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

Optimal. Leaf size=337 \[ \frac{3 d \left (-20 a^3 b^3 c d+28 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+4}}{b^{10} (n+4)}+\frac{9 a^2 d^2 \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{n+5}}{b^{10} (n+5)}-\frac{18 a d^2 \left (b^3 c-7 a^3 d\right ) (a+b x)^{n+6}}{b^{10} (n+6)}+\frac{3 d^2 \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+7}}{b^{10} (n+7)}+\frac{\left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{10} (n+1)}+\frac{9 a^2 d \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{10} (n+2)}-\frac{9 a d \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+3}}{b^{10} (n+3)}+\frac{36 a^2 d^3 (a+b x)^{n+8}}{b^{10} (n+8)}-\frac{9 a d^3 (a+b x)^{n+9}}{b^{10} (n+9)}+\frac{d^3 (a+b x)^{n+10}}{b^{10} (n+10)} \]

[Out]

((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^1
0*(9 + n)) + (d^3*(a + b*x)^(10 + n))/(b^10*(10 + n))

________________________________________________________________________________________

Rubi [A]  time = 0.208595, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {1850} \[ \frac{3 d \left (-20 a^3 b^3 c d+28 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+4}}{b^{10} (n+4)}+\frac{9 a^2 d^2 \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{n+5}}{b^{10} (n+5)}-\frac{18 a d^2 \left (b^3 c-7 a^3 d\right ) (a+b x)^{n+6}}{b^{10} (n+6)}+\frac{3 d^2 \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+7}}{b^{10} (n+7)}+\frac{\left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{10} (n+1)}+\frac{9 a^2 d \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{10} (n+2)}-\frac{9 a d \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+3}}{b^{10} (n+3)}+\frac{36 a^2 d^3 (a+b x)^{n+8}}{b^{10} (n+8)}-\frac{9 a d^3 (a+b x)^{n+9}}{b^{10} (n+9)}+\frac{d^3 (a+b x)^{n+10}}{b^{10} (n+10)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n*(c + d*x^3)^3,x]

[Out]

((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^1
0*(9 + n)) + (d^3*(a + b*x)^(10 + n))/(b^10*(10 + n))

Rule 1850

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])

Rubi steps

\begin{align*} \int (a+b x)^n \left (c+d x^3\right )^3 \, dx &=\int \left (\frac{\left (b^3 c-a^3 d\right )^3 (a+b x)^n}{b^9}+\frac{9 d \left (a b^3 c-a^4 d\right )^2 (a+b x)^{1+n}}{b^9}+\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+n}}{b^9}+\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+n}}{b^9}-\frac{9 a^2 d^2 \left (-5 b^3 c+14 a^3 d\right ) (a+b x)^{4+n}}{b^9}+\frac{18 a d^2 \left (-b^3 c+7 a^3 d\right ) (a+b x)^{5+n}}{b^9}+\frac{3 d^2 \left (b^3 c-28 a^3 d\right ) (a+b x)^{6+n}}{b^9}+\frac{36 a^2 d^3 (a+b x)^{7+n}}{b^9}-\frac{9 a d^3 (a+b x)^{8+n}}{b^9}+\frac{d^3 (a+b x)^{9+n}}{b^9}\right ) \, 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)}\\ \end{align*}

Mathematica [A]  time = 0.358814, size = 290, normalized size = 0.86 \[ \frac{(a+b x)^{n+1} \left (\frac{3 d (a+b x)^3 \left (-20 a^3 b^3 c d+28 a^6 d^2+b^6 c^2\right )}{n+4}+\frac{3 d^2 (a+b x)^6 \left (b^3 c-28 a^3 d\right )}{n+7}+\frac{18 a d^2 (a+b x)^5 \left (7 a^3 d-b^3 c\right )}{n+6}+\frac{9 a^2 d^2 (a+b x)^4 \left (5 b^3 c-14 a^3 d\right )}{n+5}-\frac{9 a d (a+b x)^2 \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right )}{n+3}+\frac{9 d (a+b x) \left (a b^3 c-a^4 d\right )^2}{n+2}+\frac{\left (b^3 c-a^3 d\right )^3}{n+1}+\frac{36 a^2 d^3 (a+b x)^7}{n+8}+\frac{d^3 (a+b x)^9}{n+10}-\frac{9 a d^3 (a+b x)^8}{n+9}\right )}{b^{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((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

________________________________________________________________________________________

Maple [B]  time = 0.018, size = 2280, normalized size = 6.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x+a)^(1+n)*(-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+202
041*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^8-15
3*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+678942*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^3*b^6*d^3*n^2*x^6-18*a^2*b^7*c^2*d*n^7*x-372150*a^2*b^7*c*d^2*n^4*x^4-9
40896*a^2*b^7*d^3*n*x^7+8748*a*b^8*c^2*d*n^6*x^2+2217024*a*b^8*c*d^2*n^3*x^5+362880*a*b^8*d^3*x^8-54*b^9*c^3*n
^8-309087*b^9*c^2*d*n^5*x^3-4846824*b^9*c*d^2*n^2*x^6+151200*a^5*b^4*d^3*n^3*x^4-1080*a^4*b^5*c*d^2*n^5*x^2-68
0400*a^4*b^5*d^3*n^2*x^5+149400*a^3*b^6*c*d^2*n^4*x^3+889056*a^3*b^6*d^3*n*x^6-828*a^2*b^7*c^2*d*n^6*x-1533960
*a^2*b^7*c*d^2*n^3*x^4-362880*a^2*b^7*d^3*x^7+96930*a*b^8*c^2*d*n^5*x^2+4167864*a*b^8*c*d^2*n^2*x^5-1266*b^9*c
^3*n^7-1469817*b^9*c^2*d*n^4*x^3-4332960*b^9*c*d^2*n*x^6-60480*a^6*b^3*d^3*n^3*x^3+529200*a^5*b^4*d^3*n^2*x^4-
32400*a^4*b^5*c*d^2*n^4*x^2-828576*a^4*b^5*d^3*n*x^5+18*a^3*b^6*c^2*d*n^6+891000*a^3*b^6*c*d^2*n^3*x^3+362880*
a^3*b^6*d^3*x^6-15840*a^2*b^7*c^2*d*n^5*x-3415320*a^2*b^7*c*d^2*n^2*x^4+636471*a*b^8*c^2*d*n^4*x^2+4073760*a*b
^8*c*d^2*n*x^5-16884*b^9*c^3*n^6-4371522*b^9*c^2*d*n^3*x^3-1555200*b^9*c*d^2*x^6-362880*a^6*b^3*d^3*n^2*x^3+21
60*a^5*b^4*c*d^2*n^4*x+756000*a^5*b^4*d^3*n*x^4-351000*a^4*b^5*c*d^2*n^3*x^2-362880*a^4*b^5*d^3*x^5+810*a^3*b^
6*c^2*d*n^5+2571840*a^3*b^6*c*d^2*n^2*x^3-162180*a^2*b^7*c^2*d*n^4*x-3762720*a^2*b^7*c*d^2*n*x^4+2500038*a*b^8
*c^2*d*n^3*x^2+1555200*a*b^8*c*d^2*x^5-140889*b^9*c^3*n^5-7742412*b^9*c^2*d*n^2*x^3+181440*a^7*b^2*d^3*n^2*x^2
-665280*a^6*b^3*d^3*n*x^3+60480*a^5*b^4*c*d^2*n^3*x+362880*a^5*b^4*d^3*x^4-1620000*a^4*b^5*c*d^2*n^2*x^2+15030
*a^3*b^6*c^2*d*n^4+3373920*a^3*b^6*c*d^2*n*x^3-948582*a^2*b^7*c^2*d*n^3*x-1555200*a^2*b^7*c*d^2*x^4+5614452*a*
b^8*c^2*d*n^2*x^2-761166*b^9*c^3*n^4-7291080*b^9*c^2*d*n*x^3+544320*a^7*b^2*d^3*n*x^2-2160*a^6*b^3*c*d^2*n^3-3
62880*a^6*b^3*d^3*x^3+581040*a^5*b^4*c*d^2*n^2*x-2855520*a^4*b^5*c*d^2*n*x^2+147150*a^3*b^6*c^2*d*n^3+1555200*
a^3*b^6*c*d^2*x^3-3102912*a^2*b^7*c^2*d*n^2*x+6383880*a*b^8*c^2*d*n*x^2-2655764*b^9*c^3*n^3-2721600*b^9*c^2*d*
x^3-362880*a^8*b*d^3*n*x+362880*a^7*b^2*d^3*x^2-58320*a^6*b^3*c*d^2*n^2+2077920*a^5*b^4*c*d^2*n*x-1555200*a^4*
b^5*c*d^2*x^2+801432*a^3*b^6*c^2*d*n^2-5023080*a^2*b^7*c^2*d*n*x+2721600*a*b^8*c^2*d*x^2-5753736*b^9*c^3*n^2-3
62880*a^8*b*d^3*x-522720*a^6*b^3*c*d^2*n+1555200*a^5*b^4*c*d^2*x+2301480*a^3*b^6*c^2*d*n-2721600*a^2*b^7*c^2*d
*x-6999840*b^9*c^3*n+362880*a^9*d^3-1555200*a^6*b^3*c*d^2+2721600*a^3*b^6*c^2*d-3628800*b^9*c^3)/b^10/(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)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 0.920244, size = 5404, normalized size = 16.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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 + 6327
3*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 + 36288
0*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 + 40320*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 + 51840
0*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*(242639*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 + 1
8*(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 + 732*a*b^9*c*d^2*n^7 + 6*(114
5*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 + 3*(46963*a*b^9*c^3 - 270*a^4*b^6*c^2*d)*n^5 - 18*(a^2*b^8*c*d
^2*n^8 + 37*a^2*b^8*c*d^2*n^7 + 547*a^2*b^8*c*d^2*n^6 + (4135*a^2*b^8*c*d^2 - 168*a^5*b^5*d^3)*n^5 + 4*(4261*a
^2*b^8*c*d^2 - 420*a^5*b^5*d^3)*n^4 + 4*(9487*a^2*b^8*c*d^2 - 1470*a^5*b^5*d^3)*n^3 + 48*(871*a^2*b^8*c*d^2 -
175*a^5*b^5*d^3)*n^2 + 576*(30*a^2*b^8*c*d^2 - 7*a^5*b^5*d^3)*n)*x^5 + 18*(42287*a*b^9*c^3 - 835*a^4*b^6*c^2*d
)*n^4 + 3*(b^10*c^2*d*n^9 + 51*b^10*c^2*d*n^8 + 907200*b^10*c^2*d + 6*(186*b^10*c^2*d + 5*a^3*b^7*c*d^2)*n^7 +
6*(2281*b^10*c^2*d + 165*a^3*b^7*c*d^2)*n^6 + 3*(34343*b^10*c^2*d + 4150*a^3*b^7*c*d^2)*n^5 + 3*(163313*b^10*
c^2*d + 24750*a^3*b^7*c*d^2 - 1680*a^6*b^4*d^3)*n^4 + 2*(728587*b^10*c^2*d + 107160*a^3*b^7*c*d^2 - 15120*a^6*
b^4*d^3)*n^3 + 36*(71689*b^10*c^2*d + 7810*a^3*b^7*c*d^2 - 1540*a^6*b^4*d^3)*n^2 + 360*(6751*b^10*c^2*d + 360*
a^3*b^7*c*d^2 - 84*a^6*b^4*d^3)*n)*x^4 + 2*(1327882*a*b^9*c^3 - 73575*a^4*b^6*c^2*d + 1080*a^7*b^3*c*d^2)*n^3
+ 3*(a*b^9*c^2*d*n^9 + 48*a*b^9*c^2*d*n^8 + 972*a*b^9*c^2*d*n^7 + 30*(359*a*b^9*c^2*d - 4*a^4*b^6*c*d^2)*n^6 +
3*(23573*a*b^9*c^2*d - 1200*a^4*b^6*c*d^2)*n^5 + 6*(46297*a*b^9*c^2*d - 6500*a^4*b^6*c*d^2)*n^4 + 4*(155957*a
*b^9*c^2*d - 45000*a^4*b^6*c*d^2 + 5040*a^7*b^3*d^3)*n^3 + 120*(5911*a*b^9*c^2*d - 2644*a^4*b^6*c*d^2 + 504*a^
7*b^3*d^3)*n^2 + 2880*(105*a*b^9*c^2*d - 60*a^4*b^6*c*d^2 + 14*a^7*b^3*d^3)*n)*x^3 + 72*(79913*a*b^9*c^3 - 111
31*a^4*b^6*c^2*d + 810*a^7*b^3*c*d^2)*n^2 - 9*(a^2*b^8*c^2*d*n^8 + 46*a^2*b^8*c^2*d*n^7 + 880*a^2*b^8*c^2*d*n^
6 + 10*(901*a^2*b^8*c^2*d - 12*a^5*b^5*c*d^2)*n^5 + (52699*a^2*b^8*c^2*d - 3360*a^5*b^5*c*d^2)*n^4 + 8*(21548*
a^2*b^8*c^2*d - 4035*a^5*b^5*c*d^2)*n^3 + 60*(4651*a^2*b^8*c^2*d - 1924*a^5*b^5*c*d^2 + 336*a^8*b^2*d^3)*n^2 +
1440*(105*a^2*b^8*c^2*d - 60*a^5*b^5*c*d^2 + 14*a^8*b^2*d^3)*n)*x^2 + 360*(19444*a*b^9*c^3 - 6393*a^4*b^6*c^2
*d + 1452*a^7*b^3*c*d^2)*n + (b^10*c^3*n^9 + 54*b^10*c^3*n^8 + 3628800*b^10*c^3 + 6*(211*b^10*c^3 + 3*a^3*b^7*
c^2*d)*n^7 + 18*(938*b^10*c^3 + 45*a^3*b^7*c^2*d)*n^6 + 3*(46963*b^10*c^3 + 5010*a^3*b^7*c^2*d)*n^5 + 18*(4228
7*b^10*c^3 + 8175*a^3*b^7*c^2*d - 120*a^6*b^4*c*d^2)*n^4 + 4*(663941*b^10*c^3 + 200358*a^3*b^7*c^2*d - 14580*a
^6*b^4*c*d^2)*n^3 + 72*(79913*b^10*c^3 + 31965*a^3*b^7*c^2*d - 7260*a^6*b^4*c*d^2)*n^2 + 1440*(4861*b^10*c^3 +
1890*a^3*b^7*c^2*d - 1080*a^6*b^4*c*d^2 + 252*a^9*b*d^3)*n)*x)*(b*x + a)^n/(b^10*n^10 + 55*b^10*n^9 + 1320*b^
10*n^8 + 18150*b^10*n^7 + 157773*b^10*n^6 + 902055*b^10*n^5 + 3416930*b^10*n^4 + 8409500*b^10*n^3 + 12753576*b
^10*n^2 + 10628640*b^10*n + 3628800*b^10)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.37085, size = 5230, normalized size = 15.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((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^10*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 + 67284*(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 + 14
4*(b*x + a)^n*a*b^9*c^2*d*n^8*x^3 + 3348*(b*x + a)^n*b^10*c^2*d*n^7*x^4 + 90*(b*x + a)^n*a^3*b^7*c*d^2*n^7*x^4
- 9846*(b*x + a)^n*a^2*b^8*c*d^2*n^6*x^5 + 113157*(b*x + a)^n*a*b^9*c*d^2*n^5*x^6 - 7560*(b*x + a)^n*a^4*b^6*
d^3*n^5*x^6 + 1048446*(b*x + a)^n*b^10*c*d^2*n^4*x^7 + 52920*(b*x + a)^n*a^3*b^7*d^3*n^4*x^7 - 118188*(b*x + a
)^n*a^2*b^8*d^3*n^3*x^8 + 109584*(b*x + a)^n*a*b^9*d^3*n^2*x^9 + 1026576*(b*x + a)^n*b^10*d^3*n*x^10 + (b*x +
a)^n*b^10*c^3*n^9*x - 9*(b*x + a)^n*a^2*b^8*c^2*d*n^8*x^2 + 2916*(b*x + a)^n*a*b^9*c^2*d*n^7*x^3 + 41058*(b*x
+ a)^n*b^10*c^2*d*n^6*x^4 + 2970*(b*x + a)^n*a^3*b^7*c*d^2*n^6*x^4 - 74430*(b*x + a)^n*a^2*b^8*c*d^2*n^5*x^5 +
3024*(b*x + a)^n*a^5*b^5*d^3*n^5*x^5 + 369504*(b*x + a)^n*a*b^9*c*d^2*n^4*x^6 - 42840*(b*x + a)^n*a^4*b^6*d^3
*n^4*x^6 + 2911668*(b*x + a)^n*b^10*c*d^2*n^3*x^7 + 116928*(b*x + a)^n*a^3*b^7*d^3*n^3*x^7 - 117612*(b*x + a)^
n*a^2*b^8*d^3*n^2*x^8 + 40320*(b*x + a)^n*a*b^9*d^3*n*x^9 + 362880*(b*x + a)^n*b^10*d^3*x^10 + (b*x + a)^n*a*b
^9*c^3*n^9 + 54*(b*x + a)^n*b^10*c^3*n^8*x - 414*(b*x + a)^n*a^2*b^8*c^2*d*n^7*x^2 + 32310*(b*x + a)^n*a*b^9*c
^2*d*n^6*x^3 - 360*(b*x + a)^n*a^4*b^6*c*d^2*n^6*x^3 + 309087*(b*x + a)^n*b^10*c^2*d*n^5*x^4 + 37350*(b*x + a)
^n*a^3*b^7*c*d^2*n^5*x^4 - 306792*(b*x + a)^n*a^2*b^8*c*d^2*n^4*x^5 + 30240*(b*x + a)^n*a^5*b^5*d^3*n^4*x^5 +
694644*(b*x + a)^n*a*b^9*c*d^2*n^3*x^6 - 113400*(b*x + a)^n*a^4*b^6*d^3*n^3*x^6 + 4846824*(b*x + a)^n*b^10*c*d
^2*n^2*x^7 + 127008*(b*x + a)^n*a^3*b^7*d^3*n^2*x^7 - 45360*(b*x + a)^n*a^2*b^8*d^3*n*x^8 + 54*(b*x + a)^n*a*b
^9*c^3*n^8 + 1266*(b*x + a)^n*b^10*c^3*n^7*x + 18*(b*x + a)^n*a^3*b^7*c^2*d*n^7*x - 7920*(b*x + a)^n*a^2*b^8*c
^2*d*n^6*x^2 + 212157*(b*x + a)^n*a*b^9*c^2*d*n^5*x^3 - 10800*(b*x + a)^n*a^4*b^6*c*d^2*n^5*x^3 + 1469817*(b*x
+ a)^n*b^10*c^2*d*n^4*x^4 + 222750*(b*x + a)^n*a^3*b^7*c*d^2*n^4*x^4 - 15120*(b*x + a)^n*a^6*b^4*d^3*n^4*x^4
- 683064*(b*x + a)^n*a^2*b^8*c*d^2*n^3*x^5 + 105840*(b*x + a)^n*a^5*b^5*d^3*n^3*x^5 + 678960*(b*x + a)^n*a*b^9
*c*d^2*n^2*x^6 - 138096*(b*x + a)^n*a^4*b^6*d^3*n^2*x^6 + 4332960*(b*x + a)^n*b^10*c*d^2*n*x^7 + 51840*(b*x +
a)^n*a^3*b^7*d^3*n*x^7 + 1266*(b*x + a)^n*a*b^9*c^3*n^7 + 16884*(b*x + a)^n*b^10*c^3*n^6*x + 810*(b*x + a)^n*a
^3*b^7*c^2*d*n^6*x - 81090*(b*x + a)^n*a^2*b^8*c^2*d*n^5*x^2 + 1080*(b*x + a)^n*a^5*b^5*c*d^2*n^5*x^2 + 833346
*(b*x + a)^n*a*b^9*c^2*d*n^4*x^3 - 117000*(b*x + a)^n*a^4*b^6*c*d^2*n^4*x^3 + 4371522*(b*x + a)^n*b^10*c^2*d*n
^3*x^4 + 642960*(b*x + a)^n*a^3*b^7*c*d^2*n^3*x^4 - 90720*(b*x + a)^n*a^6*b^4*d^3*n^3*x^4 - 752544*(b*x + a)^n
*a^2*b^8*c*d^2*n^2*x^5 + 151200*(b*x + a)^n*a^5*b^5*d^3*n^2*x^5 + 259200*(b*x + a)^n*a*b^9*c*d^2*n*x^6 - 60480
*(b*x + a)^n*a^4*b^6*d^3*n*x^6 + 1555200*(b*x + a)^n*b^10*c*d^2*x^7 + 16884*(b*x + a)^n*a*b^9*c^3*n^6 - 18*(b*
x + a)^n*a^4*b^6*c^2*d*n^6 + 140889*(b*x + a)^n*b^10*c^3*n^5*x + 15030*(b*x + a)^n*a^3*b^7*c^2*d*n^5*x - 47429
1*(b*x + a)^n*a^2*b^8*c^2*d*n^4*x^2 + 30240*(b*x + a)^n*a^5*b^5*c*d^2*n^4*x^2 + 1871484*(b*x + a)^n*a*b^9*c^2*
d*n^3*x^3 - 540000*(b*x + a)^n*a^4*b^6*c*d^2*n^3*x^3 + 60480*(b*x + a)^n*a^7*b^3*d^3*n^3*x^3 + 7742412*(b*x +
a)^n*b^10*c^2*d*n^2*x^4 + 843480*(b*x + a)^n*a^3*b^7*c*d^2*n^2*x^4 - 166320*(b*x + a)^n*a^6*b^4*d^3*n^2*x^4 -
311040*(b*x + a)^n*a^2*b^8*c*d^2*n*x^5 + 72576*(b*x + a)^n*a^5*b^5*d^3*n*x^5 + 140889*(b*x + a)^n*a*b^9*c^3*n^
5 - 810*(b*x + a)^n*a^4*b^6*c^2*d*n^5 + 761166*(b*x + a)^n*b^10*c^3*n^4*x + 147150*(b*x + a)^n*a^3*b^7*c^2*d*n
^4*x - 2160*(b*x + a)^n*a^6*b^4*c*d^2*n^4*x - 1551456*(b*x + a)^n*a^2*b^8*c^2*d*n^3*x^2 + 290520*(b*x + a)^n*a
^5*b^5*c*d^2*n^3*x^2 + 2127960*(b*x + a)^n*a*b^9*c^2*d*n^2*x^3 - 951840*(b*x + a)^n*a^4*b^6*c*d^2*n^2*x^3 + 18
1440*(b*x + a)^n*a^7*b^3*d^3*n^2*x^3 + 7291080*(b*x + a)^n*b^10*c^2*d*n*x^4 + 388800*(b*x + a)^n*a^3*b^7*c*d^2
*n*x^4 - 90720*(b*x + a)^n*a^6*b^4*d^3*n*x^4 + 761166*(b*x + a)^n*a*b^9*c^3*n^4 - 15030*(b*x + a)^n*a^4*b^6*c^
2*d*n^4 + 2655764*(b*x + a)^n*b^10*c^3*n^3*x + 801432*(b*x + a)^n*a^3*b^7*c^2*d*n^3*x - 58320*(b*x + a)^n*a^6*
b^4*c*d^2*n^3*x - 2511540*(b*x + a)^n*a^2*b^8*c^2*d*n^2*x^2 + 1038960*(b*x + a)^n*a^5*b^5*c*d^2*n^2*x^2 - 1814
40*(b*x + a)^n*a^8*b^2*d^3*n^2*x^2 + 907200*(b*x + a)^n*a*b^9*c^2*d*n*x^3 - 518400*(b*x + a)^n*a^4*b^6*c*d^2*n
*x^3 + 120960*(b*x + a)^n*a^7*b^3*d^3*n*x^3 + 2721600*(b*x + a)^n*b^10*c^2*d*x^4 + 2655764*(b*x + a)^n*a*b^9*c
^3*n^3 - 147150*(b*x + a)^n*a^4*b^6*c^2*d*n^3 + 2160*(b*x + a)^n*a^7*b^3*c*d^2*n^3 + 5753736*(b*x + a)^n*b^10*
c^3*n^2*x + 2301480*(b*x + a)^n*a^3*b^7*c^2*d*n^2*x - 522720*(b*x + a)^n*a^6*b^4*c*d^2*n^2*x - 1360800*(b*x +
a)^n*a^2*b^8*c^2*d*n*x^2 + 777600*(b*x + a)^n*a^5*b^5*c*d^2*n*x^2 - 181440*(b*x + a)^n*a^8*b^2*d^3*n*x^2 + 575
3736*(b*x + a)^n*a*b^9*c^3*n^2 - 801432*(b*x + a)^n*a^4*b^6*c^2*d*n^2 + 58320*(b*x + a)^n*a^7*b^3*c*d^2*n^2 +
6999840*(b*x + a)^n*b^10*c^3*n*x + 2721600*(b*x + a)^n*a^3*b^7*c^2*d*n*x - 1555200*(b*x + a)^n*a^6*b^4*c*d^2*n
*x + 362880*(b*x + a)^n*a^9*b*d^3*n*x + 6999840*(b*x + a)^n*a*b^9*c^3*n - 2301480*(b*x + a)^n*a^4*b^6*c^2*d*n
+ 522720*(b*x + a)^n*a^7*b^3*c*d^2*n + 3628800*(b*x + a)^n*b^10*c^3*x + 3628800*(b*x + a)^n*a*b^9*c^3 - 272160
0*(b*x + a)^n*a^4*b^6*c^2*d + 1555200*(b*x + a)^n*a^7*b^3*c*d^2 - 362880*(b*x + a)^n*a^10*d^3)/(b^10*n^10 + 55
*b^10*n^9 + 1320*b^10*n^8 + 18150*b^10*n^7 + 157773*b^10*n^6 + 902055*b^10*n^5 + 3416930*b^10*n^4 + 8409500*b^
10*n^3 + 12753576*b^10*n^2 + 10628640*b^10*n + 3628800*b^10)