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

Optimal. Leaf size=459 \[ \frac{\left (b^3 c-a^3 d\right ) \left (-29 a^3 b^3 c d+55 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+3}}{b^{12} (n+3)}+\frac{3 a^2 d \left (-56 a^3 b^3 c d+55 a^6 d^2+10 b^6 c^2\right ) (a+b x)^{n+4}}{b^{12} (n+4)}-\frac{15 a d \left (-14 a^3 b^3 c d+22 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+5}}{b^{12} (n+5)}+\frac{3 d \left (-56 a^3 b^3 c d+154 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+6}}{b^{12} (n+6)}+\frac{42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^{n+7}}{b^{12} (n+7)}-\frac{6 a d^2 \left (4 b^3 c-55 a^3 d\right ) (a+b x)^{n+8}}{b^{12} (n+8)}+\frac{3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{n+9}}{b^{12} (n+9)}+\frac{a^2 \left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{12} (n+1)}-\frac{a \left (2 b^3 c-11 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{12} (n+2)}+\frac{55 a^2 d^3 (a+b x)^{n+10}}{b^{12} (n+10)}-\frac{11 a d^3 (a+b x)^{n+11}}{b^{12} (n+11)}+\frac{d^3 (a+b x)^{n+12}}{b^{12} (n+12)} \]

[Out]

(a^2*(b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^12*(1 + n)) - (a*(2*b^3*c - 11*a^3*d)*(b^3*c - a^3*d)^2*(a + b*x)
^(2 + n))/(b^12*(2 + n)) + ((b^3*c - a^3*d)*(b^6*c^2 - 29*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^(3 + n))/(b^12*(
3 + n)) + (3*a^2*d*(10*b^6*c^2 - 56*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^(4 + n))/(b^12*(4 + n)) - (15*a*d*(b^6
*c^2 - 14*a^3*b^3*c*d + 22*a^6*d^2)*(a + b*x)^(5 + n))/(b^12*(5 + n)) + (3*d*(b^6*c^2 - 56*a^3*b^3*c*d + 154*a
^6*d^2)*(a + b*x)^(6 + n))/(b^12*(6 + n)) + (42*a^2*d^2*(2*b^3*c - 11*a^3*d)*(a + b*x)^(7 + n))/(b^12*(7 + n))
- (6*a*d^2*(4*b^3*c - 55*a^3*d)*(a + b*x)^(8 + n))/(b^12*(8 + n)) + (3*d^2*(b^3*c - 55*a^3*d)*(a + b*x)^(9 +
n))/(b^12*(9 + n)) + (55*a^2*d^3*(a + b*x)^(10 + n))/(b^12*(10 + n)) - (11*a*d^3*(a + b*x)^(11 + n))/(b^12*(11
+ n)) + (d^3*(a + b*x)^(12 + n))/(b^12*(12 + n))

________________________________________________________________________________________

Rubi [A]  time = 0.316699, antiderivative size = 459, 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} \[ \frac{\left (b^3 c-a^3 d\right ) \left (-29 a^3 b^3 c d+55 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+3}}{b^{12} (n+3)}+\frac{3 a^2 d \left (-56 a^3 b^3 c d+55 a^6 d^2+10 b^6 c^2\right ) (a+b x)^{n+4}}{b^{12} (n+4)}-\frac{15 a d \left (-14 a^3 b^3 c d+22 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+5}}{b^{12} (n+5)}+\frac{3 d \left (-56 a^3 b^3 c d+154 a^6 d^2+b^6 c^2\right ) (a+b x)^{n+6}}{b^{12} (n+6)}+\frac{42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^{n+7}}{b^{12} (n+7)}-\frac{6 a d^2 \left (4 b^3 c-55 a^3 d\right ) (a+b x)^{n+8}}{b^{12} (n+8)}+\frac{3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{n+9}}{b^{12} (n+9)}+\frac{a^2 \left (b^3 c-a^3 d\right )^3 (a+b x)^{n+1}}{b^{12} (n+1)}-\frac{a \left (2 b^3 c-11 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+2}}{b^{12} (n+2)}+\frac{55 a^2 d^3 (a+b x)^{n+10}}{b^{12} (n+10)}-\frac{11 a d^3 (a+b x)^{n+11}}{b^{12} (n+11)}+\frac{d^3 (a+b x)^{n+12}}{b^{12} (n+12)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^n*(c + d*x^3)^3,x]

[Out]

(a^2*(b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^12*(1 + n)) - (a*(2*b^3*c - 11*a^3*d)*(b^3*c - a^3*d)^2*(a + b*x)
^(2 + n))/(b^12*(2 + n)) + ((b^3*c - a^3*d)*(b^6*c^2 - 29*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^(3 + n))/(b^12*(
3 + n)) + (3*a^2*d*(10*b^6*c^2 - 56*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^(4 + n))/(b^12*(4 + n)) - (15*a*d*(b^6
*c^2 - 14*a^3*b^3*c*d + 22*a^6*d^2)*(a + b*x)^(5 + n))/(b^12*(5 + n)) + (3*d*(b^6*c^2 - 56*a^3*b^3*c*d + 154*a
^6*d^2)*(a + b*x)^(6 + n))/(b^12*(6 + n)) + (42*a^2*d^2*(2*b^3*c - 11*a^3*d)*(a + b*x)^(7 + n))/(b^12*(7 + n))
- (6*a*d^2*(4*b^3*c - 55*a^3*d)*(a + b*x)^(8 + n))/(b^12*(8 + n)) + (3*d^2*(b^3*c - 55*a^3*d)*(a + b*x)^(9 +
n))/(b^12*(9 + n)) + (55*a^2*d^3*(a + b*x)^(10 + n))/(b^12*(10 + n)) - (11*a*d^3*(a + b*x)^(11 + n))/(b^12*(11
+ n)) + (d^3*(a + b*x)^(12 + n))/(b^12*(12 + 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 )^3 \, dx &=\int \left (-\frac{a^2 \left (-b^3 c+a^3 d\right )^3 (a+b x)^n}{b^{11}}+\frac{a \left (-b^3 c+a^3 d\right )^2 \left (-2 b^3 c+11 a^3 d\right ) (a+b x)^{1+n}}{b^{11}}+\frac{\left (b^3 c-a^3 d\right ) \left (b^6 c^2-29 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{2+n}}{b^{11}}+\frac{3 a^2 d \left (10 b^6 c^2-56 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{3+n}}{b^{11}}-\frac{15 a d \left (b^6 c^2-14 a^3 b^3 c d+22 a^6 d^2\right ) (a+b x)^{4+n}}{b^{11}}+\frac{3 d \left (b^6 c^2-56 a^3 b^3 c d+154 a^6 d^2\right ) (a+b x)^{5+n}}{b^{11}}-\frac{42 a^2 d^2 \left (-2 b^3 c+11 a^3 d\right ) (a+b x)^{6+n}}{b^{11}}+\frac{6 a d^2 \left (-4 b^3 c+55 a^3 d\right ) (a+b x)^{7+n}}{b^{11}}+\frac{3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{8+n}}{b^{11}}+\frac{55 a^2 d^3 (a+b x)^{9+n}}{b^{11}}-\frac{11 a d^3 (a+b x)^{10+n}}{b^{11}}+\frac{d^3 (a+b x)^{11+n}}{b^{11}}\right ) \, dx\\ &=\frac{a^2 \left (b^3 c-a^3 d\right )^3 (a+b x)^{1+n}}{b^{12} (1+n)}-\frac{a \left (2 b^3 c-11 a^3 d\right ) \left (b^3 c-a^3 d\right )^2 (a+b x)^{2+n}}{b^{12} (2+n)}+\frac{\left (b^3 c-a^3 d\right ) \left (b^6 c^2-29 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{3+n}}{b^{12} (3+n)}+\frac{3 a^2 d \left (10 b^6 c^2-56 a^3 b^3 c d+55 a^6 d^2\right ) (a+b x)^{4+n}}{b^{12} (4+n)}-\frac{15 a d \left (b^6 c^2-14 a^3 b^3 c d+22 a^6 d^2\right ) (a+b x)^{5+n}}{b^{12} (5+n)}+\frac{3 d \left (b^6 c^2-56 a^3 b^3 c d+154 a^6 d^2\right ) (a+b x)^{6+n}}{b^{12} (6+n)}+\frac{42 a^2 d^2 \left (2 b^3 c-11 a^3 d\right ) (a+b x)^{7+n}}{b^{12} (7+n)}-\frac{6 a d^2 \left (4 b^3 c-55 a^3 d\right ) (a+b x)^{8+n}}{b^{12} (8+n)}+\frac{3 d^2 \left (b^3 c-55 a^3 d\right ) (a+b x)^{9+n}}{b^{12} (9+n)}+\frac{55 a^2 d^3 (a+b x)^{10+n}}{b^{12} (10+n)}-\frac{11 a d^3 (a+b x)^{11+n}}{b^{12} (11+n)}+\frac{d^3 (a+b x)^{12+n}}{b^{12} (12+n)}\\ \end{align*}

Mathematica [A]  time = 0.469966, size = 402, normalized size = 0.88 \[ \frac{(a+b x)^{n+1} \left (\frac{3 d (a+b x)^5 \left (-56 a^3 b^3 c d+154 a^6 d^2+b^6 c^2\right )}{n+6}-\frac{15 a d (a+b x)^4 \left (-14 a^3 b^3 c d+22 a^6 d^2+b^6 c^2\right )}{n+5}+\frac{3 a^2 d (a+b x)^3 \left (-56 a^3 b^3 c d+55 a^6 d^2+10 b^6 c^2\right )}{n+4}+\frac{(a+b x)^2 \left (b^3 c-a^3 d\right ) \left (-29 a^3 b^3 c d+55 a^6 d^2+b^6 c^2\right )}{n+3}+\frac{3 d^2 (a+b x)^8 \left (b^3 c-55 a^3 d\right )}{n+9}+\frac{6 a d^2 (a+b x)^7 \left (55 a^3 d-4 b^3 c\right )}{n+8}+\frac{42 a^2 d^2 (a+b x)^6 \left (2 b^3 c-11 a^3 d\right )}{n+7}+\frac{a (a+b x) \left (b^3 c-a^3 d\right )^2 \left (11 a^3 d-2 b^3 c\right )}{n+2}+\frac{a^2 \left (b^3 c-a^3 d\right )^3}{n+1}+\frac{55 a^2 d^3 (a+b x)^9}{n+10}+\frac{d^3 (a+b x)^{11}}{n+12}-\frac{11 a d^3 (a+b x)^{10}}{n+11}\right )}{b^{12}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + n)*((a^2*(b^3*c - a^3*d)^3)/(1 + n) + (a*(b^3*c - a^3*d)^2*(-2*b^3*c + 11*a^3*d)*(a + b*x))/(2
+ n) + ((b^3*c - a^3*d)*(b^6*c^2 - 29*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^2)/(3 + n) + (3*a^2*d*(10*b^6*c^2 -
56*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^3)/(4 + n) - (15*a*d*(b^6*c^2 - 14*a^3*b^3*c*d + 22*a^6*d^2)*(a + b*x)
^4)/(5 + n) + (3*d*(b^6*c^2 - 56*a^3*b^3*c*d + 154*a^6*d^2)*(a + b*x)^5)/(6 + n) + (42*a^2*d^2*(2*b^3*c - 11*a
^3*d)*(a + b*x)^6)/(7 + n) + (6*a*d^2*(-4*b^3*c + 55*a^3*d)*(a + b*x)^7)/(8 + n) + (3*d^2*(b^3*c - 55*a^3*d)*(
a + b*x)^8)/(9 + n) + (55*a^2*d^3*(a + b*x)^9)/(10 + n) - (11*a*d^3*(a + b*x)^10)/(11 + n) + (d^3*(a + b*x)^11
)/(12 + n)))/b^12

________________________________________________________________________________________

Maple [B]  time = 0.027, size = 3780, normalized size = 8.2 \begin{align*} \text{output 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)^3,x)

[Out]

-(b*x+a)^(1+n)*(-b^11*d^3*n^11*x^11-66*b^11*d^3*n^10*x^11+11*a*b^10*d^3*n^10*x^10-1925*b^11*d^3*n^9*x^11+605*a
*b^10*d^3*n^9*x^10-3*b^11*c*d^2*n^11*x^8-32670*b^11*d^3*n^8*x^11-110*a^2*b^9*d^3*n^9*x^9+14520*a*b^10*d^3*n^8*
x^10-207*b^11*c*d^2*n^10*x^8-357423*b^11*d^3*n^7*x^11-4950*a^2*b^9*d^3*n^8*x^9+24*a*b^10*c*d^2*n^10*x^7+199650
*a*b^10*d^3*n^7*x^10-6288*b^11*c*d^2*n^9*x^8-2637558*b^11*d^3*n^6*x^11+990*a^3*b^8*d^3*n^8*x^8-95700*a^2*b^9*d
^3*n^7*x^9+1464*a*b^10*c*d^2*n^9*x^7+1735503*a*b^10*d^3*n^6*x^10-3*b^11*c^2*d*n^11*x^5-110718*b^11*c*d^2*n^8*x
^8-13339535*b^11*d^3*n^5*x^11+35640*a^3*b^8*d^3*n^7*x^8-168*a^2*b^9*c*d^2*n^9*x^6-1039500*a^2*b^9*d^3*n^6*x^9+
38592*a*b^10*c*d^2*n^8*x^7+9922605*a*b^10*d^3*n^5*x^10-216*b^11*c^2*d*n^10*x^5-1251927*b^11*c*d^2*n^7*x^8-4599
5730*b^11*d^3*n^4*x^11-7920*a^4*b^7*d^3*n^7*x^7+540540*a^3*b^8*d^3*n^6*x^8-9072*a^2*b^9*c*d^2*n^8*x^6-6960030*
a^2*b^9*d^3*n^5*x^9+15*a*b^10*c^2*d*n^10*x^4+577008*a*b^10*c*d^2*n^7*x^7+37586230*a*b^10*d^3*n^4*x^10-6855*b^1
1*c^2*d*n^9*x^5-9512559*b^11*c*d^2*n^6*x^8-105258076*b^11*d^3*n^3*x^11-221760*a^4*b^7*d^3*n^6*x^7+1008*a^3*b^8
*c*d^2*n^8*x^5+4490640*a^3*b^8*d^3*n^5*x^8-206640*a^2*b^9*c*d^2*n^7*x^6-29625750*a^2*b^9*d^3*n^4*x^9+1005*a*b^
10*c^2*d*n^9*x^4+5399352*a*b^10*c*d^2*n^6*x^7+92504500*a*b^10*d^3*n^3*x^10-b^11*c^3*n^11*x^2-126180*b^11*c^2*d
*n^8*x^5-49357662*b^11*c*d^2*n^5*x^8-150917976*b^11*d^3*n^2*x^11+55440*a^5*b^6*d^3*n^6*x^6-2550240*a^4*b^7*d^3
*n^5*x^7+48384*a^3*b^8*c*d^2*n^7*x^5+22224510*a^3*b^8*d^3*n^4*x^8-60*a^2*b^9*c^2*d*n^9*x^3-2592576*a^2*b^9*c*d
^2*n^6*x^6-79604800*a^2*b^9*d^3*n^3*x^9+29250*a*b^10*c^2*d*n^8*x^4+32905656*a*b^10*c*d^2*n^5*x^7+140289336*a*b
^10*d^3*n^2*x^10-75*b^11*c^3*n^10*x^2-1491309*b^11*c^2*d*n^7*x^5-173991492*b^11*c*d^2*n^4*x^8-120543840*b^11*d
^3*n*x^11+1164240*a^5*b^6*d^3*n^5*x^6-5040*a^4*b^7*c*d^2*n^7*x^4-15523200*a^4*b^7*d^3*n^4*x^7+949536*a^3*b^8*c
*d^2*n^6*x^5+66611160*a^3*b^8*d^3*n^3*x^8-3780*a^2*b^9*c^2*d*n^8*x^3-19647432*a^2*b^9*c*d^2*n^5*x^6-128997000*
a^2*b^9*d^3*n^2*x^9+2*a*b^10*c^3*n^10*x+484650*a*b^10*c^2*d*n^7*x^4+131616048*a*b^10*c*d^2*n^4*x^7+116915040*a
*b^10*d^3*n*x^10-2492*b^11*c^3*n^9*x^2-11832048*b^11*c^2*d*n^6*x^5-405697080*b^11*c*d^2*n^3*x^8-39916800*b^11*
d^3*x^11-332640*a^6*b^5*d^3*n^5*x^5+9702000*a^5*b^6*d^3*n^4*x^6-216720*a^4*b^7*c*d^2*n^6*x^4-53610480*a^4*b^7*
d^3*n^3*x^7+180*a^3*b^8*c^2*d*n^8*x^2+9858240*a^3*b^8*c*d^2*n^5*x^5+116942760*a^3*b^8*d^3*n^2*x^8-101880*a^2*b
^9*c^2*d*n^7*x^3-92807568*a^2*b^9*c*d^2*n^4*x^6-112923360*a^2*b^9*d^3*n*x^9+146*a*b^10*c^3*n^9*x+5033295*a*b^1
0*c^2*d*n^6*x^4+339003552*a*b^10*c*d^2*n^3*x^7+39916800*a*b^10*d^3*x^10-48294*b^11*c^3*n^8*x^2-63978405*b^11*c
^2*d*n^5*x^5-590770944*b^11*c*d^2*n^2*x^8-4989600*a^6*b^5*d^3*n^4*x^5+20160*a^5*b^6*c*d^2*n^6*x^3+40748400*a^5
*b^6*d^3*n^3*x^6-3664080*a^4*b^7*c*d^2*n^5*x^4-104005440*a^4*b^7*d^3*n^2*x^7+10800*a^3*b^8*c^2*d*n^7*x^2+58735
152*a^3*b^8*c*d^2*n^4*x^5+108488160*a^3*b^8*d^3*n*x^8-2*a^2*b^9*c^3*n^9-1531080*a^2*b^9*c^2*d*n^6*x^3-27165936
0*a^2*b^9*c*d^2*n^3*x^6-39916800*a^2*b^9*d^3*x^9+4692*a*b^10*c^3*n^8*x+33993765*a*b^10*c^2*d*n^5*x^4+533548224
*a*b^10*c*d^2*n^2*x^7-604581*b^11*c^3*n^7*x^2-234340020*b^11*c^2*d*n^4*x^5-477740160*b^11*c*d^2*n*x^8+1663200*
a^7*b^4*d^3*n^4*x^4-28274400*a^6*b^5*d^3*n^3*x^5+786240*a^5*b^6*c*d^2*n^5*x^3+90034560*a^5*b^6*d^3*n^2*x^6-360
*a^4*b^7*c^2*d*n^7*x-30970800*a^4*b^7*c*d^2*n^4*x^4-103498560*a^4*b^7*d^3*n*x^7+273240*a^3*b^8*c^2*d*n^6*x^2+2
04434496*a^3*b^8*c*d^2*n^3*x^5+39916800*a^3*b^8*d^3*x^8-144*a^2*b^9*c^3*n^8-14008860*a^2*b^9*c^2*d*n^5*x^3-471
409344*a^2*b^9*c*d^2*n^2*x^6+87204*a*b^10*c^3*n^7*x+149923200*a*b^10*c^2*d*n^4*x^4+457781760*a*b^10*c*d^2*n*x^
7-5112891*b^11*c^3*n^6*x^2-565580388*b^11*c^2*d*n^3*x^5-159667200*b^11*c*d^2*x^8+16632000*a^7*b^4*d^3*n^3*x^4-
60480*a^6*b^5*c*d^2*n^5*x^2-74844000*a^6*b^5*d^3*n^2*x^5+11511360*a^5*b^6*c*d^2*n^4*x^3+97796160*a^5*b^6*d^3*n
*x^6-20880*a^4*b^7*c^2*d*n^6*x-138821760*a^4*b^7*c*d^2*n^3*x^4-39916800*a^4*b^7*d^3*x^7+3773520*a^3*b^8*c^2*d*
n^5*x^2+403349184*a^3*b^8*c*d^2*n^2*x^5-4548*a^2*b^9*c^3*n^7-79939620*a^2*b^9*c^2*d*n^4*x^3-434972160*a^2*b^9*
c*d^2*n*x^6+1034754*a*b^10*c^3*n^6*x+422084100*a*b^10*c^2*d*n^3*x^4+159667200*a*b^10*c*d^2*x^7-29651558*b^11*c
^3*n^5*x^2-848562336*b^11*c^2*d*n^2*x^5-6652800*a^8*b^3*d^3*n^3*x^3+58212000*a^7*b^4*d^3*n^2*x^4-2177280*a^6*b
^5*c*d^2*n^4*x^2-91143360*a^6*b^5*d^3*n*x^5+360*a^5*b^6*c^2*d*n^6+77837760*a^5*b^6*c*d^2*n^3*x^3+39916800*a^5*
b^6*d^3*x^6-504720*a^4*b^7*c^2*d*n^5*x-328063680*a^4*b^7*c*d^2*n^2*x^4+30706020*a^3*b^8*c^2*d*n^4*x^2+40836096
0*a^3*b^8*c*d^2*n*x^5-82656*a^2*b^9*c^3*n^6-279934320*a^2*b^9*c^2*d*n^3*x^3-159667200*a^2*b^9*c*d^2*x^6+815627
4*a*b^10*c^3*n^5*x+717481440*a*b^10*c^2*d*n^2*x^4-117115476*b^11*c^3*n^4*x^2-703304640*b^11*c^2*d*n*x^5-399168
00*a^8*b^3*d^3*n^2*x^3+120960*a^7*b^4*c*d^2*n^4*x+83160000*a^7*b^4*d^3*n*x^4-28002240*a^6*b^5*c*d^2*n^3*x^2-39
916800*a^6*b^5*d^3*x^5+20520*a^5*b^6*c^2*d*n^5+243936000*a^5*b^6*c*d^2*n^2*x^3-6537600*a^4*b^7*c^2*d*n^4*x-376
427520*a^4*b^7*c*d^2*n*x^4+147700800*a^3*b^8*c^2*d*n^3*x^2+159667200*a^3*b^8*c*d^2*x^5-952098*a^2*b^9*c^3*n^5-
568599120*a^2*b^9*c^2*d*n^2*x^3+42990568*a*b^10*c^3*n^4*x+655404480*a*b^10*c^2*d*n*x^4-305860408*b^11*c^3*n^3*
x^2-239500800*b^11*c^2*d*x^5+19958400*a^9*b^2*d^3*n^2*x^2-73180800*a^8*b^3*d^3*n*x^3+4112640*a^7*b^4*c*d^2*n^3
*x+39916800*a^7*b^4*d^3*x^4-149506560*a^6*b^5*c*d^2*n^2*x^2+484200*a^5*b^6*c^2*d*n^4+336510720*a^5*b^6*c*d^2*n
*x^3-48336840*a^4*b^7*c^2*d*n^3*x-159667200*a^4*b^7*c*d^2*x^4+396700560*a^3*b^8*c^2*d*n^2*x^2-7204176*a^2*b^9*
c^3*n^4-595529280*a^2*b^9*c^2*d*n*x^3+148249816*a*b^10*c^3*n^3*x+239500800*a*b^10*c^2*d*x^4-496433664*b^11*c^3
*n^2*x^2+59875200*a^9*b^2*d^3*n*x^2-120960*a^8*b^3*c*d^2*n^3-39916800*a^8*b^3*d^3*x^3+47779200*a^7*b^4*c*d^2*n
^2*x-283288320*a^6*b^5*c*d^2*n*x^2+6053400*a^5*b^6*c^2*d*n^3+159667200*a^5*b^6*c*d^2*x^3-198727920*a^4*b^7*c^2
*d*n^2*x+515695680*a^3*b^8*c^2*d*n*x^2-35786392*a^2*b^9*c^3*n^3-239500800*a^2*b^9*c^2*d*x^3+315221184*a*b^10*c
^3*n^2*x-442258560*b^11*c^3*n*x^2-39916800*a^10*b*d^3*n*x+39916800*a^9*b^2*d^3*x^2-3991680*a^8*b^3*c*d^2*n^2+2
03454720*a^7*b^4*c*d^2*n*x-159667200*a^6*b^5*c*d^2*x^2+42283440*a^5*b^6*c^2*d*n^2-395945280*a^4*b^7*c^2*d*n*x+
239500800*a^3*b^8*c^2*d*x^2-112463424*a^2*b^9*c^3*n^2+362424960*a*b^10*c^3*n*x-159667200*b^11*c^3*x^2-39916800
*a^10*b*d^3*x-43787520*a^8*b^3*c*d^2*n+159667200*a^7*b^4*c*d^2*x+156444480*a^5*b^6*c^2*d*n-239500800*a^4*b^7*c
^2*d*x-202757760*a^2*b^9*c^3*n+159667200*a*b^10*c^3*x+39916800*a^11*d^3-159667200*a^8*b^3*c*d^2+239500800*a^5*
b^6*c^2*d-159667200*a^2*b^9*c^3)/b^12/(n^12+78*n^11+2717*n^10+55770*n^9+749463*n^8+6926634*n^7+44990231*n^6+20
6070150*n^5+657206836*n^4+1414014888*n^3+1931559552*n^2+1486442880*n+479001600)

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Maxima [B]  time = 1.15203, size = 1557, normalized size = 3.39 \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)^3,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^3/((n^3 + 6*n^2 + 11*n + 6
)*b^3) + 3*((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^2*d/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n +
720)*b^6) + 3*((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 + 2
1*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 + 2
74*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
+ 40320*a^9)*(b*x + a)^n*c*d^2/((n^9 + 45*n^8 + 870*n^7 + 9450*n^6 + 63273*n^5 + 269325*n^4 + 723680*n^3 + 11
72700*n^2 + 1026576*n + 362880)*b^9) + ((n^11 + 66*n^10 + 1925*n^9 + 32670*n^8 + 357423*n^7 + 2637558*n^6 + 13
339535*n^5 + 45995730*n^4 + 105258076*n^3 + 150917976*n^2 + 120543840*n + 39916800)*b^12*x^12 + (n^11 + 55*n^1
0 + 1320*n^9 + 18150*n^8 + 157773*n^7 + 902055*n^6 + 3416930*n^5 + 8409500*n^4 + 12753576*n^3 + 10628640*n^2 +
3628800*n)*a*b^11*x^11 - 11*(n^10 + 45*n^9 + 870*n^8 + 9450*n^7 + 63273*n^6 + 269325*n^5 + 723680*n^4 + 11727
00*n^3 + 1026576*n^2 + 362880*n)*a^2*b^10*x^10 + 110*(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^3*b^9*x^9 - 990*(n^8 + 28*n^7 + 322*n^6 + 1960*n^5 + 6769*n^4 + 13132
*n^3 + 13068*n^2 + 5040*n)*a^4*b^8*x^8 + 7920*(n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)
*a^5*b^7*x^7 - 55440*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a^6*b^6*x^6 + 332640*(n^5 + 10*n^4 +
35*n^3 + 50*n^2 + 24*n)*a^7*b^5*x^5 - 1663200*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^8*b^4*x^4 + 6652800*(n^3 + 3*n^2
+ 2*n)*a^9*b^3*x^3 - 19958400*(n^2 + n)*a^10*b^2*x^2 + 39916800*a^11*b*n*x - 39916800*a^12)*(b*x + a)^n*d^3/((
n^12 + 78*n^11 + 2717*n^10 + 55770*n^9 + 749463*n^8 + 6926634*n^7 + 44990231*n^6 + 206070150*n^5 + 657206836*n
^4 + 1414014888*n^3 + 1931559552*n^2 + 1486442880*n + 479001600)*b^12)

________________________________________________________________________________________

Fricas [B]  time = 1.04659, size = 8852, normalized size = 19.29 \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)^3,x, algorithm="fricas")

[Out]

(2*a^3*b^9*c^3*n^9 + 144*a^3*b^9*c^3*n^8 + 4548*a^3*b^9*c^3*n^7 + 159667200*a^3*b^9*c^3 - 239500800*a^6*b^6*c^
2*d + 159667200*a^9*b^3*c*d^2 - 39916800*a^12*d^3 + (b^12*d^3*n^11 + 66*b^12*d^3*n^10 + 1925*b^12*d^3*n^9 + 32
670*b^12*d^3*n^8 + 357423*b^12*d^3*n^7 + 2637558*b^12*d^3*n^6 + 13339535*b^12*d^3*n^5 + 45995730*b^12*d^3*n^4
+ 105258076*b^12*d^3*n^3 + 150917976*b^12*d^3*n^2 + 120543840*b^12*d^3*n + 39916800*b^12*d^3)*x^12 + (a*b^11*d
^3*n^11 + 55*a*b^11*d^3*n^10 + 1320*a*b^11*d^3*n^9 + 18150*a*b^11*d^3*n^8 + 157773*a*b^11*d^3*n^7 + 902055*a*b
^11*d^3*n^6 + 3416930*a*b^11*d^3*n^5 + 8409500*a*b^11*d^3*n^4 + 12753576*a*b^11*d^3*n^3 + 10628640*a*b^11*d^3*
n^2 + 3628800*a*b^11*d^3*n)*x^11 - 11*(a^2*b^10*d^3*n^10 + 45*a^2*b^10*d^3*n^9 + 870*a^2*b^10*d^3*n^8 + 9450*a
^2*b^10*d^3*n^7 + 63273*a^2*b^10*d^3*n^6 + 269325*a^2*b^10*d^3*n^5 + 723680*a^2*b^10*d^3*n^4 + 1172700*a^2*b^1
0*d^3*n^3 + 1026576*a^2*b^10*d^3*n^2 + 362880*a^2*b^10*d^3*n)*x^10 + (3*b^12*c*d^2*n^11 + 207*b^12*c*d^2*n^10
+ 159667200*b^12*c*d^2 + 2*(3144*b^12*c*d^2 + 55*a^3*b^9*d^3)*n^9 + 18*(6151*b^12*c*d^2 + 220*a^3*b^9*d^3)*n^8
+ 3*(417309*b^12*c*d^2 + 20020*a^3*b^9*d^3)*n^7 + 567*(16777*b^12*c*d^2 + 880*a^3*b^9*d^3)*n^6 + 6*(8226277*b
^12*c*d^2 + 411565*a^3*b^9*d^3)*n^5 + 36*(4833097*b^12*c*d^2 + 205590*a^3*b^9*d^3)*n^4 + 40*(10142427*b^12*c*d
^2 + 324841*a^3*b^9*d^3)*n^3 + 288*(2051288*b^12*c*d^2 + 41855*a^3*b^9*d^3)*n^2 + 5760*(82941*b^12*c*d^2 + 770
*a^3*b^9*d^3)*n)*x^9 + 3*(a*b^11*c*d^2*n^11 + 61*a*b^11*c*d^2*n^10 + 1608*a*b^11*c*d^2*n^9 + 6*(4007*a*b^11*c*
d^2 - 55*a^4*b^8*d^3)*n^8 + 21*(10713*a*b^11*c*d^2 - 440*a^4*b^8*d^3)*n^7 + 21*(65289*a*b^11*c*d^2 - 5060*a^4*
b^8*d^3)*n^6 + 2*(2742001*a*b^11*c*d^2 - 323400*a^4*b^8*d^3)*n^5 + 2*(7062574*a*b^11*c*d^2 - 1116885*a^4*b^8*d
^3)*n^4 + 264*(84209*a*b^11*c*d^2 - 16415*a^4*b^8*d^3)*n^3 + 360*(52984*a*b^11*c*d^2 - 11979*a^4*b^8*d^3)*n^2
+ 1663200*(4*a*b^11*c*d^2 - a^4*b^8*d^3)*n)*x^8 - 24*(a^2*b^10*c*d^2*n^10 + 54*a^2*b^10*c*d^2*n^9 + 1230*a^2*b
^10*c*d^2*n^8 + 6*(2572*a^2*b^10*c*d^2 - 55*a^5*b^7*d^3)*n^7 + 21*(5569*a^2*b^10*c*d^2 - 330*a^5*b^7*d^3)*n^6
+ 42*(13153*a^2*b^10*c*d^2 - 1375*a^5*b^7*d^3)*n^5 + 10*(161702*a^2*b^10*c*d^2 - 24255*a^5*b^7*d^3)*n^4 + 24*(
116917*a^2*b^10*c*d^2 - 22330*a^5*b^7*d^3)*n^3 + 360*(7192*a^2*b^10*c*d^2 - 1617*a^5*b^7*d^3)*n^2 + 237600*(4*
a^2*b^10*c*d^2 - a^5*b^7*d^3)*n)*x^7 + 72*(1148*a^3*b^9*c^3 - 5*a^6*b^6*c^2*d)*n^6 + 3*(b^12*c^2*d*n^11 + 72*b
^12*c^2*d*n^10 + 79833600*b^12*c^2*d + (2285*b^12*c^2*d + 56*a^3*b^9*c*d^2)*n^9 + 12*(3505*b^12*c^2*d + 224*a^
3*b^9*c*d^2)*n^8 + 3*(165701*b^12*c^2*d + 17584*a^3*b^9*c*d^2)*n^7 + 48*(82167*b^12*c^2*d + 11410*a^3*b^9*c*d^
2 - 385*a^6*b^6*d^3)*n^6 + (21326135*b^12*c^2*d + 3263064*a^3*b^9*c*d^2 - 277200*a^6*b^6*d^3)*n^5 + 12*(650944
5*b^12*c^2*d + 946456*a^3*b^9*c*d^2 - 130900*a^6*b^6*d^3)*n^4 + 4*(47131699*b^12*c^2*d + 5602072*a^3*b^9*c*d^2
- 1039500*a^6*b^6*d^3)*n^3 + 96*(2946397*b^12*c^2*d + 236320*a^3*b^9*c*d^2 - 52745*a^6*b^6*d^3)*n^2 + 2880*(8
1401*b^12*c^2*d + 3080*a^3*b^9*c*d^2 - 770*a^6*b^6*d^3)*n)*x^6 + 6*(158683*a^3*b^9*c^3 - 3420*a^6*b^6*c^2*d)*n
^5 + 3*(a*b^11*c^2*d*n^11 + 67*a*b^11*c^2*d*n^10 + 1950*a*b^11*c^2*d*n^9 + 6*(5385*a*b^11*c^2*d - 56*a^4*b^8*c
*d^2)*n^8 + 3*(111851*a*b^11*c^2*d - 4816*a^4*b^8*c*d^2)*n^7 + 3*(755417*a*b^11*c^2*d - 81424*a^4*b^8*c*d^2)*n
^6 + 560*(17848*a*b^11*c^2*d - 3687*a^4*b^8*c*d^2 + 198*a^7*b^5*d^3)*n^5 + 4*(7034735*a*b^11*c^2*d - 2313696*a
^4*b^8*c*d^2 + 277200*a^7*b^5*d^3)*n^4 + 96*(498251*a*b^11*c^2*d - 227822*a^4*b^8*c*d^2 + 40425*a^7*b^5*d^3)*n
^3 + 576*(75857*a*b^11*c^2*d - 43568*a^4*b^8*c*d^2 + 9625*a^7*b^5*d^3)*n^2 + 2661120*(6*a*b^11*c^2*d - 4*a^4*b
^8*c*d^2 + a^7*b^5*d^3)*n)*x^5 + 72*(100058*a^3*b^9*c^3 - 6725*a^6*b^6*c^2*d)*n^4 - 15*(a^2*b^10*c^2*d*n^10 +
63*a^2*b^10*c^2*d*n^9 + 1698*a^2*b^10*c^2*d*n^8 + 6*(4253*a^2*b^10*c^2*d - 56*a^5*b^7*c*d^2)*n^7 + 3*(77827*a^
2*b^10*c^2*d - 4368*a^5*b^7*c*d^2)*n^6 + 3*(444109*a^2*b^10*c^2*d - 63952*a^5*b^7*c*d^2)*n^5 + 4*(1166393*a^2*
b^10*c^2*d - 324324*a^5*b^7*c*d^2 + 27720*a^8*b^4*d^3)*n^4 + 12*(789721*a^2*b^10*c^2*d - 338800*a^5*b^7*c*d^2
+ 55440*a^8*b^4*d^3)*n^3 + 144*(68927*a^2*b^10*c^2*d - 38948*a^5*b^7*c*d^2 + 8470*a^8*b^4*d^3)*n^2 + 665280*(6
*a^2*b^10*c^2*d - 4*a^5*b^7*c*d^2 + a^8*b^4*d^3)*n)*x^4 + 8*(4473299*a^3*b^9*c^3 - 756675*a^6*b^6*c^2*d + 1512
0*a^9*b^3*c*d^2)*n^3 + (b^12*c^3*n^11 + 75*b^12*c^3*n^10 + 159667200*b^12*c^3 + 4*(623*b^12*c^3 + 15*a^3*b^9*c
^2*d)*n^9 + 18*(2683*b^12*c^3 + 200*a^3*b^9*c^2*d)*n^8 + 3*(201527*b^12*c^3 + 30360*a^3*b^9*c^2*d)*n^7 + 9*(56
8099*b^12*c^3 + 139760*a^3*b^9*c^2*d - 2240*a^6*b^6*c*d^2)*n^6 + 2*(14825779*b^12*c^3 + 5117670*a^3*b^9*c^2*d
- 362880*a^6*b^6*c*d^2)*n^5 + 12*(9759623*b^12*c^3 + 4102800*a^3*b^9*c^2*d - 777840*a^6*b^6*c*d^2)*n^4 + 8*(38
232551*b^12*c^3 + 16529190*a^3*b^9*c^2*d - 6229440*a^6*b^6*c*d^2 + 831600*a^9*b^3*d^3)*n^3 + 576*(861864*b^12*
c^3 + 298435*a^3*b^9*c^2*d - 163940*a^6*b^6*c*d^2 + 34650*a^9*b^3*d^3)*n^2 + 5760*(76781*b^12*c^3 + 13860*a^3*
b^9*c^2*d - 9240*a^6*b^6*c*d^2 + 2310*a^9*b^3*d^3)*n)*x^3 + 144*(780996*a^3*b^9*c^3 - 293635*a^6*b^6*c^2*d + 2
7720*a^9*b^3*c*d^2)*n^2 + (a*b^11*c^3*n^11 + 73*a*b^11*c^3*n^10 + 2346*a*b^11*c^3*n^9 + 6*(7267*a*b^11*c^3 - 3
0*a^4*b^8*c^2*d)*n^8 + 3*(172459*a*b^11*c^3 - 3480*a^4*b^8*c^2*d)*n^7 + 3*(1359379*a*b^11*c^3 - 84120*a^4*b^8*
c^2*d)*n^6 + 4*(5373821*a*b^11*c^3 - 817200*a^4*b^8*c^2*d + 15120*a^7*b^5*c*d^2)*n^5 + 4*(18531227*a*b^11*c^3
- 6042105*a^4*b^8*c^2*d + 514080*a^7*b^5*c*d^2)*n^4 + 72*(2189036*a*b^11*c^3 - 1380055*a^4*b^8*c^2*d + 331800*
a^7*b^5*c*d^2)*n^3 + 1440*(125842*a*b^11*c^3 - 137481*a^4*b^8*c^2*d + 70644*a^7*b^5*c*d^2 - 13860*a^10*b^2*d^3
)*n^2 + 19958400*(4*a*b^11*c^3 - 6*a^4*b^8*c^2*d + 4*a^7*b^5*c*d^2 - a^10*b^2*d^3)*n)*x^2 + 2880*(70402*a^3*b^
9*c^3 - 54321*a^6*b^6*c^2*d + 15204*a^9*b^3*c*d^2)*n - 2*(a^2*b^10*c^3*n^10 + 72*a^2*b^10*c^3*n^9 + 2274*a^2*b
^10*c^3*n^8 + 36*(1148*a^2*b^10*c^3 - 5*a^5*b^7*c^2*d)*n^7 + 3*(158683*a^2*b^10*c^3 - 3420*a^5*b^7*c^2*d)*n^6
+ 36*(100058*a^2*b^10*c^3 - 6725*a^5*b^7*c^2*d)*n^5 + 4*(4473299*a^2*b^10*c^3 - 756675*a^5*b^7*c^2*d + 15120*a
^8*b^4*c*d^2)*n^4 + 72*(780996*a^2*b^10*c^3 - 293635*a^5*b^7*c^2*d + 27720*a^8*b^4*c*d^2)*n^3 + 1440*(70402*a^
2*b^10*c^3 - 54321*a^5*b^7*c^2*d + 15204*a^8*b^4*c*d^2)*n^2 + 19958400*(4*a^2*b^10*c^3 - 6*a^5*b^7*c^2*d + 4*a
^8*b^4*c*d^2 - a^11*b*d^3)*n)*x)*(b*x + a)^n/(b^12*n^12 + 78*b^12*n^11 + 2717*b^12*n^10 + 55770*b^12*n^9 + 749
463*b^12*n^8 + 6926634*b^12*n^7 + 44990231*b^12*n^6 + 206070150*b^12*n^5 + 657206836*b^12*n^4 + 1414014888*b^1
2*n^3 + 1931559552*b^12*n^2 + 1486442880*b^12*n + 479001600*b^12)

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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)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError