Integrand size = 31, antiderivative size = 491 \[ \int \frac {(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^2} \, dx=-\frac {d \left (A \left (4 a^2 d^2-b^2 c^2 (3-m)+a b c d (11-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) (e x)^{1+m}}{8 a^2 c (b c-a d)^3 e \left (c+d x^2\right )}+\frac {(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )}+\frac {(A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b \left (a B \left (b^2 c^2 \left (1-m^2\right )-2 a b c d \left (5+4 m-m^2\right )-a^2 d^2 \left (15-8 m+m^2\right )\right )+A b \left (a^2 d^2 \left (35-12 m+m^2\right )-2 a b c d \left (7-8 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right )\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{8 a^3 (b c-a d)^4 e (1+m)}+\frac {d^2 (b c (B c (5-m)-A d (7-m))+a d (A d (1-m)+B c (1+m))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {d x^2}{c}\right )}{2 c^2 (b c-a d)^4 e (1+m)} \]
-1/8*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)/a^2/c/(-a*d+b*c)^3/e/(d*x^2+c)+1/4*(A*b-B*a)*(e*x)^(1+m )/a/(-a*d+b*c)/e/(b*x^2+a)^2/(d*x^2+c)+1/8*(A*b*(b*c*(3-m)-a*d*(9-m))+a*B* (a*d*(5-m)+b*c*(1+m)))*(e*x)^(1+m)/a^2/(-a*d+b*c)^2/e/(b*x^2+a)/(d*x^2+c)+ 1/8*b*(a*B*(b^2*c^2*(-m^2+1)-2*a*b*c*d*(-m^2+4*m+5)-a^2*d^2*(m^2-8*m+15))+ A*b*(a^2*d^2*(m^2-12*m+35)-2*a*b*c*d*(m^2-8*m+7)+b^2*c^2*(m^2-4*m+3)))*(e* x)^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a^3/(-a*d+b*c)^4/e /(1+m)+1/2*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)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-d*x^2/c)/c^2/(-a*d+b*c)^4/e/( 1+m)
Time = 1.20 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.54 \[ \int \frac {(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^2} \, dx=\frac {x (e x)^m \left (-\frac {b d (2 b B c-3 A b d+a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a}+\frac {d^2 (2 b B c-3 A b d+a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {d x^2}{c}\right )}{c}+\frac {b (b c-a d) (b B c-2 A b d+a B d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^2}+\frac {d^2 (b c-a d) (B c-A d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {d x^2}{c}\right )}{c^2}+\frac {b (A b-a B) (b c-a d)^2 \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^3}\right )}{(b c-a d)^4 (1+m)} \]
(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)*Hyper geometric2F1[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*Hypergeometr ic2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/a^3))/((b*c - a*d)^4*(1 + m) )
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) (e x)^m}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\int \frac {(e x)^m \left (-\left ((A b-a B) d (5-m) x^2\right )+4 a A d-A b c (3-m)-a B c (m+1)\right )}{\left (b x^2+a\right )^2 \left (d x^2+c\right )^2}dx}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int -\frac {(e x)^m \left (d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (b c (1-m)-a d (7-m)) (m+1)+A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right ) (b c-a d)}-\frac {-\frac {\int -\frac {(e x)^m \left (-d (3-m) (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1))) x^2+a B c (m+1) (a d (7-m)-b (c-c m))-A \left (b^2 \left (m^2-4 m+3\right ) c^2-a b d \left (m^2-10 m+5\right ) c+8 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}-\frac {(e x)^{m+1} (A b (b c (3-m)-a d (9-m))+a B (a d (5-m)+b c (m+1)))}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}}{4 a (b c-a d)}\) |
3.1.36.3.1 Defintions of rubi rules used
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si mp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 )^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && LtQ[p, -1]
\[\int \frac {\left (e x \right )^{m} \left (x^{2} B +A \right )}{\left (b \,x^{2}+a \right )^{3} \left (d \,x^{2}+c \right )^{2}}d x\]
\[ \int \frac {(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
integral((B*x^2 + A)*(e*x)^m/(b^3*d^2*x^10 + (2*b^3*c*d + 3*a*b^2*d^2)*x^8 + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*x^6 + a^3*c^2 + (3*a*b^2*c^2 + 6* a^2*b*c*d + a^3*d^2)*x^4 + (3*a^2*b*c^2 + 2*a^3*c*d)*x^2), x)
Timed out. \[ \int \frac {(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
\[ \int \frac {(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^2} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m}{{\left (b\,x^2+a\right )}^3\,{\left (d\,x^2+c\right )}^2} \,d x \]