3.25 \(\int (a+b x)^3 (c+d x)^n (A+B x+C x^2+D x^3) \, dx\)

Optimal. Leaf size=455 \[ -\frac{(b c-a d) (c+d x)^{n+3} \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right )\right )}{d^7 (n+3)}+\frac{(c+d x)^{n+4} \left (3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right )\right )}{d^7 (n+4)}+\frac{b (c+d x)^{n+5} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{d^7 (n+5)}-\frac{(b c-a d)^3 (c+d x)^{n+1} \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^7 (n+1)}-\frac{(b c-a d)^2 (c+d x)^{n+2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right )\right )}{d^7 (n+2)}+\frac{b^2 (c+d x)^{n+6} (3 a d D-6 b c D+b C d)}{d^7 (n+6)}+\frac{b^3 D (c+d x)^{n+7}}{d^7 (n+7)} \]

[Out]

-(((b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^7*(1 + n))) - ((b*c - a*d)^2*(a*d*(
2*c*C*d - B*d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^3 - 6*c^3*D))*(c + d*x)^(2 + n))/(d^7*(2 + n)) -
((b*c - a*d)*(a^2*d^2*(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*
A*d^3 - 15*c^3*D))*(c + d*x)^(3 + n))/(d^7*(3 + n)) + ((a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c
*C*d - B*d^2 - 10*c^2*D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(4 + n))/(d^7*(4 + n)) +
(b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D) - b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(5 + n))/(d^7*(5 + n))
+ (b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^(6 + n))/(d^7*(6 + n)) + (b^3*D*(c + d*x)^(7 + n))/(d^7*(7 + n))

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Rubi [A]  time = 0.337924, antiderivative size = 455, 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} \[ -\frac{(b c-a d) (c+d x)^{n+3} \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right )\right )}{d^7 (n+3)}+\frac{(c+d x)^{n+4} \left (3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right )\right )}{d^7 (n+4)}+\frac{b (c+d x)^{n+5} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{d^7 (n+5)}-\frac{(b c-a d)^3 (c+d x)^{n+1} \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^7 (n+1)}-\frac{(b c-a d)^2 (c+d x)^{n+2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right )\right )}{d^7 (n+2)}+\frac{b^2 (c+d x)^{n+6} (3 a d D-6 b c D+b C d)}{d^7 (n+6)}+\frac{b^3 D (c+d x)^{n+7}}{d^7 (n+7)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]

[Out]

-(((b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^7*(1 + n))) - ((b*c - a*d)^2*(a*d*(
2*c*C*d - B*d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^3 - 6*c^3*D))*(c + d*x)^(2 + n))/(d^7*(2 + n)) -
((b*c - a*d)*(a^2*d^2*(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*
A*d^3 - 15*c^3*D))*(c + d*x)^(3 + n))/(d^7*(3 + n)) + ((a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c
*C*d - B*d^2 - 10*c^2*D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(4 + n))/(d^7*(4 + n)) +
(b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D) - b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(5 + n))/(d^7*(5 + n))
+ (b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^(6 + n))/(d^7*(6 + n)) + (b^3*D*(c + d*x)^(7 + n))/(d^7*(7 + 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)^3 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (\frac{(-b c+a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^n}{d^6}+\frac{(b c-a d)^2 \left (-a d \left (2 c C d-B d^2-3 c^2 D\right )+b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) (c+d x)^{1+n}}{d^6}+\frac{(b c-a d) \left (-a^2 d^2 (C d-3 c D)+a b d \left (8 c C d-3 B d^2-15 c^2 D\right )-b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{2+n}}{d^6}+\frac{\left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{3+n}}{d^6}+\frac{b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{4+n}}{d^6}+\frac{b^2 (b C d-6 b c D+3 a d D) (c+d x)^{5+n}}{d^6}+\frac{b^3 D (c+d x)^{6+n}}{d^6}\right ) \, dx\\ &=-\frac{(b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{1+n}}{d^7 (1+n)}-\frac{(b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) (c+d x)^{2+n}}{d^7 (2+n)}-\frac{(b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{3+n}}{d^7 (3+n)}+\frac{\left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{4+n}}{d^7 (4+n)}+\frac{b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{5+n}}{d^7 (5+n)}+\frac{b^2 (b C d-6 b c D+3 a d D) (c+d x)^{6+n}}{d^7 (6+n)}+\frac{b^3 D (c+d x)^{7+n}}{d^7 (7+n)}\\ \end{align*}

Mathematica [A]  time = 0.732626, size = 418, normalized size = 0.92 \[ \frac{(c+d x)^{n+1} \left (\frac{(c+d x)^3 \left (3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D+3 a b^2 d \left (B d^2+10 c^2 D-4 c C d\right )+b^3 \left (A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right )\right )}{n+4}+\frac{(c+d x)^2 (b c-a d) \left (a^2 d^2 (3 c D-C d)+a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (-3 A d^3+6 B c d^2-10 c^2 C d+15 c^3 D\right )\right )}{n+3}+\frac{b (c+d x)^4 \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (B d^2+15 c^2 D-5 c C d\right )\right )}{n+5}-\frac{(c+d x) (b c-a d)^2 \left (b \left (-3 A d^3+4 B c d^2-5 c^2 C d+6 c^3 D\right )-a d \left (B d^2+3 c^2 D-2 c C d\right )\right )}{n+2}+\frac{(b c-a d)^3 \left (-A d^3+B c d^2-c^2 C d+c^3 D\right )}{n+1}+\frac{b^2 (c+d x)^5 (3 a d D-6 b c D+b C d)}{n+6}+\frac{b^3 D (c+d x)^6}{n+7}\right )}{d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c + d*x)^(1 + n)*(((b*c - a*d)^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(1 + n) - ((b*c - a*d)^2*(-(a*d*(-2
*c*C*d + B*d^2 + 3*c^2*D)) + b*(-5*c^2*C*d + 4*B*c*d^2 - 3*A*d^3 + 6*c^3*D))*(c + d*x))/(2 + n) + ((b*c - a*d)
*(a^2*d^2*(-(C*d) + 3*c*D) + a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(-10*c^2*C*d + 6*B*c*d^2 - 3*A*d^3 + 1
5*c^3*D))*(c + d*x)^2)/(3 + n) + ((a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) + 3*a*b^2*d*(-4*c*C*d + B*d^2 + 10*c^
2*D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^3)/(4 + n) + (b*(3*a^2*d^2*D + 3*a*b*d*(C*d
- 5*c*D) + b^2*(-5*c*C*d + B*d^2 + 15*c^2*D))*(c + d*x)^4)/(5 + n) + (b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x
)^5)/(6 + n) + (b^3*D*(c + d*x)^6)/(7 + n)))/d^7

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Maple [B]  time = 0.02, size = 5003, normalized size = 11. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x)

[Out]

result too large to display

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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)^3*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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)^3*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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((b*x+a)**3*(d*x+c)**n*(D*x**3+C*x**2+B*x+A),x)

[Out]

Timed out

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Giac [B]  time = 2.99908, size = 12193, normalized size = 26.8 \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)^3*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")

[Out]

((d*x + c)^n*D*b^3*d^7*n^6*x^7 + (d*x + c)^n*D*b^3*c*d^6*n^6*x^6 + 3*(d*x + c)^n*D*a*b^2*d^7*n^6*x^6 + (d*x +
c)^n*C*b^3*d^7*n^6*x^6 + 21*(d*x + c)^n*D*b^3*d^7*n^5*x^7 + 3*(d*x + c)^n*D*a*b^2*c*d^6*n^6*x^5 + (d*x + c)^n*
C*b^3*c*d^6*n^6*x^5 + 3*(d*x + c)^n*D*a^2*b*d^7*n^6*x^5 + 3*(d*x + c)^n*C*a*b^2*d^7*n^6*x^5 + (d*x + c)^n*B*b^
3*d^7*n^6*x^5 + 15*(d*x + c)^n*D*b^3*c*d^6*n^5*x^6 + 66*(d*x + c)^n*D*a*b^2*d^7*n^5*x^6 + 22*(d*x + c)^n*C*b^3
*d^7*n^5*x^6 + 175*(d*x + c)^n*D*b^3*d^7*n^4*x^7 + 3*(d*x + c)^n*D*a^2*b*c*d^6*n^6*x^4 + 3*(d*x + c)^n*C*a*b^2
*c*d^6*n^6*x^4 + (d*x + c)^n*B*b^3*c*d^6*n^6*x^4 + (d*x + c)^n*D*a^3*d^7*n^6*x^4 + 3*(d*x + c)^n*C*a^2*b*d^7*n
^6*x^4 + 3*(d*x + c)^n*B*a*b^2*d^7*n^6*x^4 + (d*x + c)^n*A*b^3*d^7*n^6*x^4 - 6*(d*x + c)^n*D*b^3*c^2*d^5*n^5*x
^5 + 51*(d*x + c)^n*D*a*b^2*c*d^6*n^5*x^5 + 17*(d*x + c)^n*C*b^3*c*d^6*n^5*x^5 + 69*(d*x + c)^n*D*a^2*b*d^7*n^
5*x^5 + 69*(d*x + c)^n*C*a*b^2*d^7*n^5*x^5 + 23*(d*x + c)^n*B*b^3*d^7*n^5*x^5 + 85*(d*x + c)^n*D*b^3*c*d^6*n^4
*x^6 + 570*(d*x + c)^n*D*a*b^2*d^7*n^4*x^6 + 190*(d*x + c)^n*C*b^3*d^7*n^4*x^6 + 735*(d*x + c)^n*D*b^3*d^7*n^3
*x^7 + (d*x + c)^n*D*a^3*c*d^6*n^6*x^3 + 3*(d*x + c)^n*C*a^2*b*c*d^6*n^6*x^3 + 3*(d*x + c)^n*B*a*b^2*c*d^6*n^6
*x^3 + (d*x + c)^n*A*b^3*c*d^6*n^6*x^3 + (d*x + c)^n*C*a^3*d^7*n^6*x^3 + 3*(d*x + c)^n*B*a^2*b*d^7*n^6*x^3 + 3
*(d*x + c)^n*A*a*b^2*d^7*n^6*x^3 - 15*(d*x + c)^n*D*a*b^2*c^2*d^5*n^5*x^4 - 5*(d*x + c)^n*C*b^3*c^2*d^5*n^5*x^
4 + 57*(d*x + c)^n*D*a^2*b*c*d^6*n^5*x^4 + 57*(d*x + c)^n*C*a*b^2*c*d^6*n^5*x^4 + 19*(d*x + c)^n*B*b^3*c*d^6*n
^5*x^4 + 24*(d*x + c)^n*D*a^3*d^7*n^5*x^4 + 72*(d*x + c)^n*C*a^2*b*d^7*n^5*x^4 + 72*(d*x + c)^n*B*a*b^2*d^7*n^
5*x^4 + 24*(d*x + c)^n*A*b^3*d^7*n^5*x^4 - 60*(d*x + c)^n*D*b^3*c^2*d^5*n^4*x^5 + 315*(d*x + c)^n*D*a*b^2*c*d^
6*n^4*x^5 + 105*(d*x + c)^n*C*b^3*c*d^6*n^4*x^5 + 621*(d*x + c)^n*D*a^2*b*d^7*n^4*x^5 + 621*(d*x + c)^n*C*a*b^
2*d^7*n^4*x^5 + 207*(d*x + c)^n*B*b^3*d^7*n^4*x^5 + 225*(d*x + c)^n*D*b^3*c*d^6*n^3*x^6 + 2460*(d*x + c)^n*D*a
*b^2*d^7*n^3*x^6 + 820*(d*x + c)^n*C*b^3*d^7*n^3*x^6 + 1624*(d*x + c)^n*D*b^3*d^7*n^2*x^7 + (d*x + c)^n*C*a^3*
c*d^6*n^6*x^2 + 3*(d*x + c)^n*B*a^2*b*c*d^6*n^6*x^2 + 3*(d*x + c)^n*A*a*b^2*c*d^6*n^6*x^2 + (d*x + c)^n*B*a^3*
d^7*n^6*x^2 + 3*(d*x + c)^n*A*a^2*b*d^7*n^6*x^2 - 12*(d*x + c)^n*D*a^2*b*c^2*d^5*n^5*x^3 - 12*(d*x + c)^n*C*a*
b^2*c^2*d^5*n^5*x^3 - 4*(d*x + c)^n*B*b^3*c^2*d^5*n^5*x^3 + 21*(d*x + c)^n*D*a^3*c*d^6*n^5*x^3 + 63*(d*x + c)^
n*C*a^2*b*c*d^6*n^5*x^3 + 63*(d*x + c)^n*B*a*b^2*c*d^6*n^5*x^3 + 21*(d*x + c)^n*A*b^3*c*d^6*n^5*x^3 + 25*(d*x
+ c)^n*C*a^3*d^7*n^5*x^3 + 75*(d*x + c)^n*B*a^2*b*d^7*n^5*x^3 + 75*(d*x + c)^n*A*a*b^2*d^7*n^5*x^3 + 30*(d*x +
c)^n*D*b^3*c^3*d^4*n^4*x^4 - 195*(d*x + c)^n*D*a*b^2*c^2*d^5*n^4*x^4 - 65*(d*x + c)^n*C*b^3*c^2*d^5*n^4*x^4 +
393*(d*x + c)^n*D*a^2*b*c*d^6*n^4*x^4 + 393*(d*x + c)^n*C*a*b^2*c*d^6*n^4*x^4 + 131*(d*x + c)^n*B*b^3*c*d^6*n
^4*x^4 + 226*(d*x + c)^n*D*a^3*d^7*n^4*x^4 + 678*(d*x + c)^n*C*a^2*b*d^7*n^4*x^4 + 678*(d*x + c)^n*B*a*b^2*d^7
*n^4*x^4 + 226*(d*x + c)^n*A*b^3*d^7*n^4*x^4 - 210*(d*x + c)^n*D*b^3*c^2*d^5*n^3*x^5 + 885*(d*x + c)^n*D*a*b^2
*c*d^6*n^3*x^5 + 295*(d*x + c)^n*C*b^3*c*d^6*n^3*x^5 + 2775*(d*x + c)^n*D*a^2*b*d^7*n^3*x^5 + 2775*(d*x + c)^n
*C*a*b^2*d^7*n^3*x^5 + 925*(d*x + c)^n*B*b^3*d^7*n^3*x^5 + 274*(d*x + c)^n*D*b^3*c*d^6*n^2*x^6 + 5547*(d*x + c
)^n*D*a*b^2*d^7*n^2*x^6 + 1849*(d*x + c)^n*C*b^3*d^7*n^2*x^6 + 1764*(d*x + c)^n*D*b^3*d^7*n*x^7 + (d*x + c)^n*
B*a^3*c*d^6*n^6*x + 3*(d*x + c)^n*A*a^2*b*c*d^6*n^6*x + (d*x + c)^n*A*a^3*d^7*n^6*x - 3*(d*x + c)^n*D*a^3*c^2*
d^5*n^5*x^2 - 9*(d*x + c)^n*C*a^2*b*c^2*d^5*n^5*x^2 - 9*(d*x + c)^n*B*a*b^2*c^2*d^5*n^5*x^2 - 3*(d*x + c)^n*A*
b^3*c^2*d^5*n^5*x^2 + 23*(d*x + c)^n*C*a^3*c*d^6*n^5*x^2 + 69*(d*x + c)^n*B*a^2*b*c*d^6*n^5*x^2 + 69*(d*x + c)
^n*A*a*b^2*c*d^6*n^5*x^2 + 26*(d*x + c)^n*B*a^3*d^7*n^5*x^2 + 78*(d*x + c)^n*A*a^2*b*d^7*n^5*x^2 + 60*(d*x + c
)^n*D*a*b^2*c^3*d^4*n^4*x^3 + 20*(d*x + c)^n*C*b^3*c^3*d^4*n^4*x^3 - 192*(d*x + c)^n*D*a^2*b*c^2*d^5*n^4*x^3 -
192*(d*x + c)^n*C*a*b^2*c^2*d^5*n^4*x^3 - 64*(d*x + c)^n*B*b^3*c^2*d^5*n^4*x^3 + 163*(d*x + c)^n*D*a^3*c*d^6*
n^4*x^3 + 489*(d*x + c)^n*C*a^2*b*c*d^6*n^4*x^3 + 489*(d*x + c)^n*B*a*b^2*c*d^6*n^4*x^3 + 163*(d*x + c)^n*A*b^
3*c*d^6*n^4*x^3 + 247*(d*x + c)^n*C*a^3*d^7*n^4*x^3 + 741*(d*x + c)^n*B*a^2*b*d^7*n^4*x^3 + 741*(d*x + c)^n*A*
a*b^2*d^7*n^4*x^3 + 180*(d*x + c)^n*D*b^3*c^3*d^4*n^3*x^4 - 795*(d*x + c)^n*D*a*b^2*c^2*d^5*n^3*x^4 - 265*(d*x
+ c)^n*C*b^3*c^2*d^5*n^3*x^4 + 1203*(d*x + c)^n*D*a^2*b*c*d^6*n^3*x^4 + 1203*(d*x + c)^n*C*a*b^2*c*d^6*n^3*x^
4 + 401*(d*x + c)^n*B*b^3*c*d^6*n^3*x^4 + 1056*(d*x + c)^n*D*a^3*d^7*n^3*x^4 + 3168*(d*x + c)^n*C*a^2*b*d^7*n^
3*x^4 + 3168*(d*x + c)^n*B*a*b^2*d^7*n^3*x^4 + 1056*(d*x + c)^n*A*b^3*d^7*n^3*x^4 - 300*(d*x + c)^n*D*b^3*c^2*
d^5*n^2*x^5 + 1122*(d*x + c)^n*D*a*b^2*c*d^6*n^2*x^5 + 374*(d*x + c)^n*C*b^3*c*d^6*n^2*x^5 + 6432*(d*x + c)^n*
D*a^2*b*d^7*n^2*x^5 + 6432*(d*x + c)^n*C*a*b^2*d^7*n^2*x^5 + 2144*(d*x + c)^n*B*b^3*d^7*n^2*x^5 + 120*(d*x + c
)^n*D*b^3*c*d^6*n*x^6 + 6114*(d*x + c)^n*D*a*b^2*d^7*n*x^6 + 2038*(d*x + c)^n*C*b^3*d^7*n*x^6 + 720*(d*x + c)^
n*D*b^3*d^7*x^7 + (d*x + c)^n*A*a^3*c*d^6*n^6 - 2*(d*x + c)^n*C*a^3*c^2*d^5*n^5*x - 6*(d*x + c)^n*B*a^2*b*c^2*
d^5*n^5*x - 6*(d*x + c)^n*A*a*b^2*c^2*d^5*n^5*x + 25*(d*x + c)^n*B*a^3*c*d^6*n^5*x + 75*(d*x + c)^n*A*a^2*b*c*
d^6*n^5*x + 27*(d*x + c)^n*A*a^3*d^7*n^5*x + 36*(d*x + c)^n*D*a^2*b*c^3*d^4*n^4*x^2 + 36*(d*x + c)^n*C*a*b^2*c
^3*d^4*n^4*x^2 + 12*(d*x + c)^n*B*b^3*c^3*d^4*n^4*x^2 - 57*(d*x + c)^n*D*a^3*c^2*d^5*n^4*x^2 - 171*(d*x + c)^n
*C*a^2*b*c^2*d^5*n^4*x^2 - 171*(d*x + c)^n*B*a*b^2*c^2*d^5*n^4*x^2 - 57*(d*x + c)^n*A*b^3*c^2*d^5*n^4*x^2 + 20
1*(d*x + c)^n*C*a^3*c*d^6*n^4*x^2 + 603*(d*x + c)^n*B*a^2*b*c*d^6*n^4*x^2 + 603*(d*x + c)^n*A*a*b^2*c*d^6*n^4*
x^2 + 270*(d*x + c)^n*B*a^3*d^7*n^4*x^2 + 810*(d*x + c)^n*A*a^2*b*d^7*n^4*x^2 - 120*(d*x + c)^n*D*b^3*c^4*d^3*
n^3*x^3 + 600*(d*x + c)^n*D*a*b^2*c^3*d^4*n^3*x^3 + 200*(d*x + c)^n*C*b^3*c^3*d^4*n^3*x^3 - 996*(d*x + c)^n*D*
a^2*b*c^2*d^5*n^3*x^3 - 996*(d*x + c)^n*C*a*b^2*c^2*d^5*n^3*x^3 - 332*(d*x + c)^n*B*b^3*c^2*d^5*n^3*x^3 + 567*
(d*x + c)^n*D*a^3*c*d^6*n^3*x^3 + 1701*(d*x + c)^n*C*a^2*b*c*d^6*n^3*x^3 + 1701*(d*x + c)^n*B*a*b^2*c*d^6*n^3*
x^3 + 567*(d*x + c)^n*A*b^3*c*d^6*n^3*x^3 + 1219*(d*x + c)^n*C*a^3*d^7*n^3*x^3 + 3657*(d*x + c)^n*B*a^2*b*d^7*
n^3*x^3 + 3657*(d*x + c)^n*A*a*b^2*d^7*n^3*x^3 + 330*(d*x + c)^n*D*b^3*c^3*d^4*n^2*x^4 - 1245*(d*x + c)^n*D*a*
b^2*c^2*d^5*n^2*x^4 - 415*(d*x + c)^n*C*b^3*c^2*d^5*n^2*x^4 + 1620*(d*x + c)^n*D*a^2*b*c*d^6*n^2*x^4 + 1620*(d
*x + c)^n*C*a*b^2*c*d^6*n^2*x^4 + 540*(d*x + c)^n*B*b^3*c*d^6*n^2*x^4 + 2545*(d*x + c)^n*D*a^3*d^7*n^2*x^4 + 7
635*(d*x + c)^n*C*a^2*b*d^7*n^2*x^4 + 7635*(d*x + c)^n*B*a*b^2*d^7*n^2*x^4 + 2545*(d*x + c)^n*A*b^3*d^7*n^2*x^
4 - 144*(d*x + c)^n*D*b^3*c^2*d^5*n*x^5 + 504*(d*x + c)^n*D*a*b^2*c*d^6*n*x^5 + 168*(d*x + c)^n*C*b^3*c*d^6*n*
x^5 + 7236*(d*x + c)^n*D*a^2*b*d^7*n*x^5 + 7236*(d*x + c)^n*C*a*b^2*d^7*n*x^5 + 2412*(d*x + c)^n*B*b^3*d^7*n*x
^5 + 2520*(d*x + c)^n*D*a*b^2*d^7*x^6 + 840*(d*x + c)^n*C*b^3*d^7*x^6 - (d*x + c)^n*B*a^3*c^2*d^5*n^5 - 3*(d*x
+ c)^n*A*a^2*b*c^2*d^5*n^5 + 27*(d*x + c)^n*A*a^3*c*d^6*n^5 + 6*(d*x + c)^n*D*a^3*c^3*d^4*n^4*x + 18*(d*x + c
)^n*C*a^2*b*c^3*d^4*n^4*x + 18*(d*x + c)^n*B*a*b^2*c^3*d^4*n^4*x + 6*(d*x + c)^n*A*b^3*c^3*d^4*n^4*x - 44*(d*x
+ c)^n*C*a^3*c^2*d^5*n^4*x - 132*(d*x + c)^n*B*a^2*b*c^2*d^5*n^4*x - 132*(d*x + c)^n*A*a*b^2*c^2*d^5*n^4*x +
245*(d*x + c)^n*B*a^3*c*d^6*n^4*x + 735*(d*x + c)^n*A*a^2*b*c*d^6*n^4*x + 295*(d*x + c)^n*A*a^3*d^7*n^4*x - 18
0*(d*x + c)^n*D*a*b^2*c^4*d^3*n^3*x^2 - 60*(d*x + c)^n*C*b^3*c^4*d^3*n^3*x^2 + 504*(d*x + c)^n*D*a^2*b*c^3*d^4
*n^3*x^2 + 504*(d*x + c)^n*C*a*b^2*c^3*d^4*n^3*x^2 + 168*(d*x + c)^n*B*b^3*c^3*d^4*n^3*x^2 - 375*(d*x + c)^n*D
*a^3*c^2*d^5*n^3*x^2 - 1125*(d*x + c)^n*C*a^2*b*c^2*d^5*n^3*x^2 - 1125*(d*x + c)^n*B*a*b^2*c^2*d^5*n^3*x^2 - 3
75*(d*x + c)^n*A*b^3*c^2*d^5*n^3*x^2 + 817*(d*x + c)^n*C*a^3*c*d^6*n^3*x^2 + 2451*(d*x + c)^n*B*a^2*b*c*d^6*n^
3*x^2 + 2451*(d*x + c)^n*A*a*b^2*c*d^6*n^3*x^2 + 1420*(d*x + c)^n*B*a^3*d^7*n^3*x^2 + 4260*(d*x + c)^n*A*a^2*b
*d^7*n^3*x^2 - 360*(d*x + c)^n*D*b^3*c^4*d^3*n^2*x^3 + 1380*(d*x + c)^n*D*a*b^2*c^3*d^4*n^2*x^3 + 460*(d*x + c
)^n*C*b^3*c^3*d^4*n^2*x^3 - 1824*(d*x + c)^n*D*a^2*b*c^2*d^5*n^2*x^3 - 1824*(d*x + c)^n*C*a*b^2*c^2*d^5*n^2*x^
3 - 608*(d*x + c)^n*B*b^3*c^2*d^5*n^2*x^3 + 844*(d*x + c)^n*D*a^3*c*d^6*n^2*x^3 + 2532*(d*x + c)^n*C*a^2*b*c*d
^6*n^2*x^3 + 2532*(d*x + c)^n*B*a*b^2*c*d^6*n^2*x^3 + 844*(d*x + c)^n*A*b^3*c*d^6*n^2*x^3 + 3112*(d*x + c)^n*C
*a^3*d^7*n^2*x^3 + 9336*(d*x + c)^n*B*a^2*b*d^7*n^2*x^3 + 9336*(d*x + c)^n*A*a*b^2*d^7*n^2*x^3 + 180*(d*x + c)
^n*D*b^3*c^3*d^4*n*x^4 - 630*(d*x + c)^n*D*a*b^2*c^2*d^5*n*x^4 - 210*(d*x + c)^n*C*b^3*c^2*d^5*n*x^4 + 756*(d*
x + c)^n*D*a^2*b*c*d^6*n*x^4 + 756*(d*x + c)^n*C*a*b^2*c*d^6*n*x^4 + 252*(d*x + c)^n*B*b^3*c*d^6*n*x^4 + 2952*
(d*x + c)^n*D*a^3*d^7*n*x^4 + 8856*(d*x + c)^n*C*a^2*b*d^7*n*x^4 + 8856*(d*x + c)^n*B*a*b^2*d^7*n*x^4 + 2952*(
d*x + c)^n*A*b^3*d^7*n*x^4 + 3024*(d*x + c)^n*D*a^2*b*d^7*x^5 + 3024*(d*x + c)^n*C*a*b^2*d^7*x^5 + 1008*(d*x +
c)^n*B*b^3*d^7*x^5 + 2*(d*x + c)^n*C*a^3*c^3*d^4*n^4 + 6*(d*x + c)^n*B*a^2*b*c^3*d^4*n^4 + 6*(d*x + c)^n*A*a*
b^2*c^3*d^4*n^4 - 25*(d*x + c)^n*B*a^3*c^2*d^5*n^4 - 75*(d*x + c)^n*A*a^2*b*c^2*d^5*n^4 + 295*(d*x + c)^n*A*a^
3*c*d^6*n^4 - 72*(d*x + c)^n*D*a^2*b*c^4*d^3*n^3*x - 72*(d*x + c)^n*C*a*b^2*c^4*d^3*n^3*x - 24*(d*x + c)^n*B*b
^3*c^4*d^3*n^3*x + 108*(d*x + c)^n*D*a^3*c^3*d^4*n^3*x + 324*(d*x + c)^n*C*a^2*b*c^3*d^4*n^3*x + 324*(d*x + c)
^n*B*a*b^2*c^3*d^4*n^3*x + 108*(d*x + c)^n*A*b^3*c^3*d^4*n^3*x - 358*(d*x + c)^n*C*a^3*c^2*d^5*n^3*x - 1074*(d
*x + c)^n*B*a^2*b*c^2*d^5*n^3*x - 1074*(d*x + c)^n*A*a*b^2*c^2*d^5*n^3*x + 1175*(d*x + c)^n*B*a^3*c*d^6*n^3*x
+ 3525*(d*x + c)^n*A*a^2*b*c*d^6*n^3*x + 1665*(d*x + c)^n*A*a^3*d^7*n^3*x + 360*(d*x + c)^n*D*b^3*c^5*d^2*n^2*
x^2 - 1440*(d*x + c)^n*D*a*b^2*c^4*d^3*n^2*x^2 - 480*(d*x + c)^n*C*b^3*c^4*d^3*n^2*x^2 + 1980*(d*x + c)^n*D*a^
2*b*c^3*d^4*n^2*x^2 + 1980*(d*x + c)^n*C*a*b^2*c^3*d^4*n^2*x^2 + 660*(d*x + c)^n*B*b^3*c^3*d^4*n^2*x^2 - 951*(
d*x + c)^n*D*a^3*c^2*d^5*n^2*x^2 - 2853*(d*x + c)^n*C*a^2*b*c^2*d^5*n^2*x^2 - 2853*(d*x + c)^n*B*a*b^2*c^2*d^5
*n^2*x^2 - 951*(d*x + c)^n*A*b^3*c^2*d^5*n^2*x^2 + 1478*(d*x + c)^n*C*a^3*c*d^6*n^2*x^2 + 4434*(d*x + c)^n*B*a
^2*b*c*d^6*n^2*x^2 + 4434*(d*x + c)^n*A*a*b^2*c*d^6*n^2*x^2 + 3929*(d*x + c)^n*B*a^3*d^7*n^2*x^2 + 11787*(d*x
+ c)^n*A*a^2*b*d^7*n^2*x^2 - 240*(d*x + c)^n*D*b^3*c^4*d^3*n*x^3 + 840*(d*x + c)^n*D*a*b^2*c^3*d^4*n*x^3 + 280
*(d*x + c)^n*C*b^3*c^3*d^4*n*x^3 - 1008*(d*x + c)^n*D*a^2*b*c^2*d^5*n*x^3 - 1008*(d*x + c)^n*C*a*b^2*c^2*d^5*n
*x^3 - 336*(d*x + c)^n*B*b^3*c^2*d^5*n*x^3 + 420*(d*x + c)^n*D*a^3*c*d^6*n*x^3 + 1260*(d*x + c)^n*C*a^2*b*c*d^
6*n*x^3 + 1260*(d*x + c)^n*B*a*b^2*c*d^6*n*x^3 + 420*(d*x + c)^n*A*b^3*c*d^6*n*x^3 + 3796*(d*x + c)^n*C*a^3*d^
7*n*x^3 + 11388*(d*x + c)^n*B*a^2*b*d^7*n*x^3 + 11388*(d*x + c)^n*A*a*b^2*d^7*n*x^3 + 1260*(d*x + c)^n*D*a^3*d
^7*x^4 + 3780*(d*x + c)^n*C*a^2*b*d^7*x^4 + 3780*(d*x + c)^n*B*a*b^2*d^7*x^4 + 1260*(d*x + c)^n*A*b^3*d^7*x^4
- 6*(d*x + c)^n*D*a^3*c^4*d^3*n^3 - 18*(d*x + c)^n*C*a^2*b*c^4*d^3*n^3 - 18*(d*x + c)^n*B*a*b^2*c^4*d^3*n^3 -
6*(d*x + c)^n*A*b^3*c^4*d^3*n^3 + 44*(d*x + c)^n*C*a^3*c^3*d^4*n^3 + 132*(d*x + c)^n*B*a^2*b*c^3*d^4*n^3 + 132
*(d*x + c)^n*A*a*b^2*c^3*d^4*n^3 - 245*(d*x + c)^n*B*a^3*c^2*d^5*n^3 - 735*(d*x + c)^n*A*a^2*b*c^2*d^5*n^3 + 1
665*(d*x + c)^n*A*a^3*c*d^6*n^3 + 360*(d*x + c)^n*D*a*b^2*c^5*d^2*n^2*x + 120*(d*x + c)^n*C*b^3*c^5*d^2*n^2*x
- 936*(d*x + c)^n*D*a^2*b*c^4*d^3*n^2*x - 936*(d*x + c)^n*C*a*b^2*c^4*d^3*n^2*x - 312*(d*x + c)^n*B*b^3*c^4*d^
3*n^2*x + 642*(d*x + c)^n*D*a^3*c^3*d^4*n^2*x + 1926*(d*x + c)^n*C*a^2*b*c^3*d^4*n^2*x + 1926*(d*x + c)^n*B*a*
b^2*c^3*d^4*n^2*x + 642*(d*x + c)^n*A*b^3*c^3*d^4*n^2*x - 1276*(d*x + c)^n*C*a^3*c^2*d^5*n^2*x - 3828*(d*x + c
)^n*B*a^2*b*c^2*d^5*n^2*x - 3828*(d*x + c)^n*A*a*b^2*c^2*d^5*n^2*x + 2754*(d*x + c)^n*B*a^3*c*d^6*n^2*x + 8262
*(d*x + c)^n*A*a^2*b*c*d^6*n^2*x + 5104*(d*x + c)^n*A*a^3*d^7*n^2*x + 360*(d*x + c)^n*D*b^3*c^5*d^2*n*x^2 - 12
60*(d*x + c)^n*D*a*b^2*c^4*d^3*n*x^2 - 420*(d*x + c)^n*C*b^3*c^4*d^3*n*x^2 + 1512*(d*x + c)^n*D*a^2*b*c^3*d^4*
n*x^2 + 1512*(d*x + c)^n*C*a*b^2*c^3*d^4*n*x^2 + 504*(d*x + c)^n*B*b^3*c^3*d^4*n*x^2 - 630*(d*x + c)^n*D*a^3*c
^2*d^5*n*x^2 - 1890*(d*x + c)^n*C*a^2*b*c^2*d^5*n*x^2 - 1890*(d*x + c)^n*B*a*b^2*c^2*d^5*n*x^2 - 630*(d*x + c)
^n*A*b^3*c^2*d^5*n*x^2 + 840*(d*x + c)^n*C*a^3*c*d^6*n*x^2 + 2520*(d*x + c)^n*B*a^2*b*c*d^6*n*x^2 + 2520*(d*x
+ c)^n*A*a*b^2*c*d^6*n*x^2 + 5274*(d*x + c)^n*B*a^3*d^7*n*x^2 + 15822*(d*x + c)^n*A*a^2*b*d^7*n*x^2 + 1680*(d*
x + c)^n*C*a^3*d^7*x^3 + 5040*(d*x + c)^n*B*a^2*b*d^7*x^3 + 5040*(d*x + c)^n*A*a*b^2*d^7*x^3 + 72*(d*x + c)^n*
D*a^2*b*c^5*d^2*n^2 + 72*(d*x + c)^n*C*a*b^2*c^5*d^2*n^2 + 24*(d*x + c)^n*B*b^3*c^5*d^2*n^2 - 108*(d*x + c)^n*
D*a^3*c^4*d^3*n^2 - 324*(d*x + c)^n*C*a^2*b*c^4*d^3*n^2 - 324*(d*x + c)^n*B*a*b^2*c^4*d^3*n^2 - 108*(d*x + c)^
n*A*b^3*c^4*d^3*n^2 + 358*(d*x + c)^n*C*a^3*c^3*d^4*n^2 + 1074*(d*x + c)^n*B*a^2*b*c^3*d^4*n^2 + 1074*(d*x + c
)^n*A*a*b^2*c^3*d^4*n^2 - 1175*(d*x + c)^n*B*a^3*c^2*d^5*n^2 - 3525*(d*x + c)^n*A*a^2*b*c^2*d^5*n^2 + 5104*(d*
x + c)^n*A*a^3*c*d^6*n^2 - 720*(d*x + c)^n*D*b^3*c^6*d*n*x + 2520*(d*x + c)^n*D*a*b^2*c^5*d^2*n*x + 840*(d*x +
c)^n*C*b^3*c^5*d^2*n*x - 3024*(d*x + c)^n*D*a^2*b*c^4*d^3*n*x - 3024*(d*x + c)^n*C*a*b^2*c^4*d^3*n*x - 1008*(
d*x + c)^n*B*b^3*c^4*d^3*n*x + 1260*(d*x + c)^n*D*a^3*c^3*d^4*n*x + 3780*(d*x + c)^n*C*a^2*b*c^3*d^4*n*x + 378
0*(d*x + c)^n*B*a*b^2*c^3*d^4*n*x + 1260*(d*x + c)^n*A*b^3*c^3*d^4*n*x - 1680*(d*x + c)^n*C*a^3*c^2*d^5*n*x -
5040*(d*x + c)^n*B*a^2*b*c^2*d^5*n*x - 5040*(d*x + c)^n*A*a*b^2*c^2*d^5*n*x + 2520*(d*x + c)^n*B*a^3*c*d^6*n*x
+ 7560*(d*x + c)^n*A*a^2*b*c*d^6*n*x + 8028*(d*x + c)^n*A*a^3*d^7*n*x + 2520*(d*x + c)^n*B*a^3*d^7*x^2 + 7560
*(d*x + c)^n*A*a^2*b*d^7*x^2 - 360*(d*x + c)^n*D*a*b^2*c^6*d*n - 120*(d*x + c)^n*C*b^3*c^6*d*n + 936*(d*x + c)
^n*D*a^2*b*c^5*d^2*n + 936*(d*x + c)^n*C*a*b^2*c^5*d^2*n + 312*(d*x + c)^n*B*b^3*c^5*d^2*n - 642*(d*x + c)^n*D
*a^3*c^4*d^3*n - 1926*(d*x + c)^n*C*a^2*b*c^4*d^3*n - 1926*(d*x + c)^n*B*a*b^2*c^4*d^3*n - 642*(d*x + c)^n*A*b
^3*c^4*d^3*n + 1276*(d*x + c)^n*C*a^3*c^3*d^4*n + 3828*(d*x + c)^n*B*a^2*b*c^3*d^4*n + 3828*(d*x + c)^n*A*a*b^
2*c^3*d^4*n - 2754*(d*x + c)^n*B*a^3*c^2*d^5*n - 8262*(d*x + c)^n*A*a^2*b*c^2*d^5*n + 8028*(d*x + c)^n*A*a^3*c
*d^6*n + 5040*(d*x + c)^n*A*a^3*d^7*x + 720*(d*x + c)^n*D*b^3*c^7 - 2520*(d*x + c)^n*D*a*b^2*c^6*d - 840*(d*x
+ c)^n*C*b^3*c^6*d + 3024*(d*x + c)^n*D*a^2*b*c^5*d^2 + 3024*(d*x + c)^n*C*a*b^2*c^5*d^2 + 1008*(d*x + c)^n*B*
b^3*c^5*d^2 - 1260*(d*x + c)^n*D*a^3*c^4*d^3 - 3780*(d*x + c)^n*C*a^2*b*c^4*d^3 - 3780*(d*x + c)^n*B*a*b^2*c^4
*d^3 - 1260*(d*x + c)^n*A*b^3*c^4*d^3 + 1680*(d*x + c)^n*C*a^3*c^3*d^4 + 5040*(d*x + c)^n*B*a^2*b*c^3*d^4 + 50
40*(d*x + c)^n*A*a*b^2*c^3*d^4 - 2520*(d*x + c)^n*B*a^3*c^2*d^5 - 7560*(d*x + c)^n*A*a^2*b*c^2*d^5 + 5040*(d*x
+ c)^n*A*a^3*c*d^6)/(d^7*n^7 + 28*d^7*n^6 + 322*d^7*n^5 + 1960*d^7*n^4 + 6769*d^7*n^3 + 13132*d^7*n^2 + 13068
*d^7*n + 5040*d^7)