\(\int x^2 (a+b x)^n (c+d x^3)^3 \, dx\) [35]

Optimal result

Integrand size = 20, antiderivative size = 459 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^3 \, dx=\frac {a^2 \left (b^3 c-a^3 d\right )^3 (a+b x)^{1+n}}{b^{12} (1+n)}-\frac {a \left (2 b^3 c-11 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{2+n}}{b^{12} (2+n)}+\frac {\left (b^3 c-a^3 d\right ) \left (b^6 c^2-29 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{3+n}}{b^{12} (3+n)}+\frac {3 a^2 d \left (10 b^6 c^2-56 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{4+n}}{b^{12} (4+n)}-\frac {15 a d \left (b^6 c^2-14 a^3 b^3 c d+22 a^6 d^2\right ) (a+b x)^{5+n}}{b^{12} (5+n)}+\frac {3 d \left (b^6 c^2-56 a^3 b^3 c d+154 a^6 d^2\right ) (a+b x)^{6+n}}{b^{12} (6+n)}+\frac {42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^{7+n}}{b^{12} (7+n)}-\frac {6 a d^2 \left (4 b^3 c-55 a^3 d\right ) (a+b x)^{8+n}}{b^{12} (8+n)}+\frac {3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{9+n}}{b^{12} (9+n)}+\frac {55 a^2 d^3 (a+b x)^{10+n}}{b^{12} (10+n)}-\frac {11 a d^3 (a+b x)^{11+n}}{b^{12} (11+n)}+\frac {d^3 (a+b x)^{12+n}}{b^{12} (12+n)} \] Output:

a^2*(-a^3*d+b^3*c)^3*(b*x+a)^(1+n)/b^12/(1+n)-a*(-11*a^3*d+2*b^3*c)*(-a^3* 
d+b^3*c)^2*(b*x+a)^(2+n)/b^12/(2+n)+(-a^3*d+b^3*c)*(55*a^6*d^2-29*a^3*b^3* 
c*d+b^6*c^2)*(b*x+a)^(3+n)/b^12/(3+n)+3*a^2*d*(55*a^6*d^2-56*a^3*b^3*c*d+1 
0*b^6*c^2)*(b*x+a)^(4+n)/b^12/(4+n)-15*a*d*(22*a^6*d^2-14*a^3*b^3*c*d+b^6* 
c^2)*(b*x+a)^(5+n)/b^12/(5+n)+3*d*(154*a^6*d^2-56*a^3*b^3*c*d+b^6*c^2)*(b* 
x+a)^(6+n)/b^12/(6+n)+42*a^2*d^2*(-11*a^3*d+2*b^3*c)*(b*x+a)^(7+n)/b^12/(7 
+n)-6*a*d^2*(-55*a^3*d+4*b^3*c)*(b*x+a)^(8+n)/b^12/(8+n)+3*d^2*(-55*a^3*d+ 
b^3*c)*(b*x+a)^(9+n)/b^12/(9+n)+55*a^2*d^3*(b*x+a)^(10+n)/b^12/(10+n)-11*a 
*d^3*(b*x+a)^(11+n)/b^12/(11+n)+d^3*(b*x+a)^(12+n)/b^12/(12+n)
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.88 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^3 \, dx=\frac {(a+b x)^{1+n} \left (\frac {a^2 \left (b^3 c-a^3 d\right )^3}{1+n}+\frac {a \left (b^3 c-a^3 d\right )^2 \left (-2 b^3 c+11 a^3 d\right ) (a+b x)}{2+n}+\frac {\left (b^3 c-a^3 d\right ) \left (b^6 c^2-29 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^2}{3+n}+\frac {3 a^2 d \left (10 b^6 c^2-56 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^3}{4+n}-\frac {15 a d \left (b^6 c^2-14 a^3 b^3 c d+22 a^6 d^2\right ) (a+b x)^4}{5+n}+\frac {3 d \left (b^6 c^2-56 a^3 b^3 c d+154 a^6 d^2\right ) (a+b x)^5}{6+n}+\frac {42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^6}{7+n}+\frac {6 a d^2 \left (-4 b^3 c+55 a^3 d\right ) (a+b x)^7}{8+n}+\frac {3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^8}{9+n}+\frac {55 a^2 d^3 (a+b x)^9}{10+n}-\frac {11 a d^3 (a+b x)^{10}}{11+n}+\frac {d^3 (a+b x)^{11}}{12+n}\right )}{b^{12}} \] Input:

Integrate[x^2*(a + b*x)^n*(c + d*x^3)^3,x]
 

Output:

((a + b*x)^(1 + n)*((a^2*(b^3*c - a^3*d)^3)/(1 + n) + (a*(b^3*c - a^3*d)^2 
*(-2*b^3*c + 11*a^3*d)*(a + b*x))/(2 + n) + ((b^3*c - a^3*d)*(b^6*c^2 - 29 
*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^2)/(3 + n) + (3*a^2*d*(10*b^6*c^2 - 5 
6*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^3)/(4 + n) - (15*a*d*(b^6*c^2 - 14*a 
^3*b^3*c*d + 22*a^6*d^2)*(a + b*x)^4)/(5 + n) + (3*d*(b^6*c^2 - 56*a^3*b^3 
*c*d + 154*a^6*d^2)*(a + b*x)^5)/(6 + n) + (42*a^2*d^2*(2*b^3*c - 11*a^3*d 
)*(a + b*x)^6)/(7 + n) + (6*a*d^2*(-4*b^3*c + 55*a^3*d)*(a + b*x)^7)/(8 + 
n) + (3*d^2*(b^3*c - 55*a^3*d)*(a + b*x)^8)/(9 + n) + (55*a^2*d^3*(a + b*x 
)^9)/(10 + n) - (11*a*d^3*(a + b*x)^10)/(11 + n) + (d^3*(a + b*x)^11)/(12 
+ n)))/b^12
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 459, 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 = {2123, 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.

\(\displaystyle \int x^2 \left (c+d x^3\right )^3 (a+b x)^n \, dx\)

\(\Big \downarrow \) 2123

\(\displaystyle \int \left (\frac {6 a d^2 \left (55 a^3 d-4 b^3 c\right ) (a+b x)^{n+7}}{b^{11}}+\frac {3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{n+8}}{b^{11}}+\frac {a \left (a^3 d-b^3 c\right )^2 \left (11 a^3 d-2 b^3 c\right ) (a+b x)^{n+1}}{b^{11}}+\frac {55 a^2 d^3 (a+b x)^{n+9}}{b^{11}}+\frac {\left (b^3 c-a^3 d\right ) \left (55 a^6 d^2-29 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+2}}{b^{11}}-\frac {15 a d \left (22 a^6 d^2-14 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+4}}{b^{11}}+\frac {3 d \left (154 a^6 d^2-56 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+5}}{b^{11}}-\frac {42 a^2 d^2 \left (11 a^3 d-2 b^3 c\right ) (a+b x)^{n+6}}{b^{11}}-\frac {a^2 \left (a^3 d-b^3 c\right )^3 (a+b x)^n}{b^{11}}+\frac {3 a^2 d \left (55 a^6 d^2-56 a^3 b^3 c d+10 b^6 c^2\right ) (a+b x)^{n+3}}{b^{11}}-\frac {11 a d^3 (a+b x)^{n+10}}{b^{11}}+\frac {d^3 (a+b x)^{n+11}}{b^{11}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 a d^2 \left (4 b^3 c-55 a^3 d\right ) (a+b x)^{n+8}}{b^{12} (n+8)}+\frac {3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{n+9}}{b^{12} (n+9)}-\frac {a \left (2 b^3 c-11 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{12} (n+2)}+\frac {55 a^2 d^3 (a+b x)^{n+10}}{b^{12} (n+10)}+\frac {\left (b^3 c-a^3 d\right ) \left (55 a^6 d^2-29 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+3}}{b^{12} (n+3)}-\frac {15 a d \left (22 a^6 d^2-14 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+5}}{b^{12} (n+5)}+\frac {3 d \left (154 a^6 d^2-56 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+6}}{b^{12} (n+6)}+\frac {42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^{n+7}}{b^{12} (n+7)}+\frac {a^2 \left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{12} (n+1)}+\frac {3 a^2 d \left (55 a^6 d^2-56 a^3 b^3 c d+10 b^6 c^2\right ) (a+b x)^{n+4}}{b^{12} (n+4)}-\frac {11 a d^3 (a+b x)^{n+11}}{b^{12} (n+11)}+\frac {d^3 (a+b x)^{n+12}}{b^{12} (n+12)}\)

Input:

Int[x^2*(a + b*x)^n*(c + d*x^3)^3,x]
 

Output:

(a^2*(b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^12*(1 + n)) - (a*(2*b^3*c - 1 
1*a^3*d)*(b^3*c - a^3*d)^2*(a + b*x)^(2 + n))/(b^12*(2 + n)) + ((b^3*c - a 
^3*d)*(b^6*c^2 - 29*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^(3 + n))/(b^12*(3 
+ n)) + (3*a^2*d*(10*b^6*c^2 - 56*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^(4 + 
 n))/(b^12*(4 + n)) - (15*a*d*(b^6*c^2 - 14*a^3*b^3*c*d + 22*a^6*d^2)*(a + 
 b*x)^(5 + n))/(b^12*(5 + n)) + (3*d*(b^6*c^2 - 56*a^3*b^3*c*d + 154*a^6*d 
^2)*(a + b*x)^(6 + n))/(b^12*(6 + n)) + (42*a^2*d^2*(2*b^3*c - 11*a^3*d)*( 
a + b*x)^(7 + n))/(b^12*(7 + n)) - (6*a*d^2*(4*b^3*c - 55*a^3*d)*(a + b*x) 
^(8 + n))/(b^12*(8 + n)) + (3*d^2*(b^3*c - 55*a^3*d)*(a + b*x)^(9 + n))/(b 
^12*(9 + n)) + (55*a^2*d^3*(a + b*x)^(10 + n))/(b^12*(10 + n)) - (11*a*d^3 
*(a + b*x)^(11 + n))/(b^12*(11 + n)) + (d^3*(a + b*x)^(12 + n))/(b^12*(12 
+ n))
 

Defintions of rubi rules used

rule 2009

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

rule 2123

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c 
, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3779\) vs. \(2(459)=918\).

Time = 0.49 (sec) , antiderivative size = 3780, normalized size of antiderivative = 8.24

method	result	size
gosper	\(\text {Expression too large to display}\)	\(3780\)
orering	\(\text {Expression too large to display}\)	\(3783\)
risch	\(\text {Expression too large to display}\)	\(4231\)
parallelrisch	\(\text {Expression too large to display}\)	\(6192\)

Input:

int(x^2*(b*x+a)^n*(d*x^3+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/b^12*(b*x+a)^(1+n)/(n^12+78*n^11+2717*n^10+55770*n^9+749463*n^8+6926634 
*n^7+44990231*n^6+206070150*n^5+657206836*n^4+1414014888*n^3+1931559552*n^ 
2+1486442880*n[phone])*(-b^11*d^3*n^11*x^11-66*b^11*d^3*n^10*x^11+11*a* 
b^10*d^3*n^10*x^10-1925*b^11*d^3*n^9*x^11+605*a*b^10*d^3*n^9*x^10-3*b^11*c 
*d^2*n^11*x^8-32670*b^11*d^3*n^8*x^11-110*a^2*b^9*d^3*n^9*x^9+14520*a*b^10 
*d^3*n^8*x^10-207*b^11*c*d^2*n^10*x^8-357423*b^11*d^3*n^7*x^11-4950*a^2*b^ 
9*d^3*n^8*x^9+24*a*b^10*c*d^2*n^10*x^7+199650*a*b^10*d^3*n^7*x^10-6288*b^1 
1*c*d^2*n^9*x^8-2637558*b^11*d^3*n^6*x^11+990*a^3*b^8*d^3*n^8*x^8-95700*a^ 
2*b^9*d^3*n^7*x^9+1464*a*b^10*c*d^2*n^9*x^7+1735503*a*b^10*d^3*n^6*x^10-3* 
b^11*c^2*d*n^11*x^5-110718*b^11*c*d^2*n^8*x^8-13339535*b^11*d^3*n^5*x^11+3 
5640*a^3*b^8*d^3*n^7*x^8-168*a^2*b^9*c*d^2*n^9*x^6-1039500*a^2*b^9*d^3*n^6 
*x^9+38592*a*b^10*c*d^2*n^8*x^7+9922605*a*b^10*d^3*n^5*x^10-216*b^11*c^2*d 
*n^10*x^5-1251927*b^11*c*d^2*n^7*x^8-45995730*b^11*d^3*n^4*x^11-7920*a^4*b 
^7*d^3*n^7*x^7+540540*a^3*b^8*d^3*n^6*x^8-9072*a^2*b^9*c*d^2*n^8*x^6-69600 
30*a^2*b^9*d^3*n^5*x^9+15*a*b^10*c^2*d*n^10*x^4+577008*a*b^10*c*d^2*n^7*x^ 
7+37586230*a*b^10*d^3*n^4*x^10-6855*b^11*c^2*d*n^9*x^5-9512559*b^11*c*d^2* 
n^6*x^8-105258076*b^11*d^3*n^3*x^11-221760*a^4*b^7*d^3*n^6*x^7+1008*a^3*b^ 
8*c*d^2*n^8*x^5+4490640*a^3*b^8*d^3*n^5*x^8-206640*a^2*b^9*c*d^2*n^7*x^6-2 
9625750*a^2*b^9*d^3*n^4*x^9+1005*a*b^10*c^2*d*n^9*x^4+5399352*a*b^10*c*d^2 
*n^6*x^7+92504500*a*b^10*d^3*n^3*x^10-b^11*c^3*n^11*x^2-126180*b^11*c^2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3564 vs. \(2 (459) = 918\).

Time = 0.13 (sec) , antiderivative size = 3564, normalized size of antiderivative = 7.76 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

integrate(x^2*(b*x+a)^n*(d*x^3+c)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75191 vs. \(2 (439) = 878\).

Time = 40.22 (sec) , antiderivative size = 75191, normalized size of antiderivative = 163.81 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

integrate(x**2*(b*x+a)**n*(d*x**3+c)**3,x)
 

Output:

Piecewise((a**n*(c**3*x**3/3 + c**2*d*x**6/2 + c*d**2*x**9/3 + d**3*x**12/ 
12), Eq(b, 0)), (27720*a**11*d**3*log(a/b + x)/(27720*a**11*b**12 + 304920 
*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 91476 
00*a**7*b**16*x**4 + 12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 
 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 + 1524600*a**2*b**21*x* 
*9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 83711*a**11*d**3/(27720*a 
**11*b**12 + 304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8 
*b**15*x**3 + 9147600*a**7*b**16*x**4 + 12806640*a**6*b**17*x**5 + 1280664 
0*a**5*b**18*x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 + 15 
24600*a**2*b**21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 304920 
*a**10*b*d**3*x*log(a/b + x)/(27720*a**11*b**12 + 304920*a**10*b**13*x + 1 
524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 9147600*a**7*b**16*x**4 
 + 12806640*a**6*b**17*x**5 + 12806640*a**5*b**18*x**6 + 9147600*a**4*b**1 
9*x**7 + 4573800*a**3*b**20*x**8 + 1524600*a**2*b**21*x**9 + 304920*a*b**2 
2*x**10 + 27720*b**23*x**11) + 893101*a**10*b*d**3*x/(27720*a**11*b**12 + 
304920*a**10*b**13*x + 1524600*a**9*b**14*x**2 + 4573800*a**8*b**15*x**3 + 
 9147600*a**7*b**16*x**4 + 12806640*a**6*b**17*x**5 + 12806640*a**5*b**18* 
x**6 + 9147600*a**4*b**19*x**7 + 4573800*a**3*b**20*x**8 + 1524600*a**2*b* 
*21*x**9 + 304920*a*b**22*x**10 + 27720*b**23*x**11) + 1524600*a**9*b**2*d 
**3*x**2*log(a/b + x)/(27720*a**11*b**12 + 304920*a**10*b**13*x + 15246...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1153 vs. \(2 (459) = 918\).

Time = 0.06 (sec) , antiderivative size = 1153, normalized size of antiderivative = 2.51 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

integrate(x^2*(b*x+a)^n*(d*x^3+c)^3,x, algorithm="maxima")
 

Output:

((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x 
 + a)^n*c^3/((n^3 + 6*n^2 + 11*n + 6)*b^3) + 3*((n^5 + 15*n^4 + 85*n^3 + 2 
25*n^2 + 274*n + 120)*b^6*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a* 
b^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2*b^4*x^4 + 20*(n^3 + 3*n^2 + 2 
*n)*a^3*b^3*x^3 - 60*(n^2 + n)*a^4*b^2*x^2 + 120*a^5*b*n*x - 120*a^6)*(b*x 
 + a)^n*c^2*d/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720 
)*b^6) + 3*((n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 1 
18124*n^2 + 109584*n + 40320)*b^9*x^9 + (n^8 + 28*n^7 + 322*n^6 + 1960*n^5 
 + 6769*n^4 + 13132*n^3 + 13068*n^2 + 5040*n)*a*b^8*x^8 - 8*(n^7 + 21*n^6 
+ 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*a^2*b^7*x^7 + 56*(n^6 + 
 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a^3*b^6*x^6 - 336*(n^5 + 10* 
n^4 + 35*n^3 + 50*n^2 + 24*n)*a^4*b^5*x^5 + 1680*(n^4 + 6*n^3 + 11*n^2 + 6 
*n)*a^5*b^4*x^4 - 6720*(n^3 + 3*n^2 + 2*n)*a^6*b^3*x^3 + 20160*(n^2 + n)*a 
^7*b^2*x^2 - 40320*a^8*b*n*x + 40320*a^9)*(b*x + a)^n*c*d^2/((n^9 + 45*n^8 
 + 870*n^7 + 9450*n^6 + 63273*n^5 + 269325*n^4 + 723680*n^3 + 1172700*n^2 
+ 1026576*n + 362880)*b^9) + ((n^11 + 66*n^10 + 1925*n^9 + 32670*n^8 + 357 
423*n^7 + 2637558*n^6 + 13339535*n^5 + 45995730*n^4 + 105258076*n^3 + 1509 
17976*n^2 + 120543840*n + 39916800)*b^12*x^12 + (n^11 + 55*n^10 + 1320*n^9 
 + 18150*n^8 + 157773*n^7 + 902055*n^6 + 3416930*n^5 + 8409500*n^4 + 12753 
576*n^3 + 10628640*n^2 + 3628800*n)*a*b^11*x^11 - 11*(n^10 + 45*n^9 + 8...
 

Giac [F(-2)]

Exception generated. \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^2*(b*x+a)^n*(d*x^3+c)^3,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Polynomial exponent overflow. Error 
: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 26.12 (sec) , antiderivative size = 2896, normalized size of antiderivative = 6.31 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

int(x^2*(c + d*x^3)^3*(a + b*x)^n,x)
 

Output:

(2*a^3*(a + b*x)^n*(79833600*b^9*c^3 - 19958400*a^9*d^3 + 101378880*b^9*c^ 
3*n + 56231712*b^9*c^3*n^2 + 17893196*b^9*c^3*n^3 + 3602088*b^9*c^3*n^4 + 
476049*b^9*c^3*n^5 + 41328*b^9*c^3*n^6 + 2274*b^9*c^3*n^7 + 72*b^9*c^3*n^8 
 + b^9*c^3*n^9 - 119750400*a^3*b^6*c^2*d + 79833600*a^6*b^3*c*d^2 - 782222 
40*a^3*b^6*c^2*d*n + 21893760*a^6*b^3*c*d^2*n - 21141720*a^3*b^6*c^2*d*n^2 
 + 1995840*a^6*b^3*c*d^2*n^2 - 3026700*a^3*b^6*c^2*d*n^3 + 60480*a^6*b^3*c 
*d^2*n^3 - 242100*a^3*b^6*c^2*d*n^4 - 10260*a^3*b^6*c^2*d*n^5 - 180*a^3*b^ 
6*c^2*d*n^6))/(b^12*(1486442880*n + 1931559552*n^2 + 1414014888*n^3 + 6572 
06836*n^4 + 206070150*n^5 + 44990231*n^6 + 6926634*n^7 + 749463*n^8 + 5577 
0*n^9 + 2717*n^10 + 78*n^11 + n^12 + 479001600)) + (d^3*x^12*(a + b*x)^n*( 
120543840*n + 150917976*n^2 + 105258076*n^3 + 45995730*n^4 + 13339535*n^5 
+ 2637558*n^6 + 357423*n^7 + 32670*n^8 + 1925*n^9 + 66*n^10 + n^11 + 39916 
800))/(1486442880*n + 1931559552*n^2 + 1414014888*n^3 + 657206836*n^4 + 20 
6070150*n^5 + 44990231*n^6 + 6926634*n^7 + 749463*n^8 + 55770*n^9 + 2717*n 
^10 + 78*n^11 + n^12 + 479001600) + (x^3*(a + b*x)^n*(3*n + n^2 + 2)*(7983 
3600*b^9*c^3 + 6652800*a^9*d^3*n + 101378880*b^9*c^3*n + 56231712*b^9*c^3* 
n^2 + 17893196*b^9*c^3*n^3 + 3602088*b^9*c^3*n^4 + 476049*b^9*c^3*n^5 + 41 
328*b^9*c^3*n^6 + 2274*b^9*c^3*n^7 + 72*b^9*c^3*n^8 + b^9*c^3*n^9 + 399168 
00*a^3*b^6*c^2*d*n - 26611200*a^6*b^3*c*d^2*n + 26074080*a^3*b^6*c^2*d*n^2 
 - 7297920*a^6*b^3*c*d^2*n^2 + 7047240*a^3*b^6*c^2*d*n^3 - 665280*a^6*b...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 4225, normalized size of antiderivative = 9.20 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^3 \, dx =\text {Too large to display} \] Input:

int(x^2*(b*x+a)^n*(d*x^3+c)^3,x)
                                                                                    
                                                                                    
 

Output:

((a + b*x)**n*( - 39916800*a**12*d**3 + 39916800*a**11*b*d**3*n*x - 199584 
00*a**10*b**2*d**3*n**2*x**2 - 19958400*a**10*b**2*d**3*n*x**2 + 120960*a* 
*9*b**3*c*d**2*n**3 + 3991680*a**9*b**3*c*d**2*n**2 + 43787520*a**9*b**3*c 
*d**2*n + 159667200*a**9*b**3*c*d**2 + 6652800*a**9*b**3*d**3*n**3*x**3 + 
19958400*a**9*b**3*d**3*n**2*x**3 + 13305600*a**9*b**3*d**3*n*x**3 - 12096 
0*a**8*b**4*c*d**2*n**4*x - 3991680*a**8*b**4*c*d**2*n**3*x - 43787520*a** 
8*b**4*c*d**2*n**2*x - 159667200*a**8*b**4*c*d**2*n*x - 1663200*a**8*b**4* 
d**3*n**4*x**4 - 9979200*a**8*b**4*d**3*n**3*x**4 - 18295200*a**8*b**4*d** 
3*n**2*x**4 - 9979200*a**8*b**4*d**3*n*x**4 + 60480*a**7*b**5*c*d**2*n**5* 
x**2 + 2056320*a**7*b**5*c*d**2*n**4*x**2 + 23889600*a**7*b**5*c*d**2*n**3 
*x**2 + 101727360*a**7*b**5*c*d**2*n**2*x**2 + 79833600*a**7*b**5*c*d**2*n 
*x**2 + 332640*a**7*b**5*d**3*n**5*x**5 + 3326400*a**7*b**5*d**3*n**4*x**5 
 + 11642400*a**7*b**5*d**3*n**3*x**5 + 16632000*a**7*b**5*d**3*n**2*x**5 + 
 7983360*a**7*b**5*d**3*n*x**5 - 360*a**6*b**6*c**2*d*n**6 - 20520*a**6*b* 
*6*c**2*d*n**5 - 484200*a**6*b**6*c**2*d*n**4 - 6053400*a**6*b**6*c**2*d*n 
**3 - 42283440*a**6*b**6*c**2*d*n**2 - 156444480*a**6*b**6*c**2*d*n - 2395 
00800*a**6*b**6*c**2*d - 20160*a**6*b**6*c*d**2*n**6*x**3 - 725760*a**6*b* 
*6*c*d**2*n**5*x**3 - 9334080*a**6*b**6*c*d**2*n**4*x**3 - 49835520*a**6*b 
**6*c*d**2*n**3*x**3 - 94429440*a**6*b**6*c*d**2*n**2*x**3 - 53222400*a**6 
*b**6*c*d**2*n*x**3 - 55440*a**6*b**6*d**3*n**6*x**6 - 831600*a**6*b**6...