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

Optimal. Leaf size=267 \[ \frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}-\frac{3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{d^3 (c d-b e)^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{m+2}}{e^7 (m+2)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)} \]

[Out]

(d^3*(c*d - b*e)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (3*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(2 + m))/(e
^7*(2 + m)) + (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(3 + m))/(e^7*(3 + m)) - ((2*c*d -
b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^(4 + m))/(e^7*(4 + m)) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*
e^2)*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (3*c^2*(2*c*d - b*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)
^(7 + m))/(e^7*(7 + m))

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Rubi [A]  time = 0.165443, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ \frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}-\frac{3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{d^3 (c d-b e)^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{m+2}}{e^7 (m+2)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(b*x + c*x^2)^3,x]

[Out]

(d^3*(c*d - b*e)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (3*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(2 + m))/(e
^7*(2 + m)) + (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(3 + m))/(e^7*(3 + m)) - ((2*c*d -
b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^(4 + m))/(e^7*(4 + m)) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*
e^2)*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (3*c^2*(2*c*d - b*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)
^(7 + m))/(e^7*(7 + m))

Rule 698

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

Rubi steps

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

Mathematica [A]  time = 0.206048, size = 236, normalized size = 0.88 \[ \frac{(d+e x)^{m+1} \left (\frac{3 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{m+5}-\frac{(d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{m+4}+\frac{3 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{m+3}-\frac{3 c^2 (d+e x)^5 (2 c d-b e)}{m+6}-\frac{3 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)}{m+2}+\frac{d^3 (c d-b e)^3}{m+1}+\frac{c^3 (d+e x)^6}{m+7}\right )}{e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*((d^3*(c*d - b*e)^3)/(1 + m) - (3*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x))/(2 + m) + (3*d
*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2)/(3 + m) - ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e
+ b^2*e^2)*(d + e*x)^3)/(4 + m) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4)/(5 + m) - (3*c^2*(2*c*d
- b*e)*(d + e*x)^5)/(6 + m) + (c^3*(d + e*x)^6)/(7 + m)))/e^7

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Maple [B]  time = 0.055, size = 1528, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(e*x+d)^(1+m)*(-c^3*e^6*m^6*x^6-3*b*c^2*e^6*m^6*x^5-21*c^3*e^6*m^5*x^6-3*b^2*c*e^6*m^6*x^4-66*b*c^2*e^6*m^5*x
^5+6*c^3*d*e^5*m^5*x^5-175*c^3*e^6*m^4*x^6-b^3*e^6*m^6*x^3-69*b^2*c*e^6*m^5*x^4+15*b*c^2*d*e^5*m^5*x^4-570*b*c
^2*e^6*m^4*x^5+90*c^3*d*e^5*m^4*x^5-735*c^3*e^6*m^3*x^6-24*b^3*e^6*m^5*x^3+12*b^2*c*d*e^5*m^5*x^3-621*b^2*c*e^
6*m^4*x^4+255*b*c^2*d*e^5*m^4*x^4-2460*b*c^2*e^6*m^3*x^5-30*c^3*d^2*e^4*m^4*x^4+510*c^3*d*e^5*m^3*x^5-1624*c^3
*e^6*m^2*x^6+3*b^3*d*e^5*m^5*x^2-226*b^3*e^6*m^4*x^3+228*b^2*c*d*e^5*m^4*x^3-2775*b^2*c*e^6*m^3*x^4-60*b*c^2*d
^2*e^4*m^4*x^3+1575*b*c^2*d*e^5*m^3*x^4-5547*b*c^2*e^6*m^2*x^5-300*c^3*d^2*e^4*m^3*x^4+1350*c^3*d*e^5*m^2*x^5-
1764*c^3*e^6*m*x^6+63*b^3*d*e^5*m^4*x^2-1056*b^3*e^6*m^3*x^3-36*b^2*c*d^2*e^4*m^4*x^2+1572*b^2*c*d*e^5*m^3*x^3
-6432*b^2*c*e^6*m^2*x^4-780*b*c^2*d^2*e^4*m^3*x^3+4425*b*c^2*d*e^5*m^2*x^4-6114*b*c^2*e^6*m*x^5+120*c^3*d^3*e^
3*m^3*x^3-1050*c^3*d^2*e^4*m^2*x^4+1644*c^3*d*e^5*m*x^5-720*c^3*e^6*x^6-6*b^3*d^2*e^4*m^4*x+489*b^3*d*e^5*m^3*
x^2-2545*b^3*e^6*m^2*x^3-576*b^2*c*d^2*e^4*m^3*x^2+4812*b^2*c*d*e^5*m^2*x^3-7236*b^2*c*e^6*m*x^4+180*b*c^2*d^3
*e^3*m^3*x^2-3180*b*c^2*d^2*e^4*m^2*x^3+5610*b*c^2*d*e^5*m*x^4-2520*b*c^2*e^6*x^5+720*c^3*d^3*e^3*m^2*x^3-1500
*c^3*d^2*e^4*m*x^4+720*c^3*d*e^5*x^5-114*b^3*d^2*e^4*m^3*x+1701*b^3*d*e^5*m^2*x^2-2952*b^3*e^6*m*x^3+72*b^2*c*
d^3*e^3*m^3*x-2988*b^2*c*d^2*e^4*m^2*x^2+6480*b^2*c*d*e^5*m*x^3-3024*b^2*c*e^6*x^4+1800*b*c^2*d^3*e^3*m^2*x^2-
4980*b*c^2*d^2*e^4*m*x^3+2520*b*c^2*d*e^5*x^4-360*c^3*d^4*e^2*m^2*x^2+1320*c^3*d^3*e^3*m*x^3-720*c^3*d^2*e^4*x
^4+6*b^3*d^3*e^3*m^3-750*b^3*d^2*e^4*m^2*x+2532*b^3*d*e^5*m*x^2-1260*b^3*e^6*x^3+1008*b^2*c*d^3*e^3*m^2*x-5472
*b^2*c*d^2*e^4*m*x^2+3024*b^2*c*d*e^5*x^3-360*b*c^2*d^4*e^2*m^2*x+4140*b*c^2*d^3*e^3*m*x^2-2520*b*c^2*d^2*e^4*
x^3-1080*c^3*d^4*e^2*m*x^2+720*c^3*d^3*e^3*x^3+108*b^3*d^3*e^3*m^2-1902*b^3*d^2*e^4*m*x+1260*b^3*d*e^5*x^2-72*
b^2*c*d^4*e^2*m^2+3960*b^2*c*d^3*e^3*m*x-3024*b^2*c*d^2*e^4*x^2-2880*b*c^2*d^4*e^2*m*x+2520*b*c^2*d^3*e^3*x^2+
720*c^3*d^5*e*m*x-720*c^3*d^4*e^2*x^2+642*b^3*d^3*e^3*m-1260*b^3*d^2*e^4*x-936*b^2*c*d^4*e^2*m+3024*b^2*c*d^3*
e^3*x+360*b*c^2*d^5*e*m-2520*b*c^2*d^4*e^2*x+720*c^3*d^5*e*x+1260*b^3*d^3*e^3-3024*b^2*c*d^4*e^2+2520*b*c^2*d^
5*e-720*c^3*d^6)/e^7/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)

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Maxima [B]  time = 1.32643, size = 903, normalized size = 3.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*
d^4)*(e*x + d)^m*b^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^
5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24
*d^4*e*m*x + 24*d^5)*(e*x + d)^m*b^2*c/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + 3*((m^5 + 15*m^
4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x^5 - 5*(m^4 + 6*m
^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5*e*m*x
- 120*d^6)*(e*x + d)^m*b*c^2/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6) + ((m^6 + 21*
m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*
m)*d*e^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*e^4
*x^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*e^3*x^3 + 360*(m^2 + m)*d^5*e^2*x^2 - 720*d^6*e*m*x + 720*d^7)*(e*x + d)^m*
c^3/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*e^7)

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Fricas [B]  time = 2.18289, size = 3106, normalized size = 11.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6*b^3*d^4*e^3*m^3 - 720*c^3*d^7 + 2520*b*c^2*d^6*e - 3024*b^2*c*d^5*e^2 + 1260*b^3*d^4*e^3 - (c^3*e^7*m^6 +
21*c^3*e^7*m^5 + 175*c^3*e^7*m^4 + 735*c^3*e^7*m^3 + 1624*c^3*e^7*m^2 + 1764*c^3*e^7*m + 720*c^3*e^7)*x^7 - (2
520*b*c^2*e^7 + (c^3*d*e^6 + 3*b*c^2*e^7)*m^6 + 3*(5*c^3*d*e^6 + 22*b*c^2*e^7)*m^5 + 5*(17*c^3*d*e^6 + 114*b*c
^2*e^7)*m^4 + 15*(15*c^3*d*e^6 + 164*b*c^2*e^7)*m^3 + (274*c^3*d*e^6 + 5547*b*c^2*e^7)*m^2 + 6*(20*c^3*d*e^6 +
1019*b*c^2*e^7)*m)*x^6 - 3*(1008*b^2*c*e^7 + (b*c^2*d*e^6 + b^2*c*e^7)*m^6 - (2*c^3*d^2*e^5 - 17*b*c^2*d*e^6
- 23*b^2*c*e^7)*m^5 - (20*c^3*d^2*e^5 - 105*b*c^2*d*e^6 - 207*b^2*c*e^7)*m^4 - 5*(14*c^3*d^2*e^5 - 59*b*c^2*d*
e^6 - 185*b^2*c*e^7)*m^3 - 2*(50*c^3*d^2*e^5 - 187*b*c^2*d*e^6 - 1072*b^2*c*e^7)*m^2 - 12*(4*c^3*d^2*e^5 - 14*
b*c^2*d*e^6 - 201*b^2*c*e^7)*m)*x^5 - (1260*b^3*e^7 + (3*b^2*c*d*e^6 + b^3*e^7)*m^6 - 3*(5*b*c^2*d^2*e^5 - 19*
b^2*c*d*e^6 - 8*b^3*e^7)*m^5 + (30*c^3*d^3*e^4 - 195*b*c^2*d^2*e^5 + 393*b^2*c*d*e^6 + 226*b^3*e^7)*m^4 + 3*(6
0*c^3*d^3*e^4 - 265*b*c^2*d^2*e^5 + 401*b^2*c*d*e^6 + 352*b^3*e^7)*m^3 + 5*(66*c^3*d^3*e^4 - 249*b*c^2*d^2*e^5
+ 324*b^2*c*d*e^6 + 509*b^3*e^7)*m^2 + 18*(10*c^3*d^3*e^4 - 35*b*c^2*d^2*e^5 + 42*b^2*c*d*e^6 + 164*b^3*e^7)*
m)*x^4 - (b^3*d*e^6*m^6 - 3*(4*b^2*c*d^2*e^5 - 7*b^3*d*e^6)*m^5 + (60*b*c^2*d^3*e^4 - 192*b^2*c*d^2*e^5 + 163*
b^3*d*e^6)*m^4 - 3*(40*c^3*d^4*e^3 - 200*b*c^2*d^3*e^4 + 332*b^2*c*d^2*e^5 - 189*b^3*d*e^6)*m^3 - 4*(90*c^3*d^
4*e^3 - 345*b*c^2*d^3*e^4 + 456*b^2*c*d^2*e^5 - 211*b^3*d*e^6)*m^2 - 12*(20*c^3*d^4*e^3 - 70*b*c^2*d^3*e^4 + 8
4*b^2*c*d^2*e^5 - 35*b^3*d*e^6)*m)*x^3 - 36*(2*b^2*c*d^5*e^2 - 3*b^3*d^4*e^3)*m^2 + 3*(b^3*d^2*e^5*m^5 - (12*b
^2*c*d^3*e^4 - 19*b^3*d^2*e^5)*m^4 + (60*b*c^2*d^4*e^3 - 168*b^2*c*d^3*e^4 + 125*b^3*d^2*e^5)*m^3 - (120*c^3*d
^5*e^2 - 480*b*c^2*d^4*e^3 + 660*b^2*c*d^3*e^4 - 317*b^3*d^2*e^5)*m^2 - 6*(20*c^3*d^5*e^2 - 70*b*c^2*d^4*e^3 +
84*b^2*c*d^3*e^4 - 35*b^3*d^2*e^5)*m)*x^2 + 6*(60*b*c^2*d^6*e - 156*b^2*c*d^5*e^2 + 107*b^3*d^4*e^3)*m - 6*(b
^3*d^3*e^4*m^4 - 6*(2*b^2*c*d^4*e^3 - 3*b^3*d^3*e^4)*m^3 + (60*b*c^2*d^5*e^2 - 156*b^2*c*d^4*e^3 + 107*b^3*d^3
*e^4)*m^2 - 6*(20*c^3*d^6*e - 70*b*c^2*d^5*e^2 + 84*b^2*c*d^4*e^3 - 35*b^3*d^3*e^4)*m)*x)*(e*x + d)^m/(e^7*m^7
+ 28*e^7*m^6 + 322*e^7*m^5 + 1960*e^7*m^4 + 6769*e^7*m^3 + 13132*e^7*m^2 + 13068*e^7*m + 5040*e^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.38683, size = 3426, normalized size = 12.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*c^3*m^6*x^7*e^7 + (x*e + d)^m*c^3*d*m^6*x^6*e^6 + 3*(x*e + d)^m*b*c^2*m^6*x^6*e^7 + 21*(x*e + d)^
m*c^3*m^5*x^7*e^7 + 3*(x*e + d)^m*b*c^2*d*m^6*x^5*e^6 + 15*(x*e + d)^m*c^3*d*m^5*x^6*e^6 - 6*(x*e + d)^m*c^3*d
^2*m^5*x^5*e^5 + 3*(x*e + d)^m*b^2*c*m^6*x^5*e^7 + 66*(x*e + d)^m*b*c^2*m^5*x^6*e^7 + 175*(x*e + d)^m*c^3*m^4*
x^7*e^7 + 3*(x*e + d)^m*b^2*c*d*m^6*x^4*e^6 + 51*(x*e + d)^m*b*c^2*d*m^5*x^5*e^6 + 85*(x*e + d)^m*c^3*d*m^4*x^
6*e^6 - 15*(x*e + d)^m*b*c^2*d^2*m^5*x^4*e^5 - 60*(x*e + d)^m*c^3*d^2*m^4*x^5*e^5 + 30*(x*e + d)^m*c^3*d^3*m^4
*x^4*e^4 + (x*e + d)^m*b^3*m^6*x^4*e^7 + 69*(x*e + d)^m*b^2*c*m^5*x^5*e^7 + 570*(x*e + d)^m*b*c^2*m^4*x^6*e^7
+ 735*(x*e + d)^m*c^3*m^3*x^7*e^7 + (x*e + d)^m*b^3*d*m^6*x^3*e^6 + 57*(x*e + d)^m*b^2*c*d*m^5*x^4*e^6 + 315*(
x*e + d)^m*b*c^2*d*m^4*x^5*e^6 + 225*(x*e + d)^m*c^3*d*m^3*x^6*e^6 - 12*(x*e + d)^m*b^2*c*d^2*m^5*x^3*e^5 - 19
5*(x*e + d)^m*b*c^2*d^2*m^4*x^4*e^5 - 210*(x*e + d)^m*c^3*d^2*m^3*x^5*e^5 + 60*(x*e + d)^m*b*c^2*d^3*m^4*x^3*e
^4 + 180*(x*e + d)^m*c^3*d^3*m^3*x^4*e^4 - 120*(x*e + d)^m*c^3*d^4*m^3*x^3*e^3 + 24*(x*e + d)^m*b^3*m^5*x^4*e^
7 + 621*(x*e + d)^m*b^2*c*m^4*x^5*e^7 + 2460*(x*e + d)^m*b*c^2*m^3*x^6*e^7 + 1624*(x*e + d)^m*c^3*m^2*x^7*e^7
+ 21*(x*e + d)^m*b^3*d*m^5*x^3*e^6 + 393*(x*e + d)^m*b^2*c*d*m^4*x^4*e^6 + 885*(x*e + d)^m*b*c^2*d*m^3*x^5*e^6
+ 274*(x*e + d)^m*c^3*d*m^2*x^6*e^6 - 3*(x*e + d)^m*b^3*d^2*m^5*x^2*e^5 - 192*(x*e + d)^m*b^2*c*d^2*m^4*x^3*e
^5 - 795*(x*e + d)^m*b*c^2*d^2*m^3*x^4*e^5 - 300*(x*e + d)^m*c^3*d^2*m^2*x^5*e^5 + 36*(x*e + d)^m*b^2*c*d^3*m^
4*x^2*e^4 + 600*(x*e + d)^m*b*c^2*d^3*m^3*x^3*e^4 + 330*(x*e + d)^m*c^3*d^3*m^2*x^4*e^4 - 180*(x*e + d)^m*b*c^
2*d^4*m^3*x^2*e^3 - 360*(x*e + d)^m*c^3*d^4*m^2*x^3*e^3 + 360*(x*e + d)^m*c^3*d^5*m^2*x^2*e^2 + 226*(x*e + d)^
m*b^3*m^4*x^4*e^7 + 2775*(x*e + d)^m*b^2*c*m^3*x^5*e^7 + 5547*(x*e + d)^m*b*c^2*m^2*x^6*e^7 + 1764*(x*e + d)^m
*c^3*m*x^7*e^7 + 163*(x*e + d)^m*b^3*d*m^4*x^3*e^6 + 1203*(x*e + d)^m*b^2*c*d*m^3*x^4*e^6 + 1122*(x*e + d)^m*b
*c^2*d*m^2*x^5*e^6 + 120*(x*e + d)^m*c^3*d*m*x^6*e^6 - 57*(x*e + d)^m*b^3*d^2*m^4*x^2*e^5 - 996*(x*e + d)^m*b^
2*c*d^2*m^3*x^3*e^5 - 1245*(x*e + d)^m*b*c^2*d^2*m^2*x^4*e^5 - 144*(x*e + d)^m*c^3*d^2*m*x^5*e^5 + 6*(x*e + d)
^m*b^3*d^3*m^4*x*e^4 + 504*(x*e + d)^m*b^2*c*d^3*m^3*x^2*e^4 + 1380*(x*e + d)^m*b*c^2*d^3*m^2*x^3*e^4 + 180*(x
*e + d)^m*c^3*d^3*m*x^4*e^4 - 72*(x*e + d)^m*b^2*c*d^4*m^3*x*e^3 - 1440*(x*e + d)^m*b*c^2*d^4*m^2*x^2*e^3 - 24
0*(x*e + d)^m*c^3*d^4*m*x^3*e^3 + 360*(x*e + d)^m*b*c^2*d^5*m^2*x*e^2 + 360*(x*e + d)^m*c^3*d^5*m*x^2*e^2 - 72
0*(x*e + d)^m*c^3*d^6*m*x*e + 1056*(x*e + d)^m*b^3*m^3*x^4*e^7 + 6432*(x*e + d)^m*b^2*c*m^2*x^5*e^7 + 6114*(x*
e + d)^m*b*c^2*m*x^6*e^7 + 720*(x*e + d)^m*c^3*x^7*e^7 + 567*(x*e + d)^m*b^3*d*m^3*x^3*e^6 + 1620*(x*e + d)^m*
b^2*c*d*m^2*x^4*e^6 + 504*(x*e + d)^m*b*c^2*d*m*x^5*e^6 - 375*(x*e + d)^m*b^3*d^2*m^3*x^2*e^5 - 1824*(x*e + d)
^m*b^2*c*d^2*m^2*x^3*e^5 - 630*(x*e + d)^m*b*c^2*d^2*m*x^4*e^5 + 108*(x*e + d)^m*b^3*d^3*m^3*x*e^4 + 1980*(x*e
+ d)^m*b^2*c*d^3*m^2*x^2*e^4 + 840*(x*e + d)^m*b*c^2*d^3*m*x^3*e^4 - 6*(x*e + d)^m*b^3*d^4*m^3*e^3 - 936*(x*e
+ d)^m*b^2*c*d^4*m^2*x*e^3 - 1260*(x*e + d)^m*b*c^2*d^4*m*x^2*e^3 + 72*(x*e + d)^m*b^2*c*d^5*m^2*e^2 + 2520*(
x*e + d)^m*b*c^2*d^5*m*x*e^2 - 360*(x*e + d)^m*b*c^2*d^6*m*e + 720*(x*e + d)^m*c^3*d^7 + 2545*(x*e + d)^m*b^3*
m^2*x^4*e^7 + 7236*(x*e + d)^m*b^2*c*m*x^5*e^7 + 2520*(x*e + d)^m*b*c^2*x^6*e^7 + 844*(x*e + d)^m*b^3*d*m^2*x^
3*e^6 + 756*(x*e + d)^m*b^2*c*d*m*x^4*e^6 - 951*(x*e + d)^m*b^3*d^2*m^2*x^2*e^5 - 1008*(x*e + d)^m*b^2*c*d^2*m
*x^3*e^5 + 642*(x*e + d)^m*b^3*d^3*m^2*x*e^4 + 1512*(x*e + d)^m*b^2*c*d^3*m*x^2*e^4 - 108*(x*e + d)^m*b^3*d^4*
m^2*e^3 - 3024*(x*e + d)^m*b^2*c*d^4*m*x*e^3 + 936*(x*e + d)^m*b^2*c*d^5*m*e^2 - 2520*(x*e + d)^m*b*c^2*d^6*e
+ 2952*(x*e + d)^m*b^3*m*x^4*e^7 + 3024*(x*e + d)^m*b^2*c*x^5*e^7 + 420*(x*e + d)^m*b^3*d*m*x^3*e^6 - 630*(x*e
+ d)^m*b^3*d^2*m*x^2*e^5 + 1260*(x*e + d)^m*b^3*d^3*m*x*e^4 - 642*(x*e + d)^m*b^3*d^4*m*e^3 + 3024*(x*e + d)^
m*b^2*c*d^5*e^2 + 1260*(x*e + d)^m*b^3*x^4*e^7 - 1260*(x*e + d)^m*b^3*d^4*e^3)/(m^7*e^7 + 28*m^6*e^7 + 322*m^5
*e^7 + 1960*m^4*e^7 + 6769*m^3*e^7 + 13132*m^2*e^7 + 13068*m*e^7 + 5040*e^7)