3.1.44 \(\int (e x)^m (a+b x^2)^p (A+B x^2) (c+d x^2)^3 \, dx\) [44]

3.1.44.1 Optimal result

Integrand size = 31, antiderivative size = 1059 \[ \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=-\frac {\left (a^3 B d^3 \left (105+71 m+15 m^2+m^3\right )-a^2 b d^2 (5+m) \left (A d (3+m) (9+m+2 p)+2 B c \left (30+13 m+m^2+2 p+2 m p\right )\right )+a b^2 c d \left (2 A d \left (216+m^3+84 p+8 p^2+4 m^2 (5+p)+m \left (123+44 p+4 p^2\right )\right )+B c \left (267+m^3+40 p+4 p^2+m^2 (21+4 p)+m \left (143+44 p+4 p^2\right )\right )\right )-b^3 c^2 \left (48 B c+A d \left (513+m^3+366 p+92 p^2+8 p^3+m^2 (23+6 p)+m \left (183+92 p+12 p^2\right )\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^4 e (3+m+2 p) (5+m+2 p) (7+m+2 p) (9+m+2 p)}+\frac {\left (a^2 B d^2 \left (35+12 m+m^2\right )+b^2 c \left (24 B c+A d \left (99+m^2+40 p+4 p^2+4 m (5+p)\right )\right )-a b d \left (A d (5+m) (9+m+2 p)+B c \left (65+m^2+2 p+2 m (9+p)\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^3 e (5+m+2 p) (7+m+2 p) (9+m+2 p)}-\frac {(a B d (7+m)-b (6 B c+A d (9+m+2 p))) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b^2 e (7+m+2 p) (9+m+2 p)}+\frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^3}{b e (9+m+2 p)}+\frac {\left (a (1+m) \left (a^3 B d^3 \left (105+71 m+15 m^2+m^3\right )-a^2 b d^2 (5+m) \left (A d (3+m) (9+m+2 p)+2 B c \left (30+13 m+m^2+2 p+2 m p\right )\right )+a b^2 c d \left (2 A d \left (216+m^3+84 p+8 p^2+4 m^2 (5+p)+m \left (123+44 p+4 p^2\right )\right )+B c \left (267+m^3+40 p+4 p^2+m^2 (21+4 p)+m \left (143+44 p+4 p^2\right )\right )\right )-b^3 c^2 \left (48 B c+A d \left (513+m^3+366 p+92 p^2+8 p^3+m^2 (23+6 p)+m \left (183+92 p+12 p^2\right )\right )\right )\right )-b c (3+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) (b c (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)))-a (2 b c d (3+p) (a B (1+m)-A b (9+m+2 p))+d (b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p))+4 (b c-a d) (a B d (7+m)-b (6 B c+A d (9+m+2 p))))))\right ) (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{b^4 e (1+m) (3+m+2 p) (5+m+2 p) (7+m+2 p) (9+m+2 p)} \]

output

-(a^3*B*d^3*(m^3+15*m^2+71*m+105)-a^2*b*d^2*(5+m)*(A*d*(3+m)*(9+m+2*p)+2*B 
*c*(m^2+2*m*p+13*m+2*p+30))+a*b^2*c*d*(2*A*d*(216+m^3+84*p+8*p^2+4*m^2*(5+ 
p)+m*(4*p^2+44*p+123))+B*c*(267+m^3+40*p+4*p^2+m^2*(21+4*p)+m*(4*p^2+44*p+ 
143)))-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*(12* 
p^2+92*p+183))))*(e*x)^(1+m)*(b*x^2+a)^(p+1)/b^4/e/(3+m+2*p)/(5+m+2*p)/(7+ 
m+2*p)/(9+m+2*p)+(a^2*B*d^2*(m^2+12*m+35)+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)*(b*x^2+a)^(p+1)*(d*x^2+c)/b^3/e/(5+m+2*p)/(7+m+2*p)/(9+m+2*p)-( 
a*B*d*(7+m)-b*(6*B*c+A*d*(9+m+2*p)))*(e*x)^(1+m)*(b*x^2+a)^(p+1)*(d*x^2+c) 
^2/b^2/e/(7+m+2*p)/(9+m+2*p)+B*(e*x)^(1+m)*(b*x^2+a)^(p+1)*(d*x^2+c)^3/b/e 
/(9+m+2*p)+(a*(1+m)*(a^3*B*d^3*(m^3+15*m^2+71*m+105)-a^2*b*d^2*(5+m)*(A*d* 
(3+m)*(9+m+2*p)+2*B*c*(m^2+2*m*p+13*m+2*p+30))+a*b^2*c*d*(2*A*d*(216+m^3+8 
4*p+8*p^2+4*m^2*(5+p)+m*(4*p^2+44*p+123))+B*c*(267+m^3+40*p+4*p^2+m^2*(21+ 
4*p)+m*(4*p^2+44*p+143)))-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*(12*p^2+92*p+183))))-b*c*(3+m+2*p)*(2*b*c*(2+p)*(2*b*c*(3+p 
)*(a*B*(1+m)-A*b*(9+m+2*p))+(-a*d+b*c)*(1+m)*(a*B*(7+m)-A*b*(9+m+2*p)))+(1 
+m)*(b*c*(2*b*c*(3+p)*(a*B*(1+m)-A*b*(9+m+2*p))+(-a*d+b*c)*(1+m)*(a*B*(7+m 
)-A*b*(9+m+2*p)))-a*(2*b*c*d*(3+p)*(a*B*(1+m)-A*b*(9+m+2*p))+d*(-a*d+b*c)* 
(1+m)*(a*B*(7+m)-A*b*(9+m+2*p))+4*(-a*d+b*c)*(a*B*d*(7+m)-b*(6*B*c+A*d*(9+ 
m+2*p)))))))*(e*x)^(1+m)*(b*x^2+a)^p*hypergeom([-p, 1/2+1/2*m],[3/2+1/2...
 

3.1.44.2 Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.23 \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=x (e x)^m \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (\frac {A c^3 \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{1+m}+\frac {c^2 (B c+3 A d) x^2 \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},-p,\frac {5+m}{2},-\frac {b x^2}{a}\right )}{3+m}+d x^4 \left (\frac {3 c (B c+A d) \operatorname {Hypergeometric2F1}\left (\frac {5+m}{2},-p,\frac {7+m}{2},-\frac {b x^2}{a}\right )}{5+m}+d x^2 \left (\frac {(3 B c+A d) \operatorname {Hypergeometric2F1}\left (\frac {7+m}{2},-p,\frac {9+m}{2},-\frac {b x^2}{a}\right )}{7+m}+\frac {B d x^2 \operatorname {Hypergeometric2F1}\left (\frac {9+m}{2},-p,\frac {11+m}{2},-\frac {b x^2}{a}\right )}{9+m}\right )\right )\right ) \]

input

Integrate[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2)^3,x]
 

output

(x*(e*x)^m*(a + b*x^2)^p*((A*c^3*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/ 
2, -((b*x^2)/a)])/(1 + m) + (c^2*(B*c + 3*A*d)*x^2*Hypergeometric2F1[(3 + 
m)/2, -p, (5 + m)/2, -((b*x^2)/a)])/(3 + m) + d*x^4*((3*c*(B*c + A*d)*Hype 
rgeometric2F1[(5 + m)/2, -p, (7 + m)/2, -((b*x^2)/a)])/(5 + m) + d*x^2*((( 
3*B*c + A*d)*Hypergeometric2F1[(7 + m)/2, -p, (9 + m)/2, -((b*x^2)/a)])/(7 
 + m) + (B*d*x^2*Hypergeometric2F1[(9 + m)/2, -p, (11 + m)/2, -((b*x^2)/a) 
])/(9 + m)))))/(1 + (b*x^2)/a)^p
 

3.1.44.3 Rubi [A] (verified)

Time = 1.73 (sec) , antiderivative size = 1020, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {443, 25, 443, 443, 363, 279, 278}

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.

\(\displaystyle \int \left (A+B x^2\right ) \left (c+d x^2\right )^3 (e x)^m \left (a+b x^2\right )^p \, dx\)

\(\Big \downarrow \) 443

\(\displaystyle \frac {\int -(e x)^m \left (b x^2+a\right )^p \left (d x^2+c\right )^2 \left (c (a B (m+1)-A b (m+2 p+9))-(6 b B c-a B d (m+7)+A b d (m+2 p+9)) x^2\right )dx}{b (m+2 p+9)}+\frac {B \left (c+d x^2\right )^3 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+9)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {B \left (c+d x^2\right )^3 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+9)}-\frac {\int (e x)^m \left (b x^2+a\right )^p \left (d x^2+c\right )^2 \left ((a B d (m+7)-b (6 B c+A d (m+2 p+9))) x^2+c (a B (m+1)-A b (m+2 p+9))\right )dx}{b (m+2 p+9)}\)

\(\Big \downarrow \) 443

\(\displaystyle \frac {B \left (c+d x^2\right )^3 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+9)}-\frac {\frac {\int (e x)^m \left (b x^2+a\right )^p \left (d x^2+c\right ) \left ((2 b c d (p+3) (a B (m+1)-A b (m+2 p+9))+d (b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9))+4 (b c-a d) (a B d (m+7)-b (6 B c+A d (m+2 p+9)))) x^2+c (2 b c (p+3) (a B (m+1)-A b (m+2 p+9))+(b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9)))\right )dx}{b (m+2 p+7)}-\frac {\left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+7)+A b d (m+2 p+9)+6 b B c)}{b e (m+2 p+7)}}{b (m+2 p+9)}\)

\(\Big \downarrow \) 443

\(\displaystyle \frac {B \left (c+d x^2\right )^3 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+9)}-\frac {\frac {\frac {\int (e x)^m \left (b x^2+a\right )^p \left (\left (-c^2 \left (48 B c+A d \left (m^3+(6 p+23) m^2+\left (12 p^2+92 p+183\right ) m+8 p^3+92 p^2+366 p+513\right )\right ) b^3+a c d \left (2 A d \left (m^3+4 (p+5) m^2+\left (4 p^2+44 p+123\right ) m+8 p^2+84 p+216\right )+B c \left (m^3+(4 p+21) m^2+\left (4 p^2+44 p+143\right ) m+4 p^2+40 p+267\right )\right ) b^2-a^2 d^2 (m+5) \left (A d (m+3) (m+2 p+9)+2 B c \left (m^2+2 p m+13 m+2 p+30\right )\right ) b+a^3 B d^3 \left (m^3+15 m^2+71 m+105\right )\right ) x^2+c (2 b c (p+2) (2 b c (p+3) (a B (m+1)-A b (m+2 p+9))+(b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9)))+(m+1) (b c (2 b c (p+3) (a B (m+1)-A b (m+2 p+9))+(b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9)))-a (2 b c d (p+3) (a B (m+1)-A b (m+2 p+9))+d (b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9))+4 (b c-a d) (a B d (m+7)-b (6 B c+A d (m+2 p+9))))))\right )dx}{b (m+2 p+5)}-\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+12 m+35\right )-a b d \left (A d (m+5) (m+2 p+9)+B c \left (m^2+2 m (p+9)+2 p+65\right )\right )+b^2 c \left (A d \left (m^2+4 m (p+5)+4 p^2+40 p+99\right )+24 B c\right )\right )}{b e (m+2 p+5)}}{b (m+2 p+7)}-\frac {\left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+7)+A b d (m+2 p+9)+6 b B c)}{b e (m+2 p+7)}}{b (m+2 p+9)}\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {B (e x)^{m+1} \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^3}{b e (m+2 p+9)}-\frac {\frac {\frac {\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 ) (e x)^{m+1} \left (b x^2+a\right )^{p+1}}{b e (m+2 p+3)}-\left (\frac {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 (m+2 p+3)}-c (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 ) \int (e x)^m \left (b x^2+a\right )^pdx}{b (m+2 p+5)}-\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 ) (e x)^{m+1} \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )}{b e (m+2 p+5)}}{b (m+2 p+7)}-\frac {(6 b B c-a B d (m+7)+A b d (m+2 p+9)) (e x)^{m+1} \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^2}{b e (m+2 p+7)}}{b (m+2 p+9)}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {B (e x)^{m+1} \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^3}{b e (m+2 p+9)}-\frac {\frac {\frac {\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 ) (e x)^{m+1} \left (b x^2+a\right )^{p+1}}{b e (m+2 p+3)}-\left (\frac {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 (m+2 p+3)}-c (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} \int (e x)^m \left (\frac {b x^2}{a}+1\right )^pdx}{b (m+2 p+5)}-\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 ) (e x)^{m+1} \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )}{b e (m+2 p+5)}}{b (m+2 p+7)}-\frac {(6 b B c-a B d (m+7)+A b d (m+2 p+9)) (e x)^{m+1} \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^2}{b e (m+2 p+7)}}{b (m+2 p+9)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {B (e x)^{m+1} \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^3}{b e (m+2 p+9)}-\frac {\frac {\frac {\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 ) (e x)^{m+1} \left (b x^2+a\right )^{p+1}}{b e (m+2 p+3)}-\frac {\left (\frac {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 (m+2 p+3)}-c (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 ) (e x)^{m+1} \left (b x^2+a\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},-\frac {b x^2}{a}\right )}{e (m+1)}}{b (m+2 p+5)}-\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 ) (e x)^{m+1} \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )}{b e (m+2 p+5)}}{b (m+2 p+7)}-\frac {(6 b B c-a B d (m+7)+A b d (m+2 p+9)) (e x)^{m+1} \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^2}{b e (m+2 p+7)}}{b (m+2 p+9)}\)

input

Int[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2)^3,x]
 

output

(B*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2)^3)/(b*e*(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*e*(7 + 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*e*(5 + m + 2*p))) + (((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*(18 
3 + 92*p + 12*p^2))))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(b*e*(3 + m + 2*p 
)) - (((a*(1 + m)*(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)) - c*(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)*(b*c*(2*b*c*(3 + p)*(...
 

3.1.44.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 443

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + b*x^2)^(p 
 + 1)*((c + d*x^2)^q/(b*g*(m + 2*(p + q + 1) + 1))), x] + Simp[1/(b*(m + 2* 
(p + q + 1) + 1))   Int[(g*x)^m*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(( 
b*e - a*f)*(m + 1) + b*e*2*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*2*q*(b 
*c - a*d) + b*e*d*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f 
, g, m, p}, x] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^ 
2])
 

3.1.44.4 Maple [F]

\[\int \left (e x \right )^{m} \left (b \,x^{2}+a \right )^{p} \left (x^{2} B +A \right ) \left (d \,x^{2}+c \right )^{3}d x\]

input

int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^3,x)
 

output

int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^3,x)
 

3.1.44.5 Fricas [F]

\[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\int { {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{3} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \]

input

integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="fricas")
 

output

integral((B*d^3*x^8 + (3*B*c*d^2 + A*d^3)*x^6 + 3*(B*c^2*d + A*c*d^2)*x^4 
+ A*c^3 + (B*c^3 + 3*A*c^2*d)*x^2)*(b*x^2 + a)^p*(e*x)^m, x)
 

3.1.44.6 Sympy [F(-1)]

Timed out. \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\text {Timed out} \]

input

integrate((e*x)**m*(b*x**2+a)**p*(B*x**2+A)*(d*x**2+c)**3,x)
 

output

Timed out
 

3.1.44.7 Maxima [F]

\[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\int { {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{3} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \]

input

integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="maxima")
 

output

integrate((B*x^2 + A)*(d*x^2 + c)^3*(b*x^2 + a)^p*(e*x)^m, x)
 

3.1.44.8 Giac [F]

\[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\int { {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{3} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \]

input

integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="giac")
 

output

integrate((B*x^2 + A)*(d*x^2 + c)^3*(b*x^2 + a)^p*(e*x)^m, x)
 

3.1.44.9 Mupad [F(-1)]

Timed out. \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx=\int \left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^3 \,d x \]

input

int((A + B*x^2)*(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^3,x)
 

output

int((A + B*x^2)*(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^3, x)