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3.2.83 \(\int x (a+b x)^n (c+d x^3)^3 \, dx\) [183]

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

[Out]

-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)

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Rubi [A]
time = 0.19, antiderivative size = 396, normalized size of antiderivative = 1.00, 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 = {1634} \begin {gather*} -\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)} \end {gather*}

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 1634

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*}

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Mathematica [A]
time = 0.32, size = 345, normalized size = 0.87 \begin {gather*} \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}} \end {gather*}

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2971\) vs. \(2(396)=792\).
time = 0.28, size = 2972, normalized size = 7.51

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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...

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 953 vs. \(2 (396) = 792\).
time = 0.30, size = 953, normalized size = 2.41 \begin {gather*} \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{3}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {3 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} c^{2} d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} + \frac {3 \, {\left ({\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{8} x^{8} + {\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a b^{7} x^{7} - 7 \, {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{2} b^{6} x^{6} + 42 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{3} b^{5} x^{5} - 210 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{4} b^{4} x^{4} + 840 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{5} b^{3} x^{3} - 2520 \, {\left (n^{2} + n\right )} a^{6} b^{2} x^{2} + 5040 \, a^{7} b n x - 5040 \, a^{8}\right )} {\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{8} + 36 \, n^{7} + 546 \, n^{6} + 4536 \, n^{5} + 22449 \, n^{4} + 67284 \, n^{3} + 118124 \, n^{2} + 109584 \, n + 40320\right )} b^{8}} + \frac {{\left ({\left (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\right )} b^{11} x^{11} + {\left (n^{10} + 45 \, n^{9} + 870 \, n^{8} + 9450 \, n^{7} + 63273 \, n^{6} + 269325 \, n^{5} + 723680 \, n^{4} + 1172700 \, n^{3} + 1026576 \, n^{2} + 362880 \, n\right )} a b^{10} x^{10} - 10 \, {\left (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\right )} a^{2} b^{9} x^{9} + 90 \, {\left (n^{8} + 28 \, n^{7} + 322 \, n^{6} + 1960 \, n^{5} + 6769 \, n^{4} + 13132 \, n^{3} + 13068 \, n^{2} + 5040 \, n\right )} a^{3} b^{8} x^{8} - 720 \, {\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a^{4} b^{7} x^{7} + 5040 \, {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{5} b^{6} x^{6} - 30240 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{6} b^{5} x^{5} + 151200 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{7} b^{4} x^{4} - 604800 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{8} b^{3} x^{3} + 1814400 \, {\left (n^{2} + n\right )} a^{9} b^{2} x^{2} - 3628800 \, a^{10} b n x + 3628800 \, a^{11}\right )} {\left (b x + a\right )}^{n} d^{3}}{{\left (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\right )} b^{11}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 2919 vs. \(2 (396) = 792\).
time = 0.43, size = 2919, normalized size = 7.37 \begin {gather*} \text {Too large to display} \end {gather*}

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^...

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56151 vs. \(2 (374) = 748\).
time = 37.09, size = 56151, normalized size = 141.80 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + 25200*a**9*b**12*x + 113400*a**8*b**13*x**2 + 302400*a**7*b**14*x**3 + 52920
0*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) + 7381*a**10*d**3/(2520*a**10*b**11 + 25200*a**9*b**12*x + 11
3400*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) + 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 + 11
3400*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 + 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) + 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 + 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 + 302
400*a**3*b**18*x**7 + 113400*a**2*b**19*x**8 + 25200*a*b**20*x**9 + 2520*b**21*x**10) + 308205*a**8*b**2*d**3*
x**2/(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) - 21*a**7*b**3*c*d**2/(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) + 302400*a**7
*b**3*d**3*x**3*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) + 784080*a**7*b**3*d**3*x**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 + 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) - 210*a**6*b**4*c*d**2*x/(2520*a**10*b**11 + 25200*a**9*b**12*x + 113400*a**8*b**13*x**2 + 3
02400*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) + 529200*a**6*b**4*d**3*x**4*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) + 1296540*a**6*b**4*d**3*x**4/(2520*a**10*b**11 + 25200*a**9*b**1
2*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 + 5292
00*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)
- 945*a**5*b**5*c*d**2*x**2/(2520*a**10*b**11 + 25200*a**9*b**12*x + 113400*a**8*b**13*x**2 + 302400*a**7*b**1
4*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 + 1
13400*a**2*b**19*x**8 + 25200*a*b**20*x**9 + 2520*b**21*x**10) + 635040*a**5*b**5*d**3*x**5*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) + 1450008*a**5*b**5*d**3*x**5/(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) - 6*a**4*b**6*c
**2*d/(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) - 2520*a**4*b**6*c*d**2*x**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 + 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) + 529
200*a**4*b**6*d**3*x**6*log(a/b + x)/(2520*a**1...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4934 vs. \(2 (396) = 792\).
time = 4.12, size = 4934, normalized size = 12.46 \begin {gather*} \text {Too large to display} \end {gather*}

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)^...

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Mupad [B]
time = 5.60, size = 2500, normalized size = 6.31 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c + d*x^3)^3*(a + b*x)^n,x)

[Out]

(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 + 3628800))/(120543840*n + 150917976*n^2 + 105258076*n^3 + 45995730*n^4 + 1333
9535*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*(19
958400*b^9*c^3 - 3628800*a^9*d^3 + 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 - 239500
80*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
+ 45995730*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 - 226
0440*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 - 756
0*a^6*b^3*c*d^2*n^4 + 38700*a^3*b^6*c^2*d*n^5 + 1836*a^3*b^6*c^2*d*n^6 + 36*a^3*b^6*c^2*d*n^7))/(b^9*(12054384
0*n + 150917976*n^2 + 105258076*n^3 + 45995730*n^4 + 13339535*n^5 + 2637558*n^6 + 357423*n^7 + 32670*n^8 + 192
5*n^9 + 66*n^10 + n^11 + 39916800)) + (a*n*x*(a + b*x)^n*(19958400*b^9*c^3 - 3628800*a^9*d^3 + 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 - 23950080*a^3*b^6*c^2*d + 14968800*a^6*b^3*c*d^2 - 176402
88*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^10*(120543840*n + 150917976*n^2 + 105258076*n^3 + 45995730*n^4 + 13339535*n^5 + 2637558*n^6 + 3574
23*n^7 + 32670*n^8 + 1925*n^9 + 66*n^10 + n^11 + 39916800)) + (3*d*x^5*(a + b*x)^n*(50*n + 35*n^2 + 10*n^3 + n
^4 + 24)*(332640*b^6*c^2 - 10080*a^6*d^2*n + 245004*b^6*c^2*n + 74524*b^6*c^2*n^2 + 11985*b^6*c^2*n^3 + 1075*b
^6*c^2*n^4 + 51*b^6*c^2*n^5 + b^6*c^2*n^6 + 41580*a^3*b^3*c*d*n + 12558*a^3*b^3*c*d*n^2 + 1260*a^3*b^3*c*d*n^3
 + 42*a^3*b^3*c*d*n^4))/(b^6*(120543840*n + 150917976*n^2 + 105258076*n^3 + 45995730*n^4 + 13339535*n^5 + 2637
558*n^6 + 357423*n^7 + 32670*n^8 + 1925*n^9 + 66*n^10 + n^11 + 39916800)) + (3*d^2*x^8*(a + b*x)^n*(990*b^3*c
+ 30*b^3*c*n^2 + b^3*c*n^3 + 30*a^3*d*n + 299*b^3*c*n)*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 +
28*n^6 + n^7 + 5040))/(b^3*(120543840*n + 150917976*n^2 + 105258076*n^3 + 45995730*n^4 + 13339535*n^5 + 263755
8*n^6 + 357423*n^7 + 32670*n^8 + 1925*n^9 + 66*n^10 + n^11 + 39916800)) + (a*d^3*n*x^10*(a + b*x)^n*(1026576*n
 + 1172700*n^2 + 723680*n^3 + 269325*n^4 + 63273*n^5 + 9450*n^6 + 870*n^7 + 45*n^8 + n^9 + 362880))/(b*(120543
840*n + 150917976*n^2 + 105258076*n^3 + 45995730*n^4 + 13339535*n^5 + 2637558*n^6 + 357423*n^7 + 32670*n^8 + 1
925*n^9 + 66*n^10 + n^11 + 39916800)) - (10*a^2*d^3*n*x^9*(a + b*x)^n*(109584*n + 118124*n^2 + 67284*n^3 + 224
49*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320))/(b^2*(120543840*n + 150917976*n^2 + 105258076*n^3 + 45995
730*n^4 + 13339535*n^5 + 2637558*n^6 + 357423*n^7 + 32670*n^8 + 1925*n^9 + 66*n^10 + n^11 + 39916800)) + (3*a*
d^2*n*x^7*(a + b*x)^n*(990*b^3*c - 240*a^3*d + 30*b^3*c*n^2 + b^3*c*n^3 + 299*b^3*c*n)*(1764*n + 1624*n^2 + 73
5*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))/(b^4*(120543840*n + 150917976*n^2 + 105258076*n^3 + 45995730*n^4 + 1333
9535*n^5 + 2637558*n^6 + 357423*n^7 + 32670*n^8 + 1925*n^9 + 66*n^10 + n^11 + 39916800)) - (21*a^2*d^2*n*x^6*(
a + b*x)^n*(990*b^3*c - 240*a^3*d + 30*b^3*c*n^2 + b^3*c*n^3 + 299*b^3*c*n)*(274*n + 225*n^2 + 85*n^3 + 15*n^4
 + n^5 + 120))/(b^5*(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 + 39916800)) + (3*a*d*n*x^4*(a + b*x)^n*(11*n + 6*n^2 + n^
3 + 6)*(50400*a^6*d^2 + 332640*b^6*c^2 + 245004*b^6*c^2*n + 74524*b^6*c^2*n^2 + 11985*b^6*c^2*n^3 + 1075*b^6*c
^2*n^4 + 51*b^6*c^2*n^5 + b^6*c^2*n^6 - 207900*a^3*b^3*c*d - 62790*a^3*b^3*c*d*n - 6300*a^3*b^3*c*d*n^2 - 210*
a^3*b^3*c*d*n^3))/(b^7*(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 + 39916800)) - (12*a^2*d*n*x^3*(a + b*x)^n*(3*n + n^2 +
 2)*(50400*a^6*d^2 + 332640*b^6*c^2 + 245004*b^6*c^2*n + 74524*b^6*c^2*n^2 + 11985*b^6*c^2*n^3 + 1075*b^6*c^2*
n^4 + 51*b^6*c^2*n^5 + b^6*c^2*n^6 - 207900*a^3...

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