\(\int x (a+b x)^n (c+d x^3)^3 \, dx\) [36]

Optimal result

Integrand size = 18, antiderivative size = 396 \[ \int x (a+b x)^n \left (c+d x^3\right )^3 \, dx=-\frac {a \left (b^3 c-a^3 d\right )^3 (a+b x)^{1+n}}{b^{11} (1+n)}+\frac {\left (b^3 c-10 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{2+n}}{b^{11} (2+n)}+\frac {9 a^2 d \left (2 b^3 c-5 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{3+n}}{b^{11} (3+n)}-\frac {3 a d \left (4 b^6 c^2-35 a^3 b^3 c d+40 a^6 d^2\right ) (a+b x)^{4+n}}{b^{11} (4+n)}+\frac {3 d \left (b^6 c^2-35 a^3 b^3 c d+70 a^6 d^2\right ) (a+b x)^{5+n}}{b^{11} (5+n)}+\frac {63 a^2 d^2 \left (b^3 c-4 a^3 d\right ) (a+b x)^{6+n}}{b^{11} (6+n)}-\frac {21 a d^2 \left (b^3 c-10 a^3 d\right ) (a+b x)^{7+n}}{b^{11} (7+n)}+\frac {3 d^2 \left (b^3 c-40 a^3 d\right ) (a+b x)^{8+n}}{b^{11} (8+n)}+\frac {45 a^2 d^3 (a+b x)^{9+n}}{b^{11} (9+n)}-\frac {10 a d^3 (a+b x)^{10+n}}{b^{11} (10+n)}+\frac {d^3 (a+b x)^{11+n}}{b^{11} (11+n)} \] Output:

-a*(-a^3*d+b^3*c)^3*(b*x+a)^(1+n)/b^11/(1+n)+(-10*a^3*d+b^3*c)*(-a^3*d+b^3 
*c)^2*(b*x+a)^(2+n)/b^11/(2+n)+9*a^2*d*(-5*a^3*d+2*b^3*c)*(-a^3*d+b^3*c)*( 
b*x+a)^(3+n)/b^11/(3+n)-3*a*d*(40*a^6*d^2-35*a^3*b^3*c*d+4*b^6*c^2)*(b*x+a 
)^(4+n)/b^11/(4+n)+3*d*(70*a^6*d^2-35*a^3*b^3*c*d+b^6*c^2)*(b*x+a)^(5+n)/b 
^11/(5+n)+63*a^2*d^2*(-4*a^3*d+b^3*c)*(b*x+a)^(6+n)/b^11/(6+n)-21*a*d^2*(- 
10*a^3*d+b^3*c)*(b*x+a)^(7+n)/b^11/(7+n)+3*d^2*(-40*a^3*d+b^3*c)*(b*x+a)^( 
8+n)/b^11/(8+n)+45*a^2*d^3*(b*x+a)^(9+n)/b^11/(9+n)-10*a*d^3*(b*x+a)^(10+n 
)/b^11/(10+n)+d^3*(b*x+a)^(11+n)/b^11/(11+n)
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.87 \[ \int x (a+b x)^n \left (c+d x^3\right )^3 \, dx=\frac {(a+b x)^{1+n} \left (\frac {a \left (-b^3 c+a^3 d\right )^3}{1+n}+\frac {\left (b^3 c-10 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)}{2+n}+\frac {9 a^2 d \left (-b^3 c+a^3 d\right ) \left (-2 b^3 c+5 a^3 d\right ) (a+b x)^2}{3+n}-\frac {3 a d \left (4 b^6 c^2-35 a^3 b^3 c d+40 a^6 d^2\right ) (a+b x)^3}{4+n}+\frac {3 d \left (b^6 c^2-35 a^3 b^3 c d+70 a^6 d^2\right ) (a+b x)^4}{5+n}+\frac {63 a^2 d^2 \left (b^3 c-4 a^3 d\right ) (a+b x)^5}{6+n}+\frac {21 a d^2 \left (-b^3 c+10 a^3 d\right ) (a+b x)^6}{7+n}+\frac {3 d^2 \left (b^3 c-40 a^3 d\right ) (a+b x)^7}{8+n}+\frac {45 a^2 d^3 (a+b x)^8}{9+n}-\frac {10 a d^3 (a+b x)^9}{10+n}+\frac {d^3 (a+b x)^{10}}{11+n}\right )}{b^{11}} \] Input:

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

Output:

((a + b*x)^(1 + n)*((a*(-(b^3*c) + a^3*d)^3)/(1 + n) + ((b^3*c - 10*a^3*d) 
*(b^3*c - a^3*d)^2*(a + b*x))/(2 + n) + (9*a^2*d*(-(b^3*c) + a^3*d)*(-2*b^ 
3*c + 5*a^3*d)*(a + b*x)^2)/(3 + n) - (3*a*d*(4*b^6*c^2 - 35*a^3*b^3*c*d + 
 40*a^6*d^2)*(a + b*x)^3)/(4 + n) + (3*d*(b^6*c^2 - 35*a^3*b^3*c*d + 70*a^ 
6*d^2)*(a + b*x)^4)/(5 + n) + (63*a^2*d^2*(b^3*c - 4*a^3*d)*(a + b*x)^5)/( 
6 + n) + (21*a*d^2*(-(b^3*c) + 10*a^3*d)*(a + b*x)^6)/(7 + n) + (3*d^2*(b^ 
3*c - 40*a^3*d)*(a + b*x)^7)/(8 + n) + (45*a^2*d^3*(a + b*x)^8)/(9 + n) - 
(10*a*d^3*(a + b*x)^9)/(10 + n) + (d^3*(a + b*x)^10)/(11 + n)))/b^11
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

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

\(\Big \downarrow \) 2123

\(\displaystyle \int \left (\frac {21 a d^2 \left (10 a^3 d-b^3 c\right ) (a+b x)^{n+6}}{b^{10}}+\frac {3 d^2 \left (b^3 c-40 a^3 d\right ) (a+b x)^{n+7}}{b^{10}}+\frac {a \left (a^3 d-b^3 c\right )^3 (a+b x)^n}{b^{10}}+\frac {\left (b^3 c-10 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^{10}}+\frac {45 a^2 d^3 (a+b x)^{n+8}}{b^{10}}-\frac {3 a d \left (40 a^6 d^2-35 a^3 b^3 c d+4 b^6 c^2\right ) (a+b x)^{n+3}}{b^{10}}+\frac {3 d \left (70 a^6 d^2-35 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+4}}{b^{10}}-\frac {63 a^2 d^2 \left (4 a^3 d-b^3 c\right ) (a+b x)^{n+5}}{b^{10}}+\frac {9 a^2 d \left (2 b^3 c-5 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^{10}}-\frac {10 a d^3 (a+b x)^{n+9}}{b^{10}}+\frac {d^3 (a+b x)^{n+10}}{b^{10}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {21 a d^2 \left (b^3 c-10 a^3 d\right ) (a+b x)^{n+7}}{b^{11} (n+7)}+\frac {3 d^2 \left (b^3 c-40 a^3 d\right ) (a+b x)^{n+8}}{b^{11} (n+8)}-\frac {a \left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{11} (n+1)}+\frac {\left (b^3 c-10 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{11} (n+2)}+\frac {45 a^2 d^3 (a+b x)^{n+9}}{b^{11} (n+9)}-\frac {3 a d \left (40 a^6 d^2-35 a^3 b^3 c d+4 b^6 c^2\right ) (a+b x)^{n+4}}{b^{11} (n+4)}+\frac {3 d \left (70 a^6 d^2-35 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+5}}{b^{11} (n+5)}+\frac {63 a^2 d^2 \left (b^3 c-4 a^3 d\right ) (a+b x)^{n+6}}{b^{11} (n+6)}+\frac {9 a^2 d \left (2 b^3 c-5 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+3}}{b^{11} (n+3)}-\frac {10 a d^3 (a+b x)^{n+10}}{b^{11} (n+10)}+\frac {d^3 (a+b x)^{n+11}}{b^{11} (n+11)}\)

Input:

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

Output:

-((a*(b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^11*(1 + n))) + ((b^3*c - 10*a 
^3*d)*(b^3*c - a^3*d)^2*(a + b*x)^(2 + n))/(b^11*(2 + n)) + (9*a^2*d*(2*b^ 
3*c - 5*a^3*d)*(b^3*c - a^3*d)*(a + b*x)^(3 + n))/(b^11*(3 + n)) - (3*a*d* 
(4*b^6*c^2 - 35*a^3*b^3*c*d + 40*a^6*d^2)*(a + b*x)^(4 + n))/(b^11*(4 + n) 
) + (3*d*(b^6*c^2 - 35*a^3*b^3*c*d + 70*a^6*d^2)*(a + b*x)^(5 + n))/(b^11* 
(5 + n)) + (63*a^2*d^2*(b^3*c - 4*a^3*d)*(a + b*x)^(6 + n))/(b^11*(6 + n)) 
 - (21*a*d^2*(b^3*c - 10*a^3*d)*(a + b*x)^(7 + n))/(b^11*(7 + n)) + (3*d^2 
*(b^3*c - 40*a^3*d)*(a + b*x)^(8 + n))/(b^11*(8 + n)) + (45*a^2*d^3*(a + b 
*x)^(9 + n))/(b^11*(9 + n)) - (10*a*d^3*(a + b*x)^(10 + n))/(b^11*(10 + n) 
) + (d^3*(a + b*x)^(11 + n))/(b^11*(11 + 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. \(2971\) vs. \(2(396)=792\).

Time = 0.38 (sec) , antiderivative size = 2972, normalized size of antiderivative = 7.51

method	result	size
gosper	\(\text {Expression too large to display}\)	\(2972\)
orering	\(\text {Expression too large to display}\)	\(2975\)
risch	\(\text {Expression too large to display}\)	\(3409\)
parallelrisch	\(\text {Expression too large to display}\)	\(4900\)

Input:

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

Output:

1/b^11*(b*x+a)^(1+n)/(n^11+66*n^10+1925*n^9+32670*n^8+357423*n^7+2637558*n 
^6+13339535*n^5+45995730*n^4+105258076*n^3+150917976*n^2+120543840*n+39916 
800)*(b^10*d^3*n^10*x^10+55*b^10*d^3*n^9*x^10-10*a*b^9*d^3*n^9*x^9+1320*b^ 
10*d^3*n^8*x^10-450*a*b^9*d^3*n^8*x^9+3*b^10*c*d^2*n^10*x^7+18150*b^10*d^3 
*n^7*x^10+90*a^2*b^8*d^3*n^8*x^8-8700*a*b^9*d^3*n^7*x^9+174*b^10*c*d^2*n^9 
*x^7+157773*b^10*d^3*n^6*x^10+3240*a^2*b^8*d^3*n^7*x^8-21*a*b^9*c*d^2*n^9* 
x^6-94500*a*b^9*d^3*n^6*x^9+4383*b^10*c*d^2*n^8*x^7+902055*b^10*d^3*n^5*x^ 
10-720*a^3*b^7*d^3*n^7*x^7+49140*a^2*b^8*d^3*n^6*x^8-1071*a*b^9*c*d^2*n^8* 
x^6-632730*a*b^9*d^3*n^5*x^9+3*b^10*c^2*d*n^10*x^4+62946*b^10*c*d^2*n^7*x^ 
7+3416930*b^10*d^3*n^4*x^10-20160*a^3*b^7*d^3*n^6*x^7+126*a^2*b^8*c*d^2*n^ 
8*x^5+408240*a^2*b^8*d^3*n^5*x^8-23184*a*b^9*c*d^2*n^7*x^6-2693250*a*b^9*d 
^3*n^4*x^9+183*b^10*c^2*d*n^9*x^4+568701*b^10*c*d^2*n^6*x^7+8409500*b^10*d 
^3*n^3*x^10+5040*a^4*b^6*d^3*n^6*x^6-231840*a^3*b^7*d^3*n^5*x^7+5670*a^2*b 
^8*c*d^2*n^7*x^5+2020410*a^2*b^8*d^3*n^4*x^8-12*a*b^9*c^2*d*n^9*x^3-278334 
*a*b^9*c*d^2*n^6*x^6-7236800*a*b^9*d^3*n^3*x^9+4860*b^10*c^2*d*n^8*x^4+336 
3066*b^10*c*d^2*n^5*x^7+12753576*b^10*d^3*n^2*x^10+105840*a^4*b^6*d^3*n^5* 
x^6-630*a^3*b^7*c*d^2*n^7*x^4-1411200*a^3*b^7*d^3*n^4*x^7+105084*a^2*b^8*c 
*d^2*n^6*x^5+6055560*a^2*b^8*d^3*n^3*x^8-684*a*b^9*c^2*d*n^8*x^3-2032569*a 
*b^9*c*d^2*n^5*x^6-11727000*a*b^9*d^3*n^2*x^9+b^10*c^3*n^10*x+73710*b^10*c 
^2*d*n^7*x^4+13114077*b^10*c*d^2*n^4*x^7+10628640*b^10*d^3*n*x^10-30240...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2919 vs. \(2 (396) = 792\).

Time = 0.12 (sec) , antiderivative size = 2919, normalized size of antiderivative = 7.37 \[ \int x (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

-(a^2*b^9*c^3*n^9 + 63*a^2*b^9*c^3*n^8 + 1734*a^2*b^9*c^3*n^7 + 19958400*a 
^2*b^9*c^3 - 23950080*a^5*b^6*c^2*d + 14968800*a^8*b^3*c*d^2 - 3628800*a^1 
1*d^3 - (b^11*d^3*n^10 + 55*b^11*d^3*n^9 + 1320*b^11*d^3*n^8 + 18150*b^11* 
d^3*n^7 + 157773*b^11*d^3*n^6 + 902055*b^11*d^3*n^5 + 3416930*b^11*d^3*n^4 
 + 8409500*b^11*d^3*n^3 + 12753576*b^11*d^3*n^2 + 10628640*b^11*d^3*n + 36 
28800*b^11*d^3)*x^11 - (a*b^10*d^3*n^10 + 45*a*b^10*d^3*n^9 + 870*a*b^10*d 
^3*n^8 + 9450*a*b^10*d^3*n^7 + 63273*a*b^10*d^3*n^6 + 269325*a*b^10*d^3*n^ 
5 + 723680*a*b^10*d^3*n^4 + 1172700*a*b^10*d^3*n^3 + 1026576*a*b^10*d^3*n^ 
2 + 362880*a*b^10*d^3*n)*x^10 + 10*(a^2*b^9*d^3*n^9 + 36*a^2*b^9*d^3*n^8 + 
 546*a^2*b^9*d^3*n^7 + 4536*a^2*b^9*d^3*n^6 + 22449*a^2*b^9*d^3*n^5 + 6728 
4*a^2*b^9*d^3*n^4 + 118124*a^2*b^9*d^3*n^3 + 109584*a^2*b^9*d^3*n^2 + 4032 
0*a^2*b^9*d^3*n)*x^9 - 3*(b^11*c*d^2*n^10 + 58*b^11*c*d^2*n^9 + 4989600*b^ 
11*c*d^2 + 3*(487*b^11*c*d^2 + 10*a^3*b^8*d^3)*n^8 + 6*(3497*b^11*c*d^2 + 
140*a^3*b^8*d^3)*n^7 + 21*(9027*b^11*c*d^2 + 460*a^3*b^8*d^3)*n^6 + 294*(3 
813*b^11*c*d^2 + 200*a^3*b^8*d^3)*n^5 + (4371359*b^11*c*d^2 + 203070*a^3*b 
^8*d^3)*n^4 + 2*(5512429*b^11*c*d^2 + 196980*a^3*b^8*d^3)*n^3 + 36*(473867 
*b^11*c*d^2 + 10890*a^3*b^8*d^3)*n^2 + 360*(40123*b^11*c*d^2 + 420*a^3*b^8 
*d^3)*n)*x^8 - 3*(a*b^10*c*d^2*n^10 + 51*a*b^10*c*d^2*n^9 + 1104*a*b^10*c* 
d^2*n^8 + 6*(2209*a*b^10*c*d^2 - 40*a^4*b^7*d^3)*n^7 + 21*(4609*a*b^10*c*d 
^2 - 240*a^4*b^7*d^3)*n^6 + 21*(21119*a*b^10*c*d^2 - 2000*a^4*b^7*d^3)*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56151 vs. \(2 (374) = 748\).

Time = 22.05 (sec) , antiderivative size = 56151, normalized size of antiderivative = 141.80 \[ \int x (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((a**n*(c**3*x**2/2 + 3*c**2*d*x**5/5 + 3*c*d**2*x**8/8 + d**3*x* 
*11/11), Eq(b, 0)), (2520*a**10*d**3*log(a/b + x)/(2520*a**10*b**11 + 2520 
0*a**9*b**12*x + 113400*a**8*b**13*x**2 + 302400*a**7*b**14*x**3 + 529200* 
a**6*b**15*x**4 + 635040*a**5*b**16*x**5 + 529200*a**4*b**17*x**6 + 302400 
*a**3*b**18*x**7 + 113400*a**2*b**19*x**8 + 25200*a*b**20*x**9 + 2520*b**2 
1*x**10) + 7381*a**10*d**3/(2520*a**10*b**11 + 25200*a**9*b**12*x + 113400 
*a**8*b**13*x**2 + 302400*a**7*b**14*x**3 + 529200*a**6*b**15*x**4 + 63504 
0*a**5*b**16*x**5 + 529200*a**4*b**17*x**6 + 302400*a**3*b**18*x**7 + 1134 
00*a**2*b**19*x**8 + 25200*a*b**20*x**9 + 2520*b**21*x**10) + 25200*a**9*b 
*d**3*x*log(a/b + x)/(2520*a**10*b**11 + 25200*a**9*b**12*x + 113400*a**8* 
b**13*x**2 + 302400*a**7*b**14*x**3 + 529200*a**6*b**15*x**4 + 635040*a**5 
*b**16*x**5 + 529200*a**4*b**17*x**6 + 302400*a**3*b**18*x**7 + 113400*a** 
2*b**19*x**8 + 25200*a*b**20*x**9 + 2520*b**21*x**10) + 71290*a**9*b*d**3* 
x/(2520*a**10*b**11 + 25200*a**9*b**12*x + 113400*a**8*b**13*x**2 + 302400 
*a**7*b**14*x**3 + 529200*a**6*b**15*x**4 + 635040*a**5*b**16*x**5 + 52920 
0*a**4*b**17*x**6 + 302400*a**3*b**18*x**7 + 113400*a**2*b**19*x**8 + 2520 
0*a*b**20*x**9 + 2520*b**21*x**10) + 113400*a**8*b**2*d**3*x**2*log(a/b + 
x)/(2520*a**10*b**11 + 25200*a**9*b**12*x + 113400*a**8*b**13*x**2 + 30240 
0*a**7*b**14*x**3 + 529200*a**6*b**15*x**4 + 635040*a**5*b**16*x**5 + 5292 
00*a**4*b**17*x**6 + 302400*a**3*b**18*x**7 + 113400*a**2*b**19*x**8 + ...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 953 vs. \(2 (396) = 792\).

Time = 0.07 (sec) , antiderivative size = 953, normalized size of antiderivative = 2.41 \[ \int x (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^3/((n^2 + 3*n + 2)*b^2) + 
3*((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6 
*n)*a*b^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b^2*x 
^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n*c^2*d/((n^5 + 15*n^4 + 85*n^3 + 22 
5*n^2 + 274*n + 120)*b^5) + 3*((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n 
^3 + 13132*n^2 + 13068*n + 5040)*b^8*x^8 + (n^7 + 21*n^6 + 175*n^5 + 735*n 
^4 + 1624*n^3 + 1764*n^2 + 720*n)*a*b^7*x^7 - 7*(n^6 + 15*n^5 + 85*n^4 + 2 
25*n^3 + 274*n^2 + 120*n)*a^2*b^6*x^6 + 42*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 
 + 24*n)*a^3*b^5*x^5 - 210*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^4*b^4*x^4 + 840* 
(n^3 + 3*n^2 + 2*n)*a^5*b^3*x^3 - 2520*(n^2 + n)*a^6*b^2*x^2 + 5040*a^7*b* 
n*x - 5040*a^8)*(b*x + a)^n*c*d^2/((n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22 
449*n^4 + 67284*n^3 + 118124*n^2 + 109584*n + 40320)*b^8) + ((n^10 + 55*n^ 
9 + 1320*n^8 + 18150*n^7 + 157773*n^6 + 902055*n^5 + 3416930*n^4 + 8409500 
*n^3 + 12753576*n^2 + 10628640*n + 3628800)*b^11*x^11 + (n^10 + 45*n^9 + 8 
70*n^8 + 9450*n^7 + 63273*n^6 + 269325*n^5 + 723680*n^4 + 1172700*n^3 + 10 
26576*n^2 + 362880*n)*a*b^10*x^10 - 10*(n^9 + 36*n^8 + 546*n^7 + 4536*n^6 
+ 22449*n^5 + 67284*n^4 + 118124*n^3 + 109584*n^2 + 40320*n)*a^2*b^9*x^9 + 
 90*(n^8 + 28*n^7 + 322*n^6 + 1960*n^5 + 6769*n^4 + 13132*n^3 + 13068*n^2 
+ 5040*n)*a^3*b^8*x^8 - 720*(n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 
 1764*n^2 + 720*n)*a^4*b^7*x^7 + 5040*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4934 vs. \(2 (396) = 792\).

Time = 0.17 (sec) , antiderivative size = 4934, normalized size of antiderivative = 12.46 \[ \int x (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

((b*x + a)^n*b^11*d^3*n^10*x^11 + (b*x + a)^n*a*b^10*d^3*n^10*x^10 + 55*(b 
*x + a)^n*b^11*d^3*n^9*x^11 + 45*(b*x + a)^n*a*b^10*d^3*n^9*x^10 + 1320*(b 
*x + a)^n*b^11*d^3*n^8*x^11 + 3*(b*x + a)^n*b^11*c*d^2*n^10*x^8 - 10*(b*x 
+ a)^n*a^2*b^9*d^3*n^9*x^9 + 870*(b*x + a)^n*a*b^10*d^3*n^8*x^10 + 18150*( 
b*x + a)^n*b^11*d^3*n^7*x^11 + 3*(b*x + a)^n*a*b^10*c*d^2*n^10*x^7 + 174*( 
b*x + a)^n*b^11*c*d^2*n^9*x^8 - 360*(b*x + a)^n*a^2*b^9*d^3*n^8*x^9 + 9450 
*(b*x + a)^n*a*b^10*d^3*n^7*x^10 + 157773*(b*x + a)^n*b^11*d^3*n^6*x^11 + 
153*(b*x + a)^n*a*b^10*c*d^2*n^9*x^7 + 4383*(b*x + a)^n*b^11*c*d^2*n^8*x^8 
 + 90*(b*x + a)^n*a^3*b^8*d^3*n^8*x^8 - 5460*(b*x + a)^n*a^2*b^9*d^3*n^7*x 
^9 + 63273*(b*x + a)^n*a*b^10*d^3*n^6*x^10 + 902055*(b*x + a)^n*b^11*d^3*n 
^5*x^11 + 3*(b*x + a)^n*b^11*c^2*d*n^10*x^5 - 21*(b*x + a)^n*a^2*b^9*c*d^2 
*n^9*x^6 + 3312*(b*x + a)^n*a*b^10*c*d^2*n^8*x^7 + 62946*(b*x + a)^n*b^11* 
c*d^2*n^7*x^8 + 2520*(b*x + a)^n*a^3*b^8*d^3*n^7*x^8 - 45360*(b*x + a)^n*a 
^2*b^9*d^3*n^6*x^9 + 269325*(b*x + a)^n*a*b^10*d^3*n^5*x^10 + 3416930*(b*x 
 + a)^n*b^11*d^3*n^4*x^11 + 3*(b*x + a)^n*a*b^10*c^2*d*n^10*x^4 + 183*(b*x 
 + a)^n*b^11*c^2*d*n^9*x^5 - 945*(b*x + a)^n*a^2*b^9*c*d^2*n^8*x^6 + 39762 
*(b*x + a)^n*a*b^10*c*d^2*n^7*x^7 - 720*(b*x + a)^n*a^4*b^7*d^3*n^7*x^7 + 
568701*(b*x + a)^n*b^11*c*d^2*n^6*x^8 + 28980*(b*x + a)^n*a^3*b^8*d^3*n^6* 
x^8 - 224490*(b*x + a)^n*a^2*b^9*d^3*n^5*x^9 + 723680*(b*x + a)^n*a*b^10*d 
^3*n^4*x^10 + 8409500*(b*x + a)^n*b^11*d^3*n^3*x^11 + 171*(b*x + a)^n*a...
 

Mupad [B] (verification not implemented)

Time = 25.30 (sec) , antiderivative size = 2436, normalized size of antiderivative = 6.15 \[ \int x (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(d^3*x^11*(a + b*x)^n*(10628640*n + 12753576*n^2 + 8409500*n^3 + 3416930*n 
^4 + 902055*n^5 + 157773*n^6 + 18150*n^7 + 1320*n^8 + 55*n^9 + n^10 + 3628 
800))/(120543840*n + 150917976*n^2 + 105258076*n^3 + 45995730*n^4 + 133395 
35*n^5 + 2637558*n^6 + 357423*n^7 + 32670*n^8 + 1925*n^9 + 66*n^10 + n^11 
+ 39916800) - (a^2*(a + b*x)^n*(19958400*b^9*c^3 - 3628800*a^9*d^3 + 30334 
320*b^9*c^3*n + 19978308*b^9*c^3*n^2 + 7494416*b^9*c^3*n^3 + 1767087*b^9*c 
^3*n^4 + 271929*b^9*c^3*n^5 + 27342*b^9*c^3*n^6 + 1734*b^9*c^3*n^7 + 63*b^ 
9*c^3*n^8 + b^9*c^3*n^9 - 23950080*a^3*b^6*c^2*d + 14968800*a^6*b^3*c*d^2 
- 17640288*a^3*b^6*c^2*d*n + 4520880*a^6*b^3*c*d^2*n - 5365728*a^3*b^6*c^2 
*d*n^2 + 453600*a^6*b^3*c*d^2*n^2 - 862920*a^3*b^6*c^2*d*n^3 + 15120*a^6*b 
^3*c*d^2*n^3 - 77400*a^3*b^6*c^2*d*n^4 - 3672*a^3*b^6*c^2*d*n^5 - 72*a^3*b 
^6*c^2*d*n^6))/(b^11*(120543840*n + 150917976*n^2 + 105258076*n^3 + 459957 
30*n^4 + 13339535*n^5 + 2637558*n^6 + 357423*n^7 + 32670*n^8 + 1925*n^9 + 
66*n^10 + n^11 + 39916800)) + (x^2*(n + 1)*(a + b*x)^n*(19958400*b^9*c^3 + 
 1814400*a^9*d^3*n + 30334320*b^9*c^3*n + 19978308*b^9*c^3*n^2 + 7494416*b 
^9*c^3*n^3 + 1767087*b^9*c^3*n^4 + 271929*b^9*c^3*n^5 + 27342*b^9*c^3*n^6 
+ 1734*b^9*c^3*n^7 + 63*b^9*c^3*n^8 + b^9*c^3*n^9 + 11975040*a^3*b^6*c^2*d 
*n - 7484400*a^6*b^3*c*d^2*n + 8820144*a^3*b^6*c^2*d*n^2 - 2260440*a^6*b^3 
*c*d^2*n^2 + 2682864*a^3*b^6*c^2*d*n^3 - 226800*a^6*b^3*c*d^2*n^3 + 431460 
*a^3*b^6*c^2*d*n^4 - 7560*a^6*b^3*c*d^2*n^4 + 38700*a^3*b^6*c^2*d*n^5 +...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 3408, normalized size of antiderivative = 8.61 \[ \int x (a+b x)^n \left (c+d x^3\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

((a + b*x)**n*(3628800*a**11*d**3 - 3628800*a**10*b*d**3*n*x + 1814400*a** 
9*b**2*d**3*n**2*x**2 + 1814400*a**9*b**2*d**3*n*x**2 - 15120*a**8*b**3*c* 
d**2*n**3 - 453600*a**8*b**3*c*d**2*n**2 - 4520880*a**8*b**3*c*d**2*n - 14 
968800*a**8*b**3*c*d**2 - 604800*a**8*b**3*d**3*n**3*x**3 - 1814400*a**8*b 
**3*d**3*n**2*x**3 - 1209600*a**8*b**3*d**3*n*x**3 + 15120*a**7*b**4*c*d** 
2*n**4*x + 453600*a**7*b**4*c*d**2*n**3*x + 4520880*a**7*b**4*c*d**2*n**2* 
x + 14968800*a**7*b**4*c*d**2*n*x + 151200*a**7*b**4*d**3*n**4*x**4 + 9072 
00*a**7*b**4*d**3*n**3*x**4 + 1663200*a**7*b**4*d**3*n**2*x**4 + 907200*a* 
*7*b**4*d**3*n*x**4 - 7560*a**6*b**5*c*d**2*n**5*x**2 - 234360*a**6*b**5*c 
*d**2*n**4*x**2 - 2487240*a**6*b**5*c*d**2*n**3*x**2 - 9744840*a**6*b**5*c 
*d**2*n**2*x**2 - 7484400*a**6*b**5*c*d**2*n*x**2 - 30240*a**6*b**5*d**3*n 
**5*x**5 - 302400*a**6*b**5*d**3*n**4*x**5 - 1058400*a**6*b**5*d**3*n**3*x 
**5 - 1512000*a**6*b**5*d**3*n**2*x**5 - 725760*a**6*b**5*d**3*n*x**5 + 72 
*a**5*b**6*c**2*d*n**6 + 3672*a**5*b**6*c**2*d*n**5 + 77400*a**5*b**6*c**2 
*d*n**4 + 862920*a**5*b**6*c**2*d*n**3 + 5365728*a**5*b**6*c**2*d*n**2 + 1 
7640288*a**5*b**6*c**2*d*n + 23950080*a**5*b**6*c**2*d + 2520*a**5*b**6*c* 
d**2*n**6*x**3 + 83160*a**5*b**6*c*d**2*n**5*x**3 + 985320*a**5*b**6*c*d** 
2*n**4*x**3 + 4906440*a**5*b**6*c*d**2*n**3*x**3 + 8991360*a**5*b**6*c*d** 
2*n**2*x**3 + 4989600*a**5*b**6*c*d**2*n*x**3 + 5040*a**5*b**6*d**3*n**6*x 
**6 + 75600*a**5*b**6*d**3*n**5*x**6 + 428400*a**5*b**6*d**3*n**4*x**6 ...