\(\int x^m (a+b x^3)^5 (A+B x^3) \, dx\) [377]

Optimal result

Integrand size = 20, antiderivative size = 148 \[ \int x^m \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {a^5 A x^{1+m}}{1+m}+\frac {a^4 (5 A b+a B) x^{4+m}}{4+m}+\frac {5 a^3 b (2 A b+a B) x^{7+m}}{7+m}+\frac {10 a^2 b^2 (A b+a B) x^{10+m}}{10+m}+\frac {5 a b^3 (A b+2 a B) x^{13+m}}{13+m}+\frac {b^4 (A b+5 a B) x^{16+m}}{16+m}+\frac {b^5 B x^{19+m}}{19+m} \] Output:

a^5*A*x^(1+m)/(1+m)+a^4*(5*A*b+B*a)*x^(4+m)/(4+m)+5*a^3*b*(2*A*b+B*a)*x^(7 
+m)/(7+m)+10*a^2*b^2*(A*b+B*a)*x^(10+m)/(10+m)+5*a*b^3*(A*b+2*B*a)*x^(13+m 
)/(13+m)+b^4*(A*b+5*B*a)*x^(16+m)/(16+m)+b^5*B*x^(19+m)/(19+m)
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.93 \[ \int x^m \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=x^{1+m} \left (\frac {a^5 A}{1+m}+\frac {a^4 (5 A b+a B) x^3}{4+m}+\frac {5 a^3 b (2 A b+a B) x^6}{7+m}+\frac {10 a^2 b^2 (A b+a B) x^9}{10+m}+\frac {5 a b^3 (A b+2 a B) x^{12}}{13+m}+\frac {b^4 (A b+5 a B) x^{15}}{16+m}+\frac {b^5 B x^{18}}{19+m}\right ) \] Input:

Integrate[x^m*(a + b*x^3)^5*(A + B*x^3),x]
 

Output:

x^(1 + m)*((a^5*A)/(1 + m) + (a^4*(5*A*b + a*B)*x^3)/(4 + m) + (5*a^3*b*(2 
*A*b + a*B)*x^6)/(7 + m) + (10*a^2*b^2*(A*b + a*B)*x^9)/(10 + m) + (5*a*b^ 
3*(A*b + 2*a*B)*x^12)/(13 + m) + (b^4*(A*b + 5*a*B)*x^15)/(16 + m) + (b^5* 
B*x^18)/(19 + m))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

\(\displaystyle \int x^m \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx\)

\(\Big \downarrow \) 950

\(\displaystyle \int \left (a^5 A x^m+a^4 x^{m+3} (a B+5 A b)+5 a^3 b x^{m+6} (a B+2 A b)+10 a^2 b^2 x^{m+9} (a B+A b)+b^4 x^{m+15} (5 a B+A b)+5 a b^3 x^{m+12} (2 a B+A b)+b^5 B x^{m+18}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^5 A x^{m+1}}{m+1}+\frac {a^4 x^{m+4} (a B+5 A b)}{m+4}+\frac {5 a^3 b x^{m+7} (a B+2 A b)}{m+7}+\frac {10 a^2 b^2 x^{m+10} (a B+A b)}{m+10}+\frac {b^4 x^{m+16} (5 a B+A b)}{m+16}+\frac {5 a b^3 x^{m+13} (2 a B+A b)}{m+13}+\frac {b^5 B x^{m+19}}{m+19}\)

Input:

Int[x^m*(a + b*x^3)^5*(A + B*x^3),x]
 

Output:

(a^5*A*x^(1 + m))/(1 + m) + (a^4*(5*A*b + a*B)*x^(4 + m))/(4 + m) + (5*a^3 
*b*(2*A*b + a*B)*x^(7 + m))/(7 + m) + (10*a^2*b^2*(A*b + a*B)*x^(10 + m))/ 
(10 + m) + (5*a*b^3*(A*b + 2*a*B)*x^(13 + m))/(13 + m) + (b^4*(A*b + 5*a*B 
)*x^(16 + m))/(16 + m) + (b^5*B*x^(19 + m))/(19 + 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1076\) vs. \(2(148)=296\).

Time = 1.49 (sec) , antiderivative size = 1077, normalized size of antiderivative = 7.28

method	result	size
risch	\(\text {Expression too large to display}\)	\(1077\)
orering	\(\text {Expression too large to display}\)	\(1077\)
gosper	\(\text {Expression too large to display}\)	\(1078\)
parallelrisch	\(\text {Expression too large to display}\)	\(1332\)

Input:

int(x^m*(b*x^3+a)^5*(B*x^3+A),x,method=_RETURNVERBOSE)
 

Output:

x*(B*b^5*m^6*x^18+51*B*b^5*m^5*x^18+1005*B*b^5*m^4*x^18+A*b^5*m^6*x^15+5*B 
*a*b^4*m^6*x^15+9605*B*b^5*m^3*x^18+54*A*b^5*m^5*x^15+270*B*a*b^4*m^5*x^15 
+45474*B*b^5*m^2*x^18+1110*A*b^5*m^4*x^15+5550*B*a*b^4*m^4*x^15+95064*B*b^ 
5*m*x^18+5*A*a*b^4*m^6*x^12+10940*A*b^5*m^3*x^15+10*B*a^2*b^3*m^6*x^12+547 
00*B*a*b^4*m^3*x^15+58240*B*b^5*x^18+285*A*a*b^4*m^5*x^12+52929*A*b^5*m^2* 
x^15+570*B*a^2*b^3*m^5*x^12+264645*B*a*b^4*m^2*x^15+6165*A*a*b^4*m^4*x^12+ 
112206*A*b^5*m*x^15+12330*B*a^2*b^3*m^4*x^12+561030*B*a*b^4*m*x^15+10*A*a^ 
2*b^3*m^6*x^9+63355*A*a*b^4*m^3*x^12+69160*A*b^5*x^15+10*B*a^3*b^2*m^6*x^9 
+126710*B*a^2*b^3*m^3*x^12+345800*B*a*b^4*x^15+600*A*a^2*b^3*m^5*x^9+31623 
0*A*a*b^4*m^2*x^12+600*B*a^3*b^2*m^5*x^9+632460*B*a^2*b^3*m^2*x^12+13740*A 
*a^2*b^3*m^4*x^9+684360*A*a*b^4*m*x^12+13740*B*a^3*b^2*m^4*x^9+1368720*B*a 
^2*b^3*m*x^12+10*A*a^3*b^2*m^6*x^6+149600*A*a^2*b^3*m^3*x^9+425600*A*a*b^4 
*x^12+5*B*a^4*b*m^6*x^6+149600*B*a^3*b^2*m^3*x^9+851200*B*a^2*b^3*x^12+630 
*A*a^3*b^2*m^5*x^6+783690*A*a^2*b^3*m^2*x^9+315*B*a^4*b*m^5*x^6+783690*B*a 
^3*b^2*m^2*x^9+15330*A*a^3*b^2*m^4*x^6+1753800*A*a^2*b^3*m*x^9+7665*B*a^4* 
b*m^4*x^6+1753800*B*a^3*b^2*m*x^9+5*A*a^4*b*m^6*x^3+179690*A*a^3*b^2*m^3*x 
^6+1106560*A*a^2*b^3*x^9+B*a^5*m^6*x^3+89845*B*a^4*b*m^3*x^6+1106560*B*a^3 
*b^2*x^9+330*A*a^4*b*m^5*x^3+1021860*A*a^3*b^2*m^2*x^6+66*B*a^5*m^5*x^3+51 
0930*B*a^4*b*m^2*x^6+8550*A*a^4*b*m^4*x^3+2437680*A*a^3*b^2*m*x^6+1710*B*a 
^5*m^4*x^3+1218840*B*a^4*b*m*x^6+A*a^5*m^6+109300*A*a^4*b*m^3*x^3+15808...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 851 vs. \(2 (148) = 296\).

Time = 0.12 (sec) , antiderivative size = 851, normalized size of antiderivative = 5.75 \[ \int x^m \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx =\text {Too large to display} \] Input:

integrate(x^m*(b*x^3+a)^5*(B*x^3+A),x, algorithm="fricas")
 

Output:

((B*b^5*m^6 + 51*B*b^5*m^5 + 1005*B*b^5*m^4 + 9605*B*b^5*m^3 + 45474*B*b^5 
*m^2 + 95064*B*b^5*m + 58240*B*b^5)*x^19 + ((5*B*a*b^4 + A*b^5)*m^6 + 3458 
00*B*a*b^4 + 69160*A*b^5 + 54*(5*B*a*b^4 + A*b^5)*m^5 + 1110*(5*B*a*b^4 + 
A*b^5)*m^4 + 10940*(5*B*a*b^4 + A*b^5)*m^3 + 52929*(5*B*a*b^4 + A*b^5)*m^2 
 + 112206*(5*B*a*b^4 + A*b^5)*m)*x^16 + 5*((2*B*a^2*b^3 + A*a*b^4)*m^6 + 1 
70240*B*a^2*b^3 + 85120*A*a*b^4 + 57*(2*B*a^2*b^3 + A*a*b^4)*m^5 + 1233*(2 
*B*a^2*b^3 + A*a*b^4)*m^4 + 12671*(2*B*a^2*b^3 + A*a*b^4)*m^3 + 63246*(2*B 
*a^2*b^3 + A*a*b^4)*m^2 + 136872*(2*B*a^2*b^3 + A*a*b^4)*m)*x^13 + 10*((B* 
a^3*b^2 + A*a^2*b^3)*m^6 + 110656*B*a^3*b^2 + 110656*A*a^2*b^3 + 60*(B*a^3 
*b^2 + A*a^2*b^3)*m^5 + 1374*(B*a^3*b^2 + A*a^2*b^3)*m^4 + 14960*(B*a^3*b^ 
2 + A*a^2*b^3)*m^3 + 78369*(B*a^3*b^2 + A*a^2*b^3)*m^2 + 175380*(B*a^3*b^2 
 + A*a^2*b^3)*m)*x^10 + 5*((B*a^4*b + 2*A*a^3*b^2)*m^6 + 158080*B*a^4*b + 
316160*A*a^3*b^2 + 63*(B*a^4*b + 2*A*a^3*b^2)*m^5 + 1533*(B*a^4*b + 2*A*a^ 
3*b^2)*m^4 + 17969*(B*a^4*b + 2*A*a^3*b^2)*m^3 + 102186*(B*a^4*b + 2*A*a^3 
*b^2)*m^2 + 243768*(B*a^4*b + 2*A*a^3*b^2)*m)*x^7 + ((B*a^5 + 5*A*a^4*b)*m 
^6 + 276640*B*a^5 + 1383200*A*a^4*b + 66*(B*a^5 + 5*A*a^4*b)*m^5 + 1710*(B 
*a^5 + 5*A*a^4*b)*m^4 + 21860*(B*a^5 + 5*A*a^4*b)*m^3 + 140529*(B*a^5 + 5* 
A*a^4*b)*m^2 + 396954*(B*a^5 + 5*A*a^4*b)*m)*x^4 + (A*a^5*m^6 + 69*A*a^5*m 
^5 + 1905*A*a^5*m^4 + 26795*A*a^5*m^3 + 201174*A*a^5*m^2 + 757896*A*a^5*m 
+ 1106560*A*a^5)*x)*x^m/(m^7 + 70*m^6 + 1974*m^5 + 28700*m^4 + 227969*m...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5418 vs. \(2 (138) = 276\).

Time = 1.73 (sec) , antiderivative size = 5418, normalized size of antiderivative = 36.61 \[ \int x^m \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate(x**m*(b*x**3+a)**5*(B*x**3+A),x)
 

Output:

Piecewise((-A*a**5/(18*x**18) - A*a**4*b/(3*x**15) - 5*A*a**3*b**2/(6*x**1 
2) - 10*A*a**2*b**3/(9*x**9) - 5*A*a*b**4/(6*x**6) - A*b**5/(3*x**3) - B*a 
**5/(15*x**15) - 5*B*a**4*b/(12*x**12) - 10*B*a**3*b**2/(9*x**9) - 5*B*a** 
2*b**3/(3*x**6) - 5*B*a*b**4/(3*x**3) + B*b**5*log(x), Eq(m, -19)), (-A*a* 
*5/(15*x**15) - 5*A*a**4*b/(12*x**12) - 10*A*a**3*b**2/(9*x**9) - 5*A*a**2 
*b**3/(3*x**6) - 5*A*a*b**4/(3*x**3) + A*b**5*log(x) - B*a**5/(12*x**12) - 
 5*B*a**4*b/(9*x**9) - 5*B*a**3*b**2/(3*x**6) - 10*B*a**2*b**3/(3*x**3) + 
5*B*a*b**4*log(x) + B*b**5*x**3/3, Eq(m, -16)), (-A*a**5/(12*x**12) - 5*A* 
a**4*b/(9*x**9) - 5*A*a**3*b**2/(3*x**6) - 10*A*a**2*b**3/(3*x**3) + 5*A*a 
*b**4*log(x) + A*b**5*x**3/3 - B*a**5/(9*x**9) - 5*B*a**4*b/(6*x**6) - 10* 
B*a**3*b**2/(3*x**3) + 10*B*a**2*b**3*log(x) + 5*B*a*b**4*x**3/3 + B*b**5* 
x**6/6, Eq(m, -13)), (-A*a**5/(9*x**9) - 5*A*a**4*b/(6*x**6) - 10*A*a**3*b 
**2/(3*x**3) + 10*A*a**2*b**3*log(x) + 5*A*a*b**4*x**3/3 + A*b**5*x**6/6 - 
 B*a**5/(6*x**6) - 5*B*a**4*b/(3*x**3) + 10*B*a**3*b**2*log(x) + 10*B*a**2 
*b**3*x**3/3 + 5*B*a*b**4*x**6/6 + B*b**5*x**9/9, Eq(m, -10)), (-A*a**5/(6 
*x**6) - 5*A*a**4*b/(3*x**3) + 10*A*a**3*b**2*log(x) + 10*A*a**2*b**3*x**3 
/3 + 5*A*a*b**4*x**6/6 + A*b**5*x**9/9 - B*a**5/(3*x**3) + 5*B*a**4*b*log( 
x) + 10*B*a**3*b**2*x**3/3 + 5*B*a**2*b**3*x**6/3 + 5*B*a*b**4*x**9/9 + B* 
b**5*x**12/12, Eq(m, -7)), (-A*a**5/(3*x**3) + 5*A*a**4*b*log(x) + 10*A*a* 
*3*b**2*x**3/3 + 5*A*a**2*b**3*x**6/3 + 5*A*a*b**4*x**9/9 + A*b**5*x**1...
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.39 \[ \int x^m \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {B b^{5} x^{m + 19}}{m + 19} + \frac {5 \, B a b^{4} x^{m + 16}}{m + 16} + \frac {A b^{5} x^{m + 16}}{m + 16} + \frac {10 \, B a^{2} b^{3} x^{m + 13}}{m + 13} + \frac {5 \, A a b^{4} x^{m + 13}}{m + 13} + \frac {10 \, B a^{3} b^{2} x^{m + 10}}{m + 10} + \frac {10 \, A a^{2} b^{3} x^{m + 10}}{m + 10} + \frac {5 \, B a^{4} b x^{m + 7}}{m + 7} + \frac {10 \, A a^{3} b^{2} x^{m + 7}}{m + 7} + \frac {B a^{5} x^{m + 4}}{m + 4} + \frac {5 \, A a^{4} b x^{m + 4}}{m + 4} + \frac {A a^{5} x^{m + 1}}{m + 1} \] Input:

integrate(x^m*(b*x^3+a)^5*(B*x^3+A),x, algorithm="maxima")
 

Output:

B*b^5*x^(m + 19)/(m + 19) + 5*B*a*b^4*x^(m + 16)/(m + 16) + A*b^5*x^(m + 1 
6)/(m + 16) + 10*B*a^2*b^3*x^(m + 13)/(m + 13) + 5*A*a*b^4*x^(m + 13)/(m + 
 13) + 10*B*a^3*b^2*x^(m + 10)/(m + 10) + 10*A*a^2*b^3*x^(m + 10)/(m + 10) 
 + 5*B*a^4*b*x^(m + 7)/(m + 7) + 10*A*a^3*b^2*x^(m + 7)/(m + 7) + B*a^5*x^ 
(m + 4)/(m + 4) + 5*A*a^4*b*x^(m + 4)/(m + 4) + A*a^5*x^(m + 1)/(m + 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1331 vs. \(2 (148) = 296\).

Time = 0.17 (sec) , antiderivative size = 1331, normalized size of antiderivative = 8.99 \[ \int x^m \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate(x^m*(b*x^3+a)^5*(B*x^3+A),x, algorithm="giac")
 

Output:

(B*b^5*m^6*x^19*x^m + 51*B*b^5*m^5*x^19*x^m + 1005*B*b^5*m^4*x^19*x^m + 5* 
B*a*b^4*m^6*x^16*x^m + A*b^5*m^6*x^16*x^m + 9605*B*b^5*m^3*x^19*x^m + 270* 
B*a*b^4*m^5*x^16*x^m + 54*A*b^5*m^5*x^16*x^m + 45474*B*b^5*m^2*x^19*x^m + 
5550*B*a*b^4*m^4*x^16*x^m + 1110*A*b^5*m^4*x^16*x^m + 95064*B*b^5*m*x^19*x 
^m + 10*B*a^2*b^3*m^6*x^13*x^m + 5*A*a*b^4*m^6*x^13*x^m + 54700*B*a*b^4*m^ 
3*x^16*x^m + 10940*A*b^5*m^3*x^16*x^m + 58240*B*b^5*x^19*x^m + 570*B*a^2*b 
^3*m^5*x^13*x^m + 285*A*a*b^4*m^5*x^13*x^m + 264645*B*a*b^4*m^2*x^16*x^m + 
 52929*A*b^5*m^2*x^16*x^m + 12330*B*a^2*b^3*m^4*x^13*x^m + 6165*A*a*b^4*m^ 
4*x^13*x^m + 561030*B*a*b^4*m*x^16*x^m + 112206*A*b^5*m*x^16*x^m + 10*B*a^ 
3*b^2*m^6*x^10*x^m + 10*A*a^2*b^3*m^6*x^10*x^m + 126710*B*a^2*b^3*m^3*x^13 
*x^m + 63355*A*a*b^4*m^3*x^13*x^m + 345800*B*a*b^4*x^16*x^m + 69160*A*b^5* 
x^16*x^m + 600*B*a^3*b^2*m^5*x^10*x^m + 600*A*a^2*b^3*m^5*x^10*x^m + 63246 
0*B*a^2*b^3*m^2*x^13*x^m + 316230*A*a*b^4*m^2*x^13*x^m + 13740*B*a^3*b^2*m 
^4*x^10*x^m + 13740*A*a^2*b^3*m^4*x^10*x^m + 1368720*B*a^2*b^3*m*x^13*x^m 
+ 684360*A*a*b^4*m*x^13*x^m + 5*B*a^4*b*m^6*x^7*x^m + 10*A*a^3*b^2*m^6*x^7 
*x^m + 149600*B*a^3*b^2*m^3*x^10*x^m + 149600*A*a^2*b^3*m^3*x^10*x^m + 851 
200*B*a^2*b^3*x^13*x^m + 425600*A*a*b^4*x^13*x^m + 315*B*a^4*b*m^5*x^7*x^m 
 + 630*A*a^3*b^2*m^5*x^7*x^m + 783690*B*a^3*b^2*m^2*x^10*x^m + 783690*A*a^ 
2*b^3*m^2*x^10*x^m + 7665*B*a^4*b*m^4*x^7*x^m + 15330*A*a^3*b^2*m^4*x^7*x^ 
m + 1753800*B*a^3*b^2*m*x^10*x^m + 1753800*A*a^2*b^3*m*x^10*x^m + B*a^5...
 

Mupad [B] (verification not implemented)

Time = 1.47 (sec) , antiderivative size = 559, normalized size of antiderivative = 3.78 \[ \int x^m \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {B\,b^5\,x^m\,x^{19}\,\left (m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240\right )}{m^7+70\,m^6+1974\,m^5+28700\,m^4+227969\,m^3+959070\,m^2+1864456\,m+1106560}+\frac {a^4\,x^m\,x^4\,\left (5\,A\,b+B\,a\right )\,\left (m^6+66\,m^5+1710\,m^4+21860\,m^3+140529\,m^2+396954\,m+276640\right )}{m^7+70\,m^6+1974\,m^5+28700\,m^4+227969\,m^3+959070\,m^2+1864456\,m+1106560}+\frac {b^4\,x^m\,x^{16}\,\left (A\,b+5\,B\,a\right )\,\left (m^6+54\,m^5+1110\,m^4+10940\,m^3+52929\,m^2+112206\,m+69160\right )}{m^7+70\,m^6+1974\,m^5+28700\,m^4+227969\,m^3+959070\,m^2+1864456\,m+1106560}+\frac {A\,a^5\,x\,x^m\,\left (m^6+69\,m^5+1905\,m^4+26795\,m^3+201174\,m^2+757896\,m+1106560\right )}{m^7+70\,m^6+1974\,m^5+28700\,m^4+227969\,m^3+959070\,m^2+1864456\,m+1106560}+\frac {10\,a^2\,b^2\,x^m\,x^{10}\,\left (A\,b+B\,a\right )\,\left (m^6+60\,m^5+1374\,m^4+14960\,m^3+78369\,m^2+175380\,m+110656\right )}{m^7+70\,m^6+1974\,m^5+28700\,m^4+227969\,m^3+959070\,m^2+1864456\,m+1106560}+\frac {5\,a\,b^3\,x^m\,x^{13}\,\left (A\,b+2\,B\,a\right )\,\left (m^6+57\,m^5+1233\,m^4+12671\,m^3+63246\,m^2+136872\,m+85120\right )}{m^7+70\,m^6+1974\,m^5+28700\,m^4+227969\,m^3+959070\,m^2+1864456\,m+1106560}+\frac {5\,a^3\,b\,x^m\,x^7\,\left (2\,A\,b+B\,a\right )\,\left (m^6+63\,m^5+1533\,m^4+17969\,m^3+102186\,m^2+243768\,m+158080\right )}{m^7+70\,m^6+1974\,m^5+28700\,m^4+227969\,m^3+959070\,m^2+1864456\,m+1106560} \] Input:

int(x^m*(A + B*x^3)*(a + b*x^3)^5,x)
 

Output:

(B*b^5*x^m*x^19*(95064*m + 45474*m^2 + 9605*m^3 + 1005*m^4 + 51*m^5 + m^6 
+ 58240))/(1864456*m + 959070*m^2 + 227969*m^3 + 28700*m^4 + 1974*m^5 + 70 
*m^6 + m^7 + 1106560) + (a^4*x^m*x^4*(5*A*b + B*a)*(396954*m + 140529*m^2 
+ 21860*m^3 + 1710*m^4 + 66*m^5 + m^6 + 276640))/(1864456*m + 959070*m^2 + 
 227969*m^3 + 28700*m^4 + 1974*m^5 + 70*m^6 + m^7 + 1106560) + (b^4*x^m*x^ 
16*(A*b + 5*B*a)*(112206*m + 52929*m^2 + 10940*m^3 + 1110*m^4 + 54*m^5 + m 
^6 + 69160))/(1864456*m + 959070*m^2 + 227969*m^3 + 28700*m^4 + 1974*m^5 + 
 70*m^6 + m^7 + 1106560) + (A*a^5*x*x^m*(757896*m + 201174*m^2 + 26795*m^3 
 + 1905*m^4 + 69*m^5 + m^6 + 1106560))/(1864456*m + 959070*m^2 + 227969*m^ 
3 + 28700*m^4 + 1974*m^5 + 70*m^6 + m^7 + 1106560) + (10*a^2*b^2*x^m*x^10* 
(A*b + B*a)*(175380*m + 78369*m^2 + 14960*m^3 + 1374*m^4 + 60*m^5 + m^6 + 
110656))/(1864456*m + 959070*m^2 + 227969*m^3 + 28700*m^4 + 1974*m^5 + 70* 
m^6 + m^7 + 1106560) + (5*a*b^3*x^m*x^13*(A*b + 2*B*a)*(136872*m + 63246*m 
^2 + 12671*m^3 + 1233*m^4 + 57*m^5 + m^6 + 85120))/(1864456*m + 959070*m^2 
 + 227969*m^3 + 28700*m^4 + 1974*m^5 + 70*m^6 + m^7 + 1106560) + (5*a^3*b* 
x^m*x^7*(2*A*b + B*a)*(243768*m + 102186*m^2 + 17969*m^3 + 1533*m^4 + 63*m 
^5 + m^6 + 158080))/(1864456*m + 959070*m^2 + 227969*m^3 + 28700*m^4 + 197 
4*m^5 + 70*m^6 + m^7 + 1106560)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 599, normalized size of antiderivative = 4.05 \[ \int x^m \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {x^{m} x \left (b^{6} m^{6} x^{18}+51 b^{6} m^{5} x^{18}+1005 b^{6} m^{4} x^{18}+6 a \,b^{5} m^{6} x^{15}+9605 b^{6} m^{3} x^{18}+324 a \,b^{5} m^{5} x^{15}+45474 b^{6} m^{2} x^{18}+6660 a \,b^{5} m^{4} x^{15}+95064 b^{6} m \,x^{18}+15 a^{2} b^{4} m^{6} x^{12}+65640 a \,b^{5} m^{3} x^{15}+58240 b^{6} x^{18}+855 a^{2} b^{4} m^{5} x^{12}+317574 a \,b^{5} m^{2} x^{15}+18495 a^{2} b^{4} m^{4} x^{12}+673236 a \,b^{5} m \,x^{15}+20 a^{3} b^{3} m^{6} x^{9}+190065 a^{2} b^{4} m^{3} x^{12}+414960 a \,b^{5} x^{15}+1200 a^{3} b^{3} m^{5} x^{9}+948690 a^{2} b^{4} m^{2} x^{12}+27480 a^{3} b^{3} m^{4} x^{9}+2053080 a^{2} b^{4} m \,x^{12}+15 a^{4} b^{2} m^{6} x^{6}+299200 a^{3} b^{3} m^{3} x^{9}+1276800 a^{2} b^{4} x^{12}+945 a^{4} b^{2} m^{5} x^{6}+1567380 a^{3} b^{3} m^{2} x^{9}+22995 a^{4} b^{2} m^{4} x^{6}+3507600 a^{3} b^{3} m \,x^{9}+6 a^{5} b \,m^{6} x^{3}+269535 a^{4} b^{2} m^{3} x^{6}+2213120 a^{3} b^{3} x^{9}+396 a^{5} b \,m^{5} x^{3}+1532790 a^{4} b^{2} m^{2} x^{6}+10260 a^{5} b \,m^{4} x^{3}+3656520 a^{4} b^{2} m \,x^{6}+a^{6} m^{6}+131160 a^{5} b \,m^{3} x^{3}+2371200 a^{4} b^{2} x^{6}+69 a^{6} m^{5}+843174 a^{5} b \,m^{2} x^{3}+1905 a^{6} m^{4}+2381724 a^{5} b m \,x^{3}+26795 a^{6} m^{3}+1659840 a^{5} b \,x^{3}+201174 a^{6} m^{2}+757896 a^{6} m +1106560 a^{6}\right )}{m^{7}+70 m^{6}+1974 m^{5}+28700 m^{4}+227969 m^{3}+959070 m^{2}+1864456 m +1106560} \] Input:

int(x^m*(b*x^3+a)^5*(B*x^3+A),x)
 

Output:

(x**m*x*(a**6*m**6 + 69*a**6*m**5 + 1905*a**6*m**4 + 26795*a**6*m**3 + 201 
174*a**6*m**2 + 757896*a**6*m + 1106560*a**6 + 6*a**5*b*m**6*x**3 + 396*a* 
*5*b*m**5*x**3 + 10260*a**5*b*m**4*x**3 + 131160*a**5*b*m**3*x**3 + 843174 
*a**5*b*m**2*x**3 + 2381724*a**5*b*m*x**3 + 1659840*a**5*b*x**3 + 15*a**4* 
b**2*m**6*x**6 + 945*a**4*b**2*m**5*x**6 + 22995*a**4*b**2*m**4*x**6 + 269 
535*a**4*b**2*m**3*x**6 + 1532790*a**4*b**2*m**2*x**6 + 3656520*a**4*b**2* 
m*x**6 + 2371200*a**4*b**2*x**6 + 20*a**3*b**3*m**6*x**9 + 1200*a**3*b**3* 
m**5*x**9 + 27480*a**3*b**3*m**4*x**9 + 299200*a**3*b**3*m**3*x**9 + 15673 
80*a**3*b**3*m**2*x**9 + 3507600*a**3*b**3*m*x**9 + 2213120*a**3*b**3*x**9 
 + 15*a**2*b**4*m**6*x**12 + 855*a**2*b**4*m**5*x**12 + 18495*a**2*b**4*m* 
*4*x**12 + 190065*a**2*b**4*m**3*x**12 + 948690*a**2*b**4*m**2*x**12 + 205 
3080*a**2*b**4*m*x**12 + 1276800*a**2*b**4*x**12 + 6*a*b**5*m**6*x**15 + 3 
24*a*b**5*m**5*x**15 + 6660*a*b**5*m**4*x**15 + 65640*a*b**5*m**3*x**15 + 
317574*a*b**5*m**2*x**15 + 673236*a*b**5*m*x**15 + 414960*a*b**5*x**15 + b 
**6*m**6*x**18 + 51*b**6*m**5*x**18 + 1005*b**6*m**4*x**18 + 9605*b**6*m** 
3*x**18 + 45474*b**6*m**2*x**18 + 95064*b**6*m*x**18 + 58240*b**6*x**18))/ 
(m**7 + 70*m**6 + 1974*m**5 + 28700*m**4 + 227969*m**3 + 959070*m**2 + 186 
4456*m + 1106560)