\(\int (c x)^m (a+b x^n)^3 (A+B x^n) \, dx\) [403]

Optimal result

Integrand size = 22, antiderivative size = 149 \[ \int (c x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {a^3 A (c x)^{1+m}}{c (1+m)}+\frac {a^2 (3 A b+a B) x^n (c x)^{1+m}}{c (1+m+n)}+\frac {3 a b (A b+a B) x^{2 n} (c x)^{1+m}}{c (1+m+2 n)}+\frac {b^2 (A b+3 a B) x^{3 n} (c x)^{1+m}}{c (1+m+3 n)}+\frac {b^3 B x^{4 n} (c x)^{1+m}}{c (1+m+4 n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.71 \[ \int (c x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=x (c x)^m \left (\frac {a^3 A}{1+m}+\frac {a^2 (3 A b+a B) x^n}{1+m+n}+\frac {3 a b (A b+a B) x^{2 n}}{1+m+2 n}+\frac {b^2 (A b+3 a B) x^{3 n}}{1+m+3 n}+\frac {b^3 B x^{4 n}}{1+m+4 n}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {950, 2009}

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.

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

\(\Big \downarrow \) 950

\(\displaystyle \int \left (a^3 A (c x)^m+a^2 x^n (c x)^m (a B+3 A b)+b^2 x^{3 n} (c x)^m (3 a B+A b)+3 a b x^{2 n} (c x)^m (a B+A b)+b^3 B x^{4 n} (c x)^m\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 A (c x)^{m+1}}{c (m+1)}+\frac {a^2 x^{n+1} (c x)^m (a B+3 A b)}{m+n+1}+\frac {b^2 x^{3 n+1} (c x)^m (3 a B+A b)}{m+3 n+1}+\frac {3 a b x^{2 n+1} (c x)^m (a B+A b)}{m+2 n+1}+\frac {b^3 B x^{4 n+1} (c x)^m}{m+4 n+1}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 950

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^ 
n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGt 
Q[p, 0] && IGtQ[q, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.13 (sec) , antiderivative size = 1576, normalized size of antiderivative = 10.58

method	result	size
risch	\(\text {Expression too large to display}\)	\(1576\)
parallelrisch	\(\text {Expression too large to display}\)	\(2207\)
orering	\(\text {Expression too large to display}\)	\(2976\)

Input:

int((c*x)^m*(a+b*x^n)^3*(A+B*x^n),x,method=_RETURNVERBOSE)
 

Output:

x*(6*B*b^3*m^3*n*(x^n)^4+11*B*b^3*m^2*n^2*(x^n)^4+6*B*b^3*m*n^3*(x^n)^4+27 
*A*a^2*b*x^n*n+(x^n)^4*b^3*B+(x^n)^3*b^3*A+18*B*b^3*m^2*n*(x^n)^4+22*B*b^3 
*m*n^2*(x^n)^4+3*A*a*b^2*m^4*(x^n)^2+21*A*b^3*m^2*n*(x^n)^3+28*A*b^3*m*n^2 
*(x^n)^3+72*B*a^2*b*m^2*n*(x^n)^2+114*B*a^2*b*m*n^2*(x^n)^2+63*B*a*b^2*m*n 
*(x^n)^3+81*A*a^2*b*m^2*n*x^n+156*A*a^2*b*m*n^2*x^n+72*A*a*b^2*m*n*(x^n)^2 
+72*B*a^2*b*m*n*(x^n)^2+81*A*a^2*b*m*n*x^n+A*a^3*m^4+7*A*b^3*m^3*n*(x^n)^3 
+14*A*b^3*m^2*n^2*(x^n)^3+8*A*b^3*m*n^3*(x^n)^3+3*B*a*b^2*m^4*(x^n)^3+36*B 
*a^2*b*m*n^3*(x^n)^2+63*B*a*b^2*m^2*n*(x^n)^3+84*B*a*b^2*m*n^2*(x^n)^3+27* 
A*a^2*b*m^3*n*x^n+78*A*a^2*b*m^2*n^2*x^n+72*A*a^2*b*m*n^3*x^n+72*A*a*b^2*m 
^2*n*(x^n)^2+114*A*a*b^2*m*n^2*(x^n)^2+21*B*a*b^2*m^3*n*(x^n)^3+42*B*a*b^2 
*m^2*n^2*(x^n)^3+24*B*a*b^2*m*n^3*(x^n)^3+24*A*a*b^2*m^3*n*(x^n)^2+57*A*a* 
b^2*m^2*n^2*(x^n)^2+24*A*a^3*n^4+4*A*a^3*m^3+50*A*a^3*n^3+6*A*a^3*m^2+4*a^ 
3*A*m+10*a^3*A*n+10*A*a^3*m^3*n+35*A*a^3*m^2*n^2+50*A*a^3*m*n^3+30*A*a^3*m 
^2*n+70*A*a^3*m*n^2+30*A*a^3*m*n+35*A*a^3*n^2+a^3*A+x^n*B*a^3+36*A*a*b^2*m 
*n^3*(x^n)^2+24*B*a^2*b*m^3*n*(x^n)^2+57*B*a^2*b*m^2*n^2*(x^n)^2+21*B*a*b^ 
2*(x^n)^3*n+18*A*a^2*b*m^2*x^n+78*A*a^2*b*n^2*x^n+12*A*a*b^2*(x^n)^2*m+24* 
A*a*b^2*(x^n)^2*n+27*B*a^3*m*n*x^n+12*B*a^2*b*(x^n)^2*m+24*B*a^2*b*(x^n)^2 
*n+12*A*a^2*b*x^n*m+21*A*b^3*m*n*(x^n)^3+9*B*a^3*m^3*n*x^n+26*B*a^3*m^2*n^ 
2*x^n+24*B*a^3*m*n^3*x^n+12*B*a^2*b*m^3*(x^n)^2+36*B*a^2*b*n^3*(x^n)^2+3*B 
*a^2*b*m^4*(x^n)^2+12*B*a*b^2*m^3*(x^n)^3+24*B*a*b^2*n^3*(x^n)^3+18*B*b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1104 vs. \(2 (149) = 298\).

Time = 0.14 (sec) , antiderivative size = 1104, normalized size of antiderivative = 7.41 \[ \int (c x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

((B*b^3*m^4 + 4*B*b^3*m^3 + 6*B*b^3*m^2 + 4*B*b^3*m + B*b^3 + 6*(B*b^3*m + 
 B*b^3)*n^3 + 11*(B*b^3*m^2 + 2*B*b^3*m + B*b^3)*n^2 + 6*(B*b^3*m^3 + 3*B* 
b^3*m^2 + 3*B*b^3*m + B*b^3)*n)*x*x^(4*n)*e^(m*log(c) + m*log(x)) + ((3*B* 
a*b^2 + A*b^3)*m^4 + 3*B*a*b^2 + A*b^3 + 4*(3*B*a*b^2 + A*b^3)*m^3 + 8*(3* 
B*a*b^2 + A*b^3 + (3*B*a*b^2 + A*b^3)*m)*n^3 + 6*(3*B*a*b^2 + A*b^3)*m^2 + 
 14*(3*B*a*b^2 + A*b^3 + (3*B*a*b^2 + A*b^3)*m^2 + 2*(3*B*a*b^2 + A*b^3)*m 
)*n^2 + 4*(3*B*a*b^2 + A*b^3)*m + 7*(3*B*a*b^2 + A*b^3 + (3*B*a*b^2 + A*b^ 
3)*m^3 + 3*(3*B*a*b^2 + A*b^3)*m^2 + 3*(3*B*a*b^2 + A*b^3)*m)*n)*x*x^(3*n) 
*e^(m*log(c) + m*log(x)) + 3*((B*a^2*b + A*a*b^2)*m^4 + B*a^2*b + A*a*b^2 
+ 4*(B*a^2*b + A*a*b^2)*m^3 + 12*(B*a^2*b + A*a*b^2 + (B*a^2*b + A*a*b^2)* 
m)*n^3 + 6*(B*a^2*b + A*a*b^2)*m^2 + 19*(B*a^2*b + A*a*b^2 + (B*a^2*b + A* 
a*b^2)*m^2 + 2*(B*a^2*b + A*a*b^2)*m)*n^2 + 4*(B*a^2*b + A*a*b^2)*m + 8*(B 
*a^2*b + A*a*b^2 + (B*a^2*b + A*a*b^2)*m^3 + 3*(B*a^2*b + A*a*b^2)*m^2 + 3 
*(B*a^2*b + A*a*b^2)*m)*n)*x*x^(2*n)*e^(m*log(c) + m*log(x)) + ((B*a^3 + 3 
*A*a^2*b)*m^4 + B*a^3 + 3*A*a^2*b + 4*(B*a^3 + 3*A*a^2*b)*m^3 + 24*(B*a^3 
+ 3*A*a^2*b + (B*a^3 + 3*A*a^2*b)*m)*n^3 + 6*(B*a^3 + 3*A*a^2*b)*m^2 + 26* 
(B*a^3 + 3*A*a^2*b + (B*a^3 + 3*A*a^2*b)*m^2 + 2*(B*a^3 + 3*A*a^2*b)*m)*n^ 
2 + 4*(B*a^3 + 3*A*a^2*b)*m + 9*(B*a^3 + 3*A*a^2*b + (B*a^3 + 3*A*a^2*b)*m 
^3 + 3*(B*a^3 + 3*A*a^2*b)*m^2 + 3*(B*a^3 + 3*A*a^2*b)*m)*n)*x*x^n*e^(m*lo 
g(c) + m*log(x)) + (A*a^3*m^4 + 24*A*a^3*n^4 + 4*A*a^3*m^3 + 6*A*a^3*m^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 16781 vs. \(2 (134) = 268\).

Time = 6.76 (sec) , antiderivative size = 16781, normalized size of antiderivative = 112.62 \[ \int (c x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise(((A + B)*(a + b)**3*log(x)/c, Eq(m, -1) & Eq(n, 0)), ((A*a**3*lo 
g(x) + 3*A*a**2*b*x**n/n + 3*A*a*b**2*x**(2*n)/(2*n) + A*b**3*x**(3*n)/(3* 
n) + B*a**3*x**n/n + 3*B*a**2*b*x**(2*n)/(2*n) + B*a*b**2*x**(3*n)/n + B*b 
**3*x**(4*n)/(4*n))/c, Eq(m, -1)), (A*a**3*Piecewise((0**(-4*n - 1)*x, Eq( 
c, 0)), (Piecewise((-1/(4*n*(c*x)**(4*n)), Ne(n, 0)), (log(c*x), True))/c, 
 True)) + 3*A*a**2*b*Piecewise((-x*x**n*(c*x)**(-4*n - 1)/(3*n), Ne(n, 0)) 
, (x*x**n*(c*x)**(-4*n - 1)*log(x), True)) + 3*A*a*b**2*Piecewise((-x*x**( 
2*n)*(c*x)**(-4*n - 1)/(2*n), Ne(n, 0)), (x*x**(2*n)*(c*x)**(-4*n - 1)*log 
(x), True)) + A*b**3*Piecewise((-x*x**(3*n)*(c*x)**(-4*n - 1)/n, Ne(n, 0)) 
, (x*x**(3*n)*(c*x)**(-4*n - 1)*log(x), True)) + B*a**3*Piecewise((-x*x**n 
*(c*x)**(-4*n - 1)/(3*n), Ne(n, 0)), (x*x**n*(c*x)**(-4*n - 1)*log(x), Tru 
e)) + 3*B*a**2*b*Piecewise((-x*x**(2*n)*(c*x)**(-4*n - 1)/(2*n), Ne(n, 0)) 
, (x*x**(2*n)*(c*x)**(-4*n - 1)*log(x), True)) + 3*B*a*b**2*Piecewise((-x* 
x**(3*n)*(c*x)**(-4*n - 1)/n, Ne(n, 0)), (x*x**(3*n)*(c*x)**(-4*n - 1)*log 
(x), True)) + B*b**3*x*x**(4*n)*(c*x)**(-4*n - 1)*log(x), Eq(m, -4*n - 1)) 
, (A*a**3*Piecewise((0**(-3*n - 1)*x, Eq(c, 0)), (Piecewise((-1/(3*n*(c*x) 
**(3*n)), Ne(n, 0)), (log(c*x), True))/c, True)) + 3*A*a**2*b*Piecewise((- 
x*x**n*(c*x)**(-3*n - 1)/(2*n), Ne(n, 0)), (x*x**n*(c*x)**(-3*n - 1)*log(x 
), True)) + 3*A*a*b**2*Piecewise((-x*x**(2*n)*(c*x)**(-3*n - 1)/n, Ne(n, 0 
)), (x*x**(2*n)*(c*x)**(-3*n - 1)*log(x), True)) + A*b**3*x*x**(3*n)*(c...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.47 \[ \int (c x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {B b^{3} c^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {3 \, B a b^{2} c^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {A b^{3} c^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {3 \, B a^{2} b c^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {3 \, A a b^{2} c^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {B a^{3} c^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {3 \, A a^{2} b c^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (c x\right )^{m + 1} A a^{3}}{c {\left (m + 1\right )}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7893 vs. \(2 (149) = 298\).

Time = 0.18 (sec) , antiderivative size = 7893, normalized size of antiderivative = 52.97 \[ \int (c x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

(B*b^3*m^4*x*x^(4*n)*e^(m*log(c) + m*log(x)) + 6*B*b^3*m^3*n*x*x^(4*n)*e^( 
m*log(c) + m*log(x)) + 11*B*b^3*m^2*n^2*x*x^(4*n)*e^(m*log(c) + m*log(x)) 
+ 6*B*b^3*m*n^3*x*x^(4*n)*e^(m*log(c) + m*log(x)) + 3*B*a*b^2*m^4*x*x^(3*n 
)*e^(m*log(c) + m*log(x)) + A*b^3*m^4*x*x^(3*n)*e^(m*log(c) + m*log(x)) + 
B*b^3*m^4*x*x^(3*n)*e^(m*log(c) + m*log(x)) + 21*B*a*b^2*m^3*n*x*x^(3*n)*e 
^(m*log(c) + m*log(x)) + 7*A*b^3*m^3*n*x*x^(3*n)*e^(m*log(c) + m*log(x)) + 
 6*B*b^3*m^3*n*x*x^(3*n)*e^(m*log(c) + m*log(x)) + 42*B*a*b^2*m^2*n^2*x*x^ 
(3*n)*e^(m*log(c) + m*log(x)) + 14*A*b^3*m^2*n^2*x*x^(3*n)*e^(m*log(c) + m 
*log(x)) + 11*B*b^3*m^2*n^2*x*x^(3*n)*e^(m*log(c) + m*log(x)) + 24*B*a*b^2 
*m*n^3*x*x^(3*n)*e^(m*log(c) + m*log(x)) + 8*A*b^3*m*n^3*x*x^(3*n)*e^(m*lo 
g(c) + m*log(x)) + 6*B*b^3*m*n^3*x*x^(3*n)*e^(m*log(c) + m*log(x)) + 3*B*a 
^2*b*m^4*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 3*A*a*b^2*m^4*x*x^(2*n)*e^(m* 
log(c) + m*log(x)) + 3*B*a*b^2*m^4*x*x^(2*n)*e^(m*log(c) + m*log(x)) + A*b 
^3*m^4*x*x^(2*n)*e^(m*log(c) + m*log(x)) + B*b^3*m^4*x*x^(2*n)*e^(m*log(c) 
 + m*log(x)) + 24*B*a^2*b*m^3*n*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 24*A*a 
*b^2*m^3*n*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 21*B*a*b^2*m^3*n*x*x^(2*n)* 
e^(m*log(c) + m*log(x)) + 7*A*b^3*m^3*n*x*x^(2*n)*e^(m*log(c) + m*log(x)) 
+ 6*B*b^3*m^3*n*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 57*B*a^2*b*m^2*n^2*x*x 
^(2*n)*e^(m*log(c) + m*log(x)) + 57*A*a*b^2*m^2*n^2*x*x^(2*n)*e^(m*log(c) 
+ m*log(x)) + 42*B*a*b^2*m^2*n^2*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 14...
 

Mupad [B] (verification not implemented)

Time = 4.19 (sec) , antiderivative size = 563, normalized size of antiderivative = 3.78 \[ \int (c x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {A\,a^3\,x\,{\left (c\,x\right )}^m}{m+1}+\frac {b^2\,x\,x^{3\,n}\,{\left (c\,x\right )}^m\,\left (A\,b+3\,B\,a\right )\,\left (m^3+7\,m^2\,n+3\,m^2+14\,m\,n^2+14\,m\,n+3\,m+8\,n^3+14\,n^2+7\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {a^2\,x\,x^n\,{\left (c\,x\right )}^m\,\left (3\,A\,b+B\,a\right )\,\left (m^3+9\,m^2\,n+3\,m^2+26\,m\,n^2+18\,m\,n+3\,m+24\,n^3+26\,n^2+9\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {B\,b^3\,x\,x^{4\,n}\,{\left (c\,x\right )}^m\,\left (m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {3\,a\,b\,x\,x^{2\,n}\,{\left (c\,x\right )}^m\,\left (A\,b+B\,a\right )\,\left (m^3+8\,m^2\,n+3\,m^2+19\,m\,n^2+16\,m\,n+3\,m+12\,n^3+19\,n^2+8\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1} \] Input:

int((c*x)^m*(A + B*x^n)*(a + b*x^n)^3,x)
 

Output:

(A*a^3*x*(c*x)^m)/(m + 1) + (b^2*x*x^(3*n)*(c*x)^m*(A*b + 3*B*a)*(3*m + 7* 
n + 14*m*n + 14*m*n^2 + 7*m^2*n + 3*m^2 + m^3 + 14*n^2 + 8*n^3 + 1))/(4*m 
+ 10*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^ 
3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 1) + (a^2*x*x^n*(c*x)^m* 
(3*A*b + B*a)*(3*m + 9*n + 18*m*n + 26*m*n^2 + 9*m^2*n + 3*m^2 + m^3 + 26* 
n^2 + 24*n^3 + 1))/(4*m + 10*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 
 10*m^3*n + 6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 
1) + (B*b^3*x*x^(4*n)*(c*x)^m*(3*m + 6*n + 12*m*n + 11*m*n^2 + 6*m^2*n + 3 
*m^2 + m^3 + 11*n^2 + 6*n^3 + 1))/(4*m + 10*n + 30*m*n + 70*m*n^2 + 30*m^2 
*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 
+ 35*m^2*n^2 + 1) + (3*a*b*x*x^(2*n)*(c*x)^m*(A*b + B*a)*(3*m + 8*n + 16*m 
*n + 19*m*n^2 + 8*m^2*n + 3*m^2 + m^3 + 19*n^2 + 12*n^3 + 1))/(4*m + 10*n 
+ 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^3 + m^4 
 + 35*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 1)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 996, normalized size of antiderivative = 6.68 \[ \int (c x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {x^{m} c^{m} x \left (144 x^{2 n} a^{2} b^{2} m^{2} n +72 x^{2 n} a^{2} b^{2} m \,n^{3}+228 x^{2 n} a^{2} b^{2} m \,n^{2}+144 x^{2 n} a^{2} b^{2} m n +36 x^{n} a^{3} b \,m^{3} n +104 x^{n} a^{3} b \,m^{2} n^{2}+108 x^{n} a^{3} b \,m^{2} n +96 x^{n} a^{3} b m \,n^{3}+208 x^{n} a^{3} b m \,n^{2}+108 x^{n} a^{3} b m n +28 x^{3 n} a \,b^{3} m^{3} n +56 x^{3 n} a \,b^{3} m^{2} n^{2}+84 x^{3 n} a \,b^{3} m^{2} n +32 x^{3 n} a \,b^{3} m \,n^{3}+112 x^{3 n} a \,b^{3} m \,n^{2}+84 x^{3 n} a \,b^{3} m n +48 x^{2 n} a^{2} b^{2} m^{3} n +114 x^{2 n} a^{2} b^{2} m^{2} n^{2}+11 x^{4 n} b^{4} m^{2} n^{2}+18 x^{4 n} b^{4} m^{2} n +6 x^{4 n} b^{4} m \,n^{3}+22 x^{4 n} b^{4} m \,n^{2}+18 x^{4 n} b^{4} m n +4 x^{3 n} a \,b^{3} m^{4}+16 x^{3 n} a \,b^{3} m^{3}+24 x^{3 n} a \,b^{3} m^{2}+16 x^{3 n} a \,b^{3} m +6 x^{2 n} a^{2} b^{2} m^{4}+24 x^{2 n} a^{2} b^{2} m^{3}+36 x^{2 n} a^{2} b^{2} m^{2}+24 x^{2 n} a^{2} b^{2} m +4 x^{n} a^{3} b \,m^{4}+16 x^{n} a^{3} b \,m^{3}+24 x^{n} a^{3} b \,m^{2}+16 x^{n} a^{3} b m +a^{4} m^{4}+4 a^{4} m^{3}+6 a^{4} m^{2}+4 a^{4} m +4 x^{4 n} b^{4} m^{3}+6 x^{4 n} b^{4} m^{2}+4 x^{4 n} b^{4} m +10 a^{4} m^{3} n +35 a^{4} m^{2} n^{2}+30 a^{4} m^{2} n +50 a^{4} m \,n^{3}+70 a^{4} m \,n^{2}+30 a^{4} m n +6 x^{4 n} b^{4} n^{3}+11 x^{4 n} b^{4} n^{2}+6 x^{4 n} b^{4} n +4 x^{3 n} a \,b^{3}+6 x^{2 n} a^{2} b^{2}+4 x^{n} a^{3} b +a^{4}+6 x^{4 n} b^{4} m^{3} n +x^{4 n} b^{4} m^{4}+24 a^{4} n^{4}+50 a^{4} n^{3}+35 a^{4} n^{2}+10 a^{4} n +x^{4 n} b^{4}+32 x^{3 n} a \,b^{3} n^{3}+56 x^{3 n} a \,b^{3} n^{2}+28 x^{3 n} a \,b^{3} n +72 x^{2 n} a^{2} b^{2} n^{3}+114 x^{2 n} a^{2} b^{2} n^{2}+48 x^{2 n} a^{2} b^{2} n +96 x^{n} a^{3} b \,n^{3}+104 x^{n} a^{3} b \,n^{2}+36 x^{n} a^{3} b n \right )}{m^{5}+10 m^{4} n +35 m^{3} n^{2}+50 m^{2} n^{3}+24 m \,n^{4}+5 m^{4}+40 m^{3} n +105 m^{2} n^{2}+100 m \,n^{3}+24 n^{4}+10 m^{3}+60 m^{2} n +105 m \,n^{2}+50 n^{3}+10 m^{2}+40 m n +35 n^{2}+5 m +10 n +1} \] Input:

int((c*x)^m*(a+b*x^n)^3*(A+B*x^n),x)
 

Output:

(x**m*c**m*x*(x**(4*n)*b**4*m**4 + 6*x**(4*n)*b**4*m**3*n + 4*x**(4*n)*b** 
4*m**3 + 11*x**(4*n)*b**4*m**2*n**2 + 18*x**(4*n)*b**4*m**2*n + 6*x**(4*n) 
*b**4*m**2 + 6*x**(4*n)*b**4*m*n**3 + 22*x**(4*n)*b**4*m*n**2 + 18*x**(4*n 
)*b**4*m*n + 4*x**(4*n)*b**4*m + 6*x**(4*n)*b**4*n**3 + 11*x**(4*n)*b**4*n 
**2 + 6*x**(4*n)*b**4*n + x**(4*n)*b**4 + 4*x**(3*n)*a*b**3*m**4 + 28*x**( 
3*n)*a*b**3*m**3*n + 16*x**(3*n)*a*b**3*m**3 + 56*x**(3*n)*a*b**3*m**2*n** 
2 + 84*x**(3*n)*a*b**3*m**2*n + 24*x**(3*n)*a*b**3*m**2 + 32*x**(3*n)*a*b* 
*3*m*n**3 + 112*x**(3*n)*a*b**3*m*n**2 + 84*x**(3*n)*a*b**3*m*n + 16*x**(3 
*n)*a*b**3*m + 32*x**(3*n)*a*b**3*n**3 + 56*x**(3*n)*a*b**3*n**2 + 28*x**( 
3*n)*a*b**3*n + 4*x**(3*n)*a*b**3 + 6*x**(2*n)*a**2*b**2*m**4 + 48*x**(2*n 
)*a**2*b**2*m**3*n + 24*x**(2*n)*a**2*b**2*m**3 + 114*x**(2*n)*a**2*b**2*m 
**2*n**2 + 144*x**(2*n)*a**2*b**2*m**2*n + 36*x**(2*n)*a**2*b**2*m**2 + 72 
*x**(2*n)*a**2*b**2*m*n**3 + 228*x**(2*n)*a**2*b**2*m*n**2 + 144*x**(2*n)* 
a**2*b**2*m*n + 24*x**(2*n)*a**2*b**2*m + 72*x**(2*n)*a**2*b**2*n**3 + 114 
*x**(2*n)*a**2*b**2*n**2 + 48*x**(2*n)*a**2*b**2*n + 6*x**(2*n)*a**2*b**2 
+ 4*x**n*a**3*b*m**4 + 36*x**n*a**3*b*m**3*n + 16*x**n*a**3*b*m**3 + 104*x 
**n*a**3*b*m**2*n**2 + 108*x**n*a**3*b*m**2*n + 24*x**n*a**3*b*m**2 + 96*x 
**n*a**3*b*m*n**3 + 208*x**n*a**3*b*m*n**2 + 108*x**n*a**3*b*m*n + 16*x**n 
*a**3*b*m + 96*x**n*a**3*b*n**3 + 104*x**n*a**3*b*n**2 + 36*x**n*a**3*b*n 
+ 4*x**n*a**3*b + a**4*m**4 + 10*a**4*m**3*n + 4*a**4*m**3 + 35*a**4*m*...