3.183 \(\int x (a+b x)^n (c+d x^3)^3 \, dx\)

Optimal. Leaf size=396 \[ -\frac{3 a d \left (-35 a^3 b^3 c d+40 a^6 d^2+4 b^6 c^2\right ) (a+b x)^{n+4}}{b^{11} (n+4)}+\frac{3 d \left (-35 a^3 b^3 c d+70 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+5}}{b^{11} (n+5)}+\frac{63 a^2 d^2 \left (b^3 c-4 a^3 d\right ) (a+b x)^{n+6}}{b^{11} (n+6)}-\frac{21 a d^2 \left (b^3 c-10 a^3 d\right ) (a+b x)^{n+7}}{b^{11} (n+7)}+\frac{3 d^2 \left (b^3 c-40 a^3 d\right ) (a+b x)^{n+8}}{b^{11} (n+8)}-\frac{a \left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{11} (n+1)}+\frac{\left (b^3 c-10 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{11} (n+2)}+\frac{9 a^2 d \left (2 b^3 c-5 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+3}}{b^{11} (n+3)}+\frac{45 a^2 d^3 (a+b x)^{n+9}}{b^{11} (n+9)}-\frac{10 a d^3 (a+b x)^{n+10}}{b^{11} (n+10)}+\frac{d^3 (a+b x)^{n+11}}{b^{11} (n+11)} \]

[Out]

-((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*(1
0 + n)) + (d^3*(a + b*x)^(11 + n))/(b^11*(11 + n))

________________________________________________________________________________________

Rubi [A]  time = 0.264233, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {1620} \[ -\frac{3 a d \left (-35 a^3 b^3 c d+40 a^6 d^2+4 b^6 c^2\right ) (a+b x)^{n+4}}{b^{11} (n+4)}+\frac{3 d \left (-35 a^3 b^3 c d+70 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+5}}{b^{11} (n+5)}+\frac{63 a^2 d^2 \left (b^3 c-4 a^3 d\right ) (a+b x)^{n+6}}{b^{11} (n+6)}-\frac{21 a d^2 \left (b^3 c-10 a^3 d\right ) (a+b x)^{n+7}}{b^{11} (n+7)}+\frac{3 d^2 \left (b^3 c-40 a^3 d\right ) (a+b x)^{n+8}}{b^{11} (n+8)}-\frac{a \left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{11} (n+1)}+\frac{\left (b^3 c-10 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{11} (n+2)}+\frac{9 a^2 d \left (2 b^3 c-5 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+3}}{b^{11} (n+3)}+\frac{45 a^2 d^3 (a+b x)^{n+9}}{b^{11} (n+9)}-\frac{10 a d^3 (a+b x)^{n+10}}{b^{11} (n+10)}+\frac{d^3 (a+b x)^{n+11}}{b^{11} (n+11)} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^n*(c + d*x^3)^3,x]

[Out]

-((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*(1
0 + n)) + (d^3*(a + b*x)^(11 + n))/(b^11*(11 + n))

Rule 1620

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]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int x (a+b x)^n \left (c+d x^3\right )^3 \, dx &=\int \left (\frac{a \left (-b^3 c+a^3 d\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)^{1+n}}{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)^{2+n}}{b^{10}}-\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+n}}{b^{10}}+\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+n}}{b^{10}}-\frac{63 a^2 d^2 \left (-b^3 c+4 a^3 d\right ) (a+b x)^{5+n}}{b^{10}}+\frac{21 a d^2 \left (-b^3 c+10 a^3 d\right ) (a+b x)^{6+n}}{b^{10}}+\frac{3 d^2 \left (b^3 c-40 a^3 d\right ) (a+b x)^{7+n}}{b^{10}}+\frac{45 a^2 d^3 (a+b x)^{8+n}}{b^{10}}-\frac{10 a d^3 (a+b x)^{9+n}}{b^{10}}+\frac{d^3 (a+b x)^{10+n}}{b^{10}}\right ) \, 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)}\\ \end{align*}

Mathematica [A]  time = 0.380743, size = 345, normalized size = 0.87 \[ \frac{(a+b x)^{n+1} \left (\frac{3 d (a+b x)^4 \left (-35 a^3 b^3 c d+70 a^6 d^2+b^6 c^2\right )}{n+5}-\frac{3 a d (a+b x)^3 \left (-35 a^3 b^3 c d+40 a^6 d^2+4 b^6 c^2\right )}{n+4}+\frac{3 d^2 (a+b x)^7 \left (b^3 c-40 a^3 d\right )}{n+8}+\frac{21 a d^2 (a+b x)^6 \left (10 a^3 d-b^3 c\right )}{n+7}+\frac{63 a^2 d^2 (a+b x)^5 \left (b^3 c-4 a^3 d\right )}{n+6}+\frac{9 a^2 d (a+b x)^2 \left (a^3 d-b^3 c\right ) \left (5 a^3 d-2 b^3 c\right )}{n+3}+\frac{(a+b x) \left (b^3 c-10 a^3 d\right ) \left (b^3 c-a^3 d\right )^2}{n+2}+\frac{a \left (a^3 d-b^3 c\right )^3}{n+1}+\frac{45 a^2 d^3 (a+b x)^8}{n+9}+\frac{d^3 (a+b x)^{10}}{n+11}-\frac{10 a d^3 (a+b x)^9}{n+10}\right )}{b^{11}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((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

________________________________________________________________________________________

Maple [B]  time = 0.021, size = 2972, normalized size = 7.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)^(1+n)*(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^1
0*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+3363066*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^1
0*d^3*n*x^10-30240*a^5*b^5*d^3*n^5*x^5+882000*a^4*b^6*d^3*n^4*x^6-25200*a^3*b^7*c*d^2*n^6*x^4-4873680*a^3*b^7*
d^3*n^3*x^7+36*a^2*b^8*c^2*d*n^8*x^2+1039500*a^2*b^8*c*d^2*n^5*x^5+10631160*a^2*b^8*d^3*n^2*x^8-16704*a*b^9*c^
2*d*n^7*x^3-9313479*a*b^9*c*d^2*n^4*x^6-10265760*a*b^9*d^3*n*x^9+64*b^10*c^3*n^9*x+703719*b^10*c^2*d*n^6*x^4+3
3074574*b^10*c*d^2*n^3*x^7+3628800*b^10*d^3*x^10-453600*a^5*b^5*d^3*n^4*x^5+2520*a^4*b^6*c*d^2*n^6*x^3+3704400
*a^4*b^6*d^3*n^3*x^6-399420*a^3*b^7*c*d^2*n^5*x^4-9455040*a^3*b^7*d^3*n^2*x^7+1944*a^2*b^8*c^2*d*n^7*x^2+59584
14*a^2*b^8*c*d^2*n^4*x^5+9862560*a^2*b^8*d^3*n*x^8-a*b^9*c^3*n^9-228024*a*b^9*c^2*d*n^6*x^3-26604186*a*b^9*c*d
^2*n^3*x^6-3628800*a*b^9*d^3*x^9+1797*b^10*c^3*n^8*x+4394079*b^10*c^2*d*n^5*x^4+51177636*b^10*c*d^2*n^2*x^7+15
1200*a^6*b^4*d^3*n^4*x^4-2570400*a^5*b^5*d^3*n^3*x^5+90720*a^4*b^6*c*d^2*n^5*x^3+8184960*a^4*b^6*d^3*n^2*x^6-7
2*a^3*b^7*c^2*d*n^7*x-3200400*a^3*b^7*c*d^2*n^4*x^4-9408960*a^3*b^7*d^3*n*x^7+44280*a^2*b^8*c^2*d*n^6*x^2+2013
0390*a^2*b^8*c*d^2*n^3*x^5+3628800*a^2*b^8*d^3*x^8-63*a*b^9*c^3*n^8-1902780*a*b^9*c^2*d*n^5*x^3-45292716*a*b^9
*c*d^2*n^2*x^6+29076*b^10*c^3*n^7*x+18048210*b^10*c^2*d*n^4*x^4+43332840*b^10*c*d^2*n*x^7+1512000*a^6*b^4*d^3*
n^3*x^4-7560*a^5*b^5*c*d^2*n^5*x^2-6804000*a^5*b^5*d^3*n^2*x^5+1234800*a^4*b^6*c*d^2*n^4*x^3+8890560*a^4*b^6*d
^3*n*x^6-3744*a^3*b^7*c^2*d*n^6*x-13790070*a^3*b^7*c*d^2*n^3*x^4-3628800*a^3*b^7*d^3*x^7+551232*a^2*b^8*c^2*d*
n^5*x^2+38842776*a^2*b^8*c*d^2*n^2*x^5-1734*a*b^9*c^3*n^7-9965196*a*b^9*c^2*d*n^4*x^3-41194440*a*b^9*c*d^2*n*x
^6+299271*b^10*c^3*n^6*x+47746140*b^10*c^2*d*n^3*x^4+14968800*b^10*c*d^2*x^7-604800*a^7*b^3*d^3*n^3*x^3+529200
0*a^6*b^4*d^3*n^2*x^4-249480*a^5*b^5*c*d^2*n^4*x^2-8285760*a^5*b^5*d^3*n*x^5+72*a^4*b^6*c^2*d*n^6+7862400*a^4*
b^6*c*d^2*n^3*x^3+3628800*a^4*b^6*d^3*x^6-81072*a^3*b^7*c^2*d*n^5*x-31701600*a^3*b^7*c*d^2*n^2*x^4+4054644*a^2
*b^8*c^2*d*n^4*x^2+38699640*a^2*b^8*c*d^2*n*x^5-27342*a*b^9*c^3*n^6-32332056*a*b^9*c^2*d*n^3*x^3-14968800*a*b^
9*c*d^2*x^6+2039016*b^10*c^3*n^5*x+77043528*b^10*c^2*d*n^2*x^4-3628800*a^7*b^3*d^3*n^2*x^3+15120*a^6*b^4*c*d^2
*n^4*x+7560000*a^6*b^4*d^3*n*x^4-2955960*a^5*b^5*c*d^2*n^3*x^2-3628800*a^5*b^5*d^3*x^5+3672*a^4*b^6*c^2*d*n^5+
23710680*a^4*b^6*c*d^2*n^2*x^3-940320*a^3*b^7*c^2*d*n^4*x-35705880*a^3*b^7*c*d^2*n*x^4+17731656*a^2*b^8*c^2*d*
n^3*x^2+14968800*a^2*b^8*c*d^2*x^5-271929*a*b^9*c^3*n^5-61656336*a*b^9*c^2*d*n^2*x^3+9261503*b^10*c^3*n^4*x+67
536288*b^10*c^2*d*n*x^4+1814400*a^8*b^2*d^3*n^2*x^2-6652800*a^7*b^3*d^3*n*x^3+468720*a^6*b^4*c*d^2*n^3*x+36288
00*a^6*b^4*d^3*x^4-14719320*a^5*b^5*c*d^2*n^2*x^2+77400*a^4*b^6*c^2*d*n^4+31963680*a^4*b^6*c*d^2*n*x^3-6228648
*a^3*b^7*c^2*d*n^3*x-14968800*a^3*b^7*c*d^2*x^4+43801200*a^2*b^8*c^2*d*n^2*x^2-1767087*a*b^9*c^3*n^4-61548768*
a*b^9*c^2*d*n*x^3+27472724*b^10*c^3*n^3*x+23950080*b^10*c^2*d*x^4+5443200*a^8*b^2*d^3*n*x^2-15120*a^7*b^3*c*d^
2*n^3-3628800*a^7*b^3*d^3*x^3+4974480*a^6*b^4*c*d^2*n^2*x-26974080*a^5*b^5*c*d^2*n*x^2+862920*a^4*b^6*c^2*d*n^
3+14968800*a^4*b^6*c*d^2*x^3-23006016*a^3*b^7*c^2*d*n^2*x+53565408*a^2*b^8*c^2*d*n*x^2-7494416*a*b^9*c^3*n^3-2
3950080*a*b^9*c^2*d*x^3+50312628*b^10*c^3*n^2*x-3628800*a^9*b*d^3*n*x+3628800*a^8*b^2*d^3*x^2-453600*a^7*b^3*c
*d^2*n^2+19489680*a^6*b^4*c*d^2*n*x-14968800*a^5*b^5*c*d^2*x^2+5365728*a^4*b^6*c^2*d*n^2-41590368*a^3*b^7*c^2*
d*n*x+23950080*a^2*b^8*c^2*d*x^2-19978308*a*b^9*c^3*n^2+50292720*b^10*c^3*n*x-3628800*a^9*b*d^3*x-4520880*a^7*
b^3*c*d^2*n+14968800*a^6*b^4*c*d^2*x+17640288*a^4*b^6*c^2*d*n-23950080*a^3*b^7*c^2*d*x-30334320*a*b^9*c^3*n+19
958400*b^10*c^3*x+3628800*a^10*d^3-14968800*a^7*b^3*c*d^2+23950080*a^4*b^6*c^2*d-19958400*a*b^9*c^3)/b^11/(n^1
1+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+1205
43840*n+39916800)

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Maxima [B]  time = 1.09508, size = 1287, normalized size = 3.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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 + 2
4)*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 + 225*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 + 7
35*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*a*b^7*x^7 - 7*(n^6 + 15*n^5 + 85*n^4 + 225*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 + 22449*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 + 106286
40*n + 3628800)*b^11*x^11 + (n^10 + 45*n^9 + 870*n^8 + 9450*n^7 + 63273*n^6 + 269325*n^5 + 723680*n^4 + 117270
0*n^3 + 1026576*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 + 274*n^2 + 120*n)*a^5*b^6*x^6 - 30240*(n^5 + 10*n^4 + 35*n^3 +
50*n^2 + 24*n)*a^6*b^5*x^5 + 151200*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^7*b^4*x^4 - 604800*(n^3 + 3*n^2 + 2*n)*a^8
*b^3*x^3 + 1814400*(n^2 + n)*a^9*b^2*x^2 - 3628800*a^10*b*n*x + 3628800*a^11)*(b*x + a)^n*d^3/((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 + 39916800)*b^11)

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Fricas [B]  time = 1.01276, size = 7065, normalized size = 17.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(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^11*d^3 - (b^11*d^3*n^10 + 55*b^11*d^3*n^9 + 1320*b^11*d^3*n^8 + 18150*b^1
1*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 + 3628800*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 + 1172
700*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 + 67284*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 + 40320*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 + 49
89600*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*(9
027*b^11*c*d^2 + 460*a^3*b^8*d^3)*n^6 + 294*(3813*b^11*c*d^2 + 200*a^3*b^8*d^3)*n^5 + (4371359*b^11*c*d^2 + 20
3070*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 + 11
04*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)*n^5 + 2*(633433*a*b^10*c*d^2 - 88200*a^4*b^7*d^3)*n^4 + 12*(179
733*a*b^10*c*d^2 - 32480*a^4*b^7*d^3)*n^3 + 360*(5449*a*b^10*c*d^2 - 1176*a^4*b^7*d^3)*n^2 + 21600*(33*a*b^10*
c*d^2 - 8*a^4*b^7*d^3)*n)*x^7 + 18*(1519*a^2*b^9*c^3 - 4*a^5*b^6*c^2*d)*n^6 + 21*(a^2*b^9*c*d^2*n^9 + 45*a^2*b
^9*c*d^2*n^8 + 834*a^2*b^9*c*d^2*n^7 + 30*(275*a^2*b^9*c*d^2 - 8*a^5*b^6*d^3)*n^6 + 3*(15763*a^2*b^9*c*d^2 - 1
200*a^5*b^6*d^3)*n^5 + 15*(10651*a^2*b^9*c*d^2 - 1360*a^5*b^6*d^3)*n^4 + 4*(77069*a^2*b^9*c*d^2 - 13500*a^5*b^
6*d^3)*n^3 + 60*(5119*a^2*b^9*c*d^2 - 1096*a^5*b^6*d^3)*n^2 + 3600*(33*a^2*b^9*c*d^2 - 8*a^5*b^6*d^3)*n)*x^6 +
3*(90643*a^2*b^9*c^3 - 1224*a^5*b^6*c^2*d)*n^5 - 3*(b^11*c^2*d*n^10 + 61*b^11*c^2*d*n^9 + 7983360*b^11*c^2*d
+ 6*(270*b^11*c^2*d + 7*a^3*b^8*c*d^2)*n^8 + 210*(117*b^11*c^2*d + 8*a^3*b^8*c*d^2)*n^7 + 3*(78191*b^11*c^2*d
+ 8876*a^3*b^8*c*d^2)*n^6 + 3*(488231*b^11*c^2*d + 71120*a^3*b^8*c*d^2 - 3360*a^6*b^5*d^3)*n^5 + 2*(3008035*b^
11*c^2*d + 459669*a^3*b^8*c*d^2 - 50400*a^6*b^5*d^3)*n^4 + 20*(795769*b^11*c^2*d + 105672*a^3*b^8*c*d^2 - 1764
0*a^6*b^5*d^3)*n^3 + 72*(356683*b^11*c^2*d + 33061*a^3*b^8*c*d^2 - 7000*a^6*b^5*d^3)*n^2 + 288*(78167*b^11*c^2
*d + 3465*a^3*b^8*c*d^2 - 840*a^6*b^5*d^3)*n)*x^5 + 9*(196343*a^2*b^9*c^3 - 8600*a^5*b^6*c^2*d)*n^4 - 3*(a*b^1
0*c^2*d*n^10 + 57*a*b^10*c^2*d*n^9 + 1392*a*b^10*c^2*d*n^8 + 6*(3167*a*b^10*c^2*d - 35*a^4*b^7*c*d^2)*n^7 + 15
*(10571*a*b^10*c^2*d - 504*a^4*b^7*c*d^2)*n^6 + 3*(276811*a*b^10*c^2*d - 34300*a^4*b^7*c*d^2)*n^5 + 2*(1347169
*a*b^10*c^2*d - 327600*a^4*b^7*c*d^2 + 25200*a^7*b^4*d^3)*n^4 + 42*(122334*a*b^10*c^2*d - 47045*a^4*b^7*c*d^2
+ 7200*a^7*b^4*d^3)*n^3 + 72*(71237*a*b^10*c^2*d - 36995*a^4*b^7*c*d^2 + 7700*a^7*b^4*d^3)*n^2 + 7560*(264*a*b
^10*c^2*d - 165*a^4*b^7*c*d^2 + 40*a^7*b^4*d^3)*n)*x^4 + 8*(936802*a^2*b^9*c^3 - 107865*a^5*b^6*c^2*d + 1890*a
^8*b^3*c*d^2)*n^3 + 12*(a^2*b^9*c^2*d*n^9 + 54*a^2*b^9*c^2*d*n^8 + 1230*a^2*b^9*c^2*d*n^7 + 6*(2552*a^2*b^9*c^
2*d - 35*a^5*b^6*c*d^2)*n^6 + 33*(3413*a^2*b^9*c^2*d - 210*a^5*b^6*c*d^2)*n^5 + 6*(82091*a^2*b^9*c^2*d - 13685
*a^5*b^6*c*d^2)*n^4 + 10*(121670*a^2*b^9*c^2*d - 40887*a^5*b^6*c*d^2 + 5040*a^8*b^3*d^3)*n^3 + 24*(61997*a^2*b
^9*c^2*d - 31220*a^5*b^6*c*d^2 + 6300*a^8*b^3*d^3)*n^2 + 2520*(264*a^2*b^9*c^2*d - 165*a^5*b^6*c*d^2 + 40*a^8*
b^3*d^3)*n)*x^3 + 36*(554953*a^2*b^9*c^3 - 149048*a^5*b^6*c^2*d + 12600*a^8*b^3*c*d^2)*n^2 - (b^11*c^3*n^10 +
64*b^11*c^3*n^9 + 19958400*b^11*c^3 + 3*(599*b^11*c^3 + 12*a^3*b^8*c^2*d)*n^8 + 12*(2423*b^11*c^3 + 156*a^3*b^
8*c^2*d)*n^7 + 3*(99757*b^11*c^3 + 13512*a^3*b^8*c^2*d)*n^6 + 24*(84959*b^11*c^3 + 19590*a^3*b^8*c^2*d - 315*a
^6*b^5*c*d^2)*n^5 + (9261503*b^11*c^3 + 3114324*a^3*b^8*c^2*d - 234360*a^6*b^5*c*d^2)*n^4 + 4*(6868181*b^11*c^
3 + 2875752*a^3*b^8*c^2*d - 621810*a^6*b^5*c*d^2)*n^3 + 36*(1397573*b^11*c^3 + 577644*a^3*b^8*c^2*d - 270690*a
^6*b^5*c*d^2 + 50400*a^9*b^2*d^3)*n^2 + 720*(69851*b^11*c^3 + 16632*a^3*b^8*c^2*d - 10395*a^6*b^5*c*d^2 + 2520
*a^9*b^2*d^3)*n)*x^2 + 144*(210655*a^2*b^9*c^3 - 122502*a^5*b^6*c^2*d + 31395*a^8*b^3*c*d^2)*n - (a*b^10*c^3*n
^10 + 63*a*b^10*c^3*n^9 + 1734*a*b^10*c^3*n^8 + 18*(1519*a*b^10*c^3 - 4*a^4*b^7*c^2*d)*n^7 + 3*(90643*a*b^10*c
^3 - 1224*a^4*b^7*c^2*d)*n^6 + 9*(196343*a*b^10*c^3 - 8600*a^4*b^7*c^2*d)*n^5 + 8*(936802*a*b^10*c^3 - 107865*
a^4*b^7*c^2*d + 1890*a^7*b^4*c*d^2)*n^4 + 36*(554953*a*b^10*c^3 - 149048*a^4*b^7*c^2*d + 12600*a^7*b^4*c*d^2)*
n^3 + 144*(210655*a*b^10*c^3 - 122502*a^4*b^7*c^2*d + 31395*a^7*b^4*c*d^2)*n^2 + 90720*(220*a*b^10*c^3 - 264*a
^4*b^7*c^2*d + 165*a^7*b^4*c*d^2 - 40*a^10*b*d^3)*n)*x)*(b*x + a)^n/(b^11*n^11 + 66*b^11*n^10 + 1925*b^11*n^9
+ 32670*b^11*n^8 + 357423*b^11*n^7 + 2637558*b^11*n^6 + 13339535*b^11*n^5 + 45995730*b^11*n^4 + 105258076*b^11
*n^3 + 150917976*b^11*n^2 + 120543840*b^11*n + 39916800*b^11)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.34142, size = 6661, normalized size = 16.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((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*b^10*c^2*d*n^9*x^4 + 4860*(b*x + a)^n*b^11*c^2*d*n^8*x^5 + 126
*(b*x + a)^n*a^3*b^8*c*d^2*n^8*x^5 - 17514*(b*x + a)^n*a^2*b^9*c*d^2*n^7*x^6 + 290367*(b*x + a)^n*a*b^10*c*d^2
*n^6*x^7 - 15120*(b*x + a)^n*a^4*b^7*d^3*n^6*x^7 + 3363066*(b*x + a)^n*b^11*c*d^2*n^5*x^8 + 176400*(b*x + a)^n
*a^3*b^8*d^3*n^5*x^8 - 672840*(b*x + a)^n*a^2*b^9*d^3*n^4*x^9 + 1172700*(b*x + a)^n*a*b^10*d^3*n^3*x^10 + 1275
3576*(b*x + a)^n*b^11*d^3*n^2*x^11 + (b*x + a)^n*b^11*c^3*n^10*x^2 - 12*(b*x + a)^n*a^2*b^9*c^2*d*n^9*x^3 + 41
76*(b*x + a)^n*a*b^10*c^2*d*n^8*x^4 + 73710*(b*x + a)^n*b^11*c^2*d*n^7*x^5 + 5040*(b*x + a)^n*a^3*b^8*c*d^2*n^
7*x^5 - 173250*(b*x + a)^n*a^2*b^9*c*d^2*n^6*x^6 + 5040*(b*x + a)^n*a^5*b^6*d^3*n^6*x^6 + 1330497*(b*x + a)^n*
a*b^10*c*d^2*n^5*x^7 - 126000*(b*x + a)^n*a^4*b^7*d^3*n^5*x^7 + 13114077*(b*x + a)^n*b^11*c*d^2*n^4*x^8 + 6092
10*(b*x + a)^n*a^3*b^8*d^3*n^4*x^8 - 1181240*(b*x + a)^n*a^2*b^9*d^3*n^3*x^9 + 1026576*(b*x + a)^n*a*b^10*d^3*
n^2*x^10 + 10628640*(b*x + a)^n*b^11*d^3*n*x^11 + (b*x + a)^n*a*b^10*c^3*n^10*x + 64*(b*x + a)^n*b^11*c^3*n^9*
x^2 - 648*(b*x + a)^n*a^2*b^9*c^2*d*n^8*x^3 + 57006*(b*x + a)^n*a*b^10*c^2*d*n^7*x^4 - 630*(b*x + a)^n*a^4*b^7
*c*d^2*n^7*x^4 + 703719*(b*x + a)^n*b^11*c^2*d*n^6*x^5 + 79884*(b*x + a)^n*a^3*b^8*c*d^2*n^6*x^5 - 993069*(b*x
+ a)^n*a^2*b^9*c*d^2*n^5*x^6 + 75600*(b*x + a)^n*a^5*b^6*d^3*n^5*x^6 + 3800598*(b*x + a)^n*a*b^10*c*d^2*n^4*x
^7 - 529200*(b*x + a)^n*a^4*b^7*d^3*n^4*x^7 + 33074574*(b*x + a)^n*b^11*c*d^2*n^3*x^8 + 1181880*(b*x + a)^n*a^
3*b^8*d^3*n^3*x^8 - 1095840*(b*x + a)^n*a^2*b^9*d^3*n^2*x^9 + 362880*(b*x + a)^n*a*b^10*d^3*n*x^10 + 3628800*(
b*x + a)^n*b^11*d^3*x^11 + 63*(b*x + a)^n*a*b^10*c^3*n^9*x + 1797*(b*x + a)^n*b^11*c^3*n^8*x^2 + 36*(b*x + a)^
n*a^3*b^8*c^2*d*n^8*x^2 - 14760*(b*x + a)^n*a^2*b^9*c^2*d*n^7*x^3 + 475695*(b*x + a)^n*a*b^10*c^2*d*n^6*x^4 -
22680*(b*x + a)^n*a^4*b^7*c*d^2*n^6*x^4 + 4394079*(b*x + a)^n*b^11*c^2*d*n^5*x^5 + 640080*(b*x + a)^n*a^3*b^8*
c*d^2*n^5*x^5 - 30240*(b*x + a)^n*a^6*b^5*d^3*n^5*x^5 - 3355065*(b*x + a)^n*a^2*b^9*c*d^2*n^4*x^6 + 428400*(b*
x + a)^n*a^5*b^6*d^3*n^4*x^6 + 6470388*(b*x + a)^n*a*b^10*c*d^2*n^3*x^7 - 1169280*(b*x + a)^n*a^4*b^7*d^3*n^3*
x^7 + 51177636*(b*x + a)^n*b^11*c*d^2*n^2*x^8 + 1176120*(b*x + a)^n*a^3*b^8*d^3*n^2*x^8 - 403200*(b*x + a)^n*a
^2*b^9*d^3*n*x^9 - (b*x + a)^n*a^2*b^9*c^3*n^9 + 1734*(b*x + a)^n*a*b^10*c^3*n^8*x + 29076*(b*x + a)^n*b^11*c^
3*n^7*x^2 + 1872*(b*x + a)^n*a^3*b^8*c^2*d*n^7*x^2 - 183744*(b*x + a)^n*a^2*b^9*c^2*d*n^6*x^3 + 2520*(b*x + a)
^n*a^5*b^6*c*d^2*n^6*x^3 + 2491299*(b*x + a)^n*a*b^10*c^2*d*n^5*x^4 - 308700*(b*x + a)^n*a^4*b^7*c*d^2*n^5*x^4
+ 18048210*(b*x + a)^n*b^11*c^2*d*n^4*x^5 + 2758014*(b*x + a)^n*a^3*b^8*c*d^2*n^4*x^5 - 302400*(b*x + a)^n*a^
6*b^5*d^3*n^4*x^5 - 6473796*(b*x + a)^n*a^2*b^9*c*d^2*n^3*x^6 + 1134000*(b*x + a)^n*a^5*b^6*d^3*n^3*x^6 + 5884
920*(b*x + a)^n*a*b^10*c*d^2*n^2*x^7 - 1270080*(b*x + a)^n*a^4*b^7*d^3*n^2*x^7 + 43332840*(b*x + a)^n*b^11*c*d
^2*n*x^8 + 453600*(b*x + a)^n*a^3*b^8*d^3*n*x^8 - 63*(b*x + a)^n*a^2*b^9*c^3*n^8 + 27342*(b*x + a)^n*a*b^10*c^
3*n^7*x - 72*(b*x + a)^n*a^4*b^7*c^2*d*n^7*x + 299271*(b*x + a)^n*b^11*c^3*n^6*x^2 + 40536*(b*x + a)^n*a^3*b^8
*c^2*d*n^6*x^2 - 1351548*(b*x + a)^n*a^2*b^9*c^2*d*n^5*x^3 + 83160*(b*x + a)^n*a^5*b^6*c*d^2*n^5*x^3 + 8083014
*(b*x + a)^n*a*b^10*c^2*d*n^4*x^4 - 1965600*(b*x + a)^n*a^4*b^7*c*d^2*n^4*x^4 + 151200*(b*x + a)^n*a^7*b^4*d^3
*n^4*x^4 + 47746140*(b*x + a)^n*b^11*c^2*d*n^3*x^5 + 6340320*(b*x + a)^n*a^3*b^8*c*d^2*n^3*x^5 - 1058400*(b*x
+ a)^n*a^6*b^5*d^3*n^3*x^5 - 6449940*(b*x + a)^n*a^2*b^9*c*d^2*n^2*x^6 + 1380960*(b*x + a)^n*a^5*b^6*d^3*n^2*x
^6 + 2138400*(b*x + a)^n*a*b^10*c*d^2*n*x^7 - 518400*(b*x + a)^n*a^4*b^7*d^3*n*x^7 + 14968800*(b*x + a)^n*b^11
*c*d^2*x^8 - 1734*(b*x + a)^n*a^2*b^9*c^3*n^7 + 271929*(b*x + a)^n*a*b^10*c^3*n^6*x - 3672*(b*x + a)^n*a^4*b^7
*c^2*d*n^6*x + 2039016*(b*x + a)^n*b^11*c^3*n^5*x^2 + 470160*(b*x + a)^n*a^3*b^8*c^2*d*n^5*x^2 - 7560*(b*x + a
)^n*a^6*b^5*c*d^2*n^5*x^2 - 5910552*(b*x + a)^n*a^2*b^9*c^2*d*n^4*x^3 + 985320*(b*x + a)^n*a^5*b^6*c*d^2*n^4*x
^3 + 15414084*(b*x + a)^n*a*b^10*c^2*d*n^3*x^4 - 5927670*(b*x + a)^n*a^4*b^7*c*d^2*n^3*x^4 + 907200*(b*x + a)^
n*a^7*b^4*d^3*n^3*x^4 + 77043528*(b*x + a)^n*b^11*c^2*d*n^2*x^5 + 7141176*(b*x + a)^n*a^3*b^8*c*d^2*n^2*x^5 -
1512000*(b*x + a)^n*a^6*b^5*d^3*n^2*x^5 - 2494800*(b*x + a)^n*a^2*b^9*c*d^2*n*x^6 + 604800*(b*x + a)^n*a^5*b^6
*d^3*n*x^6 - 27342*(b*x + a)^n*a^2*b^9*c^3*n^6 + 72*(b*x + a)^n*a^5*b^6*c^2*d*n^6 + 1767087*(b*x + a)^n*a*b^10
*c^3*n^5*x - 77400*(b*x + a)^n*a^4*b^7*c^2*d*n^5*x + 9261503*(b*x + a)^n*b^11*c^3*n^4*x^2 + 3114324*(b*x + a)^
n*a^3*b^8*c^2*d*n^4*x^2 - 234360*(b*x + a)^n*a^6*b^5*c*d^2*n^4*x^2 - 14600400*(b*x + a)^n*a^2*b^9*c^2*d*n^3*x^
3 + 4906440*(b*x + a)^n*a^5*b^6*c*d^2*n^3*x^3 - 604800*(b*x + a)^n*a^8*b^3*d^3*n^3*x^3 + 15387192*(b*x + a)^n*
a*b^10*c^2*d*n^2*x^4 - 7990920*(b*x + a)^n*a^4*b^7*c*d^2*n^2*x^4 + 1663200*(b*x + a)^n*a^7*b^4*d^3*n^2*x^4 + 6
7536288*(b*x + a)^n*b^11*c^2*d*n*x^5 + 2993760*(b*x + a)^n*a^3*b^8*c*d^2*n*x^5 - 725760*(b*x + a)^n*a^6*b^5*d^
3*n*x^5 - 271929*(b*x + a)^n*a^2*b^9*c^3*n^5 + 3672*(b*x + a)^n*a^5*b^6*c^2*d*n^5 + 7494416*(b*x + a)^n*a*b^10
*c^3*n^4*x - 862920*(b*x + a)^n*a^4*b^7*c^2*d*n^4*x + 15120*(b*x + a)^n*a^7*b^4*c*d^2*n^4*x + 27472724*(b*x +
a)^n*b^11*c^3*n^3*x^2 + 11503008*(b*x + a)^n*a^3*b^8*c^2*d*n^3*x^2 - 2487240*(b*x + a)^n*a^6*b^5*c*d^2*n^3*x^2
- 17855136*(b*x + a)^n*a^2*b^9*c^2*d*n^2*x^3 + 8991360*(b*x + a)^n*a^5*b^6*c*d^2*n^2*x^3 - 1814400*(b*x + a)^
n*a^8*b^3*d^3*n^2*x^3 + 5987520*(b*x + a)^n*a*b^10*c^2*d*n*x^4 - 3742200*(b*x + a)^n*a^4*b^7*c*d^2*n*x^4 + 907
200*(b*x + a)^n*a^7*b^4*d^3*n*x^4 + 23950080*(b*x + a)^n*b^11*c^2*d*x^5 - 1767087*(b*x + a)^n*a^2*b^9*c^3*n^4
+ 77400*(b*x + a)^n*a^5*b^6*c^2*d*n^4 + 19978308*(b*x + a)^n*a*b^10*c^3*n^3*x - 5365728*(b*x + a)^n*a^4*b^7*c^
2*d*n^3*x + 453600*(b*x + a)^n*a^7*b^4*c*d^2*n^3*x + 50312628*(b*x + a)^n*b^11*c^3*n^2*x^2 + 20795184*(b*x + a
)^n*a^3*b^8*c^2*d*n^2*x^2 - 9744840*(b*x + a)^n*a^6*b^5*c*d^2*n^2*x^2 + 1814400*(b*x + a)^n*a^9*b^2*d^3*n^2*x^
2 - 7983360*(b*x + a)^n*a^2*b^9*c^2*d*n*x^3 + 4989600*(b*x + a)^n*a^5*b^6*c*d^2*n*x^3 - 1209600*(b*x + a)^n*a^
8*b^3*d^3*n*x^3 - 7494416*(b*x + a)^n*a^2*b^9*c^3*n^3 + 862920*(b*x + a)^n*a^5*b^6*c^2*d*n^3 - 15120*(b*x + a)
^n*a^8*b^3*c*d^2*n^3 + 30334320*(b*x + a)^n*a*b^10*c^3*n^2*x - 17640288*(b*x + a)^n*a^4*b^7*c^2*d*n^2*x + 4520
880*(b*x + a)^n*a^7*b^4*c*d^2*n^2*x + 50292720*(b*x + a)^n*b^11*c^3*n*x^2 + 11975040*(b*x + a)^n*a^3*b^8*c^2*d
*n*x^2 - 7484400*(b*x + a)^n*a^6*b^5*c*d^2*n*x^2 + 1814400*(b*x + a)^n*a^9*b^2*d^3*n*x^2 - 19978308*(b*x + a)^
n*a^2*b^9*c^3*n^2 + 5365728*(b*x + a)^n*a^5*b^6*c^2*d*n^2 - 453600*(b*x + a)^n*a^8*b^3*c*d^2*n^2 + 19958400*(b
*x + a)^n*a*b^10*c^3*n*x - 23950080*(b*x + a)^n*a^4*b^7*c^2*d*n*x + 14968800*(b*x + a)^n*a^7*b^4*c*d^2*n*x - 3
628800*(b*x + a)^n*a^10*b*d^3*n*x + 19958400*(b*x + a)^n*b^11*c^3*x^2 - 30334320*(b*x + a)^n*a^2*b^9*c^3*n + 1
7640288*(b*x + a)^n*a^5*b^6*c^2*d*n - 4520880*(b*x + a)^n*a^8*b^3*c*d^2*n - 19958400*(b*x + a)^n*a^2*b^9*c^3 +
23950080*(b*x + a)^n*a^5*b^6*c^2*d - 14968800*(b*x + a)^n*a^8*b^3*c*d^2 + 3628800*(b*x + a)^n*a^11*d^3)/(b^11
*n^11 + 66*b^11*n^10 + 1925*b^11*n^9 + 32670*b^11*n^8 + 357423*b^11*n^7 + 2637558*b^11*n^6 + 13339535*b^11*n^5
+ 45995730*b^11*n^4 + 105258076*b^11*n^3 + 150917976*b^11*n^2 + 120543840*b^11*n + 39916800*b^11)