3.1731 \(\int (d+e x)^m (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=206 \[ \frac{15 b^2 (b d-a e)^4 (d+e x)^{m+3}}{e^7 (m+3)}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{m+4}}{e^7 (m+4)}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{m+5}}{e^7 (m+5)}-\frac{6 b^5 (b d-a e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{(b d-a e)^6 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{6 b (b d-a e)^5 (d+e x)^{m+2}}{e^7 (m+2)}+\frac{b^6 (d+e x)^{m+7}}{e^7 (m+7)} \]

[Out]

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

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Rubi [A]  time = 0.109974, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac{15 b^2 (b d-a e)^4 (d+e x)^{m+3}}{e^7 (m+3)}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{m+4}}{e^7 (m+4)}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{m+5}}{e^7 (m+5)}-\frac{6 b^5 (b d-a e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{(b d-a e)^6 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{6 b (b d-a e)^5 (d+e x)^{m+2}}{e^7 (m+2)}+\frac{b^6 (d+e x)^{m+7}}{e^7 (m+7)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.179302, size = 175, normalized size = 0.85 \[ \frac{(d+e x)^{m+1} \left (\frac{15 b^2 (d+e x)^2 (b d-a e)^4}{m+3}-\frac{20 b^3 (d+e x)^3 (b d-a e)^3}{m+4}+\frac{15 b^4 (d+e x)^4 (b d-a e)^2}{m+5}-\frac{6 b^5 (d+e x)^5 (b d-a e)}{m+6}-\frac{6 b (d+e x) (b d-a e)^5}{m+2}+\frac{(b d-a e)^6}{m+1}+\frac{b^6 (d+e x)^6}{m+7}\right )}{e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.053, size = 2157, normalized size = 10.5 \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*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

(e*x+d)^(1+m)*(b^6*e^6*m^6*x^6+6*a*b^5*e^6*m^6*x^5+21*b^6*e^6*m^5*x^6+15*a^2*b^4*e^6*m^6*x^4+132*a*b^5*e^6*m^5
*x^5-6*b^6*d*e^5*m^5*x^5+175*b^6*e^6*m^4*x^6+20*a^3*b^3*e^6*m^6*x^3+345*a^2*b^4*e^6*m^5*x^4-30*a*b^5*d*e^5*m^5
*x^4+1140*a*b^5*e^6*m^4*x^5-90*b^6*d*e^5*m^4*x^5+735*b^6*e^6*m^3*x^6+15*a^4*b^2*e^6*m^6*x^2+480*a^3*b^3*e^6*m^
5*x^3-60*a^2*b^4*d*e^5*m^5*x^3+3105*a^2*b^4*e^6*m^4*x^4-510*a*b^5*d*e^5*m^4*x^4+4920*a*b^5*e^6*m^3*x^5+30*b^6*
d^2*e^4*m^4*x^4-510*b^6*d*e^5*m^3*x^5+1624*b^6*e^6*m^2*x^6+6*a^5*b*e^6*m^6*x+375*a^4*b^2*e^6*m^5*x^2-60*a^3*b^
3*d*e^5*m^5*x^2+4520*a^3*b^3*e^6*m^4*x^3-1140*a^2*b^4*d*e^5*m^4*x^3+13875*a^2*b^4*e^6*m^3*x^4+120*a*b^5*d^2*e^
4*m^4*x^3-3150*a*b^5*d*e^5*m^3*x^4+11094*a*b^5*e^6*m^2*x^5+300*b^6*d^2*e^4*m^3*x^4-1350*b^6*d*e^5*m^2*x^5+1764
*b^6*e^6*m*x^6+a^6*e^6*m^6+156*a^5*b*e^6*m^5*x-30*a^4*b^2*d*e^5*m^5*x+3705*a^4*b^2*e^6*m^4*x^2-1260*a^3*b^3*d*
e^5*m^4*x^2+21120*a^3*b^3*e^6*m^3*x^3+180*a^2*b^4*d^2*e^4*m^4*x^2-7860*a^2*b^4*d*e^5*m^3*x^3+32160*a^2*b^4*e^6
*m^2*x^4+1560*a*b^5*d^2*e^4*m^3*x^3-8850*a*b^5*d*e^5*m^2*x^4+12228*a*b^5*e^6*m*x^5-120*b^6*d^3*e^3*m^3*x^3+105
0*b^6*d^2*e^4*m^2*x^4-1644*b^6*d*e^5*m*x^5+720*b^6*e^6*x^6+27*a^6*e^6*m^5-6*a^5*b*d*e^5*m^5+1620*a^5*b*e^6*m^4
*x-690*a^4*b^2*d*e^5*m^4*x+18285*a^4*b^2*e^6*m^3*x^2+120*a^3*b^3*d^2*e^4*m^4*x-9780*a^3*b^3*d*e^5*m^3*x^2+5090
0*a^3*b^3*e^6*m^2*x^3+2880*a^2*b^4*d^2*e^4*m^3*x^2-24060*a^2*b^4*d*e^5*m^2*x^3+36180*a^2*b^4*e^6*m*x^4-360*a*b
^5*d^3*e^3*m^3*x^2+6360*a*b^5*d^2*e^4*m^2*x^3-11220*a*b^5*d*e^5*m*x^4+5040*a*b^5*e^6*x^5-720*b^6*d^3*e^3*m^2*x
^3+1500*b^6*d^2*e^4*m*x^4-720*b^6*d*e^5*x^5+295*a^6*e^6*m^4-150*a^5*b*d*e^5*m^4+8520*a^5*b*e^6*m^3*x+30*a^4*b^
2*d^2*e^4*m^4-6030*a^4*b^2*d*e^5*m^3*x+46680*a^4*b^2*e^6*m^2*x^2+2280*a^3*b^3*d^2*e^4*m^3*x-34020*a^3*b^3*d*e^
5*m^2*x^2+59040*a^3*b^3*e^6*m*x^3-360*a^2*b^4*d^3*e^3*m^3*x+14940*a^2*b^4*d^2*e^4*m^2*x^2-32400*a^2*b^4*d*e^5*
m*x^3+15120*a^2*b^4*e^6*x^4-3600*a*b^5*d^3*e^3*m^2*x^2+9960*a*b^5*d^2*e^4*m*x^3-5040*a*b^5*d*e^5*x^4+360*b^6*d
^4*e^2*m^2*x^2-1320*b^6*d^3*e^3*m*x^3+720*b^6*d^2*e^4*x^4+1665*a^6*e^6*m^3-1470*a^5*b*d*e^5*m^3+23574*a^5*b*e^
6*m^2*x+660*a^4*b^2*d^2*e^4*m^3-24510*a^4*b^2*d*e^5*m^2*x+56940*a^4*b^2*e^6*m*x^2-120*a^3*b^3*d^3*e^3*m^3+1500
0*a^3*b^3*d^2*e^4*m^2*x-50640*a^3*b^3*d*e^5*m*x^2+25200*a^3*b^3*e^6*x^3-5040*a^2*b^4*d^3*e^3*m^2*x+27360*a^2*b
^4*d^2*e^4*m*x^2-15120*a^2*b^4*d*e^5*x^3+720*a*b^5*d^4*e^2*m^2*x-8280*a*b^5*d^3*e^3*m*x^2+5040*a*b^5*d^2*e^4*x
^3+1080*b^6*d^4*e^2*m*x^2-720*b^6*d^3*e^3*x^3+5104*a^6*e^6*m^2-7050*a^5*b*d*e^5*m^2+31644*a^5*b*e^6*m*x+5370*a
^4*b^2*d^2*e^4*m^2-44340*a^4*b^2*d*e^5*m*x+25200*a^4*b^2*e^6*x^2-2160*a^3*b^3*d^3*e^3*m^2+38040*a^3*b^3*d^2*e^
4*m*x-25200*a^3*b^3*d*e^5*x^2+360*a^2*b^4*d^4*e^2*m^2-19800*a^2*b^4*d^3*e^3*m*x+15120*a^2*b^4*d^2*e^4*x^2+5760
*a*b^5*d^4*e^2*m*x-5040*a*b^5*d^3*e^3*x^2-720*b^6*d^5*e*m*x+720*b^6*d^4*e^2*x^2+8028*a^6*e^6*m-16524*a^5*b*d*e
^5*m+15120*a^5*b*e^6*x+19140*a^4*b^2*d^2*e^4*m-25200*a^4*b^2*d*e^5*x-12840*a^3*b^3*d^3*e^3*m+25200*a^3*b^3*d^2
*e^4*x+4680*a^2*b^4*d^4*e^2*m-15120*a^2*b^4*d^3*e^3*x-720*a*b^5*d^5*e*m+5040*a*b^5*d^4*e^2*x-720*b^6*d^5*e*x+5
040*a^6*e^6-15120*a^5*b*d*e^5+25200*a^4*b^2*d^2*e^4-25200*a^3*b^3*d^3*e^3+15120*a^2*b^4*d^4*e^2-5040*a*b^5*d^5
*e+720*b^6*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 [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((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.89748, size = 4810, normalized size = 23.35 \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*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

(a^6*d*e^6*m^6 + 720*b^6*d^7 - 5040*a*b^5*d^6*e + 15120*a^2*b^4*d^5*e^2 - 25200*a^3*b^3*d^4*e^3 + 25200*a^4*b^
2*d^3*e^4 - 15120*a^5*b*d^2*e^5 + 5040*a^6*d*e^6 + (b^6*e^7*m^6 + 21*b^6*e^7*m^5 + 175*b^6*e^7*m^4 + 735*b^6*e
^7*m^3 + 1624*b^6*e^7*m^2 + 1764*b^6*e^7*m + 720*b^6*e^7)*x^7 + (5040*a*b^5*e^7 + (b^6*d*e^6 + 6*a*b^5*e^7)*m^
6 + 3*(5*b^6*d*e^6 + 44*a*b^5*e^7)*m^5 + 5*(17*b^6*d*e^6 + 228*a*b^5*e^7)*m^4 + 15*(15*b^6*d*e^6 + 328*a*b^5*e
^7)*m^3 + 2*(137*b^6*d*e^6 + 5547*a*b^5*e^7)*m^2 + 12*(10*b^6*d*e^6 + 1019*a*b^5*e^7)*m)*x^6 - 3*(2*a^5*b*d^2*
e^5 - 9*a^6*d*e^6)*m^5 + 3*(5040*a^2*b^4*e^7 + (2*a*b^5*d*e^6 + 5*a^2*b^4*e^7)*m^6 - (2*b^6*d^2*e^5 - 34*a*b^5
*d*e^6 - 115*a^2*b^4*e^7)*m^5 - 5*(4*b^6*d^2*e^5 - 42*a*b^5*d*e^6 - 207*a^2*b^4*e^7)*m^4 - 5*(14*b^6*d^2*e^5 -
118*a*b^5*d*e^6 - 925*a^2*b^4*e^7)*m^3 - 4*(25*b^6*d^2*e^5 - 187*a*b^5*d*e^6 - 2680*a^2*b^4*e^7)*m^2 - 12*(4*
b^6*d^2*e^5 - 28*a*b^5*d*e^6 - 1005*a^2*b^4*e^7)*m)*x^5 + 5*(6*a^4*b^2*d^3*e^4 - 30*a^5*b*d^2*e^5 + 59*a^6*d*e
^6)*m^4 + 5*(5040*a^3*b^3*e^7 + (3*a^2*b^4*d*e^6 + 4*a^3*b^3*e^7)*m^6 - 3*(2*a*b^5*d^2*e^5 - 19*a^2*b^4*d*e^6
- 32*a^3*b^3*e^7)*m^5 + (6*b^6*d^3*e^4 - 78*a*b^5*d^2*e^5 + 393*a^2*b^4*d*e^6 + 904*a^3*b^3*e^7)*m^4 + 3*(12*b
^6*d^3*e^4 - 106*a*b^5*d^2*e^5 + 401*a^2*b^4*d*e^6 + 1408*a^3*b^3*e^7)*m^3 + 2*(33*b^6*d^3*e^4 - 249*a*b^5*d^2
*e^5 + 810*a^2*b^4*d*e^6 + 5090*a^3*b^3*e^7)*m^2 + 36*(b^6*d^3*e^4 - 7*a*b^5*d^2*e^5 + 21*a^2*b^4*d*e^6 + 328*
a^3*b^3*e^7)*m)*x^4 - 15*(8*a^3*b^3*d^4*e^3 - 44*a^4*b^2*d^3*e^4 + 98*a^5*b*d^2*e^5 - 111*a^6*d*e^6)*m^3 + 5*(
5040*a^4*b^2*e^7 + (4*a^3*b^3*d*e^6 + 3*a^4*b^2*e^7)*m^6 - 3*(4*a^2*b^4*d^2*e^5 - 28*a^3*b^3*d*e^6 - 25*a^4*b^
2*e^7)*m^5 + (24*a*b^5*d^3*e^4 - 192*a^2*b^4*d^2*e^5 + 652*a^3*b^3*d*e^6 + 741*a^4*b^2*e^7)*m^4 - 3*(8*b^6*d^4
*e^3 - 80*a*b^5*d^3*e^4 + 332*a^2*b^4*d^2*e^5 - 756*a^3*b^3*d*e^6 - 1219*a^4*b^2*e^7)*m^3 - 8*(9*b^6*d^4*e^3 -
69*a*b^5*d^3*e^4 + 228*a^2*b^4*d^2*e^5 - 422*a^3*b^3*d*e^6 - 1167*a^4*b^2*e^7)*m^2 - 12*(4*b^6*d^4*e^3 - 28*a
*b^5*d^3*e^4 + 84*a^2*b^4*d^2*e^5 - 140*a^3*b^3*d*e^6 - 949*a^4*b^2*e^7)*m)*x^3 + 2*(180*a^2*b^4*d^5*e^2 - 108
0*a^3*b^3*d^4*e^3 + 2685*a^4*b^2*d^3*e^4 - 3525*a^5*b*d^2*e^5 + 2552*a^6*d*e^6)*m^2 + 3*(5040*a^5*b*e^7 + (5*a
^4*b^2*d*e^6 + 2*a^5*b*e^7)*m^6 - (20*a^3*b^3*d^2*e^5 - 115*a^4*b^2*d*e^6 - 52*a^5*b*e^7)*m^5 + 5*(12*a^2*b^4*
d^3*e^4 - 76*a^3*b^3*d^2*e^5 + 201*a^4*b^2*d*e^6 + 108*a^5*b*e^7)*m^4 - 5*(24*a*b^5*d^4*e^3 - 168*a^2*b^4*d^3*
e^4 + 500*a^3*b^3*d^2*e^5 - 817*a^4*b^2*d*e^6 - 568*a^5*b*e^7)*m^3 + 2*(60*b^6*d^5*e^2 - 480*a*b^5*d^4*e^3 + 1
650*a^2*b^4*d^3*e^4 - 3170*a^3*b^3*d^2*e^5 + 3695*a^4*b^2*d*e^6 + 3929*a^5*b*e^7)*m^2 + 12*(10*b^6*d^5*e^2 - 7
0*a*b^5*d^4*e^3 + 210*a^2*b^4*d^3*e^4 - 350*a^3*b^3*d^2*e^5 + 350*a^4*b^2*d*e^6 + 879*a^5*b*e^7)*m)*x^2 - 12*(
60*a*b^5*d^6*e - 390*a^2*b^4*d^5*e^2 + 1070*a^3*b^3*d^4*e^3 - 1595*a^4*b^2*d^3*e^4 + 1377*a^5*b*d^2*e^5 - 669*
a^6*d*e^6)*m + (5040*a^6*e^7 + (6*a^5*b*d*e^6 + a^6*e^7)*m^6 - 3*(10*a^4*b^2*d^2*e^5 - 50*a^5*b*d*e^6 - 9*a^6*
e^7)*m^5 + 5*(24*a^3*b^3*d^3*e^4 - 132*a^4*b^2*d^2*e^5 + 294*a^5*b*d*e^6 + 59*a^6*e^7)*m^4 - 15*(24*a^2*b^4*d^
4*e^3 - 144*a^3*b^3*d^3*e^4 + 358*a^4*b^2*d^2*e^5 - 470*a^5*b*d*e^6 - 111*a^6*e^7)*m^3 + 4*(180*a*b^5*d^5*e^2
- 1170*a^2*b^4*d^4*e^3 + 3210*a^3*b^3*d^3*e^4 - 4785*a^4*b^2*d^2*e^5 + 4131*a^5*b*d*e^6 + 1276*a^6*e^7)*m^2 -
36*(20*b^6*d^6*e - 140*a*b^5*d^5*e^2 + 420*a^2*b^4*d^4*e^3 - 700*a^3*b^3*d^3*e^4 + 700*a^4*b^2*d^2*e^5 - 420*a
^5*b*d*e^6 - 223*a^6*e^7)*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)

________________________________________________________________________________________

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*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.23667, size = 5234, normalized size = 25.41 \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*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

((x*e + d)^m*b^6*m^6*x^7*e^7 + (x*e + d)^m*b^6*d*m^6*x^6*e^6 + 6*(x*e + d)^m*a*b^5*m^6*x^6*e^7 + 21*(x*e + d)^
m*b^6*m^5*x^7*e^7 + 6*(x*e + d)^m*a*b^5*d*m^6*x^5*e^6 + 15*(x*e + d)^m*b^6*d*m^5*x^6*e^6 - 6*(x*e + d)^m*b^6*d
^2*m^5*x^5*e^5 + 15*(x*e + d)^m*a^2*b^4*m^6*x^5*e^7 + 132*(x*e + d)^m*a*b^5*m^5*x^6*e^7 + 175*(x*e + d)^m*b^6*
m^4*x^7*e^7 + 15*(x*e + d)^m*a^2*b^4*d*m^6*x^4*e^6 + 102*(x*e + d)^m*a*b^5*d*m^5*x^5*e^6 + 85*(x*e + d)^m*b^6*
d*m^4*x^6*e^6 - 30*(x*e + d)^m*a*b^5*d^2*m^5*x^4*e^5 - 60*(x*e + d)^m*b^6*d^2*m^4*x^5*e^5 + 30*(x*e + d)^m*b^6
*d^3*m^4*x^4*e^4 + 20*(x*e + d)^m*a^3*b^3*m^6*x^4*e^7 + 345*(x*e + d)^m*a^2*b^4*m^5*x^5*e^7 + 1140*(x*e + d)^m
*a*b^5*m^4*x^6*e^7 + 735*(x*e + d)^m*b^6*m^3*x^7*e^7 + 20*(x*e + d)^m*a^3*b^3*d*m^6*x^3*e^6 + 285*(x*e + d)^m*
a^2*b^4*d*m^5*x^4*e^6 + 630*(x*e + d)^m*a*b^5*d*m^4*x^5*e^6 + 225*(x*e + d)^m*b^6*d*m^3*x^6*e^6 - 60*(x*e + d)
^m*a^2*b^4*d^2*m^5*x^3*e^5 - 390*(x*e + d)^m*a*b^5*d^2*m^4*x^4*e^5 - 210*(x*e + d)^m*b^6*d^2*m^3*x^5*e^5 + 120
*(x*e + d)^m*a*b^5*d^3*m^4*x^3*e^4 + 180*(x*e + d)^m*b^6*d^3*m^3*x^4*e^4 - 120*(x*e + d)^m*b^6*d^4*m^3*x^3*e^3
+ 15*(x*e + d)^m*a^4*b^2*m^6*x^3*e^7 + 480*(x*e + d)^m*a^3*b^3*m^5*x^4*e^7 + 3105*(x*e + d)^m*a^2*b^4*m^4*x^5
*e^7 + 4920*(x*e + d)^m*a*b^5*m^3*x^6*e^7 + 1624*(x*e + d)^m*b^6*m^2*x^7*e^7 + 15*(x*e + d)^m*a^4*b^2*d*m^6*x^
2*e^6 + 420*(x*e + d)^m*a^3*b^3*d*m^5*x^3*e^6 + 1965*(x*e + d)^m*a^2*b^4*d*m^4*x^4*e^6 + 1770*(x*e + d)^m*a*b^
5*d*m^3*x^5*e^6 + 274*(x*e + d)^m*b^6*d*m^2*x^6*e^6 - 60*(x*e + d)^m*a^3*b^3*d^2*m^5*x^2*e^5 - 960*(x*e + d)^m
*a^2*b^4*d^2*m^4*x^3*e^5 - 1590*(x*e + d)^m*a*b^5*d^2*m^3*x^4*e^5 - 300*(x*e + d)^m*b^6*d^2*m^2*x^5*e^5 + 180*
(x*e + d)^m*a^2*b^4*d^3*m^4*x^2*e^4 + 1200*(x*e + d)^m*a*b^5*d^3*m^3*x^3*e^4 + 330*(x*e + d)^m*b^6*d^3*m^2*x^4
*e^4 - 360*(x*e + d)^m*a*b^5*d^4*m^3*x^2*e^3 - 360*(x*e + d)^m*b^6*d^4*m^2*x^3*e^3 + 360*(x*e + d)^m*b^6*d^5*m
^2*x^2*e^2 + 6*(x*e + d)^m*a^5*b*m^6*x^2*e^7 + 375*(x*e + d)^m*a^4*b^2*m^5*x^3*e^7 + 4520*(x*e + d)^m*a^3*b^3*
m^4*x^4*e^7 + 13875*(x*e + d)^m*a^2*b^4*m^3*x^5*e^7 + 11094*(x*e + d)^m*a*b^5*m^2*x^6*e^7 + 1764*(x*e + d)^m*b
^6*m*x^7*e^7 + 6*(x*e + d)^m*a^5*b*d*m^6*x*e^6 + 345*(x*e + d)^m*a^4*b^2*d*m^5*x^2*e^6 + 3260*(x*e + d)^m*a^3*
b^3*d*m^4*x^3*e^6 + 6015*(x*e + d)^m*a^2*b^4*d*m^3*x^4*e^6 + 2244*(x*e + d)^m*a*b^5*d*m^2*x^5*e^6 + 120*(x*e +
d)^m*b^6*d*m*x^6*e^6 - 30*(x*e + d)^m*a^4*b^2*d^2*m^5*x*e^5 - 1140*(x*e + d)^m*a^3*b^3*d^2*m^4*x^2*e^5 - 4980
*(x*e + d)^m*a^2*b^4*d^2*m^3*x^3*e^5 - 2490*(x*e + d)^m*a*b^5*d^2*m^2*x^4*e^5 - 144*(x*e + d)^m*b^6*d^2*m*x^5*
e^5 + 120*(x*e + d)^m*a^3*b^3*d^3*m^4*x*e^4 + 2520*(x*e + d)^m*a^2*b^4*d^3*m^3*x^2*e^4 + 2760*(x*e + d)^m*a*b^
5*d^3*m^2*x^3*e^4 + 180*(x*e + d)^m*b^6*d^3*m*x^4*e^4 - 360*(x*e + d)^m*a^2*b^4*d^4*m^3*x*e^3 - 2880*(x*e + d)
^m*a*b^5*d^4*m^2*x^2*e^3 - 240*(x*e + d)^m*b^6*d^4*m*x^3*e^3 + 720*(x*e + d)^m*a*b^5*d^5*m^2*x*e^2 + 360*(x*e
+ d)^m*b^6*d^5*m*x^2*e^2 - 720*(x*e + d)^m*b^6*d^6*m*x*e + (x*e + d)^m*a^6*m^6*x*e^7 + 156*(x*e + d)^m*a^5*b*m
^5*x^2*e^7 + 3705*(x*e + d)^m*a^4*b^2*m^4*x^3*e^7 + 21120*(x*e + d)^m*a^3*b^3*m^3*x^4*e^7 + 32160*(x*e + d)^m*
a^2*b^4*m^2*x^5*e^7 + 12228*(x*e + d)^m*a*b^5*m*x^6*e^7 + 720*(x*e + d)^m*b^6*x^7*e^7 + (x*e + d)^m*a^6*d*m^6*
e^6 + 150*(x*e + d)^m*a^5*b*d*m^5*x*e^6 + 3015*(x*e + d)^m*a^4*b^2*d*m^4*x^2*e^6 + 11340*(x*e + d)^m*a^3*b^3*d
*m^3*x^3*e^6 + 8100*(x*e + d)^m*a^2*b^4*d*m^2*x^4*e^6 + 1008*(x*e + d)^m*a*b^5*d*m*x^5*e^6 - 6*(x*e + d)^m*a^5
*b*d^2*m^5*e^5 - 660*(x*e + d)^m*a^4*b^2*d^2*m^4*x*e^5 - 7500*(x*e + d)^m*a^3*b^3*d^2*m^3*x^2*e^5 - 9120*(x*e
+ d)^m*a^2*b^4*d^2*m^2*x^3*e^5 - 1260*(x*e + d)^m*a*b^5*d^2*m*x^4*e^5 + 30*(x*e + d)^m*a^4*b^2*d^3*m^4*e^4 + 2
160*(x*e + d)^m*a^3*b^3*d^3*m^3*x*e^4 + 9900*(x*e + d)^m*a^2*b^4*d^3*m^2*x^2*e^4 + 1680*(x*e + d)^m*a*b^5*d^3*
m*x^3*e^4 - 120*(x*e + d)^m*a^3*b^3*d^4*m^3*e^3 - 4680*(x*e + d)^m*a^2*b^4*d^4*m^2*x*e^3 - 2520*(x*e + d)^m*a*
b^5*d^4*m*x^2*e^3 + 360*(x*e + d)^m*a^2*b^4*d^5*m^2*e^2 + 5040*(x*e + d)^m*a*b^5*d^5*m*x*e^2 - 720*(x*e + d)^m
*a*b^5*d^6*m*e + 720*(x*e + d)^m*b^6*d^7 + 27*(x*e + d)^m*a^6*m^5*x*e^7 + 1620*(x*e + d)^m*a^5*b*m^4*x^2*e^7 +
18285*(x*e + d)^m*a^4*b^2*m^3*x^3*e^7 + 50900*(x*e + d)^m*a^3*b^3*m^2*x^4*e^7 + 36180*(x*e + d)^m*a^2*b^4*m*x
^5*e^7 + 5040*(x*e + d)^m*a*b^5*x^6*e^7 + 27*(x*e + d)^m*a^6*d*m^5*e^6 + 1470*(x*e + d)^m*a^5*b*d*m^4*x*e^6 +
12255*(x*e + d)^m*a^4*b^2*d*m^3*x^2*e^6 + 16880*(x*e + d)^m*a^3*b^3*d*m^2*x^3*e^6 + 3780*(x*e + d)^m*a^2*b^4*d
*m*x^4*e^6 - 150*(x*e + d)^m*a^5*b*d^2*m^4*e^5 - 5370*(x*e + d)^m*a^4*b^2*d^2*m^3*x*e^5 - 19020*(x*e + d)^m*a^
3*b^3*d^2*m^2*x^2*e^5 - 5040*(x*e + d)^m*a^2*b^4*d^2*m*x^3*e^5 + 660*(x*e + d)^m*a^4*b^2*d^3*m^3*e^4 + 12840*(
x*e + d)^m*a^3*b^3*d^3*m^2*x*e^4 + 7560*(x*e + d)^m*a^2*b^4*d^3*m*x^2*e^4 - 2160*(x*e + d)^m*a^3*b^3*d^4*m^2*e
^3 - 15120*(x*e + d)^m*a^2*b^4*d^4*m*x*e^3 + 4680*(x*e + d)^m*a^2*b^4*d^5*m*e^2 - 5040*(x*e + d)^m*a*b^5*d^6*e
+ 295*(x*e + d)^m*a^6*m^4*x*e^7 + 8520*(x*e + d)^m*a^5*b*m^3*x^2*e^7 + 46680*(x*e + d)^m*a^4*b^2*m^2*x^3*e^7
+ 59040*(x*e + d)^m*a^3*b^3*m*x^4*e^7 + 15120*(x*e + d)^m*a^2*b^4*x^5*e^7 + 295*(x*e + d)^m*a^6*d*m^4*e^6 + 70
50*(x*e + d)^m*a^5*b*d*m^3*x*e^6 + 22170*(x*e + d)^m*a^4*b^2*d*m^2*x^2*e^6 + 8400*(x*e + d)^m*a^3*b^3*d*m*x^3*
e^6 - 1470*(x*e + d)^m*a^5*b*d^2*m^3*e^5 - 19140*(x*e + d)^m*a^4*b^2*d^2*m^2*x*e^5 - 12600*(x*e + d)^m*a^3*b^3
*d^2*m*x^2*e^5 + 5370*(x*e + d)^m*a^4*b^2*d^3*m^2*e^4 + 25200*(x*e + d)^m*a^3*b^3*d^3*m*x*e^4 - 12840*(x*e + d
)^m*a^3*b^3*d^4*m*e^3 + 15120*(x*e + d)^m*a^2*b^4*d^5*e^2 + 1665*(x*e + d)^m*a^6*m^3*x*e^7 + 23574*(x*e + d)^m
*a^5*b*m^2*x^2*e^7 + 56940*(x*e + d)^m*a^4*b^2*m*x^3*e^7 + 25200*(x*e + d)^m*a^3*b^3*x^4*e^7 + 1665*(x*e + d)^
m*a^6*d*m^3*e^6 + 16524*(x*e + d)^m*a^5*b*d*m^2*x*e^6 + 12600*(x*e + d)^m*a^4*b^2*d*m*x^2*e^6 - 7050*(x*e + d)
^m*a^5*b*d^2*m^2*e^5 - 25200*(x*e + d)^m*a^4*b^2*d^2*m*x*e^5 + 19140*(x*e + d)^m*a^4*b^2*d^3*m*e^4 - 25200*(x*
e + d)^m*a^3*b^3*d^4*e^3 + 5104*(x*e + d)^m*a^6*m^2*x*e^7 + 31644*(x*e + d)^m*a^5*b*m*x^2*e^7 + 25200*(x*e + d
)^m*a^4*b^2*x^3*e^7 + 5104*(x*e + d)^m*a^6*d*m^2*e^6 + 15120*(x*e + d)^m*a^5*b*d*m*x*e^6 - 16524*(x*e + d)^m*a
^5*b*d^2*m*e^5 + 25200*(x*e + d)^m*a^4*b^2*d^3*e^4 + 8028*(x*e + d)^m*a^6*m*x*e^7 + 15120*(x*e + d)^m*a^5*b*x^
2*e^7 + 8028*(x*e + d)^m*a^6*d*m*e^6 - 15120*(x*e + d)^m*a^5*b*d^2*e^5 + 5040*(x*e + d)^m*a^6*x*e^7 + 5040*(x*
e + d)^m*a^6*d*e^6)/(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)