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

Optimal. Leaf size=379 \[ \frac{a^3 A c^3 (e x)^{m+1}}{e (m+1)}+\frac{3 a c (e x)^{m+5} \left (A \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a B c (a d+b c)\right )}{e^5 (m+5)}+\frac{3 b d (e x)^{m+11} \left (a^2 B d^2+a b d (A d+3 B c)+b^2 c (A d+B c)\right )}{e^{11} (m+11)}+\frac{a^2 c^2 (e x)^{m+3} (3 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac{(e x)^{m+9} \left (a^3 B d^3+3 a^2 b d^2 (A d+3 B c)+9 a b^2 c d (A d+B c)+b^3 c^2 (3 A d+B c)\right )}{e^9 (m+9)}+\frac{(e x)^{m+7} \left (3 a B c \left (a^2 d^2+3 a b c d+b^2 c^2\right )+A \left (a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right )\right )}{e^7 (m+7)}+\frac{b^2 d^2 (e x)^{m+13} (3 a B d+A b d+3 b B c)}{e^{13} (m+13)}+\frac{b^3 B d^3 (e x)^{m+15}}{e^{15} (m+15)} \]

[Out]

(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))

_______________________________________________________________________________________

Rubi [A]  time = 1.12521, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032 \[ \frac{a^3 A c^3 (e x)^{m+1}}{e (m+1)}+\frac{3 a c (e x)^{m+5} \left (A \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a B c (a d+b c)\right )}{e^5 (m+5)}+\frac{3 b d (e x)^{m+11} \left (a^2 B d^2+a b d (A d+3 B c)+b^2 c (A d+B c)\right )}{e^{11} (m+11)}+\frac{a^2 c^2 (e x)^{m+3} (3 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac{(e x)^{m+9} \left (a^3 B d^3+3 a^2 b d^2 (A d+3 B c)+9 a b^2 c d (A d+B c)+b^3 c^2 (3 A d+B c)\right )}{e^9 (m+9)}+\frac{(e x)^{m+7} \left (3 a B c \left (a^2 d^2+3 a b c d+b^2 c^2\right )+A \left (a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right )\right )}{e^7 (m+7)}+\frac{b^2 d^2 (e x)^{m+13} (3 a B d+A b d+3 b B c)}{e^{13} (m+13)}+\frac{b^3 B d^3 (e x)^{m+15}}{e^{15} (m+15)} \]

Antiderivative was successfully verified.

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

[Out]

(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))

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(b*x**2+a)**3*(B*x**2+A)*(d*x**2+c)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 2.12169, size = 327, normalized size = 0.86 \[ (e x)^m \left (\frac{a^3 A c^3 x}{m+1}+\frac{3 a c x^5 \left (A \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a B c (a d+b c)\right )}{m+5}+\frac{3 b d x^{11} \left (a^2 B d^2+a b d (A d+3 B c)+b^2 c (A d+B c)\right )}{m+11}+\frac{a^2 c^2 x^3 (3 A (a d+b c)+a B c)}{m+3}+\frac{x^9 \left (a^3 B d^3+3 a^2 b d^2 (A d+3 B c)+9 a b^2 c d (A d+B c)+b^3 c^2 (3 A d+B c)\right )}{m+9}+\frac{x^7 \left (3 a B c \left (a^2 d^2+3 a b c d+b^2 c^2\right )+A \left (a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right )\right )}{m+7}+\frac{b^2 d^2 x^{13} (3 a B d+A b d+3 b B c)}{m+13}+\frac{b^3 B d^3 x^{15}}{m+15}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(e*x)^m*((a^3*A*c^3*x)/(1 + m) + (a^2*c^2*(a*B*c + 3*A*(b*c + a*d))*x^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))*x^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))*x^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))*x^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))*x^11)/(11 + m) + (b^2*d^2*(3*b*B*c + A
*b*d + 3*a*B*d)*x^13)/(13 + m) + (b^3*B*d^3*x^15)/(15 + m))

_______________________________________________________________________________________

Maple [B]  time = 0.015, size = 3953, normalized size = 10.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(B*b^3*d^3*m^7*x^14+49*B*b^3*d^3*m^6*x^14+A*b^3*d^3*m^7*x^12+3*B*a*b^2*d^3*m^7
*x^12+3*B*b^3*c*d^2*m^7*x^12+973*B*b^3*d^3*m^5*x^14+51*A*b^3*d^3*m^6*x^12+153*B*
a*b^2*d^3*m^6*x^12+153*B*b^3*c*d^2*m^6*x^12+10045*B*b^3*d^3*m^4*x^14+3*A*a*b^2*d
^3*m^7*x^10+3*A*b^3*c*d^2*m^7*x^10+1045*A*b^3*d^3*m^5*x^12+3*B*a^2*b*d^3*m^7*x^1
0+9*B*a*b^2*c*d^2*m^7*x^10+3135*B*a*b^2*d^3*m^5*x^12+3*B*b^3*c^2*d*m^7*x^10+3135
*B*b^3*c*d^2*m^5*x^12+57379*B*b^3*d^3*m^3*x^14+159*A*a*b^2*d^3*m^6*x^10+159*A*b^
3*c*d^2*m^6*x^10+11055*A*b^3*d^3*m^4*x^12+159*B*a^2*b*d^3*m^6*x^10+477*B*a*b^2*c
*d^2*m^6*x^10+33165*B*a*b^2*d^3*m^4*x^12+159*B*b^3*c^2*d*m^6*x^10+33165*B*b^3*c*
d^2*m^4*x^12+177331*B*b^3*d^3*m^2*x^14+3*A*a^2*b*d^3*m^7*x^8+9*A*a*b^2*c*d^2*m^7
*x^8+3375*A*a*b^2*d^3*m^5*x^10+3*A*b^3*c^2*d*m^7*x^8+3375*A*b^3*c*d^2*m^5*x^10+6
4339*A*b^3*d^3*m^3*x^12+B*a^3*d^3*m^7*x^8+9*B*a^2*b*c*d^2*m^7*x^8+3375*B*a^2*b*d
^3*m^5*x^10+9*B*a*b^2*c^2*d*m^7*x^8+10125*B*a*b^2*c*d^2*m^5*x^10+193017*B*a*b^2*
d^3*m^3*x^12+B*b^3*c^3*m^7*x^8+3375*B*b^3*c^2*d*m^5*x^10+193017*B*b^3*c*d^2*m^3*
x^12+264207*B*b^3*d^3*m*x^14+165*A*a^2*b*d^3*m^6*x^8+495*A*a*b^2*c*d^2*m^6*x^8+3
6795*A*a*b^2*d^3*m^4*x^10+165*A*b^3*c^2*d*m^6*x^8+36795*A*b^3*c*d^2*m^4*x^10+201
609*A*b^3*d^3*m^2*x^12+55*B*a^3*d^3*m^6*x^8+495*B*a^2*b*c*d^2*m^6*x^8+36795*B*a^
2*b*d^3*m^4*x^10+495*B*a*b^2*c^2*d*m^6*x^8+110385*B*a*b^2*c*d^2*m^4*x^10+604827*
B*a*b^2*d^3*m^2*x^12+55*B*b^3*c^3*m^6*x^8+36795*B*b^3*c^2*d*m^4*x^10+604827*B*b^
3*c*d^2*m^2*x^12+135135*B*b^3*d^3*x^14+A*a^3*d^3*m^7*x^6+9*A*a^2*b*c*d^2*m^7*x^6
+3639*A*a^2*b*d^3*m^5*x^8+9*A*a*b^2*c^2*d*m^7*x^6+10917*A*a*b^2*c*d^2*m^5*x^8+21
9417*A*a*b^2*d^3*m^3*x^10+A*b^3*c^3*m^7*x^6+3639*A*b^3*c^2*d*m^5*x^8+219417*A*b^
3*c*d^2*m^3*x^10+303255*A*b^3*d^3*m*x^12+3*B*a^3*c*d^2*m^7*x^6+1213*B*a^3*d^3*m^
5*x^8+9*B*a^2*b*c^2*d*m^7*x^6+10917*B*a^2*b*c*d^2*m^5*x^8+219417*B*a^2*b*d^3*m^3
*x^10+3*B*a*b^2*c^3*m^7*x^6+10917*B*a*b^2*c^2*d*m^5*x^8+658251*B*a*b^2*c*d^2*m^3
*x^10+909765*B*a*b^2*d^3*m*x^12+1213*B*b^3*c^3*m^5*x^8+219417*B*b^3*c^2*d*m^3*x^
10+909765*B*b^3*c*d^2*m*x^12+57*A*a^3*d^3*m^6*x^6+513*A*a^2*b*c*d^2*m^6*x^6+4116
9*A*a^2*b*d^3*m^4*x^8+513*A*a*b^2*c^2*d*m^6*x^6+123507*A*a*b^2*c*d^2*m^4*x^8+700
461*A*a*b^2*d^3*m^2*x^10+57*A*b^3*c^3*m^6*x^6+41169*A*b^3*c^2*d*m^4*x^8+700461*A
*b^3*c*d^2*m^2*x^10+155925*A*b^3*d^3*x^12+171*B*a^3*c*d^2*m^6*x^6+13723*B*a^3*d^
3*m^4*x^8+513*B*a^2*b*c^2*d*m^6*x^6+123507*B*a^2*b*c*d^2*m^4*x^8+700461*B*a^2*b*
d^3*m^2*x^10+171*B*a*b^2*c^3*m^6*x^6+123507*B*a*b^2*c^2*d*m^4*x^8+2101383*B*a*b^
2*c*d^2*m^2*x^10+467775*B*a*b^2*d^3*x^12+13723*B*b^3*c^3*m^4*x^8+700461*B*b^3*c^
2*d*m^2*x^10+467775*B*b^3*c*d^2*x^12+3*A*a^3*c*d^2*m^7*x^4+1309*A*a^3*d^3*m^5*x^
6+9*A*a^2*b*c^2*d*m^7*x^4+11781*A*a^2*b*c*d^2*m^5*x^6+253641*A*a^2*b*d^3*m^3*x^8
+3*A*a*b^2*c^3*m^7*x^4+11781*A*a*b^2*c^2*d*m^5*x^6+760923*A*a*b^2*c*d^2*m^3*x^8+
1067445*A*a*b^2*d^3*m*x^10+1309*A*b^3*c^3*m^5*x^6+253641*A*b^3*c^2*d*m^3*x^8+106
7445*A*b^3*c*d^2*m*x^10+3*B*a^3*c^2*d*m^7*x^4+3927*B*a^3*c*d^2*m^5*x^6+84547*B*a
^3*d^3*m^3*x^8+3*B*a^2*b*c^3*m^7*x^4+11781*B*a^2*b*c^2*d*m^5*x^6+760923*B*a^2*b*
c*d^2*m^3*x^8+1067445*B*a^2*b*d^3*m*x^10+3927*B*a*b^2*c^3*m^5*x^6+760923*B*a*b^2
*c^2*d*m^3*x^8+3202335*B*a*b^2*c*d^2*m*x^10+84547*B*b^3*c^3*m^3*x^8+1067445*B*b^
3*c^2*d*m*x^10+177*A*a^3*c*d^2*m^6*x^4+15477*A*a^3*d^3*m^4*x^6+531*A*a^2*b*c^2*d
*m^6*x^4+139293*A*a^2*b*c*d^2*m^4*x^6+831279*A*a^2*b*d^3*m^2*x^8+177*A*a*b^2*c^3
*m^6*x^4+139293*A*a*b^2*c^2*d*m^4*x^6+2493837*A*a*b^2*c*d^2*m^2*x^8+552825*A*a*b
^2*d^3*x^10+15477*A*b^3*c^3*m^4*x^6+831279*A*b^3*c^2*d*m^2*x^8+552825*A*b^3*c*d^
2*x^10+177*B*a^3*c^2*d*m^6*x^4+46431*B*a^3*c*d^2*m^4*x^6+277093*B*a^3*d^3*m^2*x^
8+177*B*a^2*b*c^3*m^6*x^4+139293*B*a^2*b*c^2*d*m^4*x^6+2493837*B*a^2*b*c*d^2*m^2
*x^8+552825*B*a^2*b*d^3*x^10+46431*B*a*b^2*c^3*m^4*x^6+2493837*B*a*b^2*c^2*d*m^2
*x^8+1658475*B*a*b^2*c*d^2*x^10+277093*B*b^3*c^3*m^2*x^8+552825*B*b^3*c^2*d*x^10
+3*A*a^3*c^2*d*m^7*x^2+4239*A*a^3*c*d^2*m^5*x^4+99715*A*a^3*d^3*m^3*x^6+3*A*a^2*
b*c^3*m^7*x^2+12717*A*a^2*b*c^2*d*m^5*x^4+897435*A*a^2*b*c*d^2*m^3*x^6+1291005*A
*a^2*b*d^3*m*x^8+4239*A*a*b^2*c^3*m^5*x^4+897435*A*a*b^2*c^2*d*m^3*x^6+3873015*A
*a*b^2*c*d^2*m*x^8+99715*A*b^3*c^3*m^3*x^6+1291005*A*b^3*c^2*d*m*x^8+B*a^3*c^3*m
^7*x^2+4239*B*a^3*c^2*d*m^5*x^4+299145*B*a^3*c*d^2*m^3*x^6+430335*B*a^3*d^3*m*x^
8+4239*B*a^2*b*c^3*m^5*x^4+897435*B*a^2*b*c^2*d*m^3*x^6+3873015*B*a^2*b*c*d^2*m*
x^8+299145*B*a*b^2*c^3*m^3*x^6+3873015*B*a*b^2*c^2*d*m*x^8+430335*B*b^3*c^3*m*x^
8+183*A*a^3*c^2*d*m^6*x^2+52725*A*a^3*c*d^2*m^4*x^4+340011*A*a^3*d^3*m^2*x^6+183
*A*a^2*b*c^3*m^6*x^2+158175*A*a^2*b*c^2*d*m^4*x^4+3060099*A*a^2*b*c*d^2*m^2*x^6+
675675*A*a^2*b*d^3*x^8+52725*A*a*b^2*c^3*m^4*x^4+3060099*A*a*b^2*c^2*d*m^2*x^6+2
027025*A*a*b^2*c*d^2*x^8+340011*A*b^3*c^3*m^2*x^6+675675*A*b^3*c^2*d*x^8+61*B*a^
3*c^3*m^6*x^2+52725*B*a^3*c^2*d*m^4*x^4+1020033*B*a^3*c*d^2*m^2*x^6+225225*B*a^3
*d^3*x^8+52725*B*a^2*b*c^3*m^4*x^4+3060099*B*a^2*b*c^2*d*m^2*x^6+2027025*B*a^2*b
*c*d^2*x^8+1020033*B*a*b^2*c^3*m^2*x^6+2027025*B*a*b^2*c^2*d*x^8+225225*B*b^3*c^
3*x^8+A*a^3*c^3*m^7+4575*A*a^3*c^2*d*m^5*x^2+360537*A*a^3*c*d^2*m^3*x^4+544095*A
*a^3*d^3*m*x^6+4575*A*a^2*b*c^3*m^5*x^2+1081611*A*a^2*b*c^2*d*m^3*x^4+4896855*A*
a^2*b*c*d^2*m*x^6+360537*A*a*b^2*c^3*m^3*x^4+4896855*A*a*b^2*c^2*d*m*x^6+544095*
A*b^3*c^3*m*x^6+1525*B*a^3*c^3*m^5*x^2+360537*B*a^3*c^2*d*m^3*x^4+1632285*B*a^3*
c*d^2*m*x^6+360537*B*a^2*b*c^3*m^3*x^4+4896855*B*a^2*b*c^2*d*m*x^6+1632285*B*a*b
^2*c^3*m*x^6+63*A*a^3*c^3*m^6+60195*A*a^3*c^2*d*m^4*x^2+1311363*A*a^3*c*d^2*m^2*
x^4+289575*A*a^3*d^3*x^6+60195*A*a^2*b*c^3*m^4*x^2+3934089*A*a^2*b*c^2*d*m^2*x^4
+2606175*A*a^2*b*c*d^2*x^6+1311363*A*a*b^2*c^3*m^2*x^4+2606175*A*a*b^2*c^2*d*x^6
+289575*A*b^3*c^3*x^6+20065*B*a^3*c^3*m^4*x^2+1311363*B*a^3*c^2*d*m^2*x^4+868725
*B*a^3*c*d^2*x^6+1311363*B*a^2*b*c^3*m^2*x^4+2606175*B*a^2*b*c^2*d*x^6+868725*B*
a*b^2*c^3*x^6+1645*A*a^3*c^3*m^5+443577*A*a^3*c^2*d*m^3*x^2+2215701*A*a^3*c*d^2*
m*x^4+443577*A*a^2*b*c^3*m^3*x^2+6647103*A*a^2*b*c^2*d*m*x^4+2215701*A*a*b^2*c^3
*m*x^4+147859*B*a^3*c^3*m^3*x^2+2215701*B*a^3*c^2*d*m*x^4+2215701*B*a^2*b*c^3*m*
x^4+22995*A*a^3*c^3*m^4+1783317*A*a^3*c^2*d*m^2*x^2+1216215*A*a^3*c*d^2*x^4+1783
317*A*a^2*b*c^3*m^2*x^2+3648645*A*a^2*b*c^2*d*x^4+1216215*A*a*b^2*c^3*x^4+594439
*B*a^3*c^3*m^2*x^2+1216215*B*a^3*c^2*d*x^4+1216215*B*a^2*b*c^3*x^4+185059*A*a^3*
c^3*m^3+3422565*A*a^3*c^2*d*m*x^2+3422565*A*a^2*b*c^3*m*x^2+1140855*B*a^3*c^3*m*
x^2+852957*A*a^3*c^3*m^2+2027025*A*a^3*c^2*d*x^2+2027025*A*a^2*b*c^3*x^2+675675*
B*a^3*c^3*x^2+2071215*A*a^3*c^3*m+2027025*A*a^3*c^3)*(e*x)^m/(1+m)/(3+m)/(5+m)/(
7+m)/(9+m)/(11+m)/(13+m)/(15+m)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.277518, size = 3587, normalized size = 9.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((B*b^3*d^3*m^7 + 49*B*b^3*d^3*m^6 + 973*B*b^3*d^3*m^5 + 10045*B*b^3*d^3*m^4 + 5
7379*B*b^3*d^3*m^3 + 177331*B*b^3*d^3*m^2 + 264207*B*b^3*d^3*m + 135135*B*b^3*d^
3)*x^15 + ((3*B*b^3*c*d^2 + (3*B*a*b^2 + A*b^3)*d^3)*m^7 + 467775*B*b^3*c*d^2 +
51*(3*B*b^3*c*d^2 + (3*B*a*b^2 + A*b^3)*d^3)*m^6 + 1045*(3*B*b^3*c*d^2 + (3*B*a*
b^2 + A*b^3)*d^3)*m^5 + 11055*(3*B*b^3*c*d^2 + (3*B*a*b^2 + A*b^3)*d^3)*m^4 + 15
5925*(3*B*a*b^2 + A*b^3)*d^3 + 64339*(3*B*b^3*c*d^2 + (3*B*a*b^2 + A*b^3)*d^3)*m
^3 + 201609*(3*B*b^3*c*d^2 + (3*B*a*b^2 + A*b^3)*d^3)*m^2 + 303255*(3*B*b^3*c*d^
2 + (3*B*a*b^2 + A*b^3)*d^3)*m)*x^13 + 3*((B*b^3*c^2*d + (3*B*a*b^2 + A*b^3)*c*d
^2 + (B*a^2*b + A*a*b^2)*d^3)*m^7 + 184275*B*b^3*c^2*d + 53*(B*b^3*c^2*d + (3*B*
a*b^2 + A*b^3)*c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m^6 + 1125*(B*b^3*c^2*d + (3*B*a
*b^2 + A*b^3)*c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m^5 + 12265*(B*b^3*c^2*d + (3*B*a
*b^2 + A*b^3)*c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m^4 + 184275*(3*B*a*b^2 + A*b^3)*
c*d^2 + 184275*(B*a^2*b + A*a*b^2)*d^3 + 73139*(B*b^3*c^2*d + (3*B*a*b^2 + A*b^3
)*c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m^3 + 233487*(B*b^3*c^2*d + (3*B*a*b^2 + A*b^
3)*c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m^2 + 355815*(B*b^3*c^2*d + (3*B*a*b^2 + A*b
^3)*c*d^2 + (B*a^2*b + A*a*b^2)*d^3)*m)*x^11 + ((B*b^3*c^3 + 3*(3*B*a*b^2 + A*b^
3)*c^2*d + 9*(B*a^2*b + A*a*b^2)*c*d^2 + (B*a^3 + 3*A*a^2*b)*d^3)*m^7 + 225225*B
*b^3*c^3 + 55*(B*b^3*c^3 + 3*(3*B*a*b^2 + A*b^3)*c^2*d + 9*(B*a^2*b + A*a*b^2)*c
*d^2 + (B*a^3 + 3*A*a^2*b)*d^3)*m^6 + 1213*(B*b^3*c^3 + 3*(3*B*a*b^2 + A*b^3)*c^
2*d + 9*(B*a^2*b + A*a*b^2)*c*d^2 + (B*a^3 + 3*A*a^2*b)*d^3)*m^5 + 13723*(B*b^3*
c^3 + 3*(3*B*a*b^2 + A*b^3)*c^2*d + 9*(B*a^2*b + A*a*b^2)*c*d^2 + (B*a^3 + 3*A*a
^2*b)*d^3)*m^4 + 675675*(3*B*a*b^2 + A*b^3)*c^2*d + 2027025*(B*a^2*b + A*a*b^2)*
c*d^2 + 225225*(B*a^3 + 3*A*a^2*b)*d^3 + 84547*(B*b^3*c^3 + 3*(3*B*a*b^2 + A*b^3
)*c^2*d + 9*(B*a^2*b + A*a*b^2)*c*d^2 + (B*a^3 + 3*A*a^2*b)*d^3)*m^3 + 277093*(B
*b^3*c^3 + 3*(3*B*a*b^2 + A*b^3)*c^2*d + 9*(B*a^2*b + A*a*b^2)*c*d^2 + (B*a^3 +
3*A*a^2*b)*d^3)*m^2 + 430335*(B*b^3*c^3 + 3*(3*B*a*b^2 + A*b^3)*c^2*d + 9*(B*a^2
*b + A*a*b^2)*c*d^2 + (B*a^3 + 3*A*a^2*b)*d^3)*m)*x^9 + ((A*a^3*d^3 + (3*B*a*b^2
+ A*b^3)*c^3 + 9*(B*a^2*b + A*a*b^2)*c^2*d + 3*(B*a^3 + 3*A*a^2*b)*c*d^2)*m^7 +
289575*A*a^3*d^3 + 57*(A*a^3*d^3 + (3*B*a*b^2 + A*b^3)*c^3 + 9*(B*a^2*b + A*a*b
^2)*c^2*d + 3*(B*a^3 + 3*A*a^2*b)*c*d^2)*m^6 + 1309*(A*a^3*d^3 + (3*B*a*b^2 + A*
b^3)*c^3 + 9*(B*a^2*b + A*a*b^2)*c^2*d + 3*(B*a^3 + 3*A*a^2*b)*c*d^2)*m^5 + 1547
7*(A*a^3*d^3 + (3*B*a*b^2 + A*b^3)*c^3 + 9*(B*a^2*b + A*a*b^2)*c^2*d + 3*(B*a^3
+ 3*A*a^2*b)*c*d^2)*m^4 + 289575*(3*B*a*b^2 + A*b^3)*c^3 + 2606175*(B*a^2*b + A*
a*b^2)*c^2*d + 868725*(B*a^3 + 3*A*a^2*b)*c*d^2 + 99715*(A*a^3*d^3 + (3*B*a*b^2
+ A*b^3)*c^3 + 9*(B*a^2*b + A*a*b^2)*c^2*d + 3*(B*a^3 + 3*A*a^2*b)*c*d^2)*m^3 +
340011*(A*a^3*d^3 + (3*B*a*b^2 + A*b^3)*c^3 + 9*(B*a^2*b + A*a*b^2)*c^2*d + 3*(B
*a^3 + 3*A*a^2*b)*c*d^2)*m^2 + 544095*(A*a^3*d^3 + (3*B*a*b^2 + A*b^3)*c^3 + 9*(
B*a^2*b + A*a*b^2)*c^2*d + 3*(B*a^3 + 3*A*a^2*b)*c*d^2)*m)*x^7 + 3*((A*a^3*c*d^2
+ (B*a^2*b + A*a*b^2)*c^3 + (B*a^3 + 3*A*a^2*b)*c^2*d)*m^7 + 405405*A*a^3*c*d^2
+ 59*(A*a^3*c*d^2 + (B*a^2*b + A*a*b^2)*c^3 + (B*a^3 + 3*A*a^2*b)*c^2*d)*m^6 +
1413*(A*a^3*c*d^2 + (B*a^2*b + A*a*b^2)*c^3 + (B*a^3 + 3*A*a^2*b)*c^2*d)*m^5 + 1
7575*(A*a^3*c*d^2 + (B*a^2*b + A*a*b^2)*c^3 + (B*a^3 + 3*A*a^2*b)*c^2*d)*m^4 + 4
05405*(B*a^2*b + A*a*b^2)*c^3 + 405405*(B*a^3 + 3*A*a^2*b)*c^2*d + 120179*(A*a^3
*c*d^2 + (B*a^2*b + A*a*b^2)*c^3 + (B*a^3 + 3*A*a^2*b)*c^2*d)*m^3 + 437121*(A*a^
3*c*d^2 + (B*a^2*b + A*a*b^2)*c^3 + (B*a^3 + 3*A*a^2*b)*c^2*d)*m^2 + 738567*(A*a
^3*c*d^2 + (B*a^2*b + A*a*b^2)*c^3 + (B*a^3 + 3*A*a^2*b)*c^2*d)*m)*x^5 + ((3*A*a
^3*c^2*d + (B*a^3 + 3*A*a^2*b)*c^3)*m^7 + 2027025*A*a^3*c^2*d + 61*(3*A*a^3*c^2*
d + (B*a^3 + 3*A*a^2*b)*c^3)*m^6 + 1525*(3*A*a^3*c^2*d + (B*a^3 + 3*A*a^2*b)*c^3
)*m^5 + 20065*(3*A*a^3*c^2*d + (B*a^3 + 3*A*a^2*b)*c^3)*m^4 + 675675*(B*a^3 + 3*
A*a^2*b)*c^3 + 147859*(3*A*a^3*c^2*d + (B*a^3 + 3*A*a^2*b)*c^3)*m^3 + 594439*(3*
A*a^3*c^2*d + (B*a^3 + 3*A*a^2*b)*c^3)*m^2 + 1140855*(3*A*a^3*c^2*d + (B*a^3 + 3
*A*a^2*b)*c^3)*m)*x^3 + (A*a^3*c^3*m^7 + 63*A*a^3*c^3*m^6 + 1645*A*a^3*c^3*m^5 +
22995*A*a^3*c^3*m^4 + 185059*A*a^3*c^3*m^3 + 852957*A*a^3*c^3*m^2 + 2071215*A*a
^3*c^3*m + 2027025*A*a^3*c^3)*x)*(e*x)^m/(m^8 + 64*m^7 + 1708*m^6 + 24640*m^5 +
208054*m^4 + 1038016*m^3 + 2924172*m^2 + 4098240*m + 2027025)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.253886, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done