3.1655 \(\int (b+2 c x) (d+e x)^m (a+b x+c x^2)^2 \, dx\)
Optimal. Leaf size=270 \[ \frac{2 (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (m+2)}-\frac{(2 c d-b e) (d+e x)^{m+3} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (m+3)}+\frac{4 c (d+e x)^{m+4} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (m+4)}-\frac{(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}-\frac{5 c^2 (2 c d-b e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac{2 c^3 (d+e x)^{m+6}}{e^6 (m+6)} \]
[Out]
-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (2*(c*d^2 - b*d*e + a*e^2)*(5*c
^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(2 + m))/(e^6*(2 + m)) - ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2
- 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(3 + m))/(e^6*(3 + m)) + (4*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d
+ e*x)^(4 + m))/(e^6*(4 + m)) - (5*c^2*(2*c*d - b*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (2*c^3*(d + e*x)^(6 +
m))/(e^6*(6 + m))
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Rubi [A] time = 0.189151, antiderivative size = 270, 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} \[ \frac{2 (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (m+2)}-\frac{(2 c d-b e) (d+e x)^{m+3} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (m+3)}+\frac{4 c (d+e x)^{m+4} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (m+4)}-\frac{(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}-\frac{5 c^2 (2 c d-b e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac{2 c^3 (d+e x)^{m+6}}{e^6 (m+6)} \]
Antiderivative was successfully verified.
[In]
Int[(b + 2*c*x)*(d + e*x)^m*(a + b*x + c*x^2)^2,x]
[Out]
-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (2*(c*d^2 - b*d*e + a*e^2)*(5*c
^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(2 + m))/(e^6*(2 + m)) - ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2
- 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(3 + m))/(e^6*(3 + m)) + (4*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d
+ e*x)^(4 + m))/(e^6*(4 + m)) - (5*c^2*(2*c*d - b*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (2*c^3*(d + e*x)^(6 +
m))/(e^6*(6 + 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 )^2 \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^m}{e^5}+\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{1+m}}{e^5}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{2+m}}{e^5}+\frac{4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3+m}}{e^5}-\frac{5 c^2 (2 c d-b e) (d+e x)^{4+m}}{e^5}+\frac{2 c^3 (d+e x)^{5+m}}{e^5}\right ) \, dx\\ &=-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{1+m}}{e^6 (1+m)}+\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{2+m}}{e^6 (2+m)}-\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{3+m}}{e^6 (3+m)}+\frac{4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac{5 c^2 (2 c d-b e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac{2 c^3 (d+e x)^{6+m}}{e^6 (6+m)}\\ \end{align*}
Mathematica [A] time = 1.24617, size = 408, normalized size = 1.51 \[ \frac{(d+e x)^{m+1} \left (\frac{2 \left (\frac{(d+e x) \left (2 c^2 e^2 \left (4 a^2 e^2 \left (m^2+8 m+15\right )+4 a b d e \left (m^2-4 m-30\right )+b^2 d^2 \left (m^2-4 m+60\right )\right )-2 b^2 c e^3 m (3 a e (m+4)+b d (m-4))-8 c^3 d^2 e \left (a e \left (m^2-4 m-30\right )+30 b d\right )+b^4 e^4 m (m+2)+120 c^4 d^4\right )}{e^2 (m+2)}-(a+x (b+c x)) \left (-2 c^2 e \left (2 a e \left (d \left (m^2+m-15\right )+e \left (m^2+8 m+15\right ) x\right )+5 b d (d (m+12)-2 e (m+3) x)\right )+b c e^2 \left (-2 a e (7 m+30)+b d \left (m^2+11 m+60\right )+b e m (m+3) x\right )+b^3 e^3 m+20 c^3 d^2 (3 d-e (m+3) x)\right )-\frac{(2 c d-b e) \left (e (a e-b d)+c d^2\right ) \left (-4 c e \left (a e \left (m^2+m-15\right )+15 b d\right )+b^2 e^2 m (m+1)+60 c^2 d^2\right )}{e^2 (m+1)}\right )}{e^2 (m+3) (m+4)}+c (a+x (b+c x))^2 (b e (m+10)+2 c (e (m+5) x-5 d))\right )}{c e^2 (m+5) (m+6)} \]
Antiderivative was successfully verified.
[In]
Integrate[(b + 2*c*x)*(d + e*x)^m*(a + b*x + c*x^2)^2,x]
[Out]
((d + e*x)^(1 + m)*(c*(a + x*(b + c*x))^2*(b*e*(10 + m) + 2*c*(-5*d + e*(5 + m)*x)) + (2*(-(((2*c*d - b*e)*(c*
d^2 + e*(-(b*d) + a*e))*(60*c^2*d^2 + b^2*e^2*m*(1 + m) - 4*c*e*(15*b*d + a*e*(-15 + m + m^2))))/(e^2*(1 + m))
) + ((120*c^4*d^4 + b^4*e^4*m*(2 + m) - 2*b^2*c*e^3*m*(b*d*(-4 + m) + 3*a*e*(4 + m)) - 8*c^3*d^2*e*(30*b*d + a
*e*(-30 - 4*m + m^2)) + 2*c^2*e^2*(4*a*b*d*e*(-30 - 4*m + m^2) + b^2*d^2*(60 - 4*m + m^2) + 4*a^2*e^2*(15 + 8*
m + m^2)))*(d + e*x))/(e^2*(2 + m)) - (a + x*(b + c*x))*(b^3*e^3*m + 20*c^3*d^2*(3*d - e*(3 + m)*x) + b*c*e^2*
(-2*a*e*(30 + 7*m) + b*d*(60 + 11*m + m^2) + b*e*m*(3 + m)*x) - 2*c^2*e*(5*b*d*(d*(12 + m) - 2*e*(3 + m)*x) +
2*a*e*(d*(-15 + m + m^2) + e*(15 + 8*m + m^2)*x)))))/(e^2*(3 + m)*(4 + m))))/(c*e^2*(5 + m)*(6 + m))
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Maple [B] time = 0.011, size = 1852, normalized size = 6.9 \begin{align*} \text{result 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)^2,x)
[Out]
(e*x+d)^(1+m)*(2*c^3*e^5*m^5*x^5+5*b*c^2*e^5*m^5*x^4+30*c^3*e^5*m^4*x^5+4*a*c^2*e^5*m^5*x^3+4*b^2*c*e^5*m^5*x^
3+80*b*c^2*e^5*m^4*x^4-10*c^3*d*e^4*m^4*x^4+170*c^3*e^5*m^3*x^5+6*a*b*c*e^5*m^5*x^2+68*a*c^2*e^5*m^4*x^3+b^3*e
^5*m^5*x^2+68*b^2*c*e^5*m^4*x^3-20*b*c^2*d*e^4*m^4*x^3+475*b*c^2*e^5*m^3*x^4-100*c^3*d*e^4*m^3*x^4+450*c^3*e^5
*m^2*x^5+2*a^2*c*e^5*m^5*x+2*a*b^2*e^5*m^5*x+108*a*b*c*e^5*m^4*x^2-12*a*c^2*d*e^4*m^4*x^2+428*a*c^2*e^5*m^3*x^
3+18*b^3*e^5*m^4*x^2-12*b^2*c*d*e^4*m^4*x^2+428*b^2*c*e^5*m^3*x^3-240*b*c^2*d*e^4*m^3*x^3+1300*b*c^2*e^5*m^2*x
^4+40*c^3*d^2*e^3*m^3*x^3-350*c^3*d*e^4*m^2*x^4+548*c^3*e^5*m*x^5+a^2*b*e^5*m^5+38*a^2*c*e^5*m^4*x+38*a*b^2*e^
5*m^4*x-12*a*b*c*d*e^4*m^4*x+726*a*b*c*e^5*m^3*x^2-168*a*c^2*d*e^4*m^3*x^2+1228*a*c^2*e^5*m^2*x^3-2*b^3*d*e^4*
m^4*x+121*b^3*e^5*m^3*x^2-168*b^2*c*d*e^4*m^3*x^2+1228*b^2*c*e^5*m^2*x^3+60*b*c^2*d^2*e^3*m^3*x^2-940*b*c^2*d*
e^4*m^2*x^3+1620*b*c^2*e^5*m*x^4+240*c^3*d^2*e^3*m^2*x^3-500*c^3*d*e^4*m*x^4+240*c^3*e^5*x^5+20*a^2*b*e^5*m^4-
2*a^2*c*d*e^4*m^4+274*a^2*c*e^5*m^3*x-2*a*b^2*d*e^4*m^4+274*a*b^2*e^5*m^3*x-192*a*b*c*d*e^4*m^3*x+2232*a*b*c*e
^5*m^2*x^2+24*a*c^2*d^2*e^3*m^3*x-780*a*c^2*d*e^4*m^2*x^2+1584*a*c^2*e^5*m*x^3-32*b^3*d*e^4*m^3*x+372*b^3*e^5*
m^2*x^2+24*b^2*c*d^2*e^3*m^3*x-780*b^2*c*d*e^4*m^2*x^2+1584*b^2*c*e^5*m*x^3+540*b*c^2*d^2*e^3*m^2*x^2-1440*b*c
^2*d*e^4*m*x^3+720*b*c^2*e^5*x^4-120*c^3*d^3*e^2*m^2*x^2+440*c^3*d^2*e^3*m*x^3-240*c^3*d*e^4*x^4+155*a^2*b*e^5
*m^3-36*a^2*c*d*e^4*m^3+922*a^2*c*e^5*m^2*x-36*a*b^2*d*e^4*m^3+922*a*b^2*e^5*m^2*x+12*a*b*c*d^2*e^3*m^3-1068*a
*b*c*d*e^4*m^2*x+3048*a*b*c*e^5*m*x^2+288*a*c^2*d^2*e^3*m^2*x-1344*a*c^2*d*e^4*m*x^2+720*a*c^2*e^5*x^3+2*b^3*d
^2*e^3*m^3-178*b^3*d*e^4*m^2*x+508*b^3*e^5*m*x^2+288*b^2*c*d^2*e^3*m^2*x-1344*b^2*c*d*e^4*m*x^2+720*b^2*c*e^5*
x^3-120*b*c^2*d^3*e^2*m^2*x+1200*b*c^2*d^2*e^3*m*x^2-720*b*c^2*d*e^4*x^3-360*c^3*d^3*e^2*m*x^2+240*c^3*d^2*e^3
*x^3+580*a^2*b*e^5*m^2-238*a^2*c*d*e^4*m^2+1404*a^2*c*e^5*m*x-238*a*b^2*d*e^4*m^2+1404*a*b^2*e^5*m*x+180*a*b*c
*d^2*e^3*m^2-2328*a*b*c*d*e^4*m*x+1440*a*b*c*e^5*x^2-24*a*c^2*d^3*e^2*m^2+984*a*c^2*d^2*e^3*m*x-720*a*c^2*d*e^
4*x^2+30*b^3*d^2*e^3*m^2-388*b^3*d*e^4*m*x+240*b^3*e^5*x^2-24*b^2*c*d^3*e^2*m^2+984*b^2*c*d^2*e^3*m*x-720*b^2*
c*d*e^4*x^2-840*b*c^2*d^3*e^2*m*x+720*b*c^2*d^2*e^3*x^2+240*c^3*d^4*e*m*x-240*c^3*d^3*e^2*x^2+1044*a^2*b*e^5*m
-684*a^2*c*d*e^4*m+720*a^2*c*e^5*x-684*a*b^2*d*e^4*m+720*a*b^2*e^5*x+888*a*b*c*d^2*e^3*m-1440*a*b*c*d*e^4*x-26
4*a*c^2*d^3*e^2*m+720*a*c^2*d^2*e^3*x+148*b^3*d^2*e^3*m-240*b^3*d*e^4*x-264*b^2*c*d^3*e^2*m+720*b^2*c*d^2*e^3*
x+120*b*c^2*d^4*e*m-720*b*c^2*d^3*e^2*x+240*c^3*d^4*e*x+720*a^2*b*e^5-720*a^2*c*d*e^4-720*a*b^2*d*e^4+1440*a*b
*c*d^2*e^3-720*a*c^2*d^3*e^2+240*b^3*d^2*e^3-720*b^2*c*d^3*e^2+720*b*c^2*d^4*e-240*c^3*d^5)/e^6/(m^6+21*m^5+17
5*m^4+735*m^3+1624*m^2+1764*m+720)
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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((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.22236, size = 3772, normalized size = 13.97 \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)^2,x, algorithm="fricas")
[Out]
(a^2*b*d*e^5*m^5 - 240*c^3*d^6 + 720*b*c^2*d^5*e + 720*a^2*b*d*e^5 - 720*(b^2*c + a*c^2)*d^4*e^2 + 240*(b^3 +
6*a*b*c)*d^3*e^3 - 720*(a*b^2 + a^2*c)*d^2*e^4 + 2*(c^3*e^6*m^5 + 15*c^3*e^6*m^4 + 85*c^3*e^6*m^3 + 225*c^3*e^
6*m^2 + 274*c^3*e^6*m + 120*c^3*e^6)*x^6 + (720*b*c^2*e^6 + (2*c^3*d*e^5 + 5*b*c^2*e^6)*m^5 + 20*(c^3*d*e^5 +
4*b*c^2*e^6)*m^4 + 5*(14*c^3*d*e^5 + 95*b*c^2*e^6)*m^3 + 100*(c^3*d*e^5 + 13*b*c^2*e^6)*m^2 + 12*(4*c^3*d*e^5
+ 135*b*c^2*e^6)*m)*x^5 + 2*(10*a^2*b*d*e^5 - (a*b^2 + a^2*c)*d^2*e^4)*m^4 + (720*(b^2*c + a*c^2)*e^6 + (5*b*c
^2*d*e^5 + 4*(b^2*c + a*c^2)*e^6)*m^5 - 2*(5*c^3*d^2*e^4 - 30*b*c^2*d*e^5 - 34*(b^2*c + a*c^2)*e^6)*m^4 - (60*
c^3*d^2*e^4 - 235*b*c^2*d*e^5 - 428*(b^2*c + a*c^2)*e^6)*m^3 - 2*(55*c^3*d^2*e^4 - 180*b*c^2*d*e^5 - 614*(b^2*
c + a*c^2)*e^6)*m^2 - 12*(5*c^3*d^2*e^4 - 15*b*c^2*d*e^5 - 132*(b^2*c + a*c^2)*e^6)*m)*x^4 + (155*a^2*b*d*e^5
+ 2*(b^3 + 6*a*b*c)*d^3*e^3 - 36*(a*b^2 + a^2*c)*d^2*e^4)*m^3 + (240*(b^3 + 6*a*b*c)*e^6 + (4*(b^2*c + a*c^2)*
d*e^5 + (b^3 + 6*a*b*c)*e^6)*m^5 - 2*(10*b*c^2*d^2*e^4 - 28*(b^2*c + a*c^2)*d*e^5 - 9*(b^3 + 6*a*b*c)*e^6)*m^4
+ (40*c^3*d^3*e^3 - 180*b*c^2*d^2*e^4 + 260*(b^2*c + a*c^2)*d*e^5 + 121*(b^3 + 6*a*b*c)*e^6)*m^3 + 4*(30*c^3*
d^3*e^3 - 100*b*c^2*d^2*e^4 + 112*(b^2*c + a*c^2)*d*e^5 + 93*(b^3 + 6*a*b*c)*e^6)*m^2 + 4*(20*c^3*d^3*e^3 - 60
*b*c^2*d^2*e^4 + 60*(b^2*c + a*c^2)*d*e^5 + 127*(b^3 + 6*a*b*c)*e^6)*m)*x^3 + 2*(290*a^2*b*d*e^5 - 12*(b^2*c +
a*c^2)*d^4*e^2 + 15*(b^3 + 6*a*b*c)*d^3*e^3 - 119*(a*b^2 + a^2*c)*d^2*e^4)*m^2 + (720*(a*b^2 + a^2*c)*e^6 + (
(b^3 + 6*a*b*c)*d*e^5 + 2*(a*b^2 + a^2*c)*e^6)*m^5 - 2*(6*(b^2*c + a*c^2)*d^2*e^4 - 8*(b^3 + 6*a*b*c)*d*e^5 -
19*(a*b^2 + a^2*c)*e^6)*m^4 + (60*b*c^2*d^3*e^3 - 144*(b^2*c + a*c^2)*d^2*e^4 + 89*(b^3 + 6*a*b*c)*d*e^5 + 274
*(a*b^2 + a^2*c)*e^6)*m^3 - 2*(60*c^3*d^4*e^2 - 210*b*c^2*d^3*e^3 + 246*(b^2*c + a*c^2)*d^2*e^4 - 97*(b^3 + 6*
a*b*c)*d*e^5 - 461*(a*b^2 + a^2*c)*e^6)*m^2 - 12*(10*c^3*d^4*e^2 - 30*b*c^2*d^3*e^3 + 30*(b^2*c + a*c^2)*d^2*e
^4 - 10*(b^3 + 6*a*b*c)*d*e^5 - 117*(a*b^2 + a^2*c)*e^6)*m)*x^2 + 4*(30*b*c^2*d^5*e + 261*a^2*b*d*e^5 - 66*(b^
2*c + a*c^2)*d^4*e^2 + 37*(b^3 + 6*a*b*c)*d^3*e^3 - 171*(a*b^2 + a^2*c)*d^2*e^4)*m + (720*a^2*b*e^6 + (a^2*b*e
^6 + 2*(a*b^2 + a^2*c)*d*e^5)*m^5 + 2*(10*a^2*b*e^6 - (b^3 + 6*a*b*c)*d^2*e^4 + 18*(a*b^2 + a^2*c)*d*e^5)*m^4
+ (155*a^2*b*e^6 + 24*(b^2*c + a*c^2)*d^3*e^3 - 30*(b^3 + 6*a*b*c)*d^2*e^4 + 238*(a*b^2 + a^2*c)*d*e^5)*m^3 -
4*(30*b*c^2*d^4*e^2 - 145*a^2*b*e^6 - 66*(b^2*c + a*c^2)*d^3*e^3 + 37*(b^3 + 6*a*b*c)*d^2*e^4 - 171*(a*b^2 + a
^2*c)*d*e^5)*m^2 + 12*(20*c^3*d^5*e - 60*b*c^2*d^4*e^2 + 87*a^2*b*e^6 + 60*(b^2*c + a*c^2)*d^3*e^3 - 20*(b^3 +
6*a*b*c)*d^2*e^4 + 60*(a*b^2 + a^2*c)*d*e^5)*m)*x)*(e*x + d)^m/(e^6*m^6 + 21*e^6*m^5 + 175*e^6*m^4 + 735*e^6*
m^3 + 1624*e^6*m^2 + 1764*e^6*m + 720*e^6)
________________________________________________________________________________________
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)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.64162, size = 4905, normalized size = 18.17 \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)^2,x, algorithm="giac")
[Out]
(2*(x*e + d)^m*c^3*m^5*x^6*e^6 + 2*(x*e + d)^m*c^3*d*m^5*x^5*e^5 + 5*(x*e + d)^m*b*c^2*m^5*x^5*e^6 + 30*(x*e +
d)^m*c^3*m^4*x^6*e^6 + 5*(x*e + d)^m*b*c^2*d*m^5*x^4*e^5 + 20*(x*e + d)^m*c^3*d*m^4*x^5*e^5 - 10*(x*e + d)^m*
c^3*d^2*m^4*x^4*e^4 + 4*(x*e + d)^m*b^2*c*m^5*x^4*e^6 + 4*(x*e + d)^m*a*c^2*m^5*x^4*e^6 + 80*(x*e + d)^m*b*c^2
*m^4*x^5*e^6 + 170*(x*e + d)^m*c^3*m^3*x^6*e^6 + 4*(x*e + d)^m*b^2*c*d*m^5*x^3*e^5 + 4*(x*e + d)^m*a*c^2*d*m^5
*x^3*e^5 + 60*(x*e + d)^m*b*c^2*d*m^4*x^4*e^5 + 70*(x*e + d)^m*c^3*d*m^3*x^5*e^5 - 20*(x*e + d)^m*b*c^2*d^2*m^
4*x^3*e^4 - 60*(x*e + d)^m*c^3*d^2*m^3*x^4*e^4 + 40*(x*e + d)^m*c^3*d^3*m^3*x^3*e^3 + (x*e + d)^m*b^3*m^5*x^3*
e^6 + 6*(x*e + d)^m*a*b*c*m^5*x^3*e^6 + 68*(x*e + d)^m*b^2*c*m^4*x^4*e^6 + 68*(x*e + d)^m*a*c^2*m^4*x^4*e^6 +
475*(x*e + d)^m*b*c^2*m^3*x^5*e^6 + 450*(x*e + d)^m*c^3*m^2*x^6*e^6 + (x*e + d)^m*b^3*d*m^5*x^2*e^5 + 6*(x*e +
d)^m*a*b*c*d*m^5*x^2*e^5 + 56*(x*e + d)^m*b^2*c*d*m^4*x^3*e^5 + 56*(x*e + d)^m*a*c^2*d*m^4*x^3*e^5 + 235*(x*e
+ d)^m*b*c^2*d*m^3*x^4*e^5 + 100*(x*e + d)^m*c^3*d*m^2*x^5*e^5 - 12*(x*e + d)^m*b^2*c*d^2*m^4*x^2*e^4 - 12*(x
*e + d)^m*a*c^2*d^2*m^4*x^2*e^4 - 180*(x*e + d)^m*b*c^2*d^2*m^3*x^3*e^4 - 110*(x*e + d)^m*c^3*d^2*m^2*x^4*e^4
+ 60*(x*e + d)^m*b*c^2*d^3*m^3*x^2*e^3 + 120*(x*e + d)^m*c^3*d^3*m^2*x^3*e^3 - 120*(x*e + d)^m*c^3*d^4*m^2*x^2
*e^2 + 2*(x*e + d)^m*a*b^2*m^5*x^2*e^6 + 2*(x*e + d)^m*a^2*c*m^5*x^2*e^6 + 18*(x*e + d)^m*b^3*m^4*x^3*e^6 + 10
8*(x*e + d)^m*a*b*c*m^4*x^3*e^6 + 428*(x*e + d)^m*b^2*c*m^3*x^4*e^6 + 428*(x*e + d)^m*a*c^2*m^3*x^4*e^6 + 1300
*(x*e + d)^m*b*c^2*m^2*x^5*e^6 + 548*(x*e + d)^m*c^3*m*x^6*e^6 + 2*(x*e + d)^m*a*b^2*d*m^5*x*e^5 + 2*(x*e + d)
^m*a^2*c*d*m^5*x*e^5 + 16*(x*e + d)^m*b^3*d*m^4*x^2*e^5 + 96*(x*e + d)^m*a*b*c*d*m^4*x^2*e^5 + 260*(x*e + d)^m
*b^2*c*d*m^3*x^3*e^5 + 260*(x*e + d)^m*a*c^2*d*m^3*x^3*e^5 + 360*(x*e + d)^m*b*c^2*d*m^2*x^4*e^5 + 48*(x*e + d
)^m*c^3*d*m*x^5*e^5 - 2*(x*e + d)^m*b^3*d^2*m^4*x*e^4 - 12*(x*e + d)^m*a*b*c*d^2*m^4*x*e^4 - 144*(x*e + d)^m*b
^2*c*d^2*m^3*x^2*e^4 - 144*(x*e + d)^m*a*c^2*d^2*m^3*x^2*e^4 - 400*(x*e + d)^m*b*c^2*d^2*m^2*x^3*e^4 - 60*(x*e
+ d)^m*c^3*d^2*m*x^4*e^4 + 24*(x*e + d)^m*b^2*c*d^3*m^3*x*e^3 + 24*(x*e + d)^m*a*c^2*d^3*m^3*x*e^3 + 420*(x*e
+ d)^m*b*c^2*d^3*m^2*x^2*e^3 + 80*(x*e + d)^m*c^3*d^3*m*x^3*e^3 - 120*(x*e + d)^m*b*c^2*d^4*m^2*x*e^2 - 120*(
x*e + d)^m*c^3*d^4*m*x^2*e^2 + 240*(x*e + d)^m*c^3*d^5*m*x*e + (x*e + d)^m*a^2*b*m^5*x*e^6 + 38*(x*e + d)^m*a*
b^2*m^4*x^2*e^6 + 38*(x*e + d)^m*a^2*c*m^4*x^2*e^6 + 121*(x*e + d)^m*b^3*m^3*x^3*e^6 + 726*(x*e + d)^m*a*b*c*m
^3*x^3*e^6 + 1228*(x*e + d)^m*b^2*c*m^2*x^4*e^6 + 1228*(x*e + d)^m*a*c^2*m^2*x^4*e^6 + 1620*(x*e + d)^m*b*c^2*
m*x^5*e^6 + 240*(x*e + d)^m*c^3*x^6*e^6 + (x*e + d)^m*a^2*b*d*m^5*e^5 + 36*(x*e + d)^m*a*b^2*d*m^4*x*e^5 + 36*
(x*e + d)^m*a^2*c*d*m^4*x*e^5 + 89*(x*e + d)^m*b^3*d*m^3*x^2*e^5 + 534*(x*e + d)^m*a*b*c*d*m^3*x^2*e^5 + 448*(
x*e + d)^m*b^2*c*d*m^2*x^3*e^5 + 448*(x*e + d)^m*a*c^2*d*m^2*x^3*e^5 + 180*(x*e + d)^m*b*c^2*d*m*x^4*e^5 - 2*(
x*e + d)^m*a*b^2*d^2*m^4*e^4 - 2*(x*e + d)^m*a^2*c*d^2*m^4*e^4 - 30*(x*e + d)^m*b^3*d^2*m^3*x*e^4 - 180*(x*e +
d)^m*a*b*c*d^2*m^3*x*e^4 - 492*(x*e + d)^m*b^2*c*d^2*m^2*x^2*e^4 - 492*(x*e + d)^m*a*c^2*d^2*m^2*x^2*e^4 - 24
0*(x*e + d)^m*b*c^2*d^2*m*x^3*e^4 + 2*(x*e + d)^m*b^3*d^3*m^3*e^3 + 12*(x*e + d)^m*a*b*c*d^3*m^3*e^3 + 264*(x*
e + d)^m*b^2*c*d^3*m^2*x*e^3 + 264*(x*e + d)^m*a*c^2*d^3*m^2*x*e^3 + 360*(x*e + d)^m*b*c^2*d^3*m*x^2*e^3 - 24*
(x*e + d)^m*b^2*c*d^4*m^2*e^2 - 24*(x*e + d)^m*a*c^2*d^4*m^2*e^2 - 720*(x*e + d)^m*b*c^2*d^4*m*x*e^2 + 120*(x*
e + d)^m*b*c^2*d^5*m*e - 240*(x*e + d)^m*c^3*d^6 + 20*(x*e + d)^m*a^2*b*m^4*x*e^6 + 274*(x*e + d)^m*a*b^2*m^3*
x^2*e^6 + 274*(x*e + d)^m*a^2*c*m^3*x^2*e^6 + 372*(x*e + d)^m*b^3*m^2*x^3*e^6 + 2232*(x*e + d)^m*a*b*c*m^2*x^3
*e^6 + 1584*(x*e + d)^m*b^2*c*m*x^4*e^6 + 1584*(x*e + d)^m*a*c^2*m*x^4*e^6 + 720*(x*e + d)^m*b*c^2*x^5*e^6 + 2
0*(x*e + d)^m*a^2*b*d*m^4*e^5 + 238*(x*e + d)^m*a*b^2*d*m^3*x*e^5 + 238*(x*e + d)^m*a^2*c*d*m^3*x*e^5 + 194*(x
*e + d)^m*b^3*d*m^2*x^2*e^5 + 1164*(x*e + d)^m*a*b*c*d*m^2*x^2*e^5 + 240*(x*e + d)^m*b^2*c*d*m*x^3*e^5 + 240*(
x*e + d)^m*a*c^2*d*m*x^3*e^5 - 36*(x*e + d)^m*a*b^2*d^2*m^3*e^4 - 36*(x*e + d)^m*a^2*c*d^2*m^3*e^4 - 148*(x*e
+ d)^m*b^3*d^2*m^2*x*e^4 - 888*(x*e + d)^m*a*b*c*d^2*m^2*x*e^4 - 360*(x*e + d)^m*b^2*c*d^2*m*x^2*e^4 - 360*(x*
e + d)^m*a*c^2*d^2*m*x^2*e^4 + 30*(x*e + d)^m*b^3*d^3*m^2*e^3 + 180*(x*e + d)^m*a*b*c*d^3*m^2*e^3 + 720*(x*e +
d)^m*b^2*c*d^3*m*x*e^3 + 720*(x*e + d)^m*a*c^2*d^3*m*x*e^3 - 264*(x*e + d)^m*b^2*c*d^4*m*e^2 - 264*(x*e + d)^
m*a*c^2*d^4*m*e^2 + 720*(x*e + d)^m*b*c^2*d^5*e + 155*(x*e + d)^m*a^2*b*m^3*x*e^6 + 922*(x*e + d)^m*a*b^2*m^2*
x^2*e^6 + 922*(x*e + d)^m*a^2*c*m^2*x^2*e^6 + 508*(x*e + d)^m*b^3*m*x^3*e^6 + 3048*(x*e + d)^m*a*b*c*m*x^3*e^6
+ 720*(x*e + d)^m*b^2*c*x^4*e^6 + 720*(x*e + d)^m*a*c^2*x^4*e^6 + 155*(x*e + d)^m*a^2*b*d*m^3*e^5 + 684*(x*e
+ d)^m*a*b^2*d*m^2*x*e^5 + 684*(x*e + d)^m*a^2*c*d*m^2*x*e^5 + 120*(x*e + d)^m*b^3*d*m*x^2*e^5 + 720*(x*e + d)
^m*a*b*c*d*m*x^2*e^5 - 238*(x*e + d)^m*a*b^2*d^2*m^2*e^4 - 238*(x*e + d)^m*a^2*c*d^2*m^2*e^4 - 240*(x*e + d)^m
*b^3*d^2*m*x*e^4 - 1440*(x*e + d)^m*a*b*c*d^2*m*x*e^4 + 148*(x*e + d)^m*b^3*d^3*m*e^3 + 888*(x*e + d)^m*a*b*c*
d^3*m*e^3 - 720*(x*e + d)^m*b^2*c*d^4*e^2 - 720*(x*e + d)^m*a*c^2*d^4*e^2 + 580*(x*e + d)^m*a^2*b*m^2*x*e^6 +
1404*(x*e + d)^m*a*b^2*m*x^2*e^6 + 1404*(x*e + d)^m*a^2*c*m*x^2*e^6 + 240*(x*e + d)^m*b^3*x^3*e^6 + 1440*(x*e
+ d)^m*a*b*c*x^3*e^6 + 580*(x*e + d)^m*a^2*b*d*m^2*e^5 + 720*(x*e + d)^m*a*b^2*d*m*x*e^5 + 720*(x*e + d)^m*a^2
*c*d*m*x*e^5 - 684*(x*e + d)^m*a*b^2*d^2*m*e^4 - 684*(x*e + d)^m*a^2*c*d^2*m*e^4 + 240*(x*e + d)^m*b^3*d^3*e^3
+ 1440*(x*e + d)^m*a*b*c*d^3*e^3 + 1044*(x*e + d)^m*a^2*b*m*x*e^6 + 720*(x*e + d)^m*a*b^2*x^2*e^6 + 720*(x*e
+ d)^m*a^2*c*x^2*e^6 + 1044*(x*e + d)^m*a^2*b*d*m*e^5 - 720*(x*e + d)^m*a*b^2*d^2*e^4 - 720*(x*e + d)^m*a^2*c*
d^2*e^4 + 720*(x*e + d)^m*a^2*b*x*e^6 + 720*(x*e + d)^m*a^2*b*d*e^5)/(m^6*e^6 + 21*m^5*e^6 + 175*m^4*e^6 + 735
*m^3*e^6 + 1624*m^2*e^6 + 1764*m*e^6 + 720*e^6)