+b x^3\right)^{3/4} \left(\frac{d x^3}{c}+1\right)^{3/4}}","\frac{5 x \left(c+d x^3\right)^{5/12} \left(\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}\right)^{3/4} \, _2F_1\left(\frac{1}{3},\frac{3}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left(d x^3+c\right)}\right)}{9 a \left(a+b x^3\right)^{3/4}}+\frac{4 x \left(c+d x^3\right)^{5/12}}{9 a \left(a+b x^3\right)^{3/4}}",1,"(x*(1 + (b*x^3)/a)^(3/4)*(c + d*x^3)^(5/12)*Hypergeometric2F1[1/3, 7/4, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(a*(a + b*x^3)^(3/4)*(1 + (d*x^3)/c)^(3/4))","A",0
130,1,89,121,0.0366594,"\int \frac{\left(c+d x^3\right)^{11/12}}{\left(a+b x^3\right)^{9/4}} \, dx","Integrate[(c + d*x^3)^(11/12)/(a + b*x^3)^(9/4),x]","\frac{x \sqrt[4]{\frac{b x^3}{a}+1} \left(c+d x^3\right)^{11/12} \, _2F_1\left(\frac{1}{3},\frac{9}{4};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{a^2 \sqrt[4]{a+b x^3} \left(\frac{d x^3}{c}+1\right)^{5/4}}","\frac{11 x \left(c+d x^3\right)^{11/12} \left(\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}\right)^{5/4} \, _2F_1\left(\frac{1}{3},\frac{5}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left(d x^3+c\right)}\right)}{15 a \left(a+b x^3\right)^{5/4}}+\frac{4 x \left(c+d x^3\right)^{11/12}}{15 a \left(a+b x^3\right)^{5/4}}",1,"(x*(1 + (b*x^3)/a)^(1/4)*(c + d*x^3)^(11/12)*Hypergeometric2F1[1/3, 9/4, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(a^2*(a + b*x^3)^(1/4)*(1 + (d*x^3)/c)^(5/4))","A",0
131,1,90,153,0.0460126,"\int \frac{\left(c+d x^3\right)^{17/12}}{\left(a+b x^3\right)^{11/4}} \, dx","Integrate[(c + d*x^3)^(17/12)/(a + b*x^3)^(11/4),x]","\frac{c x \left(\frac{b x^3}{a}+1\right)^{3/4} \left(c+d x^3\right)^{5/12} \, _2F_1\left(\frac{1}{3},\frac{11}{4};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{a^2 \left(a+b x^3\right)^{3/4} \left(\frac{d x^3}{c}+1\right)^{3/4}}","\frac{85 c x \left(c+d x^3\right)^{5/12} \left(\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}\right)^{3/4} \, _2F_1\left(\frac{1}{3},\frac{3}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left(d x^3+c\right)}\right)}{189 a^2 \left(a+b x^3\right)^{3/4}}+\frac{68 c x \left(c+d x^3\right)^{5/12}}{189 a^2 \left(a+b x^3\right)^{3/4}}+\frac{4 x \left(c+d x^3\right)^{17/12}}{21 a \left(a+b x^3\right)^{7/4}}",1,"(c*x*(1 + (b*x^3)/a)^(3/4)*(c + d*x^3)^(5/12)*Hypergeometric2F1[1/3, 11/4, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(a^2*(a + b*x^3)^(3/4)*(1 + (d*x^3)/c)^(3/4))","A",0
132,1,90,153,0.0363173,"\int \frac{\left(c+d x^3\right)^{23/12}}{\left(a+b x^3\right)^{13/4}} \, dx","Integrate[(c + d*x^3)^(23/12)/(a + b*x^3)^(13/4),x]","\frac{c x \sqrt[4]{\frac{b x^3}{a}+1} \left(c+d x^3\right)^{11/12} \, _2F_1\left(\frac{1}{3},\frac{13}{4};\frac{4}{3};\frac{(a d-b c) x^3}{a \left(d x^3+c\right)}\right)}{a^3 \sqrt[4]{a+b x^3} \left(\frac{d x^3}{c}+1\right)^{5/4}}","\frac{253 c x \left(c+d x^3\right)^{11/12} \left(\frac{c \left(a+b x^3\right)}{a \left(c+d x^3\right)}\right)^{5/4} \, _2F_1\left(\frac{1}{3},\frac{5}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left(d x^3+c\right)}\right)}{405 a^2 \left(a+b x^3\right)^{5/4}}+\frac{92 c x \left(c+d x^3\right)^{11/12}}{405 a^2 \left(a+b x^3\right)^{5/4}}+\frac{4 x \left(c+d x^3\right)^{23/12}}{27 a \left(a+b x^3\right)^{9/4}}",1,"(c*x*(1 + (b*x^3)/a)^(1/4)*(c + d*x^3)^(11/12)*Hypergeometric2F1[1/3, 13/4, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(a^3*(a + b*x^3)^(1/4)*(1 + (d*x^3)/c)^(5/4))","A",0
133,1,172,79,0.4323122,"\int \left(a+b x^3\right)^m \left(c+d x^3\right)^p \, dx","Integrate[(a + b*x^3)^m*(c + d*x^3)^p,x]","\frac{4 a c x \left(a+b x^3\right)^m \left(c+d x^3\right)^p F_1\left(\frac{1}{3};-m,-p;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{3 x^3 \left(b c m F_1\left(\frac{4}{3};1-m,-p;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)+a d p F_1\left(\frac{4}{3};-m,1-p;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)+4 a c F_1\left(\frac{1}{3};-m,-p;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}","x \left(a+b x^3\right)^m \left(\frac{b x^3}{a}+1\right)^{-m} \left(c+d x^3\right)^p \left(\frac{d x^3}{c}+1\right)^{-p} F_1\left(\frac{1}{3};-m,-p;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)",1,"(4*a*c*x*(a + b*x^3)^m*(c + d*x^3)^p*AppellF1[1/3, -m, -p, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(4*a*c*AppellF1[1/3, -m, -p, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + 3*x^3*(b*c*m*AppellF1[4/3, 1 - m, -p, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + a*d*p*AppellF1[4/3, -m, 1 - p, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))","B",0
134,1,106,167,0.079487,"\int \left(a+b x^3\right)^2 \left(c+d x^3\right)^q \, dx","Integrate[(a + b*x^3)^2*(c + d*x^3)^q,x]","\frac{1}{14} x \left(c+d x^3\right)^q \left(\frac{d x^3}{c}+1\right)^{-q} \left(14 a^2 \, _2F_1\left(\frac{1}{3},-q;\frac{4}{3};-\frac{d x^3}{c}\right)+b x^3 \left(7 a \, _2F_1\left(\frac{4}{3},-q;\frac{7}{3};-\frac{d x^3}{c}\right)+2 b x^3 \, _2F_1\left(\frac{7}{3},-q;\frac{10}{3};-\frac{d x^3}{c}\right)\right)\right)","\frac{x \left(c+d x^3\right)^{q+1} \left(a^2 d^2 \left(9 q^2+33 q+28\right)-2 a b c d (3 q+7)+4 b^2 c^2\right) \, _2F_1\left(1,q+\frac{4}{3};\frac{4}{3};-\frac{d x^3}{c}\right)}{c d^2 (3 q+4) (3 q+7)}-\frac{b x \left(c+d x^3\right)^{q+1} (4 b c-a d (3 q+10))}{d^2 (3 q+4) (3 q+7)}+\frac{b x \left(a+b x^3\right) \left(c+d x^3\right)^{q+1}}{d (3 q+7)}",1,"(x*(c + d*x^3)^q*(14*a^2*Hypergeometric2F1[1/3, -q, 4/3, -((d*x^3)/c)] + b*x^3*(7*a*Hypergeometric2F1[4/3, -q, 7/3, -((d*x^3)/c)] + 2*b*x^3*Hypergeometric2F1[7/3, -q, 10/3, -((d*x^3)/c)])))/(14*(1 + (d*x^3)/c)^q)","A",1
135,1,90,84,0.0369231,"\int \left(a+b x^3\right) \left(c+d x^3\right)^q \, dx","Integrate[(a + b*x^3)*(c + d*x^3)^q,x]","\frac{x \left(c+d x^3\right)^q \left(\frac{d x^3}{c}+1\right)^{-q} \left((a d (3 q+4)-b c) \, _2F_1\left(\frac{1}{3},-q;\frac{4}{3};-\frac{d x^3}{c}\right)+b \left(c+d x^3\right) \left(\frac{d x^3}{c}+1\right)^q\right)}{d (3 q+4)}","\frac{b x \left(c+d x^3\right)^{q+1}}{d (3 q+4)}-\frac{x \left(c+d x^3\right)^{q+1} (b c-a d (3 q+4)) \, _2F_1\left(1,q+\frac{4}{3};\frac{4}{3};-\frac{d x^3}{c}\right)}{c d (3 q+4)}",1,"(x*(c + d*x^3)^q*(b*(c + d*x^3)*(1 + (d*x^3)/c)^q + (-(b*c) + a*d*(4 + 3*q))*Hypergeometric2F1[1/3, -q, 4/3, -((d*x^3)/c)]))/(d*(4 + 3*q)*(1 + (d*x^3)/c)^q)","A",1
136,1,162,57,0.20366,"\int \frac{\left(c+d x^3\right)^q}{a+b x^3} \, dx","Integrate[(c + d*x^3)^q/(a + b*x^3),x]","\frac{4 a c x \left(c+d x^3\right)^q F_1\left(\frac{1}{3};-q,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 q F_1\left(\frac{4}{3};1-q,1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)-b c F_1\left(\frac{4}{3};-q,2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)+4 a c F_1\left(\frac{1}{3};-q,1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right)\right)}","\frac{x \left(c+d x^3\right)^q \left(\frac{d x^3}{c}+1\right)^{-q} F_1\left(\frac{1}{3};1,-q;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a}",1,"(4*a*c*x*(c + d*x^3)^q*AppellF1[1/3, -q, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)])/((a + b*x^3)*(4*a*c*AppellF1[1/3, -q, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(a*d*q*AppellF1[4/3, 1 - q, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)] - b*c*AppellF1[4/3, -q, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)])))","B",0
137,1,162,57,0.2583517,"\int \frac{\left(c+d x^3\right)^q}{\left(a+b x^3\right)^2} \, dx","Integrate[(c + d*x^3)^q/(a + b*x^3)^2,x]","\frac{4 a c x \left(c+d x^3\right)^q F_1\left(\frac{1}{3};2,-q;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\left(a+b x^3\right)^2 \left(3 x^3 \left(a d q F_1\left(\frac{4}{3};2,1-q;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-2 b c F_1\left(\frac{4}{3};3,-q;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)+4 a c F_1\left(\frac{1}{3};2,-q;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}","\frac{x \left(c+d x^3\right)^q \left(\frac{d x^3}{c}+1\right)^{-q} F_1\left(\frac{1}{3};2,-q;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{a^2}",1,"(4*a*c*x*(c + d*x^3)^q*AppellF1[1/3, 2, -q, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((a + b*x^3)^2*(4*a*c*AppellF1[1/3, 2, -q, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + 3*x^3*(a*d*q*AppellF1[4/3, 2, 1 - q, 7/3, -((b*x^3)/a), -((d*x^3)/c)] - 2*b*c*AppellF1[4/3, 3, -q, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))","B",0
138,1,137,298,5.0766971,"\int \left(a+b x^3\right)^m \left(c+d x^3\right)^3 \, dx","Integrate[(a + b*x^3)^m*(c + d*x^3)^3,x]","\frac{1}{140} x \left(a+b x^3\right)^m \left(\frac{b x^3}{a}+1\right)^{-m} \left(140 c^3 \, _2F_1\left(\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right)+d x^3 \left(105 c^2 \, _2F_1\left(\frac{4}{3},-m;\frac{7}{3};-\frac{b x^3}{a}\right)+2 d x^3 \left(30 c \, _2F_1\left(\frac{7}{3},-m;\frac{10}{3};-\frac{b x^3}{a}\right)+7 d x^3 \, _2F_1\left(\frac{10}{3},-m;\frac{13}{3};-\frac{b x^3}{a}\right)\right)\right)\right)","\frac{d x \left(a+b x^3\right)^{m+1} \left(28 a^2 d^2-a b c d (15 m+92)+b^2 c^2 \left(9 m^2+60 m+118\right)\right)}{b^3 (3 m+4) (3 m+7) (3 m+10)}-\frac{x \left(a+b x^3\right)^m \left(\frac{b x^3}{a}+1\right)^{-m} \left(28 a^3 d^3-12 a^2 b c d^2 (3 m+10)+3 a b^2 c^2 d \left(9 m^2+51 m+70\right)-b^3 c^3 \left(27 m^3+189 m^2+414 m+280\right)\right) \, _2F_1\left(\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right)}{b^3 (3 m+4) (3 m+7) (3 m+10)}-\frac{d x \left(c+d x^3\right) \left(a+b x^3\right)^{m+1} (7 a d-b c (3 m+16))}{b^2 (3 m+7) (3 m+10)}+\frac{d x \left(c+d x^3\right)^2 \left(a+b x^3\right)^{m+1}}{b (3 m+10)}",1,"(x*(a + b*x^3)^m*(140*c^3*Hypergeometric2F1[1/3, -m, 4/3, -((b*x^3)/a)] + d*x^3*(105*c^2*Hypergeometric2F1[4/3, -m, 7/3, -((b*x^3)/a)] + 2*d*x^3*(30*c*Hypergeometric2F1[7/3, -m, 10/3, -((b*x^3)/a)] + 7*d*x^3*Hypergeometric2F1[10/3, -m, 13/3, -((b*x^3)/a)]))))/(140*(1 + (b*x^3)/a)^m)","A",1
139,1,106,176,5.0434304,"\int \left(a+b x^3\right)^m \left(c+d x^3\right)^2 \, dx","Integrate[(a + b*x^3)^m*(c + d*x^3)^2,x]","\frac{1}{14} x \left(a+b x^3\right)^m \left(\frac{b x^3}{a}+1\right)^{-m} \left(14 c^2 \, _2F_1\left(\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right)+d x^3 \left(7 c \, _2F_1\left(\frac{4}{3},-m;\frac{7}{3};-\frac{b x^3}{a}\right)+2 d x^3 \, _2F_1\left(\frac{7}{3},-m;\frac{10}{3};-\frac{b x^3}{a}\right)\right)\right)","\frac{x \left(a+b x^3\right)^m \left(\frac{b x^3}{a}+1\right)^{-m} \left(4 a^2 d^2-2 a b c d (3 m+7)+b^2 c^2 \left(9 m^2+33 m+28\right)\right) \, _2F_1\left(\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right)}{b^2 (3 m+4) (3 m+7)}-\frac{d x \left(a+b x^3\right)^{m+1} (4 a d-b c (3 m+10))}{b^2 (3 m+4) (3 m+7)}+\frac{d x \left(c+d x^3\right) \left(a+b x^3\right)^{m+1}}{b (3 m+7)}",1,"(x*(a + b*x^3)^m*(14*c^2*Hypergeometric2F1[1/3, -m, 4/3, -((b*x^3)/a)] + d*x^3*(7*c*Hypergeometric2F1[4/3, -m, 7/3, -((b*x^3)/a)] + 2*d*x^3*Hypergeometric2F1[7/3, -m, 10/3, -((b*x^3)/a)])))/(14*(1 + (b*x^3)/a)^m)","A",1
140,1,90,93,0.0460834,"\int \left(a+b x^3\right)^m \left(c+d x^3\right) \, dx","Integrate[(a + b*x^3)^m*(c + d*x^3),x]","\frac{x \left(a+b x^3\right)^m \left(\frac{b x^3}{a}+1\right)^{-m} \left((b c (3 m+4)-a d) \, _2F_1\left(\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right)+d \left(a+b x^3\right) \left(\frac{b x^3}{a}+1\right)^m\right)}{b (3 m+4)}","\frac{d x \left(a+b x^3\right)^{m+1}}{b (3 m+4)}-\frac{x \left(a+b x^3\right)^m \left(\frac{b x^3}{a}+1\right)^{-m} (a d-b c (3 m+4)) \, _2F_1\left(\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right)}{b (3 m+4)}",1,"(x*(a + b*x^3)^m*(d*(a + b*x^3)*(1 + (b*x^3)/a)^m + (-(a*d) + b*c*(4 + 3*m))*Hypergeometric2F1[1/3, -m, 4/3, -((b*x^3)/a)]))/(b*(4 + 3*m)*(1 + (b*x^3)/a)^m)","A",1
141,1,196,44,0.1984554,"\int \left(a+b x^3\right)^m \, dx","Integrate[(a + b*x^3)^m,x]","\frac{2^{-m} \left((-1)^{2/3} \sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}\right)^{-m} \left(\frac{i \left(\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1\right)}{\sqrt{3}+3 i}\right)^{-m} \left(a+b x^3\right)^m F_1\left(m+1;-m,-m;m+2;-\frac{i \left(\sqrt[3]{b} x+(-1)^{2/3} \sqrt[3]{a}\right)}{\sqrt{3} \sqrt[3]{a}},\frac{-\frac{2 i \sqrt[3]{b} x}{\sqrt[3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right)}{\sqrt[3]{b} (m+1)}","x \left(a+b x^3\right)^m \left(\frac{b x^3}{a}+1\right)^{-m} \, _2F_1\left(\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right)",1,"(((-1)^(2/3)*a^(1/3) + b^(1/3)*x)*(a + b*x^3)^m*AppellF1[1 + m, -m, -m, 2 + m, ((-I)*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(Sqrt[3]*a^(1/3)), (I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3))/(3*I + Sqrt[3])])/(2^m*b^(1/3)*(1 + m)*((a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3)))^m*((I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I + Sqrt[3]))^m)","C",0
142,1,162,57,0.2495384,"\int \frac{\left(a+b x^3\right)^m}{c+d x^3} \, dx","Integrate[(a + b*x^3)^m/(c + d*x^3),x]","-\frac{4 a c x \left(a+b x^3\right)^m F_1\left(\frac{1}{3};-m,1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\left(c+d x^3\right) \left(3 x^3 \left(a d F_1\left(\frac{4}{3};-m,2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-b c m F_1\left(\frac{4}{3};1-m,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};-m,1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}","\frac{x \left(a+b x^3\right)^m \left(\frac{b x^3}{a}+1\right)^{-m} F_1\left(\frac{1}{3};-m,1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c}",1,"(-4*a*c*x*(a + b*x^3)^m*AppellF1[1/3, -m, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((c + d*x^3)*(-4*a*c*AppellF1[1/3, -m, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + 3*x^3*(-(b*c*m*AppellF1[4/3, 1 - m, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)]) + a*d*AppellF1[4/3, -m, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))","B",0
143,1,162,57,0.2569092,"\int \frac{\left(a+b x^3\right)^m}{\left(c+d x^3\right)^2} \, dx","Integrate[(a + b*x^3)^m/(c + d*x^3)^2,x]","-\frac{4 a c x \left(a+b x^3\right)^m F_1\left(\frac{1}{3};-m,2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\left(c+d x^3\right)^2 \left(-3 x^3 \left(b c m F_1\left(\frac{4}{3};1-m,2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-2 a d F_1\left(\frac{4}{3};-m,3;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};-m,2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}","\frac{x \left(a+b x^3\right)^m \left(\frac{b x^3}{a}+1\right)^{-m} F_1\left(\frac{1}{3};-m,2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c^2}",1,"(-4*a*c*x*(a + b*x^3)^m*AppellF1[1/3, -m, 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((c + d*x^3)^2*(-4*a*c*AppellF1[1/3, -m, 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)] - 3*x^3*(b*c*m*AppellF1[4/3, 1 - m, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] - 2*a*d*AppellF1[4/3, -m, 3, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))","B",0
144,1,162,57,0.412697,"\int \frac{\left(a+b x^3\right)^m}{\left(c+d x^3\right)^3} \, dx","Integrate[(a + b*x^3)^m/(c + d*x^3)^3,x]","-\frac{4 a c x \left(a+b x^3\right)^m F_1\left(\frac{1}{3};-m,3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{\left(c+d x^3\right)^3 \left(-3 x^3 \left(b c m F_1\left(\frac{4}{3};1-m,3;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)-3 a d F_1\left(\frac{4}{3};-m,4;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)-4 a c F_1\left(\frac{1}{3};-m,3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)\right)}","\frac{x \left(a+b x^3\right)^m \left(\frac{b x^3}{a}+1\right)^{-m} F_1\left(\frac{1}{3};-m,3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right)}{c^3}",1,"(-4*a*c*x*(a + b*x^3)^m*AppellF1[1/3, -m, 3, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((c + d*x^3)^3*(-4*a*c*AppellF1[1/3, -m, 3, 4/3, -((b*x^3)/a), -((d*x^3)/c)] - 3*x^3*(b*c*m*AppellF1[4/3, 1 - m, 3, 7/3, -((b*x^3)/a), -((d*x^3)/c)] - 3*a*d*AppellF1[4/3, -m, 4, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))","B",0
145,1,52,53,0.0382273,"\int \left(a+b x^3\right)^{-1-\frac{b c}{3 b c-3 a d}} \left(c+d x^3\right)^{-1+\frac{a d}{3 b c-3 a d}} \, dx","Integrate[(a + b*x^3)^(-1 - (b*c)/(3*b*c - 3*a*d))*(c + d*x^3)^(-1 + (a*d)/(3*b*c - 3*a*d)),x]","\frac{x \left(a+b x^3\right)^{\frac{b c}{3 a d-3 b c}} \left(c+d x^3\right)^{\frac{a d}{3 b c-3 a d}}}{a c}","\frac{x \left(a+b x^3\right)^{-\frac{b c}{3 b c-3 a d}} \left(c+d x^3\right)^{\frac{a d}{3 b c-3 a d}}}{a c}",1,"(x*(a + b*x^3)^((b*c)/(-3*b*c + 3*a*d))*(c + d*x^3)^((a*d)/(3*b*c - 3*a*d)))/(a*c)","A",1
146,1,94,94,0.0242932,"\int \left(a+b x^4\right) \left(c+d x^4\right)^4 \, dx","Integrate[(a + b*x^4)*(c + d*x^4)^4,x]","\frac{1}{5} c^3 x^5 (4 a d+b c)+\frac{2}{9} c^2 d x^9 (3 a d+2 b c)+\frac{1}{17} d^3 x^{17} (a d+4 b c)+\frac{2}{13} c d^2 x^{13} (2 a d+3 b c)+a c^4 x+\frac{1}{21} b d^4 x^{21}","\frac{1}{5} c^3 x^5 (4 a d+b c)+\frac{2}{9} c^2 d x^9 (3 a d+2 b c)+\frac{1}{17} d^3 x^{17} (a d+4 b c)+\frac{2}{13} c d^2 x^{13} (2 a d+3 b c)+a c^4 x+\frac{1}{21} b d^4 x^{21}",1,"a*c^4*x + (c^3*(b*c + 4*a*d)*x^5)/5 + (2*c^2*d*(2*b*c + 3*a*d)*x^9)/9 + (2*c*d^2*(3*b*c + 2*a*d)*x^13)/13 + (d^3*(4*b*c + a*d)*x^17)/17 + (b*d^4*x^21)/21","A",1
147,1,70,70,0.0167593,"\int \left(a+b x^4\right) \left(c+d x^4\right)^3 \, dx","Integrate[(a + b*x^4)*(c + d*x^4)^3,x]","\frac{1}{5} c^2 x^5 (3 a d+b c)+\frac{1}{13} d^2 x^{13} (a d+3 b c)+\frac{1}{3} c d x^9 (a d+b c)+a c^3 x+\frac{1}{17} b d^3 x^{17}","\frac{1}{5} c^2 x^5 (3 a d+b c)+\frac{1}{13} d^2 x^{13} (a d+3 b c)+\frac{1}{3} c d x^9 (a d+b c)+a c^3 x+\frac{1}{17} b d^3 x^{17}",1,"a*c^3*x + (c^2*(b*c + 3*a*d)*x^5)/5 + (c*d*(b*c + a*d)*x^9)/3 + (d^2*(3*b*c + a*d)*x^13)/13 + (b*d^3*x^17)/17","A",1
148,1,50,50,0.0121294,"\int \left(a+b x^4\right) \left(c+d x^4\right)^2 \, dx","Integrate[(a + b*x^4)*(c + d*x^4)^2,x]","\frac{1}{9} d x^9 (a d+2 b c)+\frac{1}{5} c x^5 (2 a d+b c)+a c^2 x+\frac{1}{13} b d^2 x^{13}","\frac{1}{9} d x^9 (a d+2 b c)+\frac{1}{5} c x^5 (2 a d+b c)+a c^2 x+\frac{1}{13} b d^2 x^{13}",1,"a*c^2*x + (c*(b*c + 2*a*d)*x^5)/5 + (d*(2*b*c + a*d)*x^9)/9 + (b*d^2*x^13)/13","A",1
149,1,28,28,0.0093542,"\int \left(a+b x^4\right) \left(c+d x^4\right) \, dx","Integrate[(a + b*x^4)*(c + d*x^4),x]","\frac{1}{5} x^5 (a d+b c)+a c x+\frac{1}{9} b d x^9","\frac{1}{5} x^5 (a d+b c)+a c x+\frac{1}{9} b d x^9",1,"a*c*x + ((b*c + a*d)*x^5)/5 + (b*d*x^9)/9","A",1
150,1,196,223,0.178863,"\int \frac{a+b x^4}{c+d x^4} \, dx","Integrate[(a + b*x^4)/(c + d*x^4),x]","\frac{\sqrt{2} (b c-a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)-\sqrt{2} (b c-a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)+2 \sqrt{2} (b c-a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)-2 \sqrt{2} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)+8 b c^{3/4} \sqrt[4]{d} x}{8 c^{3/4} d^{5/4}}","\frac{(b c-a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{3/4} d^{5/4}}-\frac{(b c-a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{3/4} d^{5/4}}+\frac{(b c-a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{2 \sqrt{2} c^{3/4} d^{5/4}}-\frac{(b c-a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} c^{3/4} d^{5/4}}+\frac{b x}{d}",1,"(8*b*c^(3/4)*d^(1/4)*x + 2*Sqrt[2]*(b*c - a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - 2*Sqrt[2]*(b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + Sqrt[2]*(b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] - Sqrt[2]*(b*c - a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(8*c^(3/4)*d^(5/4))","A",1
151,1,212,245,0.2018199,"\int \frac{a+b x^4}{\left(c+d x^4\right)^2} \, dx","Integrate[(a + b*x^4)/(c + d*x^4)^2,x]","\frac{-\frac{8 c^{3/4} \sqrt[4]{d} x (b c-a d)}{c+d x^4}-\sqrt{2} (3 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)+\sqrt{2} (3 a d+b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)-2 \sqrt{2} (3 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)+2 \sqrt{2} (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{32 c^{7/4} d^{5/4}}","-\frac{(3 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{16 \sqrt{2} c^{7/4} d^{5/4}}+\frac{(3 a d+b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{16 \sqrt{2} c^{7/4} d^{5/4}}-\frac{(3 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{8 \sqrt{2} c^{7/4} d^{5/4}}+\frac{(3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{8 \sqrt{2} c^{7/4} d^{5/4}}-\frac{x (b c-a d)}{4 c d \left(c+d x^4\right)}",1,"((-8*c^(3/4)*d^(1/4)*(b*c - a*d)*x)/(c + d*x^4) - 2*Sqrt[2]*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 2*Sqrt[2]*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - Sqrt[2]*(b*c + 3*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + Sqrt[2]*(b*c + 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(32*c^(7/4)*d^(5/4))","A",1
152,1,243,273,0.2229434,"\int \frac{a+b x^4}{\left(c+d x^4\right)^3} \, dx","Integrate[(a + b*x^4)/(c + d*x^4)^3,x]","\frac{-\frac{32 c^{7/4} \sqrt[4]{d} x (b c-a d)}{\left(c+d x^4\right)^2}+\frac{8 c^{3/4} \sqrt[4]{d} x (7 a d+b c)}{c+d x^4}-3 \sqrt{2} (7 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)+3 \sqrt{2} (7 a d+b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)-6 \sqrt{2} (7 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)+6 \sqrt{2} (7 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{256 c^{11/4} d^{5/4}}","-\frac{3 (7 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{128 \sqrt{2} c^{11/4} d^{5/4}}+\frac{3 (7 a d+b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{128 \sqrt{2} c^{11/4} d^{5/4}}-\frac{3 (7 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{64 \sqrt{2} c^{11/4} d^{5/4}}+\frac{3 (7 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{64 \sqrt{2} c^{11/4} d^{5/4}}+\frac{x (7 a d+b c)}{32 c^2 d \left(c+d x^4\right)}-\frac{x (b c-a d)}{8 c d \left(c+d x^4\right)^2}",1,"((-32*c^(7/4)*d^(1/4)*(b*c - a*d)*x)/(c + d*x^4)^2 + (8*c^(3/4)*d^(1/4)*(b*c + 7*a*d)*x)/(c + d*x^4) - 6*Sqrt[2]*(b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 6*Sqrt[2]*(b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - 3*Sqrt[2]*(b*c + 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + 3*Sqrt[2]*(b*c + 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(256*c^(11/4)*d^(5/4))","A",1
153,1,154,154,0.0332278,"\int \left(a+b x^4\right)^2 \left(c+d x^4\right)^4 \, dx","Integrate[(a + b*x^4)^2*(c + d*x^4)^4,x]","\frac{1}{17} d^2 x^{17} \left(a^2 d^2+8 a b c d+6 b^2 c^2\right)+\frac{4}{13} c d x^{13} \left(a^2 d^2+3 a b c d+b^2 c^2\right)+\frac{1}{9} c^2 x^9 \left(6 a^2 d^2+8 a b c d+b^2 c^2\right)+a^2 c^4 x+\frac{2}{5} a c^3 x^5 (2 a d+b c)+\frac{2}{21} b d^3 x^{21} (a d+2 b c)+\frac{1}{25} b^2 d^4 x^{25}","\frac{1}{17} d^2 x^{17} \left(a^2 d^2+8 a b c d+6 b^2 c^2\right)+\frac{4}{13} c d x^{13} \left(a^2 d^2+3 a b c d+b^2 c^2\right)+\frac{1}{9} c^2 x^9 \left(6 a^2 d^2+8 a b c d+b^2 c^2\right)+a^2 c^4 x+\frac{2}{5} a c^3 x^5 (2 a d+b c)+\frac{2}{21} b d^3 x^{21} (a d+2 b c)+\frac{1}{25} b^2 d^4 x^{25}",1,"a^2*c^4*x + (2*a*c^3*(b*c + 2*a*d)*x^5)/5 + (c^2*(b^2*c^2 + 8*a*b*c*d + 6*a^2*d^2)*x^9)/9 + (4*c*d*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^13)/13 + (d^2*(6*b^2*c^2 + 8*a*b*c*d + a^2*d^2)*x^17)/17 + (2*b*d^3*(2*b*c + a*d)*x^21)/21 + (b^2*d^4*x^25)/25","A",1
154,1,122,122,0.0234462,"\int \left(a+b x^4\right)^2 \left(c+d x^4\right)^3 \, dx","Integrate[(a + b*x^4)^2*(c + d*x^4)^3,x]","\frac{1}{13} d x^{13} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{9} c x^9 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 x+\frac{1}{5} a c^2 x^5 (3 a d+2 b c)+\frac{1}{17} b d^2 x^{17} (2 a d+3 b c)+\frac{1}{21} b^2 d^3 x^{21}","\frac{1}{13} d x^{13} \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+\frac{1}{9} c x^9 \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a^2 c^3 x+\frac{1}{5} a c^2 x^5 (3 a d+2 b c)+\frac{1}{17} b d^2 x^{17} (2 a d+3 b c)+\frac{1}{21} b^2 d^3 x^{21}",1,"a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^5)/5 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^9)/9 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^13)/13 + (b*d^2*(3*b*c + 2*a*d)*x^17)/17 + (b^2*d^3*x^21)/21","A",1
155,1,82,82,0.017078,"\int \left(a+b x^4\right)^2 \left(c+d x^4\right)^2 \, dx","Integrate[(a + b*x^4)^2*(c + d*x^4)^2,x]","\frac{1}{9} x^9 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 x+\frac{2}{13} b d x^{13} (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{17} b^2 d^2 x^{17}","\frac{1}{9} x^9 \left(a^2 d^2+4 a b c d+b^2 c^2\right)+a^2 c^2 x+\frac{2}{13} b d x^{13} (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{17} b^2 d^2 x^{17}",1,"a^2*c^2*x + (2*a*c*(b*c + a*d)*x^5)/5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^9)/9 + (2*b*d*(b*c + a*d)*x^13)/13 + (b^2*d^2*x^17)/17","A",1
156,1,50,50,0.0084063,"\int \left(a+b x^4\right)^2 \left(c+d x^4\right) \, dx","Integrate[(a + b*x^4)^2*(c + d*x^4),x]","a^2 c x+\frac{1}{9} b x^9 (2 a d+b c)+\frac{1}{5} a x^5 (a d+2 b c)+\frac{1}{13} b^2 d x^{13}","a^2 c x+\frac{1}{9} b x^9 (2 a d+b c)+\frac{1}{5} a x^5 (a d+2 b c)+\frac{1}{13} b^2 d x^{13}",1,"a^2*c*x + (a*(2*b*c + a*d)*x^5)/5 + (b*(b*c + 2*a*d)*x^9)/9 + (b^2*d*x^13)/13","A",1
157,1,231,253,0.1310107,"\int \frac{\left(a+b x^4\right)^2}{c+d x^4} \, dx","Integrate[(a + b*x^4)^2/(c + d*x^4),x]","\frac{-40 b c^{3/4} \sqrt[4]{d} x (b c-2 a d)-5 \sqrt{2} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)+5 \sqrt{2} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)-10 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)+10 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)+8 b^2 c^{3/4} d^{5/4} x^5}{40 c^{3/4} d^{9/4}}","-\frac{(b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{3/4} d^{9/4}}+\frac{(b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{4 \sqrt{2} c^{3/4} d^{9/4}}-\frac{(b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{2 \sqrt{2} c^{3/4} d^{9/4}}+\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{2 \sqrt{2} c^{3/4} d^{9/4}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^5}{5 d}",1,"(-40*b*c^(3/4)*d^(1/4)*(b*c - 2*a*d)*x + 8*b^2*c^(3/4)*d^(5/4)*x^5 - 10*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 10*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - 5*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + 5*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(40*c^(3/4)*d^(9/4))","A",1
158,1,298,291,0.2158494,"\int \frac{\left(a+b x^4\right)^2}{\left(c+d x^4\right)^2} \, dx","Integrate[(a + b*x^4)^2/(c + d*x^4)^2,x]","\frac{\frac{\sqrt{2} \left(-3 a^2 d^2-2 a b c d+5 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{c^{7/4}}-\frac{\sqrt{2} \left(-3 a^2 d^2-2 a b c d+5 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{c^{7/4}}+\frac{2 \sqrt{2} \left(-3 a^2 d^2-2 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{c^{7/4}}-\frac{2 \sqrt{2} \left(-3 a^2 d^2-2 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{c^{7/4}}+\frac{8 \sqrt[4]{d} x (b c-a d)^2}{c \left(c+d x^4\right)}+32 b^2 \sqrt[4]{d} x}{32 d^{9/4}}","\frac{(b c-a d) (3 a d+5 b c) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{16 \sqrt{2} c^{7/4} d^{9/4}}-\frac{(b c-a d) (3 a d+5 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{16 \sqrt{2} c^{7/4} d^{9/4}}+\frac{(b c-a d) (3 a d+5 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{8 \sqrt{2} c^{7/4} d^{9/4}}-\frac{(b c-a d) (3 a d+5 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{8 \sqrt{2} c^{7/4} d^{9/4}}+\frac{x (b c-a d)^2}{4 c d^2 \left(c+d x^4\right)}+\frac{b^2 x}{d^2}",1,"(32*b^2*d^(1/4)*x + (8*d^(1/4)*(b*c - a*d)^2*x)/(c*(c + d*x^4)) + (2*Sqrt[2]*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(7/4) - (2*Sqrt[2]*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(7/4) + (Sqrt[2]*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c^(7/4) - (Sqrt[2]*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/c^(7/4))/(32*d^(9/4))","A",1
159,1,319,349,0.2271853,"\int \frac{\left(a+b x^4\right)^2}{\left(c+d x^4\right)^3} \, dx","Integrate[(a + b*x^4)^2/(c + d*x^4)^3,x]","\frac{-\sqrt{2} \left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)+\sqrt{2} \left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)-2 \sqrt{2} \left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)+2 \sqrt{2} \left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)-\frac{8 c^{3/4} \sqrt[4]{d} x \left(-7 a^2 d^2-2 a b c d+9 b^2 c^2\right)}{c+d x^4}+\frac{32 c^{7/4} \sqrt[4]{d} x (b c-a d)^2}{\left(c+d x^4\right)^2}}{256 c^{11/4} d^{9/4}}","-\frac{\left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{128 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{128 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{64 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left(21 a^2 d^2+6 a b c d+5 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{64 \sqrt{2} c^{11/4} d^{9/4}}-\frac{x (b c-a d) (7 a d+5 b c)}{32 c^2 d^2 \left(c+d x^4\right)}-\frac{x \left(a+b x^4\right) (b c-a d)}{8 c d \left(c+d x^4\right)^2}",1,"((32*c^(7/4)*d^(1/4)*(b*c - a*d)^2*x)/(c + d*x^4)^2 - (8*c^(3/4)*d^(1/4)*(9*b^2*c^2 - 2*a*b*c*d - 7*a^2*d^2)*x)/(c + d*x^4) - 2*Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 2*Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(256*c^(11/4)*d^(9/4))","A",1
160,1,322,332,0.2237989,"\int \frac{\left(c+d x^4\right)^4}{a+b x^4} \, dx","Integrate[(c + d*x^4)^4/(a + b*x^4),x]","\frac{-\frac{585 \sqrt{2} (b c-a d)^4 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{3/4}}+\frac{585 \sqrt{2} (b c-a d)^4 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{3/4}}-\frac{1170 \sqrt{2} (b c-a d)^4 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{a^{3/4}}+\frac{1170 \sqrt{2} (b c-a d)^4 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{a^{3/4}}+936 b^{5/4} d^2 x^5 \left(a^2 d^2-4 a b c d+6 b^2 c^2\right)+4680 \sqrt[4]{b} d x \left(-a^3 d^3+4 a^2 b c d^2-6 a b^2 c^2 d+4 b^3 c^3\right)+520 b^{9/4} d^3 x^9 (4 b c-a d)+360 b^{13/4} d^4 x^{13}}{4680 b^{17/4}}","-\frac{(b c-a d)^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^{17/4}}+\frac{(b c-a d)^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^{17/4}}-\frac{(b c-a d)^4 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} b^{17/4}}+\frac{d x (2 b c-a d) \left(a^2 d^2-2 a b c d+2 b^2 c^2\right)}{b^4}+\frac{d^2 x^5 \left(a^2 d^2-4 a b c d+6 b^2 c^2\right)}{5 b^3}+\frac{d^3 x^9 (4 b c-a d)}{9 b^2}+\frac{d^4 x^{13}}{13 b}",1,"(4680*b^(1/4)*d*(4*b^3*c^3 - 6*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*x + 936*b^(5/4)*d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^5 + 520*b^(9/4)*d^3*(4*b*c - a*d)*x^9 + 360*b^(13/4)*d^4*x^13 - (1170*Sqrt[2]*(b*c - a*d)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) + (1170*Sqrt[2]*(b*c - a*d)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(3/4) - (585*Sqrt[2]*(b*c - a*d)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) + (585*Sqrt[2]*(b*c - a*d)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4))/(4680*b^(17/4))","A",1
161,1,271,288,0.1655988,"\int \frac{\left(c+d x^4\right)^3}{a+b x^4} \, dx","Integrate[(c + d*x^4)^3/(a + b*x^4),x]","\frac{-72 a^{3/4} b^{5/4} d^2 x^5 (a d-3 b c)+40 a^{3/4} b^{9/4} d^3 x^9+360 a^{3/4} \sqrt[4]{b} d x \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)-45 \sqrt{2} (b c-a d)^3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+45 \sqrt{2} (b c-a d)^3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-90 \sqrt{2} (b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+90 \sqrt{2} (b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{360 a^{3/4} b^{13/4}}","-\frac{(b c-a d)^3 \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^{13/4}}+\frac{(b c-a d)^3 \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^{13/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} b^{13/4}}+\frac{d x \left(a^2 d^2-3 a b c d+3 b^2 c^2\right)}{b^3}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{d^3 x^9}{9 b}",1,"(360*a^(3/4)*b^(1/4)*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x - 72*a^(3/4)*b^(5/4)*d^2*(-3*b*c + a*d)*x^5 + 40*a^(3/4)*b^(9/4)*d^3*x^9 - 90*Sqrt[2]*(b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 90*Sqrt[2]*(b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 45*Sqrt[2]*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 45*Sqrt[2]*(b*c - a*d)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(360*a^(3/4)*b^(13/4))","A",1
162,1,231,253,0.1316705,"\int \frac{\left(c+d x^4\right)^2}{a+b x^4} \, dx","Integrate[(c + d*x^4)^2/(a + b*x^4),x]","\frac{8 a^{3/4} b^{5/4} d^2 x^5-40 a^{3/4} \sqrt[4]{b} d x (a d-2 b c)-5 \sqrt{2} (b c-a d)^2 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+5 \sqrt{2} (b c-a d)^2 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-10 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+10 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{40 a^{3/4} b^{9/4}}","-\frac{(b c-a d)^2 \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^{9/4}}+\frac{(b c-a d)^2 \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^{9/4}}-\frac{(b c-a d)^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} b^{9/4}}+\frac{(b c-a d)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} b^{9/4}}+\frac{d x (2 b c-a d)}{b^2}+\frac{d^2 x^5}{5 b}",1,"(-40*a^(3/4)*b^(1/4)*d*(-2*b*c + a*d)*x + 8*a^(3/4)*b^(5/4)*d^2*x^5 - 10*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 10*Sqrt[2]*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 5*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 5*Sqrt[2]*(b*c - a*d)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(40*a^(3/4)*b^(9/4))","A",1
163,1,196,223,0.1513753,"\int \frac{c+d x^4}{a+b x^4} \, dx","Integrate[(c + d*x^4)/(a + b*x^4),x]","\frac{8 a^{3/4} \sqrt[4]{b} d x-\sqrt{2} (b c-a d) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+\sqrt{2} (b c-a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-2 \sqrt{2} (b c-a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+2 \sqrt{2} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 a^{3/4} b^{5/4}}","-\frac{(b c-a d) \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^{5/4}}+\frac{(b c-a d) \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^{5/4}}-\frac{(b c-a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{(b c-a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{d x}{b}",1,"(8*a^(3/4)*b^(1/4)*d*x - 2*Sqrt[2]*(b*c - a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*Sqrt[2]*(b*c - a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - Sqrt[2]*(b*c - a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(b*c - a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(8*a^(3/4)*b^(5/4))","A",1
164,1,340,449,0.1432928,"\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
165,1,498,513,0.3347134,"\int \frac{1}{\left(a+b x^4\right) \left(c+d x^4\right)^2} \, dx","Integrate[1/((a + b*x^4)*(c + d*x^4)^2),x]","\frac{8 a^{3/4} c^{3/4} d x (a d-b c)-2 \sqrt{2} a^{3/4} d^{3/4} \left(c+d x^4\right) (3 a d-7 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)+2 \sqrt{2} a^{3/4} d^{3/4} \left(c+d x^4\right) (3 a d-7 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)+\sqrt{2} a^{3/4} d^{3/4} \left(c+d x^4\right) (7 b c-3 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)+\sqrt{2} a^{3/4} d^{3/4} \left(c+d x^4\right) (3 a d-7 b c) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)-8 \sqrt{2} b^{7/4} c^{7/4} \left(c+d x^4\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+8 \sqrt{2} b^{7/4} c^{7/4} \left(c+d x^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)-4 \sqrt{2} b^{7/4} c^{7/4} \left(c+d x^4\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+4 \sqrt{2} b^{7/4} c^{7/4} \left(c+d x^4\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{32 a^{3/4} c^{7/4} \left(c+d x^4\right) (b c-a d)^2}","-\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^{3/4} (b c-a d)^2}+\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^{3/4} (b c-a d)^2}-\frac{b^{7/4} \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)^2}+\frac{b^{7/4} \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)^2}+\frac{d^{3/4} (7 b c-3 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{16 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d^{3/4} (7 b c-3 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{16 \sqrt{2} c^{7/4} (b c-a d)^2}+\frac{d^{3/4} (7 b c-3 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{8 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d^{3/4} (7 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{8 \sqrt{2} c^{7/4} (b c-a d)^2}-\frac{d x}{4 c \left(c+d x^4\right) (b c-a d)}",1,"(8*a^(3/4)*c^(3/4)*d*(-(b*c) + a*d)*x - 8*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 8*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 2*Sqrt[2]*a^(3/4)*d^(3/4)*(-7*b*c + 3*a*d)*(c + d*x^4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 2*Sqrt[2]*a^(3/4)*d^(3/4)*(-7*b*c + 3*a*d)*(c + d*x^4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - 4*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 4*Sqrt[2]*b^(7/4)*c^(7/4)*(c + d*x^4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*a^(3/4)*d^(3/4)*(7*b*c - 3*a*d)*(c + d*x^4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + Sqrt[2]*a^(3/4)*d^(3/4)*(-7*b*c + 3*a*d)*(c + d*x^4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(32*a^(3/4)*c^(7/4)*(b*c - a*d)^2*(c + d*x^4))","A",1
166,1,391,407,0.4529031,"\int \frac{\left(c+d x^4\right)^5}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x^4)^5/(a + b*x^4)^2,x]","\frac{-\frac{585 \sqrt{2} (b c-a d)^4 (17 a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}+\frac{585 \sqrt{2} (b c-a d)^4 (17 a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}-\frac{1170 \sqrt{2} (b c-a d)^4 (17 a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{a^{7/4}}+\frac{1170 \sqrt{2} (b c-a d)^4 (17 a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{a^{7/4}}+3744 b^{5/4} d^3 x^5 \left(3 a^2 d^2-10 a b c d+10 b^2 c^2\right)+18720 \sqrt[4]{b} d^2 x \left(-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right)+2080 b^{9/4} d^4 x^9 (5 b c-2 a d)+\frac{4680 \sqrt[4]{b} x (b c-a d)^5}{a \left(a+b x^4\right)}+1440 b^{13/4} d^5 x^{13}}{18720 b^{21/4}}","-\frac{(b c-a d)^4 (17 a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{21/4}}+\frac{(b c-a d)^4 (17 a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{21/4}}-\frac{(b c-a d)^4 (17 a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} b^{21/4}}+\frac{(b c-a d)^4 (17 a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} b^{21/4}}+\frac{d^3 x^5 \left(3 a^2 d^2-10 a b c d+10 b^2 c^2\right)}{5 b^4}+\frac{d^2 x \left(-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right)}{b^5}+\frac{x (b c-a d)^5}{4 a b^5 \left(a+b x^4\right)}+\frac{d^4 x^9 (5 b c-2 a d)}{9 b^3}+\frac{d^5 x^{13}}{13 b^2}",1,"(18720*b^(1/4)*d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x + 3744*b^(5/4)*d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*x^5 + 2080*b^(9/4)*d^4*(5*b*c - 2*a*d)*x^9 + 1440*b^(13/4)*d^5*x^13 + (4680*b^(1/4)*(b*c - a*d)^5*x)/(a*(a + b*x^4)) - (1170*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (1170*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) - (585*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (585*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4))/(18720*b^(21/4))","A",1
167,1,341,357,0.3489001,"\int \frac{\left(c+d x^4\right)^4}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x^4)^4/(a + b*x^4)^2,x]","\frac{\frac{45 \sqrt{2} (a d-b c)^3 (13 a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}+\frac{45 \sqrt{2} (b c-a d)^3 (13 a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}+\frac{90 \sqrt{2} (a d-b c)^3 (13 a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{a^{7/4}}+\frac{90 \sqrt{2} (b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{a^{7/4}}+1440 \sqrt[4]{b} d^2 x \left(3 a^2 d^2-8 a b c d+6 b^2 c^2\right)+576 b^{5/4} d^3 x^5 (2 b c-a d)+\frac{360 \sqrt[4]{b} x (b c-a d)^4}{a \left(a+b x^4\right)}+160 b^{9/4} d^4 x^9}{1440 b^{17/4}}","-\frac{(b c-a d)^3 (13 a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (13 a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{17/4}}-\frac{(b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} b^{17/4}}+\frac{d^2 x \left(3 a^2 d^2-8 a b c d+6 b^2 c^2\right)}{b^4}+\frac{x (b c-a d)^4}{4 a b^4 \left(a+b x^4\right)}+\frac{2 d^3 x^5 (2 b c-a d)}{5 b^3}+\frac{d^4 x^9}{9 b^2}",1,"(1440*b^(1/4)*d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x + 576*b^(5/4)*d^3*(2*b*c - a*d)*x^5 + 160*b^(9/4)*d^4*x^9 + (360*b^(1/4)*(b*c - a*d)^4*x)/(a*(a + b*x^4)) + (90*Sqrt[2]*(-(b*c) + a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (90*Sqrt[2]*(b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (45*Sqrt[2]*(-(b*c) + a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (45*Sqrt[2]*(b*c - a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4))/(1440*b^(17/4))","A",1
168,1,301,317,0.2541598,"\int \frac{\left(c+d x^4\right)^3}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x^4)^3/(a + b*x^4)^2,x]","\frac{-\frac{15 \sqrt{2} (b c-a d)^2 (3 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}+\frac{15 \sqrt{2} (b c-a d)^2 (3 a d+b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}-\frac{30 \sqrt{2} (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{a^{7/4}}+\frac{30 \sqrt{2} (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{a^{7/4}}+160 \sqrt[4]{b} d^2 x (3 b c-2 a d)+\frac{40 \sqrt[4]{b} x (b c-a d)^3}{a \left(a+b x^4\right)}+32 b^{5/4} d^3 x^5}{160 b^{13/4}}","-\frac{3 (b c-a d)^2 (3 a d+b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{13/4}}-\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{4 a b^3 \left(a+b x^4\right)}+\frac{d^3 x^5}{5 b^2}",1,"(160*b^(1/4)*d^2*(3*b*c - 2*a*d)*x + 32*b^(5/4)*d^3*x^5 + (40*b^(1/4)*(b*c - a*d)^3*x)/(a*(a + b*x^4)) - (30*Sqrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (30*Sqrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) - (15*Sqrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (15*Sqrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4))/(160*b^(13/4))","A",1
169,1,297,291,0.227079,"\int \frac{\left(c+d x^4\right)^2}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x^4)^2/(a + b*x^4)^2,x]","\frac{\frac{\sqrt{2} \left(5 a^2 d^2-2 a b c d-3 b^2 c^2\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}+\frac{\sqrt{2} \left(-5 a^2 d^2+2 a b c d+3 b^2 c^2\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{a^{7/4}}+\frac{2 \sqrt{2} \left(5 a^2 d^2-2 a b c d-3 b^2 c^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{a^{7/4}}+\frac{2 \sqrt{2} \left(-5 a^2 d^2+2 a b c d+3 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{a^{7/4}}+\frac{8 \sqrt[4]{b} x (b c-a d)^2}{a \left(a+b x^4\right)}+32 \sqrt[4]{b} d^2 x}{32 b^{9/4}}","-\frac{(b c-a d) (5 a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(b c-a d) (5 a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} b^{9/4}}+\frac{x (b c-a d)^2}{4 a b^2 \left(a+b x^4\right)}+\frac{d^2 x}{b^2}",1,"(32*b^(1/4)*d^2*x + (8*b^(1/4)*(b*c - a*d)^2*x)/(a*(a + b*x^4)) + (2*Sqrt[2]*(-3*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (2*Sqrt[2]*(3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (Sqrt[2]*(-3*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (Sqrt[2]*(3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4))/(32*b^(9/4))","A",1
170,1,212,245,0.1721049,"\int \frac{c+d x^4}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x^4)/(a + b*x^4)^2,x]","\frac{-\frac{8 a^{3/4} \sqrt[4]{b} x (a d-b c)}{a+b x^4}-\sqrt{2} (a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+\sqrt{2} (a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)-2 \sqrt{2} (a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+2 \sqrt{2} (a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{32 a^{7/4} b^{5/4}}","-\frac{(a d+3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a d+3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} b^{5/4}}-\frac{(a d+3 b c) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a d+3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{x (b c-a d)}{4 a b \left(a+b x^4\right)}",1,"((-8*a^(3/4)*b^(1/4)*(-(b*c) + a*d)*x)/(a + b*x^4) - 2*Sqrt[2]*(3*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*Sqrt[2]*(3*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - Sqrt[2]*(3*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(3*b*c + a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(32*a^(7/4)*b^(5/4))","A",1
171,1,499,513,0.3469845,"\int \frac{1}{\left(a+b x^4\right)^2 \left(c+d x^4\right)} \, dx","Integrate[1/((a + b*x^4)^2*(c + d*x^4)),x]","\frac{8 a^{3/4} b c^{3/4} x (b c-a d)-8 \sqrt{2} a^{7/4} d^{7/4} \left(a+b x^4\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)+8 \sqrt{2} a^{7/4} d^{7/4} \left(a+b x^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)-4 \sqrt{2} a^{7/4} d^{7/4} \left(a+b x^4\right) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)+4 \sqrt{2} a^{7/4} d^{7/4} \left(a+b x^4\right) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)-2 \sqrt{2} b^{3/4} c^{3/4} \left(a+b x^4\right) (3 b c-7 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)+2 \sqrt{2} b^{3/4} c^{3/4} \left(a+b x^4\right) (3 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)-\sqrt{2} b^{3/4} c^{3/4} \left(a+b x^4\right) (3 b c-7 a d) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)+\sqrt{2} b^{3/4} c^{3/4} \left(a+b x^4\right) (3 b c-7 a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{32 a^{7/4} c^{3/4} \left(a+b x^4\right) (b c-a d)^2}","-\frac{b^{3/4} (3 b c-7 a d) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} (b c-a d)^2}+\frac{b^{3/4} (3 b c-7 a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} (b c-a d)^2}-\frac{b^{3/4} (3 b c-7 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^2}+\frac{b^{3/4} (3 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^2}-\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^{3/4} (b c-a d)^2}+\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^{3/4} (b c-a d)^2}-\frac{d^{7/4} \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)^2}+\frac{d^{7/4} \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)^2}+\frac{b x}{4 a \left(a+b x^4\right) (b c-a d)}",1,"(8*a^(3/4)*b*c^(3/4)*(b*c - a*d)*x - 2*Sqrt[2]*b^(3/4)*c^(3/4)*(3*b*c - 7*a*d)*(a + b*x^4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*Sqrt[2]*b^(3/4)*c^(3/4)*(3*b*c - 7*a*d)*(a + b*x^4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - 8*Sqrt[2]*a^(7/4)*d^(7/4)*(a + b*x^4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 8*Sqrt[2]*a^(7/4)*d^(7/4)*(a + b*x^4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - Sqrt[2]*b^(3/4)*c^(3/4)*(3*b*c - 7*a*d)*(a + b*x^4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*b^(3/4)*c^(3/4)*(3*b*c - 7*a*d)*(a + b*x^4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] - 4*Sqrt[2]*a^(7/4)*d^(7/4)*(a + b*x^4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + 4*Sqrt[2]*a^(7/4)*d^(7/4)*(a + b*x^4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(32*a^(7/4)*c^(3/4)*(b*c - a*d)^2*(a + b*x^4))","A",1
172,1,629,596,6.1918585,"\int \frac{1}{\left(a+b x^4\right)^2 \left(c+d x^4\right)^2} \, dx","Integrate[1/((a + b*x^4)^2*(c + d*x^4)^2),x]","\frac{b^{7/4} (11 a d-3 b c) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (11 a d-3 b c) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (11 a d-3 b c) \tan ^{-1}\left(\frac{2 \sqrt[4]{b} x-\sqrt{2} \sqrt[4]{a}}{\sqrt{2} \sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (11 a d-3 b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a}+2 \sqrt[4]{b} x}{\sqrt{2} \sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^2 x}{4 a \left(a+b x^4\right) (a d-b c)^2}+\frac{d^{7/4} (11 b c-3 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{16 \sqrt{2} c^{7/4} (a d-b c)^3}-\frac{d^{7/4} (11 b c-3 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{16 \sqrt{2} c^{7/4} (a d-b c)^3}-\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left(\frac{2 \sqrt[4]{d} x-\sqrt{2} \sqrt[4]{c}}{\sqrt{2} \sqrt[4]{c}}\right)}{8 \sqrt{2} c^{7/4} (a d-b c)^3}-\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c}+2 \sqrt[4]{d} x}{\sqrt{2} \sqrt[4]{c}}\right)}{8 \sqrt{2} c^{7/4} (a d-b c)^3}+\frac{d^2 x}{4 c \left(c+d x^4\right) (b c-a d)^2}","-\frac{b^{7/4} (3 b c-11 a d) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x+\sqrt{a}+\sqrt{b} x^2\right)}{16 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \log \left(-\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{16 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \log \left(\sqrt{2} \sqrt[4]{c} \sqrt[4]{d} x+\sqrt{c}+\sqrt{d} x^2\right)}{16 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}\right)}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} x}{\sqrt[4]{c}}+1\right)}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{b x}{4 a \left(a+b x^4\right) \left(c+d x^4\right) (b c-a d)}+\frac{d x (a d+b c)}{4 a c \left(c+d x^4\right) (b c-a d)^2}",1,"(b^2*x)/(4*a*(-(b*c) + a*d)^2*(a + b*x^4)) + (d^2*x)/(4*c*(b*c - a*d)^2*(c + d*x^4)) - (b^(7/4)*(-3*b*c + 11*a*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*x)/(Sqrt[2]*a^(1/4))])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(7/4)*(-3*b*c + 11*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*x)/(Sqrt[2]*a^(1/4))])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*ArcTan[(-(Sqrt[2]*c^(1/4)) + 2*d^(1/4)*x)/(Sqrt[2]*c^(1/4))])/(8*Sqrt[2]*c^(7/4)*(-(b*c) + a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*x)/(Sqrt[2]*c^(1/4))])/(8*Sqrt[2]*c^(7/4)*(-(b*c) + a*d)^3) + (b^(7/4)*(-3*b*c + 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(7/4)*(-3*b*c + 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (d^(7/4)*(11*b*c - 3*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*(-(b*c) + a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*(-(b*c) + a*d)^3)","A",1
173,1,290,321,0.8198021,"\int \frac{\left(a-b x^4\right)^{5/2}}{c-d x^4} \, dx","Integrate[(a - b*x^4)^(5/2)/(c - d*x^4),x]","\frac{x \left(-\frac{b x^4 \sqrt{1-\frac{b x^4}{a}} \left(47 a^2 d^2-56 a b c d+21 b^2 c^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}{c}+\frac{25 a^2 c \left(21 a^2 d^2-16 a b c d+7 b^2 c^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}{\left(c-d x^4\right) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}+5 b \left(b x^4-a\right) \left(-16 a d+7 b c+3 b d x^4\right)\right)}{105 d^2 \sqrt{a-b x^4}}","\frac{\sqrt[4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left(47 a^2 d^2-56 a b c d+21 b^2 c^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{21 d^3 \sqrt{a-b x^4}}-\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^3 \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c d^3 \sqrt{a-b x^4}}-\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^3 \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c d^3 \sqrt{a-b x^4}}-\frac{b x \sqrt{a-b x^4} (7 b c-13 a d)}{21 d^2}+\frac{b x \left(a-b x^4\right)^{3/2}}{7 d}",1,"(x*(5*b*(-a + b*x^4)*(7*b*c - 16*a*d + 3*b*d*x^4) - (b*(21*b^2*c^2 - 56*a*b*c*d + 47*a^2*d^2)*x^4*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])/c + (25*a^2*c*(7*b^2*c^2 - 16*a*b*c*d + 21*a^2*d^2)*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c])/((c - d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))))/(105*d^2*Sqrt[a - b*x^4])","C",0
174,1,341,277,0.4606481,"\int \frac{\left(a-b x^4\right)^{3/2}}{c-d x^4} \, dx","Integrate[(a - b*x^4)^(3/2)/(c - d*x^4),x]","-\frac{x \left(\frac{5 \left(5 a c \left(3 a^2 d-a b d x^4+b^2 x^4 \left(d x^4-c\right)\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+2 b x^4 \left(a-b x^4\right) \left(c-d x^4\right) \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)\right)}{\left(d x^4-c\right) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}+\frac{b x^4 \sqrt{1-\frac{b x^4}{a}} (5 a d-3 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}{c}\right)}{15 d \sqrt{a-b x^4}}","-\frac{\sqrt[4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} (3 b c-5 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{3 d^2 \sqrt{a-b x^4}}+\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^2 \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c d^2 \sqrt{a-b x^4}}+\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^2 \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c d^2 \sqrt{a-b x^4}}+\frac{b x \sqrt{a-b x^4}}{3 d}",1,"-1/15*(x*((b*(-3*b*c + 5*a*d)*x^4*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])/c + (5*(5*a*c*(3*a^2*d - a*b*d*x^4 + b^2*x^4*(-c + d*x^4))*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*b*x^4*(a - b*x^4)*(c - d*x^4)*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))/((-c + d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))))/(d*Sqrt[a - b*x^4])","C",0
175,1,155,240,0.2091842,"\int \frac{\sqrt{a-b x^4}}{c-d x^4} \, dx","Integrate[Sqrt[a - b*x^4]/(c - d*x^4),x]","-\frac{5 a c x \sqrt{a-b x^4} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}{\left(c-d x^4\right) \left(2 x^4 \left(b c F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)-2 a d F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}","\frac{\sqrt[4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{d \sqrt{a-b x^4}}-\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c d \sqrt{a-b x^4}}-\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c d \sqrt{a-b x^4}}",1,"(-5*a*c*x*Sqrt[a - b*x^4]*AppellF1[1/4, -1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c])/((c - d*x^4)*(-5*a*c*AppellF1[1/4, -1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(-2*a*d*AppellF1[5/4, -1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))","C",0
176,1,156,162,0.1006625,"\int \frac{1}{\sqrt{a-b x^4} \left(c-d x^4\right)} \, dx","Integrate[1/(Sqrt[a - b*x^4]*(c - d*x^4)),x]","-\frac{5 a c x F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}{\sqrt{a-b x^4} \left(d x^4-c\right) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}","\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c \sqrt{a-b x^4}}+\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c \sqrt{a-b x^4}}",1,"(-5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c])/(Sqrt[a - b*x^4]*(-c + d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))","C",0
177,1,381,281,0.3157402,"\int \frac{1}{\left(a-b x^4\right)^{3/2} \left(c-d x^4\right)} \, dx","Integrate[1/((a - b*x^4)^(3/2)*(c - d*x^4)),x]","\frac{5 a c x F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right) \left(b d x^4 \sqrt{1-\frac{b x^4}{a}} \left(d x^4-c\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)-5 c \left(2 a d-2 b c+b d x^4\right)\right)+2 b x^5 \left(c-d x^4\right) \left(5 c-d x^4 \sqrt{1-\frac{b x^4}{a}} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right) \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}{10 a c \sqrt{a-b x^4} \left(d x^4-c\right) (a d-b c) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}","\frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 a^{3/4} \sqrt{a-b x^4} (b c-a d)}+\frac{b x}{2 a \sqrt{a-b x^4} (b c-a d)}-\frac{\sqrt[4]{a} d \sqrt{1-\frac{b x^4}{a}} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c \sqrt{a-b x^4} (b c-a d)}-\frac{\sqrt[4]{a} d \sqrt{1-\frac{b x^4}{a}} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c \sqrt{a-b x^4} (b c-a d)}",1,"(5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c]*(-5*c*(-2*b*c + 2*a*d + b*d*x^4) + b*d*x^4*Sqrt[1 - (b*x^4)/a]*(-c + d*x^4)*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]) + 2*b*x^5*(c - d*x^4)*(5*c - d*x^4*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]))/(10*a*c*(-(b*c) + a*d)*Sqrt[a - b*x^4]*(-c + d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))","C",0
178,1,422,334,0.8530193,"\int \frac{1}{\left(a-b x^4\right)^{5/2} \left(c-d x^4\right)} \, dx","Integrate[1/((a - b*x^4)^(5/2)*(c - d*x^4)),x]","\frac{x \left(\frac{b d x^4 \sqrt{1-\frac{b x^4}{a}} (11 a d-5 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}{c}-\frac{5 \left(2 b x^4 \left(d x^4-c\right) \left(13 a^2 d-a b \left(7 c+11 d x^4\right)+5 b^2 c x^4\right) \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c \left(12 a^3 d^2+a^2 b d \left(d x^4-24 c\right)+a b^2 \left(12 c^2+15 c d x^4-11 d^2 x^8\right)+5 b^3 c x^4 \left(d x^4-2 c\right)\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}{\left(a-b x^4\right) \left(d x^4-c\right) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}\right)}{60 a^2 \sqrt{a-b x^4} (b c-a d)^2}","\frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} (5 b c-11 a d) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{12 a^{7/4} \sqrt{a-b x^4} (b c-a d)^2}+\frac{b x (5 b c-11 a d)}{12 a^2 \sqrt{a-b x^4} (b c-a d)^2}+\frac{\sqrt[4]{a} d^2 \sqrt{1-\frac{b x^4}{a}} \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c \sqrt{a-b x^4} (b c-a d)^2}+\frac{\sqrt[4]{a} d^2 \sqrt{1-\frac{b x^4}{a}} \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c \sqrt{a-b x^4} (b c-a d)^2}+\frac{b x}{6 a \left(a-b x^4\right)^{3/2} (b c-a d)}",1,"(x*((b*d*(-5*b*c + 11*a*d)*x^4*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])/c - (5*(5*a*c*(12*a^3*d^2 + a^2*b*d*(-24*c + d*x^4) + 5*b^3*c*x^4*(-2*c + d*x^4) + a*b^2*(12*c^2 + 15*c*d*x^4 - 11*d^2*x^8))*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*b*x^4*(-c + d*x^4)*(13*a^2*d + 5*b^2*c*x^4 - a*b*(7*c + 11*d*x^4))*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))/((a - b*x^4)*(-c + d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))))/(60*a^2*(b*c - a*d)^2*Sqrt[a - b*x^4])","C",0
179,1,346,926,0.5175603,"\int \frac{\left(a+b x^4\right)^{3/2}}{c+d x^4} \, dx","Integrate[(a + b*x^4)^(3/2)/(c + d*x^4),x]","\frac{x \left(\frac{5 \left(2 b x^4 \left(a+b x^4\right) \left(c+d x^4\right) \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c \left(3 a^2 d+a b d x^4+b^2 x^4 \left(c+d x^4\right)\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}{\left(c+d x^4\right) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}+\frac{b x^4 \sqrt{\frac{b x^4}{a}+1} (5 a d-3 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{c}\right)}{15 d \sqrt{a+b x^4}}","\frac{\sqrt[4]{b} \left(\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) (b c-a d)^2}{4 \sqrt[4]{a} \sqrt{-c} d^2 (b c+a d) \sqrt{b x^4+a}}+\frac{\sqrt[4]{b} \left(\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) (b c-a d)^2}{4 \sqrt[4]{a} \sqrt{-c} d^2 (b c+a d) \sqrt{b x^4+a}}+\frac{\left(\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right)^2 \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\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]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) (b c-a d)^2}{8 \sqrt[4]{a} \sqrt[4]{b} c d^2 (b c+a d) \sqrt{b x^4+a}}+\frac{\left(\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right)^2 \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\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]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) (b c-a d)^2}{8 \sqrt[4]{a} \sqrt[4]{b} c d^2 (b c+a d) \sqrt{b x^4+a}}-\frac{\tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-c} \sqrt[4]{d} \sqrt{b x^4+a}}\right) (b c-a d)^{3/2}}{4 (-c)^{3/4} d^{7/4}}-\frac{(a d-b c)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-c} \sqrt[4]{d} \sqrt{b x^4+a}}\right)}{4 (-c)^{3/4} d^{7/4}}-\frac{b^{3/4} (3 b c-5 a d) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{6 \sqrt[4]{a} d^2 \sqrt{b x^4+a}}+\frac{b x \sqrt{b x^4+a}}{3 d}",1,"(x*((b*(-3*b*c + 5*a*d)*x^4*Sqrt[1 + (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/c + (5*(-5*a*c*(3*a^2*d + a*b*d*x^4 + b^2*x^4*(c + d*x^4))*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*b*x^4*(a + b*x^4)*(c + d*x^4)*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((c + d*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(15*d*Sqrt[a + b*x^4])","C",0
180,1,161,881,0.1861759,"\int \frac{\sqrt{a+b x^4}}{c+d x^4} \, dx","Integrate[Sqrt[a + b*x^4]/(c + d*x^4),x]","\frac{5 a c x \sqrt{a+b x^4} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{\left(c+d x^4\right) \left(2 x^4 \left(b c F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)-2 a d F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}","-\frac{(b c-a d) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\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]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right)^2}{8 \sqrt[4]{a} \sqrt[4]{b} c d (b c+a d) \sqrt{b x^4+a}}-\frac{\sqrt[4]{b} (b c-a d) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right)}{4 \sqrt[4]{a} \sqrt{-c} d (b c+a d) \sqrt{b x^4+a}}+\frac{\sqrt{b c-a d} \tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-c} \sqrt[4]{d} \sqrt{b x^4+a}}\right)}{4 (-c)^{3/4} d^{3/4}}-\frac{\sqrt{a d-b c} \tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-c} \sqrt[4]{d} \sqrt{b x^4+a}}\right)}{4 (-c)^{3/4} d^{3/4}}+\frac{b^{3/4} \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} d \sqrt{b x^4+a}}-\frac{\sqrt[4]{b} \left(\sqrt{b}+\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}\right) (b c-a d) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} d (b c+a d) \sqrt{b x^4+a}}-\frac{\left(\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right)^2 (b c-a d) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\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]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{8 \sqrt[4]{a} \sqrt[4]{b} c d (b c+a d) \sqrt{b x^4+a}}",1,"(5*a*c*x*Sqrt[a + b*x^4]*AppellF1[1/4, -1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/((c + d*x^4)*(5*a*c*AppellF1[1/4, -1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*x^4*(-2*a*d*AppellF1[5/4, -1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))","C",0
181,1,161,742,0.0317616,"\int \frac{1}{\sqrt{a+b x^4} \left(c+d x^4\right)} \, dx","Integrate[1/(Sqrt[a + b*x^4]*(c + d*x^4)),x]","-\frac{5 a c x F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{\sqrt{a+b x^4} \left(c+d x^4\right) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}","-\frac{\sqrt[4]{d} \tan ^{-1}\left(\frac{x \sqrt{b c-a d}}{\sqrt[4]{-c} \sqrt[4]{d} \sqrt{a+b x^4}}\right)}{4 (-c)^{3/4} \sqrt{b c-a d}}-\frac{\sqrt[4]{d} \tan ^{-1}\left(\frac{x \sqrt{a d-b c}}{\sqrt[4]{-c} \sqrt[4]{d} \sqrt{a+b x^4}}\right)}{4 (-c)^{3/4} \sqrt{a d-b c}}+\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}+\sqrt{b}\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt{a+b x^4} (a d+b c)}+\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\right)^2}} \left(\sqrt{a} \sqrt{-c} \sqrt{d}+\sqrt{b} c\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} c \sqrt{a+b x^4} (a d+b c)}+\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\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]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{8 \sqrt[4]{a} \sqrt[4]{b} c \sqrt{a+b x^4} (a d+b c)}+\frac{\left(\sqrt{a}+\sqrt{b} x^2\right) \sqrt{\frac{a+b x^4}{\left(\sqrt{a}+\sqrt{b} x^2\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]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{8 \sqrt[4]{a} \sqrt[4]{b} c \sqrt{a+b x^4} (a d+b c)}",1,"(-5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/(Sqrt[a + b*x^4]*(c + d*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))","C",0
182,1,331,913,0.2846561,"\int \frac{1}{\left(a+b x^4\right)^{3/2} \left(c+d x^4\right)} \, dx","Integrate[1/((a + b*x^4)^(3/2)*(c + d*x^4)),x]","\frac{x \left(\frac{5 \left(2 b x^4 \left(c+d x^4\right) \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)+5 a c \left(2 a d-b \left(2 c+d x^4\right)\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}{\left(c+d x^4\right) \left(5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)-2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)\right)}-\frac{b d x^4 \sqrt{\frac{b x^4}{a}+1} F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{c}\right)}{10 a \sqrt{a+b x^4} (a d-b c)}","-\frac{d \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\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]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) \left(\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right)^2}{8 \sqrt[4]{a} \sqrt[4]{b} c (b c-a d) (b c+a d) \sqrt{b x^4+a}}+\frac{d^{5/4} \tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-c} \sqrt[4]{d} \sqrt{b x^4+a}}\right)}{4 (-c)^{3/4} (b c-a d)^{3/2}}-\frac{d^{5/4} \tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-c} \sqrt[4]{d} \sqrt{b x^4+a}}\right)}{4 (-c)^{3/4} (a d-b c)^{3/2}}+\frac{b^{3/4} \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} (b c-a d) \sqrt{b x^4+a}}-\frac{\sqrt[4]{b} \left(\sqrt{b}+\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}\right) d \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} (b c-a d) (b c+a d) \sqrt{b x^4+a}}-\frac{\sqrt[4]{b} \left(\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right) d \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} c \left(b^2 c^2-a^2 d^2\right) \sqrt{b x^4+a}}-\frac{\left(\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right)^2 d \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\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]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{8 \sqrt[4]{a} \sqrt[4]{b} c (b c-a d) (b c+a d) \sqrt{b x^4+a}}+\frac{b x}{2 a (b c-a d) \sqrt{b x^4+a}}",1,"(x*(-((b*d*x^4*Sqrt[1 + (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/c) + (5*(5*a*c*(2*a*d - b*(2*c + d*x^4))*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*b*x^4*(c + d*x^4)*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((c + d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] - 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(10*a*(-(b*c) + a*d)*Sqrt[a + b*x^4])","C",0
183,1,429,976,0.991615,"\int \frac{1}{\left(a+b x^4\right)^{5/2} \left(c+d x^4\right)} \, dx","Integrate[1/((a + b*x^4)^(5/2)*(c + d*x^4)),x]","-\frac{x \left(\frac{5 \left(2 b x^4 \left(c+d x^4\right) \left(13 a^2 d+a b \left(11 d x^4-7 c\right)-5 b^2 c x^4\right) \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)+5 a c \left(12 a^3 d^2-a^2 b d \left(24 c+d x^4\right)+a b^2 \left(12 c^2-15 c d x^4-11 d^2 x^8\right)+5 b^3 c x^4 \left(2 c+d x^4\right)\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}{\left(a+b x^4\right) \left(c+d x^4\right) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}+\frac{b d x^4 \sqrt{\frac{b x^4}{a}+1} (11 a d-5 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{c}\right)}{60 a^2 \sqrt{a+b x^4} (b c-a d)^2}","-\frac{\tan ^{-1}\left(\frac{\sqrt{b c-a d} x}{\sqrt[4]{-c} \sqrt[4]{d} \sqrt{b x^4+a}}\right) d^{9/4}}{4 (-c)^{3/4} (b c-a d)^{5/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a d-b c} x}{\sqrt[4]{-c} \sqrt[4]{d} \sqrt{b x^4+a}}\right) d^{9/4}}{4 (-c)^{3/4} (a d-b c)^{5/2}}+\frac{\sqrt[4]{b} \left(\sqrt{b} c-\sqrt{a} \sqrt{-c} \sqrt{d}\right) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) d^2}{4 \sqrt[4]{a} c (b c-a d)^2 (b c+a d) \sqrt{b x^4+a}}+\frac{\sqrt[4]{b} \left(\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) d^2}{4 \sqrt[4]{a} c (b c-a d)^2 (b c+a d) \sqrt{b x^4+a}}+\frac{\left(\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right)^2 \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\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]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) d^2}{8 \sqrt[4]{a} \sqrt[4]{b} c (b c-a d)^2 (b c+a d) \sqrt{b x^4+a}}+\frac{\left(\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right)^2 \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\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]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) d^2}{8 \sqrt[4]{a} \sqrt[4]{b} c (b c-a d)^2 (b c+a d) \sqrt{b x^4+a}}+\frac{b^{3/4} (5 b c-11 a d) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{24 a^{9/4} (b c-a d)^2 \sqrt{b x^4+a}}+\frac{b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt{b x^4+a}}+\frac{b x}{6 a (b c-a d) \left(b x^4+a\right)^{3/2}}",1,"-1/60*(x*((b*d*(-5*b*c + 11*a*d)*x^4*Sqrt[1 + (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/c + (5*(5*a*c*(12*a^3*d^2 + 5*b^3*c*x^4*(2*c + d*x^4) - a^2*b*d*(24*c + d*x^4) + a*b^2*(12*c^2 - 15*c*d*x^4 - 11*d^2*x^8))*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*b*x^4*(c + d*x^4)*(13*a^2*d - 5*b^2*c*x^4 + a*b*(-7*c + 11*d*x^4))*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((a + b*x^4)*(c + d*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(a^2*(b*c - a*d)^2*Sqrt[a + b*x^4])","C",0
184,1,477,426,0.9183613,"\int \frac{\left(a-b x^4\right)^{7/2}}{\left(c-d x^4\right)^2} \, dx","Integrate[(a - b*x^4)^(7/2)/(c - d*x^4)^2,x]","-\frac{b x^5 \sqrt{1-\frac{b x^4}{a}} \left(21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+\frac{5 c \left(2 x^5 \left(b x^4-a\right) \left(21 a^3 d^3-63 a^2 b c d^2+a b^2 c d \left(155 c-92 d x^4\right)+b^3 c \left(-77 c^2+44 c d x^4+12 d^2 x^8\right)\right) \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c x \left(-84 a^4 d^3+21 a^3 b d^3 x^4+29 a^2 b^2 c d^2 x^4+a b^3 c d x^4 \left(111 c-104 d x^4\right)+b^4 c x^4 \left(-77 c^2+44 c d x^4+12 d^2 x^8\right)\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}{\left(c-d x^4\right) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}}{420 c^2 d^3 \sqrt{a-b x^4}}","-\frac{b x \sqrt{a-b x^4} \left(21 a^2 d^2-122 a b c d+77 b^2 c^2\right)}{84 c d^3}+\frac{\sqrt[4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left(21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{84 c d^4 \sqrt{a-b x^4}}-\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d^4 \sqrt{a-b x^4}}-\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d^4 \sqrt{a-b x^4}}+\frac{b x \left(a-b x^4\right)^{3/2} (11 b c-7 a d)}{28 c d^2}-\frac{x \left(a-b x^4\right)^{5/2} (b c-a d)}{4 c d \left(c-d x^4\right)}",1,"-1/420*(b*(231*b^3*c^3 - 553*a*b^2*c^2*d + 349*a^2*b*c*d^2 + 21*a^3*d^3)*x^5*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c] + (5*c*(5*a*c*x*(-84*a^4*d^3 + 29*a^2*b^2*c*d^2*x^4 + 21*a^3*b*d^3*x^4 + a*b^3*c*d*x^4*(111*c - 104*d*x^4) + b^4*c*x^4*(-77*c^2 + 44*c*d*x^4 + 12*d^2*x^8))*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^5*(-a + b*x^4)*(-63*a^2*b*c*d^2 + 21*a^3*d^3 + a*b^2*c*d*(155*c - 92*d*x^4) + b^3*c*(-77*c^2 + 44*c*d*x^4 + 12*d^2*x^8))*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))/((c - d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]))))/(c^2*d^3*Sqrt[a - b*x^4])","C",0
185,1,396,365,0.6690974,"\int \frac{\left(a-b x^4\right)^{5/2}}{\left(c-d x^4\right)^2} \, dx","Integrate[(a - b*x^4)^(5/2)/(c - d*x^4)^2,x]","-\frac{b x^5 \sqrt{1-\frac{b x^4}{a}} \left(3 a^2 d^2+26 a b c d-21 b^2 c^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+\frac{5 c \left(2 x^5 \left(a-b x^4\right) \left(3 a^2 d^2-6 a b c d+b^2 c \left(7 c-4 d x^4\right)\right) \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c x \left(12 a^3 d^2-3 a^2 b d^2 x^4+2 a b^2 c d x^4+b^3 c x^4 \left(4 d x^4-7 c\right)\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}{\left(d x^4-c\right) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}}{60 c^2 d^2 \sqrt{a-b x^4}}","-\frac{\sqrt[4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left(-3 a^2 d^2-26 a b c d+21 b^2 c^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{12 c d^3 \sqrt{a-b x^4}}+\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+7 b c) (b c-a d)^2 \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d^3 \sqrt{a-b x^4}}+\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+7 b c) (b c-a d)^2 \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d^3 \sqrt{a-b x^4}}+\frac{b x \sqrt{a-b x^4} (7 b c-3 a d)}{12 c d^2}-\frac{x \left(a-b x^4\right)^{3/2} (b c-a d)}{4 c d \left(c-d x^4\right)}",1,"-1/60*(b*(-21*b^2*c^2 + 26*a*b*c*d + 3*a^2*d^2)*x^5*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c] + (5*c*(5*a*c*x*(12*a^3*d^2 + 2*a*b^2*c*d*x^4 - 3*a^2*b*d^2*x^4 + b^3*c*x^4*(-7*c + 4*d*x^4))*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^5*(a - b*x^4)*(-6*a*b*c*d + 3*a^2*d^2 + b^2*c*(7*c - 4*d*x^4))*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))/((-c + d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]))))/(c^2*d^2*Sqrt[a - b*x^4])","C",0
186,1,342,309,0.4360632,"\int \frac{\left(a-b x^4\right)^{3/2}}{\left(c-d x^4\right)^2} \, dx","Integrate[(a - b*x^4)^(3/2)/(c - d*x^4)^2,x]","\frac{x \left(\frac{5 c \left(-5 a c \left(4 a^2 d-a b d x^4+b^2 c x^4\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)-2 x^4 \left(a-b x^4\right) (a d-b c) \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)\right)}{2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}-b x^4 \sqrt{1-\frac{b x^4}{a}} \left(d x^4-c\right) (a d+3 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}{20 c^2 d \sqrt{a-b x^4} \left(d x^4-c\right)}","\frac{\sqrt[4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} (a d+3 b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c d^2 \sqrt{a-b x^4}}-\frac{3 \sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) (a d+b c) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d^2 \sqrt{a-b x^4}}-\frac{3 \sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) (a d+b c) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d^2 \sqrt{a-b x^4}}-\frac{x \sqrt{a-b x^4} (b c-a d)}{4 c d \left(c-d x^4\right)}",1,"(x*(-(b*(3*b*c + a*d)*x^4*Sqrt[1 - (b*x^4)/a]*(-c + d*x^4)*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]) + (5*c*(-5*a*c*(4*a^2*d + b^2*c*x^4 - a*b*d*x^4)*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] - 2*(-(b*c) + a*d)*x^4*(a - b*x^4)*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))/(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]))))/(20*c^2*d*Sqrt[a - b*x^4]*(-c + d*x^4))","C",0
187,1,233,276,0.1296345,"\int \frac{\sqrt{a-b x^4}}{\left(c-d x^4\right)^2} \, dx","Integrate[Sqrt[a - b*x^4]/(c - d*x^4)^2,x]","\frac{x \left(-\frac{75 a^2 F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}{2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}+\frac{b x^4 \sqrt{1-\frac{b x^4}{a}} \left(c-d x^4\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}{c^2}-\frac{5 \left(a-b x^4\right)}{c}\right)}{20 \sqrt{a-b x^4} \left(d x^4-c\right)}","\frac{\sqrt[4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c d \sqrt{a-b x^4}}-\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-3 a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d \sqrt{a-b x^4}}-\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-3 a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d \sqrt{a-b x^4}}+\frac{x \sqrt{a-b x^4}}{4 c \left(c-d x^4\right)}",1,"(x*((-5*(a - b*x^4))/c + (b*x^4*Sqrt[1 - (b*x^4)/a]*(c - d*x^4)*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])/c^2 - (75*a^2*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c])/(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]))))/(20*Sqrt[a - b*x^4]*(-c + d*x^4))","C",0
188,1,386,310,0.2804819,"\int \frac{1}{\sqrt{a-b x^4} \left(c-d x^4\right)^2} \, dx","Integrate[1/(Sqrt[a - b*x^4]*(c - d*x^4)^2),x]","\frac{5 a c x F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right) \left(b d x^4 \sqrt{1-\frac{b x^4}{a}} \left(d x^4-c\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)-5 c \left(-4 a d+4 b c+b d x^4\right)\right)+2 d x^5 \left(b x^4 \sqrt{1-\frac{b x^4}{a}} \left(d x^4-c\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+5 c \left(a-b x^4\right)\right) \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}{20 c^2 \sqrt{a-b x^4} \left(d x^4-c\right) (b c-a d) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}","-\frac{\sqrt[4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{4 c \sqrt{a-b x^4} (b c-a d)}+\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (5 b c-3 a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 \sqrt{a-b x^4} (b c-a d)}+\frac{\sqrt[4]{a} \sqrt{1-\frac{b x^4}{a}} (5 b c-3 a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 \sqrt{a-b x^4} (b c-a d)}-\frac{d x \sqrt{a-b x^4}}{4 c \left(c-d x^4\right) (b c-a d)}",1,"(5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c]*(-5*c*(4*b*c - 4*a*d + b*d*x^4) + b*d*x^4*Sqrt[1 - (b*x^4)/a]*(-c + d*x^4)*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]) + 2*d*x^5*(5*c*(a - b*x^4) + b*x^4*Sqrt[1 - (b*x^4)/a]*(-c + d*x^4)*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]))/(20*c^2*(b*c - a*d)*Sqrt[a - b*x^4]*(-c + d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))","C",0
189,1,374,362,0.5152026,"\int \frac{1}{\left(a-b x^4\right)^{3/2} \left(c-d x^4\right)^2} \, dx","Integrate[1/((a - b*x^4)^(3/2)*(c - d*x^4)^2),x]","\frac{x \left(\frac{c \left(25 a c \left(4 a^2 d^2-a b d \left(8 c+d x^4\right)+2 b^2 c \left(2 c-d x^4\right)\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)-10 x^4 \left(-a^2 d^2+a b d^2 x^4-2 b^2 c \left(c-d x^4\right)\right) \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)\right)}{\left(c-d x^4\right) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}-b d x^4 \sqrt{1-\frac{b x^4}{a}} (a d+2 b c) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}{20 a c^2 \sqrt{a-b x^4} (b c-a d)^2}","\frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} (a d+2 b c) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{4 a^{3/4} c \sqrt{a-b x^4} (b c-a d)^2}-\frac{3 \sqrt[4]{a} d \sqrt{1-\frac{b x^4}{a}} (3 b c-a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^2}-\frac{3 \sqrt[4]{a} d \sqrt{1-\frac{b x^4}{a}} (3 b c-a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^2}+\frac{b x (a d+2 b c)}{4 a c \sqrt{a-b x^4} (b c-a d)^2}-\frac{d x}{4 c \sqrt{a-b x^4} \left(c-d x^4\right) (b c-a d)}",1,"(x*(-(b*d*(2*b*c + a*d)*x^4*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]) + (c*(25*a*c*(4*a^2*d^2 + 2*b^2*c*(2*c - d*x^4) - a*b*d*(8*c + d*x^4))*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] - 10*x^4*(-(a^2*d^2) + a*b*d^2*x^4 - 2*b^2*c*(c - d*x^4))*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))/((c - d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))))/(20*a*c^2*(b*c - a*d)^2*Sqrt[a - b*x^4])","C",0
190,1,382,439,0.8799857,"\int \frac{1}{\left(a-b x^4\right)^{5/2} \left(c-d x^4\right)^2} \, dx","Integrate[1/((a - b*x^4)^(5/2)*(c - d*x^4)^2),x]","\frac{x \left(\frac{b d x^4 \sqrt{1-\frac{b x^4}{a}} \left(3 a^2 d^2+17 a b c d-5 b^2 c^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}{a^2 c^2}+5 \left(\frac{2 b^3 c}{a^2-a b x^4}+\frac{5 b^3 c}{a^2}+\frac{5 \left(-9 a^3 d^3+36 a^2 b c d^2-17 a b^2 c^2 d+5 b^3 c^3\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)}{a \left(c-d x^4\right) \left(2 x^4 \left(2 a d F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)+b c F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right)\right)}-\frac{2 b^2 d}{a-b x^4}-\frac{17 b^2 d}{a}-\frac{3 a d^3}{c^2-c d x^4}+\frac{3 b d^3 x^4}{c^2-c d x^4}\right)\right)}{60 \sqrt{a-b x^4} (b c-a d)^3}","\frac{b x \left(-3 a^2 d^2-17 a b c d+5 b^2 c^2\right)}{12 a^2 c \sqrt{a-b x^4} (b c-a d)^3}+\frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left(-3 a^2 d^2-17 a b c d+5 b^2 c^2\right) F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{12 a^{7/4} c \sqrt{a-b x^4} (b c-a d)^3}+\frac{\sqrt[4]{a} d^2 \sqrt{1-\frac{b x^4}{a}} (13 b c-3 a d) \Pi \left(-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^3}+\frac{\sqrt[4]{a} d^2 \sqrt{1-\frac{b x^4}{a}} (13 b c-3 a d) \Pi \left(\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^3}-\frac{d x}{4 c \left(a-b x^4\right)^{3/2} \left(c-d x^4\right) (b c-a d)}+\frac{b x (3 a d+2 b c)}{12 a c \left(a-b x^4\right)^{3/2} (b c-a d)^2}",1,"(x*((b*d*(-5*b^2*c^2 + 17*a*b*c*d + 3*a^2*d^2)*x^4*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])/(a^2*c^2) + 5*((5*b^3*c)/a^2 - (17*b^2*d)/a - (2*b^2*d)/(a - b*x^4) + (2*b^3*c)/(a^2 - a*b*x^4) - (3*a*d^3)/(c^2 - c*d*x^4) + (3*b*d^3*x^4)/(c^2 - c*d*x^4) + (5*(5*b^3*c^3 - 17*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 9*a^3*d^3)*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c])/(a*(c - d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]))))))/(60*(b*c - a*d)^3*Sqrt[a - b*x^4])","C",0
191,1,155,103,0.1705521,"\int \frac{\sqrt{a+b x^4}}{a c-b c x^4} \, dx","Integrate[Sqrt[a + b*x^4]/(a*c - b*c*x^4),x]","\frac{5 a x \sqrt{a+b x^4} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right)}{c \left(a-b x^4\right) \left(2 b x^4 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right)+F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right)\right)+5 a F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right)\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{a+b x^4}}\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{b} c}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{a+b x^4}}\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{b} c}",1,"(5*a*x*Sqrt[a + b*x^4]*AppellF1[1/4, -1/2, 1, 5/4, -((b*x^4)/a), (b*x^4)/a])/(c*(a - b*x^4)*(5*a*AppellF1[1/4, -1/2, 1, 5/4, -((b*x^4)/a), (b*x^4)/a] + 2*b*x^4*(2*AppellF1[5/4, -1/2, 2, 9/4, -((b*x^4)/a), (b*x^4)/a] + AppellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), (b*x^4)/a])))","C",0
192,1,155,116,0.167141,"\int \frac{\sqrt{a-b x^4}}{a c+b c x^4} \, dx","Integrate[Sqrt[a - b*x^4]/(a*c + b*c*x^4),x]","\frac{5 a x \sqrt{a-b x^4} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right)}{c \left(a+b x^4\right) \left(5 a F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right)-2 b x^4 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right)+F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right)\right)\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} x \left(\sqrt{a}+\sqrt{b} x^2\right)}{\sqrt[4]{a} \sqrt{a-b x^4}}\right)}{2 \sqrt[4]{a} \sqrt[4]{b} c}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} x \left(\sqrt{a}-\sqrt{b} x^2\right)}{\sqrt[4]{a} \sqrt{a-b x^4}}\right)}{2 \sqrt[4]{a} \sqrt[4]{b} c}",1,"(5*a*x*Sqrt[a - b*x^4]*AppellF1[1/4, -1/2, 1, 5/4, (b*x^4)/a, -((b*x^4)/a)])/(c*(a + b*x^4)*(5*a*AppellF1[1/4, -1/2, 1, 5/4, (b*x^4)/a, -((b*x^4)/a)] - 2*b*x^4*(2*AppellF1[5/4, -1/2, 2, 9/4, (b*x^4)/a, -((b*x^4)/a)] + AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, -((b*x^4)/a)])))","C",0
193,1,364,211,0.628532,"\int \frac{\left(a+b x^4\right)^{7/4}}{c+d x^4} \, dx","Integrate[(a + b*x^4)^(7/4)/(c + d*x^4),x]","\frac{5 \sqrt[4]{c} \left(4 a^2 d \sqrt[4]{a+b x^4} \log \left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}+\sqrt[4]{c}\right)+4 b^2 c^{3/4} x^5 \sqrt[4]{b c-a d}+4 a b c^{3/4} x \sqrt[4]{b c-a d}+a \sqrt[4]{a+b x^4} (b c-4 a d) \log \left(\sqrt[4]{c}-\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}\right)-a b c \sqrt[4]{a+b x^4} \log \left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}+\sqrt[4]{c}\right)+2 a \sqrt[4]{a+b x^4} (4 a d-b c) \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a x^4+b}}\right)\right)+4 b x^5 \sqrt[4]{\frac{b x^4}{a}+1} \sqrt[4]{b c-a d} (7 a d-4 b c) F_1\left(\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{80 c d \sqrt[4]{a+b x^4} \sqrt[4]{b c-a d}}","-\frac{b^{3/4} (4 b c-7 a d) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a+b x^4}}\right)}{8 d^2}-\frac{b^{3/4} (4 b c-7 a d) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a+b x^4}}\right)}{8 d^2}+\frac{(b c-a d)^{7/4} \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} d^2}+\frac{(b c-a d)^{7/4} \tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} d^2}+\frac{b x \left(a+b x^4\right)^{3/4}}{4 d}",1,"(4*b*(b*c - a*d)^(1/4)*(-4*b*c + 7*a*d)*x^5*(1 + (b*x^4)/a)^(1/4)*AppellF1[5/4, 1/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 5*c^(1/4)*(4*a*b*c^(3/4)*(b*c - a*d)^(1/4)*x + 4*b^2*c^(3/4)*(b*c - a*d)^(1/4)*x^5 + 2*a*(-(b*c) + 4*a*d)*(a + b*x^4)^(1/4)*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(b + a*x^4)^(1/4))] + a*(b*c - 4*a*d)*(a + b*x^4)^(1/4)*Log[c^(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] - a*b*c*(a + b*x^4)^(1/4)*Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + 4*a^2*d*(a + b*x^4)^(1/4)*Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(80*c*d*(b*c - a*d)^(1/4)*(a + b*x^4)^(1/4))","C",0
194,1,161,173,0.1802358,"\int \frac{\left(a+b x^4\right)^{3/4}}{c+d x^4} \, dx","Integrate[(a + b*x^4)^(3/4)/(c + d*x^4),x]","\frac{5 a c x \left(a+b x^4\right)^{3/4} F_1\left(\frac{1}{4};-\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{\left(c+d x^4\right) \left(x^4 \left(3 b c F_1\left(\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)-4 a d F_1\left(\frac{5}{4};-\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};-\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}","\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a+b x^4}}\right)}{2 d}+\frac{b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a+b x^4}}\right)}{2 d}-\frac{(b c-a d)^{3/4} \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} d}-\frac{(b c-a d)^{3/4} \tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} d}",1,"(5*a*c*x*(a + b*x^4)^(3/4)*AppellF1[1/4, -3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/((c + d*x^4)*(5*a*c*AppellF1[1/4, -3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(-4*a*d*AppellF1[5/4, -3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 1/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))","C",0
195,1,84,105,0.0453341,"\int \frac{1}{\sqrt[4]{a+b x^4} \left(c+d x^4\right)} \, dx","Integrate[1/((a + b*x^4)^(1/4)*(c + d*x^4)),x]","\frac{\tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)+\tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} \sqrt[4]{b c-a d}}","\frac{\tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} \sqrt[4]{b c-a d}}+\frac{\tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} \sqrt[4]{b c-a d}}",1,"(ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))] + ArcTanh[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*(b*c - a*d)^(1/4))","A",1
196,1,256,134,0.5769808,"\int \frac{1}{\left(a+b x^4\right)^{5/4} \left(c+d x^4\right)} \, dx","Integrate[1/((a + b*x^4)^(5/4)*(c + d*x^4)),x]","-\frac{45 c^3 \left(a+b x^4\right)^2 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)-45 c^3 \left(a+b x^4\right)^2+36 c^2 d x^4 \left(a+b x^4\right)^2 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)-36 c^2 d x^4 \left(a+b x^4\right)^2+4 d x^{12} (b c-a d)^2 \, _2F_1\left(2,\frac{9}{4};\frac{13}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)+4 c x^8 (b c-a d)^2 \, _2F_1\left(2,\frac{9}{4};\frac{13}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{9 c^3 x^3 \left(a+b x^4\right)^{9/4} (a d-b c)}","-\frac{d \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} (b c-a d)^{5/4}}-\frac{d \tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} (b c-a d)^{5/4}}+\frac{b x}{a \sqrt[4]{a+b x^4} (b c-a d)}",1,"-1/9*(-45*c^3*(a + b*x^4)^2 - 36*c^2*d*x^4*(a + b*x^4)^2 + 45*c^3*(a + b*x^4)^2*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 36*c^2*d*x^4*(a + b*x^4)^2*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 4*c*(b*c - a*d)^2*x^8*Hypergeometric2F1[2, 9/4, 13/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 4*d*(b*c - a*d)^2*x^12*Hypergeometric2F1[2, 9/4, 13/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(c^3*(-(b*c) + a*d)*x^3*(a + b*x^4)^(9/4))","C",0
197,1,621,180,2.3680938,"\int \frac{1}{\left(a+b x^4\right)^{9/4} \left(c+d x^4\right)} \, dx","Integrate[1/((a + b*x^4)^(9/4)*(c + d*x^4)),x]","\frac{80 c^2 x^{12} (b c-a d)^3 \, _3F_2\left(2,2,\frac{13}{4};1,\frac{17}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)+80 d^2 x^{20} (b c-a d)^3 \, _3F_2\left(2,2,\frac{13}{4};1,\frac{17}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)+160 c d x^{16} (b c-a d)^3 \, _3F_2\left(2,2,\frac{13}{4};1,\frac{17}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)+2925 c^5 \left(a+b x^4\right)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)-2925 c^5 \left(a+b x^4\right)^3+4680 c^4 d x^4 \left(a+b x^4\right)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)-4680 c^4 d x^4 \left(a+b x^4\right)^3-585 c^4 x^4 \left(a+b x^4\right)^2 (b c-a d)+2080 c^3 d^2 x^8 \left(a+b x^4\right)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)-2080 c^3 d^2 x^8 \left(a+b x^4\right)^3-936 c^3 d x^8 \left(a+b x^4\right)^2 (b c-a d)-416 c^2 d^2 x^{12} \left(a+b x^4\right)^2 (b c-a d)+280 c^2 x^{12} (b c-a d)^3 \, _2F_1\left(2,\frac{13}{4};\frac{17}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)+240 d^2 x^{20} (b c-a d)^3 \, _2F_1\left(2,\frac{13}{4};\frac{17}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)+520 c d x^{16} (b c-a d)^3 \, _2F_1\left(2,\frac{13}{4};\frac{17}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{325 c^4 x^7 \left(a+b x^4\right)^{13/4} (b c-a d)^2}","\frac{b x (4 b c-9 a d)}{5 a^2 \sqrt[4]{a+b x^4} (b c-a d)^2}+\frac{d^2 \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{d^2 \tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{b x}{5 a \left(a+b x^4\right)^{5/4} (b c-a d)}",1,"(-585*c^4*(b*c - a*d)*x^4*(a + b*x^4)^2 - 936*c^3*d*(b*c - a*d)*x^8*(a + b*x^4)^2 - 416*c^2*d^2*(b*c - a*d)*x^12*(a + b*x^4)^2 - 2925*c^5*(a + b*x^4)^3 - 4680*c^4*d*x^4*(a + b*x^4)^3 - 2080*c^3*d^2*x^8*(a + b*x^4)^3 + 2925*c^5*(a + b*x^4)^3*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 4680*c^4*d*x^4*(a + b*x^4)^3*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 2080*c^3*d^2*x^8*(a + b*x^4)^3*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 280*c^2*(b*c - a*d)^3*x^12*Hypergeometric2F1[2, 13/4, 17/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 520*c*d*(b*c - a*d)^3*x^16*Hypergeometric2F1[2, 13/4, 17/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 240*d^2*(b*c - a*d)^3*x^20*Hypergeometric2F1[2, 13/4, 17/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 80*c^2*(b*c - a*d)^3*x^12*HypergeometricPFQ[{2, 2, 13/4}, {1, 17/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 160*c*d*(b*c - a*d)^3*x^16*HypergeometricPFQ[{2, 2, 13/4}, {1, 17/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 80*d^2*(b*c - a*d)^3*x^20*HypergeometricPFQ[{2, 2, 13/4}, {1, 17/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(325*c^4*(b*c - a*d)^2*x^7*(a + b*x^4)^(13/4))","C",0
198,1,231,233,5.4394334,"\int \frac{1}{\left(a+b x^4\right)^{13/4} \left(c+d x^4\right)} \, dx","Integrate[1/((a + b*x^4)^(13/4)*(c + d*x^4)),x]","\frac{b x \left(\left(a+b x^4\right)^2 \left(113 a^2 d^2-100 a b c d+32 b^2 c^2\right)+5 a^2 (b c-a d)^2+a \left(a+b x^4\right) (a d-b c) (17 a d-8 b c)\right)}{45 a^3 \left(a+b x^4\right)^{9/4} (b c-a d)^3}-\frac{d^3 \left(-\log \left(\sqrt[4]{c}-\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}\right)+\log \left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}+\sqrt[4]{c}\right)+2 \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a x^4+b}}\right)\right)}{4 c^{3/4} (b c-a d)^{13/4}}","\frac{b x (8 b c-17 a d)}{45 a^2 \left(a+b x^4\right)^{5/4} (b c-a d)^2}+\frac{b x \left(113 a^2 d^2-100 a b c d+32 b^2 c^2\right)}{45 a^3 \sqrt[4]{a+b x^4} (b c-a d)^3}-\frac{d^3 \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} (b c-a d)^{13/4}}-\frac{d^3 \tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{2 c^{3/4} (b c-a d)^{13/4}}+\frac{b x}{9 a \left(a+b x^4\right)^{9/4} (b c-a d)}",1,"(b*x*(5*a^2*(b*c - a*d)^2 + a*(-(b*c) + a*d)*(-8*b*c + 17*a*d)*(a + b*x^4) + (32*b^2*c^2 - 100*a*b*c*d + 113*a^2*d^2)*(a + b*x^4)^2))/(45*a^3*(b*c - a*d)^3*(a + b*x^4)^(9/4)) - (d^3*(2*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(b + a*x^4)^(1/4))] - Log[c^(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(4*c^(3/4)*(b*c - a*d)^(13/4))","A",0
199,1,294,316,0.6977951,"\int \frac{\left(a+b x^4\right)^{9/4}}{c+d x^4} \, dx","Integrate[(a + b*x^4)^(9/4)/(c + d*x^4),x]","\frac{x \left(\frac{b x^4 \left(\frac{b x^4}{a}+1\right)^{3/4} \left(23 a^2 d^2-30 a b c d+12 b^2 c^2\right) F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{c}-\frac{25 a^2 c \left(12 a^2 d^2-13 a b c d+6 b^2 c^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{\left(c+d x^4\right) \left(x^4 \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}+5 b \left(a+b x^4\right) \left(13 a d-6 b c+2 b d x^4\right)\right)}{60 d^2 \left(a+b x^4\right)^{3/4}}","\frac{\sqrt{a} b^{3/2} x^3 \left(\frac{a}{b x^4}+1\right)^{3/4} (6 b c-11 a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right|2\right)}{12 d^2 \left(a+b x^4\right)^{3/4}}-\frac{b x \sqrt[4]{a+b x^4} (6 b c-11 a d)}{12 d^2}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d)^2 \Pi \left(-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c d^2}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d)^2 \Pi \left(\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c d^2}+\frac{b x \left(a+b x^4\right)^{5/4}}{6 d}",1,"(x*(5*b*(a + b*x^4)*(-6*b*c + 13*a*d + 2*b*d*x^4) + (b*(12*b^2*c^2 - 30*a*b*c*d + 23*a^2*d^2)*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/c - (25*a^2*c*(6*b^2*c^2 - 13*a*b*c*d + 12*a^2*d^2)*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/((c + d*x^4)*(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(60*d^2*(a + b*x^4)^(3/4))","C",0
200,1,346,274,0.3212591,"\int \frac{\left(a+b x^4\right)^{5/4}}{c+d x^4} \, dx","Integrate[(a + b*x^4)^(5/4)/(c + d*x^4),x]","\frac{x \left(\frac{5 \left(b x^4 \left(a+b x^4\right) \left(c+d x^4\right) \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c \left(2 a^2 d+a b d x^4+b^2 x^4 \left(c+d x^4\right)\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}{\left(c+d x^4\right) \left(x^4 \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}+\frac{b x^4 \left(\frac{b x^4}{a}+1\right)^{3/4} (3 a d-2 b c) F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{c}\right)}{10 d \left(a+b x^4\right)^{3/4}}","-\frac{\sqrt{a} b^{3/2} x^3 \left(\frac{a}{b x^4}+1\right)^{3/4} F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right|2\right)}{2 d \left(a+b x^4\right)^{3/4}}-\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) \Pi \left(-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c d}-\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) \Pi \left(\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c d}+\frac{b x \sqrt[4]{a+b x^4}}{2 d}",1,"(x*((b*(-2*b*c + 3*a*d)*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/c + (5*(-5*a*c*(2*a^2*d + a*b*d*x^4 + b^2*x^4*(c + d*x^4))*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + b*x^4*(a + b*x^4)*(c + d*x^4)*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((c + d*x^4)*(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(10*d*(a + b*x^4)^(3/4))","C",0
201,1,160,166,0.1662755,"\int \frac{\sqrt[4]{a+b x^4}}{c+d x^4} \, dx","Integrate[(a + b*x^4)^(1/4)/(c + d*x^4),x]","\frac{5 a c x \sqrt[4]{a+b x^4} F_1\left(\frac{1}{4};-\frac{1}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{\left(c+d x^4\right) \left(x^4 \left(b c F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)-4 a d F_1\left(\frac{5}{4};-\frac{1}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};-\frac{1}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}","\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left(-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left(\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c}",1,"(5*a*c*x*(a + b*x^4)^(1/4)*AppellF1[1/4, -1/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/((c + d*x^4)*(5*a*c*AppellF1[1/4, -1/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(-4*a*d*AppellF1[5/4, -1/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))","C",0
202,1,161,259,0.0386079,"\int \frac{1}{\left(a+b x^4\right)^{3/4} \left(c+d x^4\right)} \, dx","Integrate[1/((a + b*x^4)^(3/4)*(c + d*x^4)),x]","-\frac{5 a c x F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{\left(a+b x^4\right)^{3/4} \left(c+d x^4\right) \left(x^4 \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}","-\frac{b^{3/2} x^3 \left(\frac{a}{b x^4}+1\right)^{3/4} F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} \left(a+b x^4\right)^{3/4} (b c-a d)}-\frac{d \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left(-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c (b c-a d)}-\frac{d \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left(\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c (b c-a d)}",1,"(-5*a*c*x*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/((a + b*x^4)^(3/4)*(c + d*x^4)*(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))","C",0
203,1,332,304,0.273176,"\int \frac{1}{\left(a+b x^4\right)^{7/4} \left(c+d x^4\right)} \, dx","Integrate[1/((a + b*x^4)^(7/4)*(c + d*x^4)),x]","\frac{x \left(\frac{5 \left(b x^4 \left(c+d x^4\right) \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)+5 a c \left(3 a d-b \left(3 c+d x^4\right)\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}{\left(c+d x^4\right) \left(5 a c F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)-x^4 \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)\right)}-\frac{2 b d x^4 \left(\frac{b x^4}{a}+1\right)^{3/4} F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{c}\right)}{15 a \left(a+b x^4\right)^{3/4} (a d-b c)}","-\frac{b^{3/2} x^3 \left(\frac{a}{b x^4}+1\right)^{3/4} (2 b c-5 a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right|2\right)}{3 a^{3/2} \left(a+b x^4\right)^{3/4} (b c-a d)^2}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left(-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c (b c-a d)^2}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left(\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c (b c-a d)^2}+\frac{b x}{3 a \left(a+b x^4\right)^{3/4} (b c-a d)}",1,"(x*((-2*b*d*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/c + (5*(5*a*c*(3*a*d - b*(3*c + d*x^4))*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + b*x^4*(c + d*x^4)*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((c + d*x^4)*(5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] - x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(15*a*(-(b*c) + a*d)*(a + b*x^4)^(3/4))","C",0
204,1,430,357,0.7725289,"\int \frac{1}{\left(a+b x^4\right)^{11/4} \left(c+d x^4\right)} \, dx","Integrate[1/((a + b*x^4)^(11/4)*(c + d*x^4)),x]","\frac{x \left(-\frac{5 \left(b x^4 \left(c+d x^4\right) \left(16 a^2 d+a b \left(13 d x^4-9 c\right)-6 b^2 c x^4\right) \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)+5 a c \left(21 a^3 d^2+a^2 b d \left(5 d x^4-42 c\right)+a b^2 \left(21 c^2-30 c d x^4-13 d^2 x^8\right)+6 b^3 c x^4 \left(3 c+d x^4\right)\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}{\left(a+b x^4\right) \left(c+d x^4\right) \left(x^4 \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}-\frac{2 b d x^4 \left(\frac{b x^4}{a}+1\right)^{3/4} (13 a d-6 b c) F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{c}\right)}{105 a^2 \left(a+b x^4\right)^{3/4} (b c-a d)^2}","\frac{b x (6 b c-13 a d)}{21 a^2 \left(a+b x^4\right)^{3/4} (b c-a d)^2}-\frac{b^{3/2} x^3 \left(\frac{a}{b x^4}+1\right)^{3/4} \left(47 a^2 d^2-38 a b c d+12 b^2 c^2\right) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right|2\right)}{21 a^{5/2} \left(a+b x^4\right)^{3/4} (b c-a d)^3}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left(-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c (b c-a d)^3}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left(\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{2 \sqrt[4]{b} c (b c-a d)^3}+\frac{b x}{7 a \left(a+b x^4\right)^{7/4} (b c-a d)}",1,"(x*((-2*b*d*(-6*b*c + 13*a*d)*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/c - (5*(5*a*c*(21*a^3*d^2 + 6*b^3*c*x^4*(3*c + d*x^4) + a^2*b*d*(-42*c + 5*d*x^4) + a*b^2*(21*c^2 - 30*c*d*x^4 - 13*d^2*x^8))*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + b*x^4*(c + d*x^4)*(16*a^2*d - 6*b^2*c*x^4 + a*b*(-9*c + 13*d*x^4))*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((a + b*x^4)*(c + d*x^4)*(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(105*a^2*(b*c - a*d)^2*(a + b*x^4)^(3/4))","C",0
205,1,560,280,0.9558089,"\int \frac{\left(a+b x^4\right)^{11/4}}{\left(c+d x^4\right)^2} \, dx","Integrate[(a + b*x^4)^(11/4)/(c + d*x^4)^2,x]","\frac{1}{80} \left(\frac{15 a^3 \left(-\log \left(\sqrt[4]{c}-\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}\right)+\log \left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}+\sqrt[4]{c}\right)+2 \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a x^4+b}}\right)\right)}{c^{7/4} \sqrt[4]{b c-a d}}+\frac{10 a^2 b \left(-\log \left(\sqrt[4]{c}-\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}\right)+\log \left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}+\sqrt[4]{c}\right)+2 \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a x^4+b}}\right)\right)}{c^{3/4} d \sqrt[4]{b c-a d}}-\frac{32 b^3 x^5 \sqrt[4]{\frac{b x^4}{a}+1} F_1\left(\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{d^2 \sqrt[4]{a+b x^4}}+\frac{44 a b^2 x^5 \sqrt[4]{\frac{b x^4}{a}+1} F_1\left(\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{c d \sqrt[4]{a+b x^4}}+\frac{20 x \left(a+b x^4\right)^{3/4} \left(\frac{(b c-a d)^2}{c \left(c+d x^4\right)}+b^2\right)}{d^2}-\frac{10 a b^2 \sqrt[4]{c} \left(-\log \left(\sqrt[4]{c}-\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}\right)+\log \left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}+\sqrt[4]{c}\right)+2 \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a x^4+b}}\right)\right)}{d^2 \sqrt[4]{b c-a d}}\right)","-\frac{b^{7/4} (8 b c-11 a d) \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a+b x^4}}\right)}{8 d^3}-\frac{b^{7/4} (8 b c-11 a d) \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a+b x^4}}\right)}{8 d^3}+\frac{(b c-a d)^{7/4} (3 a d+8 b c) \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} d^3}+\frac{(b c-a d)^{7/4} (3 a d+8 b c) \tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} d^3}+\frac{b x \left(a+b x^4\right)^{3/4} (2 b c-a d)}{4 c d^2}-\frac{x \left(a+b x^4\right)^{7/4} (b c-a d)}{4 c d \left(c+d x^4\right)}",1,"((20*x*(a + b*x^4)^(3/4)*(b^2 + (b*c - a*d)^2/(c*(c + d*x^4))))/d^2 - (32*b^3*x^5*(1 + (b*x^4)/a)^(1/4)*AppellF1[5/4, 1/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/(d^2*(a + b*x^4)^(1/4)) + (44*a*b^2*x^5*(1 + (b*x^4)/a)^(1/4)*AppellF1[5/4, 1/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/(c*d*(a + b*x^4)^(1/4)) + (15*a^3*(2*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(b + a*x^4)^(1/4))] - Log[c^(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(c^(7/4)*(b*c - a*d)^(1/4)) - (10*a*b^2*c^(1/4)*(2*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(b + a*x^4)^(1/4))] - Log[c^(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(d^2*(b*c - a*d)^(1/4)) + (10*a^2*b*(2*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(b + a*x^4)^(1/4))] - Log[c^(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(c^(3/4)*d*(b*c - a*d)^(1/4)))/80","C",0
206,1,358,230,0.638143,"\int \frac{\left(a+b x^4\right)^{7/4}}{\left(c+d x^4\right)^2} \, dx","Integrate[(a + b*x^4)^(7/4)/(c + d*x^4)^2,x]","\frac{\frac{15 a^2 \left(-\log \left(\sqrt[4]{c}-\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}\right)+\log \left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}+\sqrt[4]{c}\right)+2 \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a x^4+b}}\right)\right)}{\sqrt[4]{b c-a d}}+\frac{16 b^2 c^{3/4} x^5 \sqrt[4]{\frac{b x^4}{a}+1} F_1\left(\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{d \sqrt[4]{a+b x^4}}-\frac{20 c^{3/4} x \left(a+b x^4\right)^{3/4} (b c-a d)}{d \left(c+d x^4\right)}+\frac{5 a b c \left(-\log \left(\sqrt[4]{c}-\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}\right)+\log \left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{a x^4+b}}+\sqrt[4]{c}\right)+2 \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a x^4+b}}\right)\right)}{d \sqrt[4]{b c-a d}}}{80 c^{7/4}}","\frac{b^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a+b x^4}}\right)}{2 d^2}+\frac{b^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a+b x^4}}\right)}{2 d^2}-\frac{(b c-a d)^{3/4} (3 a d+4 b c) \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} d^2}-\frac{(b c-a d)^{3/4} (3 a d+4 b c) \tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} d^2}-\frac{x \left(a+b x^4\right)^{3/4} (b c-a d)}{4 c d \left(c+d x^4\right)}",1,"((-20*c^(3/4)*(b*c - a*d)*x*(a + b*x^4)^(3/4))/(d*(c + d*x^4)) + (16*b^2*c^(3/4)*x^5*(1 + (b*x^4)/a)^(1/4)*AppellF1[5/4, 1/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/(d*(a + b*x^4)^(1/4)) + (15*a^2*(2*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(b + a*x^4)^(1/4))] - Log[c^(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(b*c - a*d)^(1/4) + (5*a*b*c*(2*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(b + a*x^4)^(1/4))] - Log[c^(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)] + Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(d*(b*c - a*d)^(1/4)))/(80*c^(7/4))","C",0
207,1,78,135,0.0602898,"\int \frac{\left(a+b x^4\right)^{3/4}}{\left(c+d x^4\right)^2} \, dx","Integrate[(a + b*x^4)^(3/4)/(c + d*x^4)^2,x]","\frac{x \left(a+b x^4\right)^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{1}{4};\frac{5}{4};\frac{(a d-b c) x^4}{a \left(d x^4+c\right)}\right)}{c^2 \left(\frac{b x^4}{a}+1\right)^{3/4} \sqrt[4]{\frac{d x^4}{c}+1}}","\frac{3 a \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} \sqrt[4]{b c-a d}}+\frac{3 a \tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} \sqrt[4]{b c-a d}}+\frac{x \left(a+b x^4\right)^{3/4}}{4 c \left(c+d x^4\right)}",1,"(x*(a + b*x^4)^(3/4)*Hypergeometric2F1[-3/4, 1/4, 5/4, ((-(b*c) + a*d)*x^4)/(a*(c + d*x^4))])/(c^2*(1 + (b*x^4)/a)^(3/4)*(1 + (d*x^4)/c)^(1/4))","C",0
208,1,99,162,0.1399122,"\int \frac{1}{\sqrt[4]{a+b x^4} \left(c+d x^4\right)^2} \, dx","Integrate[1/((a + b*x^4)^(1/4)*(c + d*x^4)^2),x]","\frac{x \left(\left(c+d x^4\right) (4 b c-3 a d) \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)-c d \left(a+b x^4\right)\right)}{4 c^2 \sqrt[4]{a+b x^4} \left(c+d x^4\right) (b c-a d)}","\frac{(4 b c-3 a d) \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} (b c-a d)^{5/4}}+\frac{(4 b c-3 a d) \tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} (b c-a d)^{5/4}}-\frac{d x \left(a+b x^4\right)^{3/4}}{4 c \left(c+d x^4\right) (b c-a d)}",1,"(x*(-(c*d*(a + b*x^4)) + (4*b*c - 3*a*d)*(c + d*x^4)*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))]))/(4*c^2*(b*c - a*d)*(a + b*x^4)^(1/4)*(c + d*x^4))","C",1
209,1,625,205,2.2650721,"\int \frac{1}{\left(a+b x^4\right)^{5/4} \left(c+d x^4\right)^2} \, dx","Integrate[1/((a + b*x^4)^(5/4)*(c + d*x^4)^2),x]","\frac{c \left(a+b x^4\right)^{3/4} \left(\frac{320 d^2 x^{20} (b c-a d)^3 \, _3F_2\left(2,2,\frac{9}{4};1,\frac{17}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{c^5 \left(a+b x^4\right)^3}+\frac{640 d x^{16} (b c-a d)^3 \, _3F_2\left(2,2,\frac{9}{4};1,\frac{17}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{c^4 \left(a+b x^4\right)^3}+\frac{320 x^{12} (b c-a d)^3 \, _3F_2\left(2,2,\frac{9}{4};1,\frac{17}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{c^3 \left(a+b x^4\right)^3}+\frac{16380 d^2 x^{12} (a d-b c) \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{c^3 \left(a+b x^4\right)}+\frac{7488 d^2 x^{12} (b c-a d)}{c^3 \left(a+b x^4\right)}+\frac{44460 d^2 x^8 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{c^2}+\frac{33930 d x^8 (a d-b c) \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{c^2 \left(a+b x^4\right)}+\frac{14976 d x^8 (b c-a d)}{c^2 \left(a+b x^4\right)}+\frac{94770 d x^4 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{c}-\frac{14625 x^4 (b c-a d) \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{c \left(a+b x^4\right)}+47385 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)+\frac{5148 x^4 (b c-a d)}{c \left(a+b x^4\right)}-\frac{44460 d^2 x^8}{c^2}-\frac{94770 d x^4}{c}-47385\right)}{2340 x^7 \left(c+d x^4\right) (b c-a d)^2}","-\frac{d (8 b c-3 a d) \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} (b c-a d)^{9/4}}-\frac{d (8 b c-3 a d) \tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} (b c-a d)^{9/4}}+\frac{b x (a d+4 b c)}{4 a c \sqrt[4]{a+b x^4} (b c-a d)^2}-\frac{d x}{4 c \sqrt[4]{a+b x^4} \left(c+d x^4\right) (b c-a d)}",1,"(c*(a + b*x^4)^(3/4)*(-47385 - (94770*d*x^4)/c - (44460*d^2*x^8)/c^2 + (5148*(b*c - a*d)*x^4)/(c*(a + b*x^4)) + (14976*d*(b*c - a*d)*x^8)/(c^2*(a + b*x^4)) + (7488*d^2*(b*c - a*d)*x^12)/(c^3*(a + b*x^4)) + 47385*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + (94770*d*x^4*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/c + (44460*d^2*x^8*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/c^2 - (14625*(b*c - a*d)*x^4*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(c*(a + b*x^4)) + (33930*d*(-(b*c) + a*d)*x^8*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(c^2*(a + b*x^4)) + (16380*d^2*(-(b*c) + a*d)*x^12*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(c^3*(a + b*x^4)) + (320*(b*c - a*d)^3*x^12*HypergeometricPFQ[{2, 2, 9/4}, {1, 17/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(c^3*(a + b*x^4)^3) + (640*d*(b*c - a*d)^3*x^16*HypergeometricPFQ[{2, 2, 9/4}, {1, 17/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(c^4*(a + b*x^4)^3) + (320*d^2*(b*c - a*d)^3*x^20*HypergeometricPFQ[{2, 2, 9/4}, {1, 17/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(c^5*(a + b*x^4)^3)))/(2340*(b*c - a*d)^2*x^7*(c + d*x^4))","C",0
210,1,1216,266,5.0052624,"\int \frac{1}{\left(a+b x^4\right)^{9/4} \left(c+d x^4\right)^2} \, dx","Integrate[1/((a + b*x^4)^(9/4)*(c + d*x^4)^2),x]","-\frac{69120 d^3 (b c-a d)^4 \, _3F_2\left(2,2,\frac{13}{4};1,\frac{21}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^{28}+11520 d^3 (b c-a d)^4 \, _4F_3\left(2,2,2,\frac{13}{4};1,1,\frac{21}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^{28}+216000 c d^2 (b c-a d)^4 \, _3F_2\left(2,2,\frac{13}{4};1,\frac{21}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^{24}+34560 c d^2 (b c-a d)^4 \, _4F_3\left(2,2,2,\frac{13}{4};1,1,\frac{21}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^{24}+282880 c^2 d^3 (b c-a d)^2 \left(b x^4+a\right)^2 x^{20}+224640 c^2 d (b c-a d)^4 \, _3F_2\left(2,2,\frac{13}{4};1,\frac{21}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^{20}+34560 c^2 d (b c-a d)^4 \, _4F_3\left(2,2,2,\frac{13}{4};1,1,\frac{21}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^{20}+8146944 c^3 d^3 (b c-a d) \left(b x^4+a\right)^3 x^{16}+891072 c^3 d^2 (b c-a d)^2 \left(b x^4+a\right)^2 x^{16}-11934000 c^3 d^3 (b c-a d) \left(b x^4+a\right)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^{16}+77760 c^3 (b c-a d)^4 \, _3F_2\left(2,2,\frac{13}{4};1,\frac{21}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^{16}+11520 c^3 (b c-a d)^4 \, _4F_3\left(2,2,2,\frac{13}{4};1,1,\frac{21}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^{16}-18935280 c^4 d^3 \left(b x^4+a\right)^4 x^{12}+25968384 c^4 d^2 (b c-a d) \left(b x^4+a\right)^3 x^{12}+933504 c^4 d (b c-a d)^2 \left(b x^4+a\right)^2 x^{12}+18935280 c^4 d^3 \left(b x^4+a\right)^4 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^{12}-38069460 c^4 d^2 (b c-a d) \left(b x^4+a\right)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^{12}-60505380 c^5 d^2 \left(b x^4+a\right)^4 x^8+27973296 c^5 d (b c-a d) \left(b x^4+a\right)^3 x^8+285532 c^5 (b c-a d)^2 \left(b x^4+a\right)^2 x^8+60505380 c^5 d^2 \left(b x^4+a\right)^4 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^8-41082795 c^5 d (b c-a d) \left(b x^4+a\right)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^8-65547495 c^6 d \left(b x^4+a\right)^4 x^4+9793836 c^6 (b c-a d) \left(b x^4+a\right)^3 x^4+65547495 c^6 d \left(b x^4+a\right)^4 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^4-14499810 c^6 (b c-a d) \left(b x^4+a\right)^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right) x^4-23529870 c^7 \left(b x^4+a\right)^4+23529870 c^7 \left(b x^4+a\right)^4 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left(b x^4+a\right)}\right)}{198900 c^8 \left(\frac{a d}{c}-b\right)^3 x^{11} \left(b x^4+a\right)^{13/4} \left(d x^4+c\right)}","\frac{b x \left(-5 a^2 d^2-56 a b c d+16 b^2 c^2\right)}{20 a^2 c \sqrt[4]{a+b x^4} (b c-a d)^3}+\frac{3 d^2 (4 b c-a d) \tan ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} (b c-a d)^{13/4}}+\frac{3 d^2 (4 b c-a d) \tanh ^{-1}\left(\frac{x \sqrt[4]{b c-a d}}{\sqrt[4]{c} \sqrt[4]{a+b x^4}}\right)}{8 c^{7/4} (b c-a d)^{13/4}}-\frac{d x}{4 c \left(a+b x^4\right)^{5/4} \left(c+d x^4\right) (b c-a d)}+\frac{b x (5 a d+4 b c)}{20 a c \left(a+b x^4\right)^{5/4} (b c-a d)^2}",1,"-1/198900*(285532*c^5*(b*c - a*d)^2*x^8*(a + b*x^4)^2 + 933504*c^4*d*(b*c - a*d)^2*x^12*(a + b*x^4)^2 + 891072*c^3*d^2*(b*c - a*d)^2*x^16*(a + b*x^4)^2 + 282880*c^2*d^3*(b*c - a*d)^2*x^20*(a + b*x^4)^2 + 9793836*c^6*(b*c - a*d)*x^4*(a + b*x^4)^3 + 27973296*c^5*d*(b*c - a*d)*x^8*(a + b*x^4)^3 + 25968384*c^4*d^2*(b*c - a*d)*x^12*(a + b*x^4)^3 + 8146944*c^3*d^3*(b*c - a*d)*x^16*(a + b*x^4)^3 - 23529870*c^7*(a + b*x^4)^4 - 65547495*c^6*d*x^4*(a + b*x^4)^4 - 60505380*c^5*d^2*x^8*(a + b*x^4)^4 - 18935280*c^4*d^3*x^12*(a + b*x^4)^4 - 14499810*c^6*(b*c - a*d)*x^4*(a + b*x^4)^3*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] - 41082795*c^5*d*(b*c - a*d)*x^8*(a + b*x^4)^3*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] - 38069460*c^4*d^2*(b*c - a*d)*x^12*(a + b*x^4)^3*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] - 11934000*c^3*d^3*(b*c - a*d)*x^16*(a + b*x^4)^3*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 23529870*c^7*(a + b*x^4)^4*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 65547495*c^6*d*x^4*(a + b*x^4)^4*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 60505380*c^5*d^2*x^8*(a + b*x^4)^4*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 18935280*c^4*d^3*x^12*(a + b*x^4)^4*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 77760*c^3*(b*c - a*d)^4*x^16*HypergeometricPFQ[{2, 2, 13/4}, {1, 21/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 224640*c^2*d*(b*c - a*d)^4*x^20*HypergeometricPFQ[{2, 2, 13/4}, {1, 21/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 216000*c*d^2*(b*c - a*d)^4*x^24*HypergeometricPFQ[{2, 2, 13/4}, {1, 21/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 69120*d^3*(b*c - a*d)^4*x^28*HypergeometricPFQ[{2, 2, 13/4}, {1, 21/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 11520*c^3*(b*c - a*d)^4*x^16*HypergeometricPFQ[{2, 2, 2, 13/4}, {1, 1, 21/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 34560*c^2*d*(b*c - a*d)^4*x^20*HypergeometricPFQ[{2, 2, 2, 13/4}, {1, 1, 21/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 34560*c*d^2*(b*c - a*d)^4*x^24*HypergeometricPFQ[{2, 2, 2, 13/4}, {1, 1, 21/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 11520*d^3*(b*c - a*d)^4*x^28*HypergeometricPFQ[{2, 2, 2, 13/4}, {1, 1, 21/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(c^8*(-b + (a*d)/c)^3*x^11*(a + b*x^4)^(13/4)*(c + d*x^4))","C",0
211,1,392,353,0.5907565,"\int \frac{\left(a+b x^4\right)^{9/4}}{\left(c+d x^4\right)^2} \, dx","Integrate[(a + b*x^4)^(9/4)/(c + d*x^4)^2,x]","\frac{2 b x^5 \left(\frac{b x^4}{a}+1\right)^{3/4} \left(a^2 d^2+3 a b c d-3 b^2 c^2\right) F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+\frac{5 c \left(x^5 \left(a+b x^4\right) \left(a^2 d^2-2 a b c d+b^2 c \left(3 c+2 d x^4\right)\right) \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c x \left(4 a^3 d^2+a^2 b d^2 x^4+b^3 c x^4 \left(3 c+2 d x^4\right)\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}{\left(c+d x^4\right) \left(x^4 \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}}{20 c^2 d^2 \left(a+b x^4\right)^{3/4}}","-\frac{\sqrt{a} b^{3/2} x^3 \left(\frac{a}{b x^4}+1\right)^{3/4} (3 b c-a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right|2\right)}{4 c d^2 \left(a+b x^4\right)^{3/4}}-\frac{3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) (a d+2 b c) \Pi \left(-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d^2}-\frac{3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) (a d+2 b c) \Pi \left(\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d^2}+\frac{b x \sqrt[4]{a+b x^4} (3 b c-a d)}{4 c d^2}-\frac{x \left(a+b x^4\right)^{5/4} (b c-a d)}{4 c d \left(c+d x^4\right)}",1,"(2*b*(-3*b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^5*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + (5*c*(-5*a*c*x*(4*a^3*d^2 + a^2*b*d^2*x^4 + b^3*c*x^4*(3*c + 2*d*x^4))*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^5*(a + b*x^4)*(-2*a*b*c*d + a^2*d^2 + b^2*c*(3*c + 2*d*x^4))*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((c + d*x^4)*(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)]))))/(20*c^2*d^2*(a + b*x^4)^(3/4))","C",0
212,1,341,298,0.4899084,"\int \frac{\left(a+b x^4\right)^{5/4}}{\left(c+d x^4\right)^2} \, dx","Integrate[(a + b*x^4)^(5/4)/(c + d*x^4)^2,x]","\frac{x \left(\frac{5 c \left(x^4 \left(a+b x^4\right) (a d-b c) \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c \left(4 a^2 d+a b d x^4-b^2 c x^4\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}{\left(c+d x^4\right) \left(x^4 \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}+2 b x^4 \left(\frac{b x^4}{a}+1\right)^{3/4} (a d+b c) F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}{20 c^2 d \left(a+b x^4\right)^{3/4}}","\frac{\sqrt{a} b^{3/2} x^3 \left(\frac{a}{b x^4}+1\right)^{3/4} F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right|2\right)}{4 c d \left(a+b x^4\right)^{3/4}}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (3 a d+2 b c) \Pi \left(-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (3 a d+2 b c) \Pi \left(\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 d}-\frac{x \sqrt[4]{a+b x^4} (b c-a d)}{4 c d \left(c+d x^4\right)}",1,"(x*(2*b*(b*c + a*d)*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + (5*c*(-5*a*c*(4*a^2*d - b^2*c*x^4 + a*b*d*x^4)*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + (-(b*c) + a*d)*x^4*(a + b*x^4)*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((c + d*x^4)*(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(20*c^2*d*(a + b*x^4)^(3/4))","C",0
213,1,233,308,0.3135359,"\int \frac{\sqrt[4]{a+b x^4}}{\left(c+d x^4\right)^2} \, dx","Integrate[(a + b*x^4)^(1/4)/(c + d*x^4)^2,x]","\frac{x \left(\frac{5 \left(\frac{a+b x^4}{c}-\frac{15 a^2 F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{x^4 \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}\right)}{c+d x^4}+\frac{2 b x^4 \left(\frac{b x^4}{a}+1\right)^{3/4} F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{c^2}\right)}{20 \left(a+b x^4\right)^{3/4}}","-\frac{\sqrt{a} b^{3/2} x^3 \left(\frac{a}{b x^4}+1\right)^{3/4} F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right|2\right)}{4 c \left(a+b x^4\right)^{3/4} (b c-a d)}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-3 a d) \Pi \left(-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 (b c-a d)}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-3 a d) \Pi \left(\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 (b c-a d)}+\frac{x \sqrt[4]{a+b x^4}}{4 c \left(c+d x^4\right)}",1,"(x*((2*b*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/c^2 + (5*((a + b*x^4)/c - (15*a^2*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)]))))/(c + d*x^4)))/(20*(a + b*x^4)^(3/4))","C",0
214,1,337,330,0.3442062,"\int \frac{1}{\left(a+b x^4\right)^{3/4} \left(c+d x^4\right)^2} \, dx","Integrate[1/((a + b*x^4)^(3/4)*(c + d*x^4)^2),x]","\frac{x \left(\frac{2 b d x^4 \left(\frac{b x^4}{a}+1\right)^{3/4} F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a d-b c}+\frac{c \left(25 a c \left(4 a d-4 b c+b d x^4\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)-5 d x^4 \left(a+b x^4\right) \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)\right)}{\left(c+d x^4\right) (b c-a d) \left(x^4 \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)-5 a c F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}\right)}{20 c^2 \left(a+b x^4\right)^{3/4}}","-\frac{b^{3/2} x^3 \left(\frac{a}{b x^4}+1\right)^{3/4} (4 b c-a d) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right|2\right)}{4 \sqrt{a} c \left(a+b x^4\right)^{3/4} (b c-a d)^2}-\frac{3 d \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-a d) \Pi \left(-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 (b c-a d)^2}-\frac{3 d \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-a d) \Pi \left(\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 (b c-a d)^2}-\frac{d x \sqrt[4]{a+b x^4}}{4 c \left(c+d x^4\right) (b c-a d)}",1,"(x*((2*b*d*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/(-(b*c) + a*d) + (c*(25*a*c*(-4*b*c + 4*a*d + b*d*x^4)*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] - 5*d*x^4*(a + b*x^4)*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((b*c - a*d)*(c + d*x^4)*(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(20*c^2*(a + b*x^4)^(3/4))","C",0
215,1,387,390,0.5792465,"\int \frac{1}{\left(a+b x^4\right)^{7/4} \left(c+d x^4\right)^2} \, dx","Integrate[1/((a + b*x^4)^(7/4)*(c + d*x^4)^2),x]","\frac{x \left(\frac{c \left(25 a c \left(12 a^2 d^2+3 a b d \left(d x^4-8 c\right)+4 b^2 c \left(3 c+d x^4\right)\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)-5 x^4 \left(3 a^2 d^2+3 a b d^2 x^4+4 b^2 c \left(c+d x^4\right)\right) \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)\right)}{\left(c+d x^4\right) \left(5 a c F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)-x^4 \left(4 a d F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+3 b c F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)\right)}+2 b d x^4 \left(\frac{b x^4}{a}+1\right)^{3/4} (3 a d+4 b c) F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}{60 a c^2 \left(a+b x^4\right)^{3/4} (b c-a d)^2}","-\frac{b^{3/2} x^3 \left(\frac{a}{b x^4}+1\right)^{3/4} \left(3 a^2 d^2-32 a b c d+8 b^2 c^2\right) F\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a}}\right)\right|2\right)}{12 a^{3/2} c \left(a+b x^4\right)^{3/4} (b c-a d)^3}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (10 b c-3 a d) \Pi \left(-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 (b c-a d)^3}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (10 b c-3 a d) \Pi \left(\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left.\sin ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{b x^4+a}}\right)\right|-1\right)}{8 \sqrt[4]{b} c^2 (b c-a d)^3}+\frac{b x (3 a d+4 b c)}{12 a c \left(a+b x^4\right)^{3/4} (b c-a d)^2}-\frac{d x}{4 c \left(a+b x^4\right)^{3/4} \left(c+d x^4\right) (b c-a d)}",1,"(x*(2*b*d*(4*b*c + 3*a*d)*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + (c*(25*a*c*(12*a^2*d^2 + 3*a*b*d*(-8*c + d*x^4) + 4*b^2*c*(3*c + d*x^4))*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] - 5*x^4*(3*a^2*d^2 + 3*a*b*d^2*x^4 + 4*b^2*c*(c + d*x^4))*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((c + d*x^4)*(5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] - x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(60*a*c^2*(b*c - a*d)^2*(a + b*x^4)^(3/4))","C",0
216,1,44,53,0.0193041,"\int \frac{1}{\sqrt[4]{1+x^4} \left(2+x^4\right)} \, dx","Integrate[1/((1 + x^4)^(1/4)*(2 + x^4)),x]","\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{x^4+1}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{x^4+1}}\right)}{2\ 2^{3/4}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{x^4+1}}\right)}{2\ 2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{x^4+1}}\right)}{2\ 2^{3/4}}",1,"(ArcTan[x/(2^(1/4)*(1 + x^4)^(1/4))] + ArcTanh[x/(2^(1/4)*(1 + x^4)^(1/4))])/(2*2^(3/4))","A",1
217,1,48,57,0.0251905,"\int \frac{1}{\left(a-(a-b) x^4\right) \sqrt[4]{a+b x^4}} \, dx","Integrate[1/((a - (a - b)*x^4)*(a + b*x^4)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{a+b x^4}}\right)+\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{a+b x^4}}\right)}{2 a^{5/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{a+b x^4}}\right)}{2 a^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{a+b x^4}}\right)}{2 a^{5/4}}",1,"(ArcTan[(a^(1/4)*x)/(a + b*x^4)^(1/4)] + ArcTanh[(a^(1/4)*x)/(a + b*x^4)^(1/4)])/(2*a^(5/4))","A",1
218,1,172,79,0.263175,"\int \left(a+b x^4\right)^p \left(c+d x^4\right)^q \, dx","Integrate[(a + b*x^4)^p*(c + d*x^4)^q,x]","\frac{5 a c x \left(a+b x^4\right)^p \left(c+d x^4\right)^q F_1\left(\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{4 x^4 \left(b c p F_1\left(\frac{5}{4};1-p,-q;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)+a d q F_1\left(\frac{5}{4};-p,1-q;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}","x \left(a+b x^4\right)^p \left(\frac{b x^4}{a}+1\right)^{-p} \left(c+d x^4\right)^q \left(\frac{d x^4}{c}+1\right)^{-q} F_1\left(\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)",1,"(5*a*c*x*(a + b*x^4)^p*(c + d*x^4)^q*AppellF1[1/4, -p, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/(5*a*c*AppellF1[1/4, -p, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 4*x^4*(b*c*p*AppellF1[5/4, 1 - p, -q, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + a*d*q*AppellF1[5/4, -p, 1 - q, 9/4, -((b*x^4)/a), -((d*x^4)/c)]))","B",0
219,1,106,176,0.0563856,"\int \left(a+b x^4\right)^2 \left(c+d x^4\right)^q \, dx","Integrate[(a + b*x^4)^2*(c + d*x^4)^q,x]","\frac{1}{45} x \left(c+d x^4\right)^q \left(\frac{d x^4}{c}+1\right)^{-q} \left(45 a^2 \, _2F_1\left(\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right)+b x^4 \left(18 a \, _2F_1\left(\frac{5}{4},-q;\frac{9}{4};-\frac{d x^4}{c}\right)+5 b x^4 \, _2F_1\left(\frac{9}{4},-q;\frac{13}{4};-\frac{d x^4}{c}\right)\right)\right)","\frac{x \left(c+d x^4\right)^q \left(\frac{d x^4}{c}+1\right)^{-q} \left(a^2 d^2 \left(16 q^2+56 q+45\right)-2 a b c d (4 q+9)+5 b^2 c^2\right) \, _2F_1\left(\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right)}{d^2 (4 q+5) (4 q+9)}-\frac{b x \left(c+d x^4\right)^{q+1} (5 b c-a d (4 q+13))}{d^2 (4 q+5) (4 q+9)}+\frac{b x \left(a+b x^4\right) \left(c+d x^4\right)^{q+1}}{d (4 q+9)}",1,"(x*(c + d*x^4)^q*(45*a^2*Hypergeometric2F1[1/4, -q, 5/4, -((d*x^4)/c)] + b*x^4*(18*a*Hypergeometric2F1[5/4, -q, 9/4, -((d*x^4)/c)] + 5*b*x^4*Hypergeometric2F1[9/4, -q, 13/4, -((d*x^4)/c)])))/(45*(1 + (d*x^4)/c)^q)","A",1
220,1,90,93,0.0328325,"\int \left(a+b x^4\right) \left(c+d x^4\right)^q \, dx","Integrate[(a + b*x^4)*(c + d*x^4)^q,x]","\frac{x \left(c+d x^4\right)^q \left(\frac{d x^4}{c}+1\right)^{-q} \left((a d (4 q+5)-b c) \, _2F_1\left(\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right)+b \left(c+d x^4\right) \left(\frac{d x^4}{c}+1\right)^q\right)}{d (4 q+5)}","\frac{b x \left(c+d x^4\right)^{q+1}}{d (4 q+5)}-\frac{x \left(c+d x^4\right)^q \left(\frac{d x^4}{c}+1\right)^{-q} (b c-a d (4 q+5)) \, _2F_1\left(\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right)}{d (4 q+5)}",1,"(x*(c + d*x^4)^q*(b*(c + d*x^4)*(1 + (d*x^4)/c)^q + (-(b*c) + a*d*(5 + 4*q))*Hypergeometric2F1[1/4, -q, 5/4, -((d*x^4)/c)]))/(d*(5 + 4*q)*(1 + (d*x^4)/c)^q)","A",1
221,1,162,57,0.2173306,"\int \frac{\left(c+d x^4\right)^q}{a+b x^4} \, dx","Integrate[(c + d*x^4)^q/(a + b*x^4),x]","\frac{5 a c x \left(c+d x^4\right)^q F_1\left(\frac{1}{4};-q,1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)}{\left(a+b x^4\right) \left(4 x^4 \left(a d q F_1\left(\frac{5}{4};1-q,1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)-b c F_1\left(\frac{5}{4};-q,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};-q,1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right)\right)}","\frac{x \left(c+d x^4\right)^q \left(\frac{d x^4}{c}+1\right)^{-q} F_1\left(\frac{1}{4};1,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a}",1,"(5*a*c*x*(c + d*x^4)^q*AppellF1[1/4, -q, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)])/((a + b*x^4)*(5*a*c*AppellF1[1/4, -q, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 4*x^4*(a*d*q*AppellF1[5/4, 1 - q, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)] - b*c*AppellF1[5/4, -q, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)])))","B",0
222,1,162,57,0.2424065,"\int \frac{\left(c+d x^4\right)^q}{\left(a+b x^4\right)^2} \, dx","Integrate[(c + d*x^4)^q/(a + b*x^4)^2,x]","\frac{5 a c x \left(c+d x^4\right)^q F_1\left(\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{\left(a+b x^4\right)^2 \left(4 x^4 \left(a d q F_1\left(\frac{5}{4};2,1-q;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)-2 b c F_1\left(\frac{5}{4};3,-q;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)+5 a c F_1\left(\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)\right)}","\frac{x \left(c+d x^4\right)^q \left(\frac{d x^4}{c}+1\right)^{-q} F_1\left(\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right)}{a^2}",1,"(5*a*c*x*(c + d*x^4)^q*AppellF1[1/4, 2, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/((a + b*x^4)^2*(5*a*c*AppellF1[1/4, 2, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 4*x^4*(a*d*q*AppellF1[5/4, 2, 1 - q, 9/4, -((b*x^4)/a), -((d*x^4)/c)] - 2*b*c*AppellF1[5/4, 3, -q, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))","B",0
223,1,48,545,0.0195576,"\int \frac{1}{\sqrt[5]{a+b x^5} \left(c+d x^5\right)} \, dx","Integrate[1/((a + b*x^5)^(1/5)*(c + d*x^5)),x]","\frac{x \, _2F_1\left(\frac{1}{5},1;\frac{6}{5};\frac{(b c-a d) x^5}{c \left(b x^5+a\right)}\right)}{c \sqrt[5]{a+b x^5}}","-\frac{\log \left(\sqrt[5]{c}-\frac{x \sqrt[5]{b c-a d}}{\sqrt[5]{a+b x^5}}\right)}{5 c^{4/5} \sqrt[5]{b c-a d}}-\frac{\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{1}{5} \left(5-2 \sqrt{5}\right)}-\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} x \sqrt[5]{b c-a d}}{\sqrt[5]{c} \sqrt[5]{a+b x^5}}\right)}{5 c^{4/5} \sqrt[5]{b c-a d}}+\frac{\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{2}{5} \left(5+\sqrt{5}\right)} x \sqrt[5]{b c-a d}}{\sqrt[5]{c} \sqrt[5]{a+b x^5}}+\sqrt{\frac{1}{5} \left(5+2 \sqrt{5}\right)}\right)}{5 c^{4/5} \sqrt[5]{b c-a d}}+\frac{\left(1-\sqrt{5}\right) \log \left(\frac{2 c^{2/5} \left(a+b x^5\right)^{2/5}-\sqrt{5} \sqrt[5]{c} x \sqrt[5]{a+b x^5} \sqrt[5]{b c-a d}+\sqrt[5]{c} x \sqrt[5]{a+b x^5} \sqrt[5]{b c-a d}+2 x^2 (b c-a d)^{2/5}}{\left(a+b x^5\right)^{2/5}}\right)}{20 c^{4/5} \sqrt[5]{b c-a d}}+\frac{\left(1+\sqrt{5}\right) \log \left(\frac{2 c^{2/5} \left(a+b x^5\right)^{2/5}+\sqrt{5} \sqrt[5]{c} x \sqrt[5]{a+b x^5} \sqrt[5]{b c-a d}+\sqrt[5]{c} x \sqrt[5]{a+b x^5} \sqrt[5]{b c-a d}+2 x^2 (b c-a d)^{2/5}}{\left(a+b x^5\right)^{2/5}}\right)}{20 c^{4/5} \sqrt[5]{b c-a d}}",1,"(x*Hypergeometric2F1[1/5, 1, 6/5, ((b*c - a*d)*x^5)/(c*(a + b*x^5))])/(c*(a + b*x^5)^(1/5))","C",1
224,1,118,143,0.1899521,"\int \sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right)^3 \, dx","Integrate[Sqrt[a + b/x]*(c + d/x)^3,x]","\frac{\sqrt{a+\frac{b}{x}} \left(4 a^2 d^3 x^2-2 a b d^2 x (15 c x+d)-3 b^2 \left(-5 c^3 x^3+30 c^2 d x^2+10 c d^2 x+2 d^3\right)\right)}{15 b^2 x^2}+\frac{c^2 (6 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}","-\frac{d \sqrt{a+\frac{b}{x}} \left(2 \left(-2 a^2 d^2+15 a b c d+57 b^2 c^2\right)+\frac{b d (2 a d+33 b c)}{x}\right)}{15 b^2}+\frac{c^2 (6 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}+x \sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right)^3-\frac{7}{5} d \sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right)^2",1,"(Sqrt[a + b/x]*(4*a^2*d^3*x^2 - 2*a*b*d^2*x*(d + 15*c*x) - 3*b^2*(2*d^3 + 10*c*d^2*x + 30*c^2*d*x^2 - 5*c^3*x^3)))/(15*b^2*x^2) + (c^2*(b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a]","A",1
225,1,84,99,0.1576612,"\int \sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right)^2 \, dx","Integrate[Sqrt[a + b/x]*(c + d/x)^2,x]","\frac{\sqrt{a+\frac{b}{x}} \left(b \left(3 c^2 x^2-12 c d x-2 d^2\right)-2 a d^2 x\right)}{3 b x}+\frac{c (4 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}","\frac{c^2 x \left(a+\frac{b}{x}\right)^{3/2}}{a}-\frac{c \sqrt{a+\frac{b}{x}} (4 a d+b c)}{a}+\frac{c (4 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{2 d^2 \left(a+\frac{b}{x}\right)^{3/2}}{3 b}",1,"(Sqrt[a + b/x]*(-2*a*d^2*x + b*(-2*d^2 - 12*c*d*x + 3*c^2*x^2)))/(3*b*x) + (c*(b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a]","A",1
226,1,52,74,0.0422757,"\int \sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right) \, dx","Integrate[Sqrt[a + b/x]*(c + d/x),x]","\sqrt{a+\frac{b}{x}} (c x-2 d)+\frac{(2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}","-\frac{\sqrt{a+\frac{b}{x}} (2 a d+b c)}{a}+\frac{(2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{c x \left(a+\frac{b}{x}\right)^{3/2}}{a}",1,"Sqrt[a + b/x]*(-2*d + c*x) + ((b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a]","A",1
227,1,39,39,0.0725425,"\int \sqrt{a+\frac{b}{x}} \, dx","Integrate[Sqrt[a + b/x],x]","x \sqrt{a+\frac{b}{x}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}","x \sqrt{a+\frac{b}{x}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"Sqrt[a + b/x]*x + (b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a]","A",1
228,1,100,104,0.2436433,"\int \frac{\sqrt{a+\frac{b}{x}}}{c+\frac{d}{x}} \, dx","Integrate[Sqrt[a + b/x]/(c + d/x),x]","\frac{2 \sqrt{d} \sqrt{b c-a d} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)+\frac{(b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}+c x \sqrt{a+\frac{b}{x}}}{c^2}","\frac{2 \sqrt{d} \sqrt{b c-a d} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^2}+\frac{(b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a} c^2}+\frac{x \sqrt{a+\frac{b}{x}}}{c}",1,"(c*Sqrt[a + b/x]*x + 2*Sqrt[d]*Sqrt[b*c - a*d]*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]] + ((b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a])/c^2","A",1
229,1,122,147,0.3858915,"\int \frac{\sqrt{a+\frac{b}{x}}}{\left(c+\frac{d}{x}\right)^2} \, dx","Integrate[Sqrt[a + b/x]/(c + d/x)^2,x]","\frac{\frac{c x \sqrt{a+\frac{b}{x}} (c x+2 d)}{c x+d}+\frac{\sqrt{d} (3 b c-4 a d) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}+\frac{(b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}}{c^3}","\frac{\sqrt{d} (3 b c-4 a d) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^3 \sqrt{b c-a d}}+\frac{(b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a} c^3}+\frac{2 d \sqrt{a+\frac{b}{x}}}{c^2 \left(c+\frac{d}{x}\right)}+\frac{x \sqrt{a+\frac{b}{x}}}{c \left(c+\frac{d}{x}\right)}",1,"((c*Sqrt[a + b/x]*x*(2*d + c*x))/(d + c*x) + (Sqrt[d]*(3*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d] + ((b*c - 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a])/c^3","A",1
230,1,330,213,0.8230505,"\int \frac{\sqrt{a+\frac{b}{x}}}{\left(c+\frac{d}{x}\right)^3} \, dx","Integrate[Sqrt[a + b/x]/(c + d/x)^3,x]","\frac{(c x+d) \left(\frac{1}{2} c d^{5/2} \sqrt{a+\frac{b}{x}} (a x+b) \left(12 a^2 d^2-17 a b c d+4 b^2 c^2\right)+(c x+d) \left(-\frac{1}{2} a d^2 \left(24 a^2 d^2-40 a b c d+15 b^2 c^2\right) \left(\sqrt{d} \sqrt{a+\frac{b}{x}}-\sqrt{b c-a d} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)\right)-d^{3/2} (b c-6 a d) (b c-a d)^2 \left(2 \sqrt{a+\frac{b}{x}}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)\right)\right)\right)+2 c^3 d^{3/2} x^3 \left(a+\frac{b}{x}\right)^{3/2} (b c-a d)^2+c^2 d^{5/2} x \sqrt{a+\frac{b}{x}} (a x+b) (2 b c-3 a d) (b c-a d)}{2 a c^4 d^{3/2} (c x+d)^2 (b c-a d)^2}","\frac{\sqrt{d} \left(24 a^2 d^2-40 a b c d+15 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{4 c^4 (b c-a d)^{3/2}}+\frac{(b c-6 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a} c^4}+\frac{d \sqrt{a+\frac{b}{x}} (11 b c-12 a d)}{4 c^3 \left(c+\frac{d}{x}\right) (b c-a d)}+\frac{3 d \sqrt{a+\frac{b}{x}}}{2 c^2 \left(c+\frac{d}{x}\right)^2}+\frac{x \sqrt{a+\frac{b}{x}}}{c \left(c+\frac{d}{x}\right)^2}",1,"(2*c^3*d^(3/2)*(b*c - a*d)^2*(a + b/x)^(3/2)*x^3 + c^2*d^(5/2)*(2*b*c - 3*a*d)*(b*c - a*d)*Sqrt[a + b/x]*x*(b + a*x) + (d + c*x)*((c*d^(5/2)*(4*b^2*c^2 - 17*a*b*c*d + 12*a^2*d^2)*Sqrt[a + b/x]*(b + a*x))/2 + (d + c*x)*(-1/2*(a*d^2*(15*b^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*(Sqrt[d]*Sqrt[a + b/x] - Sqrt[b*c - a*d]*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])) - d^(3/2)*(b*c - 6*a*d)*(b*c - a*d)^2*(2*Sqrt[a + b/x] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]))))/(2*a*c^4*d^(3/2)*(b*c - a*d)^2*(d + c*x)^2)","A",1
231,1,159,164,0.2130548,"\int \left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right)^3 \, dx","Integrate[(a + b/x)^(3/2)*(c + d/x)^3,x]","\frac{\sqrt{a+\frac{b}{x}} \left(4 a^3 d^3 x^3-2 a^2 b d^2 x^2 (21 c x+d)+a b^2 x \left(35 c^3 x^3-280 c^2 d x^2-84 c d^2 x-16 d^3\right)-2 b^3 \left(35 c^3 x^3+35 c^2 d x^2+21 c d^2 x+5 d^3\right)\right)}{35 b^2 x^3}+3 \sqrt{a} c^2 (2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)","-\frac{d \left(a+\frac{b}{x}\right)^{3/2} \left(\frac{3 b d (2 a d+19 b c)}{x}+2 (13 b c-a d) (2 a d+5 b c)\right)}{35 b^2}-3 c^2 \sqrt{a+\frac{b}{x}} (2 a d+b c)+3 \sqrt{a} c^2 (2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)+x \left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right)^3-\frac{9}{7} d \left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right)^2",1,"(Sqrt[a + b/x]*(4*a^3*d^3*x^3 - 2*a^2*b*d^2*x^2*(d + 21*c*x) + a*b^2*x*(-16*d^3 - 84*c*d^2*x - 280*c^2*d*x^2 + 35*c^3*x^3) - 2*b^3*(5*d^3 + 21*c*d^2*x + 35*c^2*d*x^2 + 35*c^3*x^3)))/(35*b^2*x^3) + 3*Sqrt[a]*c^2*(b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]","A",1
232,1,106,126,0.2597588,"\int \left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right)^2 \, dx","Integrate[(a + b/x)^(3/2)*(c + d/x)^2,x]","-\frac{c (4 a d+3 b c) \left(\sqrt{a+\frac{b}{x}} (4 a x+b)-3 a^{3/2} x \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)\right)}{3 a x}+\frac{c^2 x \left(a+\frac{b}{x}\right)^{5/2}}{a}-\frac{2 d^2 \left(a+\frac{b}{x}\right)^{5/2}}{5 b}","\frac{c^2 x \left(a+\frac{b}{x}\right)^{5/2}}{a}-\frac{c \left(a+\frac{b}{x}\right)^{3/2} (4 a d+3 b c)}{3 a}-c \sqrt{a+\frac{b}{x}} (4 a d+3 b c)+\sqrt{a} c (4 a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)-\frac{2 d^2 \left(a+\frac{b}{x}\right)^{5/2}}{5 b}",1,"(-2*d^2*(a + b/x)^(5/2))/(5*b) + (c^2*(a + b/x)^(5/2)*x)/a - (c*(3*b*c + 4*a*d)*(Sqrt[a + b/x]*(b + 4*a*x) - 3*a^(3/2)*x*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]))/(3*a*x)","A",1
233,1,73,100,0.0907438,"\int \left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right) \, dx","Integrate[(a + b/x)^(3/2)*(c + d/x),x]","\frac{\sqrt{a+\frac{b}{x}} (a x (3 c x-8 d)-2 b (3 c x+d))}{3 x}+\sqrt{a} (2 a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)","-\frac{\left(a+\frac{b}{x}\right)^{3/2} (2 a d+3 b c)}{3 a}-\sqrt{a+\frac{b}{x}} (2 a d+3 b c)+\sqrt{a} (2 a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)+\frac{c x \left(a+\frac{b}{x}\right)^{5/2}}{a}",1,"(Sqrt[a + b/x]*(a*x*(-8*d + 3*c*x) - 2*b*(d + 3*c*x)))/(3*x) + Sqrt[a]*(3*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]","A",1
234,1,46,54,0.0250586,"\int \left(a+\frac{b}{x}\right)^{3/2} \, dx","Integrate[(a + b/x)^(3/2),x]","\sqrt{a+\frac{b}{x}} (a x-2 b)+3 \sqrt{a} b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)","x \left(a+\frac{b}{x}\right)^{3/2}-3 b \sqrt{a+\frac{b}{x}}+3 \sqrt{a} b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)",1,"Sqrt[a + b/x]*(-2*b + a*x) + 3*Sqrt[a]*b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]","A",1
235,1,102,106,0.2355031,"\int \frac{\left(a+\frac{b}{x}\right)^{3/2}}{c+\frac{d}{x}} \, dx","Integrate[(a + b/x)^(3/2)/(c + d/x),x]","\frac{-\frac{2 (b c-a d)^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{\sqrt{d}}+\sqrt{a} (3 b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)+a c x \sqrt{a+\frac{b}{x}}}{c^2}","-\frac{2 (b c-a d)^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^2 \sqrt{d}}+\frac{\sqrt{a} (3 b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{c^2}+\frac{a x \sqrt{a+\frac{b}{x}}}{c}",1,"(a*c*Sqrt[a + b/x]*x - (2*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/Sqrt[d] + Sqrt[a]*(3*b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/c^2","A",1
236,1,143,156,0.3873244,"\int \frac{\left(a+\frac{b}{x}\right)^{3/2}}{\left(c+\frac{d}{x}\right)^2} \, dx","Integrate[(a + b/x)^(3/2)/(c + d/x)^2,x]","\frac{-\frac{\left(4 a^2 d^2-5 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{\sqrt{d} \sqrt{b c-a d}}+\frac{c x \sqrt{a+\frac{b}{x}} (a c x+2 a d-b c)}{c x+d}+\sqrt{a} (3 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{c^3}","-\frac{(b c-4 a d) \sqrt{b c-a d} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^3 \sqrt{d}}+\frac{\sqrt{a} (3 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{c^3}-\frac{\sqrt{a+\frac{b}{x}} (b c-2 a d)}{c^2 \left(c+\frac{d}{x}\right)}+\frac{a x \sqrt{a+\frac{b}{x}}}{c \left(c+\frac{d}{x}\right)}",1,"((c*Sqrt[a + b/x]*x*(-(b*c) + 2*a*d + a*c*x))/(d + c*x) - ((b^2*c^2 - 5*a*b*c*d + 4*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(Sqrt[d]*Sqrt[b*c - a*d]) + Sqrt[a]*(3*b*c - 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/c^3","A",1
237,1,168,209,0.6038615,"\int \frac{\left(a+\frac{b}{x}\right)^{3/2}}{\left(c+\frac{d}{x}\right)^3} \, dx","Integrate[(a + b/x)^(3/2)/(c + d/x)^3,x]","\frac{-\frac{3 \left(8 a^2 d^2-8 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{\sqrt{d} \sqrt{b c-a d}}+\frac{c x \sqrt{a+\frac{b}{x}} \left(2 a \left(2 c^2 x^2+9 c d x+6 d^2\right)-b c (5 c x+3 d)\right)}{(c x+d)^2}+12 \sqrt{a} (b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{4 c^4}","-\frac{3 \left(8 a^2 d^2-8 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{4 c^4 \sqrt{d} \sqrt{b c-a d}}+\frac{3 \sqrt{a} (b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{c^4}-\frac{3 \sqrt{a+\frac{b}{x}} (b c-4 a d)}{4 c^3 \left(c+\frac{d}{x}\right)}-\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d)}{2 c^2 \left(c+\frac{d}{x}\right)^2}+\frac{a x \sqrt{a+\frac{b}{x}}}{c \left(c+\frac{d}{x}\right)^2}",1,"((c*Sqrt[a + b/x]*x*(-(b*c*(3*d + 5*c*x)) + 2*a*(6*d^2 + 9*c*d*x + 2*c^2*x^2)))/(d + c*x)^2 - (3*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(Sqrt[d]*Sqrt[b*c - a*d]) + 12*Sqrt[a]*(b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*c^4)","A",1
238,1,201,198,0.2545422,"\int \left(a+\frac{b}{x}\right)^{5/2} \left(c+\frac{d}{x}\right)^3 \, dx","Integrate[(a + b/x)^(5/2)*(c + d/x)^3,x]","a^{3/2} c^2 (6 a d+5 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)+\frac{\sqrt{a+\frac{b}{x}} \left(20 a^4 d^3 x^4-10 a^3 b d^2 x^3 (27 c x+d)-3 a^2 b^2 x^2 \left(-105 c^3 x^3+966 c^2 d x^2+270 c d^2 x+50 d^3\right)-2 a b^3 x \left(735 c^3 x^3+693 c^2 d x^2+405 c d^2 x+95 d^3\right)-2 b^4 \left(105 c^3 x^3+189 c^2 d x^2+135 c d^2 x+35 d^3\right)\right)}{315 b^2 x^4}","a^{3/2} c^2 (6 a d+5 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)-\frac{d \left(a+\frac{b}{x}\right)^{5/2} \left(2 \left(-10 a^2 d^2+135 a b c d+469 b^2 c^2\right)+\frac{5 b d (10 a d+89 b c)}{x}\right)}{315 b^2}-\frac{1}{3} c^2 \left(a+\frac{b}{x}\right)^{3/2} (6 a d+5 b c)-a c^2 \sqrt{a+\frac{b}{x}} (6 a d+5 b c)+x \left(a+\frac{b}{x}\right)^{5/2} \left(c+\frac{d}{x}\right)^3-\frac{11}{9} d \left(a+\frac{b}{x}\right)^{5/2} \left(c+\frac{d}{x}\right)^2",1,"(Sqrt[a + b/x]*(20*a^4*d^3*x^4 - 10*a^3*b*d^2*x^3*(d + 27*c*x) - 3*a^2*b^2*x^2*(50*d^3 + 270*c*d^2*x + 966*c^2*d*x^2 - 105*c^3*x^3) - 2*b^4*(35*d^3 + 135*c*d^2*x + 189*c^2*d*x^2 + 105*c^3*x^3) - 2*a*b^3*x*(95*d^3 + 405*c*d^2*x + 693*c^2*d*x^2 + 735*c^3*x^3)))/(315*b^2*x^4) + a^(3/2)*c^2*(5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]","A",1
239,1,121,152,0.158205,"\int \left(a+\frac{b}{x}\right)^{5/2} \left(c+\frac{d}{x}\right)^2 \, dx","Integrate[(a + b/x)^(5/2)*(c + d/x)^2,x]","-\frac{c (4 a d+5 b c) \left(\sqrt{a+\frac{b}{x}} \left(23 a^2 x^2+11 a b x+3 b^2\right)-15 a^{5/2} x^2 \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)\right)}{15 a x^2}+\frac{c^2 x \left(a+\frac{b}{x}\right)^{7/2}}{a}-\frac{2 d^2 \left(a+\frac{b}{x}\right)^{7/2}}{7 b}","a^{3/2} c (4 a d+5 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)+\frac{c^2 x \left(a+\frac{b}{x}\right)^{7/2}}{a}-\frac{c \left(a+\frac{b}{x}\right)^{5/2} (4 a d+5 b c)}{5 a}-\frac{1}{3} c \left(a+\frac{b}{x}\right)^{3/2} (4 a d+5 b c)-a c \sqrt{a+\frac{b}{x}} (4 a d+5 b c)-\frac{2 d^2 \left(a+\frac{b}{x}\right)^{7/2}}{7 b}",1,"(-2*d^2*(a + b/x)^(7/2))/(7*b) + (c^2*(a + b/x)^(7/2)*x)/a - (c*(5*b*c + 4*a*d)*(Sqrt[a + b/x]*(3*b^2 + 11*a*b*x + 23*a^2*x^2) - 15*a^(5/2)*x^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]))/(15*a*x^2)","A",1
240,1,94,125,0.112146,"\int \left(a+\frac{b}{x}\right)^{5/2} \left(c+\frac{d}{x}\right) \, dx","Integrate[(a + b/x)^(5/2)*(c + d/x),x]","a^{3/2} (2 a d+5 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)+\frac{\sqrt{a+\frac{b}{x}} \left(a^2 x^2 (15 c x-46 d)-2 a b x (35 c x+11 d)-2 b^2 (5 c x+3 d)\right)}{15 x^2}","a^{3/2} (2 a d+5 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)-\frac{\left(a+\frac{b}{x}\right)^{5/2} (2 a d+5 b c)}{5 a}-\frac{1}{3} \left(a+\frac{b}{x}\right)^{3/2} (2 a d+5 b c)-a \sqrt{a+\frac{b}{x}} (2 a d+5 b c)+\frac{c x \left(a+\frac{b}{x}\right)^{7/2}}{a}",1,"(Sqrt[a + b/x]*(-2*b^2*(3*d + 5*c*x) + a^2*x^2*(-46*d + 15*c*x) - 2*a*b*x*(11*d + 35*c*x)))/(15*x^2) + a^(3/2)*(5*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]","A",1
241,1,64,71,0.0621515,"\int \left(a+\frac{b}{x}\right)^{5/2} \, dx","Integrate[(a + b/x)^(5/2),x]","5 a^{3/2} b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)+\frac{\sqrt{a+\frac{b}{x}} \left(3 a^2 x^2-14 a b x-2 b^2\right)}{3 x}","5 a^{3/2} b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)+x \left(a+\frac{b}{x}\right)^{5/2}-\frac{5}{3} b \left(a+\frac{b}{x}\right)^{3/2}-5 a b \sqrt{a+\frac{b}{x}}",1,"(Sqrt[a + b/x]*(-2*b^2 - 14*a*b*x + 3*a^2*x^2))/(3*x) + 5*a^(3/2)*b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]","A",1
242,1,116,134,0.2701634,"\int \frac{\left(a+\frac{b}{x}\right)^{5/2}}{c+\frac{d}{x}} \, dx","Integrate[(a + b/x)^(5/2)/(c + d/x),x]","\frac{a^{3/2} (5 b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)+\frac{c \sqrt{a+\frac{b}{x}} \left(a^2 d x-2 b^2 c\right)}{d}+\frac{2 (b c-a d)^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{d^{3/2}}}{c^2}","\frac{a^{3/2} (5 b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{c^2}+\frac{2 (b c-a d)^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^2 d^{3/2}}-\frac{b \sqrt{a+\frac{b}{x}} (a d+2 b c)}{c d}+\frac{a x \left(a+\frac{b}{x}\right)^{3/2}}{c}",1,"((c*Sqrt[a + b/x]*(-2*b^2*c + a^2*d*x))/d + (2*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/d^(3/2) + a^(3/2)*(5*b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/c^2","A",1
243,1,145,166,0.4301396,"\int \frac{\left(a+\frac{b}{x}\right)^{5/2}}{\left(c+\frac{d}{x}\right)^2} \, dx","Integrate[(a + b/x)^(5/2)/(c + d/x)^2,x]","\frac{a^{3/2} (5 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)+\frac{c x \sqrt{a+\frac{b}{x}} \left(a^2 d (c x+2 d)-2 a b c d+b^2 c^2\right)}{d (c x+d)}-\frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{d^{3/2}}}{c^3}","\frac{a^{3/2} (5 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{c^3}-\frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^3 d^{3/2}}+\frac{\sqrt{a+\frac{b}{x}} (b c-2 a d) (b c-a d)}{c^2 d \left(c+\frac{d}{x}\right)}+\frac{a x \left(a+\frac{b}{x}\right)^{3/2}}{c \left(c+\frac{d}{x}\right)}",1,"((c*Sqrt[a + b/x]*x*(b^2*c^2 - 2*a*b*c*d + a^2*d*(2*d + c*x)))/(d*(d + c*x)) - ((b*c - a*d)^(3/2)*(b*c + 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/d^(3/2) + a^(3/2)*(5*b*c - 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/c^3","A",1
244,1,191,237,0.8489638,"\int \frac{\left(a+\frac{b}{x}\right)^{5/2}}{\left(c+\frac{d}{x}\right)^3} \, dx","Integrate[(a + b/x)^(5/2)/(c + d/x)^3,x]","\frac{-4 a^{3/2} (6 a d-5 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)+\frac{c x \sqrt{a+\frac{b}{x}} \left(2 a^2 d \left(2 c^2 x^2+9 c d x+6 d^2\right)-a b c d (11 c x+7 d)+b^2 c^2 (c x-d)\right)}{d (c x+d)^2}-\frac{\sqrt{b c-a d} \left(-24 a^2 d^2+8 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{d^{3/2}}}{4 c^4}","\frac{a^{3/2} (5 b c-6 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{c^4}-\frac{\sqrt{b c-a d} \left(-24 a^2 d^2+8 a b c d+b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{4 c^4 d^{3/2}}-\frac{\sqrt{a+\frac{b}{x}} \left(-12 a^2 d^2+7 a b c d+b^2 c^2\right)}{4 c^3 d \left(c+\frac{d}{x}\right)}+\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d) (b c-a d)}{2 c^2 d \left(c+\frac{d}{x}\right)^2}+\frac{a x \left(a+\frac{b}{x}\right)^{3/2}}{c \left(c+\frac{d}{x}\right)^2}",1,"((c*Sqrt[a + b/x]*x*(b^2*c^2*(-d + c*x) - a*b*c*d*(7*d + 11*c*x) + 2*a^2*d*(6*d^2 + 9*c*d*x + 2*c^2*x^2)))/(d*(d + c*x)^2) - (Sqrt[b*c - a*d]*(b^2*c^2 + 8*a*b*c*d - 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/d^(3/2) - 4*a^(3/2)*(-5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*c^4)","A",1
245,1,95,126,0.147413,"\int \frac{\left(c+\frac{d}{x}\right)^3}{\sqrt{a+\frac{b}{x}}} \, dx","Integrate[(c + d/x)^3/Sqrt[a + b/x],x]","\frac{c^2 (6 a d-b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{\sqrt{a+\frac{b}{x}} \left(4 a^2 d^3 x-2 a b d^2 (9 c x+d)+3 b^2 c^3 x^2\right)}{3 a b^2 x}","-\frac{c^2 (b c-6 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{d \sqrt{a+\frac{b}{x}} \left(2 \left(-2 a^2 d^2+9 a b c d+3 b^2 c^2\right)+\frac{b d (2 a d+3 b c)}{x}\right)}{3 a b^2}+\frac{c x \sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right)^2}{a}",1,"(Sqrt[a + b/x]*(4*a^2*d^3*x + 3*b^2*c^3*x^2 - 2*a*b*d^2*(d + 9*c*x)))/(3*a*b^2*x) + (c^2*(-(b*c) + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)","A",1
246,1,66,73,0.0870443,"\int \frac{\left(c+\frac{d}{x}\right)^2}{\sqrt{a+\frac{b}{x}}} \, dx","Integrate[(c + d/x)^2/Sqrt[a + b/x],x]","\frac{c (4 a d-b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{\sqrt{a+\frac{b}{x}} \left(b c^2 x-2 a d^2\right)}{a b}","-\frac{c (b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{c^2 x \sqrt{a+\frac{b}{x}}}{a}-\frac{2 d^2 \sqrt{a+\frac{b}{x}}}{b}",1,"(Sqrt[a + b/x]*(-2*a*d^2 + b*c^2*x))/(a*b) + (c*(-(b*c) + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)","A",1
247,1,53,51,0.0390499,"\int \frac{c+\frac{d}{x}}{\sqrt{a+\frac{b}{x}}} \, dx","Integrate[(c + d/x)/Sqrt[a + b/x],x]","\frac{2 \left(a d-\frac{b c}{2}\right) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{c x \sqrt{a+\frac{b}{x}}}{a}","\frac{c x \sqrt{a+\frac{b}{x}}}{a}-\frac{(b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2}}",1,"(c*Sqrt[a + b/x]*x)/a + (2*(-1/2*(b*c) + a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)","A",1
248,1,43,43,0.018228,"\int \frac{1}{\sqrt{a+\frac{b}{x}}} \, dx","Integrate[1/Sqrt[a + b/x],x]","\frac{x \sqrt{a+\frac{b}{x}}}{a}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2}}","\frac{x \sqrt{a+\frac{b}{x}}}{a}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2}}",1,"(Sqrt[a + b/x]*x)/a - (b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)","A",1
249,1,104,108,0.2459998,"\int \frac{1}{\sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right)} \, dx","Integrate[1/(Sqrt[a + b/x]*(c + d/x)),x]","\frac{-\frac{(2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{2 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}+\frac{c x \sqrt{a+\frac{b}{x}}}{a}}{c^2}","-\frac{(2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2} c^2}-\frac{2 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^2 \sqrt{b c-a d}}+\frac{x \sqrt{a+\frac{b}{x}}}{a c}",1,"((c*Sqrt[a + b/x]*x)/a - (2*d^(3/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d] - ((b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2))/c^2","A",1
250,1,150,172,0.7750874,"\int \frac{1}{\sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right)^2} \, dx","Integrate[1/(Sqrt[a + b/x]*(c + d/x)^2),x]","\frac{\frac{a d^{3/2} (4 a d-5 b c) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{3/2}}+\frac{c x \sqrt{a+\frac{b}{x}} (b c (c x+d)-a d (c x+2 d))}{(c x+d) (b c-a d)}-\frac{(4 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}}{a c^3}","-\frac{(4 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2} c^3}-\frac{d^{3/2} (5 b c-4 a d) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^3 (b c-a d)^{3/2}}+\frac{d \sqrt{a+\frac{b}{x}} (b c-2 a d)}{a c^2 \left(c+\frac{d}{x}\right) (b c-a d)}+\frac{x \sqrt{a+\frac{b}{x}}}{a c \left(c+\frac{d}{x}\right)}",1,"((c*Sqrt[a + b/x]*x*(b*c*(d + c*x) - a*d*(2*d + c*x)))/((b*c - a*d)*(d + c*x)) + (a*d^(3/2)*(-5*b*c + 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2) - ((b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a])/(a*c^3)","A",1
251,1,216,250,1.7456619,"\int \frac{1}{\sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right)^3} \, dx","Integrate[1/(Sqrt[a + b/x]*(c + d/x)^3),x]","\frac{\frac{c x \sqrt{a+\frac{b}{x}} \left(2 a^2 d^2 \left(2 c^2 x^2+9 c d x+6 d^2\right)-a b c d \left(8 c^2 x^2+29 c d x+19 d^2\right)+4 b^2 c^2 (c x+d)^2\right)}{(c x+d)^2 (b c-a d)^2}-\frac{a d^{3/2} \left(24 a^2 d^2-56 a b c d+35 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{(b c-a d)^{5/2}}-\frac{4 (6 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{\sqrt{a}}}{4 a c^4}","-\frac{(6 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{3/2} c^4}-\frac{d^{3/2} \left(24 a^2 d^2-56 a b c d+35 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{4 c^4 (b c-a d)^{5/2}}+\frac{d \sqrt{a+\frac{b}{x}} (b c-4 a d) (4 b c-3 a d)}{4 a c^3 \left(c+\frac{d}{x}\right) (b c-a d)^2}+\frac{d \sqrt{a+\frac{b}{x}} (2 b c-3 a d)}{2 a c^2 \left(c+\frac{d}{x}\right)^2 (b c-a d)}+\frac{x \sqrt{a+\frac{b}{x}}}{a c \left(c+\frac{d}{x}\right)^2}",1,"((c*Sqrt[a + b/x]*x*(4*b^2*c^2*(d + c*x)^2 + 2*a^2*d^2*(6*d^2 + 9*c*d*x + 2*c^2*x^2) - a*b*c*d*(19*d^2 + 29*c*d*x + 8*c^2*x^2)))/((b*c - a*d)^2*(d + c*x)^2) - (a*d^(3/2)*(35*b^2*c^2 - 56*a*b*c*d + 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(5/2) - (4*(b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a])/(4*a*c^4)","A",1
252,1,92,132,0.0749019,"\int \frac{\left(c+\frac{d}{x}\right)^3}{\left(a+\frac{b}{x}\right)^{3/2}} \, dx","Integrate[(c + d/x)^3/(a + b/x)^(3/2),x]","\frac{a \left(-4 a^2 d^3 x-2 a b d^2 (d-3 c x)+b^2 c^3 x^2\right)+3 b^2 c^2 x (b c-2 a d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b}{a x}+1\right)}{a^2 b^2 x \sqrt{a+\frac{b}{x}}}","-\frac{3 c^2 (b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{5/2}}+\frac{(b c-2 a d) \left(2 a^2 d^2-2 a b c d+3 b^2 c^2\right)-\frac{a b d^2 (2 a d+b c)}{x}}{a^2 b^2 \sqrt{a+\frac{b}{x}}}+\frac{c x \left(c+\frac{d}{x}\right)^2}{a \sqrt{a+\frac{b}{x}}}",1,"(a*(-4*a^2*d^3*x + b^2*c^3*x^2 - 2*a*b*d^2*(d - 3*c*x)) + 3*b^2*c^2*(b*c - 2*a*d)*x*Hypergeometric2F1[-1/2, 1, 1/2, 1 + b/(a*x)])/(a^2*b^2*Sqrt[a + b/x]*x)","C",1
253,1,81,94,0.1031102,"\int \frac{\left(c+\frac{d}{x}\right)^2}{\left(a+\frac{b}{x}\right)^{3/2}} \, dx","Integrate[(c + d/x)^2/(a + b/x)^(3/2),x]","\frac{c (4 a d-3 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{5/2}}+\frac{2 a^2 d^2+a b c (c x-4 d)+3 b^2 c^2}{a^2 b \sqrt{a+\frac{b}{x}}}","-\frac{c (3 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{5/2}}+\frac{2 a^2 d^2+b c (3 b c-4 a d)}{a^2 b \sqrt{a+\frac{b}{x}}}+\frac{c^2 x}{a \sqrt{a+\frac{b}{x}}}",1,"(3*b^2*c^2 + 2*a^2*d^2 + a*b*c*(-4*d + c*x))/(a^2*b*Sqrt[a + b/x]) + (c*(-3*b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(5/2)","A",1
254,1,48,76,0.023599,"\int \frac{c+\frac{d}{x}}{\left(a+\frac{b}{x}\right)^{3/2}} \, dx","Integrate[(c + d/x)/(a + b/x)^(3/2),x]","\frac{(3 b c-2 a d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b}{a x}+1\right)+a c x}{a^2 \sqrt{a+\frac{b}{x}}}","-\frac{(3 b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{5/2}}+\frac{3 b c-2 a d}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{c x}{a \sqrt{a+\frac{b}{x}}}",1,"(a*c*x + (3*b*c - 2*a*d)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + b/(a*x)])/(a^2*Sqrt[a + b/x])","C",1
255,1,36,60,0.0126111,"\int \frac{1}{\left(a+\frac{b}{x}\right)^{3/2}} \, dx","Integrate[(a + b/x)^(-3/2),x]","\frac{2 b \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{a+\frac{b}{x}}{a}\right)}{a^2 \sqrt{a+\frac{b}{x}}}","-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{5/2}}+\frac{3 b}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{x}{a \sqrt{a+\frac{b}{x}}}",1,"(2*b*Hypergeometric2F1[-1/2, 2, 1/2, (a + b/x)/a])/(a^2*Sqrt[a + b/x])","C",1
256,1,106,147,0.0715703,"\int \frac{1}{\left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right)} \, dx","Integrate[1/((a + b/x)^(3/2)*(c + d/x)),x]","\frac{(a d-b c) \left((2 a d+3 b c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b}{a x}+1\right)+a c x\right)-2 a^2 d^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d \left(a+\frac{b}{x}\right)}{a d-b c}\right)}{a^2 c^2 \sqrt{a+\frac{b}{x}} (a d-b c)}","-\frac{(2 a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{5/2} c^2}+\frac{b (3 b c-a d)}{a^2 c \sqrt{a+\frac{b}{x}} (b c-a d)}+\frac{2 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^2 (b c-a d)^{3/2}}+\frac{x}{a c \sqrt{a+\frac{b}{x}}}",1,"(-2*a^2*d^2*Hypergeometric2F1[-1/2, 1, 1/2, (d*(a + b/x))/(-(b*c) + a*d)] + (-(b*c) + a*d)*(a*c*x + (3*b*c + 2*a*d)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + b/(a*x)]))/(a^2*c^2*(-(b*c) + a*d)*Sqrt[a + b/x])","C",1
257,1,164,224,0.1447597,"\int \frac{1}{\left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right)^2} \, dx","Integrate[1/((a + b/x)^(3/2)*(c + d/x)^2),x]","\frac{(b c-a d) \left((c x+d) \left(-4 a^2 d^2+a b c d+3 b^2 c^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b}{a x}+1\right)+a c x (b c (c x+d)-a d (c x+2 d))\right)+a^2 d^2 (c x+d) (7 b c-4 a d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d \left(a+\frac{b}{x}\right)}{a d-b c}\right)}{a^2 c^3 \sqrt{a+\frac{b}{x}} (c x+d) (b c-a d)^2}","-\frac{(4 a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{5/2} c^3}+\frac{b \left(2 a^2 d^2-2 a b c d+3 b^2 c^2\right)}{a^2 c^2 \sqrt{a+\frac{b}{x}} (b c-a d)^2}+\frac{d^{5/2} (7 b c-4 a d) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^3 (b c-a d)^{5/2}}+\frac{d (b c-2 a d)}{a c^2 \sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right) (b c-a d)}+\frac{x}{a c \sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right)}",1,"(a^2*d^2*(7*b*c - 4*a*d)*(d + c*x)*Hypergeometric2F1[-1/2, 1, 1/2, (d*(a + b/x))/(-(b*c) + a*d)] + (b*c - a*d)*(a*c*x*(b*c*(d + c*x) - a*d*(2*d + c*x)) + (3*b^2*c^2 + a*b*c*d - 4*a^2*d^2)*(d + c*x)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + b/(a*x)]))/(a^2*c^3*(b*c - a*d)^2*Sqrt[a + b/x]*(d + c*x))","C",1
258,1,239,320,0.3207915,"\int \frac{1}{\left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right)^3} \, dx","Integrate[1/((a + b/x)^(3/2)*(c + d/x)^3),x]","\frac{(c x+d) \left(2 (c x+d) \left(\frac{3}{4} a^2 d^2 \left(8 a^2 d^2-24 a b c d+21 b^2 c^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{d \left(a+\frac{b}{x}\right)}{a d-b c}\right)+3 (2 a d+b c) (b c-a d)^3 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b}{a x}+1\right)\right)-\frac{1}{2} a c d x (a d-b c) \left(12 a^2 d^2-21 a b c d+4 b^2 c^2\right)\right)+2 a c^3 x^3 (b c-a d)^3-a c^2 d x^2 (b c-a d)^2 (3 a d-2 b c)}{2 a^2 c^4 \sqrt{a+\frac{b}{x}} (c x+d)^2 (b c-a d)^3}","-\frac{3 (2 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{5/2} c^4}+\frac{3 d^{5/2} \left(8 a^2 d^2-24 a b c d+21 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{4 c^4 (b c-a d)^{7/2}}+\frac{d \left(12 a^2 d^2-21 a b c d+4 b^2 c^2\right)}{4 a c^3 \sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right) (b c-a d)^2}+\frac{3 b (2 b c-a d) \left(4 a^2 d^2-a b c d+2 b^2 c^2\right)}{4 a^2 c^3 \sqrt{a+\frac{b}{x}} (b c-a d)^3}+\frac{d (2 b c-3 a d)}{2 a c^2 \sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right)^2 (b c-a d)}+\frac{x}{a c \sqrt{a+\frac{b}{x}} \left(c+\frac{d}{x}\right)^2}",1,"(-(a*c^2*d*(b*c - a*d)^2*(-2*b*c + 3*a*d)*x^2) + 2*a*c^3*(b*c - a*d)^3*x^3 + (d + c*x)*(-1/2*(a*c*d*(-(b*c) + a*d)*(4*b^2*c^2 - 21*a*b*c*d + 12*a^2*d^2)*x) + 2*(d + c*x)*((3*a^2*d^2*(21*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*Hypergeometric2F1[-1/2, 1, 1/2, (d*(a + b/x))/(-(b*c) + a*d)])/4 + 3*(b*c - a*d)^3*(b*c + 2*a*d)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + b/(a*x)])))/(2*a^2*c^4*(b*c - a*d)^3*Sqrt[a + b/x]*(d + c*x)^2)","C",1
259,1,145,143,0.4303149,"\int \frac{\left(c+\frac{d}{x}\right)^3}{\left(a+\frac{b}{x}\right)^{5/2}} \, dx","Integrate[(c + d/x)^3/(a + b/x)^(5/2),x]","\frac{\frac{4 a^5 d^3 x}{b^2}+\frac{6 a^4 d^2 (c x+d)}{b}+3 a^3 c^2 x (c x-8 d)+2 a^2 b c^2 (10 c x-9 d)+15 a b^2 c^3+3 a c^2 \sqrt{\frac{b}{a x}+1} (a x+b) (6 a d-5 b c) \tanh ^{-1}\left(\sqrt{\frac{b}{a x}+1}\right)}{3 a^4 \sqrt{a+\frac{b}{x}} (a x+b)}","-\frac{c^2 (5 b c-6 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{7/2}}+\frac{(b c-a d) \left(-4 a^3 d^2 x-2 a^2 b d (5 c x+3 d)+a b^2 c (20 c x-3 d)+15 b^3 c^2\right)}{3 a^3 b^2 x \left(a+\frac{b}{x}\right)^{3/2}}+\frac{c x \left(c+\frac{d}{x}\right)^2}{a \left(a+\frac{b}{x}\right)^{3/2}}",1,"(15*a*b^2*c^3 + (4*a^5*d^3*x)/b^2 + 3*a^3*c^2*x*(-8*d + c*x) + (6*a^4*d^2*(d + c*x))/b + 2*a^2*b*c^2*(-9*d + 10*c*x) + 3*a*c^2*(-5*b*c + 6*a*d)*Sqrt[1 + b/(a*x)]*(b + a*x)*ArcTanh[Sqrt[1 + b/(a*x)]])/(3*a^4*Sqrt[a + b/x]*(b + a*x))","A",1
260,1,97,122,0.0815057,"\int \frac{\left(c+\frac{d}{x}\right)^2}{\left(a+\frac{b}{x}\right)^{5/2}} \, dx","Integrate[(c + d/x)^2/(a + b/x)^(5/2),x]","\frac{a x \left(2 a^2 d^2+a b c (3 c x-4 d)+5 b^2 c^2\right)+3 b c (a x+b) (5 b c-4 a d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b}{a x}+1\right)}{3 a^3 b \sqrt{a+\frac{b}{x}} (a x+b)}","-\frac{c (5 b c-4 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{7/2}}+\frac{c (5 b c-4 a d)}{a^3 \sqrt{a+\frac{b}{x}}}+\frac{2 a^2 d^2+b c (5 b c-4 a d)}{3 a^2 b \left(a+\frac{b}{x}\right)^{3/2}}+\frac{c^2 x}{a \left(a+\frac{b}{x}\right)^{3/2}}",1,"(a*x*(5*b^2*c^2 + 2*a^2*d^2 + a*b*c*(-4*d + 3*c*x)) + 3*b*c*(5*b*c - 4*a*d)*(b + a*x)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + b/(a*x)])/(3*a^3*b*Sqrt[a + b/x]*(b + a*x))","C",1
261,1,60,103,0.0354573,"\int \frac{c+\frac{d}{x}}{\left(a+\frac{b}{x}\right)^{5/2}} \, dx","Integrate[(c + d/x)/(a + b/x)^(5/2),x]","\frac{x \left((5 b c-2 a d) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b}{a x}+1\right)+3 a c x\right)}{3 a^2 \sqrt{a+\frac{b}{x}} (a x+b)}","-\frac{(5 b c-2 a d) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{7/2}}+\frac{5 b c-2 a d}{a^3 \sqrt{a+\frac{b}{x}}}+\frac{5 b c-2 a d}{3 a^2 \left(a+\frac{b}{x}\right)^{3/2}}+\frac{c x}{a \left(a+\frac{b}{x}\right)^{3/2}}",1,"(x*(3*a*c*x + (5*b*c - 2*a*d)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + b/(a*x)]))/(3*a^2*Sqrt[a + b/x]*(b + a*x))","C",1
262,1,38,79,0.0245932,"\int \frac{1}{\left(a+\frac{b}{x}\right)^{5/2}} \, dx","Integrate[(a + b/x)^(-5/2),x]","\frac{2 b \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};\frac{a+\frac{b}{x}}{a}\right)}{3 a^2 \left(a+\frac{b}{x}\right)^{3/2}}","-\frac{5 b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{7/2}}+\frac{5 b}{a^3 \sqrt{a+\frac{b}{x}}}+\frac{5 b}{3 a^2 \left(a+\frac{b}{x}\right)^{3/2}}+\frac{x}{a \left(a+\frac{b}{x}\right)^{3/2}}",1,"(2*b*Hypergeometric2F1[-3/2, 2, -1/2, (a + b/x)/a])/(3*a^2*(a + b/x)^(3/2))","C",1
263,1,118,201,0.0842122,"\int \frac{1}{\left(a+\frac{b}{x}\right)^{5/2} \left(c+\frac{d}{x}\right)} \, dx","Integrate[1/((a + b/x)^(5/2)*(c + d/x)),x]","\frac{x \left((a d-b c) \left((2 a d+5 b c) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b}{a x}+1\right)+3 a c x\right)-2 a^2 d^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{d \left(a+\frac{b}{x}\right)}{a d-b c}\right)\right)}{3 a^2 c^2 \sqrt{a+\frac{b}{x}} (a x+b) (a d-b c)}","-\frac{(2 a d+5 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{7/2} c^2}+\frac{b (5 b c-3 a d)}{3 a^2 c \left(a+\frac{b}{x}\right)^{3/2} (b c-a d)}+\frac{b \left(a^2 d^2-8 a b c d+5 b^2 c^2\right)}{a^3 c \sqrt{a+\frac{b}{x}} (b c-a d)^2}-\frac{2 d^{7/2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^2 (b c-a d)^{5/2}}+\frac{x}{a c \left(a+\frac{b}{x}\right)^{3/2}}",1,"(x*(-2*a^2*d^2*Hypergeometric2F1[-3/2, 1, -1/2, (d*(a + b/x))/(-(b*c) + a*d)] + (-(b*c) + a*d)*(3*a*c*x + (5*b*c + 2*a*d)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + b/(a*x)])))/(3*a^2*c^2*(-(b*c) + a*d)*Sqrt[a + b/x]*(b + a*x))","C",1
264,1,178,287,0.1886333,"\int \frac{1}{\left(a+\frac{b}{x}\right)^{5/2} \left(c+\frac{d}{x}\right)^2} \, dx","Integrate[1/((a + b/x)^(5/2)*(c + d/x)^2),x]","\frac{x \left((a d-b c) \left(3 a c x (a d (c x+2 d)-b c (c x+d))-(c x+d) \left(-4 a^2 d^2-a b c d+5 b^2 c^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b}{a x}+1\right)\right)+a^2 d^2 (c x+d) (9 b c-4 a d) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{d \left(a+\frac{b}{x}\right)}{a d-b c}\right)\right)}{3 a^2 c^3 \sqrt{a+\frac{b}{x}} (a x+b) (c x+d) (b c-a d)^2}","-\frac{(4 a d+5 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{7/2} c^3}+\frac{b \left(6 a^2 d^2-6 a b c d+5 b^2 c^2\right)}{3 a^2 c^2 \left(a+\frac{b}{x}\right)^{3/2} (b c-a d)^2}+\frac{b (b c-2 a d) \left(a^2 d^2-a b c d+5 b^2 c^2\right)}{a^3 c^2 \sqrt{a+\frac{b}{x}} (b c-a d)^3}-\frac{d^{7/2} (9 b c-4 a d) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{c^3 (b c-a d)^{7/2}}+\frac{d (b c-2 a d)}{a c^2 \left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right) (b c-a d)}+\frac{x}{a c \left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right)}",1,"(x*(a^2*d^2*(9*b*c - 4*a*d)*(d + c*x)*Hypergeometric2F1[-3/2, 1, -1/2, (d*(a + b/x))/(-(b*c) + a*d)] + (-(b*c) + a*d)*(3*a*c*x*(-(b*c*(d + c*x)) + a*d*(2*d + c*x)) - (5*b^2*c^2 - a*b*c*d - 4*a^2*d^2)*(d + c*x)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + b/(a*x)])))/(3*a^2*c^3*(b*c - a*d)^2*Sqrt[a + b/x]*(b + a*x)*(d + c*x))","C",1
265,1,239,409,0.3866581,"\int \frac{1}{\left(a+\frac{b}{x}\right)^{5/2} \left(c+\frac{d}{x}\right)^3} \, dx","Integrate[1/((a + b/x)^(5/2)*(c + d/x)^3),x]","\frac{(c x+d) \left(2 (c x+d) \left(\frac{1}{4} a^2 d^2 \left(24 a^2 d^2-88 a b c d+99 b^2 c^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{d \left(a+\frac{b}{x}\right)}{a d-b c}\right)+(6 a d+5 b c) (b c-a d)^3 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b}{a x}+1\right)\right)-\frac{3}{2} a c d x (a d-b c) \left(12 a^2 d^2-23 a b c d+4 b^2 c^2\right)\right)+6 a c^3 x^3 (b c-a d)^3-3 a c^2 d x^2 (b c-a d)^2 (3 a d-2 b c)}{6 a^2 c^4 \left(a+\frac{b}{x}\right)^{3/2} (c x+d)^2 (b c-a d)^3}","-\frac{(6 a d+5 b c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right)}{a^{7/2} c^4}-\frac{d^{7/2} \left(24 a^2 d^2-88 a b c d+99 b^2 c^2\right) \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)}{4 c^4 (b c-a d)^{9/2}}+\frac{d \left(12 a^2 d^2-23 a b c d+4 b^2 c^2\right)}{4 a c^3 \left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right) (b c-a d)^2}+\frac{b \left(-36 a^3 d^3+87 a^2 b c d^2-36 a b^2 c^2 d+20 b^3 c^3\right)}{12 a^2 c^3 \left(a+\frac{b}{x}\right)^{3/2} (b c-a d)^3}+\frac{b \left(12 a^4 d^4-35 a^3 b c d^3+24 a^2 b^2 c^2 d^2-56 a b^3 c^3 d+20 b^4 c^4\right)}{4 a^3 c^3 \sqrt{a+\frac{b}{x}} (b c-a d)^4}+\frac{d (2 b c-3 a d)}{2 a c^2 \left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right)^2 (b c-a d)}+\frac{x}{a c \left(a+\frac{b}{x}\right)^{3/2} \left(c+\frac{d}{x}\right)^2}",1,"(-3*a*c^2*d*(b*c - a*d)^2*(-2*b*c + 3*a*d)*x^2 + 6*a*c^3*(b*c - a*d)^3*x^3 + (d + c*x)*((-3*a*c*d*(-(b*c) + a*d)*(4*b^2*c^2 - 23*a*b*c*d + 12*a^2*d^2)*x)/2 + 2*(d + c*x)*((a^2*d^2*(99*b^2*c^2 - 88*a*b*c*d + 24*a^2*d^2)*Hypergeometric2F1[-3/2, 1, -1/2, (d*(a + b/x))/(-(b*c) + a*d)])/4 + (b*c - a*d)^3*(5*b*c + 6*a*d)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + b/(a*x)])))/(6*a^2*c^4*(b*c - a*d)^3*(a + b/x)^(3/2)*(d + c*x)^2)","C",1
266,1,167,123,1.4014859,"\int \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}} \, dx","Integrate[Sqrt[a + b/x]*Sqrt[c + d/x],x]","\frac{\sqrt{a+\frac{b}{x}} (c x+d)-2 \sqrt{d} \sqrt{b c-a d} \sqrt{\frac{b c x+b d}{b c x-a d x}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right)+\frac{\sqrt{c+\frac{d}{x}} (a d+b c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right)}{\sqrt{a} \sqrt{c}}}{\sqrt{c+\frac{d}{x}}}","x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+\frac{(a d+b c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right)}{\sqrt{a} \sqrt{c}}-2 \sqrt{b} \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b} \sqrt{c+\frac{d}{x}}}\right)",1,"(Sqrt[a + b/x]*(d + c*x) - 2*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[(b*d + b*c*x)/(b*c*x - a*d*x)]*ArcSinh[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]] + ((b*c + a*d)*Sqrt[c + d/x]*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])/(Sqrt[a]*Sqrt[c + d/x])])/(Sqrt[a]*Sqrt[c]))/Sqrt[c + d/x]","A",1
267,1,81,81,0.0678399,"\int \frac{\sqrt{a+\frac{b}{x}}}{\sqrt{c+\frac{d}{x}}} \, dx","Integrate[Sqrt[a + b/x]/Sqrt[c + d/x],x]","\frac{(b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right)}{\sqrt{a} c^{3/2}}+\frac{x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}}{c}","\frac{(b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right)}{\sqrt{a} c^{3/2}}+\frac{x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}}{c}",1,"(Sqrt[a + b/x]*Sqrt[c + d/x]*x)/c + ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])/(Sqrt[a]*Sqrt[c + d/x])])/(Sqrt[a]*c^(3/2))","A",1
268,1,87,122,0.1059079,"\int \frac{\sqrt{a+\frac{b}{x}}}{\left(c+\frac{d}{x}\right)^{3/2}} \, dx","Integrate[Sqrt[a + b/x]/(c + d/x)^(3/2),x]","\frac{(b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right)}{\sqrt{a} c^{5/2}}+\frac{\sqrt{a+\frac{b}{x}} (c x+3 d)}{c^2 \sqrt{c+\frac{d}{x}}}","\frac{(b c-3 a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right)}{\sqrt{a} c^{5/2}}-\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d)}{a c^2 \sqrt{c+\frac{d}{x}}}+\frac{x \left(a+\frac{b}{x}\right)^{3/2}}{a c \sqrt{c+\frac{d}{x}}}",1,"(Sqrt[a + b/x]*(3*d + c*x))/(c^2*Sqrt[c + d/x]) + ((b*c - 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])/(Sqrt[a]*Sqrt[c + d/x])])/(Sqrt[a]*c^(5/2))","A",1
269,1,206,96,0.3665574,"\int \left(a+\frac{b}{x}\right)^p \left(c+\frac{d}{x}\right)^q \, dx","Integrate[(a + b/x)^p*(c + d/x)^q,x]","\frac{b d x (p+q-2) \left(a+\frac{b}{x}\right)^p \left(c+\frac{d}{x}\right)^q F_1\left(-p-q+1;-p,-q;-p-q+2;-\frac{a x}{b},-\frac{c x}{d}\right)}{(p+q-1) \left(x \left(a d p F_1\left(-p-q+2;1-p,-q;-p-q+3;-\frac{a x}{b},-\frac{c x}{d}\right)+b c q F_1\left(-p-q+2;-p,1-q;-p-q+3;-\frac{a x}{b},-\frac{c x}{d}\right)\right)-b d (p+q-2) F_1\left(-p-q+1;-p,-q;-p-q+2;-\frac{a x}{b},-\frac{c x}{d}\right)\right)}","-\frac{b \left(a+\frac{b}{x}\right)^{p+1} \left(c+\frac{d}{x}\right)^q \left(\frac{b \left(c+\frac{d}{x}\right)}{b c-a d}\right)^{-q} F_1\left(p+1;-q,2;p+2;-\frac{d \left(a+\frac{b}{x}\right)}{b c-a d},\frac{a+\frac{b}{x}}{a}\right)}{a^2 (p+1)}",1,"(b*d*(-2 + p + q)*(a + b/x)^p*(c + d/x)^q*x*AppellF1[1 - p - q, -p, -q, 2 - p - q, -((a*x)/b), -((c*x)/d)])/((-1 + p + q)*(-(b*d*(-2 + p + q)*AppellF1[1 - p - q, -p, -q, 2 - p - q, -((a*x)/b), -((c*x)/d)]) + x*(a*d*p*AppellF1[2 - p - q, 1 - p, -q, 3 - p - q, -((a*x)/b), -((c*x)/d)] + b*c*q*AppellF1[2 - p - q, -p, 1 - q, 3 - p - q, -((a*x)/b), -((c*x)/d)])))","B",0
270,1,40,39,0.0315074,"\int \frac{a+\frac{b}{x^2}}{c+\frac{d}{x^2}} \, dx","Integrate[(a + b/x^2)/(c + d/x^2),x]","\frac{a x}{c}-\frac{(a d-b c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{c^{3/2} \sqrt{d}}","\frac{(b c-a d) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)}{c^{3/2} \sqrt{d}}+\frac{a x}{c}",1,"(a*x)/c - ((-(b*c) + a*d)*ArcTan[(Sqrt[c]*x)/Sqrt[d]])/(c^(3/2)*Sqrt[d])","A",1
271,1,205,233,0.3783456,"\int \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} \, dx","Integrate[Sqrt[a + b/x^2]*Sqrt[c + d/x^2],x]","-\frac{x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} \left(\sqrt{\frac{a}{b}} \left(a x^2+b\right) \left(c x^2+d\right)+i x \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} (b c-a d) F\left(i \sinh ^{-1}\left(\sqrt{\frac{a}{b}} x\right)|\frac{b c}{a d}\right)+2 i a d x \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a}{b}} x\right)|\frac{b c}{a d}\right)\right)}{\sqrt{\frac{a}{b}} \left(a x^2+b\right) \left(c x^2+d\right)}","-\frac{2 d \sqrt{a+\frac{b}{x^2}}}{x \sqrt{c+\frac{d}{x^2}}}+x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}-\frac{\sqrt{c} \sqrt{a+\frac{b}{x^2}} (a d+b c) F\left(\cot ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)|1-\frac{b c}{a d}\right)}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left(a+\frac{b}{x^2}\right)}{a \left(c+\frac{d}{x^2}\right)}}}+\frac{2 \sqrt{c} \sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left(\cot ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)|1-\frac{b c}{a d}\right)}{\sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left(a+\frac{b}{x^2}\right)}{a \left(c+\frac{d}{x^2}\right)}}}",1,"-((Sqrt[a + b/x^2]*Sqrt[c + d/x^2]*x*(Sqrt[a/b]*(b + a*x^2)*(d + c*x^2) + (2*I)*a*d*x*Sqrt[1 + (a*x^2)/b]*Sqrt[1 + (c*x^2)/d]*EllipticE[I*ArcSinh[Sqrt[a/b]*x], (b*c)/(a*d)] + I*(b*c - a*d)*x*Sqrt[1 + (a*x^2)/b]*Sqrt[1 + (c*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[a/b]*x], (b*c)/(a*d)]))/(Sqrt[a/b]*(b + a*x^2)*(d + c*x^2)))","C",1
272,1,86,232,0.073739,"\int \frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{c+\frac{d}{x^2}}} \, dx","Integrate[Sqrt[a + b/x^2]/Sqrt[c + d/x^2],x]","\frac{\sqrt{a+\frac{b}{x^2}} \sqrt{\frac{c x^2+d}{d}} E\left(\sin ^{-1}\left(\sqrt{-\frac{c}{d}} x\right)|\frac{a d}{b c}\right)}{\sqrt{-\frac{c}{d}} \sqrt{\frac{a x^2+b}{b}} \sqrt{c+\frac{d}{x^2}}}","-\frac{d \sqrt{a+\frac{b}{x^2}}}{c x \sqrt{c+\frac{d}{x^2}}}+\frac{x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c}-\frac{b \sqrt{c} \sqrt{a+\frac{b}{x^2}} F\left(\cot ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)|1-\frac{b c}{a d}\right)}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left(a+\frac{b}{x^2}\right)}{a \left(c+\frac{d}{x^2}\right)}}}+\frac{\sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left(\cot ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)|1-\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left(a+\frac{b}{x^2}\right)}{a \left(c+\frac{d}{x^2}\right)}}}",1,"(Sqrt[a + b/x^2]*Sqrt[(d + c*x^2)/d]*EllipticE[ArcSin[Sqrt[-(c/d)]*x], (a*d)/(b*c)])/(Sqrt[-(c/d)]*Sqrt[c + d/x^2]*Sqrt[(b + a*x^2)/b])","A",1
273,1,191,262,0.327799,"\int \frac{\sqrt{a+\frac{b}{x^2}}}{\left(c+\frac{d}{x^2}\right)^{3/2}} \, dx","Integrate[Sqrt[a + b/x^2]/(c + d/x^2)^(3/2),x]","-\frac{\sqrt{a+\frac{b}{x^2}} \left(i \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} (b c-2 a d) F\left(i \sinh ^{-1}\left(\sqrt{\frac{a}{b}} x\right)|\frac{b c}{a d}\right)+2 i a d \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a}{b}} x\right)|\frac{b c}{a d}\right)+c x \sqrt{\frac{a}{b}} \left(a x^2+b\right)\right)}{c^2 \sqrt{\frac{a}{b}} \left(a x^2+b\right) \sqrt{c+\frac{d}{x^2}}}","\frac{2 \sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left(\cot ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)|1-\frac{b c}{a d}\right)}{c^{3/2} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left(a+\frac{b}{x^2}\right)}{a \left(c+\frac{d}{x^2}\right)}}}-\frac{2 d \sqrt{a+\frac{b}{x^2}}}{c^2 x \sqrt{c+\frac{d}{x^2}}}+\frac{2 x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c^2}-\frac{x \sqrt{a+\frac{b}{x^2}}}{c \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{a+\frac{b}{x^2}} F\left(\cot ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d}}\right)|1-\frac{b c}{a d}\right)}{a \sqrt{c} \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left(a+\frac{b}{x^2}\right)}{a \left(c+\frac{d}{x^2}\right)}}}",1,"-((Sqrt[a + b/x^2]*(Sqrt[a/b]*c*x*(b + a*x^2) + (2*I)*a*d*Sqrt[1 + (a*x^2)/b]*Sqrt[1 + (c*x^2)/d]*EllipticE[I*ArcSinh[Sqrt[a/b]*x], (b*c)/(a*d)] + I*(b*c - 2*a*d)*Sqrt[1 + (a*x^2)/b]*Sqrt[1 + (c*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[a/b]*x], (b*c)/(a*d)]))/(Sqrt[a/b]*c^2*Sqrt[c + d/x^2]*(b + a*x^2)))","C",1
274,1,104,79,0.1494039,"\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
275,1,129,145,0.1022691,"\int \frac{a+\frac{b}{x^3}}{c+\frac{d}{x^3}} \, dx","Integrate[(a + b/x^3)/(c + d/x^3),x]","\frac{-(b c-a d) \log \left(c^{2/3} x^2-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3}\right)+2 (b c-a d) \log \left(\sqrt[3]{c} x+\sqrt[3]{d}\right)-2 \sqrt{3} (b c-a d) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{c} x}{\sqrt[3]{d}}}{\sqrt{3}}\right)+6 a \sqrt[3]{c} d^{2/3} x}{6 c^{4/3} d^{2/3}}","-\frac{(b c-a d) \log \left(c^{2/3} x^2-\sqrt[3]{c} \sqrt[3]{d} x+d^{2/3}\right)}{6 c^{4/3} d^{2/3}}+\frac{(b c-a d) \log \left(\sqrt[3]{c} x+\sqrt[3]{d}\right)}{3 c^{4/3} d^{2/3}}-\frac{(b c-a d) \tan ^{-1}\left(\frac{\sqrt[3]{d}-2 \sqrt[3]{c} x}{\sqrt{3} \sqrt[3]{d}}\right)}{\sqrt{3} c^{4/3} d^{2/3}}+\frac{a x}{c}",1,"(6*a*c^(1/3)*d^(2/3)*x - 2*Sqrt[3]*(b*c - a*d)*ArcTan[(1 - (2*c^(1/3)*x)/d^(1/3))/Sqrt[3]] + 2*(b*c - a*d)*Log[d^(1/3) + c^(1/3)*x] - (b*c - a*d)*Log[d^(2/3) - c^(1/3)*d^(1/3)*x + c^(2/3)*x^2])/(6*c^(4/3)*d^(2/3))","A",1
276,1,41,49,0.072161,"\int \frac{a+b \sqrt{x}}{c+d \sqrt{x}} \, dx","Integrate[(a + b*Sqrt[x])/(c + d*Sqrt[x]),x]","\frac{2 (a d-b c) \left(d \sqrt{x}-c \log \left(c+d \sqrt{x}\right)\right)}{d^3}+\frac{b x}{d}","\frac{2 c (b c-a d) \log \left(c+d \sqrt{x}\right)}{d^3}-\frac{2 \sqrt{x} (b c-a d)}{d^2}+\frac{b x}{d}",1,"(b*x)/d + (2*(-(b*c) + a*d)*(d*Sqrt[x] - c*Log[c + d*Sqrt[x]]))/d^3","A",1
277,1,26,26,0.0133312,"\int \frac{-1+\sqrt[3]{x}}{1+\sqrt[3]{x}} \, dx","Integrate[(-1 + x^(1/3))/(1 + x^(1/3)),x]","-3 x^{2/3}+x+6 \sqrt[3]{x}-6 \log \left(\sqrt[3]{x}+1\right)","-3 x^{2/3}+x+6 \sqrt[3]{x}-6 \log \left(\sqrt[3]{x}+1\right)",1,"6*x^(1/3) - 3*x^(2/3) + x - 6*Log[1 + x^(1/3)]","A",1
278,1,17,17,0.0056496,"\int \frac{1+x^{2/3}}{-1+x^{2/3}} \, dx","Integrate[(1 + x^(2/3))/(-1 + x^(2/3)),x]","x+6 \sqrt[3]{x}-6 \tanh ^{-1}\left(\sqrt[3]{x}\right)","x+6 \sqrt[3]{x}-6 \tanh ^{-1}\left(\sqrt[3]{x}\right)",1,"6*x^(1/3) + x - 6*ArcTanh[x^(1/3)]","A",1
279,1,22,104,0.0066493,"\int \frac{-16+x^{3/4}}{16+x^{3/4}} \, dx","Integrate[(-16 + x^(3/4))/(16 + x^(3/4)),x]","x-2 x \, _2F_1\left(1,\frac{4}{3};\frac{7}{3};-\frac{x^{3/4}}{16}\right)","x-128 \sqrt[4]{x}+\frac{256}{3} \sqrt[3]{2} \log \left(\sqrt[4]{x}+2 \sqrt[3]{2}\right)-\frac{128}{3} \sqrt[3]{2} \log \left(\sqrt{x}-2 \sqrt[3]{2} \sqrt[4]{x}+4\ 2^{2/3}\right)-\frac{256 \sqrt[3]{2} \tan ^{-1}\left(\frac{\sqrt[3]{2}-\sqrt[4]{x}}{\sqrt[3]{2} \sqrt{3}}\right)}{\sqrt{3}}",1,"x - 2*x*Hypergeometric2F1[1, 4/3, 7/3, -1/16*x^(3/4)]","C",1
280,1,30,30,0.0312977,"\int \frac{1+\frac{1}{\sqrt[3]{x}}}{-1+\frac{1}{\sqrt[3]{x}}} \, dx","Integrate[(1 + x^(-1/3))/(-1 + x^(-1/3)),x]","-3 x^{2/3}-x-6 \sqrt[3]{x}-6 \log \left(1-\sqrt[3]{x}\right)","-3 x^{2/3}-x-6 \sqrt[3]{x}-6 \log \left(1-\sqrt[3]{x}\right)",1,"-6*x^(1/3) - 3*x^(2/3) - x - 6*Log[1 - x^(1/3)]","A",1
281,1,79,79,0.0397119,"\int \left(a-b x^n\right)^{3/2} \left(a+b x^n\right)^{3/2} \, dx","Integrate[(a - b*x^n)^(3/2)*(a + b*x^n)^(3/2),x]","\frac{a^2 x \sqrt{a-b x^n} \sqrt{a+b x^n} \, _2F_1\left(-\frac{3}{2},\frac{1}{2 n};1+\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right)}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}}","\frac{a^2 x \sqrt{a-b x^n} \sqrt{a+b x^n} \, _2F_1\left(-\frac{3}{2},\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);\frac{b^2 x^{2 n}}{a^2}\right)}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}}",1,"(a^2*x*Sqrt[a - b*x^n]*Sqrt[a + b*x^n]*Hypergeometric2F1[-3/2, 1/(2*n), 1 + 1/(2*n), (b^2*x^(2*n))/a^2])/Sqrt[1 - (b^2*x^(2*n))/a^2]","A",1
282,1,76,76,0.0179301,"\int \sqrt{a-b x^n} \sqrt{a+b x^n} \, dx","Integrate[Sqrt[a - b*x^n]*Sqrt[a + b*x^n],x]","\frac{x \sqrt{a-b x^n} \sqrt{a+b x^n} \, _2F_1\left(-\frac{1}{2},\frac{1}{2 n};1+\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right)}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}}","\frac{x \sqrt{a-b x^n} \sqrt{a+b x^n} \, _2F_1\left(-\frac{1}{2},\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);\frac{b^2 x^{2 n}}{a^2}\right)}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}}",1,"(x*Sqrt[a - b*x^n]*Sqrt[a + b*x^n]*Hypergeometric2F1[-1/2, 1/(2*n), 1 + 1/(2*n), (b^2*x^(2*n))/a^2])/Sqrt[1 - (b^2*x^(2*n))/a^2]","A",1
283,1,72,72,0.0426018,"\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(a-b x^n\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(a-b x^n\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
284,1,110,132,0.2839695,"\int \left(a+b x^n\right) \left(c+d x^n\right)^4 \, dx","Integrate[(a + b*x^n)*(c + d*x^n)^4,x]","\frac{b x \left(c+d x^n\right)^5-x \left(c^4+\frac{4 c^3 d x^n}{n+1}+\frac{6 c^2 d^2 x^{2 n}}{2 n+1}+\frac{4 c d^3 x^{3 n}}{3 n+1}+\frac{d^4 x^{4 n}}{4 n+1}\right) (b c-a d (5 n+1))}{5 d n+d}","\frac{c^3 x^{n+1} (4 a d+b c)}{n+1}+\frac{2 c^2 d x^{2 n+1} (3 a d+2 b c)}{2 n+1}+\frac{d^3 x^{4 n+1} (a d+4 b c)}{4 n+1}+\frac{2 c d^2 x^{3 n+1} (2 a d+3 b c)}{3 n+1}+a c^4 x+\frac{b d^4 x^{5 n+1}}{5 n+1}",1,"(b*x*(c + d*x^n)^5 - (b*c - a*d*(1 + 5*n))*x*(c^4 + (4*c^3*d*x^n)/(1 + n) + (6*c^2*d^2*x^(2*n))/(1 + 2*n) + (4*c*d^3*x^(3*n))/(1 + 3*n) + (d^4*x^(4*n))/(1 + 4*n)))/(d + 5*d*n)","A",1
285,1,90,99,0.1283978,"\int \left(a+b x^n\right) \left(c+d x^n\right)^3 \, dx","Integrate[(a + b*x^n)*(c + d*x^n)^3,x]","\frac{b x \left(c+d x^n\right)^4-x \left(c^3+\frac{3 c^2 d x^n}{n+1}+\frac{3 c d^2 x^{2 n}}{2 n+1}+\frac{d^3 x^{3 n}}{3 n+1}\right) (b c-a d (4 n+1))}{4 d n+d}","\frac{c^2 x^{n+1} (3 a d+b c)}{n+1}+\frac{d^2 x^{3 n+1} (a d+3 b c)}{3 n+1}+\frac{3 c d x^{2 n+1} (a d+b c)}{2 n+1}+a c^3 x+\frac{b d^3 x^{4 n+1}}{4 n+1}",1,"(b*x*(c + d*x^n)^4 - (b*c - a*d*(1 + 4*n))*x*(c^3 + (3*c^2*d*x^n)/(1 + n) + (3*c*d^2*x^(2*n))/(1 + 2*n) + (d^3*x^(3*n))/(1 + 3*n)))/(d + 4*d*n)","A",1
286,1,70,70,0.1016887,"\int \left(a+b x^n\right) \left(c+d x^n\right)^2 \, dx","Integrate[(a + b*x^n)*(c + d*x^n)^2,x]","\frac{b x \left(c+d x^n\right)^3-x \left(c^2+\frac{2 c d x^n}{n+1}+\frac{d^2 x^{2 n}}{2 n+1}\right) (b c-a d (3 n+1))}{3 d n+d}","\frac{c x^{n+1} (2 a d+b c)}{n+1}+\frac{d x^{2 n+1} (a d+2 b c)}{2 n+1}+a c^2 x+\frac{b d^2 x^{3 n+1}}{3 n+1}",1,"(b*x*(c + d*x^n)^3 - (b*c - a*d*(1 + 3*n))*x*(c^2 + (2*c*d*x^n)/(1 + n) + (d^2*x^(2*n))/(1 + 2*n)))/(d + 3*d*n)","A",1
287,1,37,40,0.0751221,"\int \left(a+b x^n\right) \left(c+d x^n\right) \, dx","Integrate[(a + b*x^n)*(c + d*x^n),x]","x \left(\frac{x^n (a d+b c)}{n+1}+a c+\frac{b d x^{2 n}}{2 n+1}\right)","\frac{x^{n+1} (a d+b c)}{n+1}+a c x+\frac{b d x^{2 n+1}}{2 n+1}",1,"x*(a*c + ((b*c + a*d)*x^n)/(1 + n) + (b*d*x^(2*n))/(1 + 2*n))","A",1
288,1,40,43,0.0136102,"\int \frac{a+b x^n}{c+d x^n} \, dx","Integrate[(a + b*x^n)/(c + d*x^n),x]","\frac{x \left((a d-b c) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)+b c\right)}{c d}","\frac{b x}{d}-\frac{x (b c-a d) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c d}",1,"(x*(b*c + (-(b*c) + a*d)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)]))/(c*d)","A",1
289,1,56,73,0.0463593,"\int \frac{a+b x^n}{\left(c+d x^n\right)^2} \, dx","Integrate[(a + b*x^n)/(c + d*x^n)^2,x]","\frac{x \left(\frac{b}{c+d x^n}-\frac{(a d (n-1)+b c) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c^2}\right)}{d-d n}","\frac{x (b c-a d (1-n)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c^2 d n}-\frac{x (b c-a d)}{c d n \left(c+d x^n\right)}",1,"(x*(b/(c + d*x^n) - ((b*c + a*d*(-1 + n))*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/c^2))/(d - d*n)","A",1
290,1,58,78,0.0556647,"\int \frac{a+b x^n}{\left(c+d x^n\right)^3} \, dx","Integrate[(a + b*x^n)/(c + d*x^n)^3,x]","\frac{x \left(\frac{b}{\left(c+d x^n\right)^2}-\frac{(a d (2 n-1)+b c) \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c^3}\right)}{d-2 d n}","\frac{x (b c-a d (1-2 n)) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{2 c^3 d n}-\frac{x (b c-a d)}{2 c d n \left(c+d x^n\right)^2}",1,"(x*(b/(c + d*x^n)^2 - ((b*c + a*d*(-1 + 2*n))*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/c^3))/(d - 2*d*n)","A",1
291,1,58,78,0.065299,"\int \frac{a+b x^n}{\left(c+d x^n\right)^4} \, dx","Integrate[(a + b*x^n)/(c + d*x^n)^4,x]","\frac{x \left(\frac{b}{\left(c+d x^n\right)^3}-\frac{(a d (3 n-1)+b c) \, _2F_1\left(4,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c^4}\right)}{d-3 d n}","\frac{x (b c-a d (1-3 n)) \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{3 c^4 d n}-\frac{x (b c-a d)}{3 c d n \left(c+d x^n\right)^3}",1,"(x*(b/(c + d*x^n)^3 - ((b*c + a*d*(-1 + 3*n))*Hypergeometric2F1[4, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/c^4))/(d - 3*d*n)","A",1
292,1,149,158,0.2107104,"\int \left(a+b x^n\right)^2 \left(d+e x^n\right)^3 \, dx","Integrate[(a + b*x^n)^2*(d + e*x^n)^3,x]","x \left(\frac{d x^{2 n} \left(3 a^2 e^2+6 a b d e+b^2 d^2\right)}{2 n+1}+\frac{e x^{3 n} \left(a^2 e^2+6 a b d e+3 b^2 d^2\right)}{3 n+1}+a^2 d^3+\frac{a d^2 x^n (3 a e+2 b d)}{n+1}+\frac{b e^2 x^{4 n} (2 a e+3 b d)}{4 n+1}+\frac{b^2 e^3 x^{5 n}}{5 n+1}\right)","\frac{d x^{2 n+1} \left(3 a^2 e^2+6 a b d e+b^2 d^2\right)}{2 n+1}+\frac{e x^{3 n+1} \left(a^2 e^2+6 a b d e+3 b^2 d^2\right)}{3 n+1}+a^2 d^3 x+\frac{a d^2 x^{n+1} (3 a e+2 b d)}{n+1}+\frac{b e^2 x^{4 n+1} (2 a e+3 b d)}{4 n+1}+\frac{b^2 e^3 x^{5 n+1}}{5 n+1}",1,"x*(a^2*d^3 + (a*d^2*(2*b*d + 3*a*e)*x^n)/(1 + n) + (d*(b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2)*x^(2*n))/(1 + 2*n) + (e*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2)*x^(3*n))/(1 + 3*n) + (b*e^2*(3*b*d + 2*a*e)*x^(4*n))/(1 + 4*n) + (b^2*e^3*x^(5*n))/(1 + 5*n))","A",1
293,1,105,112,0.1965346,"\int \left(a+b x^n\right)^2 \left(d+e x^n\right)^2 \, dx","Integrate[(a + b*x^n)^2*(d + e*x^n)^2,x]","x \left(\frac{x^{2 n} \left(a^2 e^2+4 a b d e+b^2 d^2\right)}{2 n+1}+a^2 d^2+\frac{2 b e x^{3 n} (a e+b d)}{3 n+1}+\frac{2 a d x^n (a e+b d)}{n+1}+\frac{b^2 e^2 x^{4 n}}{4 n+1}\right)","\frac{x^{2 n+1} \left(a^2 e^2+4 a b d e+b^2 d^2\right)}{2 n+1}+a^2 d^2 x+\frac{2 a d x^{n+1} (a e+b d)}{n+1}+\frac{2 b e x^{3 n+1} (a e+b d)}{3 n+1}+\frac{b^2 e^2 x^{4 n+1}}{4 n+1}",1,"x*(a^2*d^2 + (2*a*d*(b*d + a*e)*x^n)/(1 + n) + ((b^2*d^2 + 4*a*b*d*e + a^2*e^2)*x^(2*n))/(1 + 2*n) + (2*b*e*(b*d + a*e)*x^(3*n))/(1 + 3*n) + (b^2*e^2*x^(4*n))/(1 + 4*n))","A",1
294,1,70,70,0.108374,"\int \left(a+b x^n\right)^2 \left(c+d x^n\right) \, dx","Integrate[(a + b*x^n)^2*(c + d*x^n),x]","\frac{d x \left(a+b x^n\right)^3-x \left(a^2+\frac{2 a b x^n}{n+1}+\frac{b^2 x^{2 n}}{2 n+1}\right) (a d-b (3 c n+c))}{3 b n+b}","a^2 c x+\frac{a x^{n+1} (a d+2 b c)}{n+1}+\frac{b x^{2 n+1} (2 a d+b c)}{2 n+1}+\frac{b^2 d x^{3 n+1}}{3 n+1}",1,"(d*x*(a + b*x^n)^3 - (a*d - b*(c + 3*c*n))*x*(a^2 + (2*a*b*x^n)/(1 + n) + (b^2*x^(2*n))/(1 + 2*n)))/(b + 3*b*n)","A",1
295,1,82,84,0.0492562,"\int \frac{\left(a+b x^n\right)^2}{c+d x^n} \, dx","Integrate[(a + b*x^n)^2/(c + d*x^n),x]","\frac{a^2 x}{c}+\frac{x (a d-b c)^2 \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c d^2}-\frac{x (b c-a d)^2}{c d^2}+\frac{b^2 x^{n+1}}{d (n+1)}","\frac{x (b c-a d)^2 \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c d^2}-\frac{b x (b c (n+1)-a d (2 n+1))}{d^2 (n+1)}+\frac{b x \left(a+b x^n\right)}{d (n+1)}",1,"(a^2*x)/c - ((b*c - a*d)^2*x)/(c*d^2) + (b^2*x^(1 + n))/(d*(1 + n)) + ((-(b*c) + a*d)^2*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/(c*d^2)","A",1
296,1,95,115,0.1554751,"\int \frac{\left(a+b x^n\right)^2}{\left(c+d x^n\right)^2} \, dx","Integrate[(a + b*x^n)^2/(c + d*x^n)^2,x]","\frac{x \left(\frac{c \left(a^2 d^2-2 a b c d+b^2 c \left(c n+c+d n x^n\right)\right)}{c+d x^n}-(b c-a d) (a d (n-1)+b c (n+1)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)\right)}{c^2 d^2 n}","\frac{x (b c-a d) (a d (1-n)-b c (n+1)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c^2 d^2 n}-\frac{b x (a d-b c (n+1))}{c d^2 n}-\frac{x (b c-a d) \left(a+b x^n\right)}{c d n \left(c+d x^n\right)}",1,"(x*((c*(-2*a*b*c*d + a^2*d^2 + b^2*c*(c + c*n + d*n*x^n)))/(c + d*x^n) - (b*c - a*d)*(a*d*(-1 + n) + b*c*(1 + n))*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)]))/(c^2*d^2*n)","A",1
297,1,133,160,0.1917861,"\int \frac{\left(a+b x^n\right)^2}{\left(c+d x^n\right)^3} \, dx","Integrate[(a + b*x^n)^2/(c + d*x^n)^3,x]","\frac{x \left(\left(a^2 d^2 \left(2 n^2-3 n+1\right)+2 a b c d (n-1)+b^2 c^2 (n+1)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)+\frac{c^2 n (b c-a d)^2}{\left(c+d x^n\right)^2}-\frac{c (b c-a d) (a d (2 n-1)+b (2 c n+c))}{c+d x^n}\right)}{2 c^3 d^2 n^2}","-\frac{x \left(-a^2 d^2 \left(2 n^2-3 n+1\right)+2 a b c d (1-n)-b^2 c^2 (n+1)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{2 c^3 d^2 n^2}+\frac{x (b c-a d) (a d (1-2 n)-b c (n+1))}{2 c^2 d^2 n^2 \left(c+d x^n\right)}-\frac{x (b c-a d) \left(a+b x^n\right)}{2 c d n \left(c+d x^n\right)^2}",1,"(x*((c^2*(b*c - a*d)^2*n)/(c + d*x^n)^2 - (c*(b*c - a*d)*(a*d*(-1 + 2*n) + b*(c + 2*c*n)))/(c + d*x^n) + (2*a*b*c*d*(-1 + n) + b^2*c^2*(1 + n) + a^2*d^2*(1 - 3*n + 2*n^2))*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)]))/(2*c^3*d^2*n^2)","A",1
298,1,133,310,4.1249518,"\int \frac{\left(c+d x^n\right)^4}{a+b x^n} \, dx","Integrate[(c + d*x^n)^4/(a + b*x^n),x]","\frac{x \left(c^4 \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+4 c^3 d x^n \Phi \left(-\frac{b x^n}{a},1,1+\frac{1}{n}\right)+6 c^2 d^2 x^{2 n} \Phi \left(-\frac{b x^n}{a},1,2+\frac{1}{n}\right)+4 c d^3 x^{3 n} \Phi \left(-\frac{b x^n}{a},1,3+\frac{1}{n}\right)+d^4 x^{4 n} \Phi \left(-\frac{b x^n}{a},1,4+\frac{1}{n}\right)\right)}{a n}","-\frac{d x \left(c+d x^n\right) \left(-a^2 d^2 \left(6 n^2+5 n+1\right)+2 a b c d (3 n+1)^2-b^2 c^2 \left(18 n^2+7 n+1\right)\right)}{b^3 (n+1) (2 n+1) (3 n+1)}-\frac{d x \left(a^3 d^3 \left(6 n^3+11 n^2+6 n+1\right)-a^2 b c d^2 \left(24 n^3+38 n^2+19 n+3\right)+a b^2 c^2 d \left(36 n^3+45 n^2+20 n+3\right)-b^3 c^3 \left(24 n^3+18 n^2+7 n+1\right)\right)}{b^4 (n+1) (2 n+1) (3 n+1)}+\frac{x (b c-a d)^4 \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a b^4}-\frac{d x \left(c+d x^n\right)^2 (a d (3 n+1)-b (6 c n+c))}{b^2 \left(6 n^2+5 n+1\right)}+\frac{d x \left(c+d x^n\right)^3}{b (3 n+1)}",1,"(x*(4*c^3*d*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, 1 + n^(-1)] + 6*c^2*d^2*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 2 + n^(-1)] + 4*c*d^3*x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + d^4*x^(4*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 4 + n^(-1)] + c^4*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)]))/(a*n)","C",1
299,1,104,173,1.5883548,"\int \frac{\left(c+d x^n\right)^3}{a+b x^n} \, dx","Integrate[(c + d*x^n)^3/(a + b*x^n),x]","\frac{x \left(c^3 \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+3 c^2 d x^n \Phi \left(-\frac{b x^n}{a},1,1+\frac{1}{n}\right)+3 c d^2 x^{2 n} \Phi \left(-\frac{b x^n}{a},1,2+\frac{1}{n}\right)+d^3 x^{3 n} \Phi \left(-\frac{b x^n}{a},1,3+\frac{1}{n}\right)\right)}{a n}","\frac{d x \left(a^2 d^2 \left(2 n^2+3 n+1\right)-a b c d \left(6 n^2+7 n+2\right)+b^2 c^2 \left(6 n^2+4 n+1\right)\right)}{b^3 (n+1) (2 n+1)}+\frac{x (b c-a d)^3 \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a b^3}-\frac{d x \left(c+d x^n\right) (a d (2 n+1)-b (4 c n+c))}{b^2 (n+1) (2 n+1)}+\frac{d x \left(c+d x^n\right)^2}{b (2 n+1)}",1,"(x*(3*c^2*d*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, 1 + n^(-1)] + 3*c*d^2*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 2 + n^(-1)] + d^3*x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + c^3*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)]))/(a*n)","C",1
300,1,75,84,0.3607866,"\int \frac{\left(c+d x^n\right)^2}{a+b x^n} \, dx","Integrate[(c + d*x^n)^2/(a + b*x^n),x]","\frac{x \left(c^2 \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+2 c d x^n \Phi \left(-\frac{b x^n}{a},1,1+\frac{1}{n}\right)+d^2 x^{2 n} \Phi \left(-\frac{b x^n}{a},1,2+\frac{1}{n}\right)\right)}{a n}","\frac{x (b c-a d)^2 \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a b^2}-\frac{d x (a d (n+1)-b (2 c n+c))}{b^2 (n+1)}+\frac{d x \left(c+d x^n\right)}{b (n+1)}",1,"(x*(2*c*d*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, 1 + n^(-1)] + d^2*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 2 + n^(-1)] + c^2*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)]))/(a*n)","C",1
301,1,40,42,0.0105655,"\int \frac{c+d x^n}{a+b x^n} \, dx","Integrate[(c + d*x^n)/(a + b*x^n),x]","\frac{x \left((b c-a d) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)+a d\right)}{a b}","\frac{x (b c-a d) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a b}+\frac{d x}{b}",1,"(x*(a*d + (b*c - a*d)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)]))/(a*b)","A",1
302,1,64,72,0.0630602,"\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
303,1,121,123,0.2166898,"\int \frac{1}{\left(a+b x^n\right) \left(c+d x^n\right)^2} \, dx","Integrate[1/((a + b*x^n)*(c + d*x^n)^2),x]","\frac{x \left(b^2 c^2 n \left(c+d x^n\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)+a d \left(\left(c+d x^n\right) (a d (n-1)+b (c-2 c n)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)+c (a d-b c)\right)\right)}{a c^2 n (b c-a d)^2 \left(c+d x^n\right)}","\frac{b^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a (b c-a d)^2}+\frac{d x (b c (1-2 n)-a d (1-n)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c^2 n (b c-a d)^2}-\frac{d x}{c n (b c-a d) \left(c+d x^n\right)}",1,"(x*(b^2*c^2*n*(c + d*x^n)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)] + a*d*(c*(-(b*c) + a*d) + (a*d*(-1 + n) + b*(c - 2*c*n))*(c + d*x^n)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])))/(a*c^2*(b*c - a*d)^2*n*(c + d*x^n))","A",1
304,1,210,210,0.2639,"\int \frac{1}{\left(a+b x^n\right) \left(c+d x^n\right)^3} \, dx","Integrate[1/((a + b*x^n)*(c + d*x^n)^3),x]","\frac{x \left(-a d \left(c+d x^n\right)^2 \left(a^2 d^2 \left(2 n^2-3 n+1\right)-2 a b c d \left(3 n^2-4 n+1\right)+b^2 c^2 \left(6 n^2-5 n+1\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)+2 b^3 c^3 n^2 \left(c+d x^n\right)^2 \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)-a c^2 d n (b c-a d)^2+a c d (b c-a d) \left(c+d x^n\right) (a d (2 n-1)+b (c-4 c n))\right)}{2 a c^3 n^2 (b c-a d)^3 \left(c+d x^n\right)^2}","-\frac{d x \left(a^2 d^2 \left(2 n^2-3 n+1\right)-2 a b c d \left(3 n^2-4 n+1\right)+b^2 c^2 \left(6 n^2-5 n+1\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{2 c^3 n^2 (b c-a d)^3}+\frac{b^3 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a (b c-a d)^3}-\frac{d x (a d (1-2 n)-b (c-4 c n))}{2 c^2 n^2 (b c-a d)^2 \left(c+d x^n\right)}-\frac{d x}{2 c n (b c-a d) \left(c+d x^n\right)^2}",1,"(x*(-(a*c^2*d*(b*c - a*d)^2*n) + a*c*d*(b*c - a*d)*(a*d*(-1 + 2*n) + b*(c - 4*c*n))*(c + d*x^n) + 2*b^3*c^3*n^2*(c + d*x^n)^2*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)] - a*d*(a^2*d^2*(1 - 3*n + 2*n^2) - 2*a*b*c*d*(1 - 4*n + 3*n^2) + b^2*c^2*(1 - 5*n + 6*n^2))*(c + d*x^n)^2*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)]))/(2*a*c^3*(b*c - a*d)^3*n^2*(c + d*x^n)^2)","A",1
305,1,217,341,5.2640713,"\int \frac{\left(c+d x^n\right)^4}{\left(a+b x^n\right)^2} \, dx","Integrate[(c + d*x^n)^4/(a + b*x^n)^2,x]","\frac{x \left(\frac{(b c-a d)^3 (a d (3 n+1)+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{(a d-b c)^3 (a d (3 n+1)+b c (n-1))}{a^2 n}+\frac{-a^4 d^4+4 a^3 b c d^3-6 a^2 b^2 c^2 d^2+4 a b^3 c^3 d+b^4 c^4 (n-1)}{a^2 n}+\frac{2 b d^3 x^n (2 b c-a d)}{n+1}+\frac{(b c-a d)^4}{a n \left(a+b x^n\right)}+\frac{b^2 d^4 x^{2 n}}{2 n+1}\right)}{b^4}","-\frac{x (b c-a d)^3 (b c (1-n)-a d (3 n+1)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2 b^4 n}-\frac{d x \left(c+d x^n\right) \left(a^2 d^2 \left(6 n^2+5 n+1\right)-2 a b c d \left(5 n^2+4 n+1\right)+b^2 c^2 \left(2 n^2+3 n+1\right)\right)}{a b^3 n (n+1) (2 n+1)}-\frac{d x \left(-a^3 d^3 \left(6 n^3+11 n^2+6 n+1\right)+a^2 b c d^2 \left(16 n^3+26 n^2+15 n+3\right)-a b^2 c^2 d \left(12 n^3+17 n^2+12 n+3\right)+b^3 c^3 \left(2 n^2+3 n+1\right)\right)}{a b^4 n (n+1) (2 n+1)}+\frac{d x \left(c+d x^n\right)^2 (a d (3 n+1)-b (2 c n+c))}{a b^2 n (2 n+1)}+\frac{x (b c-a d) \left(c+d x^n\right)^3}{a b n \left(a+b x^n\right)}",1,"(x*((4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4 + b^4*c^4*(-1 + n))/(a^2*n) + ((-(b*c) + a*d)^3*(b*c*(-1 + n) + a*d*(1 + 3*n)))/(a^2*n) + (2*b*d^3*(2*b*c - a*d)*x^n)/(1 + n) + (b^2*d^4*x^(2*n))/(1 + 2*n) + (b*c - a*d)^4/(a*n*(a + b*x^n)) + ((b*c - a*d)^3*(b*c*(-1 + n) + a*d*(1 + 3*n))*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a^2*n)))/b^4","A",1
306,1,167,200,5.3789923,"\int \frac{\left(c+d x^n\right)^3}{\left(a+b x^n\right)^2} \, dx","Integrate[(c + d*x^n)^3/(a + b*x^n)^2,x]","\frac{x \left(\frac{a \left(-a^3 d^3 \left(2 n^2+3 n+1\right)+a^2 b d^2 \left(3 c (n+1)^2-d n (2 n+1) x^n\right)+a b^2 d \left(-3 c^2 (n+1)+3 c d n (n+1) x^n+d^2 n x^{2 n}\right)+b^3 c^3 (n+1)\right)}{(n+1) \left(a+b x^n\right)}+(b c-a d)^2 (a d (2 n+1)+b c (n-1)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)\right)}{a^2 b^3 n}","-\frac{x (b c-a d)^2 (b c (1-n)-a d (2 n+1)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2 b^3 n}-\frac{d x \left(a^2 d^2 \left(2 n^2+3 n+1\right)-a b c d \left(3 n^2+4 n+2\right)+b^2 c^2 (n+1)\right)}{a b^3 n (n+1)}-\frac{d x \left(c+d x^n\right) (b c (n+1)-a d (2 n+1))}{a b^2 n (n+1)}+\frac{x (b c-a d) \left(c+d x^n\right)^2}{a b n \left(a+b x^n\right)}",1,"(x*((a*(b^3*c^3*(1 + n) - a^3*d^3*(1 + 3*n + 2*n^2) + a^2*b*d^2*(3*c*(1 + n)^2 - d*n*(1 + 2*n)*x^n) + a*b^2*d*(-3*c^2*(1 + n) + 3*c*d*n*(1 + n)*x^n + d^2*n*x^(2*n))))/((1 + n)*(a + b*x^n)) + (b*c - a*d)^2*(b*c*(-1 + n) + a*d*(1 + 2*n))*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)]))/(a^2*b^3*n)","A",1
307,1,666,115,2.5801626,"\int \frac{\left(c+d x^n\right)^2}{\left(a+b x^n\right)^2} \, dx","Integrate[(c + d*x^n)^2/(a + b*x^n)^2,x]","\frac{x \left(-2 b c^2 n^6 x^n \, _4F_3\left(2,2,2,1+\frac{1}{n};1,1,4+\frac{1}{n};-\frac{b x^n}{a}\right)-4 b c d n^6 x^{2 n} \, _4F_3\left(2,2,2,1+\frac{1}{n};1,1,4+\frac{1}{n};-\frac{b x^n}{a}\right)-2 b d^2 n^6 x^{3 n} \, _4F_3\left(2,2,2,1+\frac{1}{n};1,1,4+\frac{1}{n};-\frac{b x^n}{a}\right)-2 a \left(6 n^3+11 n^2+6 n+1\right) \left(c^2 (n+1)^3+2 c d \left(n^3+4 n^2+3 n+1\right) x^n+d^2 (n+1)^3 x^{2 n}\right) \Phi \left(-\frac{b x^n}{a},1,1+\frac{1}{n}\right)+a \left(6 n^3+11 n^2+6 n+1\right) \left(c^2 (2 n+1)^3+2 c d (2 n+1)^3 x^n+d^2 \left(6 n^3+10 n^2+6 n+1\right) x^{2 n}\right) \Phi \left(-\frac{b x^n}{a},1,2+\frac{1}{n}\right)+12 a c^2 n^6 \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+10 a c^2 n^5 \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)-10 a c^2 n^4 \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)-4 a c^2 n^3 \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+9 a c^2 n^2 \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+a c^2 \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+6 a c^2 n \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+12 a c d n^3 x^n \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+22 a c d n^2 x^n \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+2 a c d x^n \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+12 a c d n x^n \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+6 a d^2 n^3 x^{2 n} \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+11 a d^2 n^2 x^{2 n} \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+a d^2 x^{2 n} \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)+6 a d^2 n x^{2 n} \Phi \left(-\frac{b x^n}{a},1,\frac{1}{n}\right)\right)}{2 a^3 n^4 \left(6 n^3+11 n^2+6 n+1\right)}","-\frac{x (b c-a d) (b c (1-n)-a d (n+1)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2 b^2 n}-\frac{d x (b c-a d (n+1))}{a b^2 n}+\frac{x (b c-a d) \left(c+d x^n\right)}{a b n \left(a+b x^n\right)}",1,"(x*(-2*a*(1 + 6*n + 11*n^2 + 6*n^3)*(c^2*(1 + n)^3 + 2*c*d*(1 + 3*n + 4*n^2 + n^3)*x^n + d^2*(1 + n)^3*x^(2*n))*HurwitzLerchPhi[-((b*x^n)/a), 1, 1 + n^(-1)] + a*(1 + 6*n + 11*n^2 + 6*n^3)*(c^2*(1 + 2*n)^3 + 2*c*d*(1 + 2*n)^3*x^n + d^2*(1 + 6*n + 10*n^2 + 6*n^3)*x^(2*n))*HurwitzLerchPhi[-((b*x^n)/a), 1, 2 + n^(-1)] + a*c^2*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 6*a*c^2*n*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 9*a*c^2*n^2*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 4*a*c^2*n^3*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 10*a*c^2*n^4*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 10*a*c^2*n^5*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 12*a*c^2*n^6*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 2*a*c*d*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 12*a*c*d*n*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 22*a*c*d*n^2*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 12*a*c*d*n^3*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + a*d^2*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 6*a*d^2*n*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 11*a*d^2*n^2*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 6*a*d^2*n^3*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 2*b*c^2*n^6*x^n*HypergeometricPFQ[{2, 2, 2, 1 + n^(-1)}, {1, 1, 4 + n^(-1)}, -((b*x^n)/a)] - 4*b*c*d*n^6*x^(2*n)*HypergeometricPFQ[{2, 2, 2, 1 + n^(-1)}, {1, 1, 4 + n^(-1)}, -((b*x^n)/a)] - 2*b*d^2*n^6*x^(3*n)*HypergeometricPFQ[{2, 2, 2, 1 + n^(-1)}, {1, 1, 4 + n^(-1)}, -((b*x^n)/a)]))/(2*a^3*n^4*(1 + 6*n + 11*n^2 + 6*n^3))","C",0
308,1,56,72,0.043973,"\int \frac{c+d x^n}{\left(a+b x^n\right)^2} \, dx","Integrate[(c + d*x^n)/(a + b*x^n)^2,x]","\frac{x \left(\frac{d}{a+b x^n}-\frac{(a d+b c (n-1)) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2}\right)}{b-b n}","\frac{x (a d-b c (1-n)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2 b n}+\frac{x (b c-a d)}{a b n \left(a+b x^n\right)}",1,"(x*(d/(a + b*x^n) - ((a*d + b*c*(-1 + n))*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^2))/(b - b*n)","A",1
309,1,108,122,0.2008692,"\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
310,1,147,193,0.2846345,"\int \frac{1}{\left(a+b x^n\right)^2 \left(c+d x^n\right)^2} \, dx","Integrate[1/((a + b*x^n)^2*(c + d*x^n)^2),x]","\frac{x \left(\frac{b^2 (a d (1-3 n)+b c (n-1)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2}+\frac{b^2 (b c-a d)}{a \left(a+b x^n\right)}+\frac{d^2 (b c (3 n-1)-a d (n-1)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c^2}+\frac{d^2 (b c-a d)}{c \left(c+d x^n\right)}\right)}{n (b c-a d)^3}","\frac{b^2 x (a d (1-3 n)-b (c-c 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)^3}-\frac{d^2 x (b c (1-3 n)-a d (1-n)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c^2 n (b c-a d)^3}+\frac{b x}{a n (b c-a d) \left(a+b x^n\right) \left(c+d x^n\right)}+\frac{d x (a d+b c)}{a c n (b c-a d)^2 \left(c+d x^n\right)}",1,"(x*((b^2*(b*c - a*d))/(a*(a + b*x^n)) + (d^2*(b*c - a*d))/(c*(c + d*x^n)) + (b^2*(a*d*(1 - 3*n) + b*c*(-1 + n))*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^2 + (d^2*(-(a*d*(-1 + n)) + b*c*(-1 + 3*n))*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/c^2))/((b*c - a*d)^3*n)","A",1
311,1,233,299,0.4380295,"\int \frac{1}{\left(a+b x^n\right)^2 \left(c+d x^n\right)^3} \, dx","Integrate[1/((a + b*x^n)^2*(c + d*x^n)^3),x]","\frac{x \left(\frac{2 b^3 n (a d (1-4 n)+b c (n-1)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2}+\frac{d^2 \left(a^2 d^2 \left(2 n^2-3 n+1\right)-2 a b c d \left(4 n^2-5 n+1\right)+b^2 c^2 \left(12 n^2-7 n+1\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c^3}+\frac{2 b^3 n (b c-a d)}{a \left(a+b x^n\right)}+\frac{d^2 (a d-b c) (a d (2 n-1)+b (c-6 c n))}{c^2 \left(c+d x^n\right)}+\frac{d^2 n (b c-a d)^2}{c \left(c+d x^n\right)^2}\right)}{2 n^2 (b c-a d)^4}","\frac{b^3 x (a d (1-4 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)^4}-\frac{d x \left(-a^2 d^2 (1-2 n)+a b c d (1-6 n)-2 b^2 c^2 n\right)}{2 a c^2 n^2 (b c-a d)^3 \left(c+d x^n\right)}+\frac{d^2 x \left(a^2 d^2 \left(2 n^2-3 n+1\right)-2 a b c d \left(4 n^2-5 n+1\right)+b^2 c^2 \left(12 n^2-7 n+1\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{2 c^3 n^2 (b c-a d)^4}+\frac{b x}{a n (b c-a d) \left(a+b x^n\right) \left(c+d x^n\right)^2}+\frac{d x (a d+2 b c)}{2 a c n (b c-a d)^2 \left(c+d x^n\right)^2}",1,"(x*((2*b^3*(b*c - a*d)*n)/(a*(a + b*x^n)) + (d^2*(b*c - a*d)^2*n)/(c*(c + d*x^n)^2) + (d^2*(-(b*c) + a*d)*(a*d*(-1 + 2*n) + b*(c - 6*c*n)))/(c^2*(c + d*x^n)) + (2*b^3*(a*d*(1 - 4*n) + b*c*(-1 + n))*n*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^2 + (d^2*(a^2*d^2*(1 - 3*n + 2*n^2) - 2*a*b*c*d*(1 - 5*n + 4*n^2) + b^2*c^2*(1 - 7*n + 12*n^2))*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/c^3))/(2*(b*c - a*d)^4*n^2)","A",1
312,1,190,81,0.5656511,"\int \left(a+b x^n\right)^p \left(c+d x^n\right)^q \, dx","Integrate[(a + b*x^n)^p*(c + d*x^n)^q,x]","\frac{a c (n+1) x \left(a+b x^n\right)^p \left(c+d x^n\right)^q F_1\left(\frac{1}{n};-p,-q;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{b c n p x^n F_1\left(1+\frac{1}{n};1-p,-q;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)+a d n q x^n F_1\left(1+\frac{1}{n};-p,1-q;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)+a c (n+1) F_1\left(\frac{1}{n};-p,-q;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}","x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(c+d x^n\right)^q \left(\frac{d x^n}{c}+1\right)^{-q} F_1\left(\frac{1}{n};-p,-q;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)",1,"(a*c*(1 + n)*x*(a + b*x^n)^p*(c + d*x^n)^q*AppellF1[n^(-1), -p, -q, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)])/(b*c*n*p*x^n*AppellF1[1 + n^(-1), 1 - p, -q, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + a*d*n*q*x^n*AppellF1[1 + n^(-1), -p, 1 - q, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + a*c*(1 + n)*AppellF1[n^(-1), -p, -q, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)])","B",0
313,1,168,402,5.3179529,"\int \left(a+b x^n\right)^p \left(c+d x^n\right)^3 \, dx","Integrate[(a + b*x^n)^p*(c + d*x^n)^3,x]","x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(c^3 \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right)+\frac{3 c^2 d x^n \, _2F_1\left(1+\frac{1}{n},-p;2+\frac{1}{n};-\frac{b x^n}{a}\right)}{n+1}+\frac{3 c d^2 x^{2 n} \, _2F_1\left(2+\frac{1}{n},-p;3+\frac{1}{n};-\frac{b x^n}{a}\right)}{2 n+1}+\frac{d^3 x^{3 n} \, _2F_1\left(3+\frac{1}{n},-p;4+\frac{1}{n};-\frac{b x^n}{a}\right)}{3 n+1}\right)","\frac{d x \left(a+b x^n\right)^{p+1} \left(a^2 d^2 \left(2 n^2+3 n+1\right)-a b c d \left(n^2 (p+7)+n (2 p+9)+2\right)+b^2 c^2 \left(n^2 \left(p^2+6 p+11\right)+2 n (p+3)+1\right)\right)}{b^3 (n p+n+1) (n (p+2)+1) (n (p+3)+1)}-\frac{x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(a^3 d^3 \left(2 n^2+3 n+1\right)-3 a^2 b c d^2 (n+1) (n (p+3)+1)+3 a b^2 c^2 d \left(n^2 \left(p^2+5 p+6\right)+n (2 p+5)+1\right)-b^3 c^3 \left(n^3 \left(p^3+6 p^2+11 p+6\right)+n^2 \left(3 p^2+12 p+11\right)+3 n (p+2)+1\right)\right) \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right)}{b^3 (n p+n+1) (n (p+2)+1) (n (p+3)+1)}-\frac{d x \left(c+d x^n\right) \left(a+b x^n\right)^{p+1} (a d (2 n+1)-b c (n (p+5)+1))}{b^2 (n (p+2)+1) (n (p+3)+1)}+\frac{d x \left(c+d x^n\right)^2 \left(a+b x^n\right)^{p+1}}{b (n p+3 n+1)}",1,"(x*(a + b*x^n)^p*((3*c^2*d*x^n*Hypergeometric2F1[1 + n^(-1), -p, 2 + n^(-1), -((b*x^n)/a)])/(1 + n) + (3*c*d^2*x^(2*n)*Hypergeometric2F1[2 + n^(-1), -p, 3 + n^(-1), -((b*x^n)/a)])/(1 + 2*n) + (d^3*x^(3*n)*Hypergeometric2F1[3 + n^(-1), -p, 4 + n^(-1), -((b*x^n)/a)])/(1 + 3*n) + c^3*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*x^n)/a)]))/(1 + (b*x^n)/a)^p","A",1
314,1,140,202,5.2017145,"\int \left(a+b x^n\right)^p \left(c+d x^n\right)^2 \, dx","Integrate[(a + b*x^n)^p*(c + d*x^n)^2,x]","\frac{x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left((n+1) \left(c^2 (2 n+1) \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right)+d^2 x^{2 n} \, _2F_1\left(2+\frac{1}{n},-p;3+\frac{1}{n};-\frac{b x^n}{a}\right)\right)+2 c d (2 n+1) x^n \, _2F_1\left(1+\frac{1}{n},-p;2+\frac{1}{n};-\frac{b x^n}{a}\right)\right)}{(n+1) (2 n+1)}","-\frac{x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} (b c (n p+n+1) (a d-b c (n (p+2)+1))-a d (a d (n+1)-b c (n (p+3)+1))) \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right)}{b^2 (n p+n+1) (n (p+2)+1)}-\frac{d x \left(a+b x^n\right)^{p+1} (a d (n+1)-b c (n (p+3)+1))}{b^2 (n p+n+1) (n (p+2)+1)}+\frac{d x \left(c+d x^n\right) \left(a+b x^n\right)^{p+1}}{b (n p+2 n+1)}",1,"(x*(a + b*x^n)^p*(2*c*d*(1 + 2*n)*x^n*Hypergeometric2F1[1 + n^(-1), -p, 2 + n^(-1), -((b*x^n)/a)] + (1 + n)*(d^2*x^(2*n)*Hypergeometric2F1[2 + n^(-1), -p, 3 + n^(-1), -((b*x^n)/a)] + c^2*(1 + 2*n)*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*x^n)/a)])))/((1 + n)*(1 + 2*n)*(1 + (b*x^n)/a)^p)","A",1
315,1,94,98,0.0473253,"\int \left(a+b x^n\right)^p \left(c+d x^n\right) \, dx","Integrate[(a + b*x^n)^p*(c + d*x^n),x]","\frac{x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left((b c (n p+n+1)-a d) \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right)+d \left(a+b x^n\right) \left(\frac{b x^n}{a}+1\right)^p\right)}{b (n p+n+1)}","\frac{d x \left(a+b x^n\right)^{p+1}}{b (n p+n+1)}-\frac{x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} (a d-b c (n p+n+1)) \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right)}{b (n p+n+1)}",1,"(x*(a + b*x^n)^p*(d*(a + b*x^n)*(1 + (b*x^n)/a)^p + (-(a*d) + b*c*(1 + n + n*p))*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*x^n)/a)]))/(b*(1 + n + n*p)*(1 + (b*x^n)/a)^p)","A",1
316,1,46,46,0.0053448,"\int \left(a+b x^n\right)^p \, dx","Integrate[(a + b*x^n)^p,x]","x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right)","x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right)",1,"(x*(a + b*x^n)^p*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*x^n)/a)])/(1 + (b*x^n)/a)^p","A",1
317,1,180,59,0.3093168,"\int \frac{\left(a+b x^n\right)^p}{c+d x^n} \, dx","Integrate[(a + b*x^n)^p/(c + d*x^n),x]","\frac{a c (n+1) x \left(a+b x^n\right)^p F_1\left(\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{\left(c+d x^n\right) \left(b c n p x^n F_1\left(1+\frac{1}{n};1-p,1;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)-a d n x^n F_1\left(1+\frac{1}{n};-p,2;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)+a c (n+1) F_1\left(\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)\right)}","\frac{x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} F_1\left(\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{c}",1,"(a*c*(1 + n)*x*(a + b*x^n)^p*AppellF1[n^(-1), -p, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)])/((c + d*x^n)*(b*c*n*p*x^n*AppellF1[1 + n^(-1), 1 - p, 1, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] - a*d*n*x^n*AppellF1[1 + n^(-1), -p, 2, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + a*c*(1 + n)*AppellF1[n^(-1), -p, 1, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)]))","B",0
318,1,180,59,0.3417538,"\int \frac{\left(a+b x^n\right)^p}{\left(c+d x^n\right)^2} \, dx","Integrate[(a + b*x^n)^p/(c + d*x^n)^2,x]","\frac{a c (n+1) x \left(a+b x^n\right)^p F_1\left(\frac{1}{n};-p,2;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{\left(c+d x^n\right)^2 \left(b c n p x^n F_1\left(1+\frac{1}{n};1-p,2;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)-2 a d n x^n F_1\left(1+\frac{1}{n};-p,3;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)+a c (n+1) F_1\left(\frac{1}{n};-p,2;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)\right)}","\frac{x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} F_1\left(\frac{1}{n};-p,2;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{c^2}",1,"(a*c*(1 + n)*x*(a + b*x^n)^p*AppellF1[n^(-1), -p, 2, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)])/((c + d*x^n)^2*(b*c*n*p*x^n*AppellF1[1 + n^(-1), 1 - p, 2, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] - 2*a*d*n*x^n*AppellF1[1 + n^(-1), -p, 3, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + a*c*(1 + n)*AppellF1[n^(-1), -p, 2, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)]))","B",0
319,1,180,59,0.6514481,"\int \frac{\left(a+b x^n\right)^p}{\left(c+d x^n\right)^3} \, dx","Integrate[(a + b*x^n)^p/(c + d*x^n)^3,x]","\frac{a c (n+1) x \left(a+b x^n\right)^p F_1\left(\frac{1}{n};-p,3;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{\left(c+d x^n\right)^3 \left(b c n p x^n F_1\left(1+\frac{1}{n};1-p,3;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)-3 a d n x^n F_1\left(1+\frac{1}{n};-p,4;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)+a c (n+1) F_1\left(\frac{1}{n};-p,3;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)\right)}","\frac{x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} F_1\left(\frac{1}{n};-p,3;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{c^3}",1,"(a*c*(1 + n)*x*(a + b*x^n)^p*AppellF1[n^(-1), -p, 3, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)])/((c + d*x^n)^3*(b*c*n*p*x^n*AppellF1[1 + n^(-1), 1 - p, 3, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] - 3*a*d*n*x^n*AppellF1[1 + n^(-1), -p, 4, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + a*c*(1 + n)*AppellF1[n^(-1), -p, 3, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)]))","B",0
320,1,94,93,0.0884122,"\int \left(a+b x^n\right)^p \left(c+d x^n\right)^{-1-\frac{1}{n}-p} \, dx","Integrate[(a + b*x^n)^p*(c + d*x^n)^(-1 - n^(-1) - p),x]","\frac{x \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(c+d x^n\right)^{-\frac{n p+1}{n}} \left(\frac{d x^n}{c}+1\right)^p \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};\frac{(a d-b c) x^n}{a \left(d x^n+c\right)}\right)}{c}","\frac{x \left(a+b x^n\right)^p \left(c+d x^n\right)^{-\frac{1}{n}-p} \left(\frac{c \left(a+b x^n\right)}{a \left(c+d x^n\right)}\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left(d x^n+c\right)}\right)}{c}",1,"(x*(a + b*x^n)^p*(1 + (d*x^n)/c)^p*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), ((-(b*c) + a*d)*x^n)/(a*(c + d*x^n))])/(c*(1 + (b*x^n)/a)^p*(c + d*x^n)^((1 + n*p)/n))","A",0
321,1,218,178,0.14216,"\int \left(a+b x^n\right)^3 \left(c+d x^n\right)^{-4-\frac{1}{n}} \, dx","Integrate[(a + b*x^n)^3*(c + d*x^n)^(-4 - n^(-1)),x]","\frac{x \left(c+d x^n\right)^{-\frac{1}{n}-3} \left(a^3 \left(c^3 \left(6 n^3+11 n^2+6 n+1\right)+3 c^2 d n \left(6 n^2+5 n+1\right) x^n+6 c d^2 n^2 (3 n+1) x^{2 n}+6 d^3 n^3 x^{3 n}\right)+3 a^2 b c x^n \left(c^2 \left(6 n^2+5 n+1\right)+2 c d n (3 n+1) x^n+2 d^2 n^2 x^{2 n}\right)+3 a b^2 c^2 (n+1) x^{2 n} \left(3 c n+c+d n x^n\right)+b^3 c^3 \left(2 n^2+3 n+1\right) x^{3 n}\right)}{c^4 (n+1) (2 n+1) (3 n+1)}","\frac{6 a^3 n^3 x \left(c+d x^n\right)^{-1/n}}{c^4 (n+1) (2 n+1) (3 n+1)}+\frac{6 a^2 n^2 x \left(a+b x^n\right) \left(c+d x^n\right)^{-\frac{1}{n}-1}}{c^3 (n+1) (2 n+1) (3 n+1)}+\frac{3 a n x \left(a+b x^n\right)^2 \left(c+d x^n\right)^{-\frac{1}{n}-2}}{c^2 \left(6 n^2+5 n+1\right)}+\frac{x \left(a+b x^n\right)^3 \left(c+d x^n\right)^{-\frac{1}{n}-3}}{c (3 n+1)}",1,"(x*(c + d*x^n)^(-3 - n^(-1))*(b^3*c^3*(1 + 3*n + 2*n^2)*x^(3*n) + 3*a*b^2*c^2*(1 + n)*x^(2*n)*(c + 3*c*n + d*n*x^n) + 3*a^2*b*c*x^n*(c^2*(1 + 5*n + 6*n^2) + 2*c*d*n*(1 + 3*n)*x^n + 2*d^2*n^2*x^(2*n)) + a^3*(c^3*(1 + 6*n + 11*n^2 + 6*n^3) + 3*c^2*d*n*(1 + 5*n + 6*n^2)*x^n + 6*c*d^2*n^2*(1 + 3*n)*x^(2*n) + 6*d^3*n^3*x^(3*n))))/(c^4*(1 + n)*(1 + 2*n)*(1 + 3*n))","A",1
322,1,113,116,0.0830703,"\int \left(a+b x^n\right)^2 \left(c+d x^n\right)^{-3-\frac{1}{n}} \, dx","Integrate[(a + b*x^n)^2*(c + d*x^n)^(-3 - n^(-1)),x]","\frac{x \left(c+d x^n\right)^{-\frac{1}{n}-2} \left(a^2 \left(c^2 \left(2 n^2+3 n+1\right)+2 c d n (2 n+1) x^n+2 d^2 n^2 x^{2 n}\right)+2 a b c x^n \left(2 c n+c+d n x^n\right)+b^2 c^2 (n+1) x^{2 n}\right)}{c^3 (n+1) (2 n+1)}","\frac{2 a^2 n^2 x \left(c+d x^n\right)^{-1/n}}{c^3 (n+1) (2 n+1)}+\frac{2 a n x \left(a+b x^n\right) \left(c+d x^n\right)^{-\frac{1}{n}-1}}{c^2 (n+1) (2 n+1)}+\frac{x \left(a+b x^n\right)^2 \left(c+d x^n\right)^{-\frac{1}{n}-2}}{c (2 n+1)}",1,"(x*(c + d*x^n)^(-2 - n^(-1))*(b^2*c^2*(1 + n)*x^(2*n) + 2*a*b*c*x^n*(c + 2*c*n + d*n*x^n) + a^2*(c^2*(1 + 3*n + 2*n^2) + 2*c*d*n*(1 + 2*n)*x^n + 2*d^2*n^2*x^(2*n))))/(c^3*(1 + n)*(1 + 2*n))","A",1
323,1,82,58,0.1373123,"\int \left(a+b x^n\right) \left(c+d x^n\right)^{-2-\frac{1}{n}} \, dx","Integrate[(a + b*x^n)*(c + d*x^n)^(-2 - n^(-1)),x]","\frac{x \left(c+d x^n\right)^{-\frac{n+1}{n}} \left(a (n+1) \left(c+d x^n\right) \left(\frac{d x^n}{c}+1\right)^{\frac{1}{n}} \, _2F_1\left(2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)+b c x^n\right)}{c^2 (n+1)}","\frac{x \left(a+b x^n\right) \left(c+d x^n\right)^{-\frac{1}{n}-1}}{c (n+1)}+\frac{a n x \left(c+d x^n\right)^{-1/n}}{c^2 (n+1)}",1,"(x*(b*c*x^n + a*(1 + n)*(c + d*x^n)*(1 + (d*x^n)/c)^n^(-1)*Hypergeometric2F1[2 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)]))/(c^2*(1 + n)*(c + d*x^n)^((1 + n)/n))","C",1
324,1,18,18,0.035965,"\int \left(c+d x^n\right)^{-1-\frac{1}{n}} \, dx","Integrate[(c + d*x^n)^(-1 - n^(-1)),x]","\frac{x \left(c+d x^n\right)^{-1/n}}{c}","\frac{x \left(c+d x^n\right)^{-1/n}}{c}",1,"x/(c*(c + d*x^n)^n^(-1))","A",1
325,1,52,53,0.0140013,"\int \frac{\left(c+d x^n\right)^{-1/n}}{a+b x^n} \, dx","Integrate[1/((a + b*x^n)*(c + d*x^n)^n^(-1)),x]","\frac{x \left(c+d x^n\right)^{-1/n} \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};\frac{(a d-b c) x^n}{a \left(d x^n+c\right)}\right)}{a}","\frac{x \left(c+d x^n\right)^{-1/n} \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left(d x^n+c\right)}\right)}{a}",1,"(x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), ((-(b*c) + a*d)*x^n)/(a*(c + d*x^n))])/(a*(c + d*x^n)^n^(-1))","A",1
326,1,53,54,0.0145295,"\int \frac{\left(c+d x^n\right)^{1-\frac{1}{n}}}{\left(a+b x^n\right)^2} \, dx","Integrate[(c + d*x^n)^(1 - n^(-1))/(a + b*x^n)^2,x]","\frac{c x \left(c+d x^n\right)^{-1/n} \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};\frac{(a d-b c) x^n}{a \left(d x^n+c\right)}\right)}{a^2}","\frac{c x \left(c+d x^n\right)^{-1/n} \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left(d x^n+c\right)}\right)}{a^2}",1,"(c*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), ((-(b*c) + a*d)*x^n)/(a*(c + d*x^n))])/(a^2*(c + d*x^n)^n^(-1))","A",1
327,1,55,56,0.0133962,"\int \frac{\left(c+d x^n\right)^{2-\frac{1}{n}}}{\left(a+b x^n\right)^3} \, dx","Integrate[(c + d*x^n)^(2 - n^(-1))/(a + b*x^n)^3,x]","\frac{c^2 x \left(c+d x^n\right)^{-1/n} \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};\frac{(a d-b c) x^n}{a \left(d x^n+c\right)}\right)}{a^3}","\frac{c^2 x \left(c+d x^n\right)^{-1/n} \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left(d x^n+c\right)}\right)}{a^3}",1,"(c^2*x*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), ((-(b*c) + a*d)*x^n)/(a*(c + d*x^n))])/(a^3*(c + d*x^n)^n^(-1))","A",1
328,1,1414,193,51.2128106,"\int \left(a+b x^n\right)^p \left(c+d x^n\right)^{-2-\frac{1}{n}-p} \, dx","Integrate[(a + b*x^n)^p*(c + d*x^n)^(-2 - n^(-1) - p),x]","\frac{c^4 (n+1) (2 n+1) (3 n+1) x \left(b x^n+a\right)^{p+3} \left(d x^n+c\right)^{-p-\frac{1}{n}-2} \left(\frac{d x^n}{c}+1\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma (-p) \left(\frac{d n \left(\frac{(b c-a d) \Gamma \left(1+\frac{1}{n}\right) \Gamma (1-p) \, _2F_1\left(2,1-p;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n}{(2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma (-p)}+\frac{c \, _2F_1\left(1,-p;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)}{n+1}\right) x^n}{c^2}+\, _2F_1\left(1,-p;1+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right)}{-c d (3 n+1) (p n+n+1) \left(b x^n+a\right)^2 \left(d n \left((b c-a d) (n+1) \Gamma \left(1+\frac{1}{n}\right) \Gamma (1-p) \, _2F_1\left(2,1-p;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma (-p) \, _2F_1\left(1,-p;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+c^2 (n+1) (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma (-p) \, _2F_1\left(1,-p;1+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+b c n (3 n+1) p \left(b x^n+a\right) \left(d x^n+c\right) \left(d n \left((b c-a d) (n+1) \Gamma \left(1+\frac{1}{n}\right) \Gamma (1-p) \, _2F_1\left(2,1-p;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma (-p) \, _2F_1\left(1,-p;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+c^2 (n+1) (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma (-p) \, _2F_1\left(1,-p;1+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+n^2 \left(d x^n+c\right) \left(-d (b c-a d) \left(b c (n+1) (3 n+1) \left(b x^n+a\right) \Gamma \left(1+\frac{1}{n}\right) \Gamma (1-p) \, _2F_1\left(2,1-p;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n-2 a (a d-b c) n (n+1) (p-1) \Gamma \left(1+\frac{1}{n}\right) \Gamma (1-p) \, _2F_1\left(3,2-p;4+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n-c (n+1) (3 n+1) \left(b x^n+a\right)^2 \Gamma \left(1+\frac{1}{n}\right) \Gamma (1-p) \, _2F_1\left(2,1-p;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)+a c n (3 n+1) p \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma (-p) \, _2F_1\left(2,1-p;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+a c^2 (a d-b c) (2 n+1) (3 n+1) p \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma (-p) \, _2F_1\left(2,1-p;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)+c d (3 n+1) \left(b x^n+a\right)^2 \left((b c-a d) (n+1) \Gamma \left(1+\frac{1}{n}\right) \Gamma (1-p) \, _2F_1\left(2,1-p;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma (-p) \, _2F_1\left(1,-p;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right)\right) x^n+c (3 n+1) \left(b x^n+a\right)^2 \left(d x^n+c\right) \left(d n \left((b c-a d) (n+1) \Gamma \left(1+\frac{1}{n}\right) \Gamma (1-p) \, _2F_1\left(2,1-p;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma (-p) \, _2F_1\left(1,-p;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+c^2 (n+1) (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma (-p) \, _2F_1\left(1,-p;1+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right)}","\frac{x \left(a+b x^n\right)^{p+1} \left(c+d x^n\right)^{-\frac{1}{n}-p-1} (n (p+1) (b c-a d)+b c) \left(\frac{c \left(a+b x^n\right)}{a \left(c+d x^n\right)}\right)^{-p-1} \, _2F_1\left(\frac{1}{n},-p-1;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left(d x^n+c\right)}\right)}{a c n (p+1) (b c-a d)}-\frac{b x \left(a+b x^n\right)^{p+1} \left(c+d x^n\right)^{-\frac{1}{n}-p-1}}{a n (p+1) (b c-a d)}",1,"(c^4*(1 + n)*(1 + 2*n)*(1 + 3*n)*x*(a + b*x^n)^(3 + p)*(c + d*x^n)^(-2 - n^(-1) - p)*(1 + (d*x^n)/c)*Gamma[2 + n^(-1)]*Gamma[-p]*(Hypergeometric2F1[1, -p, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (d*n*x^n*((c*Hypergeometric2F1[1, -p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/(1 + n) + ((b*c - a*d)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/((1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p])))/c^2))/(-(c*d*(1 + 3*n)*(1 + n + n*p)*x^n*(a + b*x^n)^2*(c^2*(1 + n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]))) + b*c*n*(1 + 3*n)*p*x^n*(a + b*x^n)*(c + d*x^n)*(c^2*(1 + n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) + c*(1 + 3*n)*(a + b*x^n)^2*(c + d*x^n)*(c^2*(1 + n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) + n^2*x^n*(c + d*x^n)*(a*c^2*(-(b*c) + a*d)*(1 + 2*n)*(1 + 3*n)*p*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[2, 1 - p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + c*d*(1 + 3*n)*(a + b*x^n)^2*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) - d*(b*c - a*d)*x^n*(b*c*(1 + n)*(1 + 3*n)*x^n*(a + b*x^n)*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - c*(1 + n)*(1 + 3*n)*(a + b*x^n)^2*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + a*c*n*(1 + 3*n)*p*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - 2*a*(-(b*c) + a*d)*n*(1 + n)*(-1 + p)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[3, 2 - p, 4 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])))","B",0
329,1,55,57,0.0689426,"\int \left(a+b x^n\right)^{\frac{a d n-b c (1+n)}{(b c-a d) n}} \left(c+d x^n\right)^{\frac{a d-b c n+a d n}{b c n-a d n}} \, dx","Integrate[(a + b*x^n)^((a*d*n - b*c*(1 + n))/((b*c - a*d)*n))*(c + d*x^n)^((a*d - b*c*n + a*d*n)/(b*c*n - a*d*n)),x]","\frac{x \left(a+b x^n\right)^{-\frac{b c}{b c n-a d n}} \left(c+d x^n\right)^{\frac{a d}{b c n-a d n}}}{a c}","\frac{x \left(a+b x^n\right)^{-\frac{b c}{n (b c-a d)}} \left(c+d x^n\right)^{\frac{a d}{n (b c-a d)}}}{a c}",1,"(x*(c + d*x^n)^((a*d)/(b*c*n - a*d*n)))/(a*c*(a + b*x^n)^((b*c)/(b*c*n - a*d*n)))","A",1
330,1,136,327,0.4627661,"\int \left(a+b x^n\right)^2 \left(c+d x^n\right)^{-4-\frac{1}{n}} \, dx","Integrate[(a + b*x^n)^2*(c + d*x^n)^(-4 - n^(-1)),x]","\frac{x \left(c+d x^n\right)^{-1/n} \left(\frac{d x^n}{c}+1\right)^{\frac{1}{n}} \left(b^2 c^2 \, _2F_1\left(2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)-(b c-a d) \left((a d-b c) \, _2F_1\left(4+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)+2 b c \, _2F_1\left(3+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)\right)\right)}{c^4 d^2}","-\frac{2 a^2 n^2 x \left(c+d x^n\right)^{-1/n} (3 a d n-b (3 c n+c))}{c^4 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{2 a n x \left(a+b x^n\right) \left(c+d x^n\right)^{-\frac{1}{n}-1} (3 a d n-b (3 c n+c))}{c^3 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{x \left(a+b x^n\right)^2 \left(c+d x^n\right)^{-\frac{1}{n}-2} (3 a d n-b (3 c n+c))}{c^2 \left(6 n^2+5 n+1\right) (b c-a d)}-\frac{x \left(a+b x^n\right)^3 \left(c+d x^n\right)^{-\frac{1}{n}-3} (3 a d n-b (3 c n+c))}{3 a c n (3 n+1) (b c-a d)}-\frac{b x \left(a+b x^n\right)^3 \left(c+d x^n\right)^{-\frac{1}{n}-3}}{3 a n (b c-a d)}",1,"(x*(1 + (d*x^n)/c)^n^(-1)*(b^2*c^2*Hypergeometric2F1[2 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)] - (b*c - a*d)*(2*b*c*Hypergeometric2F1[3 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)] + (-(b*c) + a*d)*Hypergeometric2F1[4 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)])))/(c^4*d^2*(c + d*x^n)^n^(-1))","C",1
331,1,94,127,0.1700427,"\int \left(a+b x^n\right) \left(c+d x^n\right)^{-3-\frac{1}{n}} \, dx","Integrate[(a + b*x^n)*(c + d*x^n)^(-3 - n^(-1)),x]","\frac{x \left(c+d x^n\right)^{-1/n} \left(\frac{d x^n}{c}+1\right)^{\frac{1}{n}} \left((a d-b c) \, _2F_1\left(3+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)+b c \, _2F_1\left(2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)\right)}{c^3 d}","\frac{n x \left(c+d x^n\right)^{-1/n} (2 a d n+b c)}{c^3 d (n+1) (2 n+1)}+\frac{x \left(c+d x^n\right)^{-\frac{1}{n}-1} (2 a d n+b c)}{c^2 d (n+1) (2 n+1)}-\frac{x (b c-a d) \left(c+d x^n\right)^{-\frac{1}{n}-2}}{c d (2 n+1)}",1,"(x*(1 + (d*x^n)/c)^n^(-1)*(b*c*Hypergeometric2F1[2 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)] + (-(b*c) + a*d)*Hypergeometric2F1[3 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)]))/(c^3*d*(c + d*x^n)^n^(-1))","C",1
332,1,55,50,0.0293483,"\int \left(c+d x^n\right)^{-2-\frac{1}{n}} \, dx","Integrate[(c + d*x^n)^(-2 - n^(-1)),x]","\frac{x \left(c+d x^n\right)^{-1/n} \left(\frac{d x^n}{c}+1\right)^{\frac{1}{n}} \, _2F_1\left(2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right)}{c^2}","\frac{n x \left(c+d x^n\right)^{-1/n}}{c^2 (n+1)}+\frac{x \left(c+d x^n\right)^{-\frac{1}{n}-1}}{c (n+1)}",1,"(x*(1 + (d*x^n)/c)^n^(-1)*Hypergeometric2F1[2 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)])/(c^2*(c + d*x^n)^n^(-1))","C",1
333,1,153,95,6.9421968,"\int \frac{\left(c+d x^n\right)^{-1-\frac{1}{n}}}{a+b x^n} \, dx","Integrate[(c + d*x^n)^(-1 - n^(-1))/(a + b*x^n),x]","\frac{x \left(c+d x^n\right)^{-\frac{n+1}{n}} \left(\frac{b n x^{2 n} (a d-b c) \, _2F_1\left(2,2+\frac{1}{n};3+\frac{1}{n};\frac{(a d-b c) x^n}{a \left(d x^n+c\right)}\right)}{a^2 (2 n+1) \left(c+d x^n\right)}+\frac{b x^n \Phi \left(\frac{(a d-b c) x^n}{a \left(d x^n+c\right)},1,1+\frac{1}{n}\right)}{a}+\frac{a \left(c+d x^n\right)}{c \left(a+b x^n\right)}\right)}{a}","\frac{b x \left(c+d x^n\right)^{-1/n} \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left(d x^n+c\right)}\right)}{a (b c-a d)}-\frac{d x \left(c+d x^n\right)^{-1/n}}{c (b c-a d)}",1,"(x*((a*(c + d*x^n))/(c*(a + b*x^n)) + (b*x^n*HurwitzLerchPhi[((-(b*c) + a*d)*x^n)/(a*(c + d*x^n)), 1, 1 + n^(-1)])/a + (b*(-(b*c) + a*d)*n*x^(2*n)*Hypergeometric2F1[2, 2 + n^(-1), 3 + n^(-1), ((-(b*c) + a*d)*x^n)/(a*(c + d*x^n))])/(a^2*(1 + 2*n)*(c + d*x^n))))/(a*(c + d*x^n)^((1 + n)/n))","C",1
334,1,1070,127,63.2596732,"\int \frac{\left(c+d x^n\right)^{-1/n}}{\left(a+b x^n\right)^2} \, dx","Integrate[1/((a + b*x^n)^2*(c + d*x^n)^n^(-1)),x]","\frac{c^2 (2 n+1) (3 n+1) x \left(b x^n+a\right) \left(d x^n+c\right)^{-1/n} \left(\frac{d x^n}{c}+1\right) \Gamma \left(2+\frac{1}{n}\right) \Gamma \left(3+\frac{1}{n}\right) \left(\frac{2 (b c-a d) n \left(d x^n+c\right) \, _2F_1\left(2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n}{\left(b x^n+a\right) \Gamma \left(3+\frac{1}{n}\right)}+\frac{c \left(d n x^n+c+c n\right) \, _2F_1\left(1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)}{\Gamma \left(2+\frac{1}{n}\right)}\right)}{-c d (1-n) (2 n+1) (3 n+1) \left(b x^n+a\right)^2 \left(2 (b c-a d) n \left(d x^n+c\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c \left(b x^n+a\right) \left(d n x^n+c+c n\right) \Gamma \left(3+\frac{1}{n}\right) \, _2F_1\left(1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n-2 b c n (2 n+1) (3 n+1) \left(b x^n+a\right) \left(d x^n+c\right) \left(2 (b c-a d) n \left(d x^n+c\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c \left(b x^n+a\right) \left(d n x^n+c+c n\right) \Gamma \left(3+\frac{1}{n}\right) \, _2F_1\left(1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+n^2 \left(d x^n+c\right) \left(2 c d (b c-a d) (2 n+1) (3 n+1) \left(b x^n+a\right)^2 \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n-2 b c (b c-a d) (2 n+1) (3 n+1) \left(b x^n+a\right) \left(d x^n+c\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+12 a (b c-a d)^2 n (2 n+1) \left(d x^n+c\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(3,4;4+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c^2 d (2 n+1) (3 n+1) \left(b x^n+a\right)^3 \Gamma \left(3+\frac{1}{n}\right) \, _2F_1\left(1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)+2 c (b c-a d) (2 n+1) (3 n+1) \left(b x^n+a\right)^2 \left(d x^n+c\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)+2 a c (b c-a d) (3 n+1) \left(b x^n+a\right) \left(d n x^n+c+c n\right) \Gamma \left(3+\frac{1}{n}\right) \, _2F_1\left(2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+c (2 n+1) (3 n+1) \left(b x^n+a\right)^2 \left(d x^n+c\right) \left(2 (b c-a d) n \left(d x^n+c\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(2,3;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c \left(b x^n+a\right) \left(d n x^n+c+c n\right) \Gamma \left(3+\frac{1}{n}\right) \, _2F_1\left(1,2;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right)}","\frac{b x \left(c+d x^n\right)^{-\frac{1-n}{n}}}{a n (b c-a d) \left(a+b x^n\right)}-\frac{x \left(c+d x^n\right)^{-1/n} (a d n+b c (1-n)) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left(d x^n+c\right)}\right)}{a^2 n (b c-a d)}",1,"(c^2*(1 + 2*n)*(1 + 3*n)*x*(a + b*x^n)*(1 + (d*x^n)/c)*Gamma[2 + n^(-1)]*Gamma[3 + n^(-1)]*((c*(c + c*n + d*n*x^n)*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/Gamma[2 + n^(-1)] + (2*(b*c - a*d)*n*x^n*(c + d*x^n)*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/((a + b*x^n)*Gamma[3 + n^(-1)])))/((c + d*x^n)^n^(-1)*(-(c*d*(1 - n)*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)^2*(c*(a + b*x^n)*(c + c*n + d*n*x^n)*Gamma[3 + n^(-1)]*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*(b*c - a*d)*n*x^n*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) - 2*b*c*n*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)*(c + d*x^n)*(c*(a + b*x^n)*(c + c*n + d*n*x^n)*Gamma[3 + n^(-1)]*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*(b*c - a*d)*n*x^n*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) + c*(1 + 2*n)*(1 + 3*n)*(a + b*x^n)^2*(c + d*x^n)*(c*(a + b*x^n)*(c + c*n + d*n*x^n)*Gamma[3 + n^(-1)]*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*(b*c - a*d)*n*x^n*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) + n^2*x^n*(c + d*x^n)*(c^2*d*(1 + 2*n)*(1 + 3*n)*(a + b*x^n)^3*Gamma[3 + n^(-1)]*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*c*d*(b*c - a*d)*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)^2*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - 2*b*c*(b*c - a*d)*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*c*(b*c - a*d)*(1 + 2*n)*(1 + 3*n)*(a + b*x^n)^2*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*a*c*(b*c - a*d)*(1 + 3*n)*(a + b*x^n)*(c + c*n + d*n*x^n)*Gamma[3 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 12*a*(b*c - a*d)^2*n*(1 + 2*n)*x^n*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[3, 4, 4 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])))","B",0
335,1,1241,131,44.8631979,"\int \frac{\left(c+d x^n\right)^{1-\frac{1}{n}}}{\left(a+b x^n\right)^3} \, dx","Integrate[(c + d*x^n)^(1 - n^(-1))/(a + b*x^n)^3,x]","-\frac{c^3 (n+1) (2 n+1) (3 n+1) x \left(d x^n+c\right)^{2-\frac{1}{n}} \Gamma \left(2+\frac{1}{n}\right) \left(\frac{d n \left(\frac{3 (b c-a d) \Gamma \left(1+\frac{1}{n}\right) \, _2F_1\left(2,4;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n}{(2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right)}+\frac{c \, _2F_1\left(1,3;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)}{n+1}\right) x^n}{c^2}+\, _2F_1\left(1,3;1+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right)}{c d (1-2 n) (3 n+1) \left(b x^n+a\right)^2 \left(d n \left(3 (b c-a d) (n+1) \Gamma \left(1+\frac{1}{n}\right) \, _2F_1\left(2,4;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(1,3;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+c^2 (n+1) (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(1,3;1+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+3 b c n (3 n+1) \left(b x^n+a\right) \left(d x^n+c\right) \left(d n \left(3 (b c-a d) (n+1) \Gamma \left(1+\frac{1}{n}\right) \, _2F_1\left(2,4;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(1,3;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+c^2 (n+1) (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(1,3;1+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+n^2 \left(d x^n+c\right) \left(3 d (b c-a d) \left(b c (n+1) (3 n+1) \left(b x^n+a\right) \Gamma \left(1+\frac{1}{n}\right) \, _2F_1\left(2,4;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+8 a (a d-b c) n (n+1) \Gamma \left(1+\frac{1}{n}\right) \, _2F_1\left(3,5;4+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n-c (n+1) (3 n+1) \left(b x^n+a\right)^2 \Gamma \left(1+\frac{1}{n}\right) \, _2F_1\left(2,4;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)-a c n (3 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(2,4;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+3 a c^2 (a d-b c) (2 n+1) (3 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(2,4;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)-c d (3 n+1) \left(b x^n+a\right)^2 \left(3 (b c-a d) (n+1) \Gamma \left(1+\frac{1}{n}\right) \, _2F_1\left(2,4;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(1,3;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right)\right) x^n-c (3 n+1) \left(b x^n+a\right)^2 \left(d x^n+c\right) \left(d n \left(3 (b c-a d) (n+1) \Gamma \left(1+\frac{1}{n}\right) \, _2F_1\left(2,4;3+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right) x^n+c (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(1,3;2+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right) x^n+c^2 (n+1) (2 n+1) \left(b x^n+a\right) \Gamma \left(2+\frac{1}{n}\right) \, _2F_1\left(1,3;1+\frac{1}{n};\frac{(b c-a d) x^n}{c \left(b x^n+a\right)}\right)\right)}","\frac{b x \left(c+d x^n\right)^{2-\frac{1}{n}}}{2 a n (b c-a d) \left(a+b x^n\right)^2}-\frac{c x \left(c+d x^n\right)^{-1/n} (2 a d n+b c (1-2 n)) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left(d x^n+c\right)}\right)}{2 a^3 n (b c-a d)}",1,"-((c^3*(1 + n)*(1 + 2*n)*(1 + 3*n)*x*(c + d*x^n)^(2 - n^(-1))*Gamma[2 + n^(-1)]*(Hypergeometric2F1[1, 3, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (d*n*x^n*((c*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/(1 + n) + (3*(b*c - a*d)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/((1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)])))/c^2))/(c*d*(1 - 2*n)*(1 + 3*n)*x^n*(a + b*x^n)^2*(c^2*(1 + n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 3*(b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) + 3*b*c*n*(1 + 3*n)*x^n*(a + b*x^n)*(c + d*x^n)*(c^2*(1 + n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 3*(b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) - c*(1 + 3*n)*(a + b*x^n)^2*(c + d*x^n)*(c^2*(1 + n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 3*(b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) + n^2*x^n*(c + d*x^n)*(3*a*c^2*(-(b*c) + a*d)*(1 + 2*n)*(1 + 3*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 4, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - c*d*(1 + 3*n)*(a + b*x^n)^2*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 3*(b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) + 3*d*(b*c - a*d)*x^n*(b*c*(1 + n)*(1 + 3*n)*x^n*(a + b*x^n)*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - c*(1 + n)*(1 + 3*n)*(a + b*x^n)^2*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - a*c*n*(1 + 3*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 8*a*(-(b*c) + a*d)*n*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[3, 5, 4 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]))))","B",0
336,-1,0,133,180.0126132,"\int \frac{\left(c+d x^n\right)^{2-\frac{1}{n}}}{\left(a+b x^n\right)^4} \, dx","Integrate[(c + d*x^n)^(2 - n^(-1))/(a + b*x^n)^4,x]","\text{\$Aborted}","\frac{b x \left(c+d x^n\right)^{3-\frac{1}{n}}}{3 a n (b c-a d) \left(a+b x^n\right)^3}-\frac{c^2 x \left(c+d x^n\right)^{-1/n} (3 a d n+b c (1-3 n)) \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left(d x^n+c\right)}\right)}{3 a^4 n (b c-a d)}",1,"$Aborted","F",-1
337,1,110,152,0.103845,"\int x^5 \sqrt{-c+d x} \sqrt{c+d x} \left(a+b x^2\right) \, dx","Integrate[x^5*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]","\frac{\sqrt{d x-c} \sqrt{c+d x} \left(d^2 x^2-c^2\right) \left(3 a d^2 \left(8 c^4+12 c^2 d^2 x^2+15 d^4 x^4\right)+b \left(16 c^6+24 c^4 d^2 x^2+30 c^2 d^4 x^4+35 d^6 x^6\right)\right)}{315 d^8}","\frac{(d x-c)^{7/2} (c+d x)^{7/2} \left(a d^2+3 b c^2\right)}{7 d^8}+\frac{c^2 (d x-c)^{5/2} (c+d x)^{5/2} \left(2 a d^2+3 b c^2\right)}{5 d^8}+\frac{c^4 (d x-c)^{3/2} (c+d x)^{3/2} \left(a d^2+b c^2\right)}{3 d^8}+\frac{b (d x-c)^{9/2} (c+d x)^{9/2}}{9 d^8}",1,"(Sqrt[-c + d*x]*Sqrt[c + d*x]*(-c^2 + d^2*x^2)*(3*a*d^2*(8*c^4 + 12*c^2*d^2*x^2 + 15*d^4*x^4) + b*(16*c^6 + 24*c^4*d^2*x^2 + 30*c^2*d^4*x^4 + 35*d^6*x^6)))/(315*d^8)","A",1
338,1,88,109,0.0602851,"\int x^3 \sqrt{-c+d x} \sqrt{c+d x} \left(a+b x^2\right) \, dx","Integrate[x^3*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]","\frac{\sqrt{d x-c} \sqrt{c+d x} \left(d^2 x^2-c^2\right) \left(7 a d^2 \left(2 c^2+3 d^2 x^2\right)+b \left(8 c^4+12 c^2 d^2 x^2+15 d^4 x^4\right)\right)}{105 d^6}","\frac{(d x-c)^{5/2} (c+d x)^{5/2} \left(a d^2+2 b c^2\right)}{5 d^6}+\frac{c^2 (d x-c)^{3/2} (c+d x)^{3/2} \left(a d^2+b c^2\right)}{3 d^6}+\frac{b (d x-c)^{7/2} (c+d x)^{7/2}}{7 d^6}",1,"(Sqrt[-c + d*x]*Sqrt[c + d*x]*(-c^2 + d^2*x^2)*(7*a*d^2*(2*c^2 + 3*d^2*x^2) + b*(8*c^4 + 12*c^2*d^2*x^2 + 15*d^4*x^4)))/(105*d^6)","A",1
339,1,62,67,0.0461719,"\int x \sqrt{-c+d x} \sqrt{c+d x} \left(a+b x^2\right) \, dx","Integrate[x*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]","\frac{\sqrt{d x-c} \sqrt{c+d x} \left(d^2 x^2-c^2\right) \left(5 a d^2+2 b c^2+3 b d^2 x^2\right)}{15 d^4}","\frac{(d x-c)^{3/2} (c+d x)^{3/2} \left(a d^2+b c^2\right)}{3 d^4}+\frac{b (d x-c)^{5/2} (c+d x)^{5/2}}{5 d^4}",1,"(Sqrt[-c + d*x]*Sqrt[c + d*x]*(-c^2 + d^2*x^2)*(2*b*c^2 + 5*a*d^2 + 3*b*d^2*x^2))/(15*d^4)","A",1
340,1,85,80,0.24263,"\int \frac{\sqrt{-c+d x} \sqrt{c+d x} \left(a+b x^2\right)}{x} \, dx","Integrate[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x,x]","\frac{1}{3} \sqrt{d x-c} \sqrt{c+d x} \left(-\frac{3 a c \tan ^{-1}\left(\frac{\sqrt{d^2 x^2-c^2}}{c}\right)}{\sqrt{d^2 x^2-c^2}}+3 a+b \left(x^2-\frac{c^2}{d^2}\right)\right)","a \sqrt{d x-c} \sqrt{c+d x}-a c \tan ^{-1}\left(\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right)+\frac{b (d x-c)^{3/2} (c+d x)^{3/2}}{3 d^2}",1,"(Sqrt[-c + d*x]*Sqrt[c + d*x]*(3*a + b*(-(c^2/d^2) + x^2) - (3*a*c*ArcTan[Sqrt[-c^2 + d^2*x^2]/c])/Sqrt[-c^2 + d^2*x^2]))/3","A",1
341,1,114,96,0.056193,"\int \frac{\sqrt{-c+d x} \sqrt{c+d x} \left(a+b x^2\right)}{x^3} \, dx","Integrate[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x^3,x]","-\frac{\sqrt{d x-c} \sqrt{c+d x} \left(c \left(a-2 b x^2\right) \sqrt{d^2 x^2-c^2}+x^2 \left(2 b c^2-a d^2\right) \tan ^{-1}\left(\frac{\sqrt{d^2 x^2-c^2}}{c}\right)\right)}{2 c x^2 \sqrt{d^2 x^2-c^2}}","-\frac{\left(2 b c^2-a d^2\right) \tan ^{-1}\left(\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right)}{2 c}-\frac{a \sqrt{d x-c} \sqrt{c+d x}}{2 x^2}+b \sqrt{d x-c} \sqrt{c+d x}",1,"-1/2*(Sqrt[-c + d*x]*Sqrt[c + d*x]*(c*(a - 2*b*x^2)*Sqrt[-c^2 + d^2*x^2] + (2*b*c^2 - a*d^2)*x^2*ArcTan[Sqrt[-c^2 + d^2*x^2]/c]))/(c*x^2*Sqrt[-c^2 + d^2*x^2])","A",1
342,1,137,121,0.0916646,"\int \frac{\sqrt{-c+d x} \sqrt{c+d x} \left(a+b x^2\right)}{x^5} \, dx","Integrate[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x^5,x]","\frac{\sqrt{d x-c} \sqrt{c+d x} \left(\left(c^2-d^2 x^2\right) \left(2 a c^2-a d^2 x^2+4 b c^2 x^2\right)-d^2 x^4 \sqrt{1-\frac{d^2 x^2}{c^2}} \left(a d^2+4 b c^2\right) \tanh ^{-1}\left(\sqrt{1-\frac{d^2 x^2}{c^2}}\right)\right)}{8 c^2 d^2 x^6-8 c^4 x^4}","-\frac{\sqrt{d x-c} \sqrt{c+d x} \left(a d^2+4 b c^2\right)}{8 c^2 x^2}+\frac{d^2 \left(a d^2+4 b c^2\right) \tan ^{-1}\left(\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right)}{8 c^3}+\frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4}",1,"(Sqrt[-c + d*x]*Sqrt[c + d*x]*((c^2 - d^2*x^2)*(2*a*c^2 + 4*b*c^2*x^2 - a*d^2*x^2) - d^2*(4*b*c^2 + a*d^2)*x^4*Sqrt[1 - (d^2*x^2)/c^2]*ArcTanh[Sqrt[1 - (d^2*x^2)/c^2]]))/(-8*c^4*x^4 + 8*c^2*d^2*x^6)","A",1
343,1,161,208,0.3090893,"\int x^4 \sqrt{-c+d x} \sqrt{c+d x} \left(a+b x^2\right) \, dx","Integrate[x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]","\frac{\sqrt{d x-c} \sqrt{c+d x} \left(3 \left(8 a c^5 d^2+5 b c^7\right) \sin ^{-1}\left(\frac{d x}{c}\right)-d x \sqrt{1-\frac{d^2 x^2}{c^2}} \left(8 a d^2 \left(3 c^4+2 c^2 d^2 x^2-8 d^4 x^4\right)+b \left(15 c^6+10 c^4 d^2 x^2+8 c^2 d^4 x^4-48 d^6 x^6\right)\right)\right)}{384 d^7 \sqrt{1-\frac{d^2 x^2}{c^2}}}","\frac{c^2 x (d x-c)^{3/2} (c+d x)^{3/2} \left(8 a d^2+5 b c^2\right)}{64 d^6}+\frac{x^3 (d x-c)^{3/2} (c+d x)^{3/2} \left(8 a d^2+5 b c^2\right)}{48 d^4}-\frac{c^6 \left(8 a d^2+5 b c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)}{64 d^7}+\frac{c^4 x \sqrt{d x-c} \sqrt{c+d x} \left(8 a d^2+5 b c^2\right)}{128 d^6}+\frac{b x^5 (d x-c)^{3/2} (c+d x)^{3/2}}{8 d^2}",1,"(Sqrt[-c + d*x]*Sqrt[c + d*x]*(-(d*x*Sqrt[1 - (d^2*x^2)/c^2]*(8*a*d^2*(3*c^4 + 2*c^2*d^2*x^2 - 8*d^4*x^4) + b*(15*c^6 + 10*c^4*d^2*x^2 + 8*c^2*d^4*x^4 - 48*d^6*x^6))) + 3*(5*b*c^7 + 8*a*c^5*d^2)*ArcSin[(d*x)/c]))/(384*d^7*Sqrt[1 - (d^2*x^2)/c^2])","A",1
344,1,135,159,0.1906524,"\int x^2 \sqrt{-c+d x} \sqrt{c+d x} \left(a+b x^2\right) \, dx","Integrate[x^2*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]","\frac{\sqrt{d x-c} \sqrt{c+d x} \left(3 \left(2 a c^3 d^2+b c^5\right) \sin ^{-1}\left(\frac{d x}{c}\right)+d x \sqrt{1-\frac{d^2 x^2}{c^2}} \left(b \left(-3 c^4-2 c^2 d^2 x^2+8 d^4 x^4\right)-6 a d^2 \left(c^2-2 d^2 x^2\right)\right)\right)}{48 d^5 \sqrt{1-\frac{d^2 x^2}{c^2}}}","\frac{c^2 x \sqrt{d x-c} \sqrt{c+d x} \left(2 a d^2+b c^2\right)}{16 d^4}+\frac{x (d x-c)^{3/2} (c+d x)^{3/2} \left(2 a d^2+b c^2\right)}{8 d^4}-\frac{c^4 \left(2 a d^2+b c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)}{8 d^5}+\frac{b x^3 (d x-c)^{3/2} (c+d x)^{3/2}}{6 d^2}",1,"(Sqrt[-c + d*x]*Sqrt[c + d*x]*(d*x*Sqrt[1 - (d^2*x^2)/c^2]*(-6*a*d^2*(c^2 - 2*d^2*x^2) + b*(-3*c^4 - 2*c^2*d^2*x^2 + 8*d^4*x^4)) + 3*(b*c^5 + 2*a*c^3*d^2)*ArcSin[(d*x)/c]))/(48*d^5*Sqrt[1 - (d^2*x^2)/c^2])","A",1
345,1,129,114,0.278511,"\int \sqrt{-c+d x} \sqrt{c+d x} \left(a+b x^2\right) \, dx","Integrate[Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]","\frac{d x \left(c^2-d^2 x^2\right) \left(b \left(c^2-2 d^2 x^2\right)-4 a d^2\right)-2 c^{5/2} \sqrt{d x-c} \sqrt{\frac{d x}{c}+1} \left(4 a d^2+b c^2\right) \sinh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{2} \sqrt{c}}\right)}{8 d^3 \sqrt{d x-c} \sqrt{c+d x}}","\frac{x \sqrt{d x-c} \sqrt{c+d x} \left(4 a d^2+b c^2\right)}{8 d^2}-\frac{c^2 \left(4 a d^2+b c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)}{4 d^3}+\frac{b x (d x-c)^{3/2} (c+d x)^{3/2}}{4 d^2}",1,"(d*x*(c^2 - d^2*x^2)*(-4*a*d^2 + b*(c^2 - 2*d^2*x^2)) - 2*c^(5/2)*(b*c^2 + 4*a*d^2)*Sqrt[-c + d*x]*Sqrt[1 + (d*x)/c]*ArcSinh[Sqrt[-c + d*x]/(Sqrt[2]*Sqrt[c])])/(8*d^3*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
346,1,101,104,0.088658,"\int \frac{\sqrt{-c+d x} \sqrt{c+d x} \left(a+b x^2\right)}{x^2} \, dx","Integrate[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x^2,x]","\frac{\sqrt{d x-c} \sqrt{c+d x} \left(c d \left(b x^2-2 a\right) \sqrt{1-\frac{d^2 x^2}{c^2}}+x \left(b c^2-2 a d^2\right) \sin ^{-1}\left(\frac{d x}{c}\right)\right)}{2 c d x \sqrt{1-\frac{d^2 x^2}{c^2}}}","\frac{1}{2} x \sqrt{d x-c} \sqrt{c+d x} \left(b-\frac{2 a d^2}{c^2}\right)-\frac{\left(b c^2-2 a d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)}{d}+\frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{c^2 x}",1,"(Sqrt[-c + d*x]*Sqrt[c + d*x]*(c*d*(-2*a + b*x^2)*Sqrt[1 - (d^2*x^2)/c^2] + (b*c^2 - 2*a*d^2)*x*ArcSin[(d*x)/c]))/(2*c*d*x*Sqrt[1 - (d^2*x^2)/c^2])","A",1
347,1,105,84,0.0859925,"\int \frac{\sqrt{-c+d x} \sqrt{c+d x} \left(a+b x^2\right)}{x^4} \, dx","Integrate[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x^4,x]","-\frac{\sqrt{d x-c} \sqrt{c+d x} \left(\sqrt{1-\frac{d^2 x^2}{c^2}} \left(a \left(c^2-d^2 x^2\right)+3 b c^2 x^2\right)+3 b c d x^3 \sin ^{-1}\left(\frac{d x}{c}\right)\right)}{3 c^2 x^3 \sqrt{1-\frac{d^2 x^2}{c^2}}}","\frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}-\frac{b \sqrt{d x-c} \sqrt{c+d x}}{x}+2 b d \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)",1,"-1/3*(Sqrt[-c + d*x]*Sqrt[c + d*x]*(Sqrt[1 - (d^2*x^2)/c^2]*(3*b*c^2*x^2 + a*(c^2 - d^2*x^2)) + 3*b*c*d*x^3*ArcSin[(d*x)/c]))/(c^2*x^3*Sqrt[1 - (d^2*x^2)/c^2])","A",1
348,1,117,125,0.143459,"\int \frac{x^4 \left(a+b x^2\right)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx","Integrate[(x^4*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]","\frac{3 \sqrt{c^2 x^2-1} \left(6 a c^2+5 b\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)+c x \left(c^2 x^2-1\right) \left(6 a c^2 \left(2 c^2 x^2+3\right)+b \left(8 c^4 x^4+10 c^2 x^2+15\right)\right)}{48 c^7 \sqrt{c x-1} \sqrt{c x+1}}","\frac{\left(6 a c^2+5 b\right) \cosh ^{-1}(c x)}{16 c^7}+\frac{x \sqrt{c x-1} \sqrt{c x+1} \left(6 a c^2+5 b\right)}{16 c^6}+\frac{x^3 \sqrt{c x-1} \sqrt{c x+1} \left(6 a c^2+5 b\right)}{24 c^4}+\frac{b x^5 \sqrt{c x-1} \sqrt{c x+1}}{6 c^2}",1,"(c*x*(-1 + c^2*x^2)*(6*a*c^2*(3 + 2*c^2*x^2) + b*(15 + 10*c^2*x^2 + 8*c^4*x^4)) + 3*(5*b + 6*a*c^2)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(48*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
349,1,70,103,0.0432036,"\int \frac{x^3 \left(a+b x^2\right)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx","Integrate[(x^3*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]","\frac{\left(c^2 x^2-1\right) \left(5 a c^2 \left(c^2 x^2+2\right)+b \left(3 c^4 x^4+4 c^2 x^2+8\right)\right)}{15 c^6 \sqrt{c x-1} \sqrt{c x+1}}","\frac{2 \sqrt{c x-1} \sqrt{c x+1} \left(5 a c^2+4 b\right)}{15 c^6}+\frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \left(5 a c^2+4 b\right)}{15 c^4}+\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1}}{5 c^2}",1,"((-1 + c^2*x^2)*(5*a*c^2*(2 + c^2*x^2) + b*(8 + 4*c^2*x^2 + 3*c^4*x^4)))/(15*c^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
350,1,98,87,0.1050455,"\int \frac{x^2 \left(a+b x^2\right)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx","Integrate[(x^2*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]","\frac{c x \left(c^2 x^2-1\right) \left(4 a c^2+b \left(2 c^2 x^2+3\right)\right)+\sqrt{c^2 x^2-1} \left(4 a c^2+3 b\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{8 c^5 \sqrt{c x-1} \sqrt{c x+1}}","\frac{\left(4 a c^2+3 b\right) \cosh ^{-1}(c x)}{8 c^5}+\frac{x \sqrt{c x-1} \sqrt{c x+1} \left(4 a c^2+3 b\right)}{8 c^4}+\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1}}{4 c^2}",1,"(c*x*(-1 + c^2*x^2)*(4*a*c^2 + b*(3 + 2*c^2*x^2)) + (3*b + 4*a*c^2)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(8*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
351,1,52,65,0.0543975,"\int \frac{x \left(a+b x^2\right)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx","Integrate[(x*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]","\frac{\left(c^2 x^2-1\right) \left(3 a c^2+b \left(c^2 x^2+2\right)\right)}{3 c^4 \sqrt{c x-1} \sqrt{c x+1}}","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(3 a c^2+2 b\right)}{3 c^4}+\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{3 c^2}",1,"((-1 + c^2*x^2)*(3*a*c^2 + b*(2 + c^2*x^2)))/(3*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
352,1,101,47,0.2115268,"\int \frac{a+b x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx","Integrate[(a + b*x^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]","\frac{4 \left(a c^2+b\right) \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)+\frac{b \left(c x \sqrt{-(c x-1)^2} \sqrt{c x+1}-2 \sqrt{c x-1} \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)\right)}{\sqrt{1-c x}}}{2 c^3}","\frac{\left(2 a c^2+b\right) \cosh ^{-1}(c x)}{2 c^3}+\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{2 c^2}",1,"((b*(c*x*Sqrt[-(-1 + c*x)^2]*Sqrt[1 + c*x] - 2*Sqrt[-1 + c*x]*ArcSin[Sqrt[1 - c*x]/Sqrt[2]]))/Sqrt[1 - c*x] + 4*(b + a*c^2)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(2*c^3)","B",0
353,1,66,46,0.0332955,"\int \frac{a+b x^2}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx","Integrate[(a + b*x^2)/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]","\frac{a c^2 \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)+b \left(c^2 x^2-1\right)}{c^2 \sqrt{c x-1} \sqrt{c x+1}}","a \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)+\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c^2}",1,"(b*(-1 + c^2*x^2) + a*c^2*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
354,1,73,33,0.0302338,"\int \frac{a+b x^2}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx","Integrate[(a + b*x^2)/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]","\frac{\sqrt{c^2 x^2-1} \left(\frac{a \sqrt{c^2 x^2-1}}{x}+\frac{b \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{c}\right)}{\sqrt{c x-1} \sqrt{c x+1}}","\frac{a \sqrt{c x-1} \sqrt{c x+1}}{x}+\frac{b \cosh ^{-1}(c x)}{c}",1,"(Sqrt[-1 + c^2*x^2]*((a*Sqrt[-1 + c^2*x^2])/x + (b*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/c))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])","B",1
355,1,77,60,0.0709038,"\int \frac{a+b x^2}{x^3 \sqrt{-1+c x} \sqrt{1+c x}} \, dx","Integrate[(a + b*x^2)/(x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]","\frac{x^2 \sqrt{c^2 x^2-1} \left(a c^2+2 b\right) \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)+a \left(c^2 x^2-1\right)}{2 x^2 \sqrt{c x-1} \sqrt{c x+1}}","\frac{1}{2} \left(a c^2+2 b\right) \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)+\frac{a \sqrt{c x-1} \sqrt{c x+1}}{2 x^2}",1,"(a*(-1 + c^2*x^2) + (2*b + a*c^2)*x^2*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
356,1,51,62,0.0184915,"\int \frac{a+b x^2}{x^4 \sqrt{-1+c x} \sqrt{1+c x}} \, dx","Integrate[(a + b*x^2)/(x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]","\frac{\left(c^2 x^2-1\right) \left(2 a c^2 x^2+a+3 b x^2\right)}{3 x^3 \sqrt{c x-1} \sqrt{c x+1}}","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(2 a c^2+3 b\right)}{3 x}+\frac{a \sqrt{c x-1} \sqrt{c x+1}}{3 x^3}",1,"((-1 + c^2*x^2)*(a + 3*b*x^2 + 2*a*c^2*x^2))/(3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
357,1,102,99,0.1015229,"\int \frac{a+b x^2}{x^5 \sqrt{-1+c x} \sqrt{1+c x}} \, dx","Integrate[(a + b*x^2)/(x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]","\frac{\left(c^2 x^2-1\right) \left(a \left(3 c^2 x^2+2\right)+4 b x^2\right)-c^2 x^4 \sqrt{1-c^2 x^2} \left(3 a c^2+4 b\right) \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{8 x^4 \sqrt{c x-1} \sqrt{c x+1}}","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(3 a c^2+4 b\right)}{8 x^2}+\frac{1}{8} c^2 \left(3 a c^2+4 b\right) \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)+\frac{a \sqrt{c x-1} \sqrt{c x+1}}{4 x^4}",1,"((-1 + c^2*x^2)*(4*b*x^2 + a*(2 + 3*c^2*x^2)) - c^2*(4*b + 3*a*c^2)*x^4*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]])/(8*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
358,1,148,164,0.1242344,"\int \frac{x^4 \left(a+b x^2\right)}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx","Integrate[(x^4*(a + b*x^2))/(Sqrt[-c + d*x]*Sqrt[c + d*x]),x]","\frac{3 c^4 \sqrt{d^2 x^2-c^2} \left(6 a d^2+5 b c^2\right) \tanh ^{-1}\left(\frac{d x}{\sqrt{d^2 x^2-c^2}}\right)+d x \left(d^2 x^2-c^2\right) \left(6 a d^2 \left(3 c^2+2 d^2 x^2\right)+b \left(15 c^4+10 c^2 d^2 x^2+8 d^4 x^4\right)\right)}{48 d^7 \sqrt{d x-c} \sqrt{c+d x}}","\frac{c^2 x \sqrt{d x-c} \sqrt{c+d x} \left(6 a d^2+5 b c^2\right)}{16 d^6}+\frac{x^3 \sqrt{d x-c} \sqrt{c+d x} \left(6 a d^2+5 b c^2\right)}{24 d^4}+\frac{c^4 \left(6 a d^2+5 b c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)}{8 d^7}+\frac{b x^5 \sqrt{d x-c} \sqrt{c+d x}}{6 d^2}",1,"(d*x*(-c^2 + d^2*x^2)*(6*a*d^2*(3*c^2 + 2*d^2*x^2) + b*(15*c^4 + 10*c^2*d^2*x^2 + 8*d^4*x^4)) + 3*c^4*(5*b*c^2 + 6*a*d^2)*Sqrt[-c^2 + d^2*x^2]*ArcTanh[(d*x)/Sqrt[-c^2 + d^2*x^2]])/(48*d^7*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
359,1,87,118,0.061708,"\int \frac{x^3 \left(a+b x^2\right)}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx","Integrate[(x^3*(a + b*x^2))/(Sqrt[-c + d*x]*Sqrt[c + d*x]),x]","\frac{\left(d^2 x^2-c^2\right) \left(5 a d^2 \left(2 c^2+d^2 x^2\right)+b \left(8 c^4+4 c^2 d^2 x^2+3 d^4 x^4\right)\right)}{15 d^6 \sqrt{d x-c} \sqrt{c+d x}}","\frac{2 c^2 \sqrt{d x-c} \sqrt{c+d x} \left(5 a d^2+4 b c^2\right)}{15 d^6}+\frac{x^2 \sqrt{d x-c} \sqrt{c+d x} \left(5 a d^2+4 b c^2\right)}{15 d^4}+\frac{b x^4 \sqrt{d x-c} \sqrt{c+d x}}{5 d^2}",1,"((-c^2 + d^2*x^2)*(5*a*d^2*(2*c^2 + d^2*x^2) + b*(8*c^4 + 4*c^2*d^2*x^2 + 3*d^4*x^4)))/(15*d^6*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
360,1,121,118,0.0982932,"\int \frac{x^2 \left(a+b x^2\right)}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx","Integrate[(x^2*(a + b*x^2))/(Sqrt[-c + d*x]*Sqrt[c + d*x]),x]","\frac{d x \left(d^2 x^2-c^2\right) \left(4 a d^2+3 b c^2+2 b d^2 x^2\right)+c^2 \sqrt{d^2 x^2-c^2} \left(4 a d^2+3 b c^2\right) \tanh ^{-1}\left(\frac{d x}{\sqrt{d^2 x^2-c^2}}\right)}{8 d^5 \sqrt{d x-c} \sqrt{c+d x}}","\frac{c^2 \left(4 a d^2+3 b c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)}{4 d^5}+\frac{x \sqrt{d x-c} \sqrt{c+d x} \left(4 a d^2+3 b c^2\right)}{8 d^4}+\frac{b x^3 \sqrt{d x-c} \sqrt{c+d x}}{4 d^2}",1,"(d*x*(-c^2 + d^2*x^2)*(3*b*c^2 + 4*a*d^2 + 2*b*d^2*x^2) + c^2*(3*b*c^2 + 4*a*d^2)*Sqrt[-c^2 + d^2*x^2]*ArcTanh[(d*x)/Sqrt[-c^2 + d^2*x^2]])/(8*d^5*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
361,1,61,72,0.0375482,"\int \frac{x \left(a+b x^2\right)}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx","Integrate[(x*(a + b*x^2))/(Sqrt[-c + d*x]*Sqrt[c + d*x]),x]","\frac{\left(d^2 x^2-c^2\right) \left(3 a d^2+2 b c^2+b d^2 x^2\right)}{3 d^4 \sqrt{d x-c} \sqrt{c+d x}}","\frac{\sqrt{d x-c} \sqrt{c+d x} \left(3 a d^2+2 b c^2\right)}{3 d^4}+\frac{b x^2 \sqrt{d x-c} \sqrt{c+d x}}{3 d^2}",1,"((-c^2 + d^2*x^2)*(2*b*c^2 + 3*a*d^2 + b*d^2*x^2))/(3*d^4*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
362,1,119,68,0.349227,"\int \frac{a+b x^2}{\sqrt{-c+d x} \sqrt{c+d x}} \, dx","Integrate[(a + b*x^2)/(Sqrt[-c + d*x]*Sqrt[c + d*x]),x]","\frac{4 \left(a d^2+b c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)-\frac{2 b c^{5/2} \sqrt{\frac{d x}{c}+1} \sinh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{2} \sqrt{c}}\right)}{\sqrt{c+d x}}+b d x \sqrt{d x-c} \sqrt{c+d x}}{2 d^3}","\frac{\left(2 a d^2+b c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)}{d^3}+\frac{b x \sqrt{d x-c} \sqrt{c+d x}}{2 d^2}",1,"(b*d*x*Sqrt[-c + d*x]*Sqrt[c + d*x] - (2*b*c^(5/2)*Sqrt[1 + (d*x)/c]*ArcSinh[Sqrt[-c + d*x]/(Sqrt[2]*Sqrt[c])])/Sqrt[c + d*x] + 4*(b*c^2 + a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/(2*d^3)","A",1
363,1,87,56,0.0369664,"\int \frac{a+b x^2}{x \sqrt{-c+d x} \sqrt{c+d x}} \, dx","Integrate[(a + b*x^2)/(x*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]","\frac{a d^2 \sqrt{d^2 x^2-c^2} \tan ^{-1}\left(\frac{\sqrt{d^2 x^2-c^2}}{c}\right)-b c^3+b c d^2 x^2}{c d^2 \sqrt{d x-c} \sqrt{c+d x}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right)}{c}+\frac{b \sqrt{d x-c} \sqrt{c+d x}}{d^2}",1,"(-(b*c^3) + b*c*d^2*x^2 + a*d^2*Sqrt[-c^2 + d^2*x^2]*ArcTan[Sqrt[-c^2 + d^2*x^2]/c])/(c*d^2*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
364,1,90,57,0.0412897,"\int \frac{a+b x^2}{x^2 \sqrt{-c+d x} \sqrt{c+d x}} \, dx","Integrate[(a + b*x^2)/(x^2*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]","\frac{\sqrt{d^2 x^2-c^2} \left(\frac{a \sqrt{d^2 x^2-c^2}}{c^2 x}+\frac{b \tanh ^{-1}\left(\frac{d x}{\sqrt{d^2 x^2-c^2}}\right)}{d}\right)}{\sqrt{d x-c} \sqrt{c+d x}}","\frac{a \sqrt{d x-c} \sqrt{c+d x}}{c^2 x}+\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)}{d}",1,"(Sqrt[-c^2 + d^2*x^2]*((a*Sqrt[-c^2 + d^2*x^2])/(c^2*x) + (b*ArcTanh[(d*x)/Sqrt[-c^2 + d^2*x^2]])/d))/(Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
365,1,102,76,0.1021293,"\int \frac{a+b x^2}{x^3 \sqrt{-c+d x} \sqrt{c+d x}} \, dx","Integrate[(a + b*x^2)/(x^3*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]","\frac{x^2 \sqrt{d^2 x^2-c^2} \left(a d^2+2 b c^2\right) \tan ^{-1}\left(\frac{\sqrt{d^2 x^2-c^2}}{c}\right)+a \left(c d^2 x^2-c^3\right)}{2 c^3 x^2 \sqrt{d x-c} \sqrt{c+d x}}","\frac{\left(a d^2+2 b c^2\right) \tan ^{-1}\left(\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right)}{2 c^3}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{2 c^2 x^2}",1,"(a*(-c^3 + c*d^2*x^2) + (2*b*c^2 + a*d^2)*x^2*Sqrt[-c^2 + d^2*x^2]*ArcTan[Sqrt[-c^2 + d^2*x^2]/c])/(2*c^3*x^2*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
366,1,66,75,0.0305078,"\int \frac{a+b x^2}{x^4 \sqrt{-c+d x} \sqrt{c+d x}} \, dx","Integrate[(a + b*x^2)/(x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]","-\frac{\left(c^2-d^2 x^2\right) \left(a \left(c^2+2 d^2 x^2\right)+3 b c^2 x^2\right)}{3 c^4 x^3 \sqrt{d x-c} \sqrt{c+d x}}","\frac{\sqrt{d x-c} \sqrt{c+d x} \left(2 a d^2+3 b c^2\right)}{3 c^4 x}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{3 c^2 x^3}",1,"-1/3*((c^2 - d^2*x^2)*(3*b*c^2*x^2 + a*(c^2 + 2*d^2*x^2)))/(c^4*x^3*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
367,1,144,123,0.1084186,"\int \frac{a+b x^2}{x^5 \sqrt{-c+d x} \sqrt{c+d x}} \, dx","Integrate[(a + b*x^2)/(x^5*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]","-\frac{\left(c^2-d^2 x^2\right) \left(c^2 \sqrt{1-\frac{d^2 x^2}{c^2}} \left(2 a c^2+3 a d^2 x^2+4 b c^2 x^2\right)+d^2 x^4 \left(3 a d^2+4 b c^2\right) \tanh ^{-1}\left(\sqrt{1-\frac{d^2 x^2}{c^2}}\right)\right)}{8 c^6 x^4 \sqrt{d x-c} \sqrt{c+d x} \sqrt{1-\frac{d^2 x^2}{c^2}}}","\frac{d^2 \left(3 a d^2+4 b c^2\right) \tan ^{-1}\left(\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right)}{8 c^5}+\frac{\sqrt{d x-c} \sqrt{c+d x} \left(3 a d^2+4 b c^2\right)}{8 c^4 x^2}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{4 c^2 x^4}",1,"-1/8*((c^2 - d^2*x^2)*(c^2*(2*a*c^2 + 4*b*c^2*x^2 + 3*a*d^2*x^2)*Sqrt[1 - (d^2*x^2)/c^2] + d^2*(4*b*c^2 + 3*a*d^2)*x^4*ArcTanh[Sqrt[1 - (d^2*x^2)/c^2]]))/(c^6*x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]*Sqrt[1 - (d^2*x^2)/c^2])","A",1
368,1,119,161,0.1619394,"\int \frac{x^4 \left(a+b x^2\right)}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx","Integrate[(x^4*(a + b*x^2))/((-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]","\frac{3 c^3 \sqrt{1-\frac{d^2 x^2}{c^2}} \left(4 a d^2+5 b c^2\right) \sin ^{-1}\left(\frac{d x}{c}\right)+4 a d^3 x \left(d^2 x^2-3 c^2\right)+b d x \left(-15 c^4+5 c^2 d^2 x^2+2 d^4 x^4\right)}{8 d^7 \sqrt{d x-c} \sqrt{c+d x}}","\frac{3 c^2 \left(4 a d^2+5 b c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)}{4 d^7}+\frac{3 x \sqrt{d x-c} \sqrt{c+d x} \left(4 a d^2+5 b c^2\right)}{8 d^6}-\frac{x^3 \left(4 a d^2+5 b c^2\right)}{4 d^4 \sqrt{d x-c} \sqrt{c+d x}}+\frac{b x^5}{4 d^2 \sqrt{d x-c} \sqrt{c+d x}}",1,"(4*a*d^3*x*(-3*c^2 + d^2*x^2) + b*d*x*(-15*c^4 + 5*c^2*d^2*x^2 + 2*d^4*x^4) + 3*c^3*(5*b*c^2 + 4*a*d^2)*Sqrt[1 - (d^2*x^2)/c^2]*ArcSin[(d*x)/c])/(8*d^7*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
369,1,72,115,0.0538187,"\int \frac{x^3 \left(a+b x^2\right)}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx","Integrate[(x^3*(a + b*x^2))/((-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]","\frac{-6 a c^2 d^2+3 a d^4 x^2-8 b c^4+4 b c^2 d^2 x^2+b d^4 x^4}{3 d^6 \sqrt{d x-c} \sqrt{c+d x}}","\frac{2 \sqrt{d x-c} \sqrt{c+d x} \left(3 a d^2+4 b c^2\right)}{3 d^6}-\frac{x^2 \left(3 a d^2+4 b c^2\right)}{3 d^4 \sqrt{d x-c} \sqrt{c+d x}}+\frac{b x^4}{3 d^2 \sqrt{d x-c} \sqrt{c+d x}}",1,"(-8*b*c^4 - 6*a*c^2*d^2 + 4*b*c^2*d^2*x^2 + 3*a*d^4*x^2 + b*d^4*x^4)/(3*d^6*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
370,1,90,152,0.1208931,"\int \frac{x^2 \left(a+b x^2\right)}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx","Integrate[(x^2*(a + b*x^2))/((-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]","\frac{c \sqrt{1-\frac{d^2 x^2}{c^2}} \left(2 a d^2+3 b c^2\right) \sin ^{-1}\left(\frac{d x}{c}\right)-2 a d^3 x-3 b c^2 d x+b d^3 x^3}{2 d^5 \sqrt{d x-c} \sqrt{c+d x}}","-\frac{\sqrt{d x-c} \left(2 a d^2+3 b c^2\right)}{2 d^5 \sqrt{c+d x}}-\frac{c \left(2 a d^2+3 b c^2\right)}{2 d^5 \sqrt{d x-c} \sqrt{c+d x}}+\frac{\left(2 a d^2+3 b c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)}{d^5}+\frac{b x^3}{2 d^2 \sqrt{d x-c} \sqrt{c+d x}}",1,"(-3*b*c^2*d*x - 2*a*d^3*x + b*d^3*x^3 + c*(3*b*c^2 + 2*a*d^2)*Sqrt[1 - (d^2*x^2)/c^2]*ArcSin[(d*x)/c])/(2*d^5*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
371,1,45,76,0.0666695,"\int \frac{x \left(a+b x^2\right)}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx","Integrate[(x*(a + b*x^2))/((-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]","\frac{-a d^2-2 b c^2+b d^2 x^2}{d^4 \sqrt{d x-c} \sqrt{c+d x}}","\frac{\sqrt{d x-c} \sqrt{c+d x} \left(a d^2+2 b c^2\right)}{c^2 d^4}-\frac{x^2 \left(\frac{a}{c^2}+\frac{b}{d^2}\right)}{\sqrt{d x-c} \sqrt{c+d x}}",1,"(-2*b*c^2 - a*d^2 + b*d^2*x^2)/(d^4*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
372,1,86,63,0.2436406,"\int \frac{a+b x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx","Integrate[(a + b*x^2)/((-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]","\frac{2 b c^{5/2} \sqrt{\frac{d x}{c}+1} \sinh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{2} \sqrt{c}}\right)-\frac{d x \left(a d^2+b c^2\right)}{\sqrt{d x-c}}}{c^2 d^3 \sqrt{c+d x}}","\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right)}{d^3}-\frac{x \left(\frac{a}{c^2}+\frac{b}{d^2}\right)}{\sqrt{d x-c} \sqrt{c+d x}}",1,"(-((d*(b*c^2 + a*d^2)*x)/Sqrt[-c + d*x]) + 2*b*c^(5/2)*Sqrt[1 + (d*x)/c]*ArcSinh[Sqrt[-c + d*x]/(Sqrt[2]*Sqrt[c])])/(c^2*d^3*Sqrt[c + d*x])","A",1
373,1,84,65,0.0441782,"\int \frac{a+b x^2}{x (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx","Integrate[(a + b*x^2)/(x*(-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]","-\frac{a d^2 \sqrt{d^2 x^2-c^2} \tan ^{-1}\left(\frac{\sqrt{d^2 x^2-c^2}}{c}\right)+a c d^2+b c^3}{c^3 d^2 \sqrt{d x-c} \sqrt{c+d x}}","-\frac{\frac{a}{c^2}+\frac{b}{d^2}}{\sqrt{d x-c} \sqrt{c+d x}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right)}{c^3}",1,"-((b*c^3 + a*c*d^2 + a*d^2*Sqrt[-c^2 + d^2*x^2]*ArcTan[Sqrt[-c^2 + d^2*x^2]/c])/(c^3*d^2*Sqrt[-c + d*x]*Sqrt[c + d*x]))","A",1
374,1,51,67,0.0272649,"\int \frac{a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx","Integrate[(a + b*x^2)/(x^2*(-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]","\frac{a \left(c^2-2 d^2 x^2\right)-b c^2 x^2}{c^4 x \sqrt{d x-c} \sqrt{c+d x}}","\frac{a}{c^2 x \sqrt{d x-c} \sqrt{c+d x}}-\frac{x \left(2 a d^2+b c^2\right)}{c^4 \sqrt{d x-c} \sqrt{c+d x}}",1,"(-(b*c^2*x^2) + a*(c^2 - 2*d^2*x^2))/(c^4*x*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
375,1,75,117,0.0318555,"\int \frac{a+b x^2}{x^3 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx","Integrate[(a + b*x^2)/(x^3*(-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]","\frac{a c^2-x^2 \left(3 a d^2+2 b c^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1-\frac{d^2 x^2}{c^2}\right)}{2 c^4 x^2 \sqrt{d x-c} \sqrt{c+d x}}","-\frac{\left(3 a d^2+2 b c^2\right) \tan ^{-1}\left(\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right)}{2 c^5}-\frac{3 a d^2+2 b c^2}{2 c^4 \sqrt{d x-c} \sqrt{c+d x}}+\frac{a}{2 c^2 x^2 \sqrt{d x-c} \sqrt{c+d x}}",1,"(a*c^2 - (2*b*c^2 + 3*a*d^2)*x^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 - (d^2*x^2)/c^2])/(2*c^4*x^2*Sqrt[-c + d*x]*Sqrt[c + d*x])","C",1
376,1,77,119,0.0312481,"\int \frac{a+b x^2}{x^4 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx","Integrate[(a + b*x^2)/(x^4*(-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]","\frac{a \left(c^4+4 c^2 d^2 x^2-8 d^4 x^4\right)+3 b c^2 x^2 \left(c^2-2 d^2 x^2\right)}{3 c^6 x^3 \sqrt{d x-c} \sqrt{c+d x}}","-\frac{2 d^2 x \left(4 a d^2+3 b c^2\right)}{3 c^6 \sqrt{d x-c} \sqrt{c+d x}}+\frac{4 a d^2+3 b c^2}{3 c^4 x \sqrt{d x-c} \sqrt{c+d x}}+\frac{a}{3 c^2 x^3 \sqrt{d x-c} \sqrt{c+d x}}",1,"(3*b*c^2*x^2*(c^2 - 2*d^2*x^2) + a*(c^4 + 4*c^2*d^2*x^2 - 8*d^4*x^4))/(3*c^6*x^3*Sqrt[-c + d*x]*Sqrt[c + d*x])","A",1
377,1,78,166,0.0328403,"\int \frac{a+b x^2}{x^5 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx","Integrate[(a + b*x^2)/(x^5*(-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]","\frac{a c^4-d^2 x^4 \left(5 a d^2+4 b c^2\right) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};1-\frac{d^2 x^2}{c^2}\right)}{4 c^6 x^4 \sqrt{d x-c} \sqrt{c+d x}}","-\frac{3 d^2 \left(5 a d^2+4 b c^2\right) \tan ^{-1}\left(\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right)}{8 c^7}-\frac{3 d^2 \left(5 a d^2+4 b c^2\right)}{8 c^6 \sqrt{d x-c} \sqrt{c+d x}}+\frac{5 a d^2+4 b c^2}{8 c^4 x^2 \sqrt{d x-c} \sqrt{c+d x}}+\frac{a}{4 c^2 x^4 \sqrt{d x-c} \sqrt{c+d x}}",1,"(a*c^4 - d^2*(4*b*c^2 + 5*a*d^2)*x^4*Hypergeometric2F1[-1/2, 2, 1/2, 1 - (d^2*x^2)/c^2])/(4*c^6*x^4*Sqrt[-c + d*x]*Sqrt[c + d*x])","C",1
378,1,56,40,0.0260034,"\int \frac{1+c^2 x^2}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx","Integrate[(1 + c^2*x^2)/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]","\frac{c^2 x^2+\sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)-1}{\sqrt{c x-1} \sqrt{c x+1}}","\sqrt{c x-1} \sqrt{c x+1}+\tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)",1,"(-1 + c^2*x^2 + Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
379,1,244,53,0.3140586,"\int \frac{x^{-\frac{2 b^2 c+a^2 d}{b^2 c+a^2 d}} \left(c+d x^2\right)}{\sqrt{-a+b x} \sqrt{a+b x}} \, dx","Integrate[(c + d*x^2)/(x^((2*b^2*c + a^2*d)/(b^2*c + a^2*d))*Sqrt[-a + b*x]*Sqrt[a + b*x]),x]","\frac{\sqrt{1-\frac{b^2 x^2}{a^2}} \left(a^2 d+b^2 c\right) x^{-\frac{b^2 c}{a^2 d+b^2 c}} \left(b^2 d x^2 \, _2F_1\left(\frac{1}{2},\frac{2 d a^2+b^2 c}{2 d a^2+2 b^2 c};\frac{4 d a^2+3 b^2 c}{2 d a^2+2 b^2 c};\frac{b^2 x^2}{a^2}\right)-\left(2 a^2 d+b^2 c\right) \, _2F_1\left(\frac{1}{2},-\frac{b^2 c}{2 \left(d a^2+b^2 c\right)};\frac{2 d a^2+b^2 c}{2 d a^2+2 b^2 c};\frac{b^2 x^2}{a^2}\right)\right)}{b^2 \sqrt{b x-a} \sqrt{a+b x} \left(2 a^2 d+b^2 c\right)}","\sqrt{b x-a} \sqrt{a+b x} \left(\frac{c}{a^2}+\frac{d}{b^2}\right) x^{-\frac{b^2 c}{a^2 d+b^2 c}}",1,"((b^2*c + a^2*d)*Sqrt[1 - (b^2*x^2)/a^2]*(-((b^2*c + 2*a^2*d)*Hypergeometric2F1[1/2, -1/2*(b^2*c)/(b^2*c + a^2*d), (b^2*c + 2*a^2*d)/(2*b^2*c + 2*a^2*d), (b^2*x^2)/a^2]) + b^2*d*x^2*Hypergeometric2F1[1/2, (b^2*c + 2*a^2*d)/(2*b^2*c + 2*a^2*d), (3*b^2*c + 4*a^2*d)/(2*b^2*c + 2*a^2*d), (b^2*x^2)/a^2]))/(b^2*(b^2*c + 2*a^2*d)*x^((b^2*c)/(b^2*c + a^2*d))*Sqrt[-a + b*x]*Sqrt[a + b*x])","C",1
380,1,36,36,0.0196136,"\int \frac{1}{\sqrt{-1-\sqrt{x}} \sqrt{-1+\sqrt{x}} \sqrt{1+x}} \, dx","Integrate[1/(Sqrt[-1 - Sqrt[x]]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + x]),x]","\frac{\sqrt{1-x} \sin ^{-1}(x)}{\sqrt{-\sqrt{x}-1} \sqrt{\sqrt{x}-1}}","\frac{\sqrt{1-x} \sin ^{-1}(x)}{\sqrt{-\sqrt{x}-1} \sqrt{\sqrt{x}-1}}",1,"(Sqrt[1 - x]*ArcSin[x])/(Sqrt[-1 - Sqrt[x]]*Sqrt[-1 + Sqrt[x]])","A",1
381,1,75,75,0.0575547,"\int \frac{1}{\sqrt{a-b \sqrt{x}} \sqrt{a+b \sqrt{x}} \sqrt{a^2+b^2 x}} \, dx","Integrate[1/(Sqrt[a - b*Sqrt[x]]*Sqrt[a + b*Sqrt[x]]*Sqrt[a^2 + b^2*x]),x]","-\frac{2 \sqrt{a^2-b^2 x} \tan ^{-1}\left(\frac{\sqrt{a^2-b^2 x}}{\sqrt{a^2+b^2 x}}\right)}{b^2 \sqrt{a-b \sqrt{x}} \sqrt{a+b \sqrt{x}}}","-\frac{2 \sqrt{a^2-b^2 x} \tan ^{-1}\left(\frac{\sqrt{a^2-b^2 x}}{\sqrt{a^2+b^2 x}}\right)}{b^2 \sqrt{a-b \sqrt{x}} \sqrt{a+b \sqrt{x}}}",1,"(-2*Sqrt[a^2 - b^2*x]*ArcTan[Sqrt[a^2 - b^2*x]/Sqrt[a^2 + b^2*x]])/(b^2*Sqrt[a - b*Sqrt[x]]*Sqrt[a + b*Sqrt[x]])","A",1
382,0,0,113,0.4376301,"\int \left(a-b x^n\right)^p \left(a+b x^n\right)^p \left(c+d x^{2 n}\right)^q \, dx","Integrate[(a - b*x^n)^p*(a + b*x^n)^p*(c + d*x^(2*n))^q,x]","\int \left(a-b x^n\right)^p \left(a+b x^n\right)^p \left(c+d x^{2 n}\right)^q \, dx","x \left(a-b x^n\right)^p \left(a+b x^n\right)^p \left(1-\frac{b^2 x^{2 n}}{a^2}\right)^{-p} \left(c+d x^{2 n}\right)^q \left(\frac{d x^{2 n}}{c}+1\right)^{-q} F_1\left(\frac{1}{2 n};-p,-q;\frac{1}{2} \left(2+\frac{1}{n}\right);\frac{b^2 x^{2 n}}{a^2},-\frac{d x^{2 n}}{c}\right)",1,"Integrate[(a - b*x^n)^p*(a + b*x^n)^p*(c + d*x^(2*n))^q, x]","F",-1
383,1,87,87,0.0492564,"\int \left(a-b x^n\right)^p \left(a+b x^n\right)^p \left(a^2+b^2 x^{2 n}\right)^p \, dx","Integrate[(a - b*x^n)^p*(a + b*x^n)^p*(a^2 + b^2*x^(2*n))^p,x]","x \left(a-b x^n\right)^p \left(a+b x^n\right)^p \left(a^2+b^2 x^{2 n}\right)^p \left(1-\frac{b^4 x^{4 n}}{a^4}\right)^{-p} \, _2F_1\left(\frac{1}{4 n},-p;1+\frac{1}{4 n};\frac{b^4 x^{4 n}}{a^4}\right)","x \left(a-b x^n\right)^p \left(a+b x^n\right)^p \left(a^2+b^2 x^{2 n}\right)^p \left(1-\frac{b^4 x^{4 n}}{a^4}\right)^{-p} \, _2F_1\left(\frac{1}{4 n},-p;\frac{1}{4} \left(4+\frac{1}{n}\right);\frac{b^4 x^{4 n}}{a^4}\right)",1,"(x*(a - b*x^n)^p*(a + b*x^n)^p*(a^2 + b^2*x^(2*n))^p*Hypergeometric2F1[1/(4*n), -p, 1 + 1/(4*n), (b^4*x^(4*n))/a^4])/(1 - (b^4*x^(4*n))/a^4)^p","A",1
384,1,258,76,0.429492,"\int \frac{\left(c+d x^{2 n}\right)^p}{\left(a-b x^n\right) \left(a+b x^n\right)} \, dx","Integrate[(c + d*x^(2*n))^p/((a - b*x^n)*(a + b*x^n)),x]","\frac{a^2 c (2 n+1) x \left(c+d x^{2 n}\right)^p F_1\left(\frac{1}{2 n};-p,1;1+\frac{1}{2 n};-\frac{d x^{2 n}}{c},\frac{b^2 x^{2 n}}{a^2}\right)}{\left(a^2-b^2 x^{2 n}\right) \left(2 a^2 d n p x^{2 n} F_1\left(1+\frac{1}{2 n};1-p,1;2+\frac{1}{2 n};-\frac{d x^{2 n}}{c},\frac{b^2 x^{2 n}}{a^2}\right)+2 b^2 c n x^{2 n} F_1\left(1+\frac{1}{2 n};-p,2;2+\frac{1}{2 n};-\frac{d x^{2 n}}{c},\frac{b^2 x^{2 n}}{a^2}\right)+a^2 c (2 n+1) F_1\left(\frac{1}{2 n};-p,1;1+\frac{1}{2 n};-\frac{d x^{2 n}}{c},\frac{b^2 x^{2 n}}{a^2}\right)\right)}","\frac{x \left(c+d x^{2 n}\right)^p \left(\frac{d x^{2 n}}{c}+1\right)^{-p} F_1\left(\frac{1}{2 n};1,-p;\frac{1}{2} \left(2+\frac{1}{n}\right);\frac{b^2 x^{2 n}}{a^2},-\frac{d x^{2 n}}{c}\right)}{a^2}",1,"(a^2*c*(1 + 2*n)*x*(c + d*x^(2*n))^p*AppellF1[1/(2*n), -p, 1, 1 + 1/(2*n), -((d*x^(2*n))/c), (b^2*x^(2*n))/a^2])/((a^2 - b^2*x^(2*n))*(2*a^2*d*n*p*x^(2*n)*AppellF1[1 + 1/(2*n), 1 - p, 1, 2 + 1/(2*n), -((d*x^(2*n))/c), (b^2*x^(2*n))/a^2] + 2*b^2*c*n*x^(2*n)*AppellF1[1 + 1/(2*n), -p, 2, 2 + 1/(2*n), -((d*x^(2*n))/c), (b^2*x^(2*n))/a^2] + a^2*c*(1 + 2*n)*AppellF1[1/(2*n), -p, 1, 1 + 1/(2*n), -((d*x^(2*n))/c), (b^2*x^(2*n))/a^2]))","B",0
385,1,103,96,0.3942058,"\int \left(a-b x^{n/2}\right)^p \left(a+b x^{n/2}\right)^p \left(\frac{a^2 d (1+p)}{b^2 \left(1+\frac{-1-2 n-n p}{n}\right)}+d x^n\right)^{\frac{-1-2 n-n p}{n}} \, dx","Integrate[(a - b*x^(n/2))^p*(a + b*x^(n/2))^p*((a^2*d*(1 + p))/(b^2*(1 + (-1 - 2*n - n*p)/n)) + d*x^n)^((-1 - 2*n - n*p)/n),x]","-\frac{b^2 x (n p+n+1) \left(a^2-b^2 x^n\right) \left(a-b x^{n/2}\right)^p \left(a+b x^{n/2}\right)^p \left(d \left(x^n-\frac{a^2 n (p+1)}{b^2 (n p+n+1)}\right)\right)^{-\frac{n p+n+1}{n}}}{a^4 d n (p+1)}","-\frac{b^2 x (n p+n+1) \left(a-b x^{n/2}\right)^{p+1} \left(a+b x^{n/2}\right)^{p+1} \left(d x^n-\frac{a^2 d n (p+1)}{b^2 (n p+n+1)}\right)^{-\frac{n p+n+1}{n}}}{a^4 d n (p+1)}",1,"-((b^2*(1 + n + n*p)*x*(a - b*x^(n/2))^p*(a + b*x^(n/2))^p*(a^2 - b^2*x^n))/(a^4*d*n*(1 + p)*(d*(-((a^2*n*(1 + p))/(b^2*(1 + n + n*p))) + x^n))^((1 + n + n*p)/n)))","A",1