3.26 \(\int (a+b x)^2 (c+d x)^n (A+B x+C x^2+D x^3) \, dx\)
Optimal. Leaf size=338 \[ \frac{(c+d x)^{n+3} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )}{d^6 (n+3)}+\frac{(c+d x)^{n+4} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^6 (n+4)}+\frac{(b c-a d)^2 (c+d x)^{n+1} \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^6 (n+1)}+\frac{(b c-a d) (c+d x)^{n+2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right )\right )}{d^6 (n+2)}+\frac{b (c+d x)^{n+5} (2 a d D-5 b c D+b C d)}{d^6 (n+5)}+\frac{b^2 D (c+d x)^{n+6}}{d^6 (n+6)} \]
[Out]
((b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^6*(1 + n)) + ((b*c - a*d)*(a*d*(2*c*C
*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 + 2*A*d^3 - 5*c^3*D))*(c + d*x)^(2 + n))/(d^6*(2 + n)) + ((a^
2*d^2*(C*d - 3*c*D) - 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c
+ d*x)^(3 + n))/(d^6*(3 + n)) + ((a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c +
d*x)^(4 + n))/(d^6*(4 + n)) + (b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(5 + n))/(d^6*(5 + n)) + (b^2*D*(c + d*
x)^(6 + n))/(d^6*(6 + n))
________________________________________________________________________________________
Rubi [A] time = 0.246598, antiderivative size = 338, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used =
{1620} \[ \frac{(c+d x)^{n+3} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )}{d^6 (n+3)}+\frac{(c+d x)^{n+4} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^6 (n+4)}+\frac{(b c-a d)^2 (c+d x)^{n+1} \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^6 (n+1)}+\frac{(b c-a d) (c+d x)^{n+2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2+4 c^2 C d-5 c^3 D\right )\right )}{d^6 (n+2)}+\frac{b (c+d x)^{n+5} (2 a d D-5 b c D+b C d)}{d^6 (n+5)}+\frac{b^2 D (c+d x)^{n+6}}{d^6 (n+6)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^2*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]
[Out]
((b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^6*(1 + n)) + ((b*c - a*d)*(a*d*(2*c*C
*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 + 2*A*d^3 - 5*c^3*D))*(c + d*x)^(2 + n))/(d^6*(2 + n)) + ((a^
2*d^2*(C*d - 3*c*D) - 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c
+ d*x)^(3 + n))/(d^6*(3 + n)) + ((a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c +
d*x)^(4 + n))/(d^6*(4 + n)) + (b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(5 + n))/(d^6*(5 + n)) + (b^2*D*(c + d*
x)^(6 + n))/(d^6*(6 + 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 (a+b x)^2 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (\frac{(-b c+a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^n}{d^5}+\frac{(b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) (c+d x)^{1+n}}{d^5}+\frac{\left (a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^{2+n}}{d^5}+\frac{\left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^{3+n}}{d^5}+\frac{b (b C d-5 b c D+2 a d D) (c+d x)^{4+n}}{d^5}+\frac{b^2 D (c+d x)^{5+n}}{d^5}\right ) \, dx\\ &=\frac{(b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{1+n}}{d^6 (1+n)}+\frac{(b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) (c+d x)^{2+n}}{d^6 (2+n)}+\frac{\left (a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^{3+n}}{d^6 (3+n)}+\frac{\left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^{4+n}}{d^6 (4+n)}+\frac{b (b C d-5 b c D+2 a d D) (c+d x)^{5+n}}{d^6 (5+n)}+\frac{b^2 D (c+d x)^{6+n}}{d^6 (6+n)}\\ \end{align*}
Mathematica [A] time = 0.517371, size = 308, normalized size = 0.91 \[ \frac{(c+d x)^{n+1} \left (\frac{(c+d x)^2 \left (a^2 d^2 (C d-3 c D)+2 a b d \left (B d^2+6 c^2 D-3 c C d\right )+b^2 \left (A d^3-3 B c d^2+6 c^2 C d-10 c^3 D\right )\right )}{n+3}+\frac{(c+d x)^3 \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (B d^2+10 c^2 D-4 c C d\right )\right )}{n+4}+\frac{(c+d x) (b c-a d) \left (b \left (-2 A d^3+3 B c d^2-4 c^2 C d+5 c^3 D\right )-a d \left (B d^2+3 c^2 D-2 c C d\right )\right )}{n+2}+\frac{(b c-a d)^2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{n+1}+\frac{b (c+d x)^4 (2 a d D-5 b c D+b C d)}{n+5}+\frac{b^2 D (c+d x)^5}{n+6}\right )}{d^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^2*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]
[Out]
((c + d*x)^(1 + n)*(((b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(1 + n) + ((b*c - a*d)*(-(a*d*(-2*c*C*
d + B*d^2 + 3*c^2*D)) + b*(-4*c^2*C*d + 3*B*c*d^2 - 2*A*d^3 + 5*c^3*D))*(c + d*x))/(2 + n) + ((a^2*d^2*(C*d -
3*c*D) + 2*a*b*d*(-3*c*C*d + B*d^2 + 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c + d*x)^2)/(
3 + n) + ((a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) + b^2*(-4*c*C*d + B*d^2 + 10*c^2*D))*(c + d*x)^3)/(4 + n) + (b*(b
*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^4)/(5 + n) + (b^2*D*(c + d*x)^5)/(6 + n)))/d^6
________________________________________________________________________________________
Maple [B] time = 0.015, size = 2588, normalized size = 7.7 \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)^2*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x)
[Out]
(d*x+c)^(1+n)*(D*b^2*d^5*n^5*x^5+C*b^2*d^5*n^5*x^4+2*D*a*b*d^5*n^5*x^4+15*D*b^2*d^5*n^4*x^5+B*b^2*d^5*n^5*x^3+
2*C*a*b*d^5*n^5*x^3+16*C*b^2*d^5*n^4*x^4+D*a^2*d^5*n^5*x^3+32*D*a*b*d^5*n^4*x^4-5*D*b^2*c*d^4*n^4*x^4+85*D*b^2
*d^5*n^3*x^5+A*b^2*d^5*n^5*x^2+2*B*a*b*d^5*n^5*x^2+17*B*b^2*d^5*n^4*x^3+C*a^2*d^5*n^5*x^2+34*C*a*b*d^5*n^4*x^3
-4*C*b^2*c*d^4*n^4*x^3+95*C*b^2*d^5*n^3*x^4+17*D*a^2*d^5*n^4*x^3-8*D*a*b*c*d^4*n^4*x^3+190*D*a*b*d^5*n^3*x^4-5
0*D*b^2*c*d^4*n^3*x^4+225*D*b^2*d^5*n^2*x^5+2*A*a*b*d^5*n^5*x+18*A*b^2*d^5*n^4*x^2+B*a^2*d^5*n^5*x+36*B*a*b*d^
5*n^4*x^2-3*B*b^2*c*d^4*n^4*x^2+107*B*b^2*d^5*n^3*x^3+18*C*a^2*d^5*n^4*x^2-6*C*a*b*c*d^4*n^4*x^2+214*C*a*b*d^5
*n^3*x^3-48*C*b^2*c*d^4*n^3*x^3+260*C*b^2*d^5*n^2*x^4-3*D*a^2*c*d^4*n^4*x^2+107*D*a^2*d^5*n^3*x^3-96*D*a*b*c*d
^4*n^3*x^3+520*D*a*b*d^5*n^2*x^4+20*D*b^2*c^2*d^3*n^3*x^3-175*D*b^2*c*d^4*n^2*x^4+274*D*b^2*d^5*n*x^5+A*a^2*d^
5*n^5+38*A*a*b*d^5*n^4*x-2*A*b^2*c*d^4*n^4*x+121*A*b^2*d^5*n^3*x^2+19*B*a^2*d^5*n^4*x-4*B*a*b*c*d^4*n^4*x+242*
B*a*b*d^5*n^3*x^2-42*B*b^2*c*d^4*n^3*x^2+307*B*b^2*d^5*n^2*x^3-2*C*a^2*c*d^4*n^4*x+121*C*a^2*d^5*n^3*x^2-84*C*
a*b*c*d^4*n^3*x^2+614*C*a*b*d^5*n^2*x^3+12*C*b^2*c^2*d^3*n^3*x^2-188*C*b^2*c*d^4*n^2*x^3+324*C*b^2*d^5*n*x^4-4
2*D*a^2*c*d^4*n^3*x^2+307*D*a^2*d^5*n^2*x^3+24*D*a*b*c^2*d^3*n^3*x^2-376*D*a*b*c*d^4*n^2*x^3+648*D*a*b*d^5*n*x
^4+120*D*b^2*c^2*d^3*n^2*x^3-250*D*b^2*c*d^4*n*x^4+120*D*b^2*d^5*x^5+20*A*a^2*d^5*n^4-2*A*a*b*c*d^4*n^4+274*A*
a*b*d^5*n^3*x-32*A*b^2*c*d^4*n^3*x+372*A*b^2*d^5*n^2*x^2-B*a^2*c*d^4*n^4+137*B*a^2*d^5*n^3*x-64*B*a*b*c*d^4*n^
3*x+744*B*a*b*d^5*n^2*x^2+6*B*b^2*c^2*d^3*n^3*x-195*B*b^2*c*d^4*n^2*x^2+396*B*b^2*d^5*n*x^3-32*C*a^2*c*d^4*n^3
*x+372*C*a^2*d^5*n^2*x^2+12*C*a*b*c^2*d^3*n^3*x-390*C*a*b*c*d^4*n^2*x^2+792*C*a*b*d^5*n*x^3+108*C*b^2*c^2*d^3*
n^2*x^2-288*C*b^2*c*d^4*n*x^3+144*C*b^2*d^5*x^4+6*D*a^2*c^2*d^3*n^3*x-195*D*a^2*c*d^4*n^2*x^2+396*D*a^2*d^5*n*
x^3+216*D*a*b*c^2*d^3*n^2*x^2-576*D*a*b*c*d^4*n*x^3+288*D*a*b*d^5*x^4-60*D*b^2*c^3*d^2*n^2*x^2+220*D*b^2*c^2*d
^3*n*x^3-120*D*b^2*c*d^4*x^4+155*A*a^2*d^5*n^3-36*A*a*b*c*d^4*n^3+922*A*a*b*d^5*n^2*x+2*A*b^2*c^2*d^3*n^3-178*
A*b^2*c*d^4*n^2*x+508*A*b^2*d^5*n*x^2-18*B*a^2*c*d^4*n^3+461*B*a^2*d^5*n^2*x+4*B*a*b*c^2*d^3*n^3-356*B*a*b*c*d
^4*n^2*x+1016*B*a*b*d^5*n*x^2+72*B*b^2*c^2*d^3*n^2*x-336*B*b^2*c*d^4*n*x^2+180*B*b^2*d^5*x^3+2*C*a^2*c^2*d^3*n
^3-178*C*a^2*c*d^4*n^2*x+508*C*a^2*d^5*n*x^2+144*C*a*b*c^2*d^3*n^2*x-672*C*a*b*c*d^4*n*x^2+360*C*a*b*d^5*x^3-2
4*C*b^2*c^3*d^2*n^2*x+240*C*b^2*c^2*d^3*n*x^2-144*C*b^2*c*d^4*x^3+72*D*a^2*c^2*d^3*n^2*x-336*D*a^2*c*d^4*n*x^2
+180*D*a^2*d^5*x^3-48*D*a*b*c^3*d^2*n^2*x+480*D*a*b*c^2*d^3*n*x^2-288*D*a*b*c*d^4*x^3-180*D*b^2*c^3*d^2*n*x^2+
120*D*b^2*c^2*d^3*x^3+580*A*a^2*d^5*n^2-238*A*a*b*c*d^4*n^2+1404*A*a*b*d^5*n*x+30*A*b^2*c^2*d^3*n^2-388*A*b^2*
c*d^4*n*x+240*A*b^2*d^5*x^2-119*B*a^2*c*d^4*n^2+702*B*a^2*d^5*n*x+60*B*a*b*c^2*d^3*n^2-776*B*a*b*c*d^4*n*x+480
*B*a*b*d^5*x^2-6*B*b^2*c^3*d^2*n^2+246*B*b^2*c^2*d^3*n*x-180*B*b^2*c*d^4*x^2+30*C*a^2*c^2*d^3*n^2-388*C*a^2*c*
d^4*n*x+240*C*a^2*d^5*x^2-12*C*a*b*c^3*d^2*n^2+492*C*a*b*c^2*d^3*n*x-360*C*a*b*c*d^4*x^2-168*C*b^2*c^3*d^2*n*x
+144*C*b^2*c^2*d^3*x^2-6*D*a^2*c^3*d^2*n^2+246*D*a^2*c^2*d^3*n*x-180*D*a^2*c*d^4*x^2-336*D*a*b*c^3*d^2*n*x+288
*D*a*b*c^2*d^3*x^2+120*D*b^2*c^4*d*n*x-120*D*b^2*c^3*d^2*x^2+1044*A*a^2*d^5*n-684*A*a*b*c*d^4*n+720*A*a*b*d^5*
x+148*A*b^2*c^2*d^3*n-240*A*b^2*c*d^4*x-342*B*a^2*c*d^4*n+360*B*a^2*d^5*x+296*B*a*b*c^2*d^3*n-480*B*a*b*c*d^4*
x-66*B*b^2*c^3*d^2*n+180*B*b^2*c^2*d^3*x+148*C*a^2*c^2*d^3*n-240*C*a^2*c*d^4*x-132*C*a*b*c^3*d^2*n+360*C*a*b*c
^2*d^3*x+24*C*b^2*c^4*d*n-144*C*b^2*c^3*d^2*x-66*D*a^2*c^3*d^2*n+180*D*a^2*c^2*d^3*x+48*D*a*b*c^4*d*n-288*D*a*
b*c^3*d^2*x+120*D*b^2*c^4*d*x+720*A*a^2*d^5-720*A*a*b*c*d^4+240*A*b^2*c^2*d^3-360*B*a^2*c*d^4+480*B*a*b*c^2*d^
3-180*B*b^2*c^3*d^2+240*C*a^2*c^2*d^3-360*C*a*b*c^3*d^2+144*C*b^2*c^4*d-180*D*a^2*c^3*d^2+288*D*a*b*c^4*d-120*
D*b^2*c^5)/d^6/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+720)
________________________________________________________________________________________
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)^2*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")
[Out]
Exception raised: UnboundLocalError
________________________________________________________________________________________
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)**2*(d*x+c)**n*(D*x**3+C*x**2+B*x+A),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 2.92168, size = 6712, normalized size = 19.86 \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)^2*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
[Out]
((d*x + c)^n*D*b^2*d^6*n^5*x^6 + (d*x + c)^n*D*b^2*c*d^5*n^5*x^5 + 2*(d*x + c)^n*D*a*b*d^6*n^5*x^5 + (d*x + c)
^n*C*b^2*d^6*n^5*x^5 + 15*(d*x + c)^n*D*b^2*d^6*n^4*x^6 + 2*(d*x + c)^n*D*a*b*c*d^5*n^5*x^4 + (d*x + c)^n*C*b^
2*c*d^5*n^5*x^4 + (d*x + c)^n*D*a^2*d^6*n^5*x^4 + 2*(d*x + c)^n*C*a*b*d^6*n^5*x^4 + (d*x + c)^n*B*b^2*d^6*n^5*
x^4 + 10*(d*x + c)^n*D*b^2*c*d^5*n^4*x^5 + 32*(d*x + c)^n*D*a*b*d^6*n^4*x^5 + 16*(d*x + c)^n*C*b^2*d^6*n^4*x^5
+ 85*(d*x + c)^n*D*b^2*d^6*n^3*x^6 + (d*x + c)^n*D*a^2*c*d^5*n^5*x^3 + 2*(d*x + c)^n*C*a*b*c*d^5*n^5*x^3 + (d
*x + c)^n*B*b^2*c*d^5*n^5*x^3 + (d*x + c)^n*C*a^2*d^6*n^5*x^3 + 2*(d*x + c)^n*B*a*b*d^6*n^5*x^3 + (d*x + c)^n*
A*b^2*d^6*n^5*x^3 - 5*(d*x + c)^n*D*b^2*c^2*d^4*n^4*x^4 + 24*(d*x + c)^n*D*a*b*c*d^5*n^4*x^4 + 12*(d*x + c)^n*
C*b^2*c*d^5*n^4*x^4 + 17*(d*x + c)^n*D*a^2*d^6*n^4*x^4 + 34*(d*x + c)^n*C*a*b*d^6*n^4*x^4 + 17*(d*x + c)^n*B*b
^2*d^6*n^4*x^4 + 35*(d*x + c)^n*D*b^2*c*d^5*n^3*x^5 + 190*(d*x + c)^n*D*a*b*d^6*n^3*x^5 + 95*(d*x + c)^n*C*b^2
*d^6*n^3*x^5 + 225*(d*x + c)^n*D*b^2*d^6*n^2*x^6 + (d*x + c)^n*C*a^2*c*d^5*n^5*x^2 + 2*(d*x + c)^n*B*a*b*c*d^5
*n^5*x^2 + (d*x + c)^n*A*b^2*c*d^5*n^5*x^2 + (d*x + c)^n*B*a^2*d^6*n^5*x^2 + 2*(d*x + c)^n*A*a*b*d^6*n^5*x^2 -
8*(d*x + c)^n*D*a*b*c^2*d^4*n^4*x^3 - 4*(d*x + c)^n*C*b^2*c^2*d^4*n^4*x^3 + 14*(d*x + c)^n*D*a^2*c*d^5*n^4*x^
3 + 28*(d*x + c)^n*C*a*b*c*d^5*n^4*x^3 + 14*(d*x + c)^n*B*b^2*c*d^5*n^4*x^3 + 18*(d*x + c)^n*C*a^2*d^6*n^4*x^3
+ 36*(d*x + c)^n*B*a*b*d^6*n^4*x^3 + 18*(d*x + c)^n*A*b^2*d^6*n^4*x^3 - 30*(d*x + c)^n*D*b^2*c^2*d^4*n^3*x^4
+ 94*(d*x + c)^n*D*a*b*c*d^5*n^3*x^4 + 47*(d*x + c)^n*C*b^2*c*d^5*n^3*x^4 + 107*(d*x + c)^n*D*a^2*d^6*n^3*x^4
+ 214*(d*x + c)^n*C*a*b*d^6*n^3*x^4 + 107*(d*x + c)^n*B*b^2*d^6*n^3*x^4 + 50*(d*x + c)^n*D*b^2*c*d^5*n^2*x^5 +
520*(d*x + c)^n*D*a*b*d^6*n^2*x^5 + 260*(d*x + c)^n*C*b^2*d^6*n^2*x^5 + 274*(d*x + c)^n*D*b^2*d^6*n*x^6 + (d*
x + c)^n*B*a^2*c*d^5*n^5*x + 2*(d*x + c)^n*A*a*b*c*d^5*n^5*x + (d*x + c)^n*A*a^2*d^6*n^5*x - 3*(d*x + c)^n*D*a
^2*c^2*d^4*n^4*x^2 - 6*(d*x + c)^n*C*a*b*c^2*d^4*n^4*x^2 - 3*(d*x + c)^n*B*b^2*c^2*d^4*n^4*x^2 + 16*(d*x + c)^
n*C*a^2*c*d^5*n^4*x^2 + 32*(d*x + c)^n*B*a*b*c*d^5*n^4*x^2 + 16*(d*x + c)^n*A*b^2*c*d^5*n^4*x^2 + 19*(d*x + c)
^n*B*a^2*d^6*n^4*x^2 + 38*(d*x + c)^n*A*a*b*d^6*n^4*x^2 + 20*(d*x + c)^n*D*b^2*c^3*d^3*n^3*x^3 - 72*(d*x + c)^
n*D*a*b*c^2*d^4*n^3*x^3 - 36*(d*x + c)^n*C*b^2*c^2*d^4*n^3*x^3 + 65*(d*x + c)^n*D*a^2*c*d^5*n^3*x^3 + 130*(d*x
+ c)^n*C*a*b*c*d^5*n^3*x^3 + 65*(d*x + c)^n*B*b^2*c*d^5*n^3*x^3 + 121*(d*x + c)^n*C*a^2*d^6*n^3*x^3 + 242*(d*
x + c)^n*B*a*b*d^6*n^3*x^3 + 121*(d*x + c)^n*A*b^2*d^6*n^3*x^3 - 55*(d*x + c)^n*D*b^2*c^2*d^4*n^2*x^4 + 144*(d
*x + c)^n*D*a*b*c*d^5*n^2*x^4 + 72*(d*x + c)^n*C*b^2*c*d^5*n^2*x^4 + 307*(d*x + c)^n*D*a^2*d^6*n^2*x^4 + 614*(
d*x + c)^n*C*a*b*d^6*n^2*x^4 + 307*(d*x + c)^n*B*b^2*d^6*n^2*x^4 + 24*(d*x + c)^n*D*b^2*c*d^5*n*x^5 + 648*(d*x
+ c)^n*D*a*b*d^6*n*x^5 + 324*(d*x + c)^n*C*b^2*d^6*n*x^5 + 120*(d*x + c)^n*D*b^2*d^6*x^6 + (d*x + c)^n*A*a^2*
c*d^5*n^5 - 2*(d*x + c)^n*C*a^2*c^2*d^4*n^4*x - 4*(d*x + c)^n*B*a*b*c^2*d^4*n^4*x - 2*(d*x + c)^n*A*b^2*c^2*d^
4*n^4*x + 18*(d*x + c)^n*B*a^2*c*d^5*n^4*x + 36*(d*x + c)^n*A*a*b*c*d^5*n^4*x + 20*(d*x + c)^n*A*a^2*d^6*n^4*x
+ 24*(d*x + c)^n*D*a*b*c^3*d^3*n^3*x^2 + 12*(d*x + c)^n*C*b^2*c^3*d^3*n^3*x^2 - 36*(d*x + c)^n*D*a^2*c^2*d^4*
n^3*x^2 - 72*(d*x + c)^n*C*a*b*c^2*d^4*n^3*x^2 - 36*(d*x + c)^n*B*b^2*c^2*d^4*n^3*x^2 + 89*(d*x + c)^n*C*a^2*c
*d^5*n^3*x^2 + 178*(d*x + c)^n*B*a*b*c*d^5*n^3*x^2 + 89*(d*x + c)^n*A*b^2*c*d^5*n^3*x^2 + 137*(d*x + c)^n*B*a^
2*d^6*n^3*x^2 + 274*(d*x + c)^n*A*a*b*d^6*n^3*x^2 + 60*(d*x + c)^n*D*b^2*c^3*d^3*n^2*x^3 - 160*(d*x + c)^n*D*a
*b*c^2*d^4*n^2*x^3 - 80*(d*x + c)^n*C*b^2*c^2*d^4*n^2*x^3 + 112*(d*x + c)^n*D*a^2*c*d^5*n^2*x^3 + 224*(d*x + c
)^n*C*a*b*c*d^5*n^2*x^3 + 112*(d*x + c)^n*B*b^2*c*d^5*n^2*x^3 + 372*(d*x + c)^n*C*a^2*d^6*n^2*x^3 + 744*(d*x +
c)^n*B*a*b*d^6*n^2*x^3 + 372*(d*x + c)^n*A*b^2*d^6*n^2*x^3 - 30*(d*x + c)^n*D*b^2*c^2*d^4*n*x^4 + 72*(d*x + c
)^n*D*a*b*c*d^5*n*x^4 + 36*(d*x + c)^n*C*b^2*c*d^5*n*x^4 + 396*(d*x + c)^n*D*a^2*d^6*n*x^4 + 792*(d*x + c)^n*C
*a*b*d^6*n*x^4 + 396*(d*x + c)^n*B*b^2*d^6*n*x^4 + 288*(d*x + c)^n*D*a*b*d^6*x^5 + 144*(d*x + c)^n*C*b^2*d^6*x
^5 - (d*x + c)^n*B*a^2*c^2*d^4*n^4 - 2*(d*x + c)^n*A*a*b*c^2*d^4*n^4 + 20*(d*x + c)^n*A*a^2*c*d^5*n^4 + 6*(d*x
+ c)^n*D*a^2*c^3*d^3*n^3*x + 12*(d*x + c)^n*C*a*b*c^3*d^3*n^3*x + 6*(d*x + c)^n*B*b^2*c^3*d^3*n^3*x - 30*(d*x
+ c)^n*C*a^2*c^2*d^4*n^3*x - 60*(d*x + c)^n*B*a*b*c^2*d^4*n^3*x - 30*(d*x + c)^n*A*b^2*c^2*d^4*n^3*x + 119*(d
*x + c)^n*B*a^2*c*d^5*n^3*x + 238*(d*x + c)^n*A*a*b*c*d^5*n^3*x + 155*(d*x + c)^n*A*a^2*d^6*n^3*x - 60*(d*x +
c)^n*D*b^2*c^4*d^2*n^2*x^2 + 168*(d*x + c)^n*D*a*b*c^3*d^3*n^2*x^2 + 84*(d*x + c)^n*C*b^2*c^3*d^3*n^2*x^2 - 12
3*(d*x + c)^n*D*a^2*c^2*d^4*n^2*x^2 - 246*(d*x + c)^n*C*a*b*c^2*d^4*n^2*x^2 - 123*(d*x + c)^n*B*b^2*c^2*d^4*n^
2*x^2 + 194*(d*x + c)^n*C*a^2*c*d^5*n^2*x^2 + 388*(d*x + c)^n*B*a*b*c*d^5*n^2*x^2 + 194*(d*x + c)^n*A*b^2*c*d^
5*n^2*x^2 + 461*(d*x + c)^n*B*a^2*d^6*n^2*x^2 + 922*(d*x + c)^n*A*a*b*d^6*n^2*x^2 + 40*(d*x + c)^n*D*b^2*c^3*d
^3*n*x^3 - 96*(d*x + c)^n*D*a*b*c^2*d^4*n*x^3 - 48*(d*x + c)^n*C*b^2*c^2*d^4*n*x^3 + 60*(d*x + c)^n*D*a^2*c*d^
5*n*x^3 + 120*(d*x + c)^n*C*a*b*c*d^5*n*x^3 + 60*(d*x + c)^n*B*b^2*c*d^5*n*x^3 + 508*(d*x + c)^n*C*a^2*d^6*n*x
^3 + 1016*(d*x + c)^n*B*a*b*d^6*n*x^3 + 508*(d*x + c)^n*A*b^2*d^6*n*x^3 + 180*(d*x + c)^n*D*a^2*d^6*x^4 + 360*
(d*x + c)^n*C*a*b*d^6*x^4 + 180*(d*x + c)^n*B*b^2*d^6*x^4 + 2*(d*x + c)^n*C*a^2*c^3*d^3*n^3 + 4*(d*x + c)^n*B*
a*b*c^3*d^3*n^3 + 2*(d*x + c)^n*A*b^2*c^3*d^3*n^3 - 18*(d*x + c)^n*B*a^2*c^2*d^4*n^3 - 36*(d*x + c)^n*A*a*b*c^
2*d^4*n^3 + 155*(d*x + c)^n*A*a^2*c*d^5*n^3 - 48*(d*x + c)^n*D*a*b*c^4*d^2*n^2*x - 24*(d*x + c)^n*C*b^2*c^4*d^
2*n^2*x + 66*(d*x + c)^n*D*a^2*c^3*d^3*n^2*x + 132*(d*x + c)^n*C*a*b*c^3*d^3*n^2*x + 66*(d*x + c)^n*B*b^2*c^3*
d^3*n^2*x - 148*(d*x + c)^n*C*a^2*c^2*d^4*n^2*x - 296*(d*x + c)^n*B*a*b*c^2*d^4*n^2*x - 148*(d*x + c)^n*A*b^2*
c^2*d^4*n^2*x + 342*(d*x + c)^n*B*a^2*c*d^5*n^2*x + 684*(d*x + c)^n*A*a*b*c*d^5*n^2*x + 580*(d*x + c)^n*A*a^2*
d^6*n^2*x - 60*(d*x + c)^n*D*b^2*c^4*d^2*n*x^2 + 144*(d*x + c)^n*D*a*b*c^3*d^3*n*x^2 + 72*(d*x + c)^n*C*b^2*c^
3*d^3*n*x^2 - 90*(d*x + c)^n*D*a^2*c^2*d^4*n*x^2 - 180*(d*x + c)^n*C*a*b*c^2*d^4*n*x^2 - 90*(d*x + c)^n*B*b^2*
c^2*d^4*n*x^2 + 120*(d*x + c)^n*C*a^2*c*d^5*n*x^2 + 240*(d*x + c)^n*B*a*b*c*d^5*n*x^2 + 120*(d*x + c)^n*A*b^2*
c*d^5*n*x^2 + 702*(d*x + c)^n*B*a^2*d^6*n*x^2 + 1404*(d*x + c)^n*A*a*b*d^6*n*x^2 + 240*(d*x + c)^n*C*a^2*d^6*x
^3 + 480*(d*x + c)^n*B*a*b*d^6*x^3 + 240*(d*x + c)^n*A*b^2*d^6*x^3 - 6*(d*x + c)^n*D*a^2*c^4*d^2*n^2 - 12*(d*x
+ c)^n*C*a*b*c^4*d^2*n^2 - 6*(d*x + c)^n*B*b^2*c^4*d^2*n^2 + 30*(d*x + c)^n*C*a^2*c^3*d^3*n^2 + 60*(d*x + c)^
n*B*a*b*c^3*d^3*n^2 + 30*(d*x + c)^n*A*b^2*c^3*d^3*n^2 - 119*(d*x + c)^n*B*a^2*c^2*d^4*n^2 - 238*(d*x + c)^n*A
*a*b*c^2*d^4*n^2 + 580*(d*x + c)^n*A*a^2*c*d^5*n^2 + 120*(d*x + c)^n*D*b^2*c^5*d*n*x - 288*(d*x + c)^n*D*a*b*c
^4*d^2*n*x - 144*(d*x + c)^n*C*b^2*c^4*d^2*n*x + 180*(d*x + c)^n*D*a^2*c^3*d^3*n*x + 360*(d*x + c)^n*C*a*b*c^3
*d^3*n*x + 180*(d*x + c)^n*B*b^2*c^3*d^3*n*x - 240*(d*x + c)^n*C*a^2*c^2*d^4*n*x - 480*(d*x + c)^n*B*a*b*c^2*d
^4*n*x - 240*(d*x + c)^n*A*b^2*c^2*d^4*n*x + 360*(d*x + c)^n*B*a^2*c*d^5*n*x + 720*(d*x + c)^n*A*a*b*c*d^5*n*x
+ 1044*(d*x + c)^n*A*a^2*d^6*n*x + 360*(d*x + c)^n*B*a^2*d^6*x^2 + 720*(d*x + c)^n*A*a*b*d^6*x^2 + 48*(d*x +
c)^n*D*a*b*c^5*d*n + 24*(d*x + c)^n*C*b^2*c^5*d*n - 66*(d*x + c)^n*D*a^2*c^4*d^2*n - 132*(d*x + c)^n*C*a*b*c^4
*d^2*n - 66*(d*x + c)^n*B*b^2*c^4*d^2*n + 148*(d*x + c)^n*C*a^2*c^3*d^3*n + 296*(d*x + c)^n*B*a*b*c^3*d^3*n +
148*(d*x + c)^n*A*b^2*c^3*d^3*n - 342*(d*x + c)^n*B*a^2*c^2*d^4*n - 684*(d*x + c)^n*A*a*b*c^2*d^4*n + 1044*(d*
x + c)^n*A*a^2*c*d^5*n + 720*(d*x + c)^n*A*a^2*d^6*x - 120*(d*x + c)^n*D*b^2*c^6 + 288*(d*x + c)^n*D*a*b*c^5*d
+ 144*(d*x + c)^n*C*b^2*c^5*d - 180*(d*x + c)^n*D*a^2*c^4*d^2 - 360*(d*x + c)^n*C*a*b*c^4*d^2 - 180*(d*x + c)
^n*B*b^2*c^4*d^2 + 240*(d*x + c)^n*C*a^2*c^3*d^3 + 480*(d*x + c)^n*B*a*b*c^3*d^3 + 240*(d*x + c)^n*A*b^2*c^3*d
^3 - 360*(d*x + c)^n*B*a^2*c^2*d^4 - 720*(d*x + c)^n*A*a*b*c^2*d^4 + 720*(d*x + c)^n*A*a^2*c*d^5)/(d^6*n^6 + 2
1*d^6*n^5 + 175*d^6*n^4 + 735*d^6*n^3 + 1624*d^6*n^2 + 1764*d^6*n + 720*d^6)