3.1654 \(\int (b+2 c x) (d+e x)^m (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=449 \[ \frac{(d+e x)^{m+4} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8 (m+4)}+\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (m+2)}-\frac{3 (2 c d-b e) (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (m+3)}-\frac{5 c (2 c d-b e) (d+e x)^{m+5} \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (m+5)}+\frac{3 c^2 (d+e x)^{m+6} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (m+6)}-\frac{(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^3}{e^8 (m+1)}-\frac{7 c^3 (2 c d-b e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac{2 c^4 (d+e x)^{m+8}}{e^8 (m+8)} \]

[Out]

-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(1 + m))/(e^8*(1 + m))) + ((c*d^2 - b*d*e + a*e^2)^2*(14*
c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(2 + m))/(e^8*(2 + m)) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e
+ a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(3 + m))/(e^8*(3 + m)) + ((70*c^4*d^4 + b^4*e^4
- 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))
*(d + e*x)^(4 + m))/(e^8*(4 + m)) - (5*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(
5 + m))/(e^8*(5 + m)) + (3*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(6 + m))/(e^8*(6 + m))
- (7*c^3*(2*c*d - b*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)) + (2*c^4*(d + e*x)^(8 + m))/(e^8*(8 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.344835, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ \frac{(d+e x)^{m+4} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8 (m+4)}+\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (m+2)}-\frac{3 (2 c d-b e) (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (m+3)}-\frac{5 c (2 c d-b e) (d+e x)^{m+5} \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (m+5)}+\frac{3 c^2 (d+e x)^{m+6} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (m+6)}-\frac{(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^3}{e^8 (m+1)}-\frac{7 c^3 (2 c d-b e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac{2 c^4 (d+e x)^{m+8}}{e^8 (m+8)} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^m*(a + b*x + c*x^2)^3,x]

[Out]

-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(1 + m))/(e^8*(1 + m))) + ((c*d^2 - b*d*e + a*e^2)^2*(14*
c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(2 + m))/(e^8*(2 + m)) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e
+ a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(3 + m))/(e^8*(3 + m)) + ((70*c^4*d^4 + b^4*e^4
- 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))
*(d + e*x)^(4 + m))/(e^8*(4 + m)) - (5*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(
5 + m))/(e^8*(5 + m)) + (3*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(6 + m))/(e^8*(6 + m))
- (7*c^3*(2*c*d - b*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)) + (2*c^4*(d + e*x)^(8 + m))/(e^8*(8 + m))

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^m}{e^7}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{1+m}}{e^7}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^{2+m}}{e^7}+\frac{\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{3+m}}{e^7}+\frac{5 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^{4+m}}{e^7}+\frac{3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{5+m}}{e^7}-\frac{7 c^3 (2 c d-b e) (d+e x)^{6+m}}{e^7}+\frac{2 c^4 (d+e x)^{7+m}}{e^7}\right ) \, dx\\ &=-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{1+m}}{e^8 (1+m)}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{2+m}}{e^8 (2+m)}-\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{3+m}}{e^8 (3+m)}+\frac{\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{4+m}}{e^8 (4+m)}-\frac{5 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{5+m}}{e^8 (5+m)}+\frac{3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{6+m}}{e^8 (6+m)}-\frac{7 c^3 (2 c d-b e) (d+e x)^{7+m}}{e^8 (7+m)}+\frac{2 c^4 (d+e x)^{8+m}}{e^8 (8+m)}\\ \end{align*}

Mathematica [B]  time = 3.82152, size = 1226, normalized size = 2.73 \[ \frac{(d+e x)^{m+1} \left (\frac{3 \left (c e^4 (m+1) (m+2) (m+3) (m+4) (a+x (b+c x))^2 \left (-28 d^2 (5 d-e (m+5) x) c^3+2 e \left (7 b d (d (m+20)-2 e (m+5) x)+2 a e \left (d \left (m^2+m-35\right )+e \left (m^2+12 m+35\right ) x\right )\right ) c^2-b e^2 \left (-2 a e (11 m+70)+b d \left (m^2+15 m+140\right )+b e m (m+5) x\right ) c-2 b^3 e^3 m\right )-2 \left ((m+1) (m+2) \left (c e (m+4) \left (c e (b d-2 a e) (m+6) \left (-d e (m+14) b^2+14 \left (c d^2+a e^2\right ) b+4 a c d e m\right )-(b d (5 c d-2 b e)+a e (2 c d m-b e (m+1))) \left (28 c^2 d^2-b^2 e^2 m+4 c e (a e (m+7)-7 b d)\right )\right )-(3 c d-b e) \left (c e (2 c d-b e) (m+6) \left (-d e (m+14) b^2+14 \left (c d^2+a e^2\right ) b+4 a c d e m\right )-\left (10 c^2 d^2-b^2 e^2 (m+3)+c e (b d (m-4)+2 a e (m+5))\right ) \left (28 c^2 d^2-b^2 e^2 m+4 c e (a e (m+7)-7 b d)\right )\right )+c e (m+3) \left (c e (2 c d-b e) (m+6) \left (-d e (m+14) b^2+14 \left (c d^2+a e^2\right ) b+4 a c d e m\right )-\left (10 c^2 d^2-b^2 e^2 (m+3)+c e (b d (m-4)+2 a e (m+5))\right ) \left (28 c^2 d^2-b^2 e^2 m+4 c e (a e (m+7)-7 b d)\right )\right ) x\right ) (a+x (b+c x)) e^2+(2 c d-b e) (m+2) \left (840 c^5 d^6-40 c^4 e \left (63 b d+a e \left (2 m^2+2 m-63\right )\right ) d^4+4 c^3 e^2 \left (5 b^2 \left (m^2+m+126\right ) d^2+20 a b e \left (2 m^2+2 m-63\right ) d-2 a^2 e^2 \left (2 m^3+28 m^2+26 m-315\right )\right ) d^2+b^4 e^5 (b d-a e) m \left (m^2+4 m+3\right )-b^2 c e^4 m (m+1) \left (b^2 (m-17) d^2+8 a b e (m+8) d-4 a^2 e^2 (2 m+11)\right )-8 c^2 e^3 \left (5 b^3 \left (m^2+m+21\right ) d^3+a b^2 e \left (-m^3+m^2+2 m-315\right ) d^2-a^2 b e^2 \left (2 m^3+28 m^2+26 m-315\right ) d+a^3 e^3 \left (2 m^3+18 m^2+16 m-105\right )\right )\right )+(m+1) \left (-1680 c^6 d^6+80 c^5 e \left (63 b d+a e \left (2 m^2-5 m-63\right )\right ) d^4-4 c^4 e^2 \left (5 b^2 \left (2 m^2-5 m+252\right ) d^2+40 a b e \left (2 m^2-5 m-63\right ) d-12 a^2 e^2 \left (m^3+5 m^2-24 m-105\right )\right ) d^2+b^6 e^6 m \left (m^2+5 m+6\right )-b^4 c e^5 m (m+2) (b d (3 m-11)+a e (9 m+47))+b^2 c^2 e^4 m \left (3 b^2 \left (m^2-15 m+26\right ) d^2+8 a b e \left (3 m^2+5 m-47\right ) d+12 a^2 e^2 \left (2 m^2+20 m+47\right )\right )-8 c^3 e^3 \left (-5 b^3 \left (2 m^2-5 m+42\right ) d^3+3 a b^2 e \left (m^3-5 m^2+m+210\right ) d^2+6 a^2 b e^2 \left (m^3+5 m^2-24 m-105\right ) d+2 a^3 e^3 \left (m^3+15 m^2+71 m+105\right )\right )\right ) (d+e x)\right )\right )}{e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6)}-c^2 (14 c d-b e (m+14)-2 c e (m+7) x) (a+x (b+c x))^3\right )}{c^2 e^2 (m+7) (m+8)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)^m*(a + b*x + c*x^2)^3,x]

[Out]

((d + e*x)^(1 + m)*(-(c^2*(14*c*d - b*e*(14 + m) - 2*c*e*(7 + m)*x)*(a + x*(b + c*x))^3) + (3*(-2*((2*c*d - b*
e)*(2 + m)*(840*c^5*d^6 + b^4*e^5*(b*d - a*e)*m*(3 + 4*m + m^2) - b^2*c*e^4*m*(1 + m)*(b^2*d^2*(-17 + m) + 8*a
*b*d*e*(8 + m) - 4*a^2*e^2*(11 + 2*m)) - 40*c^4*d^4*e*(63*b*d + a*e*(-63 + 2*m + 2*m^2)) + 4*c^3*d^2*e^2*(5*b^
2*d^2*(126 + m + m^2) + 20*a*b*d*e*(-63 + 2*m + 2*m^2) - 2*a^2*e^2*(-315 + 26*m + 28*m^2 + 2*m^3)) - 8*c^2*e^3
*(5*b^3*d^3*(21 + m + m^2) + a*b^2*d^2*e*(-315 + 2*m + m^2 - m^3) + a^3*e^3*(-105 + 16*m + 18*m^2 + 2*m^3) - a
^2*b*d*e^2*(-315 + 26*m + 28*m^2 + 2*m^3))) + (1 + m)*(-1680*c^6*d^6 + b^6*e^6*m*(6 + 5*m + m^2) - b^4*c*e^5*m
*(2 + m)*(b*d*(-11 + 3*m) + a*e*(47 + 9*m)) + 80*c^5*d^4*e*(63*b*d + a*e*(-63 - 5*m + 2*m^2)) + b^2*c^2*e^4*m*
(3*b^2*d^2*(26 - 15*m + m^2) + 12*a^2*e^2*(47 + 20*m + 2*m^2) + 8*a*b*d*e*(-47 + 5*m + 3*m^2)) - 4*c^4*d^2*e^2
*(40*a*b*d*e*(-63 - 5*m + 2*m^2) + 5*b^2*d^2*(252 - 5*m + 2*m^2) - 12*a^2*e^2*(-105 - 24*m + 5*m^2 + m^3)) - 8
*c^3*e^3*(-5*b^3*d^3*(42 - 5*m + 2*m^2) + 3*a*b^2*d^2*e*(210 + m - 5*m^2 + m^3) + 6*a^2*b*d*e^2*(-105 - 24*m +
5*m^2 + m^3) + 2*a^3*e^3*(105 + 71*m + 15*m^2 + m^3)))*(d + e*x) + e^2*(1 + m)*(2 + m)*(c*e*(4 + m)*(c*e*(b*d
- 2*a*e)*(6 + m)*(14*b*(c*d^2 + a*e^2) + 4*a*c*d*e*m - b^2*d*e*(14 + m)) - (b*d*(5*c*d - 2*b*e) + a*e*(2*c*d*
m - b*e*(1 + m)))*(28*c^2*d^2 - b^2*e^2*m + 4*c*e*(-7*b*d + a*e*(7 + m)))) - (3*c*d - b*e)*(c*e*(2*c*d - b*e)*
(6 + m)*(14*b*(c*d^2 + a*e^2) + 4*a*c*d*e*m - b^2*d*e*(14 + m)) - (10*c^2*d^2 - b^2*e^2*(3 + m) + c*e*(b*d*(-4
+ m) + 2*a*e*(5 + m)))*(28*c^2*d^2 - b^2*e^2*m + 4*c*e*(-7*b*d + a*e*(7 + m)))) + c*e*(3 + m)*(c*e*(2*c*d - b
*e)*(6 + m)*(14*b*(c*d^2 + a*e^2) + 4*a*c*d*e*m - b^2*d*e*(14 + m)) - (10*c^2*d^2 - b^2*e^2*(3 + m) + c*e*(b*d
*(-4 + m) + 2*a*e*(5 + m)))*(28*c^2*d^2 - b^2*e^2*m + 4*c*e*(-7*b*d + a*e*(7 + m))))*x)*(a + x*(b + c*x))) + c
*e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(a + x*(b + c*x))^2*(-2*b^3*e^3*m - 28*c^3*d^2*(5*d - e*(5 + m)*x) - b*c*
e^2*(-2*a*e*(70 + 11*m) + b*d*(140 + 15*m + m^2) + b*e*m*(5 + m)*x) + 2*c^2*e*(7*b*d*(d*(20 + m) - 2*e*(5 + m)
*x) + 2*a*e*(d*(-35 + m + m^2) + e*(35 + 12*m + m^2)*x)))))/(e^6*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 +
m))))/(c^2*e^2*(7 + m)*(8 + m))

________________________________________________________________________________________

Maple [B]  time = 0.022, size = 5439, normalized size = 12.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a)^3,x)

[Out]

result too large to display

________________________________________________________________________________________

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((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.20976, size = 10201, normalized size = 22.72 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

(a^3*b*d*e^7*m^7 - 10080*c^4*d^8 + 40320*b*c^3*d^7*e + 40320*a^3*b*d*e^7 - 20160*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2
+ 40320*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 10080*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 40320*(a*b^3 + 3*a^2*b*c
)*d^3*e^5 - 20160*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 2*(c^4*e^8*m^7 + 28*c^4*e^8*m^6 + 322*c^4*e^8*m^5 + 1960*c^4
*e^8*m^4 + 6769*c^4*e^8*m^3 + 13132*c^4*e^8*m^2 + 13068*c^4*e^8*m + 5040*c^4*e^8)*x^8 + (40320*b*c^3*e^8 + (2*
c^4*d*e^7 + 7*b*c^3*e^8)*m^7 + 7*(6*c^4*d*e^7 + 29*b*c^3*e^8)*m^6 + 7*(50*c^4*d*e^7 + 343*b*c^3*e^8)*m^5 + 245
*(6*c^4*d*e^7 + 61*b*c^3*e^8)*m^4 + 112*(29*c^4*d*e^7 + 469*b*c^3*e^8)*m^3 + 196*(18*c^4*d*e^7 + 527*b*c^3*e^8
)*m^2 + 144*(10*c^4*d*e^7 + 721*b*c^3*e^8)*m)*x^7 + (35*a^3*b*d*e^7 - (3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m^6 + (20
160*(3*b^2*c^2 + 2*a*c^3)*e^8 + (7*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^7 - (14*c^4*d^2*e^6 - 161*b*c^
3*d*e^7 - 90*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^6 - (210*c^4*d^2*e^6 - 1435*b*c^3*d*e^7 - 1098*(3*b^2*c^2 + 2*a*c^3)
*e^8)*m^5 - 5*(238*c^4*d^2*e^6 - 1267*b*c^3*d*e^7 - 1404*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^4 - (3150*c^4*d^2*e^6 -
14518*b*c^3*d*e^7 - 25227*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^3 - 2*(1918*c^4*d^2*e^6 - 8092*b*c^3*d*e^7 - 25245*(3*b
^2*c^2 + 2*a*c^3)*e^8)*m^2 - 24*(70*c^4*d^2*e^6 - 280*b*c^3*d*e^7 - 2143*(3*b^2*c^2 + 2*a*c^3)*e^8)*m)*x^6 + (
511*a^3*b*d*e^7 + 6*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 33*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m^5 + (40320*(b^3*c + 3*a*
b*c^2)*e^8 + (3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 5*(b^3*c + 3*a*b*c^2)*e^8)*m^7 - (42*b*c^3*d^2*e^6 - 75*(3*b^2*c
^2 + 2*a*c^3)*d*e^7 - 155*(b^3*c + 3*a*b*c^2)*e^8)*m^6 + (84*c^4*d^3*e^5 - 756*b*c^3*d^2*e^6 + 723*(3*b^2*c^2
+ 2*a*c^3)*d*e^7 + 1955*(b^3*c + 3*a*b*c^2)*e^8)*m^5 + 5*(168*c^4*d^3*e^5 - 966*b*c^3*d^2*e^6 + 681*(3*b^2*c^2
+ 2*a*c^3)*d*e^7 + 2581*(b^3*c + 3*a*b*c^2)*e^8)*m^4 + 2*(1470*c^4*d^3*e^5 - 6930*b*c^3*d^2*e^6 + 4101*(3*b^2
*c^2 + 2*a*c^3)*d*e^7 + 23860*(b^3*c + 3*a*b*c^2)*e^8)*m^3 + 4*(1050*c^4*d^3*e^5 - 4452*b*c^3*d^2*e^6 + 2370*(
3*b^2*c^2 + 2*a*c^3)*d*e^7 + 24455*(b^3*c + 3*a*b*c^2)*e^8)*m^2 + 144*(14*c^4*d^3*e^5 - 56*b*c^3*d^2*e^6 + 28*
(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 705*(b^3*c + 3*a*b*c^2)*e^8)*m)*x^5 + (4025*a^3*b*d*e^7 - 6*(b^4 + 12*a*b^2*c +
6*a^2*c^2)*d^4*e^4 + 180*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 445*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m^4 + (10080*(b^4 +
12*a*b^2*c + 6*a^2*c^2)*e^8 + (5*(b^3*c + 3*a*b*c^2)*d*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*m^7 - (15*(3*
b^2*c^2 + 2*a*c^3)*d^2*e^6 - 135*(b^3*c + 3*a*b*c^2)*d*e^7 - 32*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*m^6 + (210
*b*c^3*d^3*e^5 - 315*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 1415*(b^3*c + 3*a*b*c^2)*d*e^7 + 418*(b^4 + 12*a*b^2*c +
6*a^2*c^2)*e^8)*m^5 - (420*c^4*d^4*e^4 - 2940*b*c^3*d^3*e^5 + 2355*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 7245*(b^3*c
+ 3*a*b*c^2)*d*e^7 - 2864*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*m^4 - (2520*c^4*d^4*e^4 - 12390*b*c^3*d^3*e^5 +
7605*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 18740*(b^3*c + 3*a*b*c^2)*d*e^7 - 10993*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e
^8)*m^3 - 2*(2310*c^4*d^4*e^4 - 9870*b*c^3*d^3*e^5 + 5295*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 11430*(b^3*c + 3*a*b
*c^2)*d*e^7 - 11656*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*m^2 - 36*(70*c^4*d^4*e^4 - 280*b*c^3*d^3*e^5 + 140*(3*
b^2*c^2 + 2*a*c^3)*d^2*e^6 - 280*(b^3*c + 3*a*b*c^2)*d*e^7 - 691*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*m)*x^4 +
(18424*a^3*b*d*e^7 + 120*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 156*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 2130*(a*b^
3 + 3*a^2*b*c)*d^3*e^5 - 3135*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m^3 + (40320*(a*b^3 + 3*a^2*b*c)*e^8 + ((b^4 + 12
*a*b^2*c + 6*a^2*c^2)*d*e^7 + 3*(a*b^3 + 3*a^2*b*c)*e^8)*m^7 - (20*(b^3*c + 3*a*b*c^2)*d^2*e^6 - 29*(b^4 + 12*
a*b^2*c + 6*a^2*c^2)*d*e^7 - 99*(a*b^3 + 3*a^2*b*c)*e^8)*m^6 + (60*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 480*(b^3*c
+ 3*a*b*c^2)*d^2*e^6 + 331*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 1341*(a*b^3 + 3*a^2*b*c)*e^8)*m^5 - (840*b*c
^3*d^4*e^4 - 1080*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 4220*(b^3*c + 3*a*b*c^2)*d^2*e^6 - 1871*(b^4 + 12*a*b^2*c +
6*a^2*c^2)*d*e^7 - 9585*(a*b^3 + 3*a^2*b*c)*e^8)*m^4 + 4*(420*c^4*d^5*e^3 - 2310*b*c^3*d^4*e^4 + 1545*(3*b^2*c
^2 + 2*a*c^3)*d^3*e^5 - 4080*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 1345*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 9648*(a
*b^3 + 3*a^2*b*c)*e^8)*m^3 + 4*(1260*c^4*d^5*e^3 - 5460*b*c^3*d^4*e^4 + 2970*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 6
500*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 1793*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 21519*(a*b^3 + 3*a^2*b*c)*e^8)*m
^2 + 48*(70*c^4*d^5*e^3 - 280*b*c^3*d^4*e^4 + 140*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 280*(b^3*c + 3*a*b*c^2)*d^2*
e^6 + 70*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 2003*(a*b^3 + 3*a^2*b*c)*e^8)*m)*x^3 + 2*(24430*a^3*b*d*e^7 -
180*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 1260*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 753*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*
e^4 + 6210*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 6077*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m^2 + (20160*(3*a^2*b^2 + 2*a^3*c
)*e^8 + (3*(a*b^3 + 3*a^2*b*c)*d*e^7 + (3*a^2*b^2 + 2*a^3*c)*e^8)*m^7 - (3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*
e^6 - 93*(a*b^3 + 3*a^2*b*c)*d*e^7 - 34*(3*a^2*b^2 + 2*a^3*c)*e^8)*m^6 + (60*(b^3*c + 3*a*b*c^2)*d^3*e^5 - 81*
(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 1155*(a*b^3 + 3*a^2*b*c)*d*e^7 + 478*(3*a^2*b^2 + 2*a^3*c)*e^8)*m^5 -
(180*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 1320*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 831*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^
2*e^6 - 7275*(a*b^3 + 3*a^2*b*c)*d*e^7 - 3580*(3*a^2*b^2 + 2*a^3*c)*e^8)*m^4 + (2520*b*c^3*d^5*e^3 - 2880*(3*b
^2*c^2 + 2*a*c^3)*d^4*e^4 + 10020*(b^3*c + 3*a*b*c^2)*d^3*e^5 - 3951*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 +
24042*(a*b^3 + 3*a^2*b*c)*d*e^7 + 15289*(3*a^2*b^2 + 2*a^3*c)*e^8)*m^3 - 2*(2520*c^4*d^6*e^2 - 11340*b*c^3*d^5
*e^3 + 6390*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 14460*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 4119*(b^4 + 12*a*b^2*c + 6*a^2
*c^2)*d^2*e^6 - 18996*(a*b^3 + 3*a^2*b*c)*d*e^7 - 18353*(3*a^2*b^2 + 2*a^3*c)*e^8)*m^2 - 72*(70*c^4*d^6*e^2 -
280*b*c^3*d^5*e^3 + 140*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 280*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 70*(b^4 + 12*a*b^2*c
+ 6*a^2*c^2)*d^2*e^6 - 280*(a*b^3 + 3*a^2*b*c)*d*e^7 - 621*(3*a^2*b^2 + 2*a^3*c)*e^8)*m)*x^2 + 12*(420*b*c^3*
d^7*e + 5772*a^3*b*d*e^7 - 450*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 1460*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 533*(b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 2972*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 2046*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m + (4
0320*a^3*b*e^8 + (a^3*b*e^8 + (3*a^2*b^2 + 2*a^3*c)*d*e^7)*m^7 + (35*a^3*b*e^8 - 6*(a*b^3 + 3*a^2*b*c)*d^2*e^6
+ 33*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*m^6 + (511*a^3*b*e^8 + 6*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 180*(a*b^
3 + 3*a^2*b*c)*d^2*e^6 + 445*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*m^5 + (4025*a^3*b*e^8 - 120*(b^3*c + 3*a*b*c^2)*d^4*
e^4 + 156*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 2130*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 3135*(3*a^2*b^2 + 2*a^3*
c)*d*e^7)*m^4 + 2*(9212*a^3*b*e^8 + 180*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 1260*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 753
*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 6210*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 6077*(3*a^2*b^2 + 2*a^3*c)*d*e^7)
*m^3 - 4*(1260*b*c^3*d^6*e^2 - 12215*a^3*b*e^8 - 1350*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 4380*(b^3*c + 3*a*b*c^2)
*d^4*e^4 - 1599*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 8916*(a*b^3 + 3*a^2*b*c)*d^2*e^6 - 6138*(3*a^2*b^2 +
2*a^3*c)*d*e^7)*m^2 + 144*(70*c^4*d^7*e - 280*b*c^3*d^6*e^2 + 481*a^3*b*e^8 + 140*(3*b^2*c^2 + 2*a*c^3)*d^5*e^
3 - 280*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 70*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 280*(a*b^3 + 3*a^2*b*c)*d^2*
e^6 + 140*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*m)*x)*(e*x + d)^m/(e^8*m^8 + 36*e^8*m^7 + 546*e^8*m^6 + 4536*e^8*m^5 +
22449*e^8*m^4 + 67284*e^8*m^3 + 118124*e^8*m^2 + 109584*e^8*m + 40320*e^8)

________________________________________________________________________________________

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((2*c*x+b)*(e*x+d)**m*(c*x**2+b*x+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.13022, size = 13080, normalized size = 29.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

(2*(x*e + d)^m*c^4*m^7*x^8*e^8 + 2*(x*e + d)^m*c^4*d*m^7*x^7*e^7 + 7*(x*e + d)^m*b*c^3*m^7*x^7*e^8 + 56*(x*e +
d)^m*c^4*m^6*x^8*e^8 + 7*(x*e + d)^m*b*c^3*d*m^7*x^6*e^7 + 42*(x*e + d)^m*c^4*d*m^6*x^7*e^7 - 14*(x*e + d)^m*
c^4*d^2*m^6*x^6*e^6 + 9*(x*e + d)^m*b^2*c^2*m^7*x^6*e^8 + 6*(x*e + d)^m*a*c^3*m^7*x^6*e^8 + 203*(x*e + d)^m*b*
c^3*m^6*x^7*e^8 + 644*(x*e + d)^m*c^4*m^5*x^8*e^8 + 9*(x*e + d)^m*b^2*c^2*d*m^7*x^5*e^7 + 6*(x*e + d)^m*a*c^3*
d*m^7*x^5*e^7 + 161*(x*e + d)^m*b*c^3*d*m^6*x^6*e^7 + 350*(x*e + d)^m*c^4*d*m^5*x^7*e^7 - 42*(x*e + d)^m*b*c^3
*d^2*m^6*x^5*e^6 - 210*(x*e + d)^m*c^4*d^2*m^5*x^6*e^6 + 84*(x*e + d)^m*c^4*d^3*m^5*x^5*e^5 + 5*(x*e + d)^m*b^
3*c*m^7*x^5*e^8 + 15*(x*e + d)^m*a*b*c^2*m^7*x^5*e^8 + 270*(x*e + d)^m*b^2*c^2*m^6*x^6*e^8 + 180*(x*e + d)^m*a
*c^3*m^6*x^6*e^8 + 2401*(x*e + d)^m*b*c^3*m^5*x^7*e^8 + 3920*(x*e + d)^m*c^4*m^4*x^8*e^8 + 5*(x*e + d)^m*b^3*c
*d*m^7*x^4*e^7 + 15*(x*e + d)^m*a*b*c^2*d*m^7*x^4*e^7 + 225*(x*e + d)^m*b^2*c^2*d*m^6*x^5*e^7 + 150*(x*e + d)^
m*a*c^3*d*m^6*x^5*e^7 + 1435*(x*e + d)^m*b*c^3*d*m^5*x^6*e^7 + 1470*(x*e + d)^m*c^4*d*m^4*x^7*e^7 - 45*(x*e +
d)^m*b^2*c^2*d^2*m^6*x^4*e^6 - 30*(x*e + d)^m*a*c^3*d^2*m^6*x^4*e^6 - 756*(x*e + d)^m*b*c^3*d^2*m^5*x^5*e^6 -
1190*(x*e + d)^m*c^4*d^2*m^4*x^6*e^6 + 210*(x*e + d)^m*b*c^3*d^3*m^5*x^4*e^5 + 840*(x*e + d)^m*c^4*d^3*m^4*x^5
*e^5 - 420*(x*e + d)^m*c^4*d^4*m^4*x^4*e^4 + (x*e + d)^m*b^4*m^7*x^4*e^8 + 12*(x*e + d)^m*a*b^2*c*m^7*x^4*e^8
+ 6*(x*e + d)^m*a^2*c^2*m^7*x^4*e^8 + 155*(x*e + d)^m*b^3*c*m^6*x^5*e^8 + 465*(x*e + d)^m*a*b*c^2*m^6*x^5*e^8
+ 3294*(x*e + d)^m*b^2*c^2*m^5*x^6*e^8 + 2196*(x*e + d)^m*a*c^3*m^5*x^6*e^8 + 14945*(x*e + d)^m*b*c^3*m^4*x^7*
e^8 + 13538*(x*e + d)^m*c^4*m^3*x^8*e^8 + (x*e + d)^m*b^4*d*m^7*x^3*e^7 + 12*(x*e + d)^m*a*b^2*c*d*m^7*x^3*e^7
+ 6*(x*e + d)^m*a^2*c^2*d*m^7*x^3*e^7 + 135*(x*e + d)^m*b^3*c*d*m^6*x^4*e^7 + 405*(x*e + d)^m*a*b*c^2*d*m^6*x
^4*e^7 + 2169*(x*e + d)^m*b^2*c^2*d*m^5*x^5*e^7 + 1446*(x*e + d)^m*a*c^3*d*m^5*x^5*e^7 + 6335*(x*e + d)^m*b*c^
3*d*m^4*x^6*e^7 + 3248*(x*e + d)^m*c^4*d*m^3*x^7*e^7 - 20*(x*e + d)^m*b^3*c*d^2*m^6*x^3*e^6 - 60*(x*e + d)^m*a
*b*c^2*d^2*m^6*x^3*e^6 - 945*(x*e + d)^m*b^2*c^2*d^2*m^5*x^4*e^6 - 630*(x*e + d)^m*a*c^3*d^2*m^5*x^4*e^6 - 483
0*(x*e + d)^m*b*c^3*d^2*m^4*x^5*e^6 - 3150*(x*e + d)^m*c^4*d^2*m^3*x^6*e^6 + 180*(x*e + d)^m*b^2*c^2*d^3*m^5*x
^3*e^5 + 120*(x*e + d)^m*a*c^3*d^3*m^5*x^3*e^5 + 2940*(x*e + d)^m*b*c^3*d^3*m^4*x^4*e^5 + 2940*(x*e + d)^m*c^4
*d^3*m^3*x^5*e^5 - 840*(x*e + d)^m*b*c^3*d^4*m^4*x^3*e^4 - 2520*(x*e + d)^m*c^4*d^4*m^3*x^4*e^4 + 1680*(x*e +
d)^m*c^4*d^5*m^3*x^3*e^3 + 3*(x*e + d)^m*a*b^3*m^7*x^3*e^8 + 9*(x*e + d)^m*a^2*b*c*m^7*x^3*e^8 + 32*(x*e + d)^
m*b^4*m^6*x^4*e^8 + 384*(x*e + d)^m*a*b^2*c*m^6*x^4*e^8 + 192*(x*e + d)^m*a^2*c^2*m^6*x^4*e^8 + 1955*(x*e + d)
^m*b^3*c*m^5*x^5*e^8 + 5865*(x*e + d)^m*a*b*c^2*m^5*x^5*e^8 + 21060*(x*e + d)^m*b^2*c^2*m^4*x^6*e^8 + 14040*(x
*e + d)^m*a*c^3*m^4*x^6*e^8 + 52528*(x*e + d)^m*b*c^3*m^3*x^7*e^8 + 26264*(x*e + d)^m*c^4*m^2*x^8*e^8 + 3*(x*e
+ d)^m*a*b^3*d*m^7*x^2*e^7 + 9*(x*e + d)^m*a^2*b*c*d*m^7*x^2*e^7 + 29*(x*e + d)^m*b^4*d*m^6*x^3*e^7 + 348*(x*
e + d)^m*a*b^2*c*d*m^6*x^3*e^7 + 174*(x*e + d)^m*a^2*c^2*d*m^6*x^3*e^7 + 1415*(x*e + d)^m*b^3*c*d*m^5*x^4*e^7
+ 4245*(x*e + d)^m*a*b*c^2*d*m^5*x^4*e^7 + 10215*(x*e + d)^m*b^2*c^2*d*m^4*x^5*e^7 + 6810*(x*e + d)^m*a*c^3*d*
m^4*x^5*e^7 + 14518*(x*e + d)^m*b*c^3*d*m^3*x^6*e^7 + 3528*(x*e + d)^m*c^4*d*m^2*x^7*e^7 - 3*(x*e + d)^m*b^4*d
^2*m^6*x^2*e^6 - 36*(x*e + d)^m*a*b^2*c*d^2*m^6*x^2*e^6 - 18*(x*e + d)^m*a^2*c^2*d^2*m^6*x^2*e^6 - 480*(x*e +
d)^m*b^3*c*d^2*m^5*x^3*e^6 - 1440*(x*e + d)^m*a*b*c^2*d^2*m^5*x^3*e^6 - 7065*(x*e + d)^m*b^2*c^2*d^2*m^4*x^4*e
^6 - 4710*(x*e + d)^m*a*c^3*d^2*m^4*x^4*e^6 - 13860*(x*e + d)^m*b*c^3*d^2*m^3*x^5*e^6 - 3836*(x*e + d)^m*c^4*d
^2*m^2*x^6*e^6 + 60*(x*e + d)^m*b^3*c*d^3*m^5*x^2*e^5 + 180*(x*e + d)^m*a*b*c^2*d^3*m^5*x^2*e^5 + 3240*(x*e +
d)^m*b^2*c^2*d^3*m^4*x^3*e^5 + 2160*(x*e + d)^m*a*c^3*d^3*m^4*x^3*e^5 + 12390*(x*e + d)^m*b*c^3*d^3*m^3*x^4*e^
5 + 4200*(x*e + d)^m*c^4*d^3*m^2*x^5*e^5 - 540*(x*e + d)^m*b^2*c^2*d^4*m^4*x^2*e^4 - 360*(x*e + d)^m*a*c^3*d^4
*m^4*x^2*e^4 - 9240*(x*e + d)^m*b*c^3*d^4*m^3*x^3*e^4 - 4620*(x*e + d)^m*c^4*d^4*m^2*x^4*e^4 + 2520*(x*e + d)^
m*b*c^3*d^5*m^3*x^2*e^3 + 5040*(x*e + d)^m*c^4*d^5*m^2*x^3*e^3 - 5040*(x*e + d)^m*c^4*d^6*m^2*x^2*e^2 + 3*(x*e
+ d)^m*a^2*b^2*m^7*x^2*e^8 + 2*(x*e + d)^m*a^3*c*m^7*x^2*e^8 + 99*(x*e + d)^m*a*b^3*m^6*x^3*e^8 + 297*(x*e +
d)^m*a^2*b*c*m^6*x^3*e^8 + 418*(x*e + d)^m*b^4*m^5*x^4*e^8 + 5016*(x*e + d)^m*a*b^2*c*m^5*x^4*e^8 + 2508*(x*e
+ d)^m*a^2*c^2*m^5*x^4*e^8 + 12905*(x*e + d)^m*b^3*c*m^4*x^5*e^8 + 38715*(x*e + d)^m*a*b*c^2*m^4*x^5*e^8 + 756
81*(x*e + d)^m*b^2*c^2*m^3*x^6*e^8 + 50454*(x*e + d)^m*a*c^3*m^3*x^6*e^8 + 103292*(x*e + d)^m*b*c^3*m^2*x^7*e^
8 + 26136*(x*e + d)^m*c^4*m*x^8*e^8 + 3*(x*e + d)^m*a^2*b^2*d*m^7*x*e^7 + 2*(x*e + d)^m*a^3*c*d*m^7*x*e^7 + 93
*(x*e + d)^m*a*b^3*d*m^6*x^2*e^7 + 279*(x*e + d)^m*a^2*b*c*d*m^6*x^2*e^7 + 331*(x*e + d)^m*b^4*d*m^5*x^3*e^7 +
3972*(x*e + d)^m*a*b^2*c*d*m^5*x^3*e^7 + 1986*(x*e + d)^m*a^2*c^2*d*m^5*x^3*e^7 + 7245*(x*e + d)^m*b^3*c*d*m^
4*x^4*e^7 + 21735*(x*e + d)^m*a*b*c^2*d*m^4*x^4*e^7 + 24606*(x*e + d)^m*b^2*c^2*d*m^3*x^5*e^7 + 16404*(x*e + d
)^m*a*c^3*d*m^3*x^5*e^7 + 16184*(x*e + d)^m*b*c^3*d*m^2*x^6*e^7 + 1440*(x*e + d)^m*c^4*d*m*x^7*e^7 - 6*(x*e +
d)^m*a*b^3*d^2*m^6*x*e^6 - 18*(x*e + d)^m*a^2*b*c*d^2*m^6*x*e^6 - 81*(x*e + d)^m*b^4*d^2*m^5*x^2*e^6 - 972*(x*
e + d)^m*a*b^2*c*d^2*m^5*x^2*e^6 - 486*(x*e + d)^m*a^2*c^2*d^2*m^5*x^2*e^6 - 4220*(x*e + d)^m*b^3*c*d^2*m^4*x^
3*e^6 - 12660*(x*e + d)^m*a*b*c^2*d^2*m^4*x^3*e^6 - 22815*(x*e + d)^m*b^2*c^2*d^2*m^3*x^4*e^6 - 15210*(x*e + d
)^m*a*c^3*d^2*m^3*x^4*e^6 - 17808*(x*e + d)^m*b*c^3*d^2*m^2*x^5*e^6 - 1680*(x*e + d)^m*c^4*d^2*m*x^6*e^6 + 6*(
x*e + d)^m*b^4*d^3*m^5*x*e^5 + 72*(x*e + d)^m*a*b^2*c*d^3*m^5*x*e^5 + 36*(x*e + d)^m*a^2*c^2*d^3*m^5*x*e^5 + 1
320*(x*e + d)^m*b^3*c*d^3*m^4*x^2*e^5 + 3960*(x*e + d)^m*a*b*c^2*d^3*m^4*x^2*e^5 + 18540*(x*e + d)^m*b^2*c^2*d
^3*m^3*x^3*e^5 + 12360*(x*e + d)^m*a*c^3*d^3*m^3*x^3*e^5 + 19740*(x*e + d)^m*b*c^3*d^3*m^2*x^4*e^5 + 2016*(x*e
+ d)^m*c^4*d^3*m*x^5*e^5 - 120*(x*e + d)^m*b^3*c*d^4*m^4*x*e^4 - 360*(x*e + d)^m*a*b*c^2*d^4*m^4*x*e^4 - 8640
*(x*e + d)^m*b^2*c^2*d^4*m^3*x^2*e^4 - 5760*(x*e + d)^m*a*c^3*d^4*m^3*x^2*e^4 - 21840*(x*e + d)^m*b*c^3*d^4*m^
2*x^3*e^4 - 2520*(x*e + d)^m*c^4*d^4*m*x^4*e^4 + 1080*(x*e + d)^m*b^2*c^2*d^5*m^3*x*e^3 + 720*(x*e + d)^m*a*c^
3*d^5*m^3*x*e^3 + 22680*(x*e + d)^m*b*c^3*d^5*m^2*x^2*e^3 + 3360*(x*e + d)^m*c^4*d^5*m*x^3*e^3 - 5040*(x*e + d
)^m*b*c^3*d^6*m^2*x*e^2 - 5040*(x*e + d)^m*c^4*d^6*m*x^2*e^2 + 10080*(x*e + d)^m*c^4*d^7*m*x*e + (x*e + d)^m*a
^3*b*m^7*x*e^8 + 102*(x*e + d)^m*a^2*b^2*m^6*x^2*e^8 + 68*(x*e + d)^m*a^3*c*m^6*x^2*e^8 + 1341*(x*e + d)^m*a*b
^3*m^5*x^3*e^8 + 4023*(x*e + d)^m*a^2*b*c*m^5*x^3*e^8 + 2864*(x*e + d)^m*b^4*m^4*x^4*e^8 + 34368*(x*e + d)^m*a
*b^2*c*m^4*x^4*e^8 + 17184*(x*e + d)^m*a^2*c^2*m^4*x^4*e^8 + 47720*(x*e + d)^m*b^3*c*m^3*x^5*e^8 + 143160*(x*e
+ d)^m*a*b*c^2*m^3*x^5*e^8 + 151470*(x*e + d)^m*b^2*c^2*m^2*x^6*e^8 + 100980*(x*e + d)^m*a*c^3*m^2*x^6*e^8 +
103824*(x*e + d)^m*b*c^3*m*x^7*e^8 + 10080*(x*e + d)^m*c^4*x^8*e^8 + (x*e + d)^m*a^3*b*d*m^7*e^7 + 99*(x*e + d
)^m*a^2*b^2*d*m^6*x*e^7 + 66*(x*e + d)^m*a^3*c*d*m^6*x*e^7 + 1155*(x*e + d)^m*a*b^3*d*m^5*x^2*e^7 + 3465*(x*e
+ d)^m*a^2*b*c*d*m^5*x^2*e^7 + 1871*(x*e + d)^m*b^4*d*m^4*x^3*e^7 + 22452*(x*e + d)^m*a*b^2*c*d*m^4*x^3*e^7 +
11226*(x*e + d)^m*a^2*c^2*d*m^4*x^3*e^7 + 18740*(x*e + d)^m*b^3*c*d*m^3*x^4*e^7 + 56220*(x*e + d)^m*a*b*c^2*d*
m^3*x^4*e^7 + 28440*(x*e + d)^m*b^2*c^2*d*m^2*x^5*e^7 + 18960*(x*e + d)^m*a*c^3*d*m^2*x^5*e^7 + 6720*(x*e + d)
^m*b*c^3*d*m*x^6*e^7 - 3*(x*e + d)^m*a^2*b^2*d^2*m^6*e^6 - 2*(x*e + d)^m*a^3*c*d^2*m^6*e^6 - 180*(x*e + d)^m*a
*b^3*d^2*m^5*x*e^6 - 540*(x*e + d)^m*a^2*b*c*d^2*m^5*x*e^6 - 831*(x*e + d)^m*b^4*d^2*m^4*x^2*e^6 - 9972*(x*e +
d)^m*a*b^2*c*d^2*m^4*x^2*e^6 - 4986*(x*e + d)^m*a^2*c^2*d^2*m^4*x^2*e^6 - 16320*(x*e + d)^m*b^3*c*d^2*m^3*x^3
*e^6 - 48960*(x*e + d)^m*a*b*c^2*d^2*m^3*x^3*e^6 - 31770*(x*e + d)^m*b^2*c^2*d^2*m^2*x^4*e^6 - 21180*(x*e + d)
^m*a*c^3*d^2*m^2*x^4*e^6 - 8064*(x*e + d)^m*b*c^3*d^2*m*x^5*e^6 + 6*(x*e + d)^m*a*b^3*d^3*m^5*e^5 + 18*(x*e +
d)^m*a^2*b*c*d^3*m^5*e^5 + 156*(x*e + d)^m*b^4*d^3*m^4*x*e^5 + 1872*(x*e + d)^m*a*b^2*c*d^3*m^4*x*e^5 + 936*(x
*e + d)^m*a^2*c^2*d^3*m^4*x*e^5 + 10020*(x*e + d)^m*b^3*c*d^3*m^3*x^2*e^5 + 30060*(x*e + d)^m*a*b*c^2*d^3*m^3*
x^2*e^5 + 35640*(x*e + d)^m*b^2*c^2*d^3*m^2*x^3*e^5 + 23760*(x*e + d)^m*a*c^3*d^3*m^2*x^3*e^5 + 10080*(x*e + d
)^m*b*c^3*d^3*m*x^4*e^5 - 6*(x*e + d)^m*b^4*d^4*m^4*e^4 - 72*(x*e + d)^m*a*b^2*c*d^4*m^4*e^4 - 36*(x*e + d)^m*
a^2*c^2*d^4*m^4*e^4 - 2520*(x*e + d)^m*b^3*c*d^4*m^3*x*e^4 - 7560*(x*e + d)^m*a*b*c^2*d^4*m^3*x*e^4 - 38340*(x
*e + d)^m*b^2*c^2*d^4*m^2*x^2*e^4 - 25560*(x*e + d)^m*a*c^3*d^4*m^2*x^2*e^4 - 13440*(x*e + d)^m*b*c^3*d^4*m*x^
3*e^4 + 120*(x*e + d)^m*b^3*c*d^5*m^3*e^3 + 360*(x*e + d)^m*a*b*c^2*d^5*m^3*e^3 + 16200*(x*e + d)^m*b^2*c^2*d^
5*m^2*x*e^3 + 10800*(x*e + d)^m*a*c^3*d^5*m^2*x*e^3 + 20160*(x*e + d)^m*b*c^3*d^5*m*x^2*e^3 - 1080*(x*e + d)^m
*b^2*c^2*d^6*m^2*e^2 - 720*(x*e + d)^m*a*c^3*d^6*m^2*e^2 - 40320*(x*e + d)^m*b*c^3*d^6*m*x*e^2 + 5040*(x*e + d
)^m*b*c^3*d^7*m*e - 10080*(x*e + d)^m*c^4*d^8 + 35*(x*e + d)^m*a^3*b*m^6*x*e^8 + 1434*(x*e + d)^m*a^2*b^2*m^5*
x^2*e^8 + 956*(x*e + d)^m*a^3*c*m^5*x^2*e^8 + 9585*(x*e + d)^m*a*b^3*m^4*x^3*e^8 + 28755*(x*e + d)^m*a^2*b*c*m
^4*x^3*e^8 + 10993*(x*e + d)^m*b^4*m^3*x^4*e^8 + 131916*(x*e + d)^m*a*b^2*c*m^3*x^4*e^8 + 65958*(x*e + d)^m*a^
2*c^2*m^3*x^4*e^8 + 97820*(x*e + d)^m*b^3*c*m^2*x^5*e^8 + 293460*(x*e + d)^m*a*b*c^2*m^2*x^5*e^8 + 154296*(x*e
+ d)^m*b^2*c^2*m*x^6*e^8 + 102864*(x*e + d)^m*a*c^3*m*x^6*e^8 + 40320*(x*e + d)^m*b*c^3*x^7*e^8 + 35*(x*e + d
)^m*a^3*b*d*m^6*e^7 + 1335*(x*e + d)^m*a^2*b^2*d*m^5*x*e^7 + 890*(x*e + d)^m*a^3*c*d*m^5*x*e^7 + 7275*(x*e + d
)^m*a*b^3*d*m^4*x^2*e^7 + 21825*(x*e + d)^m*a^2*b*c*d*m^4*x^2*e^7 + 5380*(x*e + d)^m*b^4*d*m^3*x^3*e^7 + 64560
*(x*e + d)^m*a*b^2*c*d*m^3*x^3*e^7 + 32280*(x*e + d)^m*a^2*c^2*d*m^3*x^3*e^7 + 22860*(x*e + d)^m*b^3*c*d*m^2*x
^4*e^7 + 68580*(x*e + d)^m*a*b*c^2*d*m^2*x^4*e^7 + 12096*(x*e + d)^m*b^2*c^2*d*m*x^5*e^7 + 8064*(x*e + d)^m*a*
c^3*d*m*x^5*e^7 - 99*(x*e + d)^m*a^2*b^2*d^2*m^5*e^6 - 66*(x*e + d)^m*a^3*c*d^2*m^5*e^6 - 2130*(x*e + d)^m*a*b
^3*d^2*m^4*x*e^6 - 6390*(x*e + d)^m*a^2*b*c*d^2*m^4*x*e^6 - 3951*(x*e + d)^m*b^4*d^2*m^3*x^2*e^6 - 47412*(x*e
+ d)^m*a*b^2*c*d^2*m^3*x^2*e^6 - 23706*(x*e + d)^m*a^2*c^2*d^2*m^3*x^2*e^6 - 26000*(x*e + d)^m*b^3*c*d^2*m^2*x
^3*e^6 - 78000*(x*e + d)^m*a*b*c^2*d^2*m^2*x^3*e^6 - 15120*(x*e + d)^m*b^2*c^2*d^2*m*x^4*e^6 - 10080*(x*e + d)
^m*a*c^3*d^2*m*x^4*e^6 + 180*(x*e + d)^m*a*b^3*d^3*m^4*e^5 + 540*(x*e + d)^m*a^2*b*c*d^3*m^4*e^5 + 1506*(x*e +
d)^m*b^4*d^3*m^3*x*e^5 + 18072*(x*e + d)^m*a*b^2*c*d^3*m^3*x*e^5 + 9036*(x*e + d)^m*a^2*c^2*d^3*m^3*x*e^5 + 2
8920*(x*e + d)^m*b^3*c*d^3*m^2*x^2*e^5 + 86760*(x*e + d)^m*a*b*c^2*d^3*m^2*x^2*e^5 + 20160*(x*e + d)^m*b^2*c^2
*d^3*m*x^3*e^5 + 13440*(x*e + d)^m*a*c^3*d^3*m*x^3*e^5 - 156*(x*e + d)^m*b^4*d^4*m^3*e^4 - 1872*(x*e + d)^m*a*
b^2*c*d^4*m^3*e^4 - 936*(x*e + d)^m*a^2*c^2*d^4*m^3*e^4 - 17520*(x*e + d)^m*b^3*c*d^4*m^2*x*e^4 - 52560*(x*e +
d)^m*a*b*c^2*d^4*m^2*x*e^4 - 30240*(x*e + d)^m*b^2*c^2*d^4*m*x^2*e^4 - 20160*(x*e + d)^m*a*c^3*d^4*m*x^2*e^4
+ 2520*(x*e + d)^m*b^3*c*d^5*m^2*e^3 + 7560*(x*e + d)^m*a*b*c^2*d^5*m^2*e^3 + 60480*(x*e + d)^m*b^2*c^2*d^5*m*
x*e^3 + 40320*(x*e + d)^m*a*c^3*d^5*m*x*e^3 - 16200*(x*e + d)^m*b^2*c^2*d^6*m*e^2 - 10800*(x*e + d)^m*a*c^3*d^
6*m*e^2 + 40320*(x*e + d)^m*b*c^3*d^7*e + 511*(x*e + d)^m*a^3*b*m^5*x*e^8 + 10740*(x*e + d)^m*a^2*b^2*m^4*x^2*
e^8 + 7160*(x*e + d)^m*a^3*c*m^4*x^2*e^8 + 38592*(x*e + d)^m*a*b^3*m^3*x^3*e^8 + 115776*(x*e + d)^m*a^2*b*c*m^
3*x^3*e^8 + 23312*(x*e + d)^m*b^4*m^2*x^4*e^8 + 279744*(x*e + d)^m*a*b^2*c*m^2*x^4*e^8 + 139872*(x*e + d)^m*a^
2*c^2*m^2*x^4*e^8 + 101520*(x*e + d)^m*b^3*c*m*x^5*e^8 + 304560*(x*e + d)^m*a*b*c^2*m*x^5*e^8 + 60480*(x*e + d
)^m*b^2*c^2*x^6*e^8 + 40320*(x*e + d)^m*a*c^3*x^6*e^8 + 511*(x*e + d)^m*a^3*b*d*m^5*e^7 + 9405*(x*e + d)^m*a^2
*b^2*d*m^4*x*e^7 + 6270*(x*e + d)^m*a^3*c*d*m^4*x*e^7 + 24042*(x*e + d)^m*a*b^3*d*m^3*x^2*e^7 + 72126*(x*e + d
)^m*a^2*b*c*d*m^3*x^2*e^7 + 7172*(x*e + d)^m*b^4*d*m^2*x^3*e^7 + 86064*(x*e + d)^m*a*b^2*c*d*m^2*x^3*e^7 + 430
32*(x*e + d)^m*a^2*c^2*d*m^2*x^3*e^7 + 10080*(x*e + d)^m*b^3*c*d*m*x^4*e^7 + 30240*(x*e + d)^m*a*b*c^2*d*m*x^4
*e^7 - 1335*(x*e + d)^m*a^2*b^2*d^2*m^4*e^6 - 890*(x*e + d)^m*a^3*c*d^2*m^4*e^6 - 12420*(x*e + d)^m*a*b^3*d^2*
m^3*x*e^6 - 37260*(x*e + d)^m*a^2*b*c*d^2*m^3*x*e^6 - 8238*(x*e + d)^m*b^4*d^2*m^2*x^2*e^6 - 98856*(x*e + d)^m
*a*b^2*c*d^2*m^2*x^2*e^6 - 49428*(x*e + d)^m*a^2*c^2*d^2*m^2*x^2*e^6 - 13440*(x*e + d)^m*b^3*c*d^2*m*x^3*e^6 -
40320*(x*e + d)^m*a*b*c^2*d^2*m*x^3*e^6 + 2130*(x*e + d)^m*a*b^3*d^3*m^3*e^5 + 6390*(x*e + d)^m*a^2*b*c*d^3*m
^3*e^5 + 6396*(x*e + d)^m*b^4*d^3*m^2*x*e^5 + 76752*(x*e + d)^m*a*b^2*c*d^3*m^2*x*e^5 + 38376*(x*e + d)^m*a^2*
c^2*d^3*m^2*x*e^5 + 20160*(x*e + d)^m*b^3*c*d^3*m*x^2*e^5 + 60480*(x*e + d)^m*a*b*c^2*d^3*m*x^2*e^5 - 1506*(x*
e + d)^m*b^4*d^4*m^2*e^4 - 18072*(x*e + d)^m*a*b^2*c*d^4*m^2*e^4 - 9036*(x*e + d)^m*a^2*c^2*d^4*m^2*e^4 - 4032
0*(x*e + d)^m*b^3*c*d^4*m*x*e^4 - 120960*(x*e + d)^m*a*b*c^2*d^4*m*x*e^4 + 17520*(x*e + d)^m*b^3*c*d^5*m*e^3 +
52560*(x*e + d)^m*a*b*c^2*d^5*m*e^3 - 60480*(x*e + d)^m*b^2*c^2*d^6*e^2 - 40320*(x*e + d)^m*a*c^3*d^6*e^2 + 4
025*(x*e + d)^m*a^3*b*m^4*x*e^8 + 45867*(x*e + d)^m*a^2*b^2*m^3*x^2*e^8 + 30578*(x*e + d)^m*a^3*c*m^3*x^2*e^8
+ 86076*(x*e + d)^m*a*b^3*m^2*x^3*e^8 + 258228*(x*e + d)^m*a^2*b*c*m^2*x^3*e^8 + 24876*(x*e + d)^m*b^4*m*x^4*e
^8 + 298512*(x*e + d)^m*a*b^2*c*m*x^4*e^8 + 149256*(x*e + d)^m*a^2*c^2*m*x^4*e^8 + 40320*(x*e + d)^m*b^3*c*x^5
*e^8 + 120960*(x*e + d)^m*a*b*c^2*x^5*e^8 + 4025*(x*e + d)^m*a^3*b*d*m^4*e^7 + 36462*(x*e + d)^m*a^2*b^2*d*m^3
*x*e^7 + 24308*(x*e + d)^m*a^3*c*d*m^3*x*e^7 + 37992*(x*e + d)^m*a*b^3*d*m^2*x^2*e^7 + 113976*(x*e + d)^m*a^2*
b*c*d*m^2*x^2*e^7 + 3360*(x*e + d)^m*b^4*d*m*x^3*e^7 + 40320*(x*e + d)^m*a*b^2*c*d*m*x^3*e^7 + 20160*(x*e + d)
^m*a^2*c^2*d*m*x^3*e^7 - 9405*(x*e + d)^m*a^2*b^2*d^2*m^3*e^6 - 6270*(x*e + d)^m*a^3*c*d^2*m^3*e^6 - 35664*(x*
e + d)^m*a*b^3*d^2*m^2*x*e^6 - 106992*(x*e + d)^m*a^2*b*c*d^2*m^2*x*e^6 - 5040*(x*e + d)^m*b^4*d^2*m*x^2*e^6 -
60480*(x*e + d)^m*a*b^2*c*d^2*m*x^2*e^6 - 30240*(x*e + d)^m*a^2*c^2*d^2*m*x^2*e^6 + 12420*(x*e + d)^m*a*b^3*d
^3*m^2*e^5 + 37260*(x*e + d)^m*a^2*b*c*d^3*m^2*e^5 + 10080*(x*e + d)^m*b^4*d^3*m*x*e^5 + 120960*(x*e + d)^m*a*
b^2*c*d^3*m*x*e^5 + 60480*(x*e + d)^m*a^2*c^2*d^3*m*x*e^5 - 6396*(x*e + d)^m*b^4*d^4*m*e^4 - 76752*(x*e + d)^m
*a*b^2*c*d^4*m*e^4 - 38376*(x*e + d)^m*a^2*c^2*d^4*m*e^4 + 40320*(x*e + d)^m*b^3*c*d^5*e^3 + 120960*(x*e + d)^
m*a*b*c^2*d^5*e^3 + 18424*(x*e + d)^m*a^3*b*m^3*x*e^8 + 110118*(x*e + d)^m*a^2*b^2*m^2*x^2*e^8 + 73412*(x*e +
d)^m*a^3*c*m^2*x^2*e^8 + 96144*(x*e + d)^m*a*b^3*m*x^3*e^8 + 288432*(x*e + d)^m*a^2*b*c*m*x^3*e^8 + 10080*(x*e
+ d)^m*b^4*x^4*e^8 + 120960*(x*e + d)^m*a*b^2*c*x^4*e^8 + 60480*(x*e + d)^m*a^2*c^2*x^4*e^8 + 18424*(x*e + d)
^m*a^3*b*d*m^3*e^7 + 73656*(x*e + d)^m*a^2*b^2*d*m^2*x*e^7 + 49104*(x*e + d)^m*a^3*c*d*m^2*x*e^7 + 20160*(x*e
+ d)^m*a*b^3*d*m*x^2*e^7 + 60480*(x*e + d)^m*a^2*b*c*d*m*x^2*e^7 - 36462*(x*e + d)^m*a^2*b^2*d^2*m^2*e^6 - 243
08*(x*e + d)^m*a^3*c*d^2*m^2*e^6 - 40320*(x*e + d)^m*a*b^3*d^2*m*x*e^6 - 120960*(x*e + d)^m*a^2*b*c*d^2*m*x*e^
6 + 35664*(x*e + d)^m*a*b^3*d^3*m*e^5 + 106992*(x*e + d)^m*a^2*b*c*d^3*m*e^5 - 10080*(x*e + d)^m*b^4*d^4*e^4 -
120960*(x*e + d)^m*a*b^2*c*d^4*e^4 - 60480*(x*e + d)^m*a^2*c^2*d^4*e^4 + 48860*(x*e + d)^m*a^3*b*m^2*x*e^8 +
134136*(x*e + d)^m*a^2*b^2*m*x^2*e^8 + 89424*(x*e + d)^m*a^3*c*m*x^2*e^8 + 40320*(x*e + d)^m*a*b^3*x^3*e^8 + 1
20960*(x*e + d)^m*a^2*b*c*x^3*e^8 + 48860*(x*e + d)^m*a^3*b*d*m^2*e^7 + 60480*(x*e + d)^m*a^2*b^2*d*m*x*e^7 +
40320*(x*e + d)^m*a^3*c*d*m*x*e^7 - 73656*(x*e + d)^m*a^2*b^2*d^2*m*e^6 - 49104*(x*e + d)^m*a^3*c*d^2*m*e^6 +
40320*(x*e + d)^m*a*b^3*d^3*e^5 + 120960*(x*e + d)^m*a^2*b*c*d^3*e^5 + 69264*(x*e + d)^m*a^3*b*m*x*e^8 + 60480
*(x*e + d)^m*a^2*b^2*x^2*e^8 + 40320*(x*e + d)^m*a^3*c*x^2*e^8 + 69264*(x*e + d)^m*a^3*b*d*m*e^7 - 60480*(x*e
+ d)^m*a^2*b^2*d^2*e^6 - 40320*(x*e + d)^m*a^3*c*d^2*e^6 + 40320*(x*e + d)^m*a^3*b*x*e^8 + 40320*(x*e + d)^m*a
^3*b*d*e^7)/(m^8*e^8 + 36*m^7*e^8 + 546*m^6*e^8 + 4536*m^5*e^8 + 22449*m^4*e^8 + 67284*m^3*e^8 + 118124*m^2*e^
8 + 109584*m*e^8 + 40320*e^8)