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

Optimal. Leaf size=232 \[ \frac{\left (12 a^2 b^2 c d+15 a^4 d^2+b^4 c^2\right ) (a+b x)^{n+3}}{b^7 (n+3)}+\frac{a^2 \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+1}}{b^7 (n+1)}-\frac{2 a \left (a^2 d+b^2 c\right ) \left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^7 (n+2)}-\frac{4 a d \left (5 a^2 d+2 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac{d \left (15 a^2 d+2 b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}-\frac{6 a d^2 (a+b x)^{n+6}}{b^7 (n+6)}+\frac{d^2 (a+b x)^{n+7}}{b^7 (n+7)} \]

[Out]

(a^2*(b^2*c + a^2*d)^2*(a + b*x)^(1 + n))/(b^7*(1 + n)) - (2*a*(b^2*c + a^2*d)*(b^2*c + 3*a^2*d)*(a + b*x)^(2
+ n))/(b^7*(2 + n)) + ((b^4*c^2 + 12*a^2*b^2*c*d + 15*a^4*d^2)*(a + b*x)^(3 + n))/(b^7*(3 + n)) - (4*a*d*(2*b^
2*c + 5*a^2*d)*(a + b*x)^(4 + n))/(b^7*(4 + n)) + (d*(2*b^2*c + 15*a^2*d)*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (
6*a*d^2*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^2*(a + b*x)^(7 + n))/(b^7*(7 + n))

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Rubi [A]  time = 0.139042, antiderivative size = 232, 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} \[ \frac{\left (12 a^2 b^2 c d+15 a^4 d^2+b^4 c^2\right ) (a+b x)^{n+3}}{b^7 (n+3)}+\frac{a^2 \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+1}}{b^7 (n+1)}-\frac{2 a \left (a^2 d+b^2 c\right ) \left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^7 (n+2)}-\frac{4 a d \left (5 a^2 d+2 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac{d \left (15 a^2 d+2 b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}-\frac{6 a d^2 (a+b x)^{n+6}}{b^7 (n+6)}+\frac{d^2 (a+b x)^{n+7}}{b^7 (n+7)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^n*(c + d*x^2)^2,x]

[Out]

(a^2*(b^2*c + a^2*d)^2*(a + b*x)^(1 + n))/(b^7*(1 + n)) - (2*a*(b^2*c + a^2*d)*(b^2*c + 3*a^2*d)*(a + b*x)^(2
+ n))/(b^7*(2 + n)) + ((b^4*c^2 + 12*a^2*b^2*c*d + 15*a^4*d^2)*(a + b*x)^(3 + n))/(b^7*(3 + n)) - (4*a*d*(2*b^
2*c + 5*a^2*d)*(a + b*x)^(4 + n))/(b^7*(4 + n)) + (d*(2*b^2*c + 15*a^2*d)*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (
6*a*d^2*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^2*(a + b*x)^(7 + n))/(b^7*(7 + 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 )^2 \, dx &=\int \left (\frac{\left (a b^2 c+a^3 d\right )^2 (a+b x)^n}{b^6}+\frac{2 a \left (-b^2 c-3 a^2 d\right ) \left (b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^6}+\frac{\left (b^4 c^2+12 a^2 b^2 c d+15 a^4 d^2\right ) (a+b x)^{2+n}}{b^6}-\frac{4 a d \left (2 b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^6}+\frac{d \left (2 b^2 c+15 a^2 d\right ) (a+b x)^{4+n}}{b^6}-\frac{6 a d^2 (a+b x)^{5+n}}{b^6}+\frac{d^2 (a+b x)^{6+n}}{b^6}\right ) \, dx\\ &=\frac{a^2 \left (b^2 c+a^2 d\right )^2 (a+b x)^{1+n}}{b^7 (1+n)}-\frac{2 a \left (b^2 c+a^2 d\right ) \left (b^2 c+3 a^2 d\right ) (a+b x)^{2+n}}{b^7 (2+n)}+\frac{\left (b^4 c^2+12 a^2 b^2 c d+15 a^4 d^2\right ) (a+b x)^{3+n}}{b^7 (3+n)}-\frac{4 a d \left (2 b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^7 (4+n)}+\frac{d \left (2 b^2 c+15 a^2 d\right ) (a+b x)^{5+n}}{b^7 (5+n)}-\frac{6 a d^2 (a+b x)^{6+n}}{b^7 (6+n)}+\frac{d^2 (a+b x)^{7+n}}{b^7 (7+n)}\\ \end{align*}

Mathematica [A]  time = 0.198186, size = 199, normalized size = 0.86 \[ \frac{(a+b x)^{n+1} \left (\frac{(a+b x)^2 \left (12 a^2 b^2 c d+15 a^4 d^2+b^4 c^2\right )}{n+3}+\frac{d (a+b x)^4 \left (15 a^2 d+2 b^2 c\right )}{n+5}-\frac{4 a d (a+b x)^3 \left (5 a^2 d+2 b^2 c\right )}{n+4}-\frac{2 a (a+b x) \left (a^2 d+b^2 c\right ) \left (3 a^2 d+b^2 c\right )}{n+2}+\frac{\left (a^3 d+a b^2 c\right )^2}{n+1}+\frac{d^2 (a+b x)^6}{n+7}-\frac{6 a d^2 (a+b x)^5}{n+6}\right )}{b^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + n)*((a*b^2*c + a^3*d)^2/(1 + n) - (2*a*(b^2*c + a^2*d)*(b^2*c + 3*a^2*d)*(a + b*x))/(2 + n) +
((b^4*c^2 + 12*a^2*b^2*c*d + 15*a^4*d^2)*(a + b*x)^2)/(3 + n) - (4*a*d*(2*b^2*c + 5*a^2*d)*(a + b*x)^3)/(4 + n
) + (d*(2*b^2*c + 15*a^2*d)*(a + b*x)^4)/(5 + n) - (6*a*d^2*(a + b*x)^5)/(6 + n) + (d^2*(a + b*x)^6)/(7 + n)))
/b^7

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Maple [B]  time = 0.054, size = 1000, normalized size = 4.3 \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^2+c)^2,x)

[Out]

(b*x+a)^(1+n)*(b^6*d^2*n^6*x^6+21*b^6*d^2*n^5*x^6-6*a*b^5*d^2*n^5*x^5+2*b^6*c*d*n^6*x^4+175*b^6*d^2*n^4*x^6-90
*a*b^5*d^2*n^4*x^5+46*b^6*c*d*n^5*x^4+735*b^6*d^2*n^3*x^6+30*a^2*b^4*d^2*n^4*x^4-8*a*b^5*c*d*n^5*x^3-510*a*b^5
*d^2*n^3*x^5+b^6*c^2*n^6*x^2+414*b^6*c*d*n^4*x^4+1624*b^6*d^2*n^2*x^6+300*a^2*b^4*d^2*n^3*x^4-152*a*b^5*c*d*n^
4*x^3-1350*a*b^5*d^2*n^2*x^5+25*b^6*c^2*n^5*x^2+1850*b^6*c*d*n^3*x^4+1764*b^6*d^2*n*x^6-120*a^3*b^3*d^2*n^3*x^
3+24*a^2*b^4*c*d*n^4*x^2+1050*a^2*b^4*d^2*n^2*x^4-2*a*b^5*c^2*n^5*x-1048*a*b^5*c*d*n^3*x^3-1644*a*b^5*d^2*n*x^
5+247*b^6*c^2*n^4*x^2+4288*b^6*c*d*n^2*x^4+720*b^6*d^2*x^6-720*a^3*b^3*d^2*n^2*x^3+384*a^2*b^4*c*d*n^3*x^2+150
0*a^2*b^4*d^2*n*x^4-46*a*b^5*c^2*n^4*x-3208*a*b^5*c*d*n^2*x^3-720*a*b^5*d^2*x^5+1219*b^6*c^2*n^3*x^2+4824*b^6*
c*d*n*x^4+360*a^4*b^2*d^2*n^2*x^2-48*a^3*b^3*c*d*n^3*x-1320*a^3*b^3*d^2*n*x^3+2*a^2*b^4*c^2*n^4+1992*a^2*b^4*c
*d*n^2*x^2+720*a^2*b^4*d^2*x^4-402*a*b^5*c^2*n^3*x-4320*a*b^5*c*d*n*x^3+3112*b^6*c^2*n^2*x^2+2016*b^6*c*d*x^4+
1080*a^4*b^2*d^2*n*x^2-672*a^3*b^3*c*d*n^2*x-720*a^3*b^3*d^2*x^3+44*a^2*b^4*c^2*n^3+3648*a^2*b^4*c*d*n*x^2-163
4*a*b^5*c^2*n^2*x-2016*a*b^5*c*d*x^3+3796*b^6*c^2*n*x^2-720*a^5*b*d^2*n*x+48*a^4*b^2*c*d*n^2+720*a^4*b^2*d^2*x
^2-2640*a^3*b^3*c*d*n*x+358*a^2*b^4*c^2*n^2+2016*a^2*b^4*c*d*x^2-2956*a*b^5*c^2*n*x+1680*b^6*c^2*x^2-720*a^5*b
*d^2*x+624*a^4*b^2*c*d*n-2016*a^3*b^3*c*d*x+1276*a^2*b^4*c^2*n-1680*a*b^5*c^2*x+720*a^6*d^2+2016*a^4*b^2*c*d+1
680*a^2*b^4*c^2)/b^7/(n^7+28*n^6+322*n^5+1960*n^4+6769*n^3+13132*n^2+13068*n+5040)

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Maxima [A]  time = 1.0588, size = 603, normalized size = 2.6 \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^{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 d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} + \frac{{\left ({\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{7} x^{7} +{\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a b^{6} x^{6} - 6 \,{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{2} b^{5} x^{5} + 30 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{3} b^{4} x^{4} - 120 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{4} b^{3} x^{3} + 360 \,{\left (n^{2} + n\right )} a^{5} b^{2} x^{2} - 720 \, a^{6} b n x + 720 \, a^{7}\right )}{\left (b x + a\right )}^{n} d^{2}}{{\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^n*(d*x^2+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^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*d/((n^5 + 15*n^4 + 85
*n^3 + 225*n^2 + 274*n + 120)*b^5) + ((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*d^2/((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2
+ 13068*n + 5040)*b^7)

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Fricas [B]  time = 1.99366, size = 2246, normalized size = 9.68 \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)^2,x, algorithm="fricas")

[Out]

(2*a^3*b^4*c^2*n^4 + 44*a^3*b^4*c^2*n^3 + 1680*a^3*b^4*c^2 + 2016*a^5*b^2*c*d + 720*a^7*d^2 + (b^7*d^2*n^6 + 2
1*b^7*d^2*n^5 + 175*b^7*d^2*n^4 + 735*b^7*d^2*n^3 + 1624*b^7*d^2*n^2 + 1764*b^7*d^2*n + 720*b^7*d^2)*x^7 + (a*
b^6*d^2*n^6 + 15*a*b^6*d^2*n^5 + 85*a*b^6*d^2*n^4 + 225*a*b^6*d^2*n^3 + 274*a*b^6*d^2*n^2 + 120*a*b^6*d^2*n)*x
^6 + 2*(b^7*c*d*n^6 + 1008*b^7*c*d + (23*b^7*c*d - 3*a^2*b^5*d^2)*n^5 + 3*(69*b^7*c*d - 10*a^2*b^5*d^2)*n^4 +
5*(185*b^7*c*d - 21*a^2*b^5*d^2)*n^3 + 2*(1072*b^7*c*d - 75*a^2*b^5*d^2)*n^2 + 36*(67*b^7*c*d - 2*a^2*b^5*d^2)
*n)*x^5 + 2*(a*b^6*c*d*n^6 + 19*a*b^6*c*d*n^5 + (131*a*b^6*c*d + 15*a^3*b^4*d^2)*n^4 + (401*a*b^6*c*d + 90*a^3
*b^4*d^2)*n^3 + 15*(36*a*b^6*c*d + 11*a^3*b^4*d^2)*n^2 + 18*(14*a*b^6*c*d + 5*a^3*b^4*d^2)*n)*x^4 + (b^7*c^2*n
^6 + 1680*b^7*c^2 + (25*b^7*c^2 - 8*a^2*b^5*c*d)*n^5 + (247*b^7*c^2 - 128*a^2*b^5*c*d)*n^4 + (1219*b^7*c^2 - 6
64*a^2*b^5*c*d - 120*a^4*b^3*d^2)*n^3 + 8*(389*b^7*c^2 - 152*a^2*b^5*c*d - 45*a^4*b^3*d^2)*n^2 + 4*(949*b^7*c^
2 - 168*a^2*b^5*c*d - 60*a^4*b^3*d^2)*n)*x^3 + 2*(179*a^3*b^4*c^2 + 24*a^5*b^2*c*d)*n^2 + (a*b^6*c^2*n^6 + 23*
a*b^6*c^2*n^5 + 3*(67*a*b^6*c^2 + 8*a^3*b^4*c*d)*n^4 + (817*a*b^6*c^2 + 336*a^3*b^4*c*d)*n^3 + 2*(739*a*b^6*c^
2 + 660*a^3*b^4*c*d + 180*a^5*b^2*d^2)*n^2 + 24*(35*a*b^6*c^2 + 42*a^3*b^4*c*d + 15*a^5*b^2*d^2)*n)*x^2 + 4*(3
19*a^3*b^4*c^2 + 156*a^5*b^2*c*d)*n - 2*(a^2*b^5*c^2*n^5 + 22*a^2*b^5*c^2*n^4 + (179*a^2*b^5*c^2 + 24*a^4*b^3*
c*d)*n^3 + 2*(319*a^2*b^5*c^2 + 156*a^4*b^3*c*d)*n^2 + 24*(35*a^2*b^5*c^2 + 42*a^4*b^3*c*d + 15*a^6*b*d^2)*n)*
x)*(b*x + a)^n/(b^7*n^7 + 28*b^7*n^6 + 322*b^7*n^5 + 1960*b^7*n^4 + 6769*b^7*n^3 + 13132*b^7*n^2 + 13068*b^7*n
+ 5040*b^7)

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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**2+c)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.15821, size = 2363, normalized size = 10.19 \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)^2,x, algorithm="giac")

[Out]

((b*x + a)^n*b^7*d^2*n^6*x^7 + (b*x + a)^n*a*b^6*d^2*n^6*x^6 + 21*(b*x + a)^n*b^7*d^2*n^5*x^7 + 2*(b*x + a)^n*
b^7*c*d*n^6*x^5 + 15*(b*x + a)^n*a*b^6*d^2*n^5*x^6 + 175*(b*x + a)^n*b^7*d^2*n^4*x^7 + 2*(b*x + a)^n*a*b^6*c*d
*n^6*x^4 + 46*(b*x + a)^n*b^7*c*d*n^5*x^5 - 6*(b*x + a)^n*a^2*b^5*d^2*n^5*x^5 + 85*(b*x + a)^n*a*b^6*d^2*n^4*x
^6 + 735*(b*x + a)^n*b^7*d^2*n^3*x^7 + (b*x + a)^n*b^7*c^2*n^6*x^3 + 38*(b*x + a)^n*a*b^6*c*d*n^5*x^4 + 414*(b
*x + a)^n*b^7*c*d*n^4*x^5 - 60*(b*x + a)^n*a^2*b^5*d^2*n^4*x^5 + 225*(b*x + a)^n*a*b^6*d^2*n^3*x^6 + 1624*(b*x
+ a)^n*b^7*d^2*n^2*x^7 + (b*x + a)^n*a*b^6*c^2*n^6*x^2 + 25*(b*x + a)^n*b^7*c^2*n^5*x^3 - 8*(b*x + a)^n*a^2*b
^5*c*d*n^5*x^3 + 262*(b*x + a)^n*a*b^6*c*d*n^4*x^4 + 30*(b*x + a)^n*a^3*b^4*d^2*n^4*x^4 + 1850*(b*x + a)^n*b^7
*c*d*n^3*x^5 - 210*(b*x + a)^n*a^2*b^5*d^2*n^3*x^5 + 274*(b*x + a)^n*a*b^6*d^2*n^2*x^6 + 1764*(b*x + a)^n*b^7*
d^2*n*x^7 + 23*(b*x + a)^n*a*b^6*c^2*n^5*x^2 + 247*(b*x + a)^n*b^7*c^2*n^4*x^3 - 128*(b*x + a)^n*a^2*b^5*c*d*n
^4*x^3 + 802*(b*x + a)^n*a*b^6*c*d*n^3*x^4 + 180*(b*x + a)^n*a^3*b^4*d^2*n^3*x^4 + 4288*(b*x + a)^n*b^7*c*d*n^
2*x^5 - 300*(b*x + a)^n*a^2*b^5*d^2*n^2*x^5 + 120*(b*x + a)^n*a*b^6*d^2*n*x^6 + 720*(b*x + a)^n*b^7*d^2*x^7 -
2*(b*x + a)^n*a^2*b^5*c^2*n^5*x + 201*(b*x + a)^n*a*b^6*c^2*n^4*x^2 + 24*(b*x + a)^n*a^3*b^4*c*d*n^4*x^2 + 121
9*(b*x + a)^n*b^7*c^2*n^3*x^3 - 664*(b*x + a)^n*a^2*b^5*c*d*n^3*x^3 - 120*(b*x + a)^n*a^4*b^3*d^2*n^3*x^3 + 10
80*(b*x + a)^n*a*b^6*c*d*n^2*x^4 + 330*(b*x + a)^n*a^3*b^4*d^2*n^2*x^4 + 4824*(b*x + a)^n*b^7*c*d*n*x^5 - 144*
(b*x + a)^n*a^2*b^5*d^2*n*x^5 - 44*(b*x + a)^n*a^2*b^5*c^2*n^4*x + 817*(b*x + a)^n*a*b^6*c^2*n^3*x^2 + 336*(b*
x + a)^n*a^3*b^4*c*d*n^3*x^2 + 3112*(b*x + a)^n*b^7*c^2*n^2*x^3 - 1216*(b*x + a)^n*a^2*b^5*c*d*n^2*x^3 - 360*(
b*x + a)^n*a^4*b^3*d^2*n^2*x^3 + 504*(b*x + a)^n*a*b^6*c*d*n*x^4 + 180*(b*x + a)^n*a^3*b^4*d^2*n*x^4 + 2016*(b
*x + a)^n*b^7*c*d*x^5 + 2*(b*x + a)^n*a^3*b^4*c^2*n^4 - 358*(b*x + a)^n*a^2*b^5*c^2*n^3*x - 48*(b*x + a)^n*a^4
*b^3*c*d*n^3*x + 1478*(b*x + a)^n*a*b^6*c^2*n^2*x^2 + 1320*(b*x + a)^n*a^3*b^4*c*d*n^2*x^2 + 360*(b*x + a)^n*a
^5*b^2*d^2*n^2*x^2 + 3796*(b*x + a)^n*b^7*c^2*n*x^3 - 672*(b*x + a)^n*a^2*b^5*c*d*n*x^3 - 240*(b*x + a)^n*a^4*
b^3*d^2*n*x^3 + 44*(b*x + a)^n*a^3*b^4*c^2*n^3 - 1276*(b*x + a)^n*a^2*b^5*c^2*n^2*x - 624*(b*x + a)^n*a^4*b^3*
c*d*n^2*x + 840*(b*x + a)^n*a*b^6*c^2*n*x^2 + 1008*(b*x + a)^n*a^3*b^4*c*d*n*x^2 + 360*(b*x + a)^n*a^5*b^2*d^2
*n*x^2 + 1680*(b*x + a)^n*b^7*c^2*x^3 + 358*(b*x + a)^n*a^3*b^4*c^2*n^2 + 48*(b*x + a)^n*a^5*b^2*c*d*n^2 - 168
0*(b*x + a)^n*a^2*b^5*c^2*n*x - 2016*(b*x + a)^n*a^4*b^3*c*d*n*x - 720*(b*x + a)^n*a^6*b*d^2*n*x + 1276*(b*x +
a)^n*a^3*b^4*c^2*n + 624*(b*x + a)^n*a^5*b^2*c*d*n + 1680*(b*x + a)^n*a^3*b^4*c^2 + 2016*(b*x + a)^n*a^5*b^2*
c*d + 720*(b*x + a)^n*a^7*d^2)/(b^7*n^7 + 28*b^7*n^6 + 322*b^7*n^5 + 1960*b^7*n^4 + 6769*b^7*n^3 + 13132*b^7*n
^2 + 13068*b^7*n + 5040*b^7)