1,1,151,189,0.3234785,"\int (e x)^m \left(a+b x^2\right)^3 \left(A+B x^2\right) \left(c+d x^2\right) \, dx","Integrate[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2),x]","x (e x)^m \left(\frac{a^3 A c}{m+1}+\frac{a^2 x^2 (a A d+a B c+3 A b c)}{m+3}+\frac{b^2 x^8 (3 a B d+A b d+b B c)}{m+9}+\frac{b x^6 (A b (3 a d+b c)+3 a B (a d+b c))}{m+7}+\frac{a x^4 (3 A b (a d+b c)+a B (a d+3 b c))}{m+5}+\frac{b^3 B d x^{10}}{m+11}\right)","\frac{a^3 A c (e x)^{m+1}}{e (m+1)}+\frac{a^2 (e x)^{m+3} (a A d+a B c+3 A b c)}{e^3 (m+3)}+\frac{b^2 (e x)^{m+9} (3 a B d+A b d+b B c)}{e^9 (m+9)}+\frac{b (e x)^{m+7} (A b (3 a d+b c)+3 a B (a d+b c))}{e^7 (m+7)}+\frac{a (e x)^{m+5} (3 A b (a d+b c)+a B (a d+3 b c))}{e^5 (m+5)}+\frac{b^3 B d (e x)^{m+11}}{e^{11} (m+11)}",1,"x*(e*x)^m*((a^3*A*c)/(1 + m) + (a^2*(3*A*b*c + a*B*c + a*A*d)*x^2)/(3 + m) + (a*(3*A*b*(b*c + a*d) + a*B*(3*b*c + a*d))*x^4)/(5 + m) + (b*(3*a*B*(b*c + a*d) + A*b*(b*c + 3*a*d))*x^6)/(7 + m) + (b^2*(b*B*c + A*b*d + 3*a*B*d)*x^8)/(9 + m) + (b^3*B*d*x^10)/(11 + m))","A",1
2,1,113,144,0.1667059,"\int (e x)^m \left(a+b x^2\right)^2 \left(A+B x^2\right) \left(c+d x^2\right) \, dx","Integrate[(e*x)^m*(a + b*x^2)^2*(A + B*x^2)*(c + d*x^2),x]","x (e x)^m \left(\frac{a^2 A c}{m+1}+\frac{b x^6 (2 a B d+A b d+b B c)}{m+7}+\frac{x^4 (A b (2 a d+b c)+a B (a d+2 b c))}{m+5}+\frac{a x^2 (a A d+a B c+2 A b c)}{m+3}+\frac{b^2 B d x^8}{m+9}\right)","\frac{a^2 A c (e x)^{m+1}}{e (m+1)}+\frac{b (e x)^{m+7} (2 a B d+A b d+b B c)}{e^7 (m+7)}+\frac{(e x)^{m+5} (A b (2 a d+b c)+a B (a d+2 b c))}{e^5 (m+5)}+\frac{a (e x)^{m+3} (a A d+a B c+2 A b c)}{e^3 (m+3)}+\frac{b^2 B d (e x)^{m+9}}{e^9 (m+9)}",1,"x*(e*x)^m*((a^2*A*c)/(1 + m) + (a*(2*A*b*c + a*B*c + a*A*d)*x^2)/(3 + m) + ((a*B*(2*b*c + a*d) + A*b*(b*c + 2*a*d))*x^4)/(5 + m) + (b*(b*B*c + A*b*d + 2*a*B*d)*x^6)/(7 + m) + (b^2*B*d*x^8)/(9 + m))","A",1
3,1,73,97,0.0780931,"\int (e x)^m \left(a+b x^2\right) \left(A+B x^2\right) \left(c+d x^2\right) \, dx","Integrate[(e*x)^m*(a + b*x^2)*(A + B*x^2)*(c + d*x^2),x]","x (e x)^m \left(\frac{x^4 (a B d+A b d+b B c)}{m+5}+\frac{x^2 (a A d+a B c+A b c)}{m+3}+\frac{a A c}{m+1}+\frac{b B d x^6}{m+7}\right)","\frac{(e x)^{m+5} (a B d+A b d+b B c)}{e^5 (m+5)}+\frac{(e x)^{m+3} (a A d+a B c+A b c)}{e^3 (m+3)}+\frac{a A c (e x)^{m+1}}{e (m+1)}+\frac{b B d (e x)^{m+7}}{e^7 (m+7)}",1,"x*(e*x)^m*((a*A*c)/(1 + m) + ((A*b*c + a*B*c + a*A*d)*x^2)/(3 + m) + ((b*B*c + A*b*d + a*B*d)*x^4)/(5 + m) + (b*B*d*x^6)/(7 + m))","A",1
4,1,43,60,0.0426775,"\int (e x)^m \left(A+B x^2\right) \left(c+d x^2\right) \, dx","Integrate[(e*x)^m*(A + B*x^2)*(c + d*x^2),x]","x (e x)^m \left(\frac{x^2 (A d+B c)}{m+3}+\frac{A c}{m+1}+\frac{B d x^4}{m+5}\right)","\frac{(e x)^{m+3} (A d+B c)}{e^3 (m+3)}+\frac{A c (e x)^{m+1}}{e (m+1)}+\frac{B d (e x)^{m+5}}{e^5 (m+5)}",1,"x*(e*x)^m*((A*c)/(1 + m) + ((B*c + A*d)*x^2)/(3 + m) + (B*d*x^4)/(5 + m))","A",1
5,1,93,118,0.1045271,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)}{a+b x^2} \, dx","Integrate[((e*x)^m*(A + B*x^2)*(c + d*x^2))/(a + b*x^2),x]","\frac{x (e x)^m \left(\frac{(a B-A b) (a d-b c) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1)}+\frac{-a B d+A b d+b B c}{m+1}+\frac{b B d x^2}{m+3}\right)}{b^2}","\frac{(e x)^{m+1} (A b-a B) (b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{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 (e x)^{m+3}}{b e^3 (m+3)}",1,"(x*(e*x)^m*((b*B*c + A*b*d - a*B*d)/(1 + m) + (b*B*d*x^2)/(3 + m) + ((-(A*b) + a*B)*(-(b*c) + a*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*(1 + m))))/b^2","A",1
6,1,108,171,0.1557541,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)}{\left(a+b x^2\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^2)*(c + d*x^2))/(a + b*x^2)^2,x]","\frac{x (e x)^m \left(a^2 B d+a \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (-2 a B d+A b d+b B c)+(A b-a B) (b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{a^2 b^2 (m+1)}","\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (A b (a d (m+1)+b (c-c m))+a B (b c (m+1)-a d (m+3)))}{2 a^2 b^2 e (m+1)}-\frac{d (e x)^{m+1} (A b (m+1)-a B (m+3))}{2 a b^2 e (m+1)}+\frac{\left(c+d x^2\right) (e x)^{m+1} (A b-a B)}{2 a b e \left(a+b x^2\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)/2, (3 + m)/2, -((b*x^2)/a)] + (A*b - a*B)*(b*c - a*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^2*b^2*(1 + m))","A",1
7,1,133,209,0.1647898,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)}{\left(a+b x^2\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^2)*(c + d*x^2))/(a + b*x^2)^3,x]","\frac{x (e x)^m \left(a^2 B d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+a \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (-2 a B d+A b d+b B c)+(A b-a B) (b c-a d) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{a^3 b^2 (m+1)}","\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (A b (1-m) (a d (m+1)+b c (3-m))+a B (m+1) (a d (m+3)+b (c-c m)))}{8 a^3 b^2 e (m+1)}-\frac{(e x)^{m+1} (A b (a d (1-m)-b c (3-m))-a B (b c (m+1)-a d (m+3)))}{8 a^2 b^2 e \left(a+b x^2\right)}+\frac{\left(c+d x^2\right) (e x)^{m+1} (A b-a B)}{4 a b e \left(a+b x^2\right)^2}",1,"(x*(e*x)^m*(a^2*B*d*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + a*(b*B*c + A*b*d - 2*a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + (A*b - a*B)*(b*c - a*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^3*b^2*(1 + m))","A",1
8,1,247,292,0.5871713,"\int (e x)^m \left(a+b x^2\right)^3 \left(A+B x^2\right) \left(c+d x^2\right)^2 \, dx","Integrate[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2)^2,x]","x (e x)^m \left(\frac{a^3 A c^2}{m+1}+\frac{x^6 \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+7}+\frac{a x^4 \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+5}+\frac{b x^8 \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+9}+\frac{a^2 c x^2 (2 a A d+a B c+3 A b c)}{m+3}+\frac{b^2 d x^{10} (3 a B d+A b d+2 b B c)}{m+11}+\frac{b^3 B d^2 x^{12}}{m+13}\right)","\frac{a^3 A c^2 (e x)^{m+1}}{e (m+1)}+\frac{(e x)^{m+7} \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)}{e^7 (m+7)}+\frac{a (e x)^{m+5} \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)}{e^5 (m+5)}+\frac{b (e x)^{m+9} \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)}{e^9 (m+9)}+\frac{a^2 c (e x)^{m+3} (2 a A d+a B c+3 A b c)}{e^3 (m+3)}+\frac{b^2 d (e x)^{m+11} (3 a B d+A b d+2 b B c)}{e^{11} (m+11)}+\frac{b^3 B d^2 (e x)^{m+13}}{e^{13} (m+13)}",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^2)/(3 + m) + (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^4)/(5 + m) + ((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^6)/(7 + m) + (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^8)/(9 + m) + (b^2*d*(2*b*B*c + A*b*d + 3*a*B*d)*x^10)/(11 + m) + (b^3*B*d^2*x^12)/(13 + m))","A",1
9,1,178,216,0.3543648,"\int (e x)^m \left(a+b x^2\right)^2 \left(A+B x^2\right) \left(c+d x^2\right)^2 \, dx","Integrate[(e*x)^m*(a + b*x^2)^2*(A + B*x^2)*(c + d*x^2)^2,x]","x (e x)^m \left(\frac{x^4 \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+5}+\frac{x^6 \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+7}+\frac{a^2 A c^2}{m+1}+\frac{b d x^8 (2 a B d+A b d+2 b B c)}{m+9}+\frac{a c x^2 (2 A (a d+b c)+a B c)}{m+3}+\frac{b^2 B d^2 x^{10}}{m+11}\right)","\frac{(e x)^{m+5} \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)}{e^5 (m+5)}+\frac{(e x)^{m+7} \left(a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right)}{e^7 (m+7)}+\frac{a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac{b d (e x)^{m+9} (2 a B d+A b d+2 b B c)}{e^9 (m+9)}+\frac{a c (e x)^{m+3} (2 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac{b^2 B d^2 (e x)^{m+11}}{e^{11} (m+11)}",1,"x*(e*x)^m*((a^2*A*c^2)/(1 + m) + (a*c*(a*B*c + 2*A*(b*c + a*d))*x^2)/(3 + m) + ((2*a*B*c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))*x^4)/(5 + m) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x^6)/(7 + m) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*x^8)/(9 + m) + (b^2*B*d^2*x^10)/(11 + m))","A",1
10,1,113,144,0.1725778,"\int (e x)^m \left(a+b x^2\right) \left(A+B x^2\right) \left(c+d x^2\right)^2 \, dx","Integrate[(e*x)^m*(a + b*x^2)*(A + B*x^2)*(c + d*x^2)^2,x]","x (e x)^m \left(\frac{d x^6 (a B d+A b d+2 b B c)}{m+7}+\frac{x^4 (a d (A d+2 B c)+b c (2 A d+B c))}{m+5}+\frac{c x^2 (2 a A d+a B c+A b c)}{m+3}+\frac{a A c^2}{m+1}+\frac{b B d^2 x^8}{m+9}\right)","\frac{d (e x)^{m+7} (a B d+A b d+2 b B c)}{e^7 (m+7)}+\frac{(e x)^{m+5} (a d (A d+2 B c)+b c (2 A d+B c))}{e^5 (m+5)}+\frac{c (e x)^{m+3} (2 a A d+a B c+A b c)}{e^3 (m+3)}+\frac{a A c^2 (e x)^{m+1}}{e (m+1)}+\frac{b B d^2 (e x)^{m+9}}{e^9 (m+9)}",1,"x*(e*x)^m*((a*A*c^2)/(1 + m) + (c*(A*b*c + a*B*c + 2*a*A*d)*x^2)/(3 + m) + ((a*d*(2*B*c + A*d) + b*c*(B*c + 2*A*d))*x^4)/(5 + m) + (d*(2*b*B*c + A*b*d + a*B*d)*x^6)/(7 + m) + (b*B*d^2*x^8)/(9 + m))","A",1
11,1,67,91,0.0535438,"\int (e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^2 \, dx","Integrate[(e*x)^m*(A + B*x^2)*(c + d*x^2)^2,x]","x (e x)^m \left(\frac{d x^4 (A d+2 B c)}{m+5}+\frac{c x^2 (2 A d+B c)}{m+3}+\frac{A c^2}{m+1}+\frac{B d^2 x^6}{m+7}\right)","\frac{d (e x)^{m+5} (A d+2 B c)}{e^5 (m+5)}+\frac{c (e x)^{m+3} (2 A d+B c)}{e^3 (m+3)}+\frac{A c^2 (e x)^{m+1}}{e (m+1)}+\frac{B d^2 (e x)^{m+7}}{e^7 (m+7)}",1,"x*(e*x)^m*((A*c^2)/(1 + m) + (c*(B*c + 2*A*d)*x^2)/(3 + m) + (d*(2*B*c + A*d)*x^4)/(5 + m) + (B*d^2*x^6)/(7 + m))","A",1
12,1,146,178,0.2635316,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^2}{a+b x^2} \, dx","Integrate[((e*x)^m*(A + B*x^2)*(c + d*x^2)^2)/(a + b*x^2),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}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1)}+\frac{b d x^2 (-a B d+A b d+2 b B c)}{m+3}+\frac{b^2 B d^2 x^4}{m+5}\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}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a b^3 e (m+1)}+\frac{d (e x)^{m+3} (-a B d+A b d+2 b B c)}{b^2 e^3 (m+3)}+\frac{B d^2 (e x)^{m+5}}{b e^5 (m+5)}",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^2)/(3 + m) + (b^2*B*d^2*x^4)/(5 + m) + ((A*b - a*B)*(b*c - a*d)^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*(1 + m))))/b^3","A",1
13,1,156,247,0.2303865,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^2}{\left(a+b x^2\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^2)*(c + d*x^2)^2)/(a + b*x^2)^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}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^2 (m+1)}+\frac{(b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (-3 a B d+2 A b d+b B c)}{a (m+1)}+\frac{d (-2 a B d+A b d+2 b B c)}{m+1}+\frac{b B d^2 x^2}{m+3}\right)}{b^3}","\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (A b (a d (m+3)+b (c-c m))+a B (b c (m+1)-a d (m+5)))}{2 a^2 b^3 e (m+1)}-\frac{d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+3))-a B (2 b c (m+3)-a d (m+5)))}{2 a b^3 e (m+1)}-\frac{d^2 (e x)^{m+3} (A b (m+3)-a B (m+5))}{2 a b^2 e^3 (m+3)}+\frac{\left(c+d x^2\right)^2 (e x)^{m+1} (A b-a B)}{2 a b e \left(a+b x^2\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^2)/(3 + m) + ((b*c - a*d)*(b*B*c + 2*A*b*d - 3*a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*(1 + m)) + ((A*b - a*B)*(b*c - a*d)^2*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a^2*(1 + m))))/b^3","A",1
14,1,165,292,0.2088782,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^2}{\left(a+b x^2\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^2)*(c + d*x^2)^2)/(a + b*x^2)^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}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^3}+\frac{(b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (-3 a B d+2 A b d+b B c)}{a^2}+\frac{d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (-3 a B d+A b d+2 b B c)}{a}+B d^2\right)}{b^3 (m+1)}","-\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (a d (A b (m+1)-a B (m+5)) (b c (m+1)-a d (m+3))-b c (a B (m+1)+A b (3-m)) (a d (m+1)+b (c-c m)))}{8 a^3 b^3 e (m+1)}+\frac{d (e x)^{m+1} (A b (m+1)-a B (m+5)) (b c (m+1)-a d (m+3))}{8 a^2 b^3 e (m+1)}+\frac{(e x)^{m+1} (b c-a d) \left(c (a B (m+1)+A b (3-m))-d x^2 (A b (m+1)-a B (m+5))\right)}{8 a^2 b^2 e \left(a+b x^2\right)}+\frac{\left(c+d x^2\right)^2 (e x)^{m+1} (A b-a B)}{4 a b e \left(a+b x^2\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)/2, (3 + m)/2, -((b*x^2)/a)])/a + ((b*c - a*d)*(b*B*c + 2*A*b*d - 3*a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^2 + ((A*b - a*B)*(b*c - a*d)^2*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^3))/(b^3*(1 + m))","A",1
15,1,327,379,0.8254462,"\int (e x)^m \left(a+b x^2\right)^3 \left(A+B x^2\right) \left(c+d x^2\right)^3 \, dx","Integrate[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2)^3,x]","x (e x)^m \left(\frac{a^3 A c^3}{m+1}+\frac{3 a c x^4 \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+5}+\frac{3 b d x^{10} \left(a^2 B d^2+a b d (A d+3 B c)+b^2 c (A d+B c)\right)}{m+11}+\frac{a^2 c^2 x^2 (3 A (a d+b c)+a B c)}{m+3}+\frac{x^8 \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+9}+\frac{x^6 \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+7}+\frac{b^2 d^2 x^{12} (3 a B d+A b d+3 b B c)}{m+13}+\frac{b^3 B d^3 x^{14}}{m+15}\right)","\frac{a^3 A c^3 (e x)^{m+1}}{e (m+1)}+\frac{3 a c (e x)^{m+5} \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)}{e^5 (m+5)}+\frac{3 b d (e x)^{m+11} \left(a^2 B d^2+a b d (A d+3 B c)+b^2 c (A d+B c)\right)}{e^{11} (m+11)}+\frac{a^2 c^2 (e x)^{m+3} (3 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac{(e x)^{m+9} \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)}{e^9 (m+9)}+\frac{(e x)^{m+7} \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)}{e^7 (m+7)}+\frac{b^2 d^2 (e x)^{m+13} (3 a B d+A b d+3 b B c)}{e^{13} (m+13)}+\frac{b^3 B d^3 (e x)^{m+15}}{e^{15} (m+15)}",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^2)/(3 + m) + (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^4)/(5 + m) + ((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^6)/(7 + m) + ((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^8)/(9 + m) + (3*b*d*(a^2*B*d^2 + b^2*c*(B*c + A*d) + a*b*d*(3*B*c + A*d))*x^10)/(11 + m) + (b^2*d^2*(3*b*B*c + A*b*d + 3*a*B*d)*x^12)/(13 + m) + (b^3*B*d^3*x^14)/(15 + m))","A",1
16,1,239,284,0.4152743,"\int (e x)^m \left(a+b x^2\right)^2 \left(A+B x^2\right) \left(c+d x^2\right)^3 \, dx","Integrate[(e*x)^m*(a + b*x^2)^2*(A + B*x^2)*(c + d*x^2)^3,x]","x (e x)^m \left(\frac{x^6 \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+7}+\frac{c x^4 \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+5}+\frac{d x^8 \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+9}+\frac{a^2 A c^3}{m+1}+\frac{a c^2 x^2 (3 a A d+a B c+2 A b c)}{m+3}+\frac{b d^2 x^{10} (2 a B d+A b d+3 b B c)}{m+11}+\frac{b^2 B d^3 x^{12}}{m+13}\right)","\frac{(e x)^{m+7} \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)}{e^7 (m+7)}+\frac{c (e x)^{m+5} \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)}{e^5 (m+5)}+\frac{d (e x)^{m+9} \left(a^2 B d^2+2 a b d (A d+3 B c)+3 b^2 c (A d+B c)\right)}{e^9 (m+9)}+\frac{a^2 A c^3 (e x)^{m+1}}{e (m+1)}+\frac{a c^2 (e x)^{m+3} (3 a A d+a B c+2 A b c)}{e^3 (m+3)}+\frac{b d^2 (e x)^{m+11} (2 a B d+A b d+3 b B c)}{e^{11} (m+11)}+\frac{b^2 B d^3 (e x)^{m+13}}{e^{13} (m+13)}",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^2)/(3 + m) + (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^4)/(5 + m) + ((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^6)/(7 + m) + (d*(a^2*B*d^2 + 3*b^2*c*(B*c + A*d) + 2*a*b*d*(3*B*c + A*d))*x^8)/(9 + m) + (b*d^2*(3*b*B*c + A*b*d + 2*a*B*d)*x^10)/(11 + m) + (b^2*B*d^3*x^12)/(13 + m))","A",1
17,1,151,189,0.2790095,"\int (e x)^m \left(a+b x^2\right) \left(A+B x^2\right) \left(c+d x^2\right)^3 \, dx","Integrate[(e*x)^m*(a + b*x^2)*(A + B*x^2)*(c + d*x^2)^3,x]","x (e x)^m \left(\frac{c^2 x^2 (3 a A d+a B c+A b c)}{m+3}+\frac{d^2 x^8 (a B d+A b d+3 b B c)}{m+9}+\frac{d x^6 (a d (A d+3 B c)+3 b c (A d+B c))}{m+7}+\frac{c x^4 (3 a d (A d+B c)+b c (3 A d+B c))}{m+5}+\frac{a A c^3}{m+1}+\frac{b B d^3 x^{10}}{m+11}\right)","\frac{c^2 (e x)^{m+3} (3 a A d+a B c+A b c)}{e^3 (m+3)}+\frac{d^2 (e x)^{m+9} (a B d+A b d+3 b B c)}{e^9 (m+9)}+\frac{d (e x)^{m+7} (a d (A d+3 B c)+3 b c (A d+B c))}{e^7 (m+7)}+\frac{c (e x)^{m+5} (3 a d (A d+B c)+b c (3 A d+B c))}{e^5 (m+5)}+\frac{a A c^3 (e x)^{m+1}}{e (m+1)}+\frac{b B d^3 (e x)^{m+11}}{e^{11} (m+11)}",1,"x*(e*x)^m*((a*A*c^3)/(1 + m) + (c^2*(A*b*c + a*B*c + 3*a*A*d)*x^2)/(3 + m) + (c*(3*a*d*(B*c + A*d) + b*c*(B*c + 3*A*d))*x^4)/(5 + m) + (d*(3*b*c*(B*c + A*d) + a*d*(3*B*c + A*d))*x^6)/(7 + m) + (d^2*(3*b*B*c + A*b*d + a*B*d)*x^8)/(9 + m) + (b*B*d^3*x^10)/(11 + m))","A",1
18,1,90,121,0.0786461,"\int (e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^3 \, dx","Integrate[(e*x)^m*(A + B*x^2)*(c + d*x^2)^3,x]","x (e x)^m \left(\frac{c^2 x^2 (3 A d+B c)}{m+3}+\frac{d^2 x^6 (A d+3 B c)}{m+7}+\frac{3 c d x^4 (A d+B c)}{m+5}+\frac{A c^3}{m+1}+\frac{B d^3 x^8}{m+9}\right)","\frac{c^2 (e x)^{m+3} (3 A d+B c)}{e^3 (m+3)}+\frac{d^2 (e x)^{m+7} (A d+3 B c)}{e^7 (m+7)}+\frac{3 c d (e x)^{m+5} (A d+B c)}{e^5 (m+5)}+\frac{A c^3 (e x)^{m+1}}{e (m+1)}+\frac{B d^3 (e x)^{m+9}}{e^9 (m+9)}",1,"x*(e*x)^m*((A*c^3)/(1 + m) + (c^2*(B*c + 3*A*d)*x^2)/(3 + m) + (3*c*d*(B*c + A*d)*x^4)/(5 + m) + (d^2*(3*B*c + A*d)*x^6)/(7 + m) + (B*d^3*x^8)/(9 + m))","A",1
19,1,217,258,0.4922953,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^3}{a+b x^2} \, dx","Integrate[((e*x)^m*(A + B*x^2)*(c + d*x^2)^3)/(a + b*x^2),x]","\frac{x (e x)^m \left(\frac{b d x^2 \left(a^2 B d^2-a b d (A d+3 B c)+3 b^2 c (A d+B c)\right)}{m+3}+\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^4 (-a B d+A b d+3 b B c)}{m+5}+\frac{(a B-A b) (a d-b c)^3 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1)}+\frac{b^3 B d^3 x^6}{m+7}\right)}{b^4}","\frac{d (e x)^{m+3} \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 e^3 (m+3)}-\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}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a b^4 e (m+1)}+\frac{d^2 (e x)^{m+5} (-a B d+A b d+3 b B c)}{b^2 e^5 (m+5)}+\frac{B d^3 (e x)^{m+7}}{b e^7 (m+7)}",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^2)/(3 + m) + (b^2*d^2*(3*b*B*c + A*b*d - a*B*d)*x^4)/(5 + m) + (b^3*B*d^3*x^6)/(7 + m) + ((-(A*b) + a*B)*(-(b*c) + a*d)^3*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*(1 + m))))/b^4","A",1
20,1,209,347,0.4339509,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^3}{\left(a+b x^2\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^2)*(c + d*x^2)^3)/(a + b*x^2)^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}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^2 (m+1)}+\frac{b d^2 x^2 (-2 a B d+A b d+3 b B c)}{m+3}+\frac{(b c-a d)^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (-4 a B d+3 A b d+b B c)}{a (m+1)}+\frac{b^2 B d^3 x^4}{m+5}\right)}{b^4}","\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (A b (a d (m+5)+b (c-c m))+a B (b c (m+1)-a d (m+7)))}{2 a^2 b^4 e (m+1)}-\frac{d (e x)^{m+1} \left(A b \left(a^2 d^2 (m+5)-3 a b c d (m+3)+3 b^2 c^2 (m+1)\right)-a B \left(a^2 d^2 (m+7)-3 a b c d (m+5)+3 b^2 c^2 (m+3)\right)\right)}{2 a b^4 e (m+1)}-\frac{d^2 (e x)^{m+3} (A b (3 b c (m+3)-a d (m+5))-a B (3 b c (m+5)-a d (m+7)))}{2 a b^3 e^3 (m+3)}-\frac{d^3 (e x)^{m+5} (A b (m+5)-a B (m+7))}{2 a b^2 e^5 (m+5)}+\frac{\left(c+d x^2\right)^3 (e x)^{m+1} (A b-a B)}{2 a b e \left(a+b x^2\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^2)/(3 + m) + (b^2*B*d^3*x^4)/(5 + m) + ((b*c - a*d)^2*(b*B*c + 3*A*b*d - 4*a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*(1 + m)) + ((-(A*b) + a*B)*(-(b*c) + a*d)^3*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a^2*(1 + m))))/b^4","A",1
21,1,218,480,0.3906733,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^3}{\left(a+b x^2\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^2)*(c + d*x^2)^3)/(a + b*x^2)^3,x]","\frac{x (e x)^m \left(\frac{(a B-A b) (a d-b c)^3 \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^3 (m+1)}+\frac{(b c-a d)^2 \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (-4 a B d+3 A b d+b B c)}{a^2 (m+1)}+\frac{d^2 (-3 a B d+A b d+3 b B c)}{m+1}+\frac{3 d (b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (-2 a B d+A b d+b B c)}{a (m+1)}+\frac{b B d^3 x^2}{m+3}\right)}{b^4}","-\frac{d^2 (e x)^{m+3} \left(A b (m+3) (a d (m+5)+b c (3-m))+a B \left(b c \left(m^2+4 m+3\right)-a d \left(m^2+12 m+35\right)\right)\right)}{8 a^2 b^3 e^3 (m+3)}+\frac{\left(c+d x^2\right)^2 (e x)^{m+1} (A b (a d (m+3)+b c (3-m))+a B (b c (m+1)-a d (m+7)))}{8 a^2 b^2 e \left(a+b x^2\right)}-\frac{d (e x)^{m+1} \left(A b \left(-a^2 d^2 \left(m^2+8 m+15\right)+3 a b c d \left(m^2+4 m+3\right)+2 b^2 c^2 \left(-m^2+2 m+3\right)\right)+a B \left(a^2 d^2 \left(m^2+12 m+35\right)-3 a b c d \left(m^2+8 m+15\right)+2 b^2 c^2 (m+1)^2\right)\right)}{8 a^2 b^4 e (m+1)}+\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) \left(A b \left(a^2 d^2 \left(m^2+8 m+15\right)+2 a b c d \left(-m^2-2 m+3\right)+b^2 c^2 \left(m^2-4 m+3\right)\right)+a B \left(-a^2 d^2 \left(m^2+12 m+35\right)+2 a b c d \left(m^2+6 m+5\right)+b^2 c^2 \left(1-m^2\right)\right)\right)}{8 a^3 b^4 e (m+1)}+\frac{\left(c+d x^2\right)^3 (e x)^{m+1} (A b-a B)}{4 a b e \left(a+b x^2\right)^2}",1,"(x*(e*x)^m*((d^2*(3*b*B*c + A*b*d - 3*a*B*d))/(1 + m) + (b*B*d^3*x^2)/(3 + m) + (3*d*(b*c - a*d)*(b*B*c + A*b*d - 2*a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*(1 + m)) + ((b*c - a*d)^2*(b*B*c + 3*A*b*d - 4*a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a^2*(1 + m)) + ((-(A*b) + a*B)*(-(b*c) + a*d)^3*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a^3*(1 + m))))/b^4","A",1
22,1,315,363,0.5176872,"\int \frac{(e x)^m \left(a+b x^2\right)^4 \left(A+B x^2\right)}{c+d x^2} \, dx","Integrate[((e*x)^m*(a + b*x^2)^4*(A + B*x^2))/(c + d*x^2),x]","\frac{x (e x)^m \left(\frac{b^2 d^2 x^4 \left(6 a^2 B d^2+4 a b d (A d-B c)+b^2 c (B c-A d)\right)}{m+5}+\frac{b d x^2 \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+3}+\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^6 (4 a B d+A b d-b B c)}{m+7}-\frac{(b c-a d)^4 (B c-A d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c (m+1)}+\frac{b^4 B d^4 x^8}{m+9}\right)}{d^5}","\frac{b^2 (e x)^{m+5} \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 e^5 (m+5)}+\frac{b (e x)^{m+3} \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 e^3 (m+3)}+\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 (e x)^{m+7} (-4 a B d-A b d+b B c)}{d^2 e^7 (m+7)}-\frac{(e x)^{m+1} (b c-a d)^4 (B c-A d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c d^5 e (m+1)}+\frac{b^4 B (e x)^{m+9}}{d e^9 (m+9)}",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^2)/(3 + m) + (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^4)/(5 + m) + (b^3*d^3*(-(b*B*c) + A*b*d + 4*a*B*d)*x^6)/(7 + m) + (b^4*B*d^4*x^8)/(9 + m) - ((b*c - a*d)^4*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(1 + m))))/d^5","A",1
23,1,219,260,0.3480893,"\int \frac{(e x)^m \left(a+b x^2\right)^3 \left(A+B x^2\right)}{c+d x^2} \, dx","Integrate[((e*x)^m*(a + b*x^2)^3*(A + B*x^2))/(c + d*x^2),x]","\frac{x (e x)^m \left(\frac{b d x^2 \left(3 a^2 B d^2+3 a b d (A d-B c)+b^2 c (B c-A d)\right)}{m+3}+\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^4 (3 a B d+A b d-b B c)}{m+5}+\frac{(b c-a d)^3 (B c-A d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c (m+1)}+\frac{b^3 B d^3 x^6}{m+7}\right)}{d^4}","\frac{b (e x)^{m+3} \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 e^3 (m+3)}+\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 (e x)^{m+5} (-3 a B d-A b d+b B c)}{d^2 e^5 (m+5)}+\frac{(e x)^{m+1} (b c-a d)^3 (B c-A d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c d^4 e (m+1)}+\frac{b^3 B (e x)^{m+7}}{d e^7 (m+7)}",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^2)/(3 + m) + (b^2*d^2*(-(b*B*c) + A*b*d + 3*a*B*d)*x^4)/(5 + m) + (b^3*B*d^3*x^6)/(7 + m) + ((b*c - a*d)^3*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(1 + m))))/d^4","A",1
24,1,147,180,0.2183828,"\int \frac{(e x)^m \left(a+b x^2\right)^2 \left(A+B x^2\right)}{c+d x^2} \, dx","Integrate[((e*x)^m*(a + b*x^2)^2*(A + B*x^2))/(c + d*x^2),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}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c (m+1)}+\frac{b d x^2 (2 a B d+A b d-b B c)}{m+3}+\frac{b^2 B d^2 x^4}{m+5}\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}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c d^3 e (m+1)}-\frac{b (e x)^{m+3} (-2 a B d-A b d+b B c)}{d^2 e^3 (m+3)}+\frac{b^2 B (e x)^{m+5}}{d e^5 (m+5)}",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^2)/(3 + m) + (b^2*B*d^2*x^4)/(5 + m) - ((b*c - a*d)^2*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(1 + m))))/d^3","A",1
25,1,93,120,0.1019854,"\int \frac{(e x)^m \left(a+b x^2\right) \left(A+B x^2\right)}{c+d x^2} \, dx","Integrate[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2),x]","\frac{x (e x)^m \left(\frac{(b c-a d) (B c-A d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c (m+1)}+\frac{a B d+A b d-b B c}{m+1}+\frac{b B d x^2}{m+3}\right)}{d^2}","\frac{(e x)^{m+1} (b c-a d) (B c-A d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{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 (e x)^{m+3}}{d e^3 (m+3)}",1,"(x*(e*x)^m*((-(b*B*c) + A*b*d + a*B*d)/(1 + m) + (b*B*d*x^2)/(3 + m) + ((b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(1 + m))))/d^2","A",1
26,1,56,77,0.0647615,"\int \frac{(e x)^m \left(A+B x^2\right)}{c+d x^2} \, dx","Integrate[((e*x)^m*(A + B*x^2))/(c + d*x^2),x]","\frac{x (e x)^m \left((A d-B c) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{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}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c d e (m+1)}",1,"(x*(e*x)^m*(B*c + (-(B*c) + A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)]))/(c*d*(1 + m))","A",1
27,1,100,125,0.1363152,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Integrate[((e*x)^m*(A + B*x^2))/((a + b*x^2)*(c + d*x^2)),x]","\frac{x (e x)^m \left((a B c-A b c) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+a (A d-B c) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{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}{2};\frac{m+3}{2};-\frac{b x^2}{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}{2};\frac{m+3}{2};-\frac{d x^2}{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)/2, (3 + m)/2, -((b*x^2)/a)] + a*(-(B*c) + A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)]))/(a*c*(-(b*c) + a*d)*(1 + m))","A",1
28,1,149,206,0.2038726,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(a+b x^2\right)^2 \left(c+d x^2\right)} \, dx","Integrate[((e*x)^m*(A + B*x^2))/((a + b*x^2)^2*(c + d*x^2)),x]","-\frac{x (e x)^m \left(a^2 d (B c-A d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+a b c (A d-B c) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)-c (A b-a B) (b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{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}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (A b (b c (1-m)-a d (3-m))+a B (a d (1-m)+b c (m+1)))}{2 a^2 e (m+1) (b c-a d)^2}-\frac{d (e x)^{m+1} (B c-A d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{2 a e \left(a+b x^2\right) (b c-a d)}",1,"-((x*(e*x)^m*(a*b*c*(-(B*c) + A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + a^2*d*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] - (A*b - a*B)*c*(b*c - a*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^2*c*(b*c - a*d)^2*(1 + m)))","A",1
29,1,195,342,0.2747781,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(a+b x^2\right)^3 \left(c+d x^2\right)} \, dx","Integrate[((e*x)^m*(A + B*x^2))/((a + b*x^2)^3*(c + d*x^2)),x]","\frac{x (e x)^m \left(\frac{(A b-a B) (b c-a d)^2 \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^3}+\frac{b (b c-a d) (B c-A d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^2}+\frac{b d (A d-B c) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a}+\frac{d^2 (B c-A d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c}\right)}{(m+1) (b c-a d)^3}","\frac{(e x)^{m+1} (A b (b c (3-m)-a d (7-m))+a B (a d (3-m)+b c (m+1)))}{8 a^2 e \left(a+b x^2\right) (b c-a d)^2}+\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) \left(A b \left(a^2 d^2 \left(m^2-8 m+15\right)-2 a b c d \left(m^2-6 m+5\right)+b^2 c^2 \left(m^2-4 m+3\right)\right)+a B \left(-a^2 d^2 \left(m^2-4 m+3\right)-2 a b c d \left(-m^2+2 m+3\right)+b^2 c^2 \left(1-m^2\right)\right)\right)}{8 a^3 e (m+1) (b c-a d)^3}+\frac{d^2 (e x)^{m+1} (B c-A d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c e (m+1) (b c-a d)^3}+\frac{(e x)^{m+1} (A b-a B)}{4 a e \left(a+b x^2\right)^2 (b c-a d)}",1,"(x*(e*x)^m*((b*d*(-(B*c) + A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a + (d^2*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c + (b*(b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^2 + ((A*b - a*B)*(b*c - a*d)^2*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^3))/((b*c - a*d)^3*(1 + m))","A",1
30,1,212,340,0.4757378,"\int \frac{(e x)^m \left(a+b x^2\right)^3 \left(A+B x^2\right)}{\left(c+d x^2\right)^2} \, dx","Integrate[((e*x)^m*(a + b*x^2)^3*(A + B*x^2))/(c + d*x^2)^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^2 (3 a B d+A b d-2 b B c)}{m+3}+\frac{(b c-a d)^3 (B c-A d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c^2 (m+1)}-\frac{(b c-a d)^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (-a B d-3 A b d+4 b B c)}{c (m+1)}+\frac{b^3 B d^2 x^4}{m+5}\right)}{d^4}","-\frac{b (e x)^{m+1} \left(3 a^2 d^2 (A d (m+1)-B c (m+3))-3 a b c d (A d (m+3)-B c (m+5))+b^2 c^2 (A d (m+5)-B c (m+7))\right)}{2 c d^4 e (m+1)}-\frac{b^2 (e x)^{m+3} (3 a d (A d (m+3)-B c (m+5))-b c (A d (m+5)-B c (m+7)))}{2 c d^3 e^3 (m+3)}+\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a d (A d (1-m)+B c (m+1))+b c (A d (m+5)-B c (m+7)))}{2 c^2 d^4 e (m+1)}-\frac{\left(a+b x^2\right)^3 (e x)^{m+1} (B c-A d)}{2 c d e \left(c+d x^2\right)}-\frac{b^3 (e x)^{m+5} (A d (m+5)-B c (m+7))}{2 c d^2 e^5 (m+5)}",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^2)/(3 + m) + (b^3*B*d^2*x^4)/(5 + m) - ((b*c - a*d)^2*(4*b*B*c - 3*A*b*d - a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(1 + m)) + ((b*c - a*d)^3*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c^2*(1 + m))))/d^4","A",1
31,1,158,246,0.3100643,"\int \frac{(e x)^m \left(a+b x^2\right)^2 \left(A+B x^2\right)}{\left(c+d x^2\right)^2} \, dx","Integrate[((e*x)^m*(a + b*x^2)^2*(A + B*x^2))/(c + d*x^2)^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}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c^2 (m+1)}+\frac{(b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (-a B d-2 A b d+3 b B c)}{c (m+1)}+\frac{b (2 a B d+A b d-2 b B c)}{m+1}+\frac{b^2 B d x^2}{m+3}\right)}{d^3}","-\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a d (A d (1-m)+B c (m+1))+b c (A d (m+3)-B c (m+5)))}{2 c^2 d^3 e (m+1)}-\frac{b (e x)^{m+1} (2 a d (A d (m+1)-B c (m+3))-b c (A d (m+3)-B c (m+5)))}{2 c d^3 e (m+1)}-\frac{\left(a+b x^2\right)^2 (e x)^{m+1} (B c-A d)}{2 c d e \left(c+d x^2\right)}-\frac{b^2 (e x)^{m+3} (A d (m+3)-B c (m+5))}{2 c d^2 e^3 (m+3)}",1,"(x*(e*x)^m*((b*(-2*b*B*c + A*b*d + 2*a*B*d))/(1 + m) + (b^2*B*d*x^2)/(3 + m) + ((b*c - a*d)*(3*b*B*c - 2*A*b*d - a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(1 + m)) - ((b*c - a*d)^2*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c^2*(1 + m))))/d^3","A",1
32,1,108,171,0.1749763,"\int \frac{(e x)^m \left(a+b x^2\right) \left(A+B x^2\right)}{\left(c+d x^2\right)^2} \, dx","Integrate[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2)^2,x]","\frac{x (e x)^m \left(c \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a B d+A b d-2 b B c)+(b c-a d) (B c-A d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{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}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a d (A d (1-m)+B c (m+1))+b c (A d (m+1)-B c (m+3)))}{2 c^2 d^2 e (m+1)}-\frac{\left(A+B x^2\right) (e x)^{m+1} (b c-a d)}{2 c d e \left(c+d x^2\right)}-\frac{B (e x)^{m+1} (a d (m+1)-b c (m+3))}{2 c d^2 e (m+1)}",1,"(x*(e*x)^m*(b*B*c^2 + c*(-2*b*B*c + A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + (b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)]))/(c^2*d^2*(1 + m))","A",1
33,1,81,103,0.082265,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(c+d x^2\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^2))/(c + d*x^2)^2,x]","\frac{x (e x)^m \left((A d-B c) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+B c \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)\right)}{c^2 d (m+1)}","\frac{(e x)^{m+1} (A d (1-m)+B c (m+1)) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{2 c^2 d e (m+1)}-\frac{(e x)^{m+1} (B c-A d)}{2 c d e \left(c+d x^2\right)}",1,"(x*(e*x)^m*(B*c*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + (-(B*c) + A*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)]))/(c^2*d*(1 + m))","A",1
34,1,147,205,0.1708075,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(a+b x^2\right) \left(c+d x^2\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^2))/((a + b*x^2)*(c + d*x^2)^2),x]","\frac{x (e x)^m \left(b c^2 (A b-a B) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)+a c d (a B-A b) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+a (b c-a d) (B c-A d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{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}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a d (A d (1-m)+B c (m+1))+b c (B c (1-m)-A d (3-m)))}{2 c^2 e (m+1) (b c-a d)^2}+\frac{b (e x)^{m+1} (A b-a B) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} (B c-A d)}{2 c e \left(c+d x^2\right) (b c-a d)}",1,"(x*(e*x)^m*(b*(A*b - a*B)*c^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + a*(-(A*b) + a*B)*c*d*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + a*(b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)]))/(a*c^2*(b*c - a*d)^2*(1 + m))","A",1
35,1,207,304,0.2676738,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^2))/((a + b*x^2)^2*(c + d*x^2)^2),x]","\frac{x (e x)^m \left(-(b c-a d) \left(a^2 d (A d-B c) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+b c^2 (A b-a B) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)+a^2 c d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a B d-2 A b d+b B c)-a b c^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (a B d-2 A b d+b B c)\right)}{a^2 c^2 (m+1) (a d-b c)^3}","\frac{b (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (A b (b c (1-m)-a d (5-m))+a B (a d (3-m)+b c (m+1)))}{2 a^2 e (m+1) (b c-a d)^3}-\frac{d (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a d (A d (1-m)+B c (m+1))+b c (B c (3-m)-A d (5-m)))}{2 c^2 e (m+1) (b c-a d)^3}+\frac{d (e x)^{m+1} (a A d-2 a B c+A b c)}{2 a c e \left(c+d x^2\right) (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{2 a e \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)}",1,"(x*(e*x)^m*(-(a*b*c^2*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]) + a^2*c*d*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] - (b*c - a*d)*(b*(A*b - a*B)*c^2*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + a^2*d*(-(B*c) + A*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])))/(a^2*c^2*(-(b*c) + a*d)^3*(1 + m))","A",1
36,1,265,491,0.3617621,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(a+b x^2\right)^3 \left(c+d x^2\right)^2} \, dx","Integrate[((e*x)^m*(A + B*x^2))/((a + b*x^2)^3*(c + d*x^2)^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}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^3}+\frac{b (b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (a B d-2 A b d+b B c)}{a^2}+\frac{d^2 (b c-a d) (B c-A d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c^2}+\frac{d^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a B d-3 A b d+2 b B c)}{c}-\frac{b d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (a B d-3 A b d+2 b B c)}{a}\right)}{(m+1) (b c-a d)^4}","-\frac{d (e x)^{m+1} \left(A \left(4 a^2 d^2+a b c d (11-m)-b^2 c^2 (3-m)\right)-a B c (a d (11-m)+b c (m+1))\right)}{8 a^2 c e \left(c+d x^2\right) (b c-a d)^3}+\frac{(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{8 a^2 e \left(a+b x^2\right) \left(c+d x^2\right) (b c-a d)^2}+\frac{b (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) \left(A b \left(a^2 d^2 \left(m^2-12 m+35\right)-2 a b c d \left(m^2-8 m+7\right)+b^2 c^2 \left(m^2-4 m+3\right)\right)+a B \left(-a^2 d^2 \left(m^2-8 m+15\right)-2 a b c d \left(-m^2+4 m+5\right)+b^2 c^2 \left(1-m^2\right)\right)\right)}{8 a^3 e (m+1) (b c-a d)^4}+\frac{d^2 (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a d (A d (1-m)+B c (m+1))+b c (B c (5-m)-A d (7-m)))}{2 c^2 e (m+1) (b c-a d)^4}+\frac{(e x)^{m+1} (A b-a B)}{4 a e \left(a+b x^2\right)^2 \left(c+d x^2\right) (b c-a d)}",1,"(x*(e*x)^m*(-((b*d*(2*b*B*c - 3*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a) + (d^2*(2*b*B*c - 3*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c + (b*(b*c - a*d)*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^2 + (d^2*(b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c^2 + (b*(A*b - a*B)*(b*c - a*d)^2*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^3))/((b*c - a*d)^4*(1 + m))","A",1
37,1,222,433,0.3511615,"\int \frac{(e x)^m \left(a+b x^2\right)^3 \left(A+B x^2\right)}{\left(c+d x^2\right)^3} \, dx","Integrate[((e*x)^m*(a + b*x^2)^3*(A + B*x^2))/(c + d*x^2)^3,x]","\frac{x (e x)^m \left(\frac{b^2 (3 a B d+A b d-3 b B c)}{m+1}+\frac{(b c-a d)^3 (B c-A d) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c^3 (m+1)}-\frac{(b c-a d)^2 \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (-a B d-3 A b d+4 b B c)}{c^2 (m+1)}+\frac{3 b (b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (-a B d-A b d+2 b B c)}{c (m+1)}+\frac{b^3 B d x^2}{m+3}\right)}{d^4}","-\frac{b (e x)^{m+1} \left(2 a^2 d^2 (m+1) (A d (3-m)+B c (m+1))+3 a b c d (m+3) (A d (m+1)-B c (m+5))-b^2 c^2 (m+5) (A d (m+3)-B c (m+7))\right)}{8 c^2 d^4 e (m+1)}-\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) \left(a^2 d^2 (1-m) (A d (3-m)+B c (m+1))+2 a b c d \left(A d \left(-m^2-2 m+3\right)+B c \left(m^2+6 m+5\right)\right)+b^2 c^2 (m+5) (A d (m+3)-B c (m+7))\right)}{8 c^3 d^4 e (m+1)}-\frac{b^2 (e x)^{m+3} (a d (m+3) (A d (3-m)+B c (m+1))+b c (m+5) (A d (m+3)-B c (m+7)))}{8 c^2 d^3 e^3 (m+3)}+\frac{\left(a+b x^2\right)^2 (e x)^{m+1} (a d (A d (3-m)+B c (m+1))+b c (A d (m+3)-B c (m+7)))}{8 c^2 d^2 e \left(c+d x^2\right)}-\frac{\left(a+b x^2\right)^3 (e x)^{m+1} (B c-A d)}{4 c d e \left(c+d x^2\right)^2}",1,"(x*(e*x)^m*((b^2*(-3*b*B*c + A*b*d + 3*a*B*d))/(1 + m) + (b^3*B*d*x^2)/(3 + m) + (3*b*(b*c - a*d)*(2*b*B*c - A*b*d - a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(1 + m)) - ((b*c - a*d)^2*(4*b*B*c - 3*A*b*d - a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c^2*(1 + m)) + ((b*c - a*d)^3*(B*c - A*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c^3*(1 + m))))/d^4","A",1
38,1,169,292,0.2010162,"\int \frac{(e x)^m \left(a+b x^2\right)^2 \left(A+B x^2\right)}{\left(c+d x^2\right)^3} \, dx","Integrate[((e*x)^m*(a + b*x^2)^2*(A + B*x^2))/(c + d*x^2)^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}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c^3}+\frac{(b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (-a B d-2 A b d+3 b B c)}{c^2}-\frac{b \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (-2 a B d-A b d+3 b B c)}{c}+b^2 B\right)}{d^3 (m+1)}","\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a d (a d (1-m)+b c (m+1)) (A d (3-m)+B c (m+1))-b c (a d (m+1)-b c (m+3)) (A d (m+1)-B c (m+5)))}{8 c^3 d^3 e (m+1)}+\frac{b (e x)^{m+1} (a d (m+1)-b c (m+3)) (A d (m+1)-B c (m+5))}{8 c^2 d^3 e (m+1)}-\frac{(e x)^{m+1} (b c-a d) \left(a (A d (3-m)+B c (m+1))-b x^2 (A d (m+1)-B c (m+5))\right)}{8 c^2 d^2 e \left(c+d x^2\right)}-\frac{\left(a+b x^2\right)^2 (e x)^{m+1} (B c-A d)}{4 c d e \left(c+d x^2\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)/2, (3 + m)/2, -((d*x^2)/c)])/c + ((b*c - a*d)*(3*b*B*c - 2*A*b*d - a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c^2 - ((b*c - a*d)^2*(B*c - A*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c^3))/(d^3*(1 + m))","A",1
39,1,133,208,0.1417604,"\int \frac{(e x)^m \left(a+b x^2\right) \left(A+B x^2\right)}{\left(c+d x^2\right)^3} \, dx","Integrate[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2)^3,x]","\frac{x (e x)^m \left(c \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a B d+A b d-2 b B c)+(b c-a d) (B c-A d) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+b B c^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)\right)}{c^3 d^2 (m+1)}","\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a d (1-m) (A d (3-m)+B c (m+1))+b c (m+1) (A d (1-m)+B c (m+3)))}{8 c^3 d^2 e (m+1)}+\frac{(e x)^{m+1} (a d (A d (3-m)-B (c-c m))+b c (A d (m+1)-B c (m+3)))}{8 c^2 d^2 e \left(c+d x^2\right)}-\frac{\left(A+B x^2\right) (e x)^{m+1} (b c-a d)}{4 c d e \left(c+d x^2\right)^2}",1,"(x*(e*x)^m*(b*B*c^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + c*(-2*b*B*c + A*b*d + a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + (b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)]))/(c^3*d^2*(1 + m))","A",1
40,1,81,103,0.0626956,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(c+d x^2\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^2))/(c + d*x^2)^3,x]","\frac{x (e x)^m \left((A d-B c) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)+B c \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)\right)}{c^3 d (m+1)}","\frac{(e x)^{m+1} (A d (3-m)+B c (m+1)) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{4 c^3 d e (m+1)}-\frac{(e x)^{m+1} (B c-A d)}{4 c d e \left(c+d x^2\right)^2}",1,"(x*(e*x)^m*(B*c*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] + (-(B*c) + A*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)]))/(c^3*d*(1 + m))","A",1
41,1,197,333,0.2297934,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(a+b x^2\right) \left(c+d x^2\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^2))/((a + b*x^2)*(c + d*x^2)^3),x]","\frac{x (e x)^m \left(\frac{b^2 (A b-a B) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a}+\frac{(b c-a d)^2 (B c-A d) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c^3}-\frac{d (A b-a B) (b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c^2}-\frac{b d (A b-a B) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c}\right)}{(m+1) (b c-a d)^3}","\frac{(e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) \left(-a^2 d^2 (1-m) (A d (3-m)+B c (m+1))+2 a b c d \left(A d \left(m^2-6 m+5\right)+B c \left(-m^2+2 m+3\right)\right)+b^2 c^2 (3-m) (B c (1-m)-A d (5-m))\right)}{8 c^3 e (m+1) (b c-a d)^3}+\frac{b^2 (e x)^{m+1} (A b-a B) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a e (m+1) (b c-a d)^3}+\frac{(e x)^{m+1} (a d (A d (3-m)+B c (m+1))+b c (B c (3-m)-A d (7-m)))}{8 c^2 e \left(c+d x^2\right) (b c-a d)^2}+\frac{(e x)^{m+1} (B c-A d)}{4 c e \left(c+d x^2\right)^2 (b c-a d)}",1,"(x*(e*x)^m*((b^2*(A*b - a*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a - (b*(A*b - a*B)*d*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c - ((A*b - a*B)*d*(b*c - a*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c^2 + ((b*c - a*d)^2*(B*c - A*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c^3))/((b*c - a*d)^3*(1 + m))","A",1
42,1,266,452,0.3549767,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(a+b x^2\right)^2 \left(c+d x^2\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^2))/((a + b*x^2)^2*(c + d*x^2)^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}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^2}+\frac{b^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (2 a B d-3 A b d+b B c)}{a}+\frac{d (b c-a d)^2 (A d-B c) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c^3}-\frac{d (b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a B d-2 A b d+b B c)}{c^2}-\frac{b d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (2 a B d-3 A b d+b B c)}{c}\right)}{(m+1) (b c-a d)^4}","\frac{d (e x)^{m+1} \left(A \left(-a^2 d^2 (3-m)+a b c d (11-m)+4 b^2 c^2\right)-a B c (a d (m+1)+b c (11-m))\right)}{8 a c^2 e \left(c+d x^2\right) (b c-a d)^3}-\frac{d (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) \left(-a^2 d^2 (1-m) (A d (3-m)+B c (m+1))+2 a b c d \left(A d \left(m^2-8 m+7\right)+B c \left(-m^2+4 m+5\right)\right)+b^2 c^2 (5-m) (B c (3-m)-A d (7-m))\right)}{8 c^3 e (m+1) (b c-a d)^4}+\frac{b^2 (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (A b (b c (1-m)-a d (7-m))+a B (a d (5-m)+b c (m+1)))}{2 a^2 e (m+1) (b c-a d)^4}+\frac{d (e x)^{m+1} (a A d-3 a B c+2 A b c)}{4 a c e \left(c+d x^2\right)^2 (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{2 a e \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)}",1,"(x*(e*x)^m*((b^2*(b*B*c - 3*A*b*d + 2*a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a - (b*d*(b*B*c - 3*A*b*d + 2*a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c + (b^2*(-(A*b) + a*B)*(-(b*c) + a*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^2 - (d*(b*c - a*d)*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c^2 + (d*(b*c - a*d)^2*(-(B*c) + A*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c^3))/((b*c - a*d)^4*(1 + m))","A",1
43,1,329,665,0.4923262,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(a+b x^2\right)^3 \left(c+d x^2\right)^3} \, dx","Integrate[((e*x)^m*(A + B*x^2))/((a + b*x^2)^3*(c + d*x^2)^3),x]","\frac{x (e x)^m \left(\frac{b^2 (A b-a B) (b c-a d)^2 \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^3}+\frac{b^2 (b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (2 a B d-3 A b d+b B c)}{a^2}-\frac{3 b^2 d \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) (a B d-2 A b d+b B c)}{a}+\frac{d^2 (b c-a d)^2 (B c-A d) \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right)}{c^3}+\frac{d^2 (b c-a d) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a B d-3 A b d+2 b B c)}{c^2}+\frac{3 b d^2 \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) (a B d-2 A b d+b B c)}{c}\right)}{(m+1) (b c-a d)^5}","\frac{d (e x)^{m+1} \left(A (a d+b c) \left(a^2 d^2 (3-m)-2 a b c d (9-m)+b^2 c^2 (3-m)\right)+a B c \left(a^2 d^2 (m+1)+2 a b c d (11-m)+b^2 c^2 (m+1)\right)\right)}{8 a^2 c^2 e \left(c+d x^2\right) (b c-a d)^4}-\frac{d (e x)^{m+1} \left(A \left(2 a^2 d^2+a b c d (13-m)-b^2 c^2 (3-m)\right)-a B c (a d (11-m)+b c (m+1))\right)}{8 a^2 c e \left(c+d x^2\right)^2 (b c-a d)^3}+\frac{d^2 (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right) \left(-a^2 d^2 (1-m) (A d (3-m)+B c (m+1))+2 a b c d \left(A d \left(m^2-10 m+9\right)+B c \left(-m^2+6 m+7\right)\right)+b^2 c^2 (7-m) (B c (5-m)-A d (9-m))\right)}{8 c^3 e (m+1) (b c-a d)^5}+\frac{(e x)^{m+1} (A b (b c (3-m)-a d (11-m))+a B (a d (7-m)+b c (m+1)))}{8 a^2 e \left(a+b x^2\right) \left(c+d x^2\right)^2 (b c-a d)^2}+\frac{b^2 (e x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right) \left(A b \left(a^2 d^2 \left(m^2-16 m+63\right)-2 a b c d \left(m^2-10 m+9\right)+b^2 c^2 \left(m^2-4 m+3\right)\right)+a B \left(-a^2 d^2 \left(m^2-12 m+35\right)-2 a b c d \left(-m^2+6 m+7\right)+b^2 c^2 \left(1-m^2\right)\right)\right)}{8 a^3 e (m+1) (b c-a d)^5}+\frac{(e x)^{m+1} (A b-a B)}{4 a e \left(a+b x^2\right)^2 \left(c+d x^2\right)^2 (b c-a d)}",1,"(x*(e*x)^m*((-3*b^2*d*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a + (3*b*d^2*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c + (b^2*(b*c - a*d)*(b*B*c - 3*A*b*d + 2*a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^2 + (d^2*(b*c - a*d)*(2*b*B*c - 3*A*b*d + a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c^2 + (b^2*(A*b - a*B)*(b*c - a*d)^2*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^3 + (d^2*(b*c - a*d)^2*(B*c - A*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/c^3))/((b*c - a*d)^5*(1 + m))","A",1
44,1,248,1059,0.4691122,"\int (e x)^m \left(a+b x^2\right)^p \left(A+B x^2\right) \left(c+d x^2\right)^3 \, dx","Integrate[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2)^3,x]","x (e x)^m \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(\frac{c^2 x^2 (3 A d+B c) \, _2F_1\left(\frac{m+3}{2},-p;\frac{m+5}{2};-\frac{b x^2}{a}\right)}{m+3}+d x^4 \left(d x^2 \left(\frac{(A d+3 B c) \, _2F_1\left(\frac{m+7}{2},-p;\frac{m+9}{2};-\frac{b x^2}{a}\right)}{m+7}+\frac{B d x^2 \, _2F_1\left(\frac{m+9}{2},-p;\frac{m+11}{2};-\frac{b x^2}{a}\right)}{m+9}\right)+\frac{3 c (A d+B c) \, _2F_1\left(\frac{m+5}{2},-p;\frac{m+7}{2};-\frac{b x^2}{a}\right)}{m+5}\right)+\frac{A c^3 \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right)}{m+1}\right)","-\frac{\left(-c^2 \left(48 B c+A d \left(m^3+(6 p+23) m^2+\left(12 p^2+92 p+183\right) m+8 p^3+92 p^2+366 p+513\right)\right) b^3+a c d \left(2 A d \left(m^3+4 (p+5) m^2+\left(4 p^2+44 p+123\right) m+8 p^2+84 p+216\right)+B c \left(m^3+(4 p+21) m^2+\left(4 p^2+44 p+143\right) m+4 p^2+40 p+267\right)\right) b^2-a^2 d^2 (m+5) \left(A d (m+3) (m+2 p+9)+2 B c \left(m^2+2 p m+13 m+2 p+30\right)\right) b+a^3 B d^3 \left(m^3+15 m^2+71 m+105\right)\right) \left(b x^2+a\right)^{p+1} (e x)^{m+1}}{b^4 e (m+2 p+3) (m+2 p+5) (m+2 p+7) (m+2 p+9)}+\frac{B \left(b x^2+a\right)^{p+1} \left(d x^2+c\right)^3 (e x)^{m+1}}{b e (m+2 p+9)}-\frac{(a B d (m+7)-b (6 B c+A d (m+2 p+9))) \left(b x^2+a\right)^{p+1} \left(d x^2+c\right)^2 (e x)^{m+1}}{b^2 e (m+2 p+7) (m+2 p+9)}+\frac{\left(c \left(24 B c+A d \left(m^2+4 (p+5) m+4 p^2+40 p+99\right)\right) b^2-a d \left(A d (m+5) (m+2 p+9)+B c \left(m^2+2 (p+9) m+2 p+65\right)\right) b+a^2 B d^2 \left(m^2+12 m+35\right)\right) \left(b x^2+a\right)^{p+1} \left(d x^2+c\right) (e x)^{m+1}}{b^3 e (m+2 p+5) (m+2 p+7) (m+2 p+9)}+\frac{\left(a (m+1) \left(-c^2 \left(48 B c+A d \left(m^3+(6 p+23) m^2+\left(12 p^2+92 p+183\right) m+8 p^3+92 p^2+366 p+513\right)\right) b^3+a c d \left(2 A d \left(m^3+4 (p+5) m^2+\left(4 p^2+44 p+123\right) m+8 p^2+84 p+216\right)+B c \left(m^3+(4 p+21) m^2+\left(4 p^2+44 p+143\right) m+4 p^2+40 p+267\right)\right) b^2-a^2 d^2 (m+5) \left(A d (m+3) (m+2 p+9)+2 B c \left(m^2+2 p m+13 m+2 p+30\right)\right) b+a^3 B d^3 \left(m^3+15 m^2+71 m+105\right)\right)-b c (m+2 p+3) (2 b c (p+2) (2 b c (p+3) (a B (m+1)-A b (m+2 p+9))+(b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9)))+(m+1) (b c (2 b c (p+3) (a B (m+1)-A b (m+2 p+9))+(b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9)))-a (2 b c d (p+3) (a B (m+1)-A b (m+2 p+9))+d (b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9))+4 (b c-a d) (a B d (m+7)-b (6 B c+A d (m+2 p+9))))))\right) \left(b x^2+a\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right) (e x)^{m+1}}{b^4 e (m+1) (m+2 p+3) (m+2 p+5) (m+2 p+7) (m+2 p+9)}",1,"(x*(e*x)^m*(a + b*x^2)^p*((A*c^3*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(1 + m) + (c^2*(B*c + 3*A*d)*x^2*Hypergeometric2F1[(3 + m)/2, -p, (5 + m)/2, -((b*x^2)/a)])/(3 + m) + d*x^4*((3*c*(B*c + A*d)*Hypergeometric2F1[(5 + m)/2, -p, (7 + m)/2, -((b*x^2)/a)])/(5 + m) + d*x^2*(((3*B*c + A*d)*Hypergeometric2F1[(7 + m)/2, -p, (9 + m)/2, -((b*x^2)/a)])/(7 + m) + (B*d*x^2*Hypergeometric2F1[(9 + m)/2, -p, (11 + m)/2, -((b*x^2)/a)])/(9 + m)))))/(1 + (b*x^2)/a)^p","A",1
45,1,198,495,0.2676728,"\int (e x)^m \left(a+b x^2\right)^p \left(A+B x^2\right) \left(c+d x^2\right)^2 \, dx","Integrate[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2)^2,x]","x (e x)^m \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(\frac{c x^2 (2 A d+B c) \, _2F_1\left(\frac{m+3}{2},-p;\frac{m+5}{2};-\frac{b x^2}{a}\right)}{m+3}+d x^4 \left(\frac{(A d+2 B c) \, _2F_1\left(\frac{m+5}{2},-p;\frac{m+7}{2};-\frac{b x^2}{a}\right)}{m+5}+\frac{B d x^2 \, _2F_1\left(\frac{m+7}{2},-p;\frac{m+9}{2};-\frac{b x^2}{a}\right)}{m+7}\right)+\frac{A c^2 \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right)}{m+1}\right)","\frac{(e x)^{m+1} \left(a+b x^2\right)^{p+1} \left(a^2 B d^2 \left(m^2+8 m+15\right)-a b d \left(A d (m+3) (m+2 p+7)+B c \left(m^2+2 m (p+6)+2 p+27\right)\right)+b^2 c \left(A d (m+2 p+7)^2+8 B c\right)\right)}{b^3 e (m+2 p+3) (m+2 p+5) (m+2 p+7)}-\frac{(e x)^{m+1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right) (b c (m+2 p+3) ((m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 b c (p+2) (a B (m+1)-A b (m+2 p+7)))-a (m+1) (2 b c d (p+2) (a B (m+1)-A b (m+2 p+7))+d (m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 (b c-a d) (a B d (m+5)-b (A d (m+2 p+7)+4 B c))))}{b^3 e (m+1) (m+2 p+3) (m+2 p+5) (m+2 p+7)}-\frac{\left(c+d x^2\right) (e x)^{m+1} \left(a+b x^2\right)^{p+1} (a B d (m+5)-b (A d (m+2 p+7)+4 B c))}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac{B \left(c+d x^2\right)^2 (e x)^{m+1} \left(a+b x^2\right)^{p+1}}{b e (m+2 p+7)}",1,"(x*(e*x)^m*(a + b*x^2)^p*((A*c^2*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(1 + m) + (c*(B*c + 2*A*d)*x^2*Hypergeometric2F1[(3 + m)/2, -p, (5 + m)/2, -((b*x^2)/a)])/(3 + m) + d*x^4*(((2*B*c + A*d)*Hypergeometric2F1[(5 + m)/2, -p, (7 + m)/2, -((b*x^2)/a)])/(5 + m) + (B*d*x^2*Hypergeometric2F1[(7 + m)/2, -p, (9 + m)/2, -((b*x^2)/a)])/(7 + m))))/(1 + (b*x^2)/a)^p","A",1
46,1,147,253,0.1312768,"\int (e x)^m \left(a+b x^2\right)^p \left(A+B x^2\right) \left(c+d x^2\right) \, dx","Integrate[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2),x]","x (e x)^m \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(\frac{x^2 (A d+B c) \, _2F_1\left(\frac{m+3}{2},-p;\frac{m+5}{2};-\frac{b x^2}{a}\right)}{m+3}+\frac{A c \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right)}{m+1}+\frac{B d x^4 \, _2F_1\left(\frac{m+5}{2},-p;\frac{m+7}{2};-\frac{b x^2}{a}\right)}{m+5}\right)","-\frac{(e x)^{m+1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right) (A b (m+2 p+3) (a d (m+1)-b c (m+2 p+5))-a (m+1) (a B d (m+3)-b (2 A d+B c (m+2 p+5))))}{b^2 e (m+1) (m+2 p+3) (m+2 p+5)}-\frac{(e x)^{m+1} \left(a+b x^2\right)^{p+1} (a B d (m+3)-b (2 A d+B c (m+2 p+5)))}{b^2 e (m+2 p+3) (m+2 p+5)}+\frac{d \left(A+B x^2\right) (e x)^{m+1} \left(a+b x^2\right)^{p+1}}{b e (m+2 p+5)}",1,"(x*(e*x)^m*(a + b*x^2)^p*((A*c*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(1 + m) + ((B*c + A*d)*x^2*Hypergeometric2F1[(3 + m)/2, -p, (5 + m)/2, -((b*x^2)/a)])/(3 + m) + (B*d*x^4*Hypergeometric2F1[(5 + m)/2, -p, (7 + m)/2, -((b*x^2)/a)])/(5 + m)))/(1 + (b*x^2)/a)^p","A",1
47,1,118,162,0.2087009,"\int \frac{(e x)^m \left(a+b x^2\right)^p \left(A+B x^2\right)}{c+d x^2} \, dx","Integrate[((e*x)^m*(a + b*x^2)^p*(A + B*x^2))/(c + d*x^2),x]","\frac{x (e x)^m \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left((A d-B c) F_1\left(\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+B c \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right)\right)}{c d (m+1)}","\frac{B (e x)^{m+1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right)}{d e (m+1)}-\frac{(e x)^{m+1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} (B c-A d) F_1\left(\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{c d e (m+1)}",1,"(x*(e*x)^m*(a + b*x^2)^p*((-(B*c) + A*d)*AppellF1[(1 + m)/2, -p, 1, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)] + B*c*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)]))/(c*d*(1 + m)*(1 + (b*x^2)/a)^p)","A",1
48,1,128,295,0.2450673,"\int \frac{(e x)^m \left(a+b x^2\right)^p \left(A+B x^2\right)}{\left(c+d x^2\right)^2} \, dx","Integrate[((e*x)^m*(a + b*x^2)^p*(A + B*x^2))/(c + d*x^2)^2,x]","\frac{x (e x)^m \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left((A d-B c) F_1\left(\frac{m+1}{2};-p,2;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+B c F_1\left(\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{c^2 d (m+1)}","-\frac{(e x)^{m+1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} (a d (A d (1-m)+B c (m+1))-b c (A d (-m-2 p+1)+B c (m+2 p+1))) F_1\left(\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{2 c^2 d e (m+1) (b c-a d)}-\frac{b (m+2 p+1) (e x)^{m+1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} (B c-A d) \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right)}{2 c d e (m+1) (b c-a d)}+\frac{(e x)^{m+1} \left(a+b x^2\right)^{p+1} (B c-A d)}{2 c e \left(c+d x^2\right) (b c-a d)}",1,"(x*(e*x)^m*(a + b*x^2)^p*(B*c*AppellF1[(1 + m)/2, -p, 1, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)] + (-(B*c) + A*d)*AppellF1[(1 + m)/2, -p, 2, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)]))/(c^2*d*(1 + m)*(1 + (b*x^2)/a)^p)","A",0
49,1,128,483,0.509376,"\int \frac{(e x)^m \left(a+b x^2\right)^p \left(A+B x^2\right)}{\left(c+d x^2\right)^3} \, dx","Integrate[((e*x)^m*(a + b*x^2)^p*(A + B*x^2))/(c + d*x^2)^3,x]","\frac{x (e x)^m \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left((A d-B c) F_1\left(\frac{m+1}{2};-p,3;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)+B c F_1\left(\frac{m+1}{2};-p,2;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)\right)}{c^3 d (m+1)}","\frac{(e x)^{m+1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a^2 d^2 (1-m) (A d (3-m)+B c (m+1))-2 a b c d (A d (1-m) (-m-2 p+3)+B c (m+1) (-m-2 p+1))+b^2 c^2 (-m-2 p+1) (A d (-m-2 p+3)+B c (m+2 p+1))\right) F_1\left(\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right)}{8 c^3 d e (m+1) (b c-a d)^2}-\frac{b (m+2 p+1) (e x)^{m+1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right) (a d (A d (3-m)+B c (m+1))+b c (B c (-m-2 p+1)-A d (-m-2 p+5)))}{8 c^2 d e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} \left(a+b x^2\right)^{p+1} (a d (A d (3-m)+B c (m+1))+b c (B c (-m-2 p+1)-A d (-m-2 p+5)))}{8 c^2 e \left(c+d x^2\right) (b c-a d)^2}+\frac{(e x)^{m+1} \left(a+b x^2\right)^{p+1} (B c-A d)}{4 c e \left(c+d x^2\right)^2 (b c-a d)}",1,"(x*(e*x)^m*(a + b*x^2)^p*(B*c*AppellF1[(1 + m)/2, -p, 2, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)] + (-(B*c) + A*d)*AppellF1[(1 + m)/2, -p, 3, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)]))/(c^3*d*(1 + m)*(1 + (b*x^2)/a)^p)","A",0
50,1,91,84,0.1404815,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right) \left(c+d x^2\right)}{x} \, dx","Integrate[(Sqrt[a + b*x^2]*(A + B*x^2)*(c + d*x^2))/x,x]","\frac{\sqrt{a+b x^2} \left(5 A b \left(a d+3 b c+b d x^2\right)-B \left(a+b x^2\right) \left(2 a d-5 b c-3 b d x^2\right)\right)}{15 b^2}-\sqrt{a} A c \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)","-\frac{\left(a+b x^2\right)^{3/2} \left(2 a B d-5 b (A d+B c)-3 b B d x^2\right)}{15 b^2}+A c \sqrt{a+b x^2}-\sqrt{a} A c \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)",1,"(Sqrt[a + b*x^2]*(-(B*(a + b*x^2)*(-5*b*c + 2*a*d - 3*b*d*x^2)) + 5*A*b*(3*b*c + a*d + b*d*x^2)))/(15*b^2) - Sqrt[a]*A*c*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
51,1,91,84,0.160116,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right) \sqrt{c+d x^2}}{x} \, dx","Integrate[((a + b*x^2)*(A + B*x^2)*Sqrt[c + d*x^2])/x,x]","\frac{\sqrt{c+d x^2} \left(5 a d \left(3 A d+B \left(c+d x^2\right)\right)-b \left(c+d x^2\right) \left(-5 A d+2 B c-3 B d x^2\right)\right)}{15 d^2}-a A \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)","-\frac{\left(c+d x^2\right)^{3/2} \left(-5 d (a B+A b)+2 b B c-3 b B d x^2\right)}{15 d^2}+a A \sqrt{c+d x^2}-a A \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right)",1,"(Sqrt[c + d*x^2]*(-(b*(c + d*x^2)*(2*B*c - 5*A*d - 3*B*d*x^2)) + 5*a*d*(3*A*d + B*(c + d*x^2))))/(15*d^2) - a*A*Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]","A",1