3.359 \(\int x^2 (a+b x)^n (c+d x^2)^3 \, dx\)
Optimal. Leaf size=343 \[ \frac{\left (a^2 d+b^2 c\right ) \left (17 a^2 b^2 c d+28 a^4 d^2+b^4 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}-\frac{4 a d \left (15 a^2 b^2 c d+14 a^4 d^2+3 b^4 c^2\right ) (a+b x)^{n+4}}{b^9 (n+4)}+\frac{d \left (45 a^2 b^2 c d+70 a^4 d^2+3 b^4 c^2\right ) (a+b x)^{n+5}}{b^9 (n+5)}-\frac{2 a d^2 \left (28 a^2 d+9 b^2 c\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{d^2 \left (28 a^2 d+3 b^2 c\right ) (a+b x)^{n+7}}{b^9 (n+7)}+\frac{a^2 \left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^9 (n+1)}-\frac{2 a \left (a^2 d+b^2 c\right )^2 \left (4 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^9 (n+2)}-\frac{8 a d^3 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^3 (a+b x)^{n+9}}{b^9 (n+9)} \]
[Out]
(a^2*(b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^9*(1 + n)) - (2*a*(b^2*c + a^2*d)^2*(b^2*c + 4*a^2*d)*(a + b*x)^(
2 + n))/(b^9*(2 + n)) + ((b^2*c + a^2*d)*(b^4*c^2 + 17*a^2*b^2*c*d + 28*a^4*d^2)*(a + b*x)^(3 + n))/(b^9*(3 +
n)) - (4*a*d*(3*b^4*c^2 + 15*a^2*b^2*c*d + 14*a^4*d^2)*(a + b*x)^(4 + n))/(b^9*(4 + n)) + (d*(3*b^4*c^2 + 45*a
^2*b^2*c*d + 70*a^4*d^2)*(a + b*x)^(5 + n))/(b^9*(5 + n)) - (2*a*d^2*(9*b^2*c + 28*a^2*d)*(a + b*x)^(6 + n))/(
b^9*(6 + n)) + (d^2*(3*b^2*c + 28*a^2*d)*(a + b*x)^(7 + n))/(b^9*(7 + n)) - (8*a*d^3*(a + b*x)^(8 + n))/(b^9*(
8 + n)) + (d^3*(a + b*x)^(9 + n))/(b^9*(9 + n))
________________________________________________________________________________________
Rubi [A] time = 0.210556, antiderivative size = 343, 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 =
{948} \[ \frac{\left (a^2 d+b^2 c\right ) \left (17 a^2 b^2 c d+28 a^4 d^2+b^4 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}-\frac{4 a d \left (15 a^2 b^2 c d+14 a^4 d^2+3 b^4 c^2\right ) (a+b x)^{n+4}}{b^9 (n+4)}+\frac{d \left (45 a^2 b^2 c d+70 a^4 d^2+3 b^4 c^2\right ) (a+b x)^{n+5}}{b^9 (n+5)}-\frac{2 a d^2 \left (28 a^2 d+9 b^2 c\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{d^2 \left (28 a^2 d+3 b^2 c\right ) (a+b x)^{n+7}}{b^9 (n+7)}+\frac{a^2 \left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^9 (n+1)}-\frac{2 a \left (a^2 d+b^2 c\right )^2 \left (4 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^9 (n+2)}-\frac{8 a d^3 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^3 (a+b x)^{n+9}}{b^9 (n+9)} \]
Antiderivative was successfully verified.
[In]
Int[x^2*(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
(a^2*(b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^9*(1 + n)) - (2*a*(b^2*c + a^2*d)^2*(b^2*c + 4*a^2*d)*(a + b*x)^(
2 + n))/(b^9*(2 + n)) + ((b^2*c + a^2*d)*(b^4*c^2 + 17*a^2*b^2*c*d + 28*a^4*d^2)*(a + b*x)^(3 + n))/(b^9*(3 +
n)) - (4*a*d*(3*b^4*c^2 + 15*a^2*b^2*c*d + 14*a^4*d^2)*(a + b*x)^(4 + n))/(b^9*(4 + n)) + (d*(3*b^4*c^2 + 45*a
^2*b^2*c*d + 70*a^4*d^2)*(a + b*x)^(5 + n))/(b^9*(5 + n)) - (2*a*d^2*(9*b^2*c + 28*a^2*d)*(a + b*x)^(6 + n))/(
b^9*(6 + n)) + (d^2*(3*b^2*c + 28*a^2*d)*(a + b*x)^(7 + n))/(b^9*(7 + n)) - (8*a*d^3*(a + b*x)^(8 + n))/(b^9*(
8 + n)) + (d^3*(a + b*x)^(9 + n))/(b^9*(9 + n))
Rule 948
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))
Rubi steps
\begin{align*} \int x^2 (a+b x)^n \left (c+d x^2\right )^3 \, dx &=\int \left (\frac{a^2 \left (b^2 c+a^2 d\right )^3 (a+b x)^n}{b^8}-\frac{2 a \left (b^2 c+a^2 d\right )^2 \left (b^2 c+4 a^2 d\right ) (a+b x)^{1+n}}{b^8}+\frac{\left (b^2 c+a^2 d\right ) \left (b^4 c^2+17 a^2 b^2 c d+28 a^4 d^2\right ) (a+b x)^{2+n}}{b^8}-\frac{4 a d \left (3 b^4 c^2+15 a^2 b^2 c d+14 a^4 d^2\right ) (a+b x)^{3+n}}{b^8}+\frac{d \left (3 b^4 c^2+45 a^2 b^2 c d+70 a^4 d^2\right ) (a+b x)^{4+n}}{b^8}-\frac{2 a d^2 \left (9 b^2 c+28 a^2 d\right ) (a+b x)^{5+n}}{b^8}+\frac{d^2 \left (3 b^2 c+28 a^2 d\right ) (a+b x)^{6+n}}{b^8}-\frac{8 a d^3 (a+b x)^{7+n}}{b^8}+\frac{d^3 (a+b x)^{8+n}}{b^8}\right ) \, dx\\ &=\frac{a^2 \left (b^2 c+a^2 d\right )^3 (a+b x)^{1+n}}{b^9 (1+n)}-\frac{2 a \left (b^2 c+a^2 d\right )^2 \left (b^2 c+4 a^2 d\right ) (a+b x)^{2+n}}{b^9 (2+n)}+\frac{\left (b^2 c+a^2 d\right ) \left (b^4 c^2+17 a^2 b^2 c d+28 a^4 d^2\right ) (a+b x)^{3+n}}{b^9 (3+n)}-\frac{4 a d \left (3 b^4 c^2+15 a^2 b^2 c d+14 a^4 d^2\right ) (a+b x)^{4+n}}{b^9 (4+n)}+\frac{d \left (3 b^4 c^2+45 a^2 b^2 c d+70 a^4 d^2\right ) (a+b x)^{5+n}}{b^9 (5+n)}-\frac{2 a d^2 \left (9 b^2 c+28 a^2 d\right ) (a+b x)^{6+n}}{b^9 (6+n)}+\frac{d^2 \left (3 b^2 c+28 a^2 d\right ) (a+b x)^{7+n}}{b^9 (7+n)}-\frac{8 a d^3 (a+b x)^{8+n}}{b^9 (8+n)}+\frac{d^3 (a+b x)^{9+n}}{b^9 (9+n)}\\ \end{align*}
Mathematica [A] time = 0.272227, size = 302, normalized size = 0.88 \[ \frac{(a+b x)^{n+1} \left (\frac{d (a+b x)^4 \left (45 a^2 b^2 c d+70 a^4 d^2+3 b^4 c^2\right )}{n+5}-\frac{4 a d (a+b x)^3 \left (15 a^2 b^2 c d+14 a^4 d^2+3 b^4 c^2\right )}{n+4}+\frac{(a+b x)^2 \left (a^2 d+b^2 c\right ) \left (17 a^2 b^2 c d+28 a^4 d^2+b^4 c^2\right )}{n+3}+\frac{d^2 (a+b x)^6 \left (28 a^2 d+3 b^2 c\right )}{n+7}-\frac{2 a d^2 (a+b x)^5 \left (28 a^2 d+9 b^2 c\right )}{n+6}-\frac{2 a (a+b x) \left (a^2 d+b^2 c\right )^2 \left (4 a^2 d+b^2 c\right )}{n+2}+\frac{a^2 \left (a^2 d+b^2 c\right )^3}{n+1}+\frac{d^3 (a+b x)^8}{n+9}-\frac{8 a d^3 (a+b x)^7}{n+8}\right )}{b^9} \]
Antiderivative was successfully verified.
[In]
Integrate[x^2*(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
((a + b*x)^(1 + n)*((a^2*(b^2*c + a^2*d)^3)/(1 + n) - (2*a*(b^2*c + a^2*d)^2*(b^2*c + 4*a^2*d)*(a + b*x))/(2 +
n) + ((b^2*c + a^2*d)*(b^4*c^2 + 17*a^2*b^2*c*d + 28*a^4*d^2)*(a + b*x)^2)/(3 + n) - (4*a*d*(3*b^4*c^2 + 15*a
^2*b^2*c*d + 14*a^4*d^2)*(a + b*x)^3)/(4 + n) + (d*(3*b^4*c^2 + 45*a^2*b^2*c*d + 70*a^4*d^2)*(a + b*x)^4)/(5 +
n) - (2*a*d^2*(9*b^2*c + 28*a^2*d)*(a + b*x)^5)/(6 + n) + (d^2*(3*b^2*c + 28*a^2*d)*(a + b*x)^6)/(7 + n) - (8
*a*d^3*(a + b*x)^7)/(8 + n) + (d^3*(a + b*x)^8)/(9 + n)))/b^9
________________________________________________________________________________________
Maple [B] time = 0.065, size = 2232, normalized size = 6.5 \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^2+c)^3,x)
[Out]
(b*x+a)^(1+n)*(b^8*d^3*n^8*x^8+36*b^8*d^3*n^7*x^8-8*a*b^7*d^3*n^7*x^7+3*b^8*c*d^2*n^8*x^6+546*b^8*d^3*n^6*x^8-
224*a*b^7*d^3*n^6*x^7+114*b^8*c*d^2*n^7*x^6+4536*b^8*d^3*n^5*x^8+56*a^2*b^6*d^3*n^6*x^6-18*a*b^7*c*d^2*n^7*x^5
-2576*a*b^7*d^3*n^5*x^7+3*b^8*c^2*d*n^8*x^4+1812*b^8*c*d^2*n^6*x^6+22449*b^8*d^3*n^4*x^8+1176*a^2*b^6*d^3*n^5*
x^6-576*a*b^7*c*d^2*n^6*x^5-15680*a*b^7*d^3*n^4*x^7+120*b^8*c^2*d*n^7*x^4+15666*b^8*c*d^2*n^5*x^6+67284*b^8*d^
3*n^3*x^8-336*a^3*b^5*d^3*n^5*x^5+90*a^2*b^6*c*d^2*n^6*x^4+9800*a^2*b^6*d^3*n^4*x^6-12*a*b^7*c^2*d*n^7*x^3-741
6*a*b^7*c*d^2*n^5*x^5-54152*a*b^7*d^3*n^3*x^7+b^8*c^3*n^8*x^2+2010*b^8*c^2*d*n^6*x^4+80157*b^8*c*d^2*n^4*x^6+1
18124*b^8*d^3*n^2*x^8-5040*a^3*b^5*d^3*n^4*x^5+2430*a^2*b^6*c*d^2*n^5*x^4+41160*a^2*b^6*d^3*n^3*x^6-432*a*b^7*
c^2*d*n^6*x^3-49500*a*b^7*c*d^2*n^4*x^5-105056*a*b^7*d^3*n^2*x^7+42*b^8*c^3*n^7*x^2+18300*b^8*c^2*d*n^5*x^4+24
6876*b^8*c*d^2*n^3*x^6+109584*b^8*d^3*n*x^8+1680*a^4*b^4*d^3*n^4*x^4-360*a^3*b^5*c*d^2*n^5*x^3-28560*a^3*b^5*d
^3*n^3*x^5+36*a^2*b^6*c^2*d*n^6*x^2+24930*a^2*b^6*c*d^2*n^4*x^4+90944*a^2*b^6*d^3*n^2*x^6-2*a*b^7*c^3*n^7*x-63
12*a*b^7*c^2*d*n^5*x^3-183942*a*b^7*c*d^2*n^3*x^5-104544*a*b^7*d^3*n*x^7+744*b^8*c^3*n^6*x^2+98319*b^8*c^2*d*n
^4*x^4+442908*b^8*c*d^2*n^2*x^6+40320*b^8*d^3*x^8+16800*a^4*b^4*d^3*n^3*x^4-8280*a^3*b^5*c*d^2*n^4*x^3-75600*a
^3*b^5*d^3*n^2*x^5+1188*a^2*b^6*c^2*d*n^5*x^2+122850*a^2*b^6*c*d^2*n^3*x^4+98784*a^2*b^6*d^3*n*x^6-80*a*b^7*c^
3*n^6*x-47952*a*b^7*c^2*d*n^4*x^3-377604*a*b^7*c*d^2*n^2*x^5-40320*a*b^7*d^3*x^7+7218*b^8*c^3*n^5*x^2+316380*b
^8*c^2*d*n^3*x^4+417744*b^8*c*d^2*n*x^6-6720*a^5*b^3*d^3*n^3*x^3+1080*a^4*b^4*c*d^2*n^4*x^2+58800*a^4*b^4*d^3*
n^2*x^4-72*a^3*b^5*c^2*d*n^5*x-66600*a^3*b^5*c*d^2*n^3*x^3-92064*a^3*b^5*d^3*n*x^5+2*a^2*b^6*c^3*n^6+15372*a^2
*b^6*c^2*d*n^4*x^2+305460*a^2*b^6*c*d^2*n^2*x^4+40320*a^2*b^6*d^3*x^6-1328*a*b^7*c^3*n^5*x-201468*a*b^7*c^2*d*
n^3*x^3-391824*a*b^7*c*d^2*n*x^5+41619*b^8*c^3*n^4*x^2+589140*b^8*c^2*d*n^2*x^4+155520*b^8*c*d^2*x^6-40320*a^5
*b^3*d^3*n^2*x^3+21600*a^4*b^4*c*d^2*n^3*x^2+84000*a^4*b^4*d^3*n*x^4-2232*a^3*b^5*c^2*d*n^4*x-225000*a^3*b^5*c
*d^2*n^2*x^3-40320*a^3*b^5*d^3*x^5+78*a^2*b^6*c^3*n^5+97740*a^2*b^6*c^2*d*n^3*x^2+360720*a^2*b^6*c*d^2*n*x^4-1
1780*a*b^7*c^3*n^4*x-459648*a*b^7*c^2*d*n^2*x^3-155520*a*b^7*c*d^2*x^5+144468*b^8*c^3*n^3*x^2+572400*b^8*c^2*d
*n*x^4+20160*a^6*b^2*d^3*n^2*x^2-2160*a^5*b^3*c*d^2*n^3*x-73920*a^5*b^3*d^3*n*x^3+72*a^4*b^4*c^2*d*n^4+135000*
a^4*b^4*c*d^2*n^2*x^2+40320*a^4*b^4*d^3*x^4-26280*a^3*b^5*c^2*d*n^3*x-321840*a^3*b^5*c*d^2*n*x^3+1250*a^2*b^6*
c^3*n^4+311184*a^2*b^6*c^2*d*n^2*x^2+155520*a^2*b^6*c*d^2*x^4-59678*a*b^7*c^3*n^3*x-517968*a*b^7*c^2*d*n*x^3+2
90276*b^8*c^3*n^2*x^2+217728*b^8*c^2*d*x^4+60480*a^6*b^2*d^3*n*x^2-38880*a^5*b^3*c*d^2*n^2*x-40320*a^5*b^3*d^3
*x^3+2160*a^4*b^4*c^2*d*n^3+270000*a^4*b^4*c*d^2*n*x^2-142920*a^3*b^5*c^2*d*n^2*x-155520*a^3*b^5*c*d^2*x^3+105
30*a^2*b^6*c^3*n^3+445392*a^2*b^6*c^2*d*n*x^2-169580*a*b^7*c^3*n^2*x-217728*a*b^7*c^2*d*x^3+301872*b^8*c^3*n*x
^2-40320*a^7*b*d^3*n*x+2160*a^6*b^2*c*d^2*n^2+40320*a^6*b^2*d^3*x^2-192240*a^5*b^3*c*d^2*n*x+24120*a^4*b^4*c^2
*d*n^2+155520*a^4*b^4*c*d^2*x^2-336528*a^3*b^5*c^2*d*n*x+49148*a^2*b^6*c^3*n^2+217728*a^2*b^6*c^2*d*x^2-241392
*a*b^7*c^3*n*x+120960*b^8*c^3*x^2-40320*a^7*b*d^3*x+36720*a^6*b^2*c*d^2*n-155520*a^5*b^3*c*d^2*x+118800*a^4*b^
4*c^2*d*n-217728*a^3*b^5*c^2*d*x+120432*a^2*b^6*c^3*n-120960*a*b^7*c^3*x+40320*a^8*d^3+155520*a^6*b^2*c*d^2+21
7728*a^4*b^4*c^2*d+120960*a^2*b^6*c^3)/b^9/(n^9+45*n^8+870*n^7+9450*n^6+63273*n^5+269325*n^4+723680*n^3+117270
0*n^2+1026576*n+362880)
________________________________________________________________________________________
Maxima [B] time = 1.0823, size = 1073, normalized size = 3.13 \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^2+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^4 + 10*n^3 + 35*n^2 + 50*n + 24)*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^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b^7*x^7
+ (n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a*b^6*x^6 - 6*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a
^2*b^5*x^5 + 30*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^3*b^4*x^4 - 120*(n^3 + 3*n^2 + 2*n)*a^4*b^3*x^3 + 360*(n^2 + n)
*a^5*b^2*x^2 - 720*a^6*b*n*x + 720*a^7)*(b*x + a)^n*c*d^2/((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 131
32*n^2 + 13068*n + 5040)*b^7) + ((n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 118124*n^2 + 109
584*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 + 40320*a^9)*(b*x + a)^n*d^3/((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)
________________________________________________________________________________________
Fricas [B] time = 2.2272, size = 4895, normalized size = 14.27 \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^2+c)^3,x, algorithm="fricas")
[Out]
(2*a^3*b^6*c^3*n^6 + 78*a^3*b^6*c^3*n^5 + 120960*a^3*b^6*c^3 + 217728*a^5*b^4*c^2*d + 155520*a^7*b^2*c*d^2 + 4
0320*a^9*d^3 + (b^9*d^3*n^8 + 36*b^9*d^3*n^7 + 546*b^9*d^3*n^6 + 4536*b^9*d^3*n^5 + 22449*b^9*d^3*n^4 + 67284*
b^9*d^3*n^3 + 118124*b^9*d^3*n^2 + 109584*b^9*d^3*n + 40320*b^9*d^3)*x^9 + (a*b^8*d^3*n^8 + 28*a*b^8*d^3*n^7 +
322*a*b^8*d^3*n^6 + 1960*a*b^8*d^3*n^5 + 6769*a*b^8*d^3*n^4 + 13132*a*b^8*d^3*n^3 + 13068*a*b^8*d^3*n^2 + 504
0*a*b^8*d^3*n)*x^8 + (3*b^9*c*d^2*n^8 + 155520*b^9*c*d^2 + 2*(57*b^9*c*d^2 - 4*a^2*b^7*d^3)*n^7 + 12*(151*b^9*
c*d^2 - 14*a^2*b^7*d^3)*n^6 + 14*(1119*b^9*c*d^2 - 100*a^2*b^7*d^3)*n^5 + 21*(3817*b^9*c*d^2 - 280*a^2*b^7*d^3
)*n^4 + 28*(8817*b^9*c*d^2 - 464*a^2*b^7*d^3)*n^3 + 36*(12303*b^9*c*d^2 - 392*a^2*b^7*d^3)*n^2 + 144*(2901*b^9
*c*d^2 - 40*a^2*b^7*d^3)*n)*x^7 + (3*a*b^8*c*d^2*n^8 + 96*a*b^8*c*d^2*n^7 + 4*(309*a*b^8*c*d^2 + 14*a^3*b^6*d^
3)*n^6 + 30*(275*a*b^8*c*d^2 + 28*a^3*b^6*d^3)*n^5 + (30657*a*b^8*c*d^2 + 4760*a^3*b^6*d^3)*n^4 + 6*(10489*a*b
^8*c*d^2 + 2100*a^3*b^6*d^3)*n^3 + 8*(8163*a*b^8*c*d^2 + 1918*a^3*b^6*d^3)*n^2 + 960*(27*a*b^8*c*d^2 + 7*a^3*b
^6*d^3)*n)*x^6 + 3*(b^9*c^2*d*n^8 + 72576*b^9*c^2*d + 2*(20*b^9*c^2*d - 3*a^2*b^7*c*d^2)*n^7 + 2*(335*b^9*c^2*
d - 81*a^2*b^7*c*d^2)*n^6 + 2*(3050*b^9*c^2*d - 831*a^2*b^7*c*d^2 - 56*a^4*b^5*d^3)*n^5 + (32773*b^9*c^2*d - 8
190*a^2*b^7*c*d^2 - 1120*a^4*b^5*d^3)*n^4 + 4*(26365*b^9*c^2*d - 5091*a^2*b^7*c*d^2 - 980*a^4*b^5*d^3)*n^3 + 4
*(49095*b^9*c^2*d - 6012*a^2*b^7*c*d^2 - 1400*a^4*b^5*d^3)*n^2 + 48*(3975*b^9*c^2*d - 216*a^2*b^7*c*d^2 - 56*a
^4*b^5*d^3)*n)*x^5 + 2*(625*a^3*b^6*c^3 + 36*a^5*b^4*c^2*d)*n^4 + 3*(a*b^8*c^2*d*n^8 + 36*a*b^8*c^2*d*n^7 + 2*
(263*a*b^8*c^2*d + 15*a^3*b^6*c*d^2)*n^6 + 6*(666*a*b^8*c^2*d + 115*a^3*b^6*c*d^2)*n^5 + (16789*a*b^8*c^2*d +
5550*a^3*b^6*c*d^2 + 560*a^5*b^4*d^3)*n^4 + 6*(6384*a*b^8*c^2*d + 3125*a^3*b^6*c*d^2 + 560*a^5*b^4*d^3)*n^3 +
4*(10791*a*b^8*c^2*d + 6705*a^3*b^6*c*d^2 + 1540*a^5*b^4*d^3)*n^2 + 96*(189*a*b^8*c^2*d + 135*a^3*b^6*c*d^2 +
35*a^5*b^4*d^3)*n)*x^4 + 270*(39*a^3*b^6*c^3 + 8*a^5*b^4*c^2*d)*n^3 + (b^9*c^3*n^8 + 120960*b^9*c^3 + 6*(7*b^9
*c^3 - 2*a^2*b^7*c^2*d)*n^7 + 12*(62*b^9*c^3 - 33*a^2*b^7*c^2*d)*n^6 + 6*(1203*b^9*c^3 - 854*a^2*b^7*c^2*d - 6
0*a^4*b^5*c*d^2)*n^5 + 3*(13873*b^9*c^3 - 10860*a^2*b^7*c^2*d - 2400*a^4*b^5*c*d^2)*n^4 + 12*(12039*b^9*c^3 -
8644*a^2*b^7*c^2*d - 3750*a^4*b^5*c*d^2 - 560*a^6*b^3*d^3)*n^3 + 4*(72569*b^9*c^3 - 37116*a^2*b^7*c^2*d - 2250
0*a^4*b^5*c*d^2 - 5040*a^6*b^3*d^3)*n^2 + 48*(6289*b^9*c^3 - 1512*a^2*b^7*c^2*d - 1080*a^4*b^5*c*d^2 - 280*a^6
*b^3*d^3)*n)*x^3 + 4*(12287*a^3*b^6*c^3 + 6030*a^5*b^4*c^2*d + 540*a^7*b^2*c*d^2)*n^2 + (a*b^8*c^3*n^8 + 40*a*
b^8*c^3*n^7 + 4*(166*a*b^8*c^3 + 9*a^3*b^6*c^2*d)*n^6 + 62*(95*a*b^8*c^3 + 18*a^3*b^6*c^2*d)*n^5 + (29839*a*b^
8*c^3 + 13140*a^3*b^6*c^2*d + 1080*a^5*b^4*c*d^2)*n^4 + 10*(8479*a*b^8*c^3 + 7146*a^3*b^6*c^2*d + 1944*a^5*b^4
*c*d^2)*n^3 + 24*(5029*a*b^8*c^3 + 7011*a^3*b^6*c^2*d + 4005*a^5*b^4*c*d^2 + 840*a^7*b^2*d^3)*n^2 + 576*(105*a
*b^8*c^3 + 189*a^3*b^6*c^2*d + 135*a^5*b^4*c*d^2 + 35*a^7*b^2*d^3)*n)*x^2 + 48*(2509*a^3*b^6*c^3 + 2475*a^5*b^
4*c^2*d + 765*a^7*b^2*c*d^2)*n - 2*(a^2*b^7*c^3*n^7 + 39*a^2*b^7*c^3*n^6 + (625*a^2*b^7*c^3 + 36*a^4*b^5*c^2*d
)*n^5 + 135*(39*a^2*b^7*c^3 + 8*a^4*b^5*c^2*d)*n^4 + 2*(12287*a^2*b^7*c^3 + 6030*a^4*b^5*c^2*d + 540*a^6*b^3*c
*d^2)*n^3 + 24*(2509*a^2*b^7*c^3 + 2475*a^4*b^5*c^2*d + 765*a^6*b^3*c*d^2)*n^2 + 576*(105*a^2*b^7*c^3 + 189*a^
4*b^5*c^2*d + 135*a^6*b^3*c*d^2 + 35*a^8*b*d^3)*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)
________________________________________________________________________________________
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**2+c)**3,x)
[Out]
Timed out
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Giac [B] time = 1.25576, size = 5013, normalized size = 14.62 \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^2+c)^3,x, algorithm="giac")
[Out]
((b*x + a)^n*b^9*d^3*n^8*x^9 + (b*x + a)^n*a*b^8*d^3*n^8*x^8 + 36*(b*x + a)^n*b^9*d^3*n^7*x^9 + 3*(b*x + a)^n*
b^9*c*d^2*n^8*x^7 + 28*(b*x + a)^n*a*b^8*d^3*n^7*x^8 + 546*(b*x + a)^n*b^9*d^3*n^6*x^9 + 3*(b*x + a)^n*a*b^8*c
*d^2*n^8*x^6 + 114*(b*x + a)^n*b^9*c*d^2*n^7*x^7 - 8*(b*x + a)^n*a^2*b^7*d^3*n^7*x^7 + 322*(b*x + a)^n*a*b^8*d
^3*n^6*x^8 + 4536*(b*x + a)^n*b^9*d^3*n^5*x^9 + 3*(b*x + a)^n*b^9*c^2*d*n^8*x^5 + 96*(b*x + a)^n*a*b^8*c*d^2*n
^7*x^6 + 1812*(b*x + a)^n*b^9*c*d^2*n^6*x^7 - 168*(b*x + a)^n*a^2*b^7*d^3*n^6*x^7 + 1960*(b*x + a)^n*a*b^8*d^3
*n^5*x^8 + 22449*(b*x + a)^n*b^9*d^3*n^4*x^9 + 3*(b*x + a)^n*a*b^8*c^2*d*n^8*x^4 + 120*(b*x + a)^n*b^9*c^2*d*n
^7*x^5 - 18*(b*x + a)^n*a^2*b^7*c*d^2*n^7*x^5 + 1236*(b*x + a)^n*a*b^8*c*d^2*n^6*x^6 + 56*(b*x + a)^n*a^3*b^6*
d^3*n^6*x^6 + 15666*(b*x + a)^n*b^9*c*d^2*n^5*x^7 - 1400*(b*x + a)^n*a^2*b^7*d^3*n^5*x^7 + 6769*(b*x + a)^n*a*
b^8*d^3*n^4*x^8 + 67284*(b*x + a)^n*b^9*d^3*n^3*x^9 + (b*x + a)^n*b^9*c^3*n^8*x^3 + 108*(b*x + a)^n*a*b^8*c^2*
d*n^7*x^4 + 2010*(b*x + a)^n*b^9*c^2*d*n^6*x^5 - 486*(b*x + a)^n*a^2*b^7*c*d^2*n^6*x^5 + 8250*(b*x + a)^n*a*b^
8*c*d^2*n^5*x^6 + 840*(b*x + a)^n*a^3*b^6*d^3*n^5*x^6 + 80157*(b*x + a)^n*b^9*c*d^2*n^4*x^7 - 5880*(b*x + a)^n
*a^2*b^7*d^3*n^4*x^7 + 13132*(b*x + a)^n*a*b^8*d^3*n^3*x^8 + 118124*(b*x + a)^n*b^9*d^3*n^2*x^9 + (b*x + a)^n*
a*b^8*c^3*n^8*x^2 + 42*(b*x + a)^n*b^9*c^3*n^7*x^3 - 12*(b*x + a)^n*a^2*b^7*c^2*d*n^7*x^3 + 1578*(b*x + a)^n*a
*b^8*c^2*d*n^6*x^4 + 90*(b*x + a)^n*a^3*b^6*c*d^2*n^6*x^4 + 18300*(b*x + a)^n*b^9*c^2*d*n^5*x^5 - 4986*(b*x +
a)^n*a^2*b^7*c*d^2*n^5*x^5 - 336*(b*x + a)^n*a^4*b^5*d^3*n^5*x^5 + 30657*(b*x + a)^n*a*b^8*c*d^2*n^4*x^6 + 476
0*(b*x + a)^n*a^3*b^6*d^3*n^4*x^6 + 246876*(b*x + a)^n*b^9*c*d^2*n^3*x^7 - 12992*(b*x + a)^n*a^2*b^7*d^3*n^3*x
^7 + 13068*(b*x + a)^n*a*b^8*d^3*n^2*x^8 + 109584*(b*x + a)^n*b^9*d^3*n*x^9 + 40*(b*x + a)^n*a*b^8*c^3*n^7*x^2
+ 744*(b*x + a)^n*b^9*c^3*n^6*x^3 - 396*(b*x + a)^n*a^2*b^7*c^2*d*n^6*x^3 + 11988*(b*x + a)^n*a*b^8*c^2*d*n^5
*x^4 + 2070*(b*x + a)^n*a^3*b^6*c*d^2*n^5*x^4 + 98319*(b*x + a)^n*b^9*c^2*d*n^4*x^5 - 24570*(b*x + a)^n*a^2*b^
7*c*d^2*n^4*x^5 - 3360*(b*x + a)^n*a^4*b^5*d^3*n^4*x^5 + 62934*(b*x + a)^n*a*b^8*c*d^2*n^3*x^6 + 12600*(b*x +
a)^n*a^3*b^6*d^3*n^3*x^6 + 442908*(b*x + a)^n*b^9*c*d^2*n^2*x^7 - 14112*(b*x + a)^n*a^2*b^7*d^3*n^2*x^7 + 5040
*(b*x + a)^n*a*b^8*d^3*n*x^8 + 40320*(b*x + a)^n*b^9*d^3*x^9 - 2*(b*x + a)^n*a^2*b^7*c^3*n^7*x + 664*(b*x + a)
^n*a*b^8*c^3*n^6*x^2 + 36*(b*x + a)^n*a^3*b^6*c^2*d*n^6*x^2 + 7218*(b*x + a)^n*b^9*c^3*n^5*x^3 - 5124*(b*x + a
)^n*a^2*b^7*c^2*d*n^5*x^3 - 360*(b*x + a)^n*a^4*b^5*c*d^2*n^5*x^3 + 50367*(b*x + a)^n*a*b^8*c^2*d*n^4*x^4 + 16
650*(b*x + a)^n*a^3*b^6*c*d^2*n^4*x^4 + 1680*(b*x + a)^n*a^5*b^4*d^3*n^4*x^4 + 316380*(b*x + a)^n*b^9*c^2*d*n^
3*x^5 - 61092*(b*x + a)^n*a^2*b^7*c*d^2*n^3*x^5 - 11760*(b*x + a)^n*a^4*b^5*d^3*n^3*x^5 + 65304*(b*x + a)^n*a*
b^8*c*d^2*n^2*x^6 + 15344*(b*x + a)^n*a^3*b^6*d^3*n^2*x^6 + 417744*(b*x + a)^n*b^9*c*d^2*n*x^7 - 5760*(b*x + a
)^n*a^2*b^7*d^3*n*x^7 - 78*(b*x + a)^n*a^2*b^7*c^3*n^6*x + 5890*(b*x + a)^n*a*b^8*c^3*n^5*x^2 + 1116*(b*x + a)
^n*a^3*b^6*c^2*d*n^5*x^2 + 41619*(b*x + a)^n*b^9*c^3*n^4*x^3 - 32580*(b*x + a)^n*a^2*b^7*c^2*d*n^4*x^3 - 7200*
(b*x + a)^n*a^4*b^5*c*d^2*n^4*x^3 + 114912*(b*x + a)^n*a*b^8*c^2*d*n^3*x^4 + 56250*(b*x + a)^n*a^3*b^6*c*d^2*n
^3*x^4 + 10080*(b*x + a)^n*a^5*b^4*d^3*n^3*x^4 + 589140*(b*x + a)^n*b^9*c^2*d*n^2*x^5 - 72144*(b*x + a)^n*a^2*
b^7*c*d^2*n^2*x^5 - 16800*(b*x + a)^n*a^4*b^5*d^3*n^2*x^5 + 25920*(b*x + a)^n*a*b^8*c*d^2*n*x^6 + 6720*(b*x +
a)^n*a^3*b^6*d^3*n*x^6 + 155520*(b*x + a)^n*b^9*c*d^2*x^7 + 2*(b*x + a)^n*a^3*b^6*c^3*n^6 - 1250*(b*x + a)^n*a
^2*b^7*c^3*n^5*x - 72*(b*x + a)^n*a^4*b^5*c^2*d*n^5*x + 29839*(b*x + a)^n*a*b^8*c^3*n^4*x^2 + 13140*(b*x + a)^
n*a^3*b^6*c^2*d*n^4*x^2 + 1080*(b*x + a)^n*a^5*b^4*c*d^2*n^4*x^2 + 144468*(b*x + a)^n*b^9*c^3*n^3*x^3 - 103728
*(b*x + a)^n*a^2*b^7*c^2*d*n^3*x^3 - 45000*(b*x + a)^n*a^4*b^5*c*d^2*n^3*x^3 - 6720*(b*x + a)^n*a^6*b^3*d^3*n^
3*x^3 + 129492*(b*x + a)^n*a*b^8*c^2*d*n^2*x^4 + 80460*(b*x + a)^n*a^3*b^6*c*d^2*n^2*x^4 + 18480*(b*x + a)^n*a
^5*b^4*d^3*n^2*x^4 + 572400*(b*x + a)^n*b^9*c^2*d*n*x^5 - 31104*(b*x + a)^n*a^2*b^7*c*d^2*n*x^5 - 8064*(b*x +
a)^n*a^4*b^5*d^3*n*x^5 + 78*(b*x + a)^n*a^3*b^6*c^3*n^5 - 10530*(b*x + a)^n*a^2*b^7*c^3*n^4*x - 2160*(b*x + a)
^n*a^4*b^5*c^2*d*n^4*x + 84790*(b*x + a)^n*a*b^8*c^3*n^3*x^2 + 71460*(b*x + a)^n*a^3*b^6*c^2*d*n^3*x^2 + 19440
*(b*x + a)^n*a^5*b^4*c*d^2*n^3*x^2 + 290276*(b*x + a)^n*b^9*c^3*n^2*x^3 - 148464*(b*x + a)^n*a^2*b^7*c^2*d*n^2
*x^3 - 90000*(b*x + a)^n*a^4*b^5*c*d^2*n^2*x^3 - 20160*(b*x + a)^n*a^6*b^3*d^3*n^2*x^3 + 54432*(b*x + a)^n*a*b
^8*c^2*d*n*x^4 + 38880*(b*x + a)^n*a^3*b^6*c*d^2*n*x^4 + 10080*(b*x + a)^n*a^5*b^4*d^3*n*x^4 + 217728*(b*x + a
)^n*b^9*c^2*d*x^5 + 1250*(b*x + a)^n*a^3*b^6*c^3*n^4 + 72*(b*x + a)^n*a^5*b^4*c^2*d*n^4 - 49148*(b*x + a)^n*a^
2*b^7*c^3*n^3*x - 24120*(b*x + a)^n*a^4*b^5*c^2*d*n^3*x - 2160*(b*x + a)^n*a^6*b^3*c*d^2*n^3*x + 120696*(b*x +
a)^n*a*b^8*c^3*n^2*x^2 + 168264*(b*x + a)^n*a^3*b^6*c^2*d*n^2*x^2 + 96120*(b*x + a)^n*a^5*b^4*c*d^2*n^2*x^2 +
20160*(b*x + a)^n*a^7*b^2*d^3*n^2*x^2 + 301872*(b*x + a)^n*b^9*c^3*n*x^3 - 72576*(b*x + a)^n*a^2*b^7*c^2*d*n*
x^3 - 51840*(b*x + a)^n*a^4*b^5*c*d^2*n*x^3 - 13440*(b*x + a)^n*a^6*b^3*d^3*n*x^3 + 10530*(b*x + a)^n*a^3*b^6*
c^3*n^3 + 2160*(b*x + a)^n*a^5*b^4*c^2*d*n^3 - 120432*(b*x + a)^n*a^2*b^7*c^3*n^2*x - 118800*(b*x + a)^n*a^4*b
^5*c^2*d*n^2*x - 36720*(b*x + a)^n*a^6*b^3*c*d^2*n^2*x + 60480*(b*x + a)^n*a*b^8*c^3*n*x^2 + 108864*(b*x + a)^
n*a^3*b^6*c^2*d*n*x^2 + 77760*(b*x + a)^n*a^5*b^4*c*d^2*n*x^2 + 20160*(b*x + a)^n*a^7*b^2*d^3*n*x^2 + 120960*(
b*x + a)^n*b^9*c^3*x^3 + 49148*(b*x + a)^n*a^3*b^6*c^3*n^2 + 24120*(b*x + a)^n*a^5*b^4*c^2*d*n^2 + 2160*(b*x +
a)^n*a^7*b^2*c*d^2*n^2 - 120960*(b*x + a)^n*a^2*b^7*c^3*n*x - 217728*(b*x + a)^n*a^4*b^5*c^2*d*n*x - 155520*(
b*x + a)^n*a^6*b^3*c*d^2*n*x - 40320*(b*x + a)^n*a^8*b*d^3*n*x + 120432*(b*x + a)^n*a^3*b^6*c^3*n + 118800*(b*
x + a)^n*a^5*b^4*c^2*d*n + 36720*(b*x + a)^n*a^7*b^2*c*d^2*n + 120960*(b*x + a)^n*a^3*b^6*c^3 + 217728*(b*x +
a)^n*a^5*b^4*c^2*d + 155520*(b*x + a)^n*a^7*b^2*c*d^2 + 40320*(b*x + a)^n*a^9*d^3)/(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)