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

Optimal. Leaf size=210 \[ \frac{c^2 x^{n+1} (e x)^m (3 a A d+a B c+A b c)}{m+n+1}+\frac{d^2 x^{4 n+1} (e x)^m (a B d+A b d+3 b B c)}{m+4 n+1}+\frac{c x^{2 n+1} (e x)^m (3 a d (A d+B c)+b c (3 A d+B c))}{m+2 n+1}+\frac{d x^{3 n+1} (e x)^m (a d (A d+3 B c)+3 b c (A d+B c))}{m+3 n+1}+\frac{a A c^3 (e x)^{m+1}}{e (m+1)}+\frac{b B d^3 x^{5 n+1} (e x)^m}{m+5 n+1} \]

[Out]

(c^2*(A*b*c + a*B*c + 3*a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (c*(3*a*d*(B*c +
A*d) + b*c*(B*c + 3*A*d))*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (d*(3*b*c*(B*c +
A*d) + a*d*(3*B*c + A*d))*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (d^2*(3*b*B*c +
A*b*d + a*B*d)*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (b*B*d^3*x^(1 + 5*n)*(e*x)^m
)/(1 + m + 5*n) + (a*A*c^3*(e*x)^(1 + m))/(e*(1 + m))

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Rubi [A]  time = 0.62886, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{c^2 x^{n+1} (e x)^m (3 a A d+a B c+A b c)}{m+n+1}+\frac{d^2 x^{4 n+1} (e x)^m (a B d+A b d+3 b B c)}{m+4 n+1}+\frac{c x^{2 n+1} (e x)^m (3 a d (A d+B c)+b c (3 A d+B c))}{m+2 n+1}+\frac{d x^{3 n+1} (e x)^m (a d (A d+3 B c)+3 b c (A d+B c))}{m+3 n+1}+\frac{a A c^3 (e x)^{m+1}}{e (m+1)}+\frac{b B d^3 x^{5 n+1} (e x)^m}{m+5 n+1} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(A*b*c + a*B*c + 3*a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (c*(3*a*d*(B*c +
A*d) + b*c*(B*c + 3*A*d))*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (d*(3*b*c*(B*c +
A*d) + a*d*(3*B*c + A*d))*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (d^2*(3*b*B*c +
A*b*d + a*B*d)*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (b*B*d^3*x^(1 + 5*n)*(e*x)^m
)/(1 + m + 5*n) + (a*A*c^3*(e*x)^(1 + m))/(e*(1 + m))

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Rubi in Sympy [A]  time = 100.451, size = 255, normalized size = 1.21 \[ \frac{A a c^{3} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B b d^{3} x^{5 n} \left (e x\right )^{- 5 n} \left (e x\right )^{m + 5 n + 1}}{e \left (m + 5 n + 1\right )} + \frac{c^{2} x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1} \left (3 A a d + A b c + B a c\right )}{e \left (m + n + 1\right )} + \frac{c x^{2 n} \left (e x\right )^{- 2 n} \left (e x\right )^{m + 2 n + 1} \left (B b c^{2} + 3 d \left (A a d + c \left (A b + B a\right )\right )\right )}{e \left (m + 2 n + 1\right )} + \frac{d^{2} x^{4 n} \left (e x\right )^{- 4 n} \left (e x\right )^{m + 4 n + 1} \left (A b d + B a d + 3 B b c\right )}{e \left (m + 4 n + 1\right )} + \frac{d x^{- m} x^{m + 3 n + 1} \left (e x\right )^{m} \left (A a d^{2} + 3 c \left (A b d + B \left (a d + b c\right )\right )\right )}{m + 3 n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*c**3*(e*x)**(m + 1)/(e*(m + 1)) + B*b*d**3*x**(5*n)*(e*x)**(-5*n)*(e*x)**(m
+ 5*n + 1)/(e*(m + 5*n + 1)) + c**2*x**n*(e*x)**(-n)*(e*x)**(m + n + 1)*(3*A*a*d
+ A*b*c + B*a*c)/(e*(m + n + 1)) + c*x**(2*n)*(e*x)**(-2*n)*(e*x)**(m + 2*n + 1
)*(B*b*c**2 + 3*d*(A*a*d + c*(A*b + B*a)))/(e*(m + 2*n + 1)) + d**2*x**(4*n)*(e*
x)**(-4*n)*(e*x)**(m + 4*n + 1)*(A*b*d + B*a*d + 3*B*b*c)/(e*(m + 4*n + 1)) + d*
x**(-m)*x**(m + 3*n + 1)*(e*x)**m*(A*a*d**2 + 3*c*(A*b*d + B*(a*d + b*c)))/(m +
3*n + 1)

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Mathematica [A]  time = 1.42561, size = 172, normalized size = 0.82 \[ x (e x)^m \left (\frac{c^2 x^n (3 a A d+a B c+A b c)}{m+n+1}+\frac{d^2 x^{4 n} (a B d+A b d+3 b B c)}{m+4 n+1}+\frac{c x^{2 n} (3 a d (A d+B c)+b c (3 A d+B c))}{m+2 n+1}+\frac{d x^{3 n} (a d (A d+3 B c)+3 b c (A d+B c))}{m+3 n+1}+\frac{a A c^3}{m+1}+\frac{b B d^3 x^{5 n}}{m+5 n+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

x*(e*x)^m*((a*A*c^3)/(1 + m) + (c^2*(A*b*c + a*B*c + 3*a*A*d)*x^n)/(1 + m + n) +
(c*(3*a*d*(B*c + A*d) + b*c*(B*c + 3*A*d))*x^(2*n))/(1 + m + 2*n) + (d*(3*b*c*(
B*c + A*d) + a*d*(3*B*c + A*d))*x^(3*n))/(1 + m + 3*n) + (d^2*(3*b*B*c + A*b*d +
a*B*d)*x^(4*n))/(1 + m + 4*n) + (b*B*d^3*x^(5*n))/(1 + m + 5*n))

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Maple [C]  time = 0.149, size = 4972, normalized size = 23.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(5*a*A*c^3*m+15*a*A*c^3*n+5*A*a*d^3*(x^n)^3*m+441*B*a*c*d^2*m*n^2*(x^n)^3+216*
B*b*c^2*d*m^2*n*(x^n)^3+441*B*b*c^2*d*m*n^2*(x^n)^3+132*B*b*c*d^2*m*n*(x^n)^4+16
8*A*a*c^2*d*m^3*n*x^n+639*A*a*c^2*d*m^2*n^2*x^n+924*A*a*c^2*d*m*n^3*x^n+5*A*b*d^
3*(x^n)^4*m+3*B*b*c*d^2*(x^n)^4+15*B*a*c*d^2*m^4*(x^n)^3+120*B*a*c*d^2*n^4*(x^n)
^3+66*B*a*d^3*m^2*n*(x^n)^4+123*B*a*d^3*m*n^2*(x^n)^4+13*B*b*c^3*m^4*n*(x^n)^2+5
9*B*b*c^3*m^3*n^2*(x^n)^2+105*B*b*d^3*m*n^2*(x^n)^5+3*A*a*c*d^2*m^5*(x^n)^2+48*A
*a*d^3*m^3*n*(x^n)^3+147*A*a*d^3*m^2*n^2*(x^n)^3+156*A*a*d^3*m*n^3*(x^n)^3+5*B*b
*c^3*(x^n)^2*m+13*B*b*c^3*(x^n)^2*n+56*B*a*c^3*m*n*x^n+15*B*a*c^2*d*(x^n)^2*m+39
*B*a*c^2*d*(x^n)^2*n+15*A*a*c^2*d*x^n*m+42*A*a*c^2*d*x^n*n+180*A*a*c*d^2*n^4*(x^
n)^2+72*A*a*d^3*m^2*n*(x^n)^3+147*A*a*d^3*m*n^2*(x^n)^3+48*A*a*d^3*m*n*(x^n)^3+5
6*A*b*c^3*m^3*n*x^n+213*A*b*c^3*m^2*n^2*x^n+30*B*a*d^3*n^4*(x^n)^4+11*B*a*d^3*(x
^n)^4*n+10*B*b*c^3*m^3*(x^n)^2+107*B*b*c^3*n^3*(x^n)^2+30*B*a*c^2*d*m^2*(x^n)^2+
177*B*a*c^2*d*n^2*(x^n)^2+15*B*a*c*d^2*(x^n)^3*m+36*B*a*c*d^2*(x^n)^3*n+52*B*b*c
^3*m*n*(x^n)^2+369*B*b*c*d^2*m^2*n^2*(x^n)^4+366*B*b*c*d^2*m*n^3*(x^n)^4+39*A*a*
c*d^2*m^4*n*(x^n)^2+177*A*a*c*d^2*m^3*n^2*(x^n)^2+36*B*a*c*d^2*m^4*n*(x^n)^3+147
*B*a*c*d^2*m^3*n^2*(x^n)^3+234*B*a*c*d^2*m^2*n^3*(x^n)^3+30*B*b*c^2*d*m^2*(x^n)^
3+147*B*b*c^2*d*n^2*(x^n)^3+15*B*b*c*d^2*(x^n)^4*m+33*B*b*c*d^2*(x^n)^4*n+30*A*a
*c^2*d*m^3*x^n+462*A*a*c^2*d*n^3*x^n+30*A*a*c*d^2*m^2*(x^n)^2+177*B*a*c^2*d*m^3*
n^2*(x^n)^2+321*B*a*c^2*d*m^2*n^3*(x^n)^2+71*A*b*c^3*n^2*x^n+10*B*a*c^3*m^2*x^n+
71*B*a*c^3*n^2*x^n+107*B*b*c^3*m^2*n^3*(x^n)^2+60*B*b*c^3*m*n^4*(x^n)^2+15*B*b*c
^2*d*m^4*(x^n)^3+120*B*b*c^2*d*n^4*(x^n)^3+30*B*b*c*d^2*m^3*(x^n)^4+183*B*b*c*d^
2*n^3*(x^n)^4+40*B*b*d^3*m*n*(x^n)^5+3*A*a*c^2*d*m^5*x^n+15*A*a*c*d^2*m^4*(x^n)^
2+30*B*a*d^3*m*n^4*(x^n)^4+3*B*b*c*d^2*m^5*(x^n)^4+40*B*b*d^3*m^3*n*(x^n)^5+105*
B*b*d^3*m^2*n^2*(x^n)^5+100*B*b*d^3*m*n^3*(x^n)^5+12*A*a*d^3*m^4*n*(x^n)^3+10*B*
b*c^3*m^2*(x^n)^2+59*B*b*c^3*n^2*(x^n)^2+154*A*b*c^3*n^3*x^n+10*B*a*c^3*m^3*x^n+
154*B*a*c^3*n^3*x^n+5*A*b*c^3*x^n*m+14*A*b*c^3*x^n*n+3*A*b*c^2*d*m^5*(x^n)^2+15*
A*b*c*d^2*m^4*(x^n)^3+120*A*b*c*d^2*n^4*(x^n)^3+66*A*b*d^3*m^2*n*(x^n)^4+123*A*b
*d^3*m*n^2*(x^n)^4+3*B*a*c^2*d*m^5*(x^n)^2+180*A*b*c^2*d*m*n^4*(x^n)^2+144*A*b*c
*d^2*m^3*n*(x^n)^3+441*A*b*c*d^2*m^2*n^2*(x^n)^3+468*A*b*c*d^2*m*n^3*(x^n)^3+39*
B*a*c^2*d*m^4*n*(x^n)^2+120*B*a*c*d^2*m*n^4*(x^n)^3+36*B*b*c^2*d*m^4*n*(x^n)^3+1
47*B*b*c^2*d*m^3*n^2*(x^n)^3+234*B*b*c^2*d*m^2*n^3*(x^n)^3+120*B*b*c^2*d*m*n^4*(
x^n)^3+132*B*b*c*d^2*m^3*n*(x^n)^4+36*A*b*c*d^2*(x^n)^3*n+84*B*a*c^3*m^2*n*x^n+2
13*B*a*c^3*m*n^2*x^n+10*B*b*d^3*m^4*n*(x^n)^5+35*B*b*d^3*m^3*n^2*(x^n)^5+50*B*b*
d^3*m^2*n^3*(x^n)^5+24*B*b*d^3*m*n^4*(x^n)^5+10*B*b*d^3*(x^n)^5*n+40*A*a*d^3*n^4
*(x^n)^3+10*A*b*d^3*m^3*(x^n)^4+3*A*b*c*d^2*(x^n)^3+60*A*a*c^3*m*n+30*A*b*d^3*n^
4*(x^n)^4+5*B*a*d^3*m^4*(x^n)^4+10*B*b*d^3*m^2*(x^n)^5+308*A*b*c^3*m*n^3*x^n+30*
A*b*c^2*d*m^3*(x^n)^2+321*A*b*c^2*d*n^3*(x^n)^2+30*A*b*c*d^2*m^2*(x^n)^3+147*A*b
*c*d^2*n^2*(x^n)^3+56*B*a*c^3*m^3*n*x^n+213*B*a*c^3*m^2*n^2*x^n+308*B*a*c^3*m*n^
3*x^n+30*B*a*c^2*d*m^3*(x^n)^2+321*B*a*c^2*d*n^3*(x^n)^2+30*B*a*c*d^2*m^2*(x^n)^
3+147*B*a*c*d^2*n^2*(x^n)^3+78*B*b*c^3*m^2*n*(x^n)^2+177*B*b*c^3*m*n^2*(x^n)^2+5
0*B*b*d^3*n^3*(x^n)^5+5*A*b*d^3*m^4*(x^n)^4+B*b*d^3*(x^n)^5+A*b*d^3*(x^n)^4+B*a*
d^3*(x^n)^4+a*A*d^3*(x^n)^3+3*B*a*c*d^2*m^5*(x^n)^3+44*B*a*d^3*m^3*n*(x^n)^4+123
*B*a*d^3*m^2*n^2*(x^n)^4+122*B*a*d^3*m*n^3*(x^n)^4+3*B*b*c^2*d*m^5*(x^n)^3+15*B*
b*c*d^2*m^4*(x^n)^4+90*B*b*c*d^2*n^4*(x^n)^4+60*B*b*d^3*m^2*n*(x^n)^5+10*A*b*c^3
*m^3*x^n+10*A*a*d^3*m^3*(x^n)^3+78*A*a*d^3*n^3*(x^n)^3+A*b*c^3*m^5*x^n+10*A*b*d^
3*m^2*(x^n)^4+234*B*b*c^2*d*n^3*(x^n)^3+30*B*b*c*d^2*m^2*(x^n)^4+123*B*b*c*d^2*n
^2*(x^n)^4+15*A*a*c^2*d*m^4*x^n+360*A*a*c^2*d*n^4*x^n+30*A*a*c*d^2*m^3*(x^n)^2+3
21*A*a*c*d^2*n^3*(x^n)^2+11*A*b*d^3*(x^n)^4*n+5*B*a*c^3*m^4*x^n+120*B*a*c^3*n^4*
x^n+5*B*a*d^3*(x^n)^4*m+120*A*b*c*d^2*m*n^4*(x^n)^3+360*A*a*c^2*d*m*n^4*x^n+156*
A*a*c*d^2*m^3*n*(x^n)^2+531*A*a*c*d^2*m^2*n^2*(x^n)^2+642*B*a*c^2*d*m*n^3*(x^n)^
2+216*B*a*c*d^2*m^2*n*(x^n)^3+41*A*b*d^3*n^2*(x^n)^4+B*a*c^3*m^5*x^n+10*B*a*d^3*
m^2*(x^n)^4+41*B*a*d^3*n^2*(x^n)^4+5*B*b*c^3*m^4*(x^n)^2+60*B*b*c^3*n^4*(x^n)^2+
234*A*a*c*d^2*m^2*n*(x^n)^2+531*A*a*c*d^2*m*n^2*(x^n)^2+234*A*b*c^2*d*m^2*n*(x^n
)^2+531*A*b*c^2*d*m*n^2*(x^n)^2+144*A*b*c*d^2*m*n*(x^n)^3+234*B*a*c^2*d*m^2*n*(x
^n)^2+B*a*c^3*x^n+5*A*b*c^3*m^4*x^n+120*A*b*c^3*n^4*x^n+180*B*a*c^2*d*m*n^4*(x^n
)^2+144*B*a*c*d^2*m^3*n*(x^n)^3+441*B*a*c*d^2*m^2*n^2*(x^n)^3+468*B*a*c*d^2*m*n^
3*(x^n)^3+144*B*b*c^2*d*m^3*n*(x^n)^3+3*a*A*c*d^2*(x^n)^2+3*A*b*c^2*d*(x^n)^2+3*
B*a*c^2*d*(x^n)^2+3*a*A*c^2*d*x^n+11*A*b*d^3*m^4*n*(x^n)^4+41*A*b*d^3*m^3*n^2*(x
^n)^4+61*A*b*d^3*m^2*n^3*(x^n)^4+30*A*b*d^3*m*n^4*(x^n)^4+11*B*a*d^3*m^4*n*(x^n)
^4+41*B*a*d^3*m^3*n^2*(x^n)^4+61*B*a*d^3*m^2*n^3*(x^n)^4+10*B*b*d^3*m^3*(x^n)^5+
120*A*a*c^3*n^5+A*a*c^3*m^5+5*A*a*d^3*m^4*(x^n)^3+35*B*b*d^3*n^2*(x^n)^5+3*B*a*c
*d^2*(x^n)^3+3*B*b*c^2*d*(x^n)^3+123*B*b*c*d^2*m^3*n^2*(x^n)^4+183*B*b*c*d^2*m^2
*n^3*(x^n)^4+90*B*b*c*d^2*m*n^4*(x^n)^4+36*A*b*c*d^2*m^4*n*(x^n)^3+147*A*b*c*d^2
*m^3*n^2*(x^n)^3+234*A*b*c*d^2*m^2*n^3*(x^n)^3+321*A*a*c*d^2*m^2*n^3*(x^n)^2+180
*A*a*c*d^2*m*n^4*(x^n)^2+39*A*b*c^2*d*m^4*n*(x^n)^2+177*A*b*c^2*d*m^3*n^2*(x^n)^
2+321*A*b*c^2*d*m^2*n^3*(x^n)^2+B*b*d^3*m^5*(x^n)^5+49*A*a*d^3*n^2*(x^n)^3+462*A
*a*c^2*d*m^2*n^3*x^n+14*A*b*c^3*m^4*n*x^n+71*A*b*c^3*m^3*n^2*x^n+154*A*b*c^3*m^2
*n^3*x^n+120*A*b*c^3*m*n^4*x^n+15*A*b*c^2*d*m^4*(x^n)^2+180*A*b*c^2*d*n^4*(x^n)^
2+30*A*b*c*d^2*m^3*(x^n)^3+234*A*b*c*d^2*n^3*(x^n)^3+5*A*a*c^3*m^4+274*A*a*c^3*n
^4+10*A*a*c^3*m^3+225*A*a*c^3*n^3+10*A*a*c^3*m^2+85*A*a*c^3*n^2+441*B*b*c^2*d*m^
2*n^2*(x^n)^3+468*B*b*c^2*d*m*n^3*(x^n)^3+198*B*b*c*d^2*m^2*n*(x^n)^4+369*B*b*c*
d^2*m*n^2*(x^n)^4+42*A*a*c^2*d*m^4*n*x^n+213*A*a*c^2*d*m^3*n^2*x^n+61*A*b*d^3*n^
3*(x^n)^4+a*A*c^3+10*A*a*d^3*m^2*(x^n)^3+A*a*d^3*m^5*(x^n)^3+30*B*a*c*d^2*m^3*(x
^n)^3+234*B*a*c*d^2*n^3*(x^n)^3+44*B*a*d^3*m*n*(x^n)^4+52*B*b*c^3*m^3*n*(x^n)^2+
177*B*b*c^3*m^2*n^2*(x^n)^2+214*B*b*c^3*m*n^3*(x^n)^2+30*B*b*c^2*d*m^3*(x^n)^3+1
56*B*a*c^2*d*m^3*n*(x^n)^2+531*B*a*c^2*d*m^2*n^2*(x^n)^2+10*B*a*d^3*m^3*(x^n)^4+
61*B*a*d^3*n^3*(x^n)^4+B*b*c^3*m^5*(x^n)^2+531*B*a*c^2*d*m*n^2*(x^n)^2+144*B*a*c
*d^2*m*n*(x^n)^3+144*B*b*c^2*d*m*n*(x^n)^3+252*A*a*c^2*d*m^2*n*x^n+639*A*a*c^2*d
*m*n^2*x^n+156*A*a*c*d^2*m*n*(x^n)^2+156*A*b*c^2*d*m*n*(x^n)^2+177*A*a*c*d^2*n^2
*(x^n)^2+84*A*b*c^3*m^2*n*x^n+213*A*b*c^3*m*n^2*x^n+30*A*b*c^2*d*m^2*(x^n)^2+177
*A*b*c^2*d*n^2*(x^n)^2+15*A*b*c*d^2*(x^n)^3*m+44*A*b*d^3*m*n*(x^n)^4+14*B*a*c^3*
m^4*n*x^n+71*B*a*c^3*m^3*n^2*x^n+154*B*a*c^3*m^2*n^3*x^n+120*B*a*c^3*m*n^4*x^n+1
5*B*a*c^2*d*m^4*(x^n)^2+180*B*a*c^2*d*n^4*(x^n)^2+156*B*a*c^2*d*m*n*(x^n)^2+168*
A*a*c^2*d*m*n*x^n+15*A*a*c^3*m^4*n+85*A*a*c^3*m^3*n^2+225*A*a*c^3*m^2*n^3+274*A*
a*c^3*m*n^4+60*A*a*c^3*m^3*n+255*A*a*c^3*m^2*n^2+450*A*a*c^3*m*n^3+90*A*a*c^3*m^
2*n+255*A*a*c^3*m*n^2+642*A*a*c*d^2*m*n^3*(x^n)^2+33*B*b*c*d^2*m^4*n*(x^n)^4+156
*A*b*c^2*d*m^3*n*(x^n)^2+531*A*b*c^2*d*m^2*n^2*(x^n)^2+642*A*b*c^2*d*m*n^3*(x^n)
^2+216*A*b*c*d^2*m^2*n*(x^n)^3+441*A*b*c*d^2*m*n^2*(x^n)^3+15*B*b*c^2*d*(x^n)^3*
m+36*B*b*c^2*d*(x^n)^3*n+30*A*a*c^2*d*m^2*x^n+213*A*a*c^2*d*n^2*x^n+15*A*a*c*d^2
*(x^n)^2*m+39*A*a*c*d^2*(x^n)^2*n+56*A*b*c^3*m*n*x^n+15*A*b*c^2*d*(x^n)^2*m+39*A
*b*c^2*d*(x^n)^2*n+10*A*b*c^3*m^2*x^n+5*B*a*c^3*x^n*m+14*B*a*c^3*x^n*n+49*A*a*d^
3*m^3*n^2*(x^n)^3+78*A*a*d^3*m^2*n^3*(x^n)^3+40*A*a*d^3*m*n^4*(x^n)^3+3*A*b*c*d^
2*m^5*(x^n)^3+44*A*b*d^3*m^3*n*(x^n)^4+123*A*b*d^3*m^2*n^2*(x^n)^4+122*A*b*d^3*m
*n^3*(x^n)^4+A*b*d^3*m^5*(x^n)^4+B*a*d^3*m^5*(x^n)^4+5*B*b*d^3*m^4*(x^n)^5+24*B*
b*d^3*n^4*(x^n)^5+12*A*a*d^3*(x^n)^3*n+5*m*B*b*d^3*(x^n)^5+B*b*c^3*(x^n)^2+A*b*c
^3*x^n)/(1+m)/(1+m+n)/(1+m+2*n)/(1+m+3*n)/(1+m+4*n)/(1+m+5*n)*exp(1/2*m*(-I*Pi*c
sgn(I*e*x)^3+I*Pi*csgn(I*e*x)^2*csgn(I*e)+I*Pi*csgn(I*e*x)^2*csgn(I*x)-I*Pi*csgn
(I*e*x)*csgn(I*e)*csgn(I*x)+2*ln(e)+2*ln(x)))

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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^n + A)*(b*x^n + a)*(d*x^n + c)^3*(e*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278139, size = 3825, normalized size = 18.21 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((B*b*d^3*m^5 + 5*B*b*d^3*m^4 + 10*B*b*d^3*m^3 + 10*B*b*d^3*m^2 + 5*B*b*d^3*m +
B*b*d^3 + 24*(B*b*d^3*m + B*b*d^3)*n^4 + 50*(B*b*d^3*m^2 + 2*B*b*d^3*m + B*b*d^3
)*n^3 + 35*(B*b*d^3*m^3 + 3*B*b*d^3*m^2 + 3*B*b*d^3*m + B*b*d^3)*n^2 + 10*(B*b*d
^3*m^4 + 4*B*b*d^3*m^3 + 6*B*b*d^3*m^2 + 4*B*b*d^3*m + B*b*d^3)*n)*x*x^(5*n)*e^(
m*log(e) + m*log(x)) + ((3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^5 + 3*B*b*c*d^2 + 5*(3
*B*b*c*d^2 + (B*a + A*b)*d^3)*m^4 + 30*(3*B*b*c*d^2 + (B*a + A*b)*d^3 + (3*B*b*c
*d^2 + (B*a + A*b)*d^3)*m)*n^4 + (B*a + A*b)*d^3 + 10*(3*B*b*c*d^2 + (B*a + A*b)
*d^3)*m^3 + 61*(3*B*b*c*d^2 + (B*a + A*b)*d^3 + (3*B*b*c*d^2 + (B*a + A*b)*d^3)*
m^2 + 2*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m)*n^3 + 10*(3*B*b*c*d^2 + (B*a + A*b)*d
^3)*m^2 + 41*(3*B*b*c*d^2 + (B*a + A*b)*d^3 + (3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^
3 + 3*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^2 + 3*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m)
*n^2 + 5*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m + 11*(3*B*b*c*d^2 + (3*B*b*c*d^2 + (B
*a + A*b)*d^3)*m^4 + (B*a + A*b)*d^3 + 4*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^3 + 6
*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^2 + 4*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m)*n)*x
*x^(4*n)*e^(m*log(e) + m*log(x)) + ((3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2
)*m^5 + 3*B*b*c^2*d + A*a*d^3 + 5*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*
m^4 + 40*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2 + (3*B*b*c^2*d + A*a*d^3 +
3*(B*a + A*b)*c*d^2)*m)*n^4 + 3*(B*a + A*b)*c*d^2 + 10*(3*B*b*c^2*d + A*a*d^3 +
3*(B*a + A*b)*c*d^2)*m^3 + 78*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2 + (3
*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^2 + 2*(3*B*b*c^2*d + A*a*d^3 + 3*(
B*a + A*b)*c*d^2)*m)*n^3 + 10*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^2
+ 49*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2 + (3*B*b*c^2*d + A*a*d^3 + 3*(
B*a + A*b)*c*d^2)*m^3 + 3*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^2 + 3*
(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m)*n^2 + 5*(3*B*b*c^2*d + A*a*d^3
+ 3*(B*a + A*b)*c*d^2)*m + 12*(3*B*b*c^2*d + A*a*d^3 + (3*B*b*c^2*d + A*a*d^3 +
3*(B*a + A*b)*c*d^2)*m^4 + 3*(B*a + A*b)*c*d^2 + 4*(3*B*b*c^2*d + A*a*d^3 + 3*(B
*a + A*b)*c*d^2)*m^3 + 6*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^2 + 4*(
3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m)*n)*x*x^(3*n)*e^(m*log(e) + m*log
(x)) + ((B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^5 + B*b*c^3 + 3*A*a*c*d^
2 + 5*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^4 + 60*(B*b*c^3 + 3*A*a*c*
d^2 + 3*(B*a + A*b)*c^2*d + (B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m)*n^4
+ 3*(B*a + A*b)*c^2*d + 10*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^3 +
107*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d + (B*b*c^3 + 3*A*a*c*d^2 + 3*(B
*a + A*b)*c^2*d)*m^2 + 2*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m)*n^3 +
10*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^2 + 59*(B*b*c^3 + 3*A*a*c*d^2
+ 3*(B*a + A*b)*c^2*d + (B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^3 + 3*(
B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^2 + 3*(B*b*c^3 + 3*A*a*c*d^2 + 3*
(B*a + A*b)*c^2*d)*m)*n^2 + 5*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m +
13*(B*b*c^3 + 3*A*a*c*d^2 + (B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^4 +
3*(B*a + A*b)*c^2*d + 4*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^3 + 6*(B
*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^2 + 4*(B*b*c^3 + 3*A*a*c*d^2 + 3*(
B*a + A*b)*c^2*d)*m)*n)*x*x^(2*n)*e^(m*log(e) + m*log(x)) + ((3*A*a*c^2*d + (B*a
+ A*b)*c^3)*m^5 + 3*A*a*c^2*d + 5*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^4 + 120*(3*
A*a*c^2*d + (B*a + A*b)*c^3 + (3*A*a*c^2*d + (B*a + A*b)*c^3)*m)*n^4 + (B*a + A*
b)*c^3 + 10*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^3 + 154*(3*A*a*c^2*d + (B*a + A*b)
*c^3 + (3*A*a*c^2*d + (B*a + A*b)*c^3)*m^2 + 2*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m
)*n^3 + 10*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^2 + 71*(3*A*a*c^2*d + (B*a + A*b)*c
^3 + (3*A*a*c^2*d + (B*a + A*b)*c^3)*m^3 + 3*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^2
+ 3*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m)*n^2 + 5*(3*A*a*c^2*d + (B*a + A*b)*c^3)*
m + 14*(3*A*a*c^2*d + (3*A*a*c^2*d + (B*a + A*b)*c^3)*m^4 + (B*a + A*b)*c^3 + 4*
(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^3 + 6*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^2 + 4*
(3*A*a*c^2*d + (B*a + A*b)*c^3)*m)*n)*x*x^n*e^(m*log(e) + m*log(x)) + (A*a*c^3*m
^5 + 120*A*a*c^3*n^5 + 5*A*a*c^3*m^4 + 10*A*a*c^3*m^3 + 10*A*a*c^3*m^2 + 5*A*a*c
^3*m + A*a*c^3 + 274*(A*a*c^3*m + A*a*c^3)*n^4 + 225*(A*a*c^3*m^2 + 2*A*a*c^3*m
+ A*a*c^3)*n^3 + 85*(A*a*c^3*m^3 + 3*A*a*c^3*m^2 + 3*A*a*c^3*m + A*a*c^3)*n^2 +
15*(A*a*c^3*m^4 + 4*A*a*c^3*m^3 + 6*A*a*c^3*m^2 + 4*A*a*c^3*m + A*a*c^3)*n)*x*e^
(m*log(e) + m*log(x)))/(m^6 + 120*(m + 1)*n^5 + 6*m^5 + 274*(m^2 + 2*m + 1)*n^4
+ 15*m^4 + 225*(m^3 + 3*m^2 + 3*m + 1)*n^3 + 20*m^3 + 85*(m^4 + 4*m^3 + 6*m^2 +
4*m + 1)*n^2 + 15*m^2 + 15*(m^5 + 5*m^4 + 10*m^3 + 10*m^2 + 5*m + 1)*n + 6*m + 1
)

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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*(a+b*x**n)*(A+B*x**n)*(c+d*x**n)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229425, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done