1,1,189,0,0.2270666,"\int (e x)^m \left(a+b x^2\right)^3 \left(A+B x^2\right) \left(c+d x^2\right) \, dx","Int[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2),x]","\frac{a^2 (e x)^{m+3} (a A d+a B c+3 A b c)}{e^3 (m+3)}+\frac{a^3 A c (e x)^{m+1}}{e (m+1)}+\frac{b^2 (e x)^{m+9} (3 a B d+A b d+b B c)}{e^9 (m+9)}+\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 (e x)^{m+7} (A b (3 a d+b c)+3 a B (a d+b c))}{e^7 (m+7)}+\frac{b^3 B d (e x)^{m+11}}{e^{11} (m+11)}","\frac{a^2 (e x)^{m+3} (a A d+a B c+3 A b c)}{e^3 (m+3)}+\frac{a^3 A c (e x)^{m+1}}{e (m+1)}+\frac{b^2 (e x)^{m+9} (3 a B d+A b d+b B c)}{e^9 (m+9)}+\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 (e x)^{m+7} (A b (3 a d+b c)+3 a B (a d+b c))}{e^7 (m+7)}+\frac{b^3 B d (e x)^{m+11}}{e^{11} (m+11)}",1,"(a^3*A*c*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*(3*A*b*c + a*B*c + a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + (a*(3*A*b*(b*c + a*d) + a*B*(3*b*c + a*d))*(e*x)^(5 + m))/(e^5*(5 + m)) + (b*(3*a*B*(b*c + a*d) + A*b*(b*c + 3*a*d))*(e*x)^(7 + m))/(e^7*(7 + m)) + (b^2*(b*B*c + A*b*d + 3*a*B*d)*(e*x)^(9 + m))/(e^9*(9 + m)) + (b^3*B*d*(e*x)^(11 + m))/(e^11*(11 + m))","A",2,1,29,0.03448,1,"{570}"
2,1,144,0,0.1214118,"\int (e x)^m \left(a+b x^2\right)^2 \left(A+B x^2\right) \left(c+d x^2\right) \, dx","Int[(e*x)^m*(a + b*x^2)^2*(A + B*x^2)*(c + d*x^2),x]","\frac{a^2 A c (e x)^{m+1}}{e (m+1)}+\frac{a (e x)^{m+3} (a A d+a B c+2 A b c)}{e^3 (m+3)}+\frac{(e x)^{m+5} (A b (2 a d+b c)+a B (a d+2 b c))}{e^5 (m+5)}+\frac{b (e x)^{m+7} (2 a B d+A b d+b B c)}{e^7 (m+7)}+\frac{b^2 B d (e x)^{m+9}}{e^9 (m+9)}","\frac{a^2 A c (e x)^{m+1}}{e (m+1)}+\frac{a (e x)^{m+3} (a A d+a B c+2 A b c)}{e^3 (m+3)}+\frac{(e x)^{m+5} (A b (2 a d+b c)+a B (a d+2 b c))}{e^5 (m+5)}+\frac{b (e x)^{m+7} (2 a B d+A b d+b B c)}{e^7 (m+7)}+\frac{b^2 B d (e x)^{m+9}}{e^9 (m+9)}",1,"(a^2*A*c*(e*x)^(1 + m))/(e*(1 + m)) + (a*(2*A*b*c + a*B*c + a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + ((a*B*(2*b*c + a*d) + A*b*(b*c + 2*a*d))*(e*x)^(5 + m))/(e^5*(5 + m)) + (b*(b*B*c + A*b*d + 2*a*B*d)*(e*x)^(7 + m))/(e^7*(7 + m)) + (b^2*B*d*(e*x)^(9 + m))/(e^9*(9 + m))","A",2,1,29,0.03448,1,"{570}"
3,1,97,0,0.0605376,"\int (e x)^m \left(a+b x^2\right) \left(A+B x^2\right) \left(c+d x^2\right) \, dx","Int[(e*x)^m*(a + b*x^2)*(A + B*x^2)*(c + d*x^2),x]","\frac{(e x)^{m+3} (a A d+a B c+A b c)}{e^3 (m+3)}+\frac{(e x)^{m+5} (a B d+A b d+b B c)}{e^5 (m+5)}+\frac{a A c (e x)^{m+1}}{e (m+1)}+\frac{b B d (e x)^{m+7}}{e^7 (m+7)}","\frac{(e x)^{m+3} (a A d+a B c+A b c)}{e^3 (m+3)}+\frac{(e x)^{m+5} (a B d+A b d+b B c)}{e^5 (m+5)}+\frac{a A c (e x)^{m+1}}{e (m+1)}+\frac{b B d (e x)^{m+7}}{e^7 (m+7)}",1,"(a*A*c*(e*x)^(1 + m))/(e*(1 + m)) + ((A*b*c + a*B*c + a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + ((b*B*c + A*b*d + a*B*d)*(e*x)^(5 + m))/(e^5*(5 + m)) + (b*B*d*(e*x)^(7 + m))/(e^7*(7 + m))","A",2,1,27,0.03704,1,"{570}"
4,1,60,0,0.0314444,"\int (e x)^m \left(A+B x^2\right) \left(c+d x^2\right) \, dx","Int[(e*x)^m*(A + B*x^2)*(c + d*x^2),x]","\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)}","\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,"(A*c*(e*x)^(1 + m))/(e*(1 + m)) + ((B*c + A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + (B*d*(e*x)^(5 + m))/(e^5*(5 + m))","A",2,1,20,0.05000,1,"{448}"
5,1,118,0,0.0980765,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)}{a+b x^2} \, dx","Int[((e*x)^m*(A + B*x^2)*(c + d*x^2))/(a + b*x^2),x]","\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)}","\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,"((b*B*c + A*b*d - a*B*d)*(e*x)^(1 + m))/(b^2*e*(1 + m)) + (B*d*(e*x)^(3 + m))/(b*e^3*(3 + m)) + ((A*b - a*B)*(b*c - a*d)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*b^2*e*(1 + m))","A",3,2,29,0.06897,1,"{570, 364}"
6,1,171,0,0.2409192,"\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","Int[((e*x)^m*(A + B*x^2)*(c + d*x^2))/(a + b*x^2)^2,x]","\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)}","\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,"-(d*(A*b*(1 + m) - a*B*(3 + m))*(e*x)^(1 + m))/(2*a*b^2*e*(1 + m)) + ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2))/(2*a*b*e*(a + b*x^2)) + ((a*B*(b*c*(1 + m) - a*d*(3 + m)) + A*b*(a*d*(1 + m) + b*(c - c*m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*b^2*e*(1 + m))","A",3,3,29,0.1034,1,"{577, 459, 364}"
7,1,209,0,0.2960045,"\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","Int[((e*x)^m*(A + B*x^2)*(c + d*x^2))/(a + b*x^2)^3,x]","\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}","\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,"-((A*b*(a*d*(1 - m) - b*c*(3 - m)) - a*B*(b*c*(1 + m) - a*d*(3 + m)))*(e*x)^(1 + m))/(8*a^2*b^2*e*(a + b*x^2)) + ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2))/(4*a*b*e*(a + b*x^2)^2) + ((A*b*(1 - m)*(b*c*(3 - m) + a*d*(1 + m)) + a*B*(1 + m)*(a*d*(3 + m) + b*(c - c*m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(8*a^3*b^2*e*(1 + m))","A",3,3,29,0.1034,1,"{577, 457, 364}"
8,1,292,0,0.2866715,"\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","Int[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2)^2,x]","\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{(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{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{a^3 A c^2 (e x)^{m+1}}{e (m+1)}+\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)}","\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{(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{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{a^3 A c^2 (e x)^{m+1}}{e (m+1)}+\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,"(a^3*A*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*c*(3*A*b*c + a*B*c + 2*a*A*d)*(e*x)^(3 + m))/(e^3*(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))*(e*x)^(5 + m))/(e^5*(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))*(e*x)^(7 + m))/(e^7*(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))*(e*x)^(9 + m))/(e^9*(9 + m)) + (b^2*d*(2*b*B*c + A*b*d + 3*a*B*d)*(e*x)^(11 + m))/(e^11*(11 + m)) + (b^3*B*d^2*(e*x)^(13 + m))/(e^13*(13 + m))","A",2,1,31,0.03226,1,"{570}"
9,1,216,0,0.2468246,"\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","Int[(e*x)^m*(a + b*x^2)^2*(A + B*x^2)*(c + d*x^2)^2,x]","\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{a c (e x)^{m+3} (2 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac{b d (e x)^{m+9} (2 a B d+A b d+2 b B c)}{e^9 (m+9)}+\frac{b^2 B d^2 (e x)^{m+11}}{e^{11} (m+11)}","\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{a c (e x)^{m+3} (2 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac{b d (e x)^{m+9} (2 a B d+A b d+2 b B c)}{e^9 (m+9)}+\frac{b^2 B d^2 (e x)^{m+11}}{e^{11} (m+11)}",1,"(a^2*A*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (a*c*(a*B*c + 2*A*(b*c + a*d))*(e*x)^(3 + m))/(e^3*(3 + m)) + ((2*a*B*c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))*(e*x)^(5 + m))/(e^5*(5 + m)) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*(e*x)^(7 + m))/(e^7*(7 + m)) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*(e*x)^(9 + m))/(e^9*(9 + m)) + (b^2*B*d^2*(e*x)^(11 + m))/(e^11*(11 + m))","A",2,1,31,0.03226,1,"{570}"
10,1,144,0,0.1343428,"\int (e x)^m \left(a+b x^2\right) \left(A+B x^2\right) \left(c+d x^2\right)^2 \, dx","Int[(e*x)^m*(a + b*x^2)*(A + B*x^2)*(c + d*x^2)^2,x]","\frac{c (e x)^{m+3} (2 a A d+a B c+A b c)}{e^3 (m+3)}+\frac{(e x)^{m+5} (a d (A d+2 B c)+b c (2 A d+B c))}{e^5 (m+5)}+\frac{d (e x)^{m+7} (a B d+A b d+2 b B c)}{e^7 (m+7)}+\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)}","\frac{c (e x)^{m+3} (2 a A d+a B c+A b c)}{e^3 (m+3)}+\frac{(e x)^{m+5} (a d (A d+2 B c)+b c (2 A d+B c))}{e^5 (m+5)}+\frac{d (e x)^{m+7} (a B d+A b d+2 b B c)}{e^7 (m+7)}+\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,"(a*A*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (c*(A*b*c + a*B*c + 2*a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + ((a*d*(2*B*c + A*d) + b*c*(B*c + 2*A*d))*(e*x)^(5 + m))/(e^5*(5 + m)) + (d*(2*b*B*c + A*b*d + a*B*d)*(e*x)^(7 + m))/(e^7*(7 + m)) + (b*B*d^2*(e*x)^(9 + m))/(e^9*(9 + m))","A",2,1,29,0.03448,1,"{570}"
11,1,91,0,0.0686443,"\int (e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^2 \, dx","Int[(e*x)^m*(A + B*x^2)*(c + d*x^2)^2,x]","\frac{c (e x)^{m+3} (2 A d+B c)}{e^3 (m+3)}+\frac{d (e x)^{m+5} (A d+2 B c)}{e^5 (m+5)}+\frac{A c^2 (e x)^{m+1}}{e (m+1)}+\frac{B d^2 (e x)^{m+7}}{e^7 (m+7)}","\frac{c (e x)^{m+3} (2 A d+B c)}{e^3 (m+3)}+\frac{d (e x)^{m+5} (A d+2 B c)}{e^5 (m+5)}+\frac{A c^2 (e x)^{m+1}}{e (m+1)}+\frac{B d^2 (e x)^{m+7}}{e^7 (m+7)}",1,"(A*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (c*(B*c + 2*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + (d*(2*B*c + A*d)*(e*x)^(5 + m))/(e^5*(5 + m)) + (B*d^2*(e*x)^(7 + m))/(e^7*(7 + m))","A",2,1,22,0.04545,1,"{448}"
12,1,178,0,0.1890752,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^2}{a+b x^2} \, dx","Int[((e*x)^m*(A + B*x^2)*(c + d*x^2)^2)/(a + b*x^2),x]","\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{d (e x)^{m+3} (-a B d+A b d+2 b B c)}{b^2 e^3 (m+3)}+\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{B d^2 (e x)^{m+5}}{b e^5 (m+5)}","\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{d (e x)^{m+3} (-a B d+A b d+2 b B c)}{b^2 e^3 (m+3)}+\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{B d^2 (e x)^{m+5}}{b e^5 (m+5)}",1,"((a^2*B*d^2 - a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*(e*x)^(1 + m))/(b^3*e*(1 + m)) + (d*(2*b*B*c + A*b*d - a*B*d)*(e*x)^(3 + m))/(b^2*e^3*(3 + m)) + (B*d^2*(e*x)^(5 + m))/(b*e^5*(5 + m)) + ((A*b - a*B)*(b*c - a*d)^2*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*b^3*e*(1 + m))","A",3,2,31,0.06452,1,"{570, 364}"
13,1,247,0,0.4421443,"\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","Int[((e*x)^m*(A + B*x^2)*(c + d*x^2)^2)/(a + b*x^2)^2,x]","\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)}","\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,"-(d*(A*b*(2*b*c*(1 + m) - a*d*(3 + m)) - a*B*(2*b*c*(3 + m) - a*d*(5 + m)))*(e*x)^(1 + m))/(2*a*b^3*e*(1 + m)) - (d^2*(A*b*(3 + m) - a*B*(5 + m))*(e*x)^(3 + m))/(2*a*b^2*e^3*(3 + m)) + ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2)^2)/(2*a*b*e*(a + b*x^2)) + ((b*c - a*d)*(a*B*(b*c*(1 + m) - a*d*(5 + m)) + A*b*(a*d*(3 + m) + b*(c - c*m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*b^3*e*(1 + m))","A",4,3,31,0.09677,1,"{577, 570, 364}"
14,1,292,0,0.4084852,"\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","Int[((e*x)^m*(A + B*x^2)*(c + d*x^2)^2)/(a + b*x^2)^3,x]","-\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{(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{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{\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}","-\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{(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{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{\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,"(d*(b*c*(1 + m) - a*d*(3 + m))*(A*b*(1 + m) - a*B*(5 + m))*(e*x)^(1 + m))/(8*a^2*b^3*e*(1 + m)) + ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2)^2)/(4*a*b*e*(a + b*x^2)^2) + ((b*c - a*d)*(e*x)^(1 + m)*(c*(A*b*(3 - m) + a*B*(1 + m)) - d*(A*b*(1 + m) - a*B*(5 + m))*x^2))/(8*a^2*b^2*e*(a + b*x^2)) - ((a*d*(b*c*(1 + m) - a*d*(3 + m))*(A*b*(1 + m) - a*B*(5 + m)) - b*c*(A*b*(3 - m) + a*B*(1 + m))*(a*d*(1 + m) + b*(c - c*m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(8*a^3*b^3*e*(1 + m))","A",4,3,31,0.09677,1,"{577, 459, 364}"
15,1,379,0,0.4007743,"\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","Int[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2)^3,x]","\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{(e x)^{m+7} \left(A \left(9 a^2 b c d^2+a^3 d^3+9 a b^2 c^2 d+b^3 c^3\right)+3 a B c \left(a^2 d^2+3 a b c d+b^2 c^2\right)\right)}{e^7 (m+7)}+\frac{(e x)^{m+9} \left(3 a^2 b d^2 (A d+3 B c)+a^3 B d^3+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{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{a^3 A c^3 (e x)^{m+1}}{e (m+1)}+\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)}","\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{(e x)^{m+7} \left(A \left(9 a^2 b c d^2+a^3 d^3+9 a b^2 c^2 d+b^3 c^3\right)+3 a B c \left(a^2 d^2+3 a b c d+b^2 c^2\right)\right)}{e^7 (m+7)}+\frac{(e x)^{m+9} \left(3 a^2 b d^2 (A d+3 B c)+a^3 B d^3+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{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{a^3 A c^3 (e x)^{m+1}}{e (m+1)}+\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,"(a^3*A*c^3*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*c^2*(a*B*c + 3*A*(b*c + a*d))*(e*x)^(3 + m))/(e^3*(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))*(e*x)^(5 + m))/(e^5*(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))*(e*x)^(7 + m))/(e^7*(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))*(e*x)^(9 + m))/(e^9*(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))*(e*x)^(11 + m))/(e^11*(11 + m)) + (b^2*d^2*(3*b*B*c + A*b*d + 3*a*B*d)*(e*x)^(13 + m))/(e^13*(13 + m)) + (b^3*B*d^3*(e*x)^(15 + m))/(e^15*(15 + m))","A",2,1,31,0.03226,1,"{570}"
16,1,284,0,0.2837007,"\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","Int[(e*x)^m*(a + b*x^2)^2*(A + B*x^2)*(c + d*x^2)^3,x]","\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{(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{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)}","\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{(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{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,"(a^2*A*c^3*(e*x)^(1 + m))/(e*(1 + m)) + (a*c^2*(2*A*b*c + a*B*c + 3*a*A*d)*(e*x)^(3 + m))/(e^3*(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))*(e*x)^(5 + m))/(e^5*(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))*(e*x)^(7 + m))/(e^7*(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))*(e*x)^(9 + m))/(e^9*(9 + m)) + (b*d^2*(3*b*B*c + A*b*d + 2*a*B*d)*(e*x)^(11 + m))/(e^11*(11 + m)) + (b^2*B*d^3*(e*x)^(13 + m))/(e^13*(13 + m))","A",2,1,31,0.03226,1,"{570}"
17,1,189,0,0.175998,"\int (e x)^m \left(a+b x^2\right) \left(A+B x^2\right) \left(c+d x^2\right)^3 \, dx","Int[(e*x)^m*(a + b*x^2)*(A + B*x^2)*(c + d*x^2)^3,x]","\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{c (e x)^{m+5} (3 a d (A d+B c)+b c (3 A d+B c))}{e^5 (m+5)}+\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{a A c^3 (e x)^{m+1}}{e (m+1)}+\frac{b B d^3 (e x)^{m+11}}{e^{11} (m+11)}","\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{c (e x)^{m+5} (3 a d (A d+B c)+b c (3 A d+B c))}{e^5 (m+5)}+\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{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,"(a*A*c^3*(e*x)^(1 + m))/(e*(1 + m)) + (c^2*(A*b*c + a*B*c + 3*a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + (c*(3*a*d*(B*c + A*d) + b*c*(B*c + 3*A*d))*(e*x)^(5 + m))/(e^5*(5 + m)) + (d*(3*b*c*(B*c + A*d) + a*d*(3*B*c + A*d))*(e*x)^(7 + m))/(e^7*(7 + m)) + (d^2*(3*b*B*c + A*b*d + a*B*d)*(e*x)^(9 + m))/(e^9*(9 + m)) + (b*B*d^3*(e*x)^(11 + m))/(e^11*(11 + m))","A",2,1,29,0.03448,1,"{570}"
18,1,121,0,0.0794761,"\int (e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^3 \, dx","Int[(e*x)^m*(A + B*x^2)*(c + d*x^2)^3,x]","\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)}","\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,"(A*c^3*(e*x)^(1 + m))/(e*(1 + m)) + (c^2*(B*c + 3*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + (3*c*d*(B*c + A*d)*(e*x)^(5 + m))/(e^5*(5 + m)) + (d^2*(3*B*c + A*d)*(e*x)^(7 + m))/(e^7*(7 + m)) + (B*d^3*(e*x)^(9 + m))/(e^9*(9 + m))","A",2,1,22,0.04545,1,"{448}"
19,1,258,0,0.2821532,"\int \frac{(e x)^m \left(A+B x^2\right) \left(c+d x^2\right)^3}{a+b x^2} \, dx","Int[((e*x)^m*(A + B*x^2)*(c + d*x^2)^3)/(a + b*x^2),x]","-\frac{(e x)^{m+1} \left(-a^2 b d^2 (A d+3 B c)+a^3 B d^3+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{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{d^2 (e x)^{m+5} (-a B d+A b d+3 b B c)}{b^2 e^5 (m+5)}+\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{B d^3 (e x)^{m+7}}{b e^7 (m+7)}","-\frac{(e x)^{m+1} \left(-a^2 b d^2 (A d+3 B c)+a^3 B d^3+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{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{d^2 (e x)^{m+5} (-a B d+A b d+3 b B c)}{b^2 e^5 (m+5)}+\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{B d^3 (e x)^{m+7}}{b e^7 (m+7)}",1,"-(((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))*(e*x)^(1 + m))/(b^4*e*(1 + m))) + (d*(a^2*B*d^2 + 3*b^2*c*(B*c + A*d) - a*b*d*(3*B*c + A*d))*(e*x)^(3 + m))/(b^3*e^3*(3 + m)) + (d^2*(3*b*B*c + A*b*d - a*B*d)*(e*x)^(5 + m))/(b^2*e^5*(5 + m)) + (B*d^3*(e*x)^(7 + m))/(b*e^7*(7 + m)) + ((A*b - a*B)*(b*c - a*d)^3*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*b^4*e*(1 + m))","A",3,2,31,0.06452,1,"{570, 364}"
20,1,347,0,0.6608906,"\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","Int[((e*x)^m*(A + B*x^2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]","-\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{(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^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)}","-\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{(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^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,"-(d*(A*b*(3*b^2*c^2*(1 + m) - 3*a*b*c*d*(3 + m) + a^2*d^2*(5 + m)) - a*B*(3*b^2*c^2*(3 + m) - 3*a*b*c*d*(5 + m) + a^2*d^2*(7 + m)))*(e*x)^(1 + m))/(2*a*b^4*e*(1 + m)) - (d^2*(A*b*(3*b*c*(3 + m) - a*d*(5 + m)) - a*B*(3*b*c*(5 + m) - a*d*(7 + m)))*(e*x)^(3 + m))/(2*a*b^3*e^3*(3 + m)) - (d^3*(A*b*(5 + m) - a*B*(7 + m))*(e*x)^(5 + m))/(2*a*b^2*e^5*(5 + m)) + ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2)^3)/(2*a*b*e*(a + b*x^2)) + ((b*c - a*d)^2*(a*B*(b*c*(1 + m) - a*d*(7 + m)) + A*b*(a*d*(5 + m) + b*(c - c*m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*b^4*e*(1 + m))","A",4,3,31,0.09677,1,"{577, 570, 364}"
21,1,480,0,1.0728221,"\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","Int[((e*x)^m*(A + B*x^2)*(c + d*x^2)^3)/(a + b*x^2)^3,x]","\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{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{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{\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}","\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{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{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{\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,"-(d*(A*b*(2*b^2*c^2*(3 + 2*m - m^2) + 3*a*b*c*d*(3 + 4*m + m^2) - a^2*d^2*(15 + 8*m + m^2)) + a*B*(2*b^2*c^2*(1 + m)^2 - 3*a*b*c*d*(15 + 8*m + m^2) + a^2*d^2*(35 + 12*m + m^2)))*(e*x)^(1 + m))/(8*a^2*b^4*e*(1 + m)) - (d^2*(A*b*(3 + m)*(b*c*(3 - m) + a*d*(5 + m)) + a*B*(b*c*(3 + 4*m + m^2) - a*d*(35 + 12*m + m^2)))*(e*x)^(3 + m))/(8*a^2*b^3*e^3*(3 + m)) + ((A*b*(b*c*(3 - m) + a*d*(3 + m)) + a*B*(b*c*(1 + m) - a*d*(7 + m)))*(e*x)^(1 + m)*(c + d*x^2)^2)/(8*a^2*b^2*e*(a + b*x^2)) + ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2)^3)/(4*a*b*e*(a + b*x^2)^2) + ((b*c - a*d)*(A*b*(2*a*b*c*d*(3 - 2*m - m^2) + b^2*c^2*(3 - 4*m + m^2) + a^2*d^2*(15 + 8*m + m^2)) + a*B*(b^2*c^2*(1 - m^2) + 2*a*b*c*d*(5 + 6*m + m^2) - a^2*d^2*(35 + 12*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(8*a^3*b^4*e*(1 + m))","A",5,3,31,0.09677,1,"{577, 570, 364}"
22,1,363,0,0.3709229,"\int \frac{(e x)^m \left(a+b x^2\right)^4 \left(A+B x^2\right)}{c+d x^2} \, dx","Int[((e*x)^m*(a + b*x^2)^4*(A + B*x^2))/(c + d*x^2),x]","\frac{b (e x)^{m+3} \left(-6 a^2 b d^2 (B c-A d)+4 a^3 B d^3+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(6 a^2 b^2 c d^2 (B c-A d)-4 a^3 b d^3 (B c-A d)+a^4 B d^4-4 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^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^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)}","\frac{b (e x)^{m+3} \left(-6 a^2 b d^2 (B c-A d)+4 a^3 B d^3+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(6 a^2 b^2 c d^2 (B c-A d)-4 a^3 b d^3 (B c-A d)+a^4 B d^4-4 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^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^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,"((a^4*B*d^4 + b^4*c^3*(B*c - A*d) - 4*a*b^3*c^2*d*(B*c - A*d) + 6*a^2*b^2*c*d^2*(B*c - A*d) - 4*a^3*b*d^3*(B*c - A*d))*(e*x)^(1 + m))/(d^5*e*(1 + m)) + (b*(4*a^3*B*d^3 - b^3*c^2*(B*c - A*d) + 4*a*b^2*c*d*(B*c - A*d) - 6*a^2*b*d^2*(B*c - A*d))*(e*x)^(3 + m))/(d^4*e^3*(3 + m)) + (b^2*(6*a^2*B*d^2 + b^2*c*(B*c - A*d) - 4*a*b*d*(B*c - A*d))*(e*x)^(5 + m))/(d^3*e^5*(5 + m)) - (b^3*(b*B*c - A*b*d - 4*a*B*d)*(e*x)^(7 + m))/(d^2*e^7*(7 + m)) + (b^4*B*(e*x)^(9 + m))/(d*e^9*(9 + m)) - ((b*c - a*d)^4*(B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*d^5*e*(1 + m))","A",3,2,31,0.06452,1,"{570, 364}"
23,1,260,0,0.2950167,"\int \frac{(e x)^m \left(a+b x^2\right)^3 \left(A+B x^2\right)}{c+d x^2} \, dx","Int[((e*x)^m*(a + b*x^2)^3*(A + B*x^2))/(c + d*x^2),x]","\frac{(e x)^{m+1} \left(-3 a^2 b d^2 (B c-A d)+a^3 B d^3+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 (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{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)}","\frac{(e x)^{m+1} \left(-3 a^2 b d^2 (B c-A d)+a^3 B d^3+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 (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{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,"((a^3*B*d^3 - b^3*c^2*(B*c - A*d) + 3*a*b^2*c*d*(B*c - A*d) - 3*a^2*b*d^2*(B*c - A*d))*(e*x)^(1 + m))/(d^4*e*(1 + m)) + (b*(3*a^2*B*d^2 + b^2*c*(B*c - A*d) - 3*a*b*d*(B*c - A*d))*(e*x)^(3 + m))/(d^3*e^3*(3 + m)) - (b^2*(b*B*c - A*b*d - 3*a*B*d)*(e*x)^(5 + m))/(d^2*e^5*(5 + m)) + (b^3*B*(e*x)^(7 + m))/(d*e^7*(7 + m)) + ((b*c - a*d)^3*(B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*d^4*e*(1 + m))","A",3,2,31,0.06452,1,"{570, 364}"
24,1,180,0,0.1875031,"\int \frac{(e x)^m \left(a+b x^2\right)^2 \left(A+B x^2\right)}{c+d x^2} \, dx","Int[((e*x)^m*(a + b*x^2)^2*(A + B*x^2))/(c + d*x^2),x]","\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{b (e x)^{m+3} (-2 a B d-A b d+b B c)}{d^2 e^3 (m+3)}-\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^2 B (e x)^{m+5}}{d e^5 (m+5)}","\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{b (e x)^{m+3} (-2 a B d-A b d+b B c)}{d^2 e^3 (m+3)}-\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^2 B (e x)^{m+5}}{d e^5 (m+5)}",1,"((a^2*B*d^2 + b^2*c*(B*c - A*d) - 2*a*b*d*(B*c - A*d))*(e*x)^(1 + m))/(d^3*e*(1 + m)) - (b*(b*B*c - A*b*d - 2*a*B*d)*(e*x)^(3 + m))/(d^2*e^3*(3 + m)) + (b^2*B*(e*x)^(5 + m))/(d*e^5*(5 + m)) - ((b*c - a*d)^2*(B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*d^3*e*(1 + m))","A",3,2,31,0.06452,1,"{570, 364}"
25,1,120,0,0.0956785,"\int \frac{(e x)^m \left(a+b x^2\right) \left(A+B x^2\right)}{c+d x^2} \, dx","Int[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2),x]","\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)}","\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,"-(((b*B*c - A*b*d - a*B*d)*(e*x)^(1 + m))/(d^2*e*(1 + m))) + (b*B*(e*x)^(3 + m))/(d*e^3*(3 + m)) + ((b*c - a*d)*(B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*d^2*e*(1 + m))","A",3,2,29,0.06897,1,"{570, 364}"
26,1,77,0,0.0383385,"\int \frac{(e x)^m \left(A+B x^2\right)}{c+d x^2} \, dx","Int[((e*x)^m*(A + B*x^2))/(c + d*x^2),x]","\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)}","\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,"(B*(e*x)^(1 + m))/(d*e*(1 + m)) - ((B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*d*e*(1 + m))","A",2,2,22,0.09091,1,"{459, 364}"
27,1,125,0,0.1371937,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(a+b x^2\right) \left(c+d x^2\right)} \, dx","Int[((e*x)^m*(A + B*x^2))/((a + b*x^2)*(c + d*x^2)),x]","\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)}","\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,"((A*b - a*B)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*(b*c - a*d)*e*(1 + m)) + ((B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(b*c - a*d)*e*(1 + m))","A",4,2,31,0.06452,1,"{584, 364}"
28,1,206,0,0.3853956,"\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","Int[((e*x)^m*(A + B*x^2))/((a + b*x^2)^2*(c + d*x^2)),x]","\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)}","\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,"((A*b - a*B)*(e*x)^(1 + m))/(2*a*(b*c - a*d)*e*(a + b*x^2)) + ((A*b*(b*c*(1 - m) - a*d*(3 - m)) + a*B*(a*d*(1 - m) + b*c*(1 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*(b*c - a*d)^2*e*(1 + m)) - (d*(B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(b*c - a*d)^2*e*(1 + m))","A",5,3,31,0.09677,1,"{579, 584, 364}"
29,1,342,0,0.7749349,"\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","Int[((e*x)^m*(A + B*x^2))/((a + b*x^2)^3*(c + d*x^2)),x]","\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{(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{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)}","\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{(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{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,"((A*b - a*B)*(e*x)^(1 + m))/(4*a*(b*c - a*d)*e*(a + b*x^2)^2) + ((A*b*(b*c*(3 - m) - a*d*(7 - m)) + a*B*(a*d*(3 - m) + b*c*(1 + m)))*(e*x)^(1 + m))/(8*a^2*(b*c - a*d)^2*e*(a + b*x^2)) + ((A*b*(a^2*d^2*(15 - 8*m + m^2) - 2*a*b*c*d*(5 - 6*m + m^2) + b^2*c^2*(3 - 4*m + m^2)) + a*B*(b^2*c^2*(1 - m^2) - 2*a*b*c*d*(3 + 2*m - m^2) - a^2*d^2*(3 - 4*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(8*a^3*(b*c - a*d)^3*e*(1 + m)) + (d^2*(B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*(b*c - a*d)^3*e*(1 + m))","A",6,3,31,0.09677,1,"{579, 584, 364}"
30,1,340,0,0.7155921,"\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","Int[((e*x)^m*(a + b*x^2)^3*(A + B*x^2))/(c + d*x^2)^2,x]","-\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)}","-\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,"-(b*(3*a^2*d^2*(A*d*(1 + m) - B*c*(3 + m)) - 3*a*b*c*d*(A*d*(3 + m) - B*c*(5 + m)) + b^2*c^2*(A*d*(5 + m) - B*c*(7 + m)))*(e*x)^(1 + m))/(2*c*d^4*e*(1 + m)) - (b^2*(3*a*d*(A*d*(3 + m) - B*c*(5 + m)) - b*c*(A*d*(5 + m) - B*c*(7 + m)))*(e*x)^(3 + m))/(2*c*d^3*e^3*(3 + m)) - (b^3*(A*d*(5 + m) - B*c*(7 + m))*(e*x)^(5 + m))/(2*c*d^2*e^5*(5 + m)) - ((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^2)^3)/(2*c*d*e*(c + d*x^2)) + ((b*c - a*d)^2*(a*d*(A*d*(1 - m) + B*c*(1 + m)) + b*c*(A*d*(5 + m) - B*c*(7 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*d^4*e*(1 + m))","A",4,3,31,0.09677,1,"{577, 570, 364}"
31,1,246,0,0.404804,"\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","Int[((e*x)^m*(a + b*x^2)^2*(A + B*x^2))/(c + d*x^2)^2,x]","-\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)}","-\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,"-(b*(2*a*d*(A*d*(1 + m) - B*c*(3 + m)) - b*c*(A*d*(3 + m) - B*c*(5 + m)))*(e*x)^(1 + m))/(2*c*d^3*e*(1 + m)) - (b^2*(A*d*(3 + m) - B*c*(5 + m))*(e*x)^(3 + m))/(2*c*d^2*e^3*(3 + m)) - ((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^2)^2)/(2*c*d*e*(c + d*x^2)) - ((b*c - a*d)*(a*d*(A*d*(1 - m) + B*c*(1 + m)) + b*c*(A*d*(3 + m) - B*c*(5 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*d^3*e*(1 + m))","A",4,3,31,0.09677,1,"{577, 570, 364}"
32,1,171,0,0.2267894,"\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","Int[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2)^2,x]","\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)}","\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,"-(B*(a*d*(1 + m) - b*c*(3 + m))*(e*x)^(1 + m))/(2*c*d^2*e*(1 + m)) - ((b*c - a*d)*(e*x)^(1 + m)*(A + B*x^2))/(2*c*d*e*(c + d*x^2)) + ((a*d*(A*d*(1 - m) + B*c*(1 + m)) + b*c*(A*d*(1 + m) - B*c*(3 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*d^2*e*(1 + m))","A",3,3,29,0.1034,1,"{577, 459, 364}"
33,1,103,0,0.0469854,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(c+d x^2\right)^2} \, dx","Int[((e*x)^m*(A + B*x^2))/(c + d*x^2)^2,x]","\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)}","\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,"-((B*c - A*d)*(e*x)^(1 + m))/(2*c*d*e*(c + d*x^2)) + ((A*d*(1 - m) + B*c*(1 + m))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*d*e*(1 + m))","A",2,2,22,0.09091,1,"{457, 364}"
34,1,205,0,0.3832967,"\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","Int[((e*x)^m*(A + B*x^2))/((a + b*x^2)*(c + d*x^2)^2),x]","\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)}","\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,"((B*c - A*d)*(e*x)^(1 + m))/(2*c*(b*c - a*d)*e*(c + d*x^2)) + (b*(A*b - a*B)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*(b*c - a*d)^2*e*(1 + m)) + ((b*c*(B*c*(1 - m) - A*d*(3 - m)) + a*d*(A*d*(1 - m) + B*c*(1 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*(b*c - a*d)^2*e*(1 + m))","A",5,3,31,0.09677,1,"{579, 584, 364}"
35,1,304,0,0.8024006,"\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","Int[((e*x)^m*(A + B*x^2))/((a + b*x^2)^2*(c + d*x^2)^2),x]","\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)}","\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,"(d*(A*b*c - 2*a*B*c + a*A*d)*(e*x)^(1 + m))/(2*a*c*(b*c - a*d)^2*e*(c + d*x^2)) + ((A*b - a*B)*(e*x)^(1 + m))/(2*a*(b*c - a*d)*e*(a + b*x^2)*(c + d*x^2)) + (b*(A*b*(b*c*(1 - m) - a*d*(5 - m)) + a*B*(a*d*(3 - m) + b*c*(1 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*(b*c - a*d)^3*e*(1 + m)) - (d*(b*c*(B*c*(3 - m) - A*d*(5 - m)) + a*d*(A*d*(1 - m) + B*c*(1 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*(b*c - a*d)^3*e*(1 + m))","A",6,3,31,0.09677,1,"{579, 584, 364}"
36,1,491,0,1.4252058,"\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","Int[((e*x)^m*(A + B*x^2))/((a + b*x^2)^3*(c + d*x^2)^2),x]","\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 (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{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)}","\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 (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{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,"-(d*(A*(4*a^2*d^2 - b^2*c^2*(3 - m) + a*b*c*d*(11 - m)) - a*B*c*(a*d*(11 - m) + b*c*(1 + m)))*(e*x)^(1 + m))/(8*a^2*c*(b*c - a*d)^3*e*(c + d*x^2)) + ((A*b - a*B)*(e*x)^(1 + m))/(4*a*(b*c - a*d)*e*(a + b*x^2)^2*(c + d*x^2)) + ((A*b*(b*c*(3 - m) - a*d*(9 - m)) + a*B*(a*d*(5 - m) + b*c*(1 + m)))*(e*x)^(1 + m))/(8*a^2*(b*c - a*d)^2*e*(a + b*x^2)*(c + d*x^2)) + (b*(a*B*(b^2*c^2*(1 - m^2) - 2*a*b*c*d*(5 + 4*m - m^2) - a^2*d^2*(15 - 8*m + m^2)) + A*b*(a^2*d^2*(35 - 12*m + m^2) - 2*a*b*c*d*(7 - 8*m + m^2) + b^2*c^2*(3 - 4*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(8*a^3*(b*c - a*d)^4*e*(1 + m)) + (d^2*(b*c*(B*c*(5 - m) - A*d*(7 - m)) + a*d*(A*d*(1 - m) + B*c*(1 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*(b*c - a*d)^4*e*(1 + m))","A",7,3,31,0.09677,1,"{579, 584, 364}"
37,1,433,0,1.1376485,"\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","Int[((e*x)^m*(a + b*x^2)^3*(A + B*x^2))/(c + d*x^2)^3,x]","-\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 (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{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}","-\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 (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{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,"-(b*(2*a^2*d^2*(1 + m)*(A*d*(3 - m) + B*c*(1 + m)) + 3*a*b*c*d*(3 + m)*(A*d*(1 + m) - B*c*(5 + m)) - b^2*c^2*(5 + m)*(A*d*(3 + m) - B*c*(7 + m)))*(e*x)^(1 + m))/(8*c^2*d^4*e*(1 + m)) - (b^2*(a*d*(3 + m)*(A*d*(3 - m) + B*c*(1 + m)) + b*c*(5 + m)*(A*d*(3 + m) - B*c*(7 + m)))*(e*x)^(3 + m))/(8*c^2*d^3*e^3*(3 + m)) - ((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^2)^3)/(4*c*d*e*(c + d*x^2)^2) + ((a*d*(A*d*(3 - m) + B*c*(1 + m)) + b*c*(A*d*(3 + m) - B*c*(7 + m)))*(e*x)^(1 + m)*(a + b*x^2)^2)/(8*c^2*d^2*e*(c + d*x^2)) - ((b*c - a*d)*(a^2*d^2*(1 - m)*(A*d*(3 - m) + B*c*(1 + m)) + b^2*c^2*(5 + m)*(A*d*(3 + m) - B*c*(7 + m)) + 2*a*b*c*d*(A*d*(3 - 2*m - m^2) + B*c*(5 + 6*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*d^4*e*(1 + m))","A",5,3,31,0.09677,1,"{577, 570, 364}"
38,1,292,0,0.4067222,"\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","Int[((e*x)^m*(a + b*x^2)^2*(A + B*x^2))/(c + d*x^2)^3,x]","\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{(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{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{\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}","\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{(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{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{\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,"(b*(a*d*(1 + m) - b*c*(3 + m))*(A*d*(1 + m) - B*c*(5 + m))*(e*x)^(1 + m))/(8*c^2*d^3*e*(1 + m)) - ((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^2)^2)/(4*c*d*e*(c + d*x^2)^2) - ((b*c - a*d)*(e*x)^(1 + m)*(a*(A*d*(3 - m) + B*c*(1 + m)) - b*(A*d*(1 + m) - B*c*(5 + m))*x^2))/(8*c^2*d^2*e*(c + d*x^2)) + ((a*d*(a*d*(1 - m) + b*c*(1 + m))*(A*d*(3 - m) + B*c*(1 + m)) - b*c*(a*d*(1 + m) - b*c*(3 + m))*(A*d*(1 + m) - B*c*(5 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*d^3*e*(1 + m))","A",4,3,31,0.09677,1,"{577, 459, 364}"
39,1,208,0,0.2977858,"\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","Int[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2)^3,x]","\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}","\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,"-((b*c - a*d)*(e*x)^(1 + m)*(A + B*x^2))/(4*c*d*e*(c + d*x^2)^2) + ((b*c*(A*d*(1 + m) - B*c*(3 + m)) + a*d*(A*d*(3 - m) - B*(c - c*m)))*(e*x)^(1 + m))/(8*c^2*d^2*e*(c + d*x^2)) + ((a*d*(1 - m)*(A*d*(3 - m) + B*c*(1 + m)) + b*c*(1 + m)*(A*d*(1 - m) + B*c*(3 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*d^2*e*(1 + m))","A",3,3,29,0.1034,1,"{577, 457, 364}"
40,1,103,0,0.0474653,"\int \frac{(e x)^m \left(A+B x^2\right)}{\left(c+d x^2\right)^3} \, dx","Int[((e*x)^m*(A + B*x^2))/(c + d*x^2)^3,x]","\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}","\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,"-((B*c - A*d)*(e*x)^(1 + m))/(4*c*d*e*(c + d*x^2)^2) + ((A*d*(3 - m) + B*c*(1 + m))*(e*x)^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(4*c^3*d*e*(1 + m))","A",2,2,22,0.09091,1,"{457, 364}"
41,1,333,0,0.7194349,"\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","Int[((e*x)^m*(A + B*x^2))/((a + b*x^2)*(c + d*x^2)^3),x]","\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)}","\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,"((B*c - A*d)*(e*x)^(1 + m))/(4*c*(b*c - a*d)*e*(c + d*x^2)^2) + ((b*c*(B*c*(3 - m) - A*d*(7 - m)) + a*d*(A*d*(3 - m) + B*c*(1 + m)))*(e*x)^(1 + m))/(8*c^2*(b*c - a*d)^2*e*(c + d*x^2)) + (b^2*(A*b - a*B)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*(b*c - a*d)^3*e*(1 + m)) + ((b^2*c^2*(B*c*(1 - m) - A*d*(5 - m))*(3 - m) - a^2*d^2*(1 - m)*(A*d*(3 - m) + B*c*(1 + m)) + 2*a*b*c*d*(B*c*(3 + 2*m - m^2) + A*d*(5 - 6*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*(b*c - a*d)^3*e*(1 + m))","A",6,3,31,0.09677,1,"{579, 584, 364}"
42,1,452,0,1.3375789,"\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","Int[((e*x)^m*(A + B*x^2))/((a + b*x^2)^2*(c + d*x^2)^3),x]","-\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{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{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)}","-\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{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{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,"(d*(2*A*b*c - 3*a*B*c + a*A*d)*(e*x)^(1 + m))/(4*a*c*(b*c - a*d)^2*e*(c + d*x^2)^2) + ((A*b - a*B)*(e*x)^(1 + m))/(2*a*(b*c - a*d)*e*(a + b*x^2)*(c + d*x^2)^2) + (d*(A*(4*b^2*c^2 - a^2*d^2*(3 - m) + a*b*c*d*(11 - m)) - a*B*c*(b*c*(11 - m) + a*d*(1 + m)))*(e*x)^(1 + m))/(8*a*c^2*(b*c - a*d)^3*e*(c + d*x^2)) + (b^2*(A*b*(b*c*(1 - m) - a*d*(7 - m)) + a*B*(a*d*(5 - m) + b*c*(1 + m)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*(b*c - a*d)^4*e*(1 + m)) - (d*(b^2*c^2*(B*c*(3 - m) - A*d*(7 - m))*(5 - m) - a^2*d^2*(1 - m)*(A*d*(3 - m) + B*c*(1 + m)) + 2*a*b*c*d*(B*c*(5 + 4*m - m^2) + A*d*(7 - 8*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*(b*c - a*d)^4*e*(1 + m))","A",7,3,31,0.09677,1,"{579, 584, 364}"
43,1,665,0,2.178635,"\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","Int[((e*x)^m*(A + B*x^2))/((a + b*x^2)^3*(c + d*x^2)^3),x]","\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{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{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{(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{(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)}","\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{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{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{(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{(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,"-(d*(A*(2*a^2*d^2 - b^2*c^2*(3 - m) + a*b*c*d*(13 - m)) - a*B*c*(a*d*(11 - m) + b*c*(1 + m)))*(e*x)^(1 + m))/(8*a^2*c*(b*c - a*d)^3*e*(c + d*x^2)^2) + ((A*b - a*B)*(e*x)^(1 + m))/(4*a*(b*c - a*d)*e*(a + b*x^2)^2*(c + d*x^2)^2) + ((A*b*(b*c*(3 - m) - a*d*(11 - m)) + a*B*(a*d*(7 - m) + b*c*(1 + m)))*(e*x)^(1 + m))/(8*a^2*(b*c - a*d)^2*e*(a + b*x^2)*(c + d*x^2)^2) + (d*(A*(b*c + a*d)*(b^2*c^2*(3 - m) + a^2*d^2*(3 - m) - 2*a*b*c*d*(9 - m)) + a*B*c*(2*a*b*c*d*(11 - m) + b^2*c^2*(1 + m) + a^2*d^2*(1 + m)))*(e*x)^(1 + m))/(8*a^2*c^2*(b*c - a*d)^4*e*(c + d*x^2)) + (b^2*(a*B*(b^2*c^2*(1 - m^2) - 2*a*b*c*d*(7 + 6*m - m^2) - a^2*d^2*(35 - 12*m + m^2)) + A*b*(a^2*d^2*(63 - 16*m + m^2) - 2*a*b*c*d*(9 - 10*m + m^2) + b^2*c^2*(3 - 4*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(8*a^3*(b*c - a*d)^5*e*(1 + m)) + (d^2*(b^2*c^2*(B*c*(5 - m) - A*d*(9 - m))*(7 - m) - a^2*d^2*(1 - m)*(A*d*(3 - m) + B*c*(1 + m)) + 2*a*b*c*d*(B*c*(7 + 6*m - m^2) + A*d*(9 - 10*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*(b*c - a*d)^5*e*(1 + m))","A",8,3,31,0.09677,1,"{579, 584, 364}"
44,1,1047,0,2.5122718,"\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","Int[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2)^3,x]","-\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{(6 b B c-a B d (m+7)+A b 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(c \left(2 b^2 (p+3) (a B (m+1)-A b (m+2 p+9)) c^2-2 a b d (p+3) (a B (m+1)-A b (m+2 p+9)) c+b (b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9)) c+\frac{2 b (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))) c}{m+1}-a d (b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9))+4 a (b c-a d) (6 b B c-a B d (m+7)+A b d (m+2 p+9))\right)-\frac{a \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 (m+2 p+3)}\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^3 e (m+2 p+5) (m+2 p+7) (m+2 p+9)}","-\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,"-(((a^3*B*d^3*(105 + 71*m + 15*m^2 + m^3) - a^2*b*d^2*(5 + m)*(A*d*(3 + m)*(9 + m + 2*p) + 2*B*c*(30 + 13*m + m^2 + 2*p + 2*m*p)) + a*b^2*c*d*(2*A*d*(216 + m^3 + 84*p + 8*p^2 + 4*m^2*(5 + p) + m*(123 + 44*p + 4*p^2)) + B*c*(267 + m^3 + 40*p + 4*p^2 + m^2*(21 + 4*p) + m*(143 + 44*p + 4*p^2))) - b^3*c^2*(48*B*c + A*d*(513 + m^3 + 366*p + 92*p^2 + 8*p^3 + m^2*(23 + 6*p) + m*(183 + 92*p + 12*p^2))))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(b^4*e*(3 + m + 2*p)*(5 + m + 2*p)*(7 + m + 2*p)*(9 + m + 2*p))) + ((a^2*B*d^2*(35 + 12*m + m^2) + b^2*c*(24*B*c + A*d*(99 + m^2 + 40*p + 4*p^2 + 4*m*(5 + p))) - a*b*d*(A*d*(5 + m)*(9 + m + 2*p) + B*c*(65 + m^2 + 2*p + 2*m*(9 + p))))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2))/(b^3*e*(5 + m + 2*p)*(7 + m + 2*p)*(9 + m + 2*p)) + ((6*b*B*c - a*B*d*(7 + m) + A*b*d*(9 + m + 2*p))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(b^2*e*(7 + m + 2*p)*(9 + m + 2*p)) + (B*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2)^3)/(b*e*(9 + m + 2*p)) - ((c*(2*b^2*c^2*(3 + p)*(a*B*(1 + m) - A*b*(9 + m + 2*p)) - 2*a*b*c*d*(3 + p)*(a*B*(1 + m) - A*b*(9 + m + 2*p)) + b*c*(b*c - a*d)*(1 + m)*(a*B*(7 + m) - A*b*(9 + m + 2*p)) - a*d*(b*c - a*d)*(1 + m)*(a*B*(7 + m) - A*b*(9 + m + 2*p)) + 4*a*(b*c - a*d)*(6*b*B*c - a*B*d*(7 + m) + A*b*d*(9 + m + 2*p)) + (2*b*c*(2 + p)*(2*b*c*(3 + p)*(a*B*(1 + m) - A*b*(9 + m + 2*p)) + (b*c - a*d)*(1 + m)*(a*B*(7 + m) - A*b*(9 + m + 2*p))))/(1 + m)) - (a*(a^3*B*d^3*(105 + 71*m + 15*m^2 + m^3) - a^2*b*d^2*(5 + m)*(A*d*(3 + m)*(9 + m + 2*p) + 2*B*c*(30 + 13*m + m^2 + 2*p + 2*m*p)) + a*b^2*c*d*(2*A*d*(216 + m^3 + 84*p + 8*p^2 + 4*m^2*(5 + p) + m*(123 + 44*p + 4*p^2)) + B*c*(267 + m^3 + 40*p + 4*p^2 + m^2*(21 + 4*p) + m*(143 + 44*p + 4*p^2))) - b^3*c^2*(48*B*c + A*d*(513 + m^3 + 366*p + 92*p^2 + 8*p^3 + m^2*(23 + 6*p) + m*(183 + 92*p + 12*p^2)))))/(b*(3 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(b^3*e*(5 + m + 2*p)*(7 + m + 2*p)*(9 + m + 2*p)*(1 + (b*x^2)/a)^p)","A",6,4,31,0.1290,1,"{581, 459, 365, 364}"
45,1,464,0,0.7486447,"\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","Int[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2)^2,x]","-\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) \left(\frac{a \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 (m+2 p+3)}+c \left((b c-a d) (a B (m+5)-A b (m+2 p+7))+\frac{2 b c (p+2) (a B (m+1)-A b (m+2 p+7))}{m+1}\right)\right)}{b^2 e (m+2 p+5) (m+2 p+7)}+\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{\left(c+d x^2\right) (e x)^{m+1} \left(a+b x^2\right)^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b 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)}","\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,"((a^2*B*d^2*(15 + 8*m + m^2) + b^2*c*(8*B*c + A*d*(7 + m + 2*p)^2) - a*b*d*(A*d*(3 + m)*(7 + m + 2*p) + B*c*(27 + m^2 + 2*p + 2*m*(6 + p))))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(b^3*e*(3 + m + 2*p)*(5 + m + 2*p)*(7 + m + 2*p)) + ((4*b*B*c - a*B*d*(5 + m) + A*b*d*(7 + m + 2*p))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2))/(b^2*e*(5 + m + 2*p)*(7 + m + 2*p)) + (B*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(b*e*(7 + m + 2*p)) - ((c*((2*b*c*(2 + p)*(a*B*(1 + m) - A*b*(7 + m + 2*p)))/(1 + m) + (b*c - a*d)*(a*B*(5 + m) - A*b*(7 + m + 2*p))) + (a*(a^2*B*d^2*(15 + 8*m + m^2) + b^2*c*(8*B*c + A*d*(7 + m + 2*p)^2) - a*b*d*(A*d*(3 + m)*(7 + m + 2*p) + B*c*(27 + m^2 + 2*p + 2*m*(6 + p)))))/(b*(3 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(b^2*e*(5 + m + 2*p)*(7 + m + 2*p)*(1 + (b*x^2)/a)^p)","A",5,4,31,0.1290,1,"{581, 459, 365, 364}"
46,1,238,0,0.2269538,"\int (e x)^m \left(a+b x^2\right)^p \left(A+B x^2\right) \left(c+d x^2\right) \, dx","Int[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2),x]","\frac{(e x)^{m+1} \left(a+b x^2\right)^{p+1} (-a B d (m+3)+2 A b d+b B c (m+2 p+5))}{b^2 e (m+2 p+3) (m+2 p+5)}-\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) \left(\frac{a (-a B d (m+3)+2 A b d+b B c (m+2 p+5))}{b (m+2 p+3)}+a A d-\frac{A b c (m+2 p+5)}{m+1}\right)}{b e (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)}","-\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,"((2*A*b*d - a*B*d*(3 + m) + b*B*c*(5 + m + 2*p))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(b^2*e*(3 + m + 2*p)*(5 + m + 2*p)) + (d*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(A + B*x^2))/(b*e*(5 + m + 2*p)) - ((a*A*d - (A*b*c*(5 + m + 2*p))/(1 + m) + (a*(2*A*b*d - a*B*d*(3 + m) + b*B*c*(5 + m + 2*p)))/(b*(3 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(b*e*(5 + m + 2*p)*(1 + (b*x^2)/a)^p)","A",4,4,29,0.1379,1,"{581, 459, 365, 364}"
47,1,162,0,0.1593539,"\int \frac{(e x)^m \left(a+b x^2\right)^p \left(A+B x^2\right)}{c+d x^2} \, dx","Int[((e*x)^m*(a + b*x^2)^p*(A + B*x^2))/(c + d*x^2),x]","\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)}","\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,"-(((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^2)^p*AppellF1[(1 + m)/2, -p, 1, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)])/(c*d*e*(1 + m)*(1 + (b*x^2)/a)^p)) + (B*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(d*e*(1 + m)*(1 + (b*x^2)/a)^p)","A",6,5,31,0.1613,1,"{584, 365, 364, 511, 510}"
48,1,295,0,0.4134567,"\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","Int[((e*x)^m*(a + b*x^2)^p*(A + B*x^2))/(c + d*x^2)^2,x]","-\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)}","-\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,"((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(2*c*(b*c - a*d)*e*(c + d*x^2)) - ((a*d*(A*d*(1 - m) + B*c*(1 + m)) - b*c*(A*d*(1 - m - 2*p) + B*c*(1 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^p*AppellF1[(1 + m)/2, -p, 1, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)])/(2*c^2*d*(b*c - a*d)*e*(1 + m)*(1 + (b*x^2)/a)^p) - (b*(B*c - A*d)*(1 + m + 2*p)*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(2*c*d*(b*c - a*d)*e*(1 + m)*(1 + (b*x^2)/a)^p)","A",7,6,31,0.1935,1,"{579, 584, 365, 364, 511, 510}"
49,1,483,0,1.0391427,"\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","Int[((e*x)^m*(a + b*x^2)^p*(A + B*x^2))/(c + d*x^2)^3,x]","\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)}","\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,"((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(4*c*(b*c - a*d)*e*(c + d*x^2)^2) + ((a*d*(A*d*(3 - m) + B*c*(1 + m)) + b*c*(B*c*(1 - m - 2*p) - A*d*(5 - m - 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(8*c^2*(b*c - a*d)^2*e*(c + d*x^2)) + ((a^2*d^2*(1 - m)*(A*d*(3 - m) + B*c*(1 + m)) - 2*a*b*c*d*(B*c*(1 + m)*(1 - m - 2*p) + A*d*(1 - m)*(3 - m - 2*p)) + b^2*c^2*(1 - m - 2*p)*(A*d*(3 - m - 2*p) + B*c*(1 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^p*AppellF1[(1 + m)/2, -p, 1, (3 + m)/2, -((b*x^2)/a), -((d*x^2)/c)])/(8*c^3*d*(b*c - a*d)^2*e*(1 + m)*(1 + (b*x^2)/a)^p) - (b*(a*d*(A*d*(3 - m) + B*c*(1 + m)) + b*c*(B*c*(1 - m - 2*p) - A*d*(5 - m - 2*p)))*(1 + m + 2*p)*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(8*c^2*d*(b*c - a*d)^2*e*(1 + m)*(1 + (b*x^2)/a)^p)","A",8,6,31,0.1935,1,"{579, 584, 365, 364, 511, 510}"
50,1,84,0,0.0774443,"\int \frac{\sqrt{a+b x^2} \left(A+B x^2\right) \left(c+d x^2\right)}{x} \, dx","Int[(Sqrt[a + b*x^2]*(A + B*x^2)*(c + d*x^2))/x,x]","-\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)","-\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,"A*c*Sqrt[a + b*x^2] - ((a + b*x^2)^(3/2)*(2*a*B*d - 5*b*(B*c + A*d) - 3*b*B*d*x^2))/(15*b^2) - Sqrt[a]*A*c*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",5,5,29,0.1724,1,"{573, 147, 50, 63, 208}"
51,1,84,0,0.0767331,"\int \frac{\left(a+b x^2\right) \left(A+B x^2\right) \sqrt{c+d x^2}}{x} \, dx","Int[((a + b*x^2)*(A + B*x^2)*Sqrt[c + d*x^2])/x,x]","-\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)","-\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,"a*A*Sqrt[c + d*x^2] - ((c + d*x^2)^(3/2)*(2*b*B*c - 5*(A*b + a*B)*d - 3*b*B*d*x^2))/(15*d^2) - a*A*Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]","A",5,5,29,0.1724,1,"{573, 147, 50, 63, 208}"