3.178 \(\int x^2 (a+b x)^n (c+d x^3)^2 \, dx\)

Optimal. Leaf size=294 \[ \frac{\left (-20 a^3 b^3 c d+28 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}+\frac{a^2 \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^9 (n+1)}-\frac{2 a \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^9 (n+2)}+\frac{4 a^2 d \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{n+4}}{b^9 (n+4)}-\frac{10 a d \left (b^3 c-7 a^3 d\right ) (a+b x)^{n+5}}{b^9 (n+5)}+\frac{2 d \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{28 a^2 d^2 (a+b x)^{n+7}}{b^9 (n+7)}-\frac{8 a d^2 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^2 (a+b x)^{n+9}}{b^9 (n+9)} \]

[Out]

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

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Rubi [A]  time = 0.201699, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1620} \[ \frac{\left (-20 a^3 b^3 c d+28 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}+\frac{a^2 \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^9 (n+1)}-\frac{2 a \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^9 (n+2)}+\frac{4 a^2 d \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{n+4}}{b^9 (n+4)}-\frac{10 a d \left (b^3 c-7 a^3 d\right ) (a+b x)^{n+5}}{b^9 (n+5)}+\frac{2 d \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{28 a^2 d^2 (a+b x)^{n+7}}{b^9 (n+7)}-\frac{8 a d^2 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^2 (a+b x)^{n+9}}{b^9 (n+9)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^n*(c + d*x^3)^2,x]

[Out]

(a^2*(b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^9*(1 + n)) - (2*a*(b^3*c - 4*a^3*d)*(b^3*c - a^3*d)*(a + b*x)^(2
+ n))/(b^9*(2 + n)) + ((b^6*c^2 - 20*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^(3 + n))/(b^9*(3 + n)) + (4*a^2*d*(5*
b^3*c - 14*a^3*d)*(a + b*x)^(4 + n))/(b^9*(4 + n)) - (10*a*d*(b^3*c - 7*a^3*d)*(a + b*x)^(5 + n))/(b^9*(5 + n)
) + (2*d*(b^3*c - 28*a^3*d)*(a + b*x)^(6 + n))/(b^9*(6 + n)) + (28*a^2*d^2*(a + b*x)^(7 + n))/(b^9*(7 + n)) -
(8*a*d^2*(a + b*x)^(8 + n))/(b^9*(8 + n)) + (d^2*(a + b*x)^(9 + n))/(b^9*(9 + 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^2 (a+b x)^n \left (c+d x^3\right )^2 \, dx &=\int \left (\frac{\left (a b^3 c-a^4 d\right )^2 (a+b x)^n}{b^8}-\frac{2 \left (a b^6 c^2-5 a^4 b^3 c d+4 a^7 d^2\right ) (a+b x)^{1+n}}{b^8}+\frac{\left (b^6 c^2-20 a^3 b^3 c d+28 a^6 d^2\right ) (a+b x)^{2+n}}{b^8}-\frac{4 a^2 d \left (-5 b^3 c+14 a^3 d\right ) (a+b x)^{3+n}}{b^8}+\frac{10 a d \left (-b^3 c+7 a^3 d\right ) (a+b x)^{4+n}}{b^8}+\frac{2 d \left (b^3 c-28 a^3 d\right ) (a+b x)^{5+n}}{b^8}+\frac{28 a^2 d^2 (a+b x)^{6+n}}{b^8}-\frac{8 a d^2 (a+b x)^{7+n}}{b^8}+\frac{d^2 (a+b x)^{8+n}}{b^8}\right ) \, dx\\ &=\frac{a^2 \left (b^3 c-a^3 d\right )^2 (a+b x)^{1+n}}{b^9 (1+n)}-\frac{2 a \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{2+n}}{b^9 (2+n)}+\frac{\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 (3+n)}+\frac{4 a^2 d \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{4+n}}{b^9 (4+n)}-\frac{10 a d \left (b^3 c-7 a^3 d\right ) (a+b x)^{5+n}}{b^9 (5+n)}+\frac{2 d \left (b^3 c-28 a^3 d\right ) (a+b x)^{6+n}}{b^9 (6+n)}+\frac{28 a^2 d^2 (a+b x)^{7+n}}{b^9 (7+n)}-\frac{8 a d^2 (a+b x)^{8+n}}{b^9 (8+n)}+\frac{d^2 (a+b x)^{9+n}}{b^9 (9+n)}\\ \end{align*}

Mathematica [A]  time = 0.278485, size = 252, normalized size = 0.86 \[ \frac{(a+b x)^{n+1} \left (\frac{(a+b x)^2 \left (-20 a^3 b^3 c d+28 a^6 d^2+b^6 c^2\right )}{n+3}+\frac{2 d (a+b x)^5 \left (b^3 c-28 a^3 d\right )}{n+6}+\frac{10 a d (a+b x)^4 \left (7 a^3 d-b^3 c\right )}{n+5}+\frac{4 a^2 d (a+b x)^3 \left (5 b^3 c-14 a^3 d\right )}{n+4}-\frac{2 a (a+b x) \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right )}{n+2}+\frac{\left (a b^3 c-a^4 d\right )^2}{n+1}+\frac{28 a^2 d^2 (a+b x)^6}{n+7}+\frac{d^2 (a+b x)^8}{n+9}-\frac{8 a d^2 (a+b x)^7}{n+8}\right )}{b^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^n*(c + d*x^3)^2,x]

[Out]

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

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Maple [B]  time = 0.014, size = 1565, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)^n*(d*x^3+c)^2,x)

[Out]

(b*x+a)^(1+n)*(b^8*d^2*n^8*x^8+36*b^8*d^2*n^7*x^8-8*a*b^7*d^2*n^7*x^7+546*b^8*d^2*n^6*x^8-224*a*b^7*d^2*n^6*x^
7+2*b^8*c*d*n^8*x^5+4536*b^8*d^2*n^5*x^8+56*a^2*b^6*d^2*n^6*x^6-2576*a*b^7*d^2*n^5*x^7+78*b^8*c*d*n^7*x^5+2244
9*b^8*d^2*n^4*x^8+1176*a^2*b^6*d^2*n^5*x^6-10*a*b^7*c*d*n^7*x^4-15680*a*b^7*d^2*n^4*x^7+1272*b^8*c*d*n^6*x^5+6
7284*b^8*d^2*n^3*x^8-336*a^3*b^5*d^2*n^5*x^5+9800*a^2*b^6*d^2*n^4*x^6-340*a*b^7*c*d*n^6*x^4-54152*a*b^7*d^2*n^
3*x^7+b^8*c^2*n^8*x^2+11268*b^8*c*d*n^5*x^5+118124*b^8*d^2*n^2*x^8-5040*a^3*b^5*d^2*n^4*x^5+40*a^2*b^6*c*d*n^6
*x^3+41160*a^2*b^6*d^2*n^3*x^6-4660*a*b^7*c*d*n^5*x^4-105056*a*b^7*d^2*n^2*x^7+42*b^8*c^2*n^7*x^2+58938*b^8*c*
d*n^4*x^5+109584*b^8*d^2*n*x^8+1680*a^4*b^4*d^2*n^4*x^4-28560*a^3*b^5*d^2*n^3*x^5+1200*a^2*b^6*c*d*n^5*x^3+909
44*a^2*b^6*d^2*n^2*x^6-2*a*b^7*c^2*n^7*x-33040*a*b^7*c*d*n^4*x^4-104544*a*b^7*d^2*n*x^7+744*b^8*c^2*n^6*x^2+18
5022*b^8*c*d*n^3*x^5+40320*b^8*d^2*x^8+16800*a^4*b^4*d^2*n^3*x^4-120*a^3*b^5*c*d*n^5*x^2-75600*a^3*b^5*d^2*n^2
*x^5+13840*a^2*b^6*c*d*n^4*x^3+98784*a^2*b^6*d^2*n*x^6-80*a*b^7*c^2*n^6*x-129490*a*b^7*c*d*n^3*x^4-40320*a*b^7
*d^2*x^7+7218*b^8*c^2*n^5*x^2+337228*b^8*c*d*n^2*x^5-6720*a^5*b^3*d^2*n^3*x^3+58800*a^4*b^4*d^2*n^2*x^4-3240*a
^3*b^5*c*d*n^4*x^2-92064*a^3*b^5*d^2*n*x^5+2*a^2*b^6*c^2*n^6+76800*a^2*b^6*c*d*n^3*x^3+40320*a^2*b^6*d^2*x^6-1
328*a*b^7*c^2*n^5*x-277660*a*b^7*c*d*n^2*x^4+41619*b^8*c^2*n^4*x^2+322032*b^8*c*d*n*x^5-40320*a^5*b^3*d^2*n^2*
x^3+240*a^4*b^4*c*d*n^4*x+84000*a^4*b^4*d^2*n*x^4-31800*a^3*b^5*c*d*n^3*x^2-40320*a^3*b^5*d^2*x^5+78*a^2*b^6*c
^2*n^5+210760*a^2*b^6*c*d*n^2*x^3-11780*a*b^7*c^2*n^4*x-297840*a*b^7*c*d*n*x^4+144468*b^8*c^2*n^3*x^2+120960*b
^8*c*d*x^5+20160*a^6*b^2*d^2*n^2*x^2-73920*a^5*b^3*d^2*n*x^3+6000*a^4*b^4*c*d*n^3*x+40320*a^4*b^4*d^2*x^4-1350
00*a^3*b^5*c*d*n^2*x^2+1250*a^2*b^6*c^2*n^4+267600*a^2*b^6*c*d*n*x^3-59678*a*b^7*c^2*n^3*x-120960*a*b^7*c*d*x^
4+290276*b^8*c^2*n^2*x^2+60480*a^6*b^2*d^2*n*x^2-240*a^5*b^3*c*d*n^3-40320*a^5*b^3*d^2*x^3+51600*a^4*b^4*c*d*n
^2*x-227280*a^3*b^5*c*d*n*x^2+10530*a^2*b^6*c^2*n^3+120960*a^2*b^6*c*d*x^3-169580*a*b^7*c^2*n^2*x+301872*b^8*c
^2*n*x^2-40320*a^7*b*d^2*n*x+40320*a^6*b^2*d^2*x^2-5760*a^5*b^3*c*d*n^2+166800*a^4*b^4*c*d*n*x-120960*a^3*b^5*
c*d*x^2+49148*a^2*b^6*c^2*n^2-241392*a*b^7*c^2*n*x+120960*b^8*c^2*x^2-40320*a^7*b*d^2*x-45840*a^5*b^3*c*d*n+12
0960*a^4*b^4*c*d*x+120432*a^2*b^6*c^2*n-120960*a*b^7*c^2*x+40320*a^8*d^2-120960*a^5*b^3*c*d+120960*a^2*b^6*c^2
)/b^9/(n^9+45*n^8+870*n^7+9450*n^6+63273*n^5+269325*n^4+723680*n^3+1172700*n^2+1026576*n+362880)

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Maxima [B]  time = 1.19546, size = 811, normalized size = 2.76 \begin{align*} \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} +{\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )}{\left (b x + a\right )}^{n} c^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac{2 \,{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} +{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \,{\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )}{\left (b x + a\right )}^{n} c d}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} + \frac{{\left ({\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^{9} x^{9} +{\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 b^{8} x^{8} - 8 \,{\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a^{2} b^{7} x^{7} + 56 \,{\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{3} b^{6} x^{6} - 336 \,{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{4} b^{5} x^{5} + 1680 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{5} b^{4} x^{4} - 6720 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{6} b^{3} x^{3} + 20160 \,{\left (n^{2} + n\right )} a^{7} b^{2} x^{2} - 40320 \, a^{8} b n x + 40320 \, a^{9}\right )}{\left (b x + a\right )}^{n} d^{2}}{{\left (n^{9} + 45 \, n^{8} + 870 \, n^{7} + 9450 \, n^{6} + 63273 \, n^{5} + 269325 \, n^{4} + 723680 \, n^{3} + 1172700 \, n^{2} + 1026576 \, n + 362880\right )} b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="maxima")

[Out]

((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^n*c^2/((n^3 + 6*n^2 + 11*n + 6
)*b^3) + 2*((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^6*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*
a*b^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2*b^4*x^4 + 20*(n^3 + 3*n^2 + 2*n)*a^3*b^3*x^3 - 60*(n^2 + n)*a^4
*b^2*x^2 + 120*a^5*b*n*x - 120*a^6)*(b*x + a)^n*c*d/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 7
20)*b^6) + ((n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 118124*n^2 + 109584*n + 40320)*b^9*x^
9 + (n^8 + 28*n^7 + 322*n^6 + 1960*n^5 + 6769*n^4 + 13132*n^3 + 13068*n^2 + 5040*n)*a*b^8*x^8 - 8*(n^7 + 21*n^
6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*a^2*b^7*x^7 + 56*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n
^2 + 120*n)*a^3*b^6*x^6 - 336*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^4*b^5*x^5 + 1680*(n^4 + 6*n^3 + 11*n^2
+ 6*n)*a^5*b^4*x^4 - 6720*(n^3 + 3*n^2 + 2*n)*a^6*b^3*x^3 + 20160*(n^2 + n)*a^7*b^2*x^2 - 40320*a^8*b*n*x + 4
0320*a^9)*(b*x + a)^n*d^2/((n^9 + 45*n^8 + 870*n^7 + 9450*n^6 + 63273*n^5 + 269325*n^4 + 723680*n^3 + 1172700*
n^2 + 1026576*n + 362880)*b^9)

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Fricas [B]  time = 1.05735, size = 3567, normalized size = 12.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="fricas")

[Out]

(2*a^3*b^6*c^2*n^6 + 78*a^3*b^6*c^2*n^5 + 1250*a^3*b^6*c^2*n^4 + 120960*a^3*b^6*c^2 - 120960*a^6*b^3*c*d + 403
20*a^9*d^2 + (b^9*d^2*n^8 + 36*b^9*d^2*n^7 + 546*b^9*d^2*n^6 + 4536*b^9*d^2*n^5 + 22449*b^9*d^2*n^4 + 67284*b^
9*d^2*n^3 + 118124*b^9*d^2*n^2 + 109584*b^9*d^2*n + 40320*b^9*d^2)*x^9 + (a*b^8*d^2*n^8 + 28*a*b^8*d^2*n^7 + 3
22*a*b^8*d^2*n^6 + 1960*a*b^8*d^2*n^5 + 6769*a*b^8*d^2*n^4 + 13132*a*b^8*d^2*n^3 + 13068*a*b^8*d^2*n^2 + 5040*
a*b^8*d^2*n)*x^8 - 8*(a^2*b^7*d^2*n^7 + 21*a^2*b^7*d^2*n^6 + 175*a^2*b^7*d^2*n^5 + 735*a^2*b^7*d^2*n^4 + 1624*
a^2*b^7*d^2*n^3 + 1764*a^2*b^7*d^2*n^2 + 720*a^2*b^7*d^2*n)*x^7 + 2*(b^9*c*d*n^8 + 39*b^9*c*d*n^7 + 60480*b^9*
c*d + 4*(159*b^9*c*d + 7*a^3*b^6*d^2)*n^6 + 6*(939*b^9*c*d + 70*a^3*b^6*d^2)*n^5 + (29469*b^9*c*d + 2380*a^3*b
^6*d^2)*n^4 + 9*(10279*b^9*c*d + 700*a^3*b^6*d^2)*n^3 + 2*(84307*b^9*c*d + 3836*a^3*b^6*d^2)*n^2 + 24*(6709*b^
9*c*d + 140*a^3*b^6*d^2)*n)*x^6 + 2*(a*b^8*c*d*n^8 + 34*a*b^8*c*d*n^7 + 466*a*b^8*c*d*n^6 + 56*(59*a*b^8*c*d -
3*a^4*b^5*d^2)*n^5 + (12949*a*b^8*c*d - 1680*a^4*b^5*d^2)*n^4 + 2*(13883*a*b^8*c*d - 2940*a^4*b^5*d^2)*n^3 +
24*(1241*a*b^8*c*d - 350*a^4*b^5*d^2)*n^2 + 4032*(3*a*b^8*c*d - a^4*b^5*d^2)*n)*x^5 - 10*(a^2*b^7*c*d*n^7 + 30
*a^2*b^7*c*d*n^6 + 346*a^2*b^7*c*d*n^5 + 24*(80*a^2*b^7*c*d - 7*a^5*b^4*d^2)*n^4 + (5269*a^2*b^7*c*d - 1008*a^
5*b^4*d^2)*n^3 + 6*(1115*a^2*b^7*c*d - 308*a^5*b^4*d^2)*n^2 + 1008*(3*a^2*b^7*c*d - a^5*b^4*d^2)*n)*x^4 + 30*(
351*a^3*b^6*c^2 - 8*a^6*b^3*c*d)*n^3 + (b^9*c^2*n^8 + 42*b^9*c^2*n^7 + 120960*b^9*c^2 + 8*(93*b^9*c^2 + 5*a^3*
b^6*c*d)*n^6 + 18*(401*b^9*c^2 + 60*a^3*b^6*c*d)*n^5 + (41619*b^9*c^2 + 10600*a^3*b^6*c*d)*n^4 + 12*(12039*b^9
*c^2 + 3750*a^3*b^6*c*d - 560*a^6*b^3*d^2)*n^3 + 4*(72569*b^9*c^2 + 18940*a^3*b^6*c*d - 5040*a^6*b^3*d^2)*n^2
+ 48*(6289*b^9*c^2 + 840*a^3*b^6*c*d - 280*a^6*b^3*d^2)*n)*x^3 + 4*(12287*a^3*b^6*c^2 - 1440*a^6*b^3*c*d)*n^2
+ (a*b^8*c^2*n^8 + 40*a*b^8*c^2*n^7 + 664*a*b^8*c^2*n^6 + 10*(589*a*b^8*c^2 - 12*a^4*b^5*c*d)*n^5 + (29839*a*b
^8*c^2 - 3000*a^4*b^5*c*d)*n^4 + 10*(8479*a*b^8*c^2 - 2580*a^4*b^5*c*d)*n^3 + 24*(5029*a*b^8*c^2 - 3475*a^4*b^
5*c*d + 840*a^7*b^2*d^2)*n^2 + 20160*(3*a*b^8*c^2 - 3*a^4*b^5*c*d + a^7*b^2*d^2)*n)*x^2 + 48*(2509*a^3*b^6*c^2
- 955*a^6*b^3*c*d)*n - 2*(a^2*b^7*c^2*n^7 + 39*a^2*b^7*c^2*n^6 + 625*a^2*b^7*c^2*n^5 + 15*(351*a^2*b^7*c^2 -
8*a^5*b^4*c*d)*n^4 + 2*(12287*a^2*b^7*c^2 - 1440*a^5*b^4*c*d)*n^3 + 24*(2509*a^2*b^7*c^2 - 955*a^5*b^4*c*d)*n^
2 + 20160*(3*a^2*b^7*c^2 - 3*a^5*b^4*c*d + a^8*b*d^2)*n)*x)*(b*x + a)^n/(b^9*n^9 + 45*b^9*n^8 + 870*b^9*n^7 +
9450*b^9*n^6 + 63273*b^9*n^5 + 269325*b^9*n^4 + 723680*b^9*n^3 + 1172700*b^9*n^2 + 1026576*b^9*n + 362880*b^9)

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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**2*(b*x+a)**n*(d*x**3+c)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.25503, size = 3591, normalized size = 12.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="giac")

[Out]

((b*x + a)^n*b^9*d^2*n^8*x^9 + (b*x + a)^n*a*b^8*d^2*n^8*x^8 + 36*(b*x + a)^n*b^9*d^2*n^7*x^9 + 28*(b*x + a)^n
*a*b^8*d^2*n^7*x^8 + 546*(b*x + a)^n*b^9*d^2*n^6*x^9 + 2*(b*x + a)^n*b^9*c*d*n^8*x^6 - 8*(b*x + a)^n*a^2*b^7*d
^2*n^7*x^7 + 322*(b*x + a)^n*a*b^8*d^2*n^6*x^8 + 4536*(b*x + a)^n*b^9*d^2*n^5*x^9 + 2*(b*x + a)^n*a*b^8*c*d*n^
8*x^5 + 78*(b*x + a)^n*b^9*c*d*n^7*x^6 - 168*(b*x + a)^n*a^2*b^7*d^2*n^6*x^7 + 1960*(b*x + a)^n*a*b^8*d^2*n^5*
x^8 + 22449*(b*x + a)^n*b^9*d^2*n^4*x^9 + 68*(b*x + a)^n*a*b^8*c*d*n^7*x^5 + 1272*(b*x + a)^n*b^9*c*d*n^6*x^6
+ 56*(b*x + a)^n*a^3*b^6*d^2*n^6*x^6 - 1400*(b*x + a)^n*a^2*b^7*d^2*n^5*x^7 + 6769*(b*x + a)^n*a*b^8*d^2*n^4*x
^8 + 67284*(b*x + a)^n*b^9*d^2*n^3*x^9 + (b*x + a)^n*b^9*c^2*n^8*x^3 - 10*(b*x + a)^n*a^2*b^7*c*d*n^7*x^4 + 93
2*(b*x + a)^n*a*b^8*c*d*n^6*x^5 + 11268*(b*x + a)^n*b^9*c*d*n^5*x^6 + 840*(b*x + a)^n*a^3*b^6*d^2*n^5*x^6 - 58
80*(b*x + a)^n*a^2*b^7*d^2*n^4*x^7 + 13132*(b*x + a)^n*a*b^8*d^2*n^3*x^8 + 118124*(b*x + a)^n*b^9*d^2*n^2*x^9
+ (b*x + a)^n*a*b^8*c^2*n^8*x^2 + 42*(b*x + a)^n*b^9*c^2*n^7*x^3 - 300*(b*x + a)^n*a^2*b^7*c*d*n^6*x^4 + 6608*
(b*x + a)^n*a*b^8*c*d*n^5*x^5 - 336*(b*x + a)^n*a^4*b^5*d^2*n^5*x^5 + 58938*(b*x + a)^n*b^9*c*d*n^4*x^6 + 4760
*(b*x + a)^n*a^3*b^6*d^2*n^4*x^6 - 12992*(b*x + a)^n*a^2*b^7*d^2*n^3*x^7 + 13068*(b*x + a)^n*a*b^8*d^2*n^2*x^8
+ 109584*(b*x + a)^n*b^9*d^2*n*x^9 + 40*(b*x + a)^n*a*b^8*c^2*n^7*x^2 + 744*(b*x + a)^n*b^9*c^2*n^6*x^3 + 40*
(b*x + a)^n*a^3*b^6*c*d*n^6*x^3 - 3460*(b*x + a)^n*a^2*b^7*c*d*n^5*x^4 + 25898*(b*x + a)^n*a*b^8*c*d*n^4*x^5 -
3360*(b*x + a)^n*a^4*b^5*d^2*n^4*x^5 + 185022*(b*x + a)^n*b^9*c*d*n^3*x^6 + 12600*(b*x + a)^n*a^3*b^6*d^2*n^3
*x^6 - 14112*(b*x + a)^n*a^2*b^7*d^2*n^2*x^7 + 5040*(b*x + a)^n*a*b^8*d^2*n*x^8 + 40320*(b*x + a)^n*b^9*d^2*x^
9 - 2*(b*x + a)^n*a^2*b^7*c^2*n^7*x + 664*(b*x + a)^n*a*b^8*c^2*n^6*x^2 + 7218*(b*x + a)^n*b^9*c^2*n^5*x^3 + 1
080*(b*x + a)^n*a^3*b^6*c*d*n^5*x^3 - 19200*(b*x + a)^n*a^2*b^7*c*d*n^4*x^4 + 1680*(b*x + a)^n*a^5*b^4*d^2*n^4
*x^4 + 55532*(b*x + a)^n*a*b^8*c*d*n^3*x^5 - 11760*(b*x + a)^n*a^4*b^5*d^2*n^3*x^5 + 337228*(b*x + a)^n*b^9*c*
d*n^2*x^6 + 15344*(b*x + a)^n*a^3*b^6*d^2*n^2*x^6 - 5760*(b*x + a)^n*a^2*b^7*d^2*n*x^7 - 78*(b*x + a)^n*a^2*b^
7*c^2*n^6*x + 5890*(b*x + a)^n*a*b^8*c^2*n^5*x^2 - 120*(b*x + a)^n*a^4*b^5*c*d*n^5*x^2 + 41619*(b*x + a)^n*b^9
*c^2*n^4*x^3 + 10600*(b*x + a)^n*a^3*b^6*c*d*n^4*x^3 - 52690*(b*x + a)^n*a^2*b^7*c*d*n^3*x^4 + 10080*(b*x + a)
^n*a^5*b^4*d^2*n^3*x^4 + 59568*(b*x + a)^n*a*b^8*c*d*n^2*x^5 - 16800*(b*x + a)^n*a^4*b^5*d^2*n^2*x^5 + 322032*
(b*x + a)^n*b^9*c*d*n*x^6 + 6720*(b*x + a)^n*a^3*b^6*d^2*n*x^6 + 2*(b*x + a)^n*a^3*b^6*c^2*n^6 - 1250*(b*x + a
)^n*a^2*b^7*c^2*n^5*x + 29839*(b*x + a)^n*a*b^8*c^2*n^4*x^2 - 3000*(b*x + a)^n*a^4*b^5*c*d*n^4*x^2 + 144468*(b
*x + a)^n*b^9*c^2*n^3*x^3 + 45000*(b*x + a)^n*a^3*b^6*c*d*n^3*x^3 - 6720*(b*x + a)^n*a^6*b^3*d^2*n^3*x^3 - 669
00*(b*x + a)^n*a^2*b^7*c*d*n^2*x^4 + 18480*(b*x + a)^n*a^5*b^4*d^2*n^2*x^4 + 24192*(b*x + a)^n*a*b^8*c*d*n*x^5
- 8064*(b*x + a)^n*a^4*b^5*d^2*n*x^5 + 120960*(b*x + a)^n*b^9*c*d*x^6 + 78*(b*x + a)^n*a^3*b^6*c^2*n^5 - 1053
0*(b*x + a)^n*a^2*b^7*c^2*n^4*x + 240*(b*x + a)^n*a^5*b^4*c*d*n^4*x + 84790*(b*x + a)^n*a*b^8*c^2*n^3*x^2 - 25
800*(b*x + a)^n*a^4*b^5*c*d*n^3*x^2 + 290276*(b*x + a)^n*b^9*c^2*n^2*x^3 + 75760*(b*x + a)^n*a^3*b^6*c*d*n^2*x
^3 - 20160*(b*x + a)^n*a^6*b^3*d^2*n^2*x^3 - 30240*(b*x + a)^n*a^2*b^7*c*d*n*x^4 + 10080*(b*x + a)^n*a^5*b^4*d
^2*n*x^4 + 1250*(b*x + a)^n*a^3*b^6*c^2*n^4 - 49148*(b*x + a)^n*a^2*b^7*c^2*n^3*x + 5760*(b*x + a)^n*a^5*b^4*c
*d*n^3*x + 120696*(b*x + a)^n*a*b^8*c^2*n^2*x^2 - 83400*(b*x + a)^n*a^4*b^5*c*d*n^2*x^2 + 20160*(b*x + a)^n*a^
7*b^2*d^2*n^2*x^2 + 301872*(b*x + a)^n*b^9*c^2*n*x^3 + 40320*(b*x + a)^n*a^3*b^6*c*d*n*x^3 - 13440*(b*x + a)^n
*a^6*b^3*d^2*n*x^3 + 10530*(b*x + a)^n*a^3*b^6*c^2*n^3 - 240*(b*x + a)^n*a^6*b^3*c*d*n^3 - 120432*(b*x + a)^n*
a^2*b^7*c^2*n^2*x + 45840*(b*x + a)^n*a^5*b^4*c*d*n^2*x + 60480*(b*x + a)^n*a*b^8*c^2*n*x^2 - 60480*(b*x + a)^
n*a^4*b^5*c*d*n*x^2 + 20160*(b*x + a)^n*a^7*b^2*d^2*n*x^2 + 120960*(b*x + a)^n*b^9*c^2*x^3 + 49148*(b*x + a)^n
*a^3*b^6*c^2*n^2 - 5760*(b*x + a)^n*a^6*b^3*c*d*n^2 - 120960*(b*x + a)^n*a^2*b^7*c^2*n*x + 120960*(b*x + a)^n*
a^5*b^4*c*d*n*x - 40320*(b*x + a)^n*a^8*b*d^2*n*x + 120432*(b*x + a)^n*a^3*b^6*c^2*n - 45840*(b*x + a)^n*a^6*b
^3*c*d*n + 120960*(b*x + a)^n*a^3*b^6*c^2 - 120960*(b*x + a)^n*a^6*b^3*c*d + 40320*(b*x + a)^n*a^9*d^2)/(b^9*n
^9 + 45*b^9*n^8 + 870*b^9*n^7 + 9450*b^9*n^6 + 63273*b^9*n^5 + 269325*b^9*n^4 + 723680*b^9*n^3 + 1172700*b^9*n
^2 + 1026576*b^9*n + 362880*b^9)