1,1,172,210,0.9936877,"\int (e x)^m \left(a+b x^n\right)^3 \left(A+B x^n\right) \left(c+d x^n\right) \, dx","Integrate[(e*x)^m*(a + b*x^n)^3*(A + B*x^n)*(c + d*x^n),x]","x (e x)^m \left(\frac{a^3 A c}{m+1}+\frac{a^2 x^n (a A d+a B c+3 A b c)}{m+n+1}+\frac{b^2 x^{4 n} (3 a B d+A b d+b B c)}{m+4 n+1}+\frac{a x^{2 n} (3 A b (a d+b c)+a B (a d+3 b c))}{m+2 n+1}+\frac{b x^{3 n} (A b (3 a d+b c)+3 a B (a d+b c))}{m+3 n+1}+\frac{b^3 B d x^{5 n}}{m+5 n+1}\right)","\frac{a^3 A c (e x)^{m+1}}{e (m+1)}+\frac{a^2 x^{n+1} (e x)^m (a A d+a B c+3 A b c)}{m+n+1}+\frac{b^2 x^{4 n+1} (e x)^m (3 a B d+A b d+b B c)}{m+4 n+1}+\frac{a x^{2 n+1} (e x)^m (3 A b (a d+b c)+a B (a d+3 b c))}{m+2 n+1}+\frac{b x^{3 n+1} (e x)^m (A b (3 a d+b c)+3 a B (a d+b c))}{m+3 n+1}+\frac{b^3 B d x^{5 n+1} (e x)^m}{m+5 n+1}",1,"x*(e*x)^m*((a^3*A*c)/(1 + m) + (a^2*(3*A*b*c + a*B*c + a*A*d)*x^n)/(1 + m + n) + (a*(3*A*b*(b*c + a*d) + a*B*(3*b*c + a*d))*x^(2*n))/(1 + m + 2*n) + (b*(3*a*B*(b*c + a*d) + A*b*(b*c + 3*a*d))*x^(3*n))/(1 + m + 3*n) + (b^2*(b*B*c + A*b*d + 3*a*B*d)*x^(4*n))/(1 + m + 4*n) + (b^3*B*d*x^(5*n))/(1 + m + 5*n))","A",1
2,1,129,160,0.5251622,"\int (e x)^m \left(a+b x^n\right)^2 \left(A+B x^n\right) \left(c+d x^n\right) \, dx","Integrate[(e*x)^m*(a + b*x^n)^2*(A + B*x^n)*(c + d*x^n),x]","x (e x)^m \left(\frac{a^2 A c}{m+1}+\frac{x^{2 n} (A b (2 a d+b c)+a B (a d+2 b c))}{m+2 n+1}+\frac{b x^{3 n} (2 a B d+A b d+b B c)}{m+3 n+1}+\frac{a x^n (a A d+a B c+2 A b c)}{m+n+1}+\frac{b^2 B d x^{4 n}}{m+4 n+1}\right)","\frac{a^2 A c (e x)^{m+1}}{e (m+1)}+\frac{a x^{n+1} (e x)^m (a A d+a B c+2 A b c)}{m+n+1}+\frac{x^{2 n+1} (e x)^m (A b (2 a d+b c)+a B (a d+2 b c))}{m+2 n+1}+\frac{b x^{3 n+1} (e x)^m (2 a B d+A b d+b B c)}{m+3 n+1}+\frac{b^2 B d x^{4 n+1} (e x)^m}{m+4 n+1}",1,"x*(e*x)^m*((a^2*A*c)/(1 + m) + (a*(2*A*b*c + a*B*c + a*A*d)*x^n)/(1 + m + n) + ((a*B*(2*b*c + a*d) + A*b*(b*c + 2*a*d))*x^(2*n))/(1 + m + 2*n) + (b*(b*B*c + A*b*d + 2*a*B*d)*x^(3*n))/(1 + m + 3*n) + (b^2*B*d*x^(4*n))/(1 + m + 4*n))","A",1
3,1,84,108,0.2466086,"\int (e x)^m \left(a+b x^n\right) \left(A+B x^n\right) \left(c+d x^n\right) \, dx","Integrate[(e*x)^m*(a + b*x^n)*(A + B*x^n)*(c + d*x^n),x]","x (e x)^m \left(\frac{x^{2 n} (a B d+A b d+b B c)}{m+2 n+1}+\frac{x^n (a A d+a B c+A b c)}{m+n+1}+\frac{a A c}{m+1}+\frac{b B d x^{3 n}}{m+3 n+1}\right)","\frac{x^{n+1} (e x)^m (a A d+a B c+A b c)}{m+n+1}+\frac{x^{2 n+1} (e x)^m (a B d+A b d+b B c)}{m+2 n+1}+\frac{a A c (e x)^{m+1}}{e (m+1)}+\frac{b B d x^{3 n+1} (e x)^m}{m+3 n+1}",1,"x*(e*x)^m*((a*A*c)/(1 + m) + ((A*b*c + a*B*c + a*A*d)*x^n)/(1 + m + n) + ((b*B*c + A*b*d + a*B*d)*x^(2*n))/(1 + m + 2*n) + (b*B*d*x^(3*n))/(1 + m + 3*n))","A",1
4,1,49,66,0.0667565,"\int (e x)^m \left(A+B x^n\right) \left(c+d x^n\right) \, dx","Integrate[(e*x)^m*(A + B*x^n)*(c + d*x^n),x]","x (e x)^m \left(\frac{x^n (A d+B c)}{m+n+1}+\frac{A c}{m+1}+\frac{B d x^{2 n}}{m+2 n+1}\right)","\frac{x^{n+1} (e x)^m (A d+B c)}{m+n+1}+\frac{A c (e x)^{m+1}}{e (m+1)}+\frac{B d x^{2 n+1} (e x)^m}{m+2 n+1}",1,"x*(e*x)^m*((A*c)/(1 + m) + ((B*c + A*d)*x^n)/(1 + m + n) + (B*d*x^(2*n))/(1 + m + 2*n))","A",1
5,1,95,120,0.218025,"\int \frac{(e x)^m \left(A+B x^n\right) \left(c+d x^n\right)}{a+b x^n} \, dx","Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n),x]","\frac{x (e x)^m \left(\frac{(a B-A b) (a d-b c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a (m+1)}+\frac{-a B d+A b d+b B c}{m+1}+\frac{b B d x^n}{m+n+1}\right)}{b^2}","\frac{(e x)^{m+1} (A b-a B) (b c-a d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a b^2 e (m+1)}+\frac{(e x)^{m+1} (-a B d+A b d+b B c)}{b^2 e (m+1)}+\frac{B d x^{n+1} (e x)^m}{b (m+n+1)}",1,"(x*(e*x)^m*((b*B*c + A*b*d - a*B*d)/(1 + m) + (b*B*d*x^n)/(1 + m + n) + ((-(A*b) + a*B)*(-(b*c) + a*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(1 + m))))/b^2","A",1
6,1,110,177,0.2375545,"\int \frac{(e x)^m \left(A+B x^n\right) \left(c+d x^n\right)}{\left(a+b x^n\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n)^2,x]","\frac{x (e x)^m \left(a^2 B d+a (-2 a B d+A b d+b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)+(A b-a B) (b c-a d) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)\right)}{a^2 b^2 (m+1)}","\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right) (b c (a B (m+1)-A b (m-n+1))+a d (A b (m+1)-a B (m+n+1)))}{a^2 b^2 e (m+1) n}-\frac{d (e x)^{m+1} (A b (m+1)-a B (m+n+1))}{a b^2 e (m+1) n}+\frac{(e x)^{m+1} (A b-a B) \left(c+d x^n\right)}{a b e n \left(a+b x^n\right)}",1,"(x*(e*x)^m*(a^2*B*d + a*(b*B*c + A*b*d - 2*a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + (A*b - a*B)*(b*c - a*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]))/(a^2*b^2*(1 + m))","A",1
7,1,136,228,0.1897993,"\int \frac{(e x)^m \left(A+B x^n\right) \left(c+d x^n\right)}{\left(a+b x^n\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n)^3,x]","\frac{x (e x)^m \left(a^2 B d \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)+a (-2 a B d+A b d+b B c) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)+(A b-a B) (b c-a d) \, _2F_1\left(3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)\right)}{a^3 b^2 (m+1)}","-\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right) (b c (m-n+1) (a B (m+1)-A b (m-2 n+1))+a d (m+1) (A b (m-n+1)-a B (m+n+1)))}{2 a^3 b^2 e (m+1) n^2}-\frac{(e x)^{m+1} (A b (b c (m-2 n+1)-a d (m-n+1))-a B (b c (m+1)-a d (m+n+1)))}{2 a^2 b^2 e n^2 \left(a+b x^n\right)}+\frac{(e x)^{m+1} (A b-a B) \left(c+d x^n\right)}{2 a b e n \left(a+b x^n\right)^2}",1,"(x*(e*x)^m*(a^2*B*d*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a*(b*B*c + A*b*d - 2*a*B*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + (A*b - a*B)*(b*c - a*d)*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]))/(a^3*b^2*(1 + m))","A",1
8,1,273,318,1.4777461,"\int (e x)^m \left(a+b x^n\right)^3 \left(A+B x^n\right) \left(c+d x^n\right)^2 \, dx","Integrate[(e*x)^m*(a + b*x^n)^3*(A + B*x^n)*(c + d*x^n)^2,x]","x (e x)^m \left(\frac{a^3 A c^2}{m+1}+\frac{a x^{2 n} \left(A \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+a B c (2 a d+3 b c)\right)}{m+2 n+1}+\frac{x^{3 n} \left(A b \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a B \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)\right)}{m+3 n+1}+\frac{b x^{4 n} \left(3 a^2 B d^2+3 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right)}{m+4 n+1}+\frac{a^2 c x^n (2 a A d+a B c+3 A b c)}{m+n+1}+\frac{b^2 d x^{5 n} (3 a B d+A b d+2 b B c)}{m+5 n+1}+\frac{b^3 B d^2 x^{6 n}}{m+6 n+1}\right)","\frac{a^3 A c^2 (e x)^{m+1}}{e (m+1)}+\frac{a x^{2 n+1} (e x)^m \left(A \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)+a B c (2 a d+3 b c)\right)}{m+2 n+1}+\frac{x^{3 n+1} (e x)^m \left(A b \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a B \left(a^2 d^2+6 a b c d+3 b^2 c^2\right)\right)}{m+3 n+1}+\frac{b x^{4 n+1} (e x)^m \left(3 a^2 B d^2+3 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right)}{m+4 n+1}+\frac{a^2 c x^{n+1} (e x)^m (2 a A d+a B c+3 A b c)}{m+n+1}+\frac{b^2 d x^{5 n+1} (e x)^m (3 a B d+A b d+2 b B c)}{m+5 n+1}+\frac{b^3 B d^2 x^{6 n+1} (e x)^m}{m+6 n+1}",1,"x*(e*x)^m*((a^3*A*c^2)/(1 + m) + (a^2*c*(3*A*b*c + a*B*c + 2*a*A*d)*x^n)/(1 + m + n) + (a*(a*B*c*(3*b*c + 2*a*d) + A*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2))*x^(2*n))/(1 + m + 2*n) + ((a*B*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2) + A*b*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2))*x^(3*n))/(1 + m + 3*n) + (b*(3*a^2*B*d^2 + 3*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x^(4*n))/(1 + m + 4*n) + (b^2*d*(2*b*B*c + A*b*d + 3*a*B*d)*x^(5*n))/(1 + m + 5*n) + (b^3*B*d^2*x^(6*n))/(1 + m + 6*n))","A",1
9,1,199,237,0.6049958,"\int (e x)^m \left(a+b x^n\right)^2 \left(A+B x^n\right) \left(c+d x^n\right)^2 \, dx","Integrate[(e*x)^m*(a + b*x^n)^2*(A + B*x^n)*(c + d*x^n)^2,x]","x (e x)^m \left(\frac{x^{2 n} \left(A \left(a^2 d^2+4 a b c d+b^2 c^2\right)+2 a B c (a d+b c)\right)}{m+2 n+1}+\frac{x^{3 n} \left(a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right)}{m+3 n+1}+\frac{a^2 A c^2}{m+1}+\frac{b d x^{4 n} (2 a B d+A b d+2 b B c)}{m+4 n+1}+\frac{a c x^n (2 A (a d+b c)+a B c)}{m+n+1}+\frac{b^2 B d^2 x^{5 n}}{m+5 n+1}\right)","\frac{x^{2 n+1} (e x)^m \left(A \left(a^2 d^2+4 a b c d+b^2 c^2\right)+2 a B c (a d+b c)\right)}{m+2 n+1}+\frac{x^{3 n+1} (e x)^m \left(a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right)}{m+3 n+1}+\frac{a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac{a c x^{n+1} (e x)^m (2 A (a d+b c)+a B c)}{m+n+1}+\frac{b d x^{4 n+1} (e x)^m (2 a B d+A b d+2 b B c)}{m+4 n+1}+\frac{b^2 B d^2 x^{5 n+1} (e x)^m}{m+5 n+1}",1,"x*(e*x)^m*((a^2*A*c^2)/(1 + m) + (a*c*(a*B*c + 2*A*(b*c + a*d))*x^n)/(1 + m + n) + ((2*a*B*c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))*x^(2*n))/(1 + m + 2*n) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x^(3*n))/(1 + m + 3*n) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*x^(4*n))/(1 + m + 4*n) + (b^2*B*d^2*x^(5*n))/(1 + m + 5*n))","A",1
10,1,129,160,0.3125056,"\int (e x)^m \left(a+b x^n\right) \left(A+B x^n\right) \left(c+d x^n\right)^2 \, dx","Integrate[(e*x)^m*(a + b*x^n)*(A + B*x^n)*(c + d*x^n)^2,x]","x (e x)^m \left(\frac{x^{2 n} (a d (A d+2 B c)+b c (2 A d+B c))}{m+2 n+1}+\frac{d x^{3 n} (a B d+A b d+2 b B c)}{m+3 n+1}+\frac{c x^n (2 a A d+a B c+A b c)}{m+n+1}+\frac{a A c^2}{m+1}+\frac{b B d^2 x^{4 n}}{m+4 n+1}\right)","\frac{c x^{n+1} (e x)^m (2 a A d+a B c+A b c)}{m+n+1}+\frac{x^{2 n+1} (e x)^m (a d (A d+2 B c)+b c (2 A d+B c))}{m+2 n+1}+\frac{d x^{3 n+1} (e x)^m (a B d+A b d+2 b B c)}{m+3 n+1}+\frac{a A c^2 (e x)^{m+1}}{e (m+1)}+\frac{b B d^2 x^{4 n+1} (e x)^m}{m+4 n+1}",1,"x*(e*x)^m*((a*A*c^2)/(1 + m) + (c*(A*b*c + a*B*c + 2*a*A*d)*x^n)/(1 + m + n) + ((a*d*(2*B*c + A*d) + b*c*(B*c + 2*A*d))*x^(2*n))/(1 + m + 2*n) + (d*(2*b*B*c + A*b*d + a*B*d)*x^(3*n))/(1 + m + 3*n) + (b*B*d^2*x^(4*n))/(1 + m + 4*n))","A",1
11,1,78,102,0.1752724,"\int (e x)^m \left(A+B x^n\right) \left(c+d x^n\right)^2 \, dx","Integrate[(e*x)^m*(A + B*x^n)*(c + d*x^n)^2,x]","x (e x)^m \left(\frac{d x^{2 n} (A d+2 B c)}{m+2 n+1}+\frac{c x^n (2 A d+B c)}{m+n+1}+\frac{A c^2}{m+1}+\frac{B d^2 x^{3 n}}{m+3 n+1}\right)","\frac{c x^{n+1} (e x)^m (2 A d+B c)}{m+n+1}+\frac{d x^{2 n+1} (e x)^m (A d+2 B c)}{m+2 n+1}+\frac{A c^2 (e x)^{m+1}}{e (m+1)}+\frac{B d^2 x^{3 n+1} (e x)^m}{m+3 n+1}",1,"x*(e*x)^m*((A*c^2)/(1 + m) + (c*(B*c + 2*A*d)*x^n)/(1 + m + n) + (d*(2*B*c + A*d)*x^(2*n))/(1 + m + 2*n) + (B*d^2*x^(3*n))/(1 + m + 3*n))","A",1
12,1,153,185,0.3393306,"\int \frac{(e x)^m \left(A+B x^n\right) \left(c+d x^n\right)^2}{a+b x^n} \, dx","Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n)^2)/(a + b*x^n),x]","\frac{x (e x)^m \left(\frac{a^2 B d^2-a b d (A d+2 B c)+b^2 c (2 A d+B c)}{m+1}+\frac{(A b-a B) (b c-a d)^2 \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a (m+1)}+\frac{b d x^n (-a B d+A b d+2 b B c)}{m+n+1}+\frac{b^2 B d^2 x^{2 n}}{m+2 n+1}\right)}{b^3}","\frac{(e x)^{m+1} \left(a^2 B d^2-a b d (A d+2 B c)+b^2 c (2 A d+B c)\right)}{b^3 e (m+1)}+\frac{(e x)^{m+1} (A b-a B) (b c-a d)^2 \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a b^3 e (m+1)}+\frac{d x^{n+1} (e x)^m (-a B d+A b d+2 b B c)}{b^2 (m+n+1)}+\frac{B d^2 x^{2 n+1} (e x)^m}{b (m+2 n+1)}",1,"(x*(e*x)^m*((a^2*B*d^2 - a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))/(1 + m) + (b*d*(2*b*B*c + A*b*d - a*B*d)*x^n)/(1 + m + n) + (b^2*B*d^2*x^(2*n))/(1 + m + 2*n) + ((A*b - a*B)*(b*c - a*d)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(1 + m))))/b^3","A",1
13,1,159,268,0.3181494,"\int \frac{(e x)^m \left(A+B x^n\right) \left(c+d x^n\right)^2}{\left(a+b x^n\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n)^2)/(a + b*x^n)^2,x]","\frac{x (e x)^m \left(\frac{(A b-a B) (b c-a d)^2 \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^2 (m+1)}+\frac{(b c-a d) (-3 a B d+2 A b d+b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a (m+1)}+\frac{d (-2 a B d+A b d+2 b B c)}{m+1}+\frac{b B d^2 x^n}{m+n+1}\right)}{b^3}","-\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right) (A b (b c (m-n+1)-a d (m+n+1))-a B (b c (m+1)-a d (m+2 n+1)))}{a^2 b^3 e (m+1) n}-\frac{d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+n+1))-a B (2 b c (m+n+1)-a d (m+2 n+1)))}{a b^3 e (m+1) n}-\frac{d^2 x^{n+1} (e x)^m (A b (m+n+1)-a B (m+2 n+1))}{a b^2 n (m+n+1)}+\frac{(e x)^{m+1} (A b-a B) \left(c+d x^n\right)^2}{a b e n \left(a+b x^n\right)}",1,"(x*(e*x)^m*((d*(2*b*B*c + A*b*d - 2*a*B*d))/(1 + m) + (b*B*d^2*x^n)/(1 + m + n) + ((b*c - a*d)*(b*B*c + 2*A*b*d - 3*a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(1 + m)) + ((A*b - a*B)*(b*c - a*d)^2*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a^2*(1 + m))))/b^3","A",1
14,1,168,322,0.3120317,"\int \frac{(e x)^m \left(A+B x^n\right) \left(c+d x^n\right)^2}{\left(a+b x^n\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n)^2)/(a + b*x^n)^3,x]","\frac{x (e x)^m \left(\frac{(A b-a B) (b c-a d)^2 \, _2F_1\left(3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^3}+\frac{(b c-a d) (-3 a B d+2 A b d+b B c) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^2}+\frac{d (-3 a B d+A b d+2 b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a}+B d^2\right)}{b^3 (m+1)}","\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right) (b c (a B (m+1)-A b (m-2 n+1)) (a d (m+1)-b c (m-n+1))-a d (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1)))}{2 a^3 b^3 e (m+1) n^2}+\frac{d (e x)^{m+1} (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1))}{2 a^2 b^3 e (m+1) n^2}+\frac{(e x)^{m+1} (b c-a d) \left(c (a B (m+1)-A b (m-2 n+1))-d x^n (A b (m+1)-a B (m+2 n+1))\right)}{2 a^2 b^2 e n^2 \left(a+b x^n\right)}+\frac{(e x)^{m+1} (A b-a B) \left(c+d x^n\right)^2}{2 a b e n \left(a+b x^n\right)^2}",1,"(x*(e*x)^m*(B*d^2 + (d*(2*b*B*c + A*b*d - 3*a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a + ((b*c - a*d)*(b*B*c + 2*A*b*d - 3*a*B*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a^2 + ((A*b - a*B)*(b*c - a*d)^2*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a^3))/(b^3*(1 + m))","A",1
15,1,358,410,1.4303515,"\int (e x)^m \left(a+b x^n\right)^3 \left(A+B x^n\right) \left(c+d x^n\right)^3 \, dx","Integrate[(e*x)^m*(a + b*x^n)^3*(A + B*x^n)*(c + d*x^n)^3,x]","x (e x)^m \left(\frac{a^3 A c^3}{m+1}+\frac{3 a c x^{2 n} \left(A \left(a^2 d^2+3 a b c d+b^2 c^2\right)+a B c (a d+b c)\right)}{m+2 n+1}+\frac{3 b d x^{5 n} \left(a^2 B d^2+a b d (A d+3 B c)+b^2 c (A d+B c)\right)}{m+5 n+1}+\frac{a^2 c^2 x^n (3 A (a d+b c)+a B c)}{m+n+1}+\frac{x^{4 n} \left(a^3 B d^3+3 a^2 b d^2 (A d+3 B c)+9 a b^2 c d (A d+B c)+b^3 c^2 (3 A d+B c)\right)}{m+4 n+1}+\frac{x^{3 n} \left(3 a B c \left(a^2 d^2+3 a b c d+b^2 c^2\right)+A \left(a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right)\right)}{m+3 n+1}+\frac{b^2 d^2 x^{6 n} (3 a B d+A b d+3 b B c)}{m+6 n+1}+\frac{b^3 B d^3 x^{7 n}}{m+7 n+1}\right)","\frac{a^3 A c^3 (e x)^{m+1}}{e (m+1)}+\frac{3 a c x^{2 n+1} (e x)^m \left(A \left(a^2 d^2+3 a b c d+b^2 c^2\right)+a B c (a d+b c)\right)}{m+2 n+1}+\frac{3 b d x^{5 n+1} (e x)^m \left(a^2 B d^2+a b d (A d+3 B c)+b^2 c (A d+B c)\right)}{m+5 n+1}+\frac{a^2 c^2 x^{n+1} (e x)^m (3 A (a d+b c)+a B c)}{m+n+1}+\frac{x^{4 n+1} (e x)^m \left(a^3 B d^3+3 a^2 b d^2 (A d+3 B c)+9 a b^2 c d (A d+B c)+b^3 c^2 (3 A d+B c)\right)}{m+4 n+1}+\frac{x^{3 n+1} (e x)^m \left(3 a B c \left(a^2 d^2+3 a b c d+b^2 c^2\right)+A \left(a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right)\right)}{m+3 n+1}+\frac{b^2 d^2 x^{6 n+1} (e x)^m (3 a B d+A b d+3 b B c)}{m+6 n+1}+\frac{b^3 B d^3 x^{7 n+1} (e x)^m}{m+7 n+1}",1,"x*(e*x)^m*((a^3*A*c^3)/(1 + m) + (a^2*c^2*(a*B*c + 3*A*(b*c + a*d))*x^n)/(1 + m + n) + (3*a*c*(a*B*c*(b*c + a*d) + A*(b^2*c^2 + 3*a*b*c*d + a^2*d^2))*x^(2*n))/(1 + m + 2*n) + ((3*a*B*c*(b^2*c^2 + 3*a*b*c*d + a^2*d^2) + A*(b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3))*x^(3*n))/(1 + m + 3*n) + ((a^3*B*d^3 + 9*a*b^2*c*d*(B*c + A*d) + 3*a^2*b*d^2*(3*B*c + A*d) + b^3*c^2*(B*c + 3*A*d))*x^(4*n))/(1 + m + 4*n) + (3*b*d*(a^2*B*d^2 + b^2*c*(B*c + A*d) + a*b*d*(3*B*c + A*d))*x^(5*n))/(1 + m + 5*n) + (b^2*d^2*(3*b*B*c + A*b*d + 3*a*B*d)*x^(6*n))/(1 + m + 6*n) + (b^3*B*d^3*x^(7*n))/(1 + m + 7*n))","A",1
16,1,265,310,1.5201211,"\int (e x)^m \left(a+b x^n\right)^2 \left(A+B x^n\right) \left(c+d x^n\right)^3 \, dx","Integrate[(e*x)^m*(a + b*x^n)^2*(A + B*x^n)*(c + d*x^n)^3,x]","x (e x)^m \left(\frac{c x^{2 n} \left(A \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a B c (3 a d+2 b c)\right)}{m+2 n+1}+\frac{x^{3 n} \left(a^2 d^2 (A d+3 B c)+6 a b c d (A d+B c)+b^2 c^2 (3 A d+B c)\right)}{m+3 n+1}+\frac{d x^{4 n} \left(a^2 B d^2+2 a b d (A d+3 B c)+3 b^2 c (A d+B c)\right)}{m+4 n+1}+\frac{a^2 A c^3}{m+1}+\frac{a c^2 x^n (3 a A d+a B c+2 A b c)}{m+n+1}+\frac{b d^2 x^{5 n} (2 a B d+A b d+3 b B c)}{m+5 n+1}+\frac{b^2 B d^3 x^{6 n}}{m+6 n+1}\right)","\frac{c x^{2 n+1} (e x)^m \left(A \left(3 a^2 d^2+6 a b c d+b^2 c^2\right)+a B c (3 a d+2 b c)\right)}{m+2 n+1}+\frac{x^{3 n+1} (e x)^m \left(a^2 d^2 (A d+3 B c)+6 a b c d (A d+B c)+b^2 c^2 (3 A d+B c)\right)}{m+3 n+1}+\frac{d x^{4 n+1} (e x)^m \left(a^2 B d^2+2 a b d (A d+3 B c)+3 b^2 c (A d+B c)\right)}{m+4 n+1}+\frac{a^2 A c^3 (e x)^{m+1}}{e (m+1)}+\frac{a c^2 x^{n+1} (e x)^m (3 a A d+a B c+2 A b c)}{m+n+1}+\frac{b d^2 x^{5 n+1} (e x)^m (2 a B d+A b d+3 b B c)}{m+5 n+1}+\frac{b^2 B d^3 x^{6 n+1} (e x)^m}{m+6 n+1}",1,"x*(e*x)^m*((a^2*A*c^3)/(1 + m) + (a*c^2*(2*A*b*c + a*B*c + 3*a*A*d)*x^n)/(1 + m + n) + (c*(a*B*c*(2*b*c + 3*a*d) + A*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2))*x^(2*n))/(1 + m + 2*n) + ((6*a*b*c*d*(B*c + A*d) + a^2*d^2*(3*B*c + A*d) + b^2*c^2*(B*c + 3*A*d))*x^(3*n))/(1 + m + 3*n) + (d*(a^2*B*d^2 + 3*b^2*c*(B*c + A*d) + 2*a*b*d*(3*B*c + A*d))*x^(4*n))/(1 + m + 4*n) + (b*d^2*(3*b*B*c + A*b*d + 2*a*B*d)*x^(5*n))/(1 + m + 5*n) + (b^2*B*d^3*x^(6*n))/(1 + m + 6*n))","A",1
17,1,172,210,0.7530356,"\int (e x)^m \left(a+b x^n\right) \left(A+B x^n\right) \left(c+d x^n\right)^3 \, dx","Integrate[(e*x)^m*(a + b*x^n)*(A + B*x^n)*(c + d*x^n)^3,x]","x (e x)^m \left(\frac{c^2 x^n (3 a A d+a B c+A b c)}{m+n+1}+\frac{d^2 x^{4 n} (a B d+A b d+3 b B c)}{m+4 n+1}+\frac{c x^{2 n} (3 a d (A d+B c)+b c (3 A d+B c))}{m+2 n+1}+\frac{d x^{3 n} (a d (A d+3 B c)+3 b c (A d+B c))}{m+3 n+1}+\frac{a A c^3}{m+1}+\frac{b B d^3 x^{5 n}}{m+5 n+1}\right)","\frac{c^2 x^{n+1} (e x)^m (3 a A d+a B c+A b c)}{m+n+1}+\frac{d^2 x^{4 n+1} (e x)^m (a B d+A b d+3 b B c)}{m+4 n+1}+\frac{c x^{2 n+1} (e x)^m (3 a d (A d+B c)+b c (3 A d+B c))}{m+2 n+1}+\frac{d x^{3 n+1} (e x)^m (a d (A d+3 B c)+3 b c (A d+B c))}{m+3 n+1}+\frac{a A c^3 (e x)^{m+1}}{e (m+1)}+\frac{b B d^3 x^{5 n+1} (e x)^m}{m+5 n+1}",1,"x*(e*x)^m*((a*A*c^3)/(1 + m) + (c^2*(A*b*c + a*B*c + 3*a*A*d)*x^n)/(1 + m + n) + (c*(3*a*d*(B*c + A*d) + b*c*(B*c + 3*A*d))*x^(2*n))/(1 + m + 2*n) + (d*(3*b*c*(B*c + A*d) + a*d*(3*B*c + A*d))*x^(3*n))/(1 + m + 3*n) + (d^2*(3*b*B*c + A*b*d + a*B*d)*x^(4*n))/(1 + m + 4*n) + (b*B*d^3*x^(5*n))/(1 + m + 5*n))","A",1
18,1,106,137,0.1771744,"\int (e x)^m \left(A+B x^n\right) \left(c+d x^n\right)^3 \, dx","Integrate[(e*x)^m*(A + B*x^n)*(c + d*x^n)^3,x]","x (e x)^m \left(\frac{c^2 x^n (3 A d+B c)}{m+n+1}+\frac{d^2 x^{3 n} (A d+3 B c)}{m+3 n+1}+\frac{3 c d x^{2 n} (A d+B c)}{m+2 n+1}+\frac{A c^3}{m+1}+\frac{B d^3 x^{4 n}}{m+4 n+1}\right)","\frac{c^2 x^{n+1} (e x)^m (3 A d+B c)}{m+n+1}+\frac{d^2 x^{3 n+1} (e x)^m (A d+3 B c)}{m+3 n+1}+\frac{3 c d x^{2 n+1} (e x)^m (A d+B c)}{m+2 n+1}+\frac{A c^3 (e x)^{m+1}}{e (m+1)}+\frac{B d^3 x^{4 n+1} (e x)^m}{m+4 n+1}",1,"x*(e*x)^m*((A*c^3)/(1 + m) + (c^2*(B*c + 3*A*d)*x^n)/(1 + m + n) + (3*c*d*(B*c + A*d)*x^(2*n))/(1 + m + 2*n) + (d^2*(3*B*c + A*d)*x^(3*n))/(1 + m + 3*n) + (B*d^3*x^(4*n))/(1 + m + 4*n))","A",1
19,1,229,270,0.6608182,"\int \frac{(e x)^m \left(A+B x^n\right) \left(c+d x^n\right)^3}{a+b x^n} \, dx","Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n)^3)/(a + b*x^n),x]","\frac{x (e x)^m \left(\frac{b d x^n \left(a^2 B d^2-a b d (A d+3 B c)+3 b^2 c (A d+B c)\right)}{m+n+1}+\frac{-a^3 B d^3+a^2 b d^2 (A d+3 B c)-3 a b^2 c d (A d+B c)+b^3 c^2 (3 A d+B c)}{m+1}+\frac{b^2 d^2 x^{2 n} (-a B d+A b d+3 b B c)}{m+2 n+1}+\frac{(a B-A b) (a d-b c)^3 \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a (m+1)}+\frac{b^3 B d^3 x^{3 n}}{m+3 n+1}\right)}{b^4}","\frac{d x^{n+1} (e x)^m \left(a^2 B d^2-a b d (A d+3 B c)+3 b^2 c (A d+B c)\right)}{b^3 (m+n+1)}-\frac{(e x)^{m+1} \left(a^3 B d^3-a^2 b d^2 (A d+3 B c)+3 a b^2 c d (A d+B c)+b^3 \left(-c^2\right) (3 A d+B c)\right)}{b^4 e (m+1)}+\frac{(e x)^{m+1} (A b-a B) (b c-a d)^3 \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a b^4 e (m+1)}+\frac{d^2 x^{2 n+1} (e x)^m (-a B d+A b d+3 b B c)}{b^2 (m+2 n+1)}+\frac{B d^3 x^{3 n+1} (e x)^m}{b (m+3 n+1)}",1,"(x*(e*x)^m*((-(a^3*B*d^3) - 3*a*b^2*c*d*(B*c + A*d) + a^2*b*d^2*(3*B*c + A*d) + b^3*c^2*(B*c + 3*A*d))/(1 + m) + (b*d*(a^2*B*d^2 + 3*b^2*c*(B*c + A*d) - a*b*d*(3*B*c + A*d))*x^n)/(1 + m + n) + (b^2*d^2*(3*b*B*c + A*b*d - a*B*d)*x^(2*n))/(1 + m + 2*n) + (b^3*B*d^3*x^(3*n))/(1 + m + 3*n) + ((-(A*b) + a*B)*(-(b*c) + a*d)^3*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(1 + m))))/b^4","A",1
20,1,217,394,0.5709163,"\int \frac{(e x)^m \left(A+B x^n\right) \left(c+d x^n\right)^3}{\left(a+b x^n\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n)^3)/(a + b*x^n)^2,x]","\frac{x (e x)^m \left(\frac{d \left(3 a^2 B d^2-2 a b d (A d+3 B c)+3 b^2 c (A d+B c)\right)}{m+1}+\frac{(a B-A b) (a d-b c)^3 \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^2 (m+1)}+\frac{b d^2 x^n (-2 a B d+A b d+3 b B c)}{m+n+1}+\frac{(b c-a d)^2 (-4 a B d+3 A b d+b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a (m+1)}+\frac{b^2 B d^3 x^{2 n}}{m+2 n+1}\right)}{b^4}","-\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right) (A b (b c (m-n+1)-a d (m+2 n+1))-a B (b c (m+1)-a d (m+3 n+1)))}{a^2 b^4 e (m+1) n}-\frac{d (e x)^{m+1} \left(A b \left(a^2 d^2 (m+2 n+1)-3 a b c d (m+n+1)+3 b^2 c^2 (m+1)\right)-a B \left(a^2 d^2 (m+3 n+1)-3 a b c d (m+2 n+1)+3 b^2 c^2 (m+n+1)\right)\right)}{a b^4 e (m+1) n}-\frac{d^2 x^{n+1} (e x)^m (A b (3 b c (m+n+1)-a d (m+2 n+1))-a B (3 b c (m+2 n+1)-a d (m+3 n+1)))}{a b^3 n (m+n+1)}-\frac{d^3 x^{2 n+1} (e x)^m (A b (m+2 n+1)-a B (m+3 n+1))}{a b^2 n (m+2 n+1)}+\frac{(e x)^{m+1} (A b-a B) \left(c+d x^n\right)^3}{a b e n \left(a+b x^n\right)}",1,"(x*(e*x)^m*((d*(3*a^2*B*d^2 + 3*b^2*c*(B*c + A*d) - 2*a*b*d*(3*B*c + A*d)))/(1 + m) + (b*d^2*(3*b*B*c + A*b*d - 2*a*B*d)*x^n)/(1 + m + n) + (b^2*B*d^3*x^(2*n))/(1 + m + 2*n) + ((b*c - a*d)^2*(b*B*c + 3*A*b*d - 4*a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(1 + m)) + ((-(A*b) + a*B)*(-(b*c) + a*d)^3*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a^2*(1 + m))))/b^4","A",1
21,1,332,380,1.1949215,"\int \frac{(e x)^m \left(a+b x^n\right)^4 \left(A+B x^n\right)}{c+d x^n} \, dx","Integrate[((e*x)^m*(a + b*x^n)^4*(A + B*x^n))/(c + d*x^n),x]","\frac{x (e x)^m \left(\frac{b^2 d^2 x^{2 n} \left(6 a^2 B d^2+4 a b d (A d-B c)+b^2 c (B c-A d)\right)}{m+2 n+1}+\frac{b d x^n \left(4 a^3 B d^3+6 a^2 b d^2 (A d-B c)+4 a b^2 c d (B c-A d)+b^3 c^2 (A d-B c)\right)}{m+n+1}+\frac{a^4 B d^4+4 a^3 b d^3 (A d-B c)+6 a^2 b^2 c d^2 (B c-A d)+4 a b^3 c^2 d (A d-B c)+b^4 c^3 (B c-A d)}{m+1}+\frac{b^3 d^3 x^{3 n} (4 a B d+A b d-b B c)}{m+3 n+1}-\frac{(b c-a d)^4 (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c (m+1)}+\frac{b^4 B d^4 x^{4 n}}{m+4 n+1}\right)}{d^5}","\frac{b^2 x^{2 n+1} (e x)^m \left(6 a^2 B d^2-4 a b d (B c-A d)+b^2 c (B c-A d)\right)}{d^3 (m+2 n+1)}+\frac{b x^{n+1} (e x)^m \left(4 a^3 B d^3-6 a^2 b d^2 (B c-A d)+4 a b^2 c d (B c-A d)+b^3 \left(-c^2\right) (B c-A d)\right)}{d^4 (m+n+1)}+\frac{(e x)^{m+1} \left(a^4 B d^4-4 a^3 b d^3 (B c-A d)+6 a^2 b^2 c d^2 (B c-A d)-4 a b^3 c^2 d (B c-A d)+b^4 c^3 (B c-A d)\right)}{d^5 e (m+1)}-\frac{b^3 x^{3 n+1} (e x)^m (-4 a B d-A b d+b B c)}{d^2 (m+3 n+1)}-\frac{(e x)^{m+1} (b c-a d)^4 (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c d^5 e (m+1)}+\frac{b^4 B x^{4 n+1} (e x)^m}{d (m+4 n+1)}",1,"(x*(e*x)^m*((a^4*B*d^4 + b^4*c^3*(B*c - A*d) + 6*a^2*b^2*c*d^2*(B*c - A*d) + 4*a*b^3*c^2*d*(-(B*c) + A*d) + 4*a^3*b*d^3*(-(B*c) + A*d))/(1 + m) + (b*d*(4*a^3*B*d^3 + 4*a*b^2*c*d*(B*c - A*d) + b^3*c^2*(-(B*c) + A*d) + 6*a^2*b*d^2*(-(B*c) + A*d))*x^n)/(1 + m + n) + (b^2*d^2*(6*a^2*B*d^2 + b^2*c*(B*c - A*d) + 4*a*b*d*(-(B*c) + A*d))*x^(2*n))/(1 + m + 2*n) + (b^3*d^3*(-(b*B*c) + A*b*d + 4*a*B*d)*x^(3*n))/(1 + m + 3*n) + (b^4*B*d^4*x^(4*n))/(1 + m + 4*n) - ((b*c - a*d)^4*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*(1 + m))))/d^5","A",1
22,1,231,272,0.7639149,"\int \frac{(e x)^m \left(a+b x^n\right)^3 \left(A+B x^n\right)}{c+d x^n} \, dx","Integrate[((e*x)^m*(a + b*x^n)^3*(A + B*x^n))/(c + d*x^n),x]","\frac{x (e x)^m \left(\frac{b d x^n \left(3 a^2 B d^2+3 a b d (A d-B c)+b^2 c (B c-A d)\right)}{m+n+1}+\frac{a^3 B d^3+3 a^2 b d^2 (A d-B c)+3 a b^2 c d (B c-A d)+b^3 c^2 (A d-B c)}{m+1}+\frac{b^2 d^2 x^{2 n} (3 a B d+A b d-b B c)}{m+2 n+1}+\frac{(b c-a d)^3 (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c (m+1)}+\frac{b^3 B d^3 x^{3 n}}{m+3 n+1}\right)}{d^4}","\frac{b x^{n+1} (e x)^m \left(3 a^2 B d^2-3 a b d (B c-A d)+b^2 c (B c-A d)\right)}{d^3 (m+n+1)}+\frac{(e x)^{m+1} \left(a^3 B d^3-3 a^2 b d^2 (B c-A d)+3 a b^2 c d (B c-A d)+b^3 \left(-c^2\right) (B c-A d)\right)}{d^4 e (m+1)}-\frac{b^2 x^{2 n+1} (e x)^m (-3 a B d-A b d+b B c)}{d^2 (m+2 n+1)}+\frac{(e x)^{m+1} (b c-a d)^3 (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c d^4 e (m+1)}+\frac{b^3 B x^{3 n+1} (e x)^m}{d (m+3 n+1)}",1,"(x*(e*x)^m*((a^3*B*d^3 + 3*a*b^2*c*d*(B*c - A*d) + b^3*c^2*(-(B*c) + A*d) + 3*a^2*b*d^2*(-(B*c) + A*d))/(1 + m) + (b*d*(3*a^2*B*d^2 + b^2*c*(B*c - A*d) + 3*a*b*d*(-(B*c) + A*d))*x^n)/(1 + m + n) + (b^2*d^2*(-(b*B*c) + A*b*d + 3*a*B*d)*x^(2*n))/(1 + m + 2*n) + (b^3*B*d^3*x^(3*n))/(1 + m + 3*n) + ((b*c - a*d)^3*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*(1 + m))))/d^4","A",1
23,1,154,187,0.3651093,"\int \frac{(e x)^m \left(a+b x^n\right)^2 \left(A+B x^n\right)}{c+d x^n} \, dx","Integrate[((e*x)^m*(a + b*x^n)^2*(A + B*x^n))/(c + d*x^n),x]","\frac{x (e x)^m \left(\frac{a^2 B d^2+2 a b d (A d-B c)+b^2 c (B c-A d)}{m+1}-\frac{(b c-a d)^2 (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c (m+1)}+\frac{b d x^n (2 a B d+A b d-b B c)}{m+n+1}+\frac{b^2 B d^2 x^{2 n}}{m+2 n+1}\right)}{d^3}","\frac{(e x)^{m+1} \left(a^2 B d^2-2 a b d (B c-A d)+b^2 c (B c-A d)\right)}{d^3 e (m+1)}-\frac{(e x)^{m+1} (b c-a d)^2 (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c d^3 e (m+1)}-\frac{b x^{n+1} (e x)^m (-2 a B d-A b d+b B c)}{d^2 (m+n+1)}+\frac{b^2 B x^{2 n+1} (e x)^m}{d (m+2 n+1)}",1,"(x*(e*x)^m*((a^2*B*d^2 + b^2*c*(B*c - A*d) + 2*a*b*d*(-(B*c) + A*d))/(1 + m) + (b*d*(-(b*B*c) + A*b*d + 2*a*B*d)*x^n)/(1 + m + n) + (b^2*B*d^2*x^(2*n))/(1 + m + 2*n) - ((b*c - a*d)^2*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*(1 + m))))/d^3","A",1
24,1,95,122,0.1658458,"\int \frac{(e x)^m \left(a+b x^n\right) \left(A+B x^n\right)}{c+d x^n} \, dx","Integrate[((e*x)^m*(a + b*x^n)*(A + B*x^n))/(c + d*x^n),x]","\frac{x (e x)^m \left(\frac{(b c-a d) (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c (m+1)}+\frac{a B d+A b d-b B c}{m+1}+\frac{b B d x^n}{m+n+1}\right)}{d^2}","\frac{(e x)^{m+1} (b c-a d) (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c d^2 e (m+1)}-\frac{(e x)^{m+1} (-a B d-A b d+b B c)}{d^2 e (m+1)}+\frac{b B x^{n+1} (e x)^m}{d (m+n+1)}",1,"(x*(e*x)^m*((-(b*B*c) + A*b*d + a*B*d)/(1 + m) + (b*B*d*x^n)/(1 + m + n) + ((b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*(1 + m))))/d^2","A",1
25,1,57,78,0.0721038,"\int \frac{(e x)^m \left(A+B x^n\right)}{c+d x^n} \, dx","Integrate[((e*x)^m*(A + B*x^n))/(c + d*x^n),x]","\frac{x (e x)^m \left((A d-B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)+B c\right)}{c d (m+1)}","\frac{B (e x)^{m+1}}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c d e (m+1)}",1,"(x*(e*x)^m*(B*c + (-(B*c) + A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(c*d*(1 + m))","A",1
26,1,102,127,0.1489406,"\int \frac{(e x)^m \left(A+B x^n\right)}{\left(a+b x^n\right) \left(c+d x^n\right)} \, dx","Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)*(c + d*x^n)),x]","\frac{x (e x)^m \left((a B c-A b c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)+a (A d-B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)\right)}{a c (m+1) (a d-b c)}","\frac{(e x)^{m+1} (A b-a B) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a e (m+1) (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c e (m+1) (b c-a d)}",1,"(x*(e*x)^m*((-(A*b*c) + a*B*c)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a*(-(B*c) + A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(a*c*(-(b*c) + a*d)*(1 + m))","A",1
27,1,152,212,0.2655086,"\int \frac{(e x)^m \left(A+B x^n\right)}{\left(a+b x^n\right)^2 \left(c+d x^n\right)} \, dx","Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)^2*(c + d*x^n)),x]","-\frac{x (e x)^m \left(a^2 d (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)+a b c (A d-B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)-c (A b-a B) (b c-a d) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)\right)}{a^2 c (m+1) (b c-a d)^2}","\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right) (A b (a d (m-2 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-n+1)))}{a^2 e (m+1) n (b c-a d)^2}-\frac{d (e x)^{m+1} (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left(a+b x^n\right)}",1,"-((x*(e*x)^m*(a*b*c*(-(B*c) + A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a^2*d*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - (A*b - a*B)*c*(b*c - a*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]))/(a^2*c*(b*c - a*d)^2*(1 + m)))","A",1
28,1,199,407,0.3444178,"\int \frac{(e x)^m \left(A+B x^n\right)}{\left(a+b x^n\right)^3 \left(c+d x^n\right)} \, dx","Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)^3*(c + d*x^n)),x]","\frac{x (e x)^m \left(\frac{(A b-a B) (b c-a d)^2 \, _2F_1\left(3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^3}+\frac{b (b c-a d) (B c-A d) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^2}+\frac{b d (A d-B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a}+\frac{d^2 (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c}\right)}{(m+1) (b c-a d)^3}","\frac{(e x)^{m+1} (A b (a d (m-4 n+1)-b c (m-2 n+1))+a B (b c (m+1)-a d (m-2 n+1)))}{2 a^2 e n^2 (b c-a d)^2 \left(a+b x^n\right)}+\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right) \left(A b \left(a^2 d^2 \left(m^2+m (2-5 n)+6 n^2-5 n+1\right)-2 a b c d \left(m^2+m (2-4 n)+3 n^2-4 n+1\right)+b^2 c^2 \left(m^2+m (2-3 n)+2 n^2-3 n+1\right)\right)+a B \left(-a^2 d^2 \left(m^2+m (2-3 n)+2 n^2-3 n+1\right)+2 a b c d (m+1) (m-2 n+1)-b^2 c^2 (m+1) (m-n+1)\right)\right)}{2 a^3 e (m+1) n^2 (b c-a d)^3}+\frac{d^2 (e x)^{m+1} (B c-A d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c e (m+1) (b c-a d)^3}+\frac{(e x)^{m+1} (A b-a B)}{2 a e n (b c-a d) \left(a+b x^n\right)^2}",1,"(x*(e*x)^m*((b*d*(-(B*c) + A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a + (d^2*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c + (b*(b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a^2 + ((A*b - a*B)*(b*c - a*d)^2*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a^3))/((b*c - a*d)^3*(1 + m))","A",1
29,1,220,386,0.6744888,"\int \frac{(e x)^m \left(a+b x^n\right)^3 \left(A+B x^n\right)}{\left(c+d x^n\right)^2} \, dx","Integrate[((e*x)^m*(a + b*x^n)^3*(A + B*x^n))/(c + d*x^n)^2,x]","\frac{x (e x)^m \left(\frac{b \left(3 a^2 B d^2+3 a b d (A d-2 B c)+b^2 c (3 B c-2 A d)\right)}{m+1}+\frac{b^2 d x^n (3 a B d+A b d-2 b B c)}{m+n+1}+\frac{(b c-a d)^3 (B c-A d) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c^2 (m+1)}-\frac{(b c-a d)^2 (-a B d-3 A b d+4 b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c (m+1)}+\frac{b^3 B d^2 x^{2 n}}{m+2 n+1}\right)}{d^4}","-\frac{b (e x)^{m+1} \left(3 a^2 d^2 (A d (m+1)-B c (m+n+1))-3 a b c d (A d (m+n+1)-B c (m+2 n+1))+b^2 c^2 (A d (m+2 n+1)-B c (m+3 n+1))\right)}{c d^4 e (m+1) n}-\frac{b^2 x^{n+1} (e x)^m (3 a d (A d (m+n+1)-B c (m+2 n+1))-b c (A d (m+2 n+1)-B c (m+3 n+1)))}{c d^3 n (m+n+1)}+\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m+2 n+1)-B c (m+3 n+1)))}{c^2 d^4 e (m+1) n}-\frac{(e x)^{m+1} \left(a+b x^n\right)^3 (B c-A d)}{c d e n \left(c+d x^n\right)}-\frac{b^3 x^{2 n+1} (e x)^m (A d (m+2 n+1)-B c (m+3 n+1))}{c d^2 n (m+2 n+1)}",1,"(x*(e*x)^m*((b*(3*a^2*B*d^2 + b^2*c*(3*B*c - 2*A*d) + 3*a*b*d*(-2*B*c + A*d)))/(1 + m) + (b^2*d*(-2*b*B*c + A*b*d + 3*a*B*d)*x^n)/(1 + m + n) + (b^3*B*d^2*x^(2*n))/(1 + m + 2*n) - ((b*c - a*d)^2*(4*b*B*c - 3*A*b*d - a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*(1 + m)) + ((b*c - a*d)^3*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c^2*(1 + m))))/d^4","A",1
30,1,161,267,0.3477485,"\int \frac{(e x)^m \left(a+b x^n\right)^2 \left(A+B x^n\right)}{\left(c+d x^n\right)^2} \, dx","Integrate[((e*x)^m*(a + b*x^n)^2*(A + B*x^n))/(c + d*x^n)^2,x]","\frac{x (e x)^m \left(-\frac{(b c-a d)^2 (B c-A d) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c^2 (m+1)}+\frac{(b c-a d) (-a B d-2 A b d+3 b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c (m+1)}+\frac{b (2 a B d+A b d-2 b B c)}{m+1}+\frac{b^2 B d x^n}{m+n+1}\right)}{d^3}","-\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m+n+1)-B c (m+2 n+1)))}{c^2 d^3 e (m+1) n}-\frac{b (e x)^{m+1} (2 a d (A d (m+1)-B c (m+n+1))-b c (A d (m+n+1)-B c (m+2 n+1)))}{c d^3 e (m+1) n}-\frac{(e x)^{m+1} \left(a+b x^n\right)^2 (B c-A d)}{c d e n \left(c+d x^n\right)}-\frac{b^2 x^{n+1} (e x)^m (A d (m+n+1)-B c (m+2 n+1))}{c d^2 n (m+n+1)}",1,"(x*(e*x)^m*((b*(-2*b*B*c + A*b*d + 2*a*B*d))/(1 + m) + (b^2*B*d*x^n)/(1 + m + n) + ((b*c - a*d)*(3*b*B*c - 2*A*b*d - a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*(1 + m)) - ((b*c - a*d)^2*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c^2*(1 + m))))/d^3","A",1
31,1,110,178,0.1892399,"\int \frac{(e x)^m \left(a+b x^n\right) \left(A+B x^n\right)}{\left(c+d x^n\right)^2} \, dx","Integrate[((e*x)^m*(a + b*x^n)*(A + B*x^n))/(c + d*x^n)^2,x]","\frac{x (e x)^m \left(c (a B d+A b d-2 b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)+(b c-a d) (B c-A d) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)+b B c^2\right)}{c^2 d^2 (m+1)}","\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right) (A d (b c (m+1)-a d (m-n+1))+B c (a d (m+1)-b c (m+n+1)))}{c^2 d^2 e (m+1) n}-\frac{(e x)^{m+1} (b c-a d) \left(A+B x^n\right)}{c d e n \left(c+d x^n\right)}-\frac{B (e x)^{m+1} (a d (m+1)-b c (m+n+1))}{c d^2 e (m+1) n}",1,"(x*(e*x)^m*(b*B*c^2 + c*(-2*b*B*c + A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + (b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(c^2*d^2*(1 + m))","A",1
32,1,83,107,0.108718,"\int \frac{(e x)^m \left(A+B x^n\right)}{\left(c+d x^n\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^n))/(c + d*x^n)^2,x]","\frac{x (e x)^m \left((A d-B c) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)+B c \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)\right)}{c^2 d (m+1)}","\frac{(e x)^{m+1} (B c (m+1)-A d (m-n+1)) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c^2 d e (m+1) n}-\frac{(e x)^{m+1} (B c-A d)}{c d e n \left(c+d x^n\right)}",1,"(x*(e*x)^m*(B*c*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + (-(B*c) + A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(c^2*d*(1 + m))","A",1
33,1,150,211,0.2524919,"\int \frac{(e x)^m \left(A+B x^n\right)}{\left(a+b x^n\right) \left(c+d x^n\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)*(c + d*x^n)^2),x]","\frac{x (e x)^m \left(b c^2 (A b-a B) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)+a c d (a B-A b) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)+a (b c-a d) (B c-A d) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)\right)}{a c^2 (m+1) (b c-a d)^2}","\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-2 n+1)-B c (m-n+1)))}{c^2 e (m+1) n (b c-a d)^2}+\frac{b (e x)^{m+1} (A b-a B) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} (B c-A d)}{c e n (b c-a d) \left(c+d x^n\right)}",1,"(x*(e*x)^m*(b*(A*b - a*B)*c^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)] + a*(-(A*b) + a*B)*c*d*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + a*(b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(a*c^2*(b*c - a*d)^2*(1 + m))","A",1
34,1,209,315,0.3630358,"\int \frac{(e x)^m \left(A+B x^n\right)}{\left(a+b x^n\right)^2 \left(c+d x^n\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)^2*(c + d*x^n)^2),x]","\frac{x (e x)^m \left(\frac{b (a B-A b) (a d-b c) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^2}-\frac{d (b c-a d) (B c-A d) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c^2}+\frac{b (a B d-2 A b d+b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a}-\frac{d (a B d-2 A b d+b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c}\right)}{(m+1) (b c-a d)^3}","\frac{b (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right) (A b (a d (m-3 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-2 n+1)))}{a^2 e (m+1) n (b c-a d)^3}-\frac{d (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-3 n+1)-B c (m-2 n+1)))}{c^2 e (m+1) n (b c-a d)^3}+\frac{d (e x)^{m+1} (a A d-2 a B c+A b c)}{a c e n (b c-a d)^2 \left(c+d x^n\right)}+\frac{(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left(a+b x^n\right) \left(c+d x^n\right)}",1,"(x*(e*x)^m*((b*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a - (d*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c + (b*(-(A*b) + a*B)*(-(b*c) + a*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a^2 - (d*(b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c^2))/((b*c - a*d)^3*(1 + m))","A",1
35,1,270,567,0.5344276,"\int \frac{(e x)^m \left(A+B x^n\right)}{\left(a+b x^n\right)^3 \left(c+d x^n\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)^3*(c + d*x^n)^2),x]","\frac{x (e x)^m \left(\frac{b (A b-a B) (b c-a d)^2 \, _2F_1\left(3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^3}+\frac{b (b c-a d) (a B d-2 A b d+b B c) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^2}+\frac{d^2 (b c-a d) (B c-A d) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c^2}+\frac{d^2 (a B d-3 A b d+2 b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c}-\frac{b d (a B d-3 A b d+2 b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a}\right)}{(m+1) (b c-a d)^4}","\frac{d (e x)^{m+1} \left(A \left(-2 a^2 d^2 n+a b c d (m-6 n+1)-b^2 c^2 (m-2 n+1)\right)+a B c (b c (m+1)-a d (m-6 n+1))\right)}{2 a^2 c e n^2 (b c-a d)^3 \left(c+d x^n\right)}+\frac{(e x)^{m+1} (A b (a d (m-5 n+1)-b c (m-2 n+1))+a B (b c (m+1)-a d (m-3 n+1)))}{2 a^2 e n^2 (b c-a d)^2 \left(a+b x^n\right) \left(c+d x^n\right)}+\frac{b (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right) \left(A b \left(a^2 d^2 \left(m^2+m (2-7 n)+12 n^2-7 n+1\right)-2 a b c d \left(m^2+m (2-5 n)+4 n^2-5 n+1\right)+b^2 c^2 \left(m^2+m (2-3 n)+2 n^2-3 n+1\right)\right)+a B \left(-a^2 d^2 \left(m^2+m (2-5 n)+6 n^2-5 n+1\right)+2 a b c d (m+1) (m-3 n+1)-b^2 c^2 (m+1) (m-n+1)\right)\right)}{2 a^3 e (m+1) n^2 (b c-a d)^4}+\frac{d^2 (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-4 n+1)-B c (m-3 n+1)))}{c^2 e (m+1) n (b c-a d)^4}+\frac{(e x)^{m+1} (A b-a B)}{2 a e n (b c-a d) \left(a+b x^n\right)^2 \left(c+d x^n\right)}",1,"(x*(e*x)^m*(-((b*d*(2*b*B*c - 3*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a) + (d^2*(2*b*B*c - 3*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c + (b*(b*c - a*d)*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a^2 + (d^2*(b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c^2 + (b*(A*b - a*B)*(b*c - a*d)^2*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a^3))/((b*c - a*d)^4*(1 + m))","A",1
36,1,172,322,0.3218417,"\int \frac{(e x)^m \left(a+b x^n\right)^2 \left(A+B x^n\right)}{\left(c+d x^n\right)^3} \, dx","Integrate[((e*x)^m*(a + b*x^n)^2*(A + B*x^n))/(c + d*x^n)^3,x]","\frac{x (e x)^m \left(-\frac{(b c-a d)^2 (B c-A d) \, _2F_1\left(3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c^3}+\frac{(b c-a d) (-a B d-2 A b d+3 b B c) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c^2}-\frac{b (-2 a B d-A b d+3 b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c}+b^2 B\right)}{d^3 (m+1)}","\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right) (a d (b c (m+1)-a d (m-n+1)) (B c (m+1)-A d (m-2 n+1))-b c (a d (m+1)-b c (m+n+1)) (A d (m+1)-B c (m+2 n+1)))}{2 c^3 d^3 e (m+1) n^2}+\frac{b (e x)^{m+1} (a d (m+1)-b c (m+n+1)) (A d (m+1)-B c (m+2 n+1))}{2 c^2 d^3 e (m+1) n^2}-\frac{(e x)^{m+1} (b c-a d) \left(a (B c (m+1)-A d (m-2 n+1))-b x^n (A d (m+1)-B c (m+2 n+1))\right)}{2 c^2 d^2 e n^2 \left(c+d x^n\right)}-\frac{(e x)^{m+1} \left(a+b x^n\right)^2 (B c-A d)}{2 c d e n \left(c+d x^n\right)^2}",1,"(x*(e*x)^m*(b^2*B - (b*(3*b*B*c - A*b*d - 2*a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c + ((b*c - a*d)*(3*b*B*c - 2*A*b*d - a*B*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c^2 - ((b*c - a*d)^2*(B*c - A*d)*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c^3))/(d^3*(1 + m))","A",1
37,1,136,228,0.206114,"\int \frac{(e x)^m \left(a+b x^n\right) \left(A+B x^n\right)}{\left(c+d x^n\right)^3} \, dx","Integrate[((e*x)^m*(a + b*x^n)*(A + B*x^n))/(c + d*x^n)^3,x]","\frac{x (e x)^m \left(c (a B d+A b d-2 b B c) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)+(b c-a d) (B c-A d) \, _2F_1\left(3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)+b B c^2 \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)\right)}{c^3 d^2 (m+1)}","-\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right) (A d (m-n+1) (b c (m+1)-a d (m-2 n+1))+B c (m+1) (a d (m-n+1)-b c (m+n+1)))}{2 c^3 d^2 e (m+1) n^2}-\frac{(e x)^{m+1} (a d (A d (m-2 n+1)-B c (m-n+1))-b c (A d (m+1)-B c (m+n+1)))}{2 c^2 d^2 e n^2 \left(c+d x^n\right)}-\frac{(e x)^{m+1} (b c-a d) \left(A+B x^n\right)}{2 c d e n \left(c+d x^n\right)^2}",1,"(x*(e*x)^m*(b*B*c^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + c*(-2*b*B*c + A*b*d + a*B*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + (b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(c^3*d^2*(1 + m))","A",1
38,1,83,112,0.123831,"\int \frac{(e x)^m \left(A+B x^n\right)}{\left(c+d x^n\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^n))/(c + d*x^n)^3,x]","\frac{x (e x)^m \left((A d-B c) \, _2F_1\left(3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)+B c \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)\right)}{c^3 d (m+1)}","\frac{(e x)^{m+1} (B c (m+1)-A d (m-2 n+1)) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{2 c^3 d e (m+1) n}-\frac{(e x)^{m+1} (B c-A d)}{2 c d e n \left(c+d x^n\right)^2}",1,"(x*(e*x)^m*(B*c*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + (-(B*c) + A*d)*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(c^3*d*(1 + m))","A",1
39,1,201,366,0.3121902,"\int \frac{(e x)^m \left(A+B x^n\right)}{\left(a+b x^n\right) \left(c+d x^n\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)*(c + d*x^n)^3),x]","\frac{x (e x)^m \left(\frac{b^2 (A b-a B) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a}+\frac{(b c-a d)^2 (B c-A d) \, _2F_1\left(3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c^3}-\frac{d (A b-a B) (b c-a d) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c^2}-\frac{b d (A b-a B) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c}\right)}{(m+1) (b c-a d)^3}","-\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right) \left(-a^2 d^2 (m-n+1) (B c (m+1)-A d (m-2 n+1))+2 a b c d \left(B c (m+1) (m-2 n+1)-A d \left(m^2+m (2-4 n)+3 n^2-4 n+1\right)\right)+b^2 c^2 (m-2 n+1) (A d (m-3 n+1)-B c (m-n+1))\right)}{2 c^3 e (m+1) n^2 (b c-a d)^3}+\frac{b^2 (e x)^{m+1} (A b-a B) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a e (m+1) (b c-a d)^3}+\frac{(e x)^{m+1} (a d (B c (m+1)-A d (m-2 n+1))+b c (A d (m-4 n+1)-B c (m-2 n+1)))}{2 c^2 e n^2 (b c-a d)^2 \left(c+d x^n\right)}+\frac{(e x)^{m+1} (B c-A d)}{2 c e n (b c-a d) \left(c+d x^n\right)^2}",1,"(x*(e*x)^m*((b^2*(A*b - a*B)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a - (b*(A*b - a*B)*d*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c - ((A*b - a*B)*d*(b*c - a*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c^2 + ((b*c - a*d)^2*(B*c - A*d)*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c^3))/((b*c - a*d)^3*(1 + m))","A",1
40,1,271,482,0.8449655,"\int \frac{(e x)^m \left(A+B x^n\right)}{\left(a+b x^n\right)^2 \left(c+d x^n\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)^2*(c + d*x^n)^3),x]","\frac{x (e x)^m \left(\frac{b^2 (a B-A b) (a d-b c) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^2}+\frac{b^2 (2 a B d-3 A b d+b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a}+\frac{d (b c-a d)^2 (A d-B c) \, _2F_1\left(3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c^3}-\frac{d (b c-a d) (a B d-2 A b d+b B c) \, _2F_1\left(2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c^2}-\frac{b d (2 a B d-3 A b d+b B c) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right)}{c}\right)}{(m+1) (b c-a d)^4}","-\frac{d (e x)^{m+1} \left(a^2 d (B c (m+1)-A d (m-2 n+1))-a b c (m-6 n+1) (B c-A d)-2 A b^2 c^2 n\right)}{2 a c^2 e n^2 (b c-a d)^3 \left(c+d x^n\right)}+\frac{d (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right) \left(-a^2 d^2 (m-n+1) (B c (m+1)-A d (m-2 n+1))+2 a b c d \left(B c (m+1) (m-3 n+1)-A d \left(m^2+m (2-5 n)+4 n^2-5 n+1\right)\right)+b^2 c^2 (m-3 n+1) (A d (m-4 n+1)-B c (m-2 n+1))\right)}{2 c^3 e (m+1) n^2 (b c-a d)^4}+\frac{b^2 (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right) (A b (a d (m-4 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-3 n+1)))}{a^2 e (m+1) n (b c-a d)^4}+\frac{d (e x)^{m+1} (a A d-3 a B c+2 A b c)}{2 a c e n (b c-a d)^2 \left(c+d x^n\right)^2}+\frac{(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left(a+b x^n\right) \left(c+d x^n\right)^2}",1,"(x*(e*x)^m*((b^2*(b*B*c - 3*A*b*d + 2*a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a - (b*d*(b*B*c - 3*A*b*d + 2*a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c + (b^2*(-(A*b) + a*B)*(-(b*c) + a*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/a^2 - (d*(b*c - a*d)*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c^2 + (d*(b*c - a*d)^2*(-(B*c) + A*d)*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c^3))/((b*c - a*d)^4*(1 + m))","A",1
41,1,162,211,0.5890265,"\int (e x)^m \left(a+b x^n\right)^p \left(A+B x^n\right) \left(c+d x^n\right)^q \, dx","Integrate[(e*x)^m*(a + b*x^n)^p*(A + B*x^n)*(c + d*x^n)^q,x]","\frac{x (e x)^m \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} \left(A (m+n+1) F_1\left(\frac{m+1}{n};-p,-q;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)+B (m+1) x^n F_1\left(\frac{m+n+1}{n};-p,-q;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)\right)}{(m+1) (m+n+1)}","\frac{A (e x)^{m+1} \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{m+1}{n};-p,-q;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{e (m+1)}+\frac{B x^{n+1} (e x)^m \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{m+n+1}{n};-p,-q;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{m+n+1}",1,"(x*(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(A*(1 + m + n)*AppellF1[(1 + m)/n, -p, -q, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)] + B*(1 + m)*x^n*AppellF1[(1 + m + n)/n, -p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)]))/((1 + m)*(1 + m + n)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c)^q)","A",1
42,1,164,271,0.2862371,"\int (e x)^m \left(a+b x^n\right)^p \left(A+B x^n\right) \left(c+d x^n\right) \, dx","Integrate[(e*x)^m*(a + b*x^n)^p*(A + B*x^n)*(c + d*x^n),x]","x (e x)^m \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(x^n \left(\frac{(A d+B c) \, _2F_1\left(\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right)}{m+n+1}+\frac{B d x^n \, _2F_1\left(\frac{m+2 n+1}{n},-p;\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right)}{m+2 n+1}\right)+\frac{A c \, _2F_1\left(\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{m+1}\right)","-\frac{(e x)^{m+1} \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right) (A b (m+n p+n+1) (a d (m+1)-b c (m+n (p+2)+1))-a (m+1) (a B d (m+n+1)-b (A d n+B c (m+n (p+2)+1))))}{b^2 e (m+1) (m+n p+n+1) (m+n (p+2)+1)}-\frac{(e x)^{m+1} \left(a+b x^n\right)^{p+1} (a B d (m+n+1)-b (A d n+B c (m+n (p+2)+1)))}{b^2 e (m+n p+n+1) (m+n (p+2)+1)}+\frac{d (e x)^{m+1} \left(A+B x^n\right) \left(a+b x^n\right)^{p+1}}{b e (m+n (p+2)+1)}",1,"(x*(e*x)^m*(a + b*x^n)^p*((A*c*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(1 + m) + x^n*(((B*c + A*d)*Hypergeometric2F1[(1 + m + n)/n, -p, (1 + m + 2*n)/n, -((b*x^n)/a)])/(1 + m + n) + (B*d*x^n*Hypergeometric2F1[(1 + m + 2*n)/n, -p, (1 + m + 3*n)/n, -((b*x^n)/a)])/(1 + m + 2*n))))/(1 + (b*x^n)/a)^p","A",1
43,1,138,164,0.2812789,"\int \frac{(e x)^m \left(a+b x^n\right)^p \left(A+B x^n\right)}{c+d x^n} \, dx","Integrate[((e*x)^m*(a + b*x^n)^p*(A + B*x^n))/(c + d*x^n),x]","\frac{x (e x)^m \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(A (m+n+1) F_1\left(\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)+B (m+1) x^n F_1\left(\frac{m+n+1}{n};-p,1;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)\right)}{c (m+1) (m+n+1)}","\frac{B (e x)^{m+1} \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} F_1\left(\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{c d e (m+1)}",1,"(x*(e*x)^m*(a + b*x^n)^p*(A*(1 + m + n)*AppellF1[(1 + m)/n, -p, 1, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)] + B*(1 + m)*x^n*AppellF1[(1 + m + n)/n, -p, 1, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)]))/(c*(1 + m)*(1 + m + n)*(1 + (b*x^n)/a)^p)","A",0
44,1,138,304,0.531572,"\int \frac{(e x)^m \left(a+b x^n\right)^p \left(A+B x^n\right)}{\left(c+d x^n\right)^2} \, dx","Integrate[((e*x)^m*(a + b*x^n)^p*(A + B*x^n))/(c + d*x^n)^2,x]","\frac{x (e x)^m \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \left(A (m+n+1) F_1\left(\frac{m+1}{n};-p,2;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)+B (m+1) x^n F_1\left(\frac{m+n+1}{n};-p,2;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)\right)}{c^2 (m+1) (m+n+1)}","-\frac{(e x)^{m+1} \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-n (1-p)+1)-B c (m+n p+1))) F_1\left(\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right)}{c^2 d e (m+1) n (b c-a d)}-\frac{b (e x)^{m+1} (m+n p+1) (B c-A d) \left(a+b x^n\right)^p \left(\frac{b x^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{c d e (m+1) n (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \left(a+b x^n\right)^{p+1}}{c e n (b c-a d) \left(c+d x^n\right)}",1,"(x*(e*x)^m*(a + b*x^n)^p*(A*(1 + m + n)*AppellF1[(1 + m)/n, -p, 2, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)] + B*(1 + m)*x^n*AppellF1[(1 + m + n)/n, -p, 2, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)]))/(c^2*(1 + m)*(1 + m + n)*(1 + (b*x^n)/a)^p)","A",0
45,1,124,139,0.1832419,"\int \frac{\left(-a+b x^{n/2}\right)^{-1+\frac{1}{n}} \left(a+b x^{n/2}\right)^{-1+\frac{1}{n}} \left(c+d x^n\right)}{x^2} \, dx","Integrate[((-a + b*x^(n/2))^(-1 + n^(-1))*(a + b*x^(n/2))^(-1 + n^(-1))*(c + d*x^n))/x^2,x]","\frac{\left(b x^{n/2}-a\right)^{\frac{1}{n}} \left(a+b x^{n/2}\right)^{\frac{1}{n}} \left(1-\frac{b^2 x^n}{a^2}\right)^{-1/n} \left(c (n-1) \left(1-\frac{b^2 x^n}{a^2}\right)^{\frac{1}{n}}-d x^n \, _2F_1\left(\frac{n-1}{n},\frac{n-1}{n};2-\frac{1}{n};\frac{b^2 x^n}{a^2}\right)\right)}{a^2 (n-1) x}","\frac{\left(\frac{c}{a^2}+\frac{d}{b^2}\right) \left(b x^{n/2}-a\right)^{\frac{1}{n}} \left(a+b x^{n/2}\right)^{\frac{1}{n}}}{x}-\frac{d \left(b x^{n/2}-a\right)^{\frac{1}{n}} \left(a+b x^{n/2}\right)^{\frac{1}{n}} \left(1-\frac{b^2 x^n}{a^2}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};-\frac{1-n}{n};\frac{b^2 x^n}{a^2}\right)}{b^2 x}",1,"((-a + b*x^(n/2))^n^(-1)*(a + b*x^(n/2))^n^(-1)*(c*(-1 + n)*(1 - (b^2*x^n)/a^2)^n^(-1) - d*x^n*Hypergeometric2F1[(-1 + n)/n, (-1 + n)/n, 2 - n^(-1), (b^2*x^n)/a^2]))/(a^2*(-1 + n)*x*(1 - (b^2*x^n)/a^2)^n^(-1))","A",1
46,1,124,139,0.0618258,"\int \frac{\left(-a+b x^{n/2}\right)^{\frac{1-n}{n}} \left(a+b x^{n/2}\right)^{\frac{1-n}{n}} \left(c+d x^n\right)}{x^2} \, dx","Integrate[((-a + b*x^(n/2))^((1 - n)/n)*(a + b*x^(n/2))^((1 - n)/n)*(c + d*x^n))/x^2,x]","\frac{\left(b x^{n/2}-a\right)^{\frac{1}{n}} \left(a+b x^{n/2}\right)^{\frac{1}{n}} \left(1-\frac{b^2 x^n}{a^2}\right)^{-1/n} \left(c (n-1) \left(1-\frac{b^2 x^n}{a^2}\right)^{\frac{1}{n}}-d x^n \, _2F_1\left(\frac{n-1}{n},\frac{n-1}{n};2-\frac{1}{n};\frac{b^2 x^n}{a^2}\right)\right)}{a^2 (n-1) x}","\frac{\left(\frac{c}{a^2}+\frac{d}{b^2}\right) \left(b x^{n/2}-a\right)^{\frac{1}{n}} \left(a+b x^{n/2}\right)^{\frac{1}{n}}}{x}-\frac{d \left(b x^{n/2}-a\right)^{\frac{1}{n}} \left(a+b x^{n/2}\right)^{\frac{1}{n}} \left(1-\frac{b^2 x^n}{a^2}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};-\frac{1-n}{n};\frac{b^2 x^n}{a^2}\right)}{b^2 x}",1,"((-a + b*x^(n/2))^n^(-1)*(a + b*x^(n/2))^n^(-1)*(c*(-1 + n)*(1 - (b^2*x^n)/a^2)^n^(-1) - d*x^n*Hypergeometric2F1[(-1 + n)/n, (-1 + n)/n, 2 - n^(-1), (b^2*x^n)/a^2]))/(a^2*(-1 + n)*x*(1 - (b^2*x^n)/a^2)^n^(-1))","A",1