3.361 \(\int (a+b x)^n (c+d x^2)^3 \, dx\)
Optimal. Leaf size=223 \[ -\frac{4 a d^2 \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac{3 d^2 \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}+\frac{\left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^7 (n+1)}-\frac{6 a d \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+2}}{b^7 (n+2)}+\frac{3 d \left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^7 (n+3)}-\frac{6 a d^3 (a+b x)^{n+6}}{b^7 (n+6)}+\frac{d^3 (a+b x)^{n+7}}{b^7 (n+7)} \]
[Out]
((b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^7*(1 + n)) - (6*a*d*(b^2*c + a^2*d)^2*(a + b*x)^(2 + n))/(b^7*(2 + n)
) + (3*d*(b^2*c + a^2*d)*(b^2*c + 5*a^2*d)*(a + b*x)^(3 + n))/(b^7*(3 + n)) - (4*a*d^2*(3*b^2*c + 5*a^2*d)*(a
+ b*x)^(4 + n))/(b^7*(4 + n)) + (3*d^2*(b^2*c + 5*a^2*d)*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (6*a*d^3*(a + b*x)
^(6 + n))/(b^7*(6 + n)) + (d^3*(a + b*x)^(7 + n))/(b^7*(7 + n))
________________________________________________________________________________________
Rubi [A] time = 0.125572, antiderivative size = 223, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used =
{697} \[ -\frac{4 a d^2 \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac{3 d^2 \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}+\frac{\left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^7 (n+1)}-\frac{6 a d \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+2}}{b^7 (n+2)}+\frac{3 d \left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^7 (n+3)}-\frac{6 a d^3 (a+b x)^{n+6}}{b^7 (n+6)}+\frac{d^3 (a+b x)^{n+7}}{b^7 (n+7)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
((b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^7*(1 + n)) - (6*a*d*(b^2*c + a^2*d)^2*(a + b*x)^(2 + n))/(b^7*(2 + n)
) + (3*d*(b^2*c + a^2*d)*(b^2*c + 5*a^2*d)*(a + b*x)^(3 + n))/(b^7*(3 + n)) - (4*a*d^2*(3*b^2*c + 5*a^2*d)*(a
+ b*x)^(4 + n))/(b^7*(4 + n)) + (3*d^2*(b^2*c + 5*a^2*d)*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (6*a*d^3*(a + b*x)
^(6 + n))/(b^7*(6 + n)) + (d^3*(a + b*x)^(7 + n))/(b^7*(7 + n))
Rule 697
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]
Rubi steps
\begin{align*} \int (a+b x)^n \left (c+d x^2\right )^3 \, dx &=\int \left (\frac{\left (b^2 c+a^2 d\right )^3 (a+b x)^n}{b^6}-\frac{6 a d \left (b^2 c+a^2 d\right )^2 (a+b x)^{1+n}}{b^6}+\frac{3 d \left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{2+n}}{b^6}-\frac{4 a d^2 \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^6}+\frac{3 d^2 \left (b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^6}-\frac{6 a d^3 (a+b x)^{5+n}}{b^6}+\frac{d^3 (a+b x)^{6+n}}{b^6}\right ) \, dx\\ &=\frac{\left (b^2 c+a^2 d\right )^3 (a+b x)^{1+n}}{b^7 (1+n)}-\frac{6 a d \left (b^2 c+a^2 d\right )^2 (a+b x)^{2+n}}{b^7 (2+n)}+\frac{3 d \left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^7 (3+n)}-\frac{4 a d^2 \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^7 (4+n)}+\frac{3 d^2 \left (b^2 c+5 a^2 d\right ) (a+b x)^{5+n}}{b^7 (5+n)}-\frac{6 a d^3 (a+b x)^{6+n}}{b^7 (6+n)}+\frac{d^3 (a+b x)^{7+n}}{b^7 (7+n)}\\ \end{align*}
Mathematica [A] time = 0.529307, size = 347, normalized size = 1.56 \[ \frac{(a+b x)^{n+1} \left (\frac{6 \left ((n+6) \left (a^2 d+b^2 c\right ) \left (4 (n+4) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left (c (n+3)+d (n+1) x^2\right )\right )-4 a d (n+1) (a+b x) \left (2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left (c (n+4)+d (n+2) x^2\right )\right )+b^4 (n+1) (n+2) (n+3) (n+4) \left (c+d x^2\right )^2\right )-a d (n+1) (a+b x) \left (4 (n+5) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left (c (n+4)+d (n+2) x^2\right )\right )-4 a d (n+2) (a+b x) \left (2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left (c (n+5)+d (n+3) x^2\right )\right )+b^4 (n+2) (n+3) (n+4) (n+5) \left (c+d x^2\right )^2\right )\right )}{b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)}+\left (c+d x^2\right )^3\right )}{b (n+7)} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
((a + b*x)^(1 + n)*((c + d*x^2)^3 + (6*((b^2*c + a^2*d)*(6 + n)*(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(c + d*x^
2)^2 + 4*(b^2*c + a^2*d)*(4 + n)*(2*a^2*d - 2*a*b*d*(1 + n)*x + b^2*(2 + n)*(c*(3 + n) + d*(1 + n)*x^2)) - 4*a
*d*(1 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2))) - a*d*(1 + n)*(a
+ b*x)*(b^4*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(5 + n)*(2*a^2*d - 2*a*b*d*(2 +
n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2)) - 4*a*d*(2 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(3 + n)*x + b^2*
(4 + n)*(c*(5 + n) + d*(3 + n)*x^2)))))/(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n))))/(b*(7 + n))
________________________________________________________________________________________
Maple [B] time = 0.058, size = 1140, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^n*(d*x^2+c)^3,x)
[Out]
(b*x+a)^(1+n)*(b^6*d^3*n^6*x^6+21*b^6*d^3*n^5*x^6-6*a*b^5*d^3*n^5*x^5+3*b^6*c*d^2*n^6*x^4+175*b^6*d^3*n^4*x^6-
90*a*b^5*d^3*n^4*x^5+69*b^6*c*d^2*n^5*x^4+735*b^6*d^3*n^3*x^6+30*a^2*b^4*d^3*n^4*x^4-12*a*b^5*c*d^2*n^5*x^3-51
0*a*b^5*d^3*n^3*x^5+3*b^6*c^2*d*n^6*x^2+621*b^6*c*d^2*n^4*x^4+1624*b^6*d^3*n^2*x^6+300*a^2*b^4*d^3*n^3*x^4-228
*a*b^5*c*d^2*n^4*x^3-1350*a*b^5*d^3*n^2*x^5+75*b^6*c^2*d*n^5*x^2+2775*b^6*c*d^2*n^3*x^4+1764*b^6*d^3*n*x^6-120
*a^3*b^3*d^3*n^3*x^3+36*a^2*b^4*c*d^2*n^4*x^2+1050*a^2*b^4*d^3*n^2*x^4-6*a*b^5*c^2*d*n^5*x-1572*a*b^5*c*d^2*n^
3*x^3-1644*a*b^5*d^3*n*x^5+b^6*c^3*n^6+741*b^6*c^2*d*n^4*x^2+6432*b^6*c*d^2*n^2*x^4+720*b^6*d^3*x^6-720*a^3*b^
3*d^3*n^2*x^3+576*a^2*b^4*c*d^2*n^3*x^2+1500*a^2*b^4*d^3*n*x^4-138*a*b^5*c^2*d*n^4*x-4812*a*b^5*c*d^2*n^2*x^3-
720*a*b^5*d^3*x^5+27*b^6*c^3*n^5+3657*b^6*c^2*d*n^3*x^2+7236*b^6*c*d^2*n*x^4+360*a^4*b^2*d^3*n^2*x^2-72*a^3*b^
3*c*d^2*n^3*x-1320*a^3*b^3*d^3*n*x^3+6*a^2*b^4*c^2*d*n^4+2988*a^2*b^4*c*d^2*n^2*x^2+720*a^2*b^4*d^3*x^4-1206*a
*b^5*c^2*d*n^3*x-6480*a*b^5*c*d^2*n*x^3+295*b^6*c^3*n^4+9336*b^6*c^2*d*n^2*x^2+3024*b^6*c*d^2*x^4+1080*a^4*b^2
*d^3*n*x^2-1008*a^3*b^3*c*d^2*n^2*x-720*a^3*b^3*d^3*x^3+132*a^2*b^4*c^2*d*n^3+5472*a^2*b^4*c*d^2*n*x^2-4902*a*
b^5*c^2*d*n^2*x-3024*a*b^5*c*d^2*x^3+1665*b^6*c^3*n^3+11388*b^6*c^2*d*n*x^2-720*a^5*b*d^3*n*x+72*a^4*b^2*c*d^2
*n^2+720*a^4*b^2*d^3*x^2-3960*a^3*b^3*c*d^2*n*x+1074*a^2*b^4*c^2*d*n^2+3024*a^2*b^4*c*d^2*x^2-8868*a*b^5*c^2*d
*n*x+5104*b^6*c^3*n^2+5040*b^6*c^2*d*x^2-720*a^5*b*d^3*x+936*a^4*b^2*c*d^2*n-3024*a^3*b^3*c*d^2*x+3828*a^2*b^4
*c^2*d*n-5040*a*b^5*c^2*d*x+8028*b^6*c^3*n+720*a^6*d^3+3024*a^4*b^2*c*d^2+5040*a^2*b^4*c^2*d+5040*b^6*c^3)/b^7
/(n^7+28*n^6+322*n^5+1960*n^4+6769*n^3+13132*n^2+13068*n+5040)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 2.15924, size = 2692, normalized size = 12.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^3,x, algorithm="fricas")
[Out]
(a*b^6*c^3*n^6 + 27*a*b^6*c^3*n^5 + 5040*a*b^6*c^3 + 5040*a^3*b^4*c^2*d + 3024*a^5*b^2*c*d^2 + 720*a^7*d^3 + (
b^7*d^3*n^6 + 21*b^7*d^3*n^5 + 175*b^7*d^3*n^4 + 735*b^7*d^3*n^3 + 1624*b^7*d^3*n^2 + 1764*b^7*d^3*n + 720*b^7
*d^3)*x^7 + (a*b^6*d^3*n^6 + 15*a*b^6*d^3*n^5 + 85*a*b^6*d^3*n^4 + 225*a*b^6*d^3*n^3 + 274*a*b^6*d^3*n^2 + 120
*a*b^6*d^3*n)*x^6 + 3*(b^7*c*d^2*n^6 + 1008*b^7*c*d^2 + (23*b^7*c*d^2 - 2*a^2*b^5*d^3)*n^5 + (207*b^7*c*d^2 -
20*a^2*b^5*d^3)*n^4 + 5*(185*b^7*c*d^2 - 14*a^2*b^5*d^3)*n^3 + 4*(536*b^7*c*d^2 - 25*a^2*b^5*d^3)*n^2 + 12*(20
1*b^7*c*d^2 - 4*a^2*b^5*d^3)*n)*x^5 + (295*a*b^6*c^3 + 6*a^3*b^4*c^2*d)*n^4 + 3*(a*b^6*c*d^2*n^6 + 19*a*b^6*c*
d^2*n^5 + (131*a*b^6*c*d^2 + 10*a^3*b^4*d^3)*n^4 + (401*a*b^6*c*d^2 + 60*a^3*b^4*d^3)*n^3 + 10*(54*a*b^6*c*d^2
+ 11*a^3*b^4*d^3)*n^2 + 12*(21*a*b^6*c*d^2 + 5*a^3*b^4*d^3)*n)*x^4 + 3*(555*a*b^6*c^3 + 44*a^3*b^4*c^2*d)*n^3
+ 3*(b^7*c^2*d*n^6 + 1680*b^7*c^2*d + (25*b^7*c^2*d - 4*a^2*b^5*c*d^2)*n^5 + (247*b^7*c^2*d - 64*a^2*b^5*c*d^
2)*n^4 + (1219*b^7*c^2*d - 332*a^2*b^5*c*d^2 - 40*a^4*b^3*d^3)*n^3 + 8*(389*b^7*c^2*d - 76*a^2*b^5*c*d^2 - 15*
a^4*b^3*d^3)*n^2 + 4*(949*b^7*c^2*d - 84*a^2*b^5*c*d^2 - 20*a^4*b^3*d^3)*n)*x^3 + 2*(2552*a*b^6*c^3 + 537*a^3*
b^4*c^2*d + 36*a^5*b^2*c*d^2)*n^2 + 3*(a*b^6*c^2*d*n^6 + 23*a*b^6*c^2*d*n^5 + 3*(67*a*b^6*c^2*d + 4*a^3*b^4*c*
d^2)*n^4 + (817*a*b^6*c^2*d + 168*a^3*b^4*c*d^2)*n^3 + 2*(739*a*b^6*c^2*d + 330*a^3*b^4*c*d^2 + 60*a^5*b^2*d^3
)*n^2 + 24*(35*a*b^6*c^2*d + 21*a^3*b^4*c*d^2 + 5*a^5*b^2*d^3)*n)*x^2 + 12*(669*a*b^6*c^3 + 319*a^3*b^4*c^2*d
+ 78*a^5*b^2*c*d^2)*n + (b^7*c^3*n^6 + 5040*b^7*c^3 + 3*(9*b^7*c^3 - 2*a^2*b^5*c^2*d)*n^5 + (295*b^7*c^3 - 132
*a^2*b^5*c^2*d)*n^4 + 3*(555*b^7*c^3 - 358*a^2*b^5*c^2*d - 24*a^4*b^3*c*d^2)*n^3 + 4*(1276*b^7*c^3 - 957*a^2*b
^5*c^2*d - 234*a^4*b^3*c*d^2)*n^2 + 36*(223*b^7*c^3 - 140*a^2*b^5*c^2*d - 84*a^4*b^3*c*d^2 - 20*a^6*b*d^3)*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)
________________________________________________________________________________________
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((b*x+a)**n*(d*x**2+c)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.2109, size = 2815, normalized size = 12.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^3,x, algorithm="giac")
[Out]
((b*x + a)^n*b^7*d^3*n^6*x^7 + (b*x + a)^n*a*b^6*d^3*n^6*x^6 + 21*(b*x + a)^n*b^7*d^3*n^5*x^7 + 3*(b*x + a)^n*
b^7*c*d^2*n^6*x^5 + 15*(b*x + a)^n*a*b^6*d^3*n^5*x^6 + 175*(b*x + a)^n*b^7*d^3*n^4*x^7 + 3*(b*x + a)^n*a*b^6*c
*d^2*n^6*x^4 + 69*(b*x + a)^n*b^7*c*d^2*n^5*x^5 - 6*(b*x + a)^n*a^2*b^5*d^3*n^5*x^5 + 85*(b*x + a)^n*a*b^6*d^3
*n^4*x^6 + 735*(b*x + a)^n*b^7*d^3*n^3*x^7 + 3*(b*x + a)^n*b^7*c^2*d*n^6*x^3 + 57*(b*x + a)^n*a*b^6*c*d^2*n^5*
x^4 + 621*(b*x + a)^n*b^7*c*d^2*n^4*x^5 - 60*(b*x + a)^n*a^2*b^5*d^3*n^4*x^5 + 225*(b*x + a)^n*a*b^6*d^3*n^3*x
^6 + 1624*(b*x + a)^n*b^7*d^3*n^2*x^7 + 3*(b*x + a)^n*a*b^6*c^2*d*n^6*x^2 + 75*(b*x + a)^n*b^7*c^2*d*n^5*x^3 -
12*(b*x + a)^n*a^2*b^5*c*d^2*n^5*x^3 + 393*(b*x + a)^n*a*b^6*c*d^2*n^4*x^4 + 30*(b*x + a)^n*a^3*b^4*d^3*n^4*x
^4 + 2775*(b*x + a)^n*b^7*c*d^2*n^3*x^5 - 210*(b*x + a)^n*a^2*b^5*d^3*n^3*x^5 + 274*(b*x + a)^n*a*b^6*d^3*n^2*
x^6 + 1764*(b*x + a)^n*b^7*d^3*n*x^7 + (b*x + a)^n*b^7*c^3*n^6*x + 69*(b*x + a)^n*a*b^6*c^2*d*n^5*x^2 + 741*(b
*x + a)^n*b^7*c^2*d*n^4*x^3 - 192*(b*x + a)^n*a^2*b^5*c*d^2*n^4*x^3 + 1203*(b*x + a)^n*a*b^6*c*d^2*n^3*x^4 + 1
80*(b*x + a)^n*a^3*b^4*d^3*n^3*x^4 + 6432*(b*x + a)^n*b^7*c*d^2*n^2*x^5 - 300*(b*x + a)^n*a^2*b^5*d^3*n^2*x^5
+ 120*(b*x + a)^n*a*b^6*d^3*n*x^6 + 720*(b*x + a)^n*b^7*d^3*x^7 + (b*x + a)^n*a*b^6*c^3*n^6 + 27*(b*x + a)^n*b
^7*c^3*n^5*x - 6*(b*x + a)^n*a^2*b^5*c^2*d*n^5*x + 603*(b*x + a)^n*a*b^6*c^2*d*n^4*x^2 + 36*(b*x + a)^n*a^3*b^
4*c*d^2*n^4*x^2 + 3657*(b*x + a)^n*b^7*c^2*d*n^3*x^3 - 996*(b*x + a)^n*a^2*b^5*c*d^2*n^3*x^3 - 120*(b*x + a)^n
*a^4*b^3*d^3*n^3*x^3 + 1620*(b*x + a)^n*a*b^6*c*d^2*n^2*x^4 + 330*(b*x + a)^n*a^3*b^4*d^3*n^2*x^4 + 7236*(b*x
+ a)^n*b^7*c*d^2*n*x^5 - 144*(b*x + a)^n*a^2*b^5*d^3*n*x^5 + 27*(b*x + a)^n*a*b^6*c^3*n^5 + 295*(b*x + a)^n*b^
7*c^3*n^4*x - 132*(b*x + a)^n*a^2*b^5*c^2*d*n^4*x + 2451*(b*x + a)^n*a*b^6*c^2*d*n^3*x^2 + 504*(b*x + a)^n*a^3
*b^4*c*d^2*n^3*x^2 + 9336*(b*x + a)^n*b^7*c^2*d*n^2*x^3 - 1824*(b*x + a)^n*a^2*b^5*c*d^2*n^2*x^3 - 360*(b*x +
a)^n*a^4*b^3*d^3*n^2*x^3 + 756*(b*x + a)^n*a*b^6*c*d^2*n*x^4 + 180*(b*x + a)^n*a^3*b^4*d^3*n*x^4 + 3024*(b*x +
a)^n*b^7*c*d^2*x^5 + 295*(b*x + a)^n*a*b^6*c^3*n^4 + 6*(b*x + a)^n*a^3*b^4*c^2*d*n^4 + 1665*(b*x + a)^n*b^7*c
^3*n^3*x - 1074*(b*x + a)^n*a^2*b^5*c^2*d*n^3*x - 72*(b*x + a)^n*a^4*b^3*c*d^2*n^3*x + 4434*(b*x + a)^n*a*b^6*
c^2*d*n^2*x^2 + 1980*(b*x + a)^n*a^3*b^4*c*d^2*n^2*x^2 + 360*(b*x + a)^n*a^5*b^2*d^3*n^2*x^2 + 11388*(b*x + a)
^n*b^7*c^2*d*n*x^3 - 1008*(b*x + a)^n*a^2*b^5*c*d^2*n*x^3 - 240*(b*x + a)^n*a^4*b^3*d^3*n*x^3 + 1665*(b*x + a)
^n*a*b^6*c^3*n^3 + 132*(b*x + a)^n*a^3*b^4*c^2*d*n^3 + 5104*(b*x + a)^n*b^7*c^3*n^2*x - 3828*(b*x + a)^n*a^2*b
^5*c^2*d*n^2*x - 936*(b*x + a)^n*a^4*b^3*c*d^2*n^2*x + 2520*(b*x + a)^n*a*b^6*c^2*d*n*x^2 + 1512*(b*x + a)^n*a
^3*b^4*c*d^2*n*x^2 + 360*(b*x + a)^n*a^5*b^2*d^3*n*x^2 + 5040*(b*x + a)^n*b^7*c^2*d*x^3 + 5104*(b*x + a)^n*a*b
^6*c^3*n^2 + 1074*(b*x + a)^n*a^3*b^4*c^2*d*n^2 + 72*(b*x + a)^n*a^5*b^2*c*d^2*n^2 + 8028*(b*x + a)^n*b^7*c^3*
n*x - 5040*(b*x + a)^n*a^2*b^5*c^2*d*n*x - 3024*(b*x + a)^n*a^4*b^3*c*d^2*n*x - 720*(b*x + a)^n*a^6*b*d^3*n*x
+ 8028*(b*x + a)^n*a*b^6*c^3*n + 3828*(b*x + a)^n*a^3*b^4*c^2*d*n + 936*(b*x + a)^n*a^5*b^2*c*d^2*n + 5040*(b*
x + a)^n*b^7*c^3*x + 5040*(b*x + a)^n*a*b^6*c^3 + 5040*(b*x + a)^n*a^3*b^4*c^2*d + 3024*(b*x + a)^n*a^5*b^2*c*
d^2 + 720*(b*x + a)^n*a^7*d^3)/(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)