\(\int (a+b x)^n (c+d x^3)^3 \, dx\) [37]

Optimal result

Integrand size = 17, antiderivative size = 337 \[ \int (a+b x)^n \left (c+d x^3\right )^3 \, dx=\frac {\left (b^3 c-a^3 d\right )^3 (a+b x)^{1+n}}{b^{10} (1+n)}+\frac {9 a^2 d \left (b^3 c-a^3 d\right )^2 (a+b x)^{2+n}}{b^{10} (2+n)}-\frac {9 a d \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{3+n}}{b^{10} (3+n)}+\frac {3 d \left (b^6 c^2-20 a^3 b^3 c d+28 a^6 d^2\right ) (a+b x)^{4+n}}{b^{10} (4+n)}+\frac {9 a^2 d^2 \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{5+n}}{b^{10} (5+n)}-\frac {18 a d^2 \left (b^3 c-7 a^3 d\right ) (a+b x)^{6+n}}{b^{10} (6+n)}+\frac {3 d^2 \left (b^3 c-28 a^3 d\right ) (a+b x)^{7+n}}{b^{10} (7+n)}+\frac {36 a^2 d^3 (a+b x)^{8+n}}{b^{10} (8+n)}-\frac {9 a d^3 (a+b x)^{9+n}}{b^{10} (9+n)}+\frac {d^3 (a+b x)^{10+n}}{b^{10} (10+n)} \] Output:

(-a^3*d+b^3*c)^3*(b*x+a)^(1+n)/b^10/(1+n)+9*a^2*d*(-a^3*d+b^3*c)^2*(b*x+a) 
^(2+n)/b^10/(2+n)-9*a*d*(-4*a^3*d+b^3*c)*(-a^3*d+b^3*c)*(b*x+a)^(3+n)/b^10 
/(3+n)+3*d*(28*a^6*d^2-20*a^3*b^3*c*d+b^6*c^2)*(b*x+a)^(4+n)/b^10/(4+n)+9* 
a^2*d^2*(-14*a^3*d+5*b^3*c)*(b*x+a)^(5+n)/b^10/(5+n)-18*a*d^2*(-7*a^3*d+b^ 
3*c)*(b*x+a)^(6+n)/b^10/(6+n)+3*d^2*(-28*a^3*d+b^3*c)*(b*x+a)^(7+n)/b^10/( 
7+n)+36*a^2*d^3*(b*x+a)^(8+n)/b^10/(8+n)-9*a*d^3*(b*x+a)^(9+n)/b^10/(9+n)+ 
d^3*(b*x+a)^(10+n)/b^10/(10+n)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.86 \[ \int (a+b x)^n \left (c+d x^3\right )^3 \, dx=\frac {(a+b x)^{1+n} \left (\frac {\left (b^3 c-a^3 d\right )^3}{1+n}+\frac {9 d \left (a b^3 c-a^4 d\right )^2 (a+b x)}{2+n}-\frac {9 a d \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^2}{3+n}+\frac {3 d \left (b^6 c^2-20 a^3 b^3 c d+28 a^6 d^2\right ) (a+b x)^3}{4+n}+\frac {9 a^2 d^2 \left (5 b^3 c-14 a^3 d\right ) (a+b x)^4}{5+n}+\frac {18 a d^2 \left (-b^3 c+7 a^3 d\right ) (a+b x)^5}{6+n}+\frac {3 d^2 \left (b^3 c-28 a^3 d\right ) (a+b x)^6}{7+n}+\frac {36 a^2 d^3 (a+b x)^7}{8+n}-\frac {9 a d^3 (a+b x)^8}{9+n}+\frac {d^3 (a+b x)^9}{10+n}\right )}{b^{10}} \] Input:

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

Output:

((a + b*x)^(1 + n)*((b^3*c - a^3*d)^3/(1 + n) + (9*d*(a*b^3*c - a^4*d)^2*( 
a + b*x))/(2 + n) - (9*a*d*(b^3*c - 4*a^3*d)*(b^3*c - a^3*d)*(a + b*x)^2)/ 
(3 + n) + (3*d*(b^6*c^2 - 20*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^3)/(4 + n 
) + (9*a^2*d^2*(5*b^3*c - 14*a^3*d)*(a + b*x)^4)/(5 + n) + (18*a*d^2*(-(b^ 
3*c) + 7*a^3*d)*(a + b*x)^5)/(6 + n) + (3*d^2*(b^3*c - 28*a^3*d)*(a + b*x) 
^6)/(7 + n) + (36*a^2*d^3*(a + b*x)^7)/(8 + n) - (9*a*d^3*(a + b*x)^8)/(9 
+ n) + (d^3*(a + b*x)^9)/(10 + n)))/b^10
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2389, 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.

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

\(\Big \downarrow \) 2389

\(\displaystyle \int \left (\frac {9 d \left (a b^3 c-a^4 d\right )^2 (a+b x)^{n+1}}{b^9}+\frac {18 a d^2 \left (7 a^3 d-b^3 c\right ) (a+b x)^{n+5}}{b^9}+\frac {3 d^2 \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+6}}{b^9}+\frac {\left (b^3 c-a^3 d\right )^3 (a+b x)^n}{b^9}+\frac {9 a d \left (b^3 c-4 a^3 d\right ) \left (a^3 d-b^3 c\right ) (a+b x)^{n+2}}{b^9}+\frac {36 a^2 d^3 (a+b x)^{n+7}}{b^9}+\frac {3 d \left (28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+3}}{b^9}-\frac {9 a^2 d^2 \left (14 a^3 d-5 b^3 c\right ) (a+b x)^{n+4}}{b^9}-\frac {9 a d^3 (a+b x)^{n+8}}{b^9}+\frac {d^3 (a+b x)^{n+9}}{b^9}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {18 a d^2 \left (b^3 c-7 a^3 d\right ) (a+b x)^{n+6}}{b^{10} (n+6)}+\frac {3 d^2 \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+7}}{b^{10} (n+7)}+\frac {\left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{10} (n+1)}-\frac {9 a d \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+3}}{b^{10} (n+3)}+\frac {36 a^2 d^3 (a+b x)^{n+8}}{b^{10} (n+8)}+\frac {3 d \left (28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+4}}{b^{10} (n+4)}+\frac {9 a^2 d^2 \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{n+5}}{b^{10} (n+5)}+\frac {9 a^2 d \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{10} (n+2)}-\frac {9 a d^3 (a+b x)^{n+9}}{b^{10} (n+9)}+\frac {d^3 (a+b x)^{n+10}}{b^{10} (n+10)}\)

Input:

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

Output:

((b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^10*(1 + n)) + (9*a^2*d*(b^3*c - a 
^3*d)^2*(a + b*x)^(2 + n))/(b^10*(2 + n)) - (9*a*d*(b^3*c - 4*a^3*d)*(b^3* 
c - a^3*d)*(a + b*x)^(3 + n))/(b^10*(3 + n)) + (3*d*(b^6*c^2 - 20*a^3*b^3* 
c*d + 28*a^6*d^2)*(a + b*x)^(4 + n))/(b^10*(4 + n)) + (9*a^2*d^2*(5*b^3*c 
- 14*a^3*d)*(a + b*x)^(5 + n))/(b^10*(5 + n)) - (18*a*d^2*(b^3*c - 7*a^3*d 
)*(a + b*x)^(6 + n))/(b^10*(6 + n)) + (3*d^2*(b^3*c - 28*a^3*d)*(a + b*x)^ 
(7 + n))/(b^10*(7 + n)) + (36*a^2*d^3*(a + b*x)^(8 + n))/(b^10*(8 + n)) - 
(9*a*d^3*(a + b*x)^(9 + n))/(b^10*(9 + n)) + (d^3*(a + b*x)^(10 + n))/(b^1 
0*(10 + n))
 

Defintions of rubi rules used

rule 2009

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

rule 2389

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand 
[Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p 
, 0] || EqQ[n, 1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2279\) vs. \(2(337)=674\).

Time = 0.33 (sec) , antiderivative size = 2280, normalized size of antiderivative = 6.77

method	result	size
gosper	\(\text {Expression too large to display}\)	\(2280\)
orering	\(\text {Expression too large to display}\)	\(2283\)
risch	\(\text {Expression too large to display}\)	\(2665\)
parallelrisch	\(\text {Expression too large to display}\)	\(3960\)

Input:

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

Output:

-1/b^10*(b*x+a)^(1+n)/(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^9*d^3*n^9*x 
^9-45*b^9*d^3*n^8*x^9+9*a*b^8*d^3*n^8*x^8-870*b^9*d^3*n^7*x^9+324*a*b^8*d^ 
3*n^7*x^8-3*b^9*c*d^2*n^9*x^6-9450*b^9*d^3*n^6*x^9-72*a^2*b^7*d^3*n^7*x^7+ 
4914*a*b^8*d^3*n^6*x^8-144*b^9*c*d^2*n^8*x^6-63273*b^9*d^3*n^5*x^9-2016*a^ 
2*b^7*d^3*n^6*x^7+18*a*b^8*c*d^2*n^8*x^5+40824*a*b^8*d^3*n^5*x^8-2952*b^9* 
c*d^2*n^7*x^6-269325*b^9*d^3*n^4*x^9+504*a^3*b^6*d^3*n^6*x^6-23184*a^2*b^7 
*d^3*n^5*x^7+756*a*b^8*c*d^2*n^7*x^5+202041*a*b^8*d^3*n^4*x^8-3*b^9*c^2*d* 
n^9*x^3-33786*b^9*c*d^2*n^6*x^6-723680*b^9*d^3*n^3*x^9+10584*a^3*b^6*d^3*n 
^5*x^6-90*a^2*b^7*c*d^2*n^7*x^4-141120*a^2*b^7*d^3*n^4*x^7+13176*a*b^8*c*d 
^2*n^6*x^5+605556*a*b^8*d^3*n^3*x^8-153*b^9*c^2*d*n^8*x^3-236817*b^9*c*d^2 
*n^5*x^6-1172700*b^9*d^3*n^2*x^9-3024*a^4*b^5*d^3*n^5*x^5+88200*a^3*b^6*d^ 
3*n^4*x^6-3330*a^2*b^7*c*d^2*n^6*x^4-487368*a^2*b^7*d^3*n^3*x^7+9*a*b^8*c^ 
2*d*n^8*x^2+123660*a*b^8*c*d^2*n^5*x^5+1063116*a*b^8*d^3*n^2*x^8-3348*b^9* 
c^2*d*n^7*x^3-1048446*b^9*c*d^2*n^4*x^6-1026576*b^9*d^3*n*x^9-45360*a^4*b^ 
5*d^3*n^4*x^5+360*a^3*b^6*c*d^2*n^6*x^3+370440*a^3*b^6*d^3*n^3*x^6-49230*a 
^2*b^7*c*d^2*n^5*x^4-945504*a^2*b^7*d^3*n^2*x^7+432*a*b^8*c^2*d*n^7*x^2+67 
8942*a*b^8*c*d^2*n^4*x^5+986256*a*b^8*d^3*n*x^8-b^9*c^3*n^9-41058*b^9*c^2* 
d*n^6*x^3-2911668*b^9*c*d^2*n^3*x^6-362880*b^9*d^3*x^9+15120*a^5*b^4*d^3*n 
^4*x^4-257040*a^4*b^5*d^3*n^3*x^5+11880*a^3*b^6*c*d^2*n^5*x^3+818496*a^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2313 vs. \(2 (337) = 674\).

Time = 0.11 (sec) , antiderivative size = 2313, normalized size of antiderivative = 6.86 \[ \int (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(a*b^9*c^3*n^9 + 54*a*b^9*c^3*n^8 + 1266*a*b^9*c^3*n^7 + 3628800*a*b^9*c^3 
 - 2721600*a^4*b^6*c^2*d + 1555200*a^7*b^3*c*d^2 - 362880*a^10*d^3 + (b^10 
*d^3*n^9 + 45*b^10*d^3*n^8 + 870*b^10*d^3*n^7 + 9450*b^10*d^3*n^6 + 63273* 
b^10*d^3*n^5 + 269325*b^10*d^3*n^4 + 723680*b^10*d^3*n^3 + 1172700*b^10*d^ 
3*n^2 + 1026576*b^10*d^3*n + 362880*b^10*d^3)*x^10 + (a*b^9*d^3*n^9 + 36*a 
*b^9*d^3*n^8 + 546*a*b^9*d^3*n^7 + 4536*a*b^9*d^3*n^6 + 22449*a*b^9*d^3*n^ 
5 + 67284*a*b^9*d^3*n^4 + 118124*a*b^9*d^3*n^3 + 109584*a*b^9*d^3*n^2 + 40 
320*a*b^9*d^3*n)*x^9 - 9*(a^2*b^8*d^3*n^8 + 28*a^2*b^8*d^3*n^7 + 322*a^2*b 
^8*d^3*n^6 + 1960*a^2*b^8*d^3*n^5 + 6769*a^2*b^8*d^3*n^4 + 13132*a^2*b^8*d 
^3*n^3 + 13068*a^2*b^8*d^3*n^2 + 5040*a^2*b^8*d^3*n)*x^8 + 3*(b^10*c*d^2*n 
^9 + 48*b^10*c*d^2*n^8 + 518400*b^10*c*d^2 + 24*(41*b^10*c*d^2 + a^3*b^7*d 
^3)*n^7 + 6*(1877*b^10*c*d^2 + 84*a^3*b^7*d^3)*n^6 + 21*(3759*b^10*c*d^2 + 
 200*a^3*b^7*d^3)*n^5 + 42*(8321*b^10*c*d^2 + 420*a^3*b^7*d^3)*n^4 + 4*(24 
2639*b^10*c*d^2 + 9744*a^3*b^7*d^3)*n^3 + 72*(22439*b^10*c*d^2 + 588*a^3*b 
^7*d^3)*n^2 + 1440*(1003*b^10*c*d^2 + 12*a^3*b^7*d^3)*n)*x^7 + 18*(938*a*b 
^9*c^3 - a^4*b^6*c^2*d)*n^6 + 3*(a*b^9*c*d^2*n^9 + 42*a*b^9*c*d^2*n^8 + 73 
2*a*b^9*c*d^2*n^7 + 6*(1145*a*b^9*c*d^2 - 28*a^4*b^6*d^3)*n^6 + 9*(4191*a* 
b^9*c*d^2 - 280*a^4*b^6*d^3)*n^5 + 24*(5132*a*b^9*c*d^2 - 595*a^4*b^6*d^3) 
*n^4 + 4*(57887*a*b^9*c*d^2 - 9450*a^4*b^6*d^3)*n^3 + 48*(4715*a*b^9*c*d^2 
 - 959*a^4*b^6*d^3)*n^2 + 2880*(30*a*b^9*c*d^2 - 7*a^4*b^6*d^3)*n)*x^6 ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40536 vs. \(2 (316) = 632\).

Time = 44.62 (sec) , antiderivative size = 40536, normalized size of antiderivative = 120.28 \[ \int (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((a**n*(c**3*x + 3*c**2*d*x**4/4 + 3*c*d**2*x**7/7 + d**3*x**10/1 
0), Eq(b, 0)), (2520*a**9*d**3*log(a/b + x)/(2520*a**9*b**10 + 22680*a**8* 
b**11*x + 90720*a**7*b**12*x**2 + 211680*a**6*b**13*x**3 + 317520*a**5*b** 
14*x**4 + 317520*a**4*b**15*x**5 + 211680*a**3*b**16*x**6 + 90720*a**2*b** 
17*x**7 + 22680*a*b**18*x**8 + 2520*b**19*x**9) + 7129*a**9*d**3/(2520*a** 
9*b**10 + 22680*a**8*b**11*x + 90720*a**7*b**12*x**2 + 211680*a**6*b**13*x 
**3 + 317520*a**5*b**14*x**4 + 317520*a**4*b**15*x**5 + 211680*a**3*b**16* 
x**6 + 90720*a**2*b**17*x**7 + 22680*a*b**18*x**8 + 2520*b**19*x**9) + 226 
80*a**8*b*d**3*x*log(a/b + x)/(2520*a**9*b**10 + 22680*a**8*b**11*x + 9072 
0*a**7*b**12*x**2 + 211680*a**6*b**13*x**3 + 317520*a**5*b**14*x**4 + 3175 
20*a**4*b**15*x**5 + 211680*a**3*b**16*x**6 + 90720*a**2*b**17*x**7 + 2268 
0*a*b**18*x**8 + 2520*b**19*x**9) + 61641*a**8*b*d**3*x/(2520*a**9*b**10 + 
 22680*a**8*b**11*x + 90720*a**7*b**12*x**2 + 211680*a**6*b**13*x**3 + 317 
520*a**5*b**14*x**4 + 317520*a**4*b**15*x**5 + 211680*a**3*b**16*x**6 + 90 
720*a**2*b**17*x**7 + 22680*a*b**18*x**8 + 2520*b**19*x**9) + 90720*a**7*b 
**2*d**3*x**2*log(a/b + x)/(2520*a**9*b**10 + 22680*a**8*b**11*x + 90720*a 
**7*b**12*x**2 + 211680*a**6*b**13*x**3 + 317520*a**5*b**14*x**4 + 317520* 
a**4*b**15*x**5 + 211680*a**3*b**16*x**6 + 90720*a**2*b**17*x**7 + 22680*a 
*b**18*x**8 + 2520*b**19*x**9) + 235224*a**7*b**2*d**3*x**2/(2520*a**9*b** 
10 + 22680*a**8*b**11*x + 90720*a**7*b**12*x**2 + 211680*a**6*b**13*x**...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (337) = 674\).

Time = 0.05 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.28 \[ \int (a+b x)^n \left (c+d x^3\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(b*x + a)^(n + 1)*c^3/(b*(n + 1)) + 3*((n^3 + 6*n^2 + 11*n + 6)*b^4*x^4 + 
(n^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n^2 + n)*a^2*b^2*x^2 + 6*a^3*b*n*x - 6* 
a^4)*(b*x + a)^n*c^2*d/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4) + 3*((n^6 
 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b^7*x^7 + (n^6 + 
15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a*b^6*x^6 - 6*(n^5 + 10*n^4 + 
 35*n^3 + 50*n^2 + 24*n)*a^2*b^5*x^5 + 30*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^3 
*b^4*x^4 - 120*(n^3 + 3*n^2 + 2*n)*a^4*b^3*x^3 + 360*(n^2 + n)*a^5*b^2*x^2 
 - 720*a^6*b*n*x + 720*a^7)*(b*x + a)^n*c*d^2/((n^7 + 28*n^6 + 322*n^5 + 1 
960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*b^7) + ((n^9 + 45*n^8 + 8 
70*n^7 + 9450*n^6 + 63273*n^5 + 269325*n^4 + 723680*n^3 + 1172700*n^2 + 10 
26576*n + 362880)*b^10*x^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*b^9*x^9 - 9*(n^8 + 2 
8*n^7 + 322*n^6 + 1960*n^5 + 6769*n^4 + 13132*n^3 + 13068*n^2 + 5040*n)*a^ 
2*b^8*x^8 + 72*(n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 7 
20*n)*a^3*b^7*x^7 - 504*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n 
)*a^4*b^6*x^6 + 3024*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^5*b^5*x^5 - 
 15120*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^6*b^4*x^4 + 60480*(n^3 + 3*n^2 + 2*n 
)*a^7*b^3*x^3 - 181440*(n^2 + n)*a^8*b^2*x^2 + 362880*a^9*b*n*x - 362880*a 
^10)*(b*x + a)^n*d^3/((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 + 3...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3874 vs. \(2 (337) = 674\).

Time = 0.16 (sec) , antiderivative size = 3874, normalized size of antiderivative = 11.50 \[ \int (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

((b*x + a)^n*b^10*d^3*n^9*x^10 + (b*x + a)^n*a*b^9*d^3*n^9*x^9 + 45*(b*x + 
 a)^n*b^10*d^3*n^8*x^10 + 36*(b*x + a)^n*a*b^9*d^3*n^8*x^9 + 870*(b*x + a) 
^n*b^10*d^3*n^7*x^10 + 3*(b*x + a)^n*b^10*c*d^2*n^9*x^7 - 9*(b*x + a)^n*a^ 
2*b^8*d^3*n^8*x^8 + 546*(b*x + a)^n*a*b^9*d^3*n^7*x^9 + 9450*(b*x + a)^n*b 
^10*d^3*n^6*x^10 + 3*(b*x + a)^n*a*b^9*c*d^2*n^9*x^6 + 144*(b*x + a)^n*b^1 
0*c*d^2*n^8*x^7 - 252*(b*x + a)^n*a^2*b^8*d^3*n^7*x^8 + 4536*(b*x + a)^n*a 
*b^9*d^3*n^6*x^9 + 63273*(b*x + a)^n*b^10*d^3*n^5*x^10 + 126*(b*x + a)^n*a 
*b^9*c*d^2*n^8*x^6 + 2952*(b*x + a)^n*b^10*c*d^2*n^7*x^7 + 72*(b*x + a)^n* 
a^3*b^7*d^3*n^7*x^7 - 2898*(b*x + a)^n*a^2*b^8*d^3*n^6*x^8 + 22449*(b*x + 
a)^n*a*b^9*d^3*n^5*x^9 + 269325*(b*x + a)^n*b^10*d^3*n^4*x^10 + 3*(b*x + a 
)^n*b^10*c^2*d*n^9*x^4 - 18*(b*x + a)^n*a^2*b^8*c*d^2*n^8*x^5 + 2196*(b*x 
+ a)^n*a*b^9*c*d^2*n^7*x^6 + 33786*(b*x + a)^n*b^10*c*d^2*n^6*x^7 + 1512*( 
b*x + a)^n*a^3*b^7*d^3*n^6*x^7 - 17640*(b*x + a)^n*a^2*b^8*d^3*n^5*x^8 + 6 
7284*(b*x + a)^n*a*b^9*d^3*n^4*x^9 + 723680*(b*x + a)^n*b^10*d^3*n^3*x^10 
+ 3*(b*x + a)^n*a*b^9*c^2*d*n^9*x^3 + 153*(b*x + a)^n*b^10*c^2*d*n^8*x^4 - 
 666*(b*x + a)^n*a^2*b^8*c*d^2*n^7*x^5 + 20610*(b*x + a)^n*a*b^9*c*d^2*n^6 
*x^6 - 504*(b*x + a)^n*a^4*b^6*d^3*n^6*x^6 + 236817*(b*x + a)^n*b^10*c*d^2 
*n^5*x^7 + 12600*(b*x + a)^n*a^3*b^7*d^3*n^5*x^7 - 60921*(b*x + a)^n*a^2*b 
^8*d^3*n^4*x^8 + 118124*(b*x + a)^n*a*b^9*d^3*n^3*x^9 + 1172700*(b*x + a)^ 
n*b^10*d^3*n^2*x^10 + 144*(b*x + a)^n*a*b^9*c^2*d*n^8*x^3 + 3348*(b*x +...
 

Mupad [B] (verification not implemented)

Time = 24.09 (sec) , antiderivative size = 2001, normalized size of antiderivative = 5.94 \[ \int (a+b x)^n \left (c+d x^3\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

((a + b*x)^n*(3628800*a*b^9*c^3 - 362880*a^10*d^3 - 2721600*a^4*b^6*c^2*d 
+ 1555200*a^7*b^3*c*d^2 + 5753736*a*b^9*c^3*n^2 + 2655764*a*b^9*c^3*n^3 + 
761166*a*b^9*c^3*n^4 + 140889*a*b^9*c^3*n^5 + 16884*a*b^9*c^3*n^6 + 1266*a 
*b^9*c^3*n^7 + 54*a*b^9*c^3*n^8 + a*b^9*c^3*n^9 + 6999840*a*b^9*c^3*n - 23 
01480*a^4*b^6*c^2*d*n + 522720*a^7*b^3*c*d^2*n - 801432*a^4*b^6*c^2*d*n^2 
+ 58320*a^7*b^3*c*d^2*n^2 - 147150*a^4*b^6*c^2*d*n^3 + 2160*a^7*b^3*c*d^2* 
n^3 - 15030*a^4*b^6*c^2*d*n^4 - 810*a^4*b^6*c^2*d*n^5 - 18*a^4*b^6*c^2*d*n 
^6))/(b^10*(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 + 3628800)) + (x* 
(a + b*x)^n*(3628800*b^10*c^3 + 6999840*b^10*c^3*n + 5753736*b^10*c^3*n^2 
+ 2655764*b^10*c^3*n^3 + 761166*b^10*c^3*n^4 + 140889*b^10*c^3*n^5 + 16884 
*b^10*c^3*n^6 + 1266*b^10*c^3*n^7 + 54*b^10*c^3*n^8 + b^10*c^3*n^9 + 36288 
0*a^9*b*d^3*n + 2721600*a^3*b^7*c^2*d*n - 1555200*a^6*b^4*c*d^2*n + 230148 
0*a^3*b^7*c^2*d*n^2 - 522720*a^6*b^4*c*d^2*n^2 + 801432*a^3*b^7*c^2*d*n^3 
- 58320*a^6*b^4*c*d^2*n^3 + 147150*a^3*b^7*c^2*d*n^4 - 2160*a^6*b^4*c*d^2* 
n^4 + 15030*a^3*b^7*c^2*d*n^5 + 810*a^3*b^7*c^2*d*n^6 + 18*a^3*b^7*c^2*d*n 
^7))/(b^10*(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 + 3628800)) + (d^ 
3*x^10*(a + b*x)^n*(1026576*n + 1172700*n^2 + 723680*n^3 + 269325*n^4 + 63 
273*n^5 + 9450*n^6 + 870*n^7 + 45*n^8 + n^9 + 362880))/(10628640*n + 12...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 2659, normalized size of antiderivative = 7.89 \[ \int (a+b x)^n \left (c+d x^3\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

((a + b*x)**n*( - 362880*a**10*d**3 + 362880*a**9*b*d**3*n*x - 181440*a**8 
*b**2*d**3*n**2*x**2 - 181440*a**8*b**2*d**3*n*x**2 + 2160*a**7*b**3*c*d** 
2*n**3 + 58320*a**7*b**3*c*d**2*n**2 + 522720*a**7*b**3*c*d**2*n + 1555200 
*a**7*b**3*c*d**2 + 60480*a**7*b**3*d**3*n**3*x**3 + 181440*a**7*b**3*d**3 
*n**2*x**3 + 120960*a**7*b**3*d**3*n*x**3 - 2160*a**6*b**4*c*d**2*n**4*x - 
 58320*a**6*b**4*c*d**2*n**3*x - 522720*a**6*b**4*c*d**2*n**2*x - 1555200* 
a**6*b**4*c*d**2*n*x - 15120*a**6*b**4*d**3*n**4*x**4 - 90720*a**6*b**4*d* 
*3*n**3*x**4 - 166320*a**6*b**4*d**3*n**2*x**4 - 90720*a**6*b**4*d**3*n*x* 
*4 + 1080*a**5*b**5*c*d**2*n**5*x**2 + 30240*a**5*b**5*c*d**2*n**4*x**2 + 
290520*a**5*b**5*c*d**2*n**3*x**2 + 1038960*a**5*b**5*c*d**2*n**2*x**2 + 7 
77600*a**5*b**5*c*d**2*n*x**2 + 3024*a**5*b**5*d**3*n**5*x**5 + 30240*a**5 
*b**5*d**3*n**4*x**5 + 105840*a**5*b**5*d**3*n**3*x**5 + 151200*a**5*b**5* 
d**3*n**2*x**5 + 72576*a**5*b**5*d**3*n*x**5 - 18*a**4*b**6*c**2*d*n**6 - 
810*a**4*b**6*c**2*d*n**5 - 15030*a**4*b**6*c**2*d*n**4 - 147150*a**4*b**6 
*c**2*d*n**3 - 801432*a**4*b**6*c**2*d*n**2 - 2301480*a**4*b**6*c**2*d*n - 
 2721600*a**4*b**6*c**2*d - 360*a**4*b**6*c*d**2*n**6*x**3 - 10800*a**4*b* 
*6*c*d**2*n**5*x**3 - 117000*a**4*b**6*c*d**2*n**4*x**3 - 540000*a**4*b**6 
*c*d**2*n**3*x**3 - 951840*a**4*b**6*c*d**2*n**2*x**3 - 518400*a**4*b**6*c 
*d**2*n*x**3 - 504*a**4*b**6*d**3*n**6*x**6 - 7560*a**4*b**6*d**3*n**5*x** 
6 - 42840*a**4*b**6*d**3*n**4*x**6 - 113400*a**4*b**6*d**3*n**3*x**6 - ...