\(\int \frac {1}{(d+e x)^3 (a+b x+c x^2)^{5/2}} \, dx\) [2399]

Optimal result

Integrand size = 22, antiderivative size = 621 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {5 e^4 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{9/2}} \]

[Out]

-2/3*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2/(c*x^2+b*x+a)^(3/2)+5/8
*e^4*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*d))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)
/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(9/2)-2/3*(8*a*c*e*(-b*e+2*c*d)^2-(2*a*c*e-b^2*e+b*c*d)*(20*a*c*e^2-
7*b^2*e^2+8*c^2*d^2)-c*(-b*e+2*c*d)*(8*c^2*d^2-7*b^2*e^2-4*c*e*(-9*a*e+2*b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+
c*d^2)^2/(e*x+d)^2/(c*x^2+b*x+a)^(1/2)+1/6*e*(64*c^4*d^4-35*b^4*e^4-128*c^3*d^2*e*(-3*a*e+b*d)-48*a*c^2*e^3*(5
*a*e+8*b*d)+8*b^2*c*e^3*(27*a*e+8*b*d))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^2+1/1
2*e*(-b*e+2*c*d)*(64*c^4*d^4-105*b^4*e^4-64*c^3*d^2*e*(-7*a*e+2*b*d)+40*b^2*c*e^3*(19*a*e+2*b*d)-16*c^2*e^2*(8
1*a^2*e^2+28*a*b*d*e+b^2*d^2))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^4/(e*x+d)

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {754, 836, 848, 820, 738, 212} \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {e \sqrt {a+b x+c x^2} (2 c d-b e) \left (-16 c^2 e^2 \left (81 a^2 e^2+28 a b d e+b^2 d^2\right )+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right )}{12 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^4}+\frac {5 e^4 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-9 a e)-7 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (20 a c e^2-7 b^2 e^2+8 c^2 d^2\right )+8 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac {e \sqrt {a+b x+c x^2} \left (8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right )}{6 \left (b^2-4 a c\right )^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3} \]

[In]

Int[1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*(a + b
*x + c*x^2)^(3/2)) - (2*(8*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 7*b^2*e^2 + 20*a*c*e
^2) - c*(2*c*d - b*e)*(8*c^2*d^2 - 7*b^2*e^2 - 4*c*e*(2*b*d - 9*a*e))*x))/(3*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e +
a*e^2)^2*(d + e*x)^2*Sqrt[a + b*x + c*x^2]) + (e*(64*c^4*d^4 - 35*b^4*e^4 - 128*c^3*d^2*e*(b*d - 3*a*e) - 48*a
*c^2*e^3*(8*b*d + 5*a*e) + 8*b^2*c*e^3*(8*b*d + 27*a*e))*Sqrt[a + b*x + c*x^2])/(6*(b^2 - 4*a*c)^2*(c*d^2 - b*
d*e + a*e^2)^3*(d + e*x)^2) + (e*(2*c*d - b*e)*(64*c^4*d^4 - 105*b^4*e^4 - 64*c^3*d^2*e*(2*b*d - 7*a*e) + 40*b
^2*c*e^3*(2*b*d + 19*a*e) - 16*c^2*e^2*(b^2*d^2 + 28*a*b*d*e + 81*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(12*(b^2 -
4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)) + (5*e^4*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[
(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*(c*d^2 - b*d*e + a*
e^2)^(9/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), 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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )+4 c e (2 c d-b e) x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {\frac {1}{4} e \left (8 c e (2 c d-b e) \left (b^2 d-12 a c d+4 a b e\right )+\left (4 b c d-5 b^2 e+12 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )\right )+c e (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {2 \int \frac {\frac {1}{8} \left (-220 b^4 c d e^4+105 b^5 e^5+8 b^3 c e^3 \left (6 c d^2-95 a e^2\right )+64 a c^3 d e^2 \left (2 c d^2-33 a e^2\right )+96 b^2 c^2 d e^2 \left (c d^2+16 a e^2\right )-16 b c^2 e \left (4 c^2 d^4+36 a c d^2 e^2-81 a^2 e^4\right )\right )-\frac {1}{4} c e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {\left (5 e^4 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^4} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4 (d+e x)}-\frac {\left (5 e^4 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{4 \left (c d^2-b d e+a e^2\right )^4} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {5 e^4 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.59 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (\frac {b^2 e-2 c (a e+c d x)+b c (-d+e x)}{(d+e x)^2 (a+x (b+c x))^{3/2}}+\frac {-7 b^4 e^3+7 b^3 c e^2 (d-e x)-4 b c^2 \left (a e^2 (13 d-9 e x)+2 c d^2 (d-3 e x)\right )+2 b^2 c e \left (21 a e^2+c d (4 d+3 e x)\right )-8 c^2 \left (5 a^2 e^3+2 c^2 d^3 x+a c d e (-2 d+9 e x)\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (d+e x)^2 \sqrt {a+x (b+c x)}}+\frac {e \left (\frac {4 \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+x (b+c x)}}{(d+e x)^2}+\frac {2 (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+x (b+c x)}}{\left (c d^2+e (-b d+a e)\right ) (d+e x)}-\frac {15 \left (b^2-4 a c\right )^2 e^3 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{3/2}}\right )}{16 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \]

[In]

Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*((b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)^2*(a + x*(b + c*x))^(3/2)) + (-7*b^4*e^3 + 7*b^3*c
*e^2*(d - e*x) - 4*b*c^2*(a*e^2*(13*d - 9*e*x) + 2*c*d^2*(d - 3*e*x)) + 2*b^2*c*e*(21*a*e^2 + c*d*(4*d + 3*e*x
)) - 8*c^2*(5*a^2*e^3 + 2*c^2*d^3*x + a*c*d*e*(-2*d + 9*e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(d +
e*x)^2*Sqrt[a + x*(b + c*x)]) + (e*((4*(64*c^4*d^4 - 35*b^4*e^4 - 128*c^3*d^2*e*(b*d - 3*a*e) - 48*a*c^2*e^3*(
8*b*d + 5*a*e) + 8*b^2*c*e^3*(8*b*d + 27*a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^2 + (2*(2*c*d - b*e)*(64*c^4*d
^4 - 105*b^4*e^4 - 64*c^3*d^2*e*(2*b*d - 7*a*e) + 40*b^2*c*e^3*(2*b*d + 19*a*e) - 16*c^2*e^2*(b^2*d^2 + 28*a*b
*d*e + 81*a^2*e^2))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) - (15*(b^2 - 4*a*c)^2*e^3*(2
4*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b
*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/(16*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) +
a*e))^2)))/(3*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2042\) vs. \(2(595)=1190\).

Time = 0.49 (sec) , antiderivative size = 2043, normalized size of antiderivative = 3.29

method	result	size
default	\(\text {Expression too large to display}\)	\(2043\)

[In]

int(1/(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/e^3*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2
)-7/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)
+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-5/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1/3/(a*e^2-b*d*e+c*d^2)*e^2/((x+d/e)^2*
c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2/3*(2*c*(x+d/e)
+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*
e+c*d^2)/e^2)^(3/2)+16/3*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^2*(2*c*(x+d/e)+(b*e-2*c*d)/e)/((x+d
/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))+1/(a*e^2-b*d*e+c*d^2)*e^2*(1/(a*e^2-b*d*e+c*d^2)
*e^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*
(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e
^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)
/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c
*d^2)/e^2)^(1/2))/(x+d/e))))-4*c/(a*e^2-b*d*e+c*d^2)*e^2*(2/3*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*
d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+16/3*c/(4*c*(a*e
^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^2*(2*c*(x+d/e)+(b*e-2*c*d)/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2
-b*d*e+c*d^2)/e^2)^(1/2)))-5/2*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/3/(a*e^2-b*d*e+c*d^2)*e^2/((x+d/e)^2*c+(b*e-2*c*d)
/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2/3*(2*c*(x+d/e)+(b*e-2*c*d)/
e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)
^(3/2)+16/3*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^2*(2*c*(x+d/e)+(b*e-2*c*d)/e)/((x+d/e)^2*c+(b*e-
2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))+1/(a*e^2-b*d*e+c*d^2)*e^2*(1/(a*e^2-b*d*e+c*d^2)*e^2/((x+d/e)
^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-
2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c
*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1
/2))/(x+d/e)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7199 vs. \(2 (595) = 1190\).

Time = 37.63 (sec) , antiderivative size = 14440, normalized size of antiderivative = 23.25 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(e*x+d)**3/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/((d + e*x)**3*(a + b*x + c*x**2)**(5/2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26254 vs. \(2 (595) = 1190\).

Time = 1.00 (sec) , antiderivative size = 26254, normalized size of antiderivative = 42.28 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

5/4*(24*c^2*d^2*e^4 - 24*b*c*d*e^5 + 7*b^2*e^6 - 4*a*c*e^6)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + s
qrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^
3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12
*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*sqrt(-c*d^2 + b*d*e - a*e^2)
) + 2/3*((((16*c^19*d^29 - 232*b*c^18*d^28*e + 1548*b^2*c^17*d^27*e^2 + 304*a*c^18*d^27*e^2 - 6282*b^3*c^16*d^
26*e^3 - 4104*a*b*c^17*d^26*e^3 + 17220*b^4*c^15*d^25*e^4 + 25572*a*b^2*c^16*d^25*e^4 + 2208*a^2*c^17*d^25*e^4
- 33315*b^5*c^14*d^24*e^5 - 97350*a*b^3*c^15*d^24*e^5 - 27600*a^2*b*c^16*d^24*e^5 + 45608*b^6*c^13*d^23*e^6 +
252264*a*b^4*c^14*d^23*e^6 + 159144*a^2*b^2*c^15*d^23*e^6 + 8608*a^3*c^16*d^23*e^6 - 41492*b^7*c^12*d^22*e^7
- 468096*a*b^5*c^13*d^22*e^7 - 560556*a^2*b^3*c^14*d^22*e^7 - 98992*a^3*b*c^15*d^22*e^7 + 17424*b^8*c^11*d^21*
e^8 + 634040*a*b^6*c^12*d^21*e^8 + 1344816*a^2*b^4*c^13*d^21*e^8 + 524568*a^3*b^2*c^14*d^21*e^8 + 19888*a^4*c^
15*d^21*e^8 + 14135*b^9*c^10*d^20*e^9 - 620334*a*b^7*c^11*d^20*e^9 - 2315082*a^2*b^5*c^12*d^20*e^9 - 1696772*a
^3*b^3*c^13*d^20*e^9 - 208824*a^4*b*c^14*d^20*e^9 - 34628*b^10*c^9*d^19*e^10 + 409860*a*b^8*c^10*d^19*e^10 + 2
924460*a^2*b^6*c^11*d^19*e^10 + 3736040*a^3*b^4*c^12*d^19*e^10 + 1011780*a^4*b^2*c^13*d^19*e^10 + 25872*a^5*c^
14*d^19*e^10 + 35910*b^11*c^8*d^18*e^11 - 132088*a*b^9*c^9*d^18*e^11 - 2704878*a^2*b^7*c^10*d^18*e^11 - 589881
6*a^3*b^5*c^11*d^18*e^11 - 2999150*a^4*b^3*c^12*d^18*e^11 - 245784*a^5*b*c^13*d^18*e^11 - 24540*b^12*c^7*d^17*
e^12 - 57420*a*b^10*c^8*d^17*e^12 + 1762992*a^2*b^8*c^9*d^17*e^12 + 6826644*a^3*b^6*c^10*d^17*e^12 + 6064740*a
^4*b^4*c^11*d^17*e^12 + 1093356*a^5*b^2*c^12*d^17*e^12 + 8448*a^6*c^13*d^17*e^12 + 11883*b^13*c^6*d^16*e^13 +
108222*a*b^11*c^7*d^16*e^13 - 702372*a^2*b^9*c^8*d^16*e^13 - 5776056*a^3*b^7*c^9*d^16*e^13 - 8797041*a^4*b^5*c
^10*d^16*e^13 - 3026034*a^5*b^3*c^11*d^16*e^13 - 71808*a^6*b*c^12*d^16*e^13 - 4080*b^14*c^5*d^15*e^14 - 75888*
a*b^12*c^6*d^15*e^14 + 44880*a^2*b^10*c^7*d^15*e^14 + 3446784*a^3*b^8*c^8*d^15*e^14 + 9317088*a^4*b^6*c^9*d^15
*e^14 + 5789520*a^5*b^4*c^10*d^15*e^14 + 350064*a^6*b^2*c^11*d^15*e^14 - 35904*a^7*c^12*d^15*e^14 + 952*b^15*c
^4*d^14*e^15 + 32640*a*b^13*c^5*d^14*e^15 + 144840*a^2*b^11*c^6*d^14*e^15 - 1286560*a^3*b^9*c^7*d^14*e^15 - 71
35920*a^4*b^7*c^8*d^14*e^15 - 7970688*a^5*b^5*c^9*d^14*e^15 - 1189320*a^6*b^3*c^10*d^14*e^15 + 269280*a^7*b*c^
11*d^14*e^15 - 136*b^16*c^3*d^13*e^16 - 8976*a*b^14*c^4*d^13*e^16 - 102816*a^2*b^12*c^5*d^13*e^16 + 146608*a^3
*b^10*c^6*d^13*e^16 + 3769920*a^4*b^8*c^7*d^13*e^16 + 7916832*a^5*b^6*c^8*d^13*e^16 + 2764608*a^6*b^4*c^9*d^13
*e^16 - 780912*a^7*b^2*c^10*d^13*e^16 - 80784*a^8*c^11*d^13*e^16 + 9*b^17*c^2*d^12*e^17 + 1462*a*b^15*c^3*d^12
*e^17 + 36414*a^2*b^13*c^4*d^12*e^17 + 129948*a^3*b^11*c^5*d^12*e^17 - 1191190*a^4*b^9*c^6*d^12*e^17 - 5513508
*a^5*b^7*c^7*d^12*e^17 - 4288284*a^6*b^5*c^8*d^12*e^17 + 991848*a^7*b^3*c^9*d^12*e^17 + 525096*a^8*b*c^10*d^12
*e^17 - 108*a*b^16*c^2*d^11*e^18 - 7044*a^2*b^14*c^3*d^11*e^18 - 79912*a^3*b^12*c^4*d^11*e^18 + 89628*a^4*b^10
*c^5*d^11*e^18 + 2500344*a^5*b^8*c^6*d^11*e^18 + 4359432*a^6*b^6*c^7*d^11*e^18 - 121968*a^7*b^4*c^8*d^11*e^18
- 1365804*a^8*b^2*c^9*d^11*e^18 - 93104*a^9*c^10*d^11*e^18 + 594*a^2*b^15*c^2*d^10*e^19 + 19888*a^3*b^13*c^3*d
^10*e^19 + 90486*a^4*b^11*c^4*d^10*e^19 - 595320*a^5*b^9*c^5*d^10*e^19 - 2798004*a^6*b^7*c^6*d^10*e^19 - 12545
28*a^7*b^5*c^7*d^10*e^19 + 1735866*a^8*b^3*c^8*d^10*e^19 + 512072*a^9*b*c^9*d^10*e^19 - 1980*a^3*b^14*c^2*d^9*
e^20 - 35860*a^4*b^12*c^3*d^9*e^20 - 8844*a^5*b^10*c^4*d^9*e^20 + 1021680*a^6*b^8*c^5*d^9*e^20 + 1661880*a^7*b
^6*c^6*d^9*e^20 - 924660*a^8*b^4*c^7*d^9*e^20 - 1106820*a^9*b^2*c^8*d^9*e^20 - 69344*a^10*c^9*d^9*e^20 + 4455*
a^4*b^13*c^2*d^8*e^21 + 41382*a^5*b^11*c^3*d^8*e^21 - 138468*a^6*b^9*c^4*d^8*e^21 - 957528*a^7*b^7*c^5*d^8*e^2
1 - 193941*a^8*b^5*c^6*d^8*e^21 + 1140150*a^9*b^3*c^7*d^8*e^21 + 312048*a^10*b*c^8*d^8*e^21 - 7128*a^5*b^12*c^
2*d^7*e^22 - 26664*a^6*b^10*c^3*d^7*e^22 + 234432*a^7*b^8*c^4*d^7*e^22 + 488664*a^8*b^6*c^5*d^7*e^22 - 479160*
a^9*b^4*c^6*d^7*e^22 - 528792*a^10*b^2*c^7*d^7*e^22 - 34656*a^11*c^8*d^7*e^22 + 8316*a^6*b^11*c^2*d^6*e^23 + 5
28*a^7*b^9*c^3*d^6*e^23 - 206316*a^8*b^7*c^4*d^6*e^23 - 59136*a^9*b^5*c^5*d^6*e^23 + 394548*a^10*b^3*c^6*d^6*e
^23 + 121296*a^11*b*c^7*d^6*e^23 - 7128*a^7*b^10*c^2*d^5*e^24 + 17424*a^8*b^8*c^3*d^5*e^24 + 106568*a^9*b^6*c^
4*d^5*e^24 - 92400*a^10*b^4*c^5*d^5*e^24 - 148008*a^11*b^2*c^6*d^5*e^24 - 11312*a^12*c^7*d^5*e^24 + 4455*a^8*b
^9*c^2*d^4*e^25 - 18590*a^9*b^7*c^3*d^4*e^25 - 27258*a^10*b^5*c^4*d^4*e^25 + 66780*a^11*b^3*c^5*d^4*e^25 + 282
80*a^12*b*c^6*d^4*e^25 - 1980*a^9*b^8*c^2*d^3*e^26 + 10604*a^10*b^6*c^3*d^3*e^26 - 1656*a^11*b^4*c^4*d^3*e^26
- 21156*a^12*b^2*c^5*d^3*e^26 - 2192*a^13*c^6*d^3*e^26 + 594*a^10*b^7*c^2*d^2*e^27 - 3648*a^11*b^5*c^3*d^2*e^2
7 + 3454*a^12*b^3*c^4*d^2*e^27 + 3288*a^13*b*c^5*d^2*e^27 - 108*a^11*b^6*c^2*d*e^28 + 716*a^12*b^4*c^3*d*e^28
- 972*a^13*b^2*c^4*d*e^28 - 192*a^14*c^5*d*e^28 + 9*a^12*b^5*c^2*e^29 - 62*a^13*b^3*c^3*e^29 + 96*a^14*b*c^4*e
^29)*x/(b^4*c^16*d^32 - 8*a*b^2*c^17*d^32 + 16*a^2*c^18*d^32 - 16*b^5*c^15*d^31*e + 128*a*b^3*c^16*d^31*e - 25
6*a^2*b*c^17*d^31*e + 120*b^6*c^14*d^30*e^2 - 944*a*b^4*c^15*d^30*e^2 + 1792*a^2*b^2*c^16*d^30*e^2 + 256*a^3*c
^17*d^30*e^2 - 560*b^7*c^13*d^29*e^3 + 4240*a*b^5*c^14*d^29*e^3 - 7040*a^2*b^3*c^15*d^29*e^3 - 3840*a^3*b*c^16
*d^29*e^3 + 1820*b^8*c^12*d^28*e^4 - 12880*a*b^6*c^13*d^28*e^4 + 15800*a^2*b^4*c^14*d^28*e^4 + 25920*a^3*b^2*c
^15*d^28*e^4 + 1920*a^4*c^16*d^28*e^4 - 4368*b^9*c^11*d^27*e^5 + 27664*a*b^7*c^12*d^27*e^5 - 13328*a^2*b^5*c^1
3*d^27*e^5 - 103040*a^3*b^3*c^14*d^27*e^5 - 26880*a^4*b*c^15*d^27*e^5 + 8008*b^10*c^10*d^26*e^6 - 42224*a*b^8*
c^11*d^26*e^6 - 35672*a^2*b^6*c^12*d^26*e^6 + 262640*a^3*b^4*c^13*d^26*e^6 + 170240*a^4*b^2*c^14*d^26*e^6 + 89
60*a^5*c^15*d^26*e^6 - 11440*b^11*c^9*d^25*e^7 + 43472*a*b^9*c^10*d^25*e^7 + 157664*a^2*b^7*c^11*d^25*e^7 - 42
6608*a^3*b^5*c^12*d^25*e^7 - 640640*a^4*b^3*c^13*d^25*e^7 - 116480*a^5*b*c^14*d^25*e^7 + 12870*b^12*c^8*d^24*e
^8 - 22880*a*b^10*c^9*d^24*e^8 - 314600*a^2*b^8*c^10*d^24*e^8 + 364000*a^3*b^6*c^11*d^24*e^8 + 1574300*a^4*b^4
*c^12*d^24*e^8 + 684320*a^5*b^2*c^13*d^24*e^8 + 29120*a^6*c^14*d^24*e^8 - 11440*b^13*c^7*d^23*e^9 - 11440*a*b^
11*c^8*d^23*e^9 + 400400*a^2*b^9*c^9*d^23*e^9 + 114400*a^3*b^7*c^10*d^23*e^9 - 2584400*a^4*b^5*c^11*d^23*e^9 -
2387840*a^5*b^3*c^12*d^23*e^9 - 349440*a^6*b*c^13*d^23*e^9 + 8008*b^14*c^6*d^22*e^10 + 38896*a*b^12*c^7*d^22*
e^10 - 335192*a^2*b^10*c^8*d^22*e^10 - 835120*a^3*b^8*c^9*d^22*e^10 + 2682680*a^4*b^6*c^10*d^22*e^10 + 5449808
*a^5*b^4*c^11*d^22*e^10 + 1886976*a^6*b^2*c^12*d^22*e^10 + 69888*a^7*c^13*d^22*e^10 - 4368*b^15*c^5*d^21*e^11
- 45136*a*b^13*c^6*d^21*e^11 + 158912*a^2*b^11*c^7*d^21*e^11 + 1292720*a^3*b^9*c^8*d^21*e^11 - 1224080*a^4*b^7
*c^9*d^21*e^11 - 8376368*a^5*b^5*c^10*d^21*e^11 - 6022016*a^6*b^3*c^11*d^21*e^11 - 768768*a^7*b*c^12*d^21*e^11
+ 1820*b^16*c^4*d^20*e^12 + 33488*a*b^14*c^5*d^20*e^12 + 5096*a^2*b^12*c^6*d^20*e^12 - 1153152*a^3*b^10*c^7*d
^20*e^12 - 1021020*a^4*b^8*c^8*d^20*e^12 + 8408400*a^5*b^6*c^9*d^20*e^12 + 12500488*a^6*b^4*c^10*d^20*e^12 + 3
779776*a^7*b^2*c^11*d^20*e^12 + 128128*a^8*c^12*d^20*e^12 - 560*b^17*c^3*d^19*e^13 - 17360*a*b^15*c^4*d^19*e^1
3 - 74480*a^2*b^13*c^5*d^19*e^13 + 611520*a^3*b^11*c^6*d^19*e^13 + 2402400*a^4*b^9*c^7*d^19*e^13 - 4564560*a^5
*b^7*c^8*d^19*e^13 - 17377360*a^6*b^5*c^9*d^19*e^13 - 10890880*a^7*b^3*c^10*d^19*e^13 - 1281280*a^8*b*c^11*d^1
9*e^13 + 120*b^18*c^2*d^18*e^14 + 6320*a*b^16*c^3*d^18*e^14 + 63800*a^2*b^14*c^4*d^18*e^14 - 123760*a^3*b^12*c
^5*d^18*e^14 - 2162160*a^4*b^10*c^6*d^18*e^14 - 480480*a^5*b^8*c^7*d^18*e^14 + 15735720*a^6*b^6*c^8*d^18*e^14
+ 20191600*a^7*b^4*c^9*d^18*e^14 + 5674240*a^8*b^2*c^10*d^18*e^14 + 183040*a^9*c^11*d^18*e^14 - 16*b^19*c*d^17
*e^15 - 1552*a*b^17*c^2*d^17*e^15 - 30496*a^2*b^15*c^3*d^17*e^15 - 77840*a^3*b^13*c^4*d^17*e^15 + 1062880*a^4*
b^11*c^5*d^17*e^15 + 3107104*a^5*b^9*c^6*d^17*e^15 - 7879872*a^6*b^7*c^7*d^17*e^15 - 24703536*a^7*b^5*c^8*d^17
*e^15 - 14551680*a^8*b^3*c^9*d^17*e^15 - 1647360*a^9*b*c^10*d^17*e^15 + b^20*d^16*e^16 + 232*a*b^18*c*d^16*e^1
6 + 9016*a^2*b^16*c^2*d^16*e^16 + 76640*a^3*b^14*c^3*d^16*e^16 - 205660*a^4*b^12*c^4*d^16*e^16 - 2626624*a^5*b
^10*c^5*d^16*e^16 - 48048*a^6*b^8*c^6*d^16*e^16 + 19246656*a^7*b^6*c^7*d^16*e^16 + 23625030*a^8*b^4*c^8*d^16*e
^16 + 6486480*a^9*b^2*c^9*d^16*e^16 + 205920*a^10*c^10*d^16*e^16 - 16*a*b^19*d^15*e^17 - 1552*a^2*b^17*c*d^15*
e^17 - 30496*a^3*b^15*c^2*d^15*e^17 - 77840*a^4*b^13*c^3*d^15*e^17 + 1062880*a^5*b^11*c^4*d^15*e^17 + 3107104*
a^6*b^9*c^5*d^15*e^17 - 7879872*a^7*b^7*c^6*d^15*e^17 - 24703536*a^8*b^5*c^7*d^15*e^17 - 14551680*a^9*b^3*c^8*
d^15*e^17 - 1647360*a^10*b*c^9*d^15*e^17 + 120*a^2*b^18*d^14*e^18 + 6320*a^3*b^16*c*d^14*e^18 + 63800*a^4*b^14
*c^2*d^14*e^18 - 123760*a^5*b^12*c^3*d^14*e^18 - 2162160*a^6*b^10*c^4*d^14*e^18 - 480480*a^7*b^8*c^5*d^14*e^18
+ 15735720*a^8*b^6*c^6*d^14*e^18 + 20191600*a^9*b^4*c^7*d^14*e^18 + 5674240*a^10*b^2*c^8*d^14*e^18 + 183040*a
^11*c^9*d^14*e^18 - 560*a^3*b^17*d^13*e^19 - 17360*a^4*b^15*c*d^13*e^19 - 74480*a^5*b^13*c^2*d^13*e^19 + 61152
0*a^6*b^11*c^3*d^13*e^19 + 2402400*a^7*b^9*c^4*d^13*e^19 - 4564560*a^8*b^7*c^5*d^13*e^19 - 17377360*a^9*b^5*c^
6*d^13*e^19 - 10890880*a^10*b^3*c^7*d^13*e^19 - 1281280*a^11*b*c^8*d^13*e^19 + 1820*a^4*b^16*d^12*e^20 + 33488
*a^5*b^14*c*d^12*e^20 + 5096*a^6*b^12*c^2*d^12*e^20 - 1153152*a^7*b^10*c^3*d^12*e^20 - 1021020*a^8*b^8*c^4*d^1
2*e^20 + 8408400*a^9*b^6*c^5*d^12*e^20 + 12500488*a^10*b^4*c^6*d^12*e^20 + 3779776*a^11*b^2*c^7*d^12*e^20 + 12
8128*a^12*c^8*d^12*e^20 - 4368*a^5*b^15*d^11*e^21 - 45136*a^6*b^13*c*d^11*e^21 + 158912*a^7*b^11*c^2*d^11*e^21
+ 1292720*a^8*b^9*c^3*d^11*e^21 - 1224080*a^9*b^7*c^4*d^11*e^21 - 8376368*a^10*b^5*c^5*d^11*e^21 - 6022016*a^
11*b^3*c^6*d^11*e^21 - 768768*a^12*b*c^7*d^11*e^21 + 8008*a^6*b^14*d^10*e^22 + 38896*a^7*b^12*c*d^10*e^22 - 33
5192*a^8*b^10*c^2*d^10*e^22 - 835120*a^9*b^8*c^3*d^10*e^22 + 2682680*a^10*b^6*c^4*d^10*e^22 + 5449808*a^11*b^4
*c^5*d^10*e^22 + 1886976*a^12*b^2*c^6*d^10*e^22 + 69888*a^13*c^7*d^10*e^22 - 11440*a^7*b^13*d^9*e^23 - 11440*a
^8*b^11*c*d^9*e^23 + 400400*a^9*b^9*c^2*d^9*e^23 + 114400*a^10*b^7*c^3*d^9*e^23 - 2584400*a^11*b^5*c^4*d^9*e^2
3 - 2387840*a^12*b^3*c^5*d^9*e^23 - 349440*a^13*b*c^6*d^9*e^23 + 12870*a^8*b^12*d^8*e^24 - 22880*a^9*b^10*c*d^
8*e^24 - 314600*a^10*b^8*c^2*d^8*e^24 + 364000*a^11*b^6*c^3*d^8*e^24 + 1574300*a^12*b^4*c^4*d^8*e^24 + 684320*
a^13*b^2*c^5*d^8*e^24 + 29120*a^14*c^6*d^8*e^24 - 11440*a^9*b^11*d^7*e^25 + 43472*a^10*b^9*c*d^7*e^25 + 157664
*a^11*b^7*c^2*d^7*e^25 - 426608*a^12*b^5*c^3*d^7*e^25 - 640640*a^13*b^3*c^4*d^7*e^25 - 116480*a^14*b*c^5*d^7*e
^25 + 8008*a^10*b^10*d^6*e^26 - 42224*a^11*b^8*c*d^6*e^26 - 35672*a^12*b^6*c^2*d^6*e^26 + 262640*a^13*b^4*c^3*
d^6*e^26 + 170240*a^14*b^2*c^4*d^6*e^26 + 8960*a^15*c^5*d^6*e^26 - 4368*a^11*b^9*d^5*e^27 + 27664*a^12*b^7*c*d
^5*e^27 - 13328*a^13*b^5*c^2*d^5*e^27 - 103040*a^14*b^3*c^3*d^5*e^27 - 26880*a^15*b*c^4*d^5*e^27 + 1820*a^12*b
^8*d^4*e^28 - 12880*a^13*b^6*c*d^4*e^28 + 15800*a^14*b^4*c^2*d^4*e^28 + 25920*a^15*b^2*c^3*d^4*e^28 + 1920*a^1
6*c^4*d^4*e^28 - 560*a^13*b^7*d^3*e^29 + 4240*a^14*b^5*c*d^3*e^29 - 7040*a^15*b^3*c^2*d^3*e^29 - 3840*a^16*b*c
^3*d^3*e^29 + 120*a^14*b^6*d^2*e^30 - 944*a^15*b^4*c*d^2*e^30 + 1792*a^16*b^2*c^2*d^2*e^30 + 256*a^17*c^3*d^2*
e^30 - 16*a^15*b^5*d*e^31 + 128*a^16*b^3*c*d*e^31 - 256*a^17*b*c^2*d*e^31 + a^16*b^4*e^32 - 8*a^17*b^2*c*e^32
+ 16*a^18*c^2*e^32) + 3*(8*b*c^18*d^29 - 116*b^2*c^17*d^28*e + 774*b^3*c^16*d^27*e^2 + 152*a*b*c^17*d^27*e^2 -
3136*b^4*c^15*d^26*e^3 - 2092*a*b^2*c^16*d^26*e^3 + 80*a^2*c^17*d^26*e^3 + 8545*b^5*c^14*d^25*e^4 + 13306*a*b
^3*c^15*d^25*e^4 + 64*a^2*b*c^16*d^25*e^4 - 16266*b^6*c^13*d^24*e^5 - 51748*a*b^4*c^14*d^24*e^5 - 8008*a^2*b^2
*c^15*d^24*e^5 + 944*a^3*c^16*d^24*e^5 + 21356*b^7*c^12*d^23*e^6 + 137008*a*b^5*c^13*d^23*e^6 + 62068*a^2*b^3*
c^14*d^23*e^6 - 7024*a^3*b*c^15*d^23*e^6 - 17072*b^8*c^11*d^22*e^7 - 259528*a*b^6*c^12*d^22*e^7 - 252472*a^2*b
^4*c^13*d^22*e^7 + 10552*a^3*b^2*c^14*d^22*e^7 + 5088*a^4*c^15*d^22*e^7 + 1947*b^9*c^10*d^21*e^8 + 357962*a*b^
7*c^11*d^21*e^8 + 666094*a^2*b^5*c^12*d^21*e^8 + 79420*a^3*b^3*c^13*d^21*e^8 - 46024*a^4*b*c^14*d^21*e^8 + 163
90*b^10*c^9*d^20*e^9 - 354552*a*b^8*c^10*d^20*e^9 - 1232352*a^2*b^6*c^11*d^20*e^9 - 504944*a^3*b^4*c^12*d^20*e
^9 + 168740*a^4*b^2*c^13*d^20*e^9 + 16544*a^5*c^14*d^20*e^9 - 27082*b^11*c^8*d^19*e^10 + 233376*a*b^9*c^9*d^19
*e^10 + 1650066*a^2*b^7*c^10*d^19*e^10 + 1490192*a^3*b^5*c^11*d^19*e^10 - 266750*a^4*b^3*c^12*d^19*e^10 - 1525
04*a^5*b*c^13*d^19*e^10 + 25776*b^12*c^7*d^18*e^11 - 68156*a*b^10*c^8*d^18*e^11 - 1601556*a^2*b^8*c^9*d^18*e^1
1 - 2810412*a^3*b^6*c^10*d^18*e^11 - 121880*a^4*b^4*c^11*d^18*e^11 + 608828*a^5*b^2*c^12*d^18*e^11 + 36080*a^6
*c^13*d^18*e^11 - 17033*b^13*c^6*d^17*e^12 - 45650*a*b^11*c^7*d^17*e^12 + 1086844*a^2*b^9*c^8*d^17*e^12 + 3675
936*a^3*b^7*c^9*d^17*e^12 + 1487739*a^4*b^5*c^10*d^17*e^12 - 1323762*a^5*b^3*c^11*d^17*e^12 - 320496*a^6*b*c^1
2*d^17*e^12 + 8114*b^14*c^5*d^16*e^13 + 74252*a*b^12*c^6*d^16*e^13 - 448888*a^2*b^10*c^7*d^16*e^13 - 3400320*a
^3*b^8*c^8*d^16*e^13 - 3465462*a^4*b^6*c^9*d^16*e^13 + 1499388*a^5*b^4*c^10*d^16*e^13 + 1247136*a^6*b^2*c^11*d
^16*e^13 + 55440*a^7*c^12*d^16*e^13 - 2760*b^15*c^4*d^15*e^14 - 51104*a*b^13*c^5*d^15*e^14 + 32392*a^2*b^11*c^
6*d^15*e^14 + 2171488*a^3*b^9*c^7*d^15*e^14 + 4691280*a^4*b^7*c^8*d^15*e^14 - 182688*a^5*b^5*c^9*d^15*e^14 - 2
728968*a^6*b^3*c^10*d^15*e^14 - 461472*a^7*b*c^11*d^15*e^14 + 640*b^16*c^3*d^14*e^15 + 21872*a*b^14*c^4*d^14*e
^15 + 93392*a^2*b^12*c^5*d^14*e^15 - 860816*a^3*b^10*c^6*d^14*e^15 - 4160640*a^4*b^8*c^7*d^14*e^15 - 2186976*a
^5*b^6*c^8*d^14*e^15 + 3502224*a^6*b^4*c^9*d^14*e^15 + 1675344*a^7*b^2*c^10*d^14*e^15 + 61248*a^8*c^11*d^14*e^
15 - 91*b^17*c^2*d^13*e^16 - 6002*a*b^15*c^3*d^13*e^16 - 67562*a^2*b^13*c^4*d^13*e^16 + 115436*a^3*b^11*c^5*d^
13*e^16 + 2414610*a^4*b^9*c^6*d^13*e^16 + 3711180*a^5*b^7*c^7*d^13*e^16 - 2237004*a^6*b^5*c^8*d^13*e^16 - 3413
784*a^7*b^3*c^9*d^13*e^16 - 469128*a^8*b*c^10*d^13*e^16 + 6*b^18*c*d^12*e^17 + 976*a*b^16*c^2*d^12*e^17 + 2412
8*a^2*b^14*c^3*d^12*e^17 + 79712*a^3*b^12*c^4*d^12*e^17 - 820952*a^4*b^10*c^5*d^12*e^17 - 3232416*a^5*b^8*c^6*
d^12*e^17 - 340032*a^6*b^6*c^7*d^12*e^17 + 4124736*a^7*b^4*c^8*d^12*e^17 + 1546644*a^8*b^2*c^9*d^12*e^17 + 485
76*a^9*c^10*d^12*e^17 - 72*a*b^17*c*d^11*e^18 - 4686*a^2*b^15*c^2*d^11*e^18 - 52000*a^3*b^13*c^3*d^11*e^18 + 7
8886*a^4*b^11*c^4*d^11*e^18 + 1641112*a^5*b^9*c^5*d^11*e^18 + 1958220*a^6*b^7*c^6*d^11*e^18 - 2710752*a^7*b^5*
c^7*d^11*e^18 - 2813910*a^8*b^3*c^8*d^11*e^18 - 338008*a^9*b*c^9*d^11*e^18 + 396*a^2*b^16*c*d^10*e^19 + 13156*
a^3*b^14*c^2*d^10*e^19 + 55880*a^4*b^12*c^3*d^10*e^19 - 423676*a^5*b^10*c^4*d^10*e^19 - 1695672*a^6*b^8*c^5*d^
10*e^19 + 398904*a^7*b^6*c^6*d^10*e^19 + 2972112*a^8*b^4*c^7*d^10*e^19 + 990220*a^9*b^2*c^8*d^10*e^19 + 26928*
a^10*c^9*d^10*e^19 - 1320*a^3*b^15*c*d^9*e^20 - 23485*a^4*b^13*c^2*d^9*e^20 + 3190*a^5*b^11*c^3*d^9*e^20 + 692
956*a^6*b^9*c^4*d^9*e^20 + 786456*a^7*b^7*c^5*d^9*e^20 - 1667193*a^8*b^5*c^6*d^9*e^20 - 1552650*a^9*b^3*c^7*d^
9*e^20 - 169312*a^10*b*c^8*d^9*e^20 + 2970*a^4*b^14*c*d^8*e^21 + 26532*a^5*b^12*c^2*d^8*e^21 - 104016*a^6*b^10
*c^3*d^8*e^21 - 613536*a^7*b^8*c^4*d^8*e^21 + 222618*a^8*b^6*c^5*d^8*e^21 + 1348820*a^9*b^4*c^6*d^8*e^21 + 432
344*a^10*b^2*c^7*d^8*e^21 + 9680*a^11*c^8*d^8*e^21 - 4752*a^5*b^13*c*d^7*e^22 - 15972*a^6*b^11*c^2*d^7*e^22 +
165264*a^7*b^9*c^3*d^7*e^22 + 270996*a^8*b^7*c^4*d^7*e^22 - 565136*a^9*b^5*c^5*d^7*e^22 - 561836*a^10*b^3*c^6*
d^7*e^22 - 56048*a^11*b*c^7*d^7*e^22 + 5544*a^6*b^12*c*d^6*e^23 - 1848*a^7*b^10*c^2*d^6*e^23 - 139920*a^8*b^8*
c^3*d^6*e^23 + 15048*a^9*b^6*c^4*d^6*e^23 + 371624*a^10*b^4*c^5*d^6*e^23 + 123128*a^11*b^2*c^6*d^6*e^23 + 1760
*a^12*c^7*d^6*e^23 - 4752*a^7*b^11*c*d^5*e^24 + 13563*a^8*b^9*c^2*d^5*e^24 + 68090*a^9*b^7*c^3*d^5*e^24 - 9369
8*a^10*b^5*c^4*d^5*e^24 - 125924*a^11*b^3*c^5*d^5*e^24 - 10936*a^12*b*c^6*d^5*e^24 + 2970*a^8*b^10*c*d^4*e^25
- 13640*a^9*b^8*c^2*d^4*e^25 - 14080*a^10*b^6*c^3*d^4*e^25 + 55472*a^11*b^4*c^4*d^4*e^25 + 21052*a^12*b^2*c^5*
d^4*e^25 - 96*a^13*c^6*d^4*e^25 - 1320*a^9*b^9*c*d^3*e^26 + 7634*a^10*b^7*c^2*d^3*e^26 - 3632*a^11*b^5*c^3*d^3
*e^26 - 15602*a^12*b^3*c^4*d^3*e^26 - 904*a^13*b*c^5*d^3*e^26 + 396*a^10*b^8*c*d^2*e^27 - 2604*a^11*b^6*c^2*d^
2*e^27 + 3208*a^12*b^4*c^3*d^2*e^27 + 1892*a^13*b^2*c^4*d^2*e^27 - 112*a^14*c^5*d^2*e^27 - 72*a^11*b^7*c*d*e^2
8 + 509*a^12*b^5*c^2*d*e^28 - 830*a^13*b^3*c^3*d*e^28 + 16*a^14*b*c^4*d*e^28 + 6*a^12*b^6*c*e^29 - 44*a^13*b^4
*c^2*e^29 + 80*a^14*b^2*c^3*e^29 - 16*a^15*c^4*e^29)/(b^4*c^16*d^32 - 8*a*b^2*c^17*d^32 + 16*a^2*c^18*d^32 - 1
6*b^5*c^15*d^31*e + 128*a*b^3*c^16*d^31*e - 256*a^2*b*c^17*d^31*e + 120*b^6*c^14*d^30*e^2 - 944*a*b^4*c^15*d^3
0*e^2 + 1792*a^2*b^2*c^16*d^30*e^2 + 256*a^3*c^17*d^30*e^2 - 560*b^7*c^13*d^29*e^3 + 4240*a*b^5*c^14*d^29*e^3
- 7040*a^2*b^3*c^15*d^29*e^3 - 3840*a^3*b*c^16*d^29*e^3 + 1820*b^8*c^12*d^28*e^4 - 12880*a*b^6*c^13*d^28*e^4 +
15800*a^2*b^4*c^14*d^28*e^4 + 25920*a^3*b^2*c^15*d^28*e^4 + 1920*a^4*c^16*d^28*e^4 - 4368*b^9*c^11*d^27*e^5 +
27664*a*b^7*c^12*d^27*e^5 - 13328*a^2*b^5*c^13*d^27*e^5 - 103040*a^3*b^3*c^14*d^27*e^5 - 26880*a^4*b*c^15*d^2
7*e^5 + 8008*b^10*c^10*d^26*e^6 - 42224*a*b^8*c^11*d^26*e^6 - 35672*a^2*b^6*c^12*d^26*e^6 + 262640*a^3*b^4*c^1
3*d^26*e^6 + 170240*a^4*b^2*c^14*d^26*e^6 + 8960*a^5*c^15*d^26*e^6 - 11440*b^11*c^9*d^25*e^7 + 43472*a*b^9*c^1
0*d^25*e^7 + 157664*a^2*b^7*c^11*d^25*e^7 - 426608*a^3*b^5*c^12*d^25*e^7 - 640640*a^4*b^3*c^13*d^25*e^7 - 1164
80*a^5*b*c^14*d^25*e^7 + 12870*b^12*c^8*d^24*e^8 - 22880*a*b^10*c^9*d^24*e^8 - 314600*a^2*b^8*c^10*d^24*e^8 +
364000*a^3*b^6*c^11*d^24*e^8 + 1574300*a^4*b^4*c^12*d^24*e^8 + 684320*a^5*b^2*c^13*d^24*e^8 + 29120*a^6*c^14*d
^24*e^8 - 11440*b^13*c^7*d^23*e^9 - 11440*a*b^11*c^8*d^23*e^9 + 400400*a^2*b^9*c^9*d^23*e^9 + 114400*a^3*b^7*c
^10*d^23*e^9 - 2584400*a^4*b^5*c^11*d^23*e^9 - 2387840*a^5*b^3*c^12*d^23*e^9 - 349440*a^6*b*c^13*d^23*e^9 + 80
08*b^14*c^6*d^22*e^10 + 38896*a*b^12*c^7*d^22*e^10 - 335192*a^2*b^10*c^8*d^22*e^10 - 835120*a^3*b^8*c^9*d^22*e
^10 + 2682680*a^4*b^6*c^10*d^22*e^10 + 5449808*a^5*b^4*c^11*d^22*e^10 + 1886976*a^6*b^2*c^12*d^22*e^10 + 69888
*a^7*c^13*d^22*e^10 - 4368*b^15*c^5*d^21*e^11 - 45136*a*b^13*c^6*d^21*e^11 + 158912*a^2*b^11*c^7*d^21*e^11 + 1
292720*a^3*b^9*c^8*d^21*e^11 - 1224080*a^4*b^7*c^9*d^21*e^11 - 8376368*a^5*b^5*c^10*d^21*e^11 - 6022016*a^6*b^
3*c^11*d^21*e^11 - 768768*a^7*b*c^12*d^21*e^11 + 1820*b^16*c^4*d^20*e^12 + 33488*a*b^14*c^5*d^20*e^12 + 5096*a
^2*b^12*c^6*d^20*e^12 - 1153152*a^3*b^10*c^7*d^20*e^12 - 1021020*a^4*b^8*c^8*d^20*e^12 + 8408400*a^5*b^6*c^9*d
^20*e^12 + 12500488*a^6*b^4*c^10*d^20*e^12 + 3779776*a^7*b^2*c^11*d^20*e^12 + 128128*a^8*c^12*d^20*e^12 - 560*
b^17*c^3*d^19*e^13 - 17360*a*b^15*c^4*d^19*e^13 - 74480*a^2*b^13*c^5*d^19*e^13 + 611520*a^3*b^11*c^6*d^19*e^13
+ 2402400*a^4*b^9*c^7*d^19*e^13 - 4564560*a^5*b^7*c^8*d^19*e^13 - 17377360*a^6*b^5*c^9*d^19*e^13 - 10890880*a
^7*b^3*c^10*d^19*e^13 - 1281280*a^8*b*c^11*d^19*e^13 + 120*b^18*c^2*d^18*e^14 + 6320*a*b^16*c^3*d^18*e^14 + 63
800*a^2*b^14*c^4*d^18*e^14 - 123760*a^3*b^12*c^5*d^18*e^14 - 2162160*a^4*b^10*c^6*d^18*e^14 - 480480*a^5*b^8*c
^7*d^18*e^14 + 15735720*a^6*b^6*c^8*d^18*e^14 + 20191600*a^7*b^4*c^9*d^18*e^14 + 5674240*a^8*b^2*c^10*d^18*e^1
4 + 183040*a^9*c^11*d^18*e^14 - 16*b^19*c*d^17*e^15 - 1552*a*b^17*c^2*d^17*e^15 - 30496*a^2*b^15*c^3*d^17*e^15
- 77840*a^3*b^13*c^4*d^17*e^15 + 1062880*a^4*b^11*c^5*d^17*e^15 + 3107104*a^5*b^9*c^6*d^17*e^15 - 7879872*a^6
*b^7*c^7*d^17*e^15 - 24703536*a^7*b^5*c^8*d^17*e^15 - 14551680*a^8*b^3*c^9*d^17*e^15 - 1647360*a^9*b*c^10*d^17
*e^15 + b^20*d^16*e^16 + 232*a*b^18*c*d^16*e^16 + 9016*a^2*b^16*c^2*d^16*e^16 + 76640*a^3*b^14*c^3*d^16*e^16 -
205660*a^4*b^12*c^4*d^16*e^16 - 2626624*a^5*b^10*c^5*d^16*e^16 - 48048*a^6*b^8*c^6*d^16*e^16 + 19246656*a^7*b
^6*c^7*d^16*e^16 + 23625030*a^8*b^4*c^8*d^16*e^16 + 6486480*a^9*b^2*c^9*d^16*e^16 + 205920*a^10*c^10*d^16*e^16
- 16*a*b^19*d^15*e^17 - 1552*a^2*b^17*c*d^15*e^17 - 30496*a^3*b^15*c^2*d^15*e^17 - 77840*a^4*b^13*c^3*d^15*e^
17 + 1062880*a^5*b^11*c^4*d^15*e^17 + 3107104*a^6*b^9*c^5*d^15*e^17 - 7879872*a^7*b^7*c^6*d^15*e^17 - 24703536
*a^8*b^5*c^7*d^15*e^17 - 14551680*a^9*b^3*c^8*d^15*e^17 - 1647360*a^10*b*c^9*d^15*e^17 + 120*a^2*b^18*d^14*e^1
8 + 6320*a^3*b^16*c*d^14*e^18 + 63800*a^4*b^14*c^2*d^14*e^18 - 123760*a^5*b^12*c^3*d^14*e^18 - 2162160*a^6*b^1
0*c^4*d^14*e^18 - 480480*a^7*b^8*c^5*d^14*e^18 + 15735720*a^8*b^6*c^6*d^14*e^18 + 20191600*a^9*b^4*c^7*d^14*e^
18 + 5674240*a^10*b^2*c^8*d^14*e^18 + 183040*a^11*c^9*d^14*e^18 - 560*a^3*b^17*d^13*e^19 - 17360*a^4*b^15*c*d^
13*e^19 - 74480*a^5*b^13*c^2*d^13*e^19 + 611520*a^6*b^11*c^3*d^13*e^19 + 2402400*a^7*b^9*c^4*d^13*e^19 - 45645
60*a^8*b^7*c^5*d^13*e^19 - 17377360*a^9*b^5*c^6*d^13*e^19 - 10890880*a^10*b^3*c^7*d^13*e^19 - 1281280*a^11*b*c
^8*d^13*e^19 + 1820*a^4*b^16*d^12*e^20 + 33488*a^5*b^14*c*d^12*e^20 + 5096*a^6*b^12*c^2*d^12*e^20 - 1153152*a^
7*b^10*c^3*d^12*e^20 - 1021020*a^8*b^8*c^4*d^12*e^20 + 8408400*a^9*b^6*c^5*d^12*e^20 + 12500488*a^10*b^4*c^6*d
^12*e^20 + 3779776*a^11*b^2*c^7*d^12*e^20 + 128128*a^12*c^8*d^12*e^20 - 4368*a^5*b^15*d^11*e^21 - 45136*a^6*b^
13*c*d^11*e^21 + 158912*a^7*b^11*c^2*d^11*e^21 + 1292720*a^8*b^9*c^3*d^11*e^21 - 1224080*a^9*b^7*c^4*d^11*e^21
- 8376368*a^10*b^5*c^5*d^11*e^21 - 6022016*a^11*b^3*c^6*d^11*e^21 - 768768*a^12*b*c^7*d^11*e^21 + 8008*a^6*b^
14*d^10*e^22 + 38896*a^7*b^12*c*d^10*e^22 - 335192*a^8*b^10*c^2*d^10*e^22 - 835120*a^9*b^8*c^3*d^10*e^22 + 268
2680*a^10*b^6*c^4*d^10*e^22 + 5449808*a^11*b^4*c^5*d^10*e^22 + 1886976*a^12*b^2*c^6*d^10*e^22 + 69888*a^13*c^7
*d^10*e^22 - 11440*a^7*b^13*d^9*e^23 - 11440*a^8*b^11*c*d^9*e^23 + 400400*a^9*b^9*c^2*d^9*e^23 + 114400*a^10*b
^7*c^3*d^9*e^23 - 2584400*a^11*b^5*c^4*d^9*e^23 - 2387840*a^12*b^3*c^5*d^9*e^23 - 349440*a^13*b*c^6*d^9*e^23 +
12870*a^8*b^12*d^8*e^24 - 22880*a^9*b^10*c*d^8*e^24 - 314600*a^10*b^8*c^2*d^8*e^24 + 364000*a^11*b^6*c^3*d^8*
e^24 + 1574300*a^12*b^4*c^4*d^8*e^24 + 684320*a^13*b^2*c^5*d^8*e^24 + 29120*a^14*c^6*d^8*e^24 - 11440*a^9*b^11
*d^7*e^25 + 43472*a^10*b^9*c*d^7*e^25 + 157664*a^11*b^7*c^2*d^7*e^25 - 426608*a^12*b^5*c^3*d^7*e^25 - 640640*a
^13*b^3*c^4*d^7*e^25 - 116480*a^14*b*c^5*d^7*e^25 + 8008*a^10*b^10*d^6*e^26 - 42224*a^11*b^8*c*d^6*e^26 - 3567
2*a^12*b^6*c^2*d^6*e^26 + 262640*a^13*b^4*c^3*d^6*e^26 + 170240*a^14*b^2*c^4*d^6*e^26 + 8960*a^15*c^5*d^6*e^26
- 4368*a^11*b^9*d^5*e^27 + 27664*a^12*b^7*c*d^5*e^27 - 13328*a^13*b^5*c^2*d^5*e^27 - 103040*a^14*b^3*c^3*d^5*
e^27 - 26880*a^15*b*c^4*d^5*e^27 + 1820*a^12*b^8*d^4*e^28 - 12880*a^13*b^6*c*d^4*e^28 + 15800*a^14*b^4*c^2*d^4
*e^28 + 25920*a^15*b^2*c^3*d^4*e^28 + 1920*a^16*c^4*d^4*e^28 - 560*a^13*b^7*d^3*e^29 + 4240*a^14*b^5*c*d^3*e^2
9 - 7040*a^15*b^3*c^2*d^3*e^29 - 3840*a^16*b*c^3*d^3*e^29 + 120*a^14*b^6*d^2*e^30 - 944*a^15*b^4*c*d^2*e^30 +
1792*a^16*b^2*c^2*d^2*e^30 + 256*a^17*c^3*d^2*e^30 - 16*a^15*b^5*d*e^31 + 128*a^16*b^3*c*d*e^31 - 256*a^17*b*c
^2*d*e^31 + a^16*b^4*e^32 - 8*a^17*b^2*c*e^32 + 16*a^18*c^2*e^32))*x + 3*(2*b^2*c^17*d^29 + 8*a*c^18*d^29 - 29
*b^3*c^16*d^28*e - 116*a*b*c^17*d^28*e + 192*b^4*c^15*d^27*e^2 + 824*a*b^2*c^16*d^27*e^2 + 128*a^2*c^17*d^27*e
^2 - 760*b^5*c^14*d^26*e^3 - 3856*a*b^3*c^15*d^26*e^3 - 1648*a^2*b*c^16*d^26*e^3 + 1960*b^6*c^13*d^25*e^4 + 13
330*a*b^4*c^14*d^25*e^4 + 10114*a^2*b^2*c^15*d^25*e^4 + 840*a^3*c^16*d^25*e^4 - 3276*b^7*c^12*d^24*e^5 - 35668
*a*b^5*c^13*d^24*e^5 - 40033*a^2*b^3*c^14*d^24*e^5 - 9556*a^3*b*c^15*d^24*e^5 + 2912*b^8*c^11*d^23*e^6 + 74744
*a*b^6*c^12*d^23*e^6 + 116560*a^2*b^4*c^13*d^23*e^6 + 50816*a^3*b^2*c^14*d^23*e^6 + 3008*a^4*c^15*d^23*e^6 + 1
144*b^9*c^10*d^22*e^7 - 121712*a*b^7*c^11*d^22*e^7 - 267100*a^2*b^5*c^12*d^22*e^7 - 171608*a^3*b^3*c^13*d^22*e
^7 - 29504*a^4*b*c^14*d^22*e^7 - 8580*b^10*c^9*d^21*e^8 + 150326*a*b^8*c^10*d^21*e^8 + 494186*a^2*b^6*c^11*d^2
1*e^8 + 426052*a^3*b^4*c^12*d^21*e^8 + 131450*a^4*b^2*c^13*d^21*e^8 + 6248*a^5*c^14*d^21*e^8 + 15730*b^11*c^8*
d^20*e^9 - 133100*a*b^9*c^9*d^20*e^9 - 735075*a^2*b^7*c^10*d^20*e^9 - 850520*a^3*b^5*c^11*d^20*e^9 - 362945*a^
4*b^3*c^12*d^20*e^9 - 49060*a^5*b*c^13*d^20*e^9 - 18304*b^12*c^7*d^19*e^10 + 70532*a*b^10*c^8*d^19*e^10 + 8590
56*a^2*b^8*c^9*d^19*e^10 + 1418912*a^3*b^6*c^10*d^19*e^10 + 740960*a^4*b^4*c^11*d^19*e^10 + 159544*a^5*b^2*c^1
2*d^19*e^10 + 6336*a^6*c^13*d^19*e^10 + 15288*b^13*c^6*d^18*e^11 + 1840*a*b^11*c^7*d^18*e^11 - 758450*a^2*b^9*
c^8*d^18*e^11 - 1957692*a^3*b^7*c^9*d^18*e^11 - 1293116*a^4*b^5*c^10*d^18*e^11 - 278168*a^5*b^3*c^11*d^18*e^11
- 24112*a^6*b*c^12*d^18*e^11 - 9464*b^14*c^5*d^17*e^12 - 44258*a*b^12*c^6*d^17*e^12 + 468886*a^2*b^10*c^7*d^1
7*e^12 + 2149356*a^3*b^8*c^8*d^17*e^12 + 2050158*a^4*b^6*c^9*d^17*e^12 + 329934*a^5*b^4*c^10*d^17*e^12 - 46662
*a^6*b^2*c^11*d^17*e^12 - 2904*a^7*c^12*d^17*e^12 + 4340*b^15*c^4*d^16*e^13 + 46916*a*b^13*c^5*d^16*e^13 - 159
463*a^2*b^11*c^6*d^16*e^13 - 1786884*a^3*b^9*c^7*d^16*e^13 - 2793945*a^4*b^7*c^8*d^16*e^13 - 533676*a^5*b^5*c^
9*d^16*e^13 + 454443*a^6*b^3*c^10*d^16*e^13 + 80124*a^7*b*c^11*d^16*e^13 - 1440*b^16*c^3*d^15*e^14 - 28880*a*b
^14*c^4*d^15*e^14 - 22112*a^2*b^12*c^5*d^15*e^14 + 1048960*a^3*b^10*c^6*d^15*e^14 + 2988480*a^4*b^8*c^7*d^15*e
^14 + 1254000*a^5*b^6*c^8*d^15*e^14 - 1145760*a^6*b^4*c^9*d^15*e^14 - 508992*a^7*b^2*c^10*d^15*e^14 - 21120*a^
8*c^11*d^15*e^14 + 328*b^17*c^2*d^14*e^15 + 11728*a*b^15*c^3*d^14*e^15 + 62552*a^2*b^13*c^4*d^14*e^15 - 369296
*a^3*b^11*c^5*d^14*e^15 - 2332880*a^4*b^9*c^6*d^14*e^15 - 2231328*a^5*b^7*c^7*d^14*e^15 + 1342440*a^6*b^5*c^8*
d^14*e^15 + 1538064*a^7*b^3*c^9*d^14*e^15 + 219648*a^8*b*c^10*d^14*e^15 - 46*b^18*c*d^13*e^16 - 3118*a*b^16*c^
2*d^13*e^16 - 38210*a^2*b^14*c^3*d^13*e^16 + 19676*a^3*b^12*c^4*d^13*e^16 + 1232198*a^4*b^10*c^5*d^13*e^16 + 2
569116*a^5*b^8*c^6*d^13*e^16 - 407484*a^6*b^6*c^7*d^13*e^16 - 2625480*a^7*b^4*c^8*d^13*e^16 - 919578*a^8*b^2*c
^9*d^13*e^16 - 37224*a^9*c^10*d^13*e^16 + 3*b^19*d^12*e^17 + 496*a*b^17*c*d^12*e^17 + 12811*a^2*b^15*c^2*d^12*
e^17 + 53552*a^3*b^13*c^3*d^12*e^17 - 372179*a^4*b^11*c^4*d^12*e^17 - 1894508*a^5*b^9*c^5*d^12*e^17 - 960366*a
^6*b^7*c^6*d^12*e^17 + 2580336*a^7*b^5*c^7*d^12*e^17 + 2072169*a^8*b^3*c^8*d^12*e^17 + 290532*a^9*b*c^9*d^12*e
^17 - 36*a*b^18*d^11*e^18 - 2400*a^2*b^16*c*d^11*e^18 - 28768*a^3*b^14*c^2*d^11*e^18 + 14720*a^4*b^12*c^3*d^11
*e^18 + 854128*a^5*b^10*c^4*d^11*e^18 + 1488960*a^6*b^8*c^5*d^11*e^18 - 1197504*a^7*b^6*c^6*d^11*e^18 - 275193
6*a^8*b^4*c^7*d^11*e^18 - 942040*a^9*b^2*c^8*d^11*e^18 - 39424*a^10*c^9*d^11*e^18 + 198*a^2*b^17*d^10*e^19 + 6
820*a^3*b^15*c*d^10*e^19 + 34540*a^4*b^13*c^2*d^10*e^19 - 189640*a^5*b^11*c^3*d^10*e^19 - 1011780*a^6*b^9*c^4*
d^10*e^19 - 222552*a^7*b^7*c^5*d^10*e^19 + 2135760*a^8*b^5*c^6*d^10*e^19 + 1650880*a^9*b^3*c^7*d^10*e^19 + 243
760*a^10*b*c^8*d^10*e^19 - 660*a^3*b^16*d^9*e^20 - 12430*a^4*b^14*c*d^9*e^20 - 8866*a^5*b^12*c^2*d^9*e^20 + 35
2594*a^6*b^10*c^3*d^9*e^20 + 635976*a^7*b^8*c^4*d^9*e^20 - 797148*a^8*b^6*c^5*d^9*e^20 - 1680690*a^9*b^4*c^6*d
^9*e^20 - 616858*a^10*b^2*c^7*d^9*e^20 - 28072*a^11*c^8*d^9*e^20 + 1485*a^4*b^15*d^8*e^21 + 14652*a^5*b^13*c*d
^8*e^21 - 41349*a^6*b^11*c^2*d^8*e^21 - 353100*a^7*b^9*c^3*d^8*e^21 - 74382*a^8*b^7*c^4*d^8*e^21 + 962324*a^9*
b^5*c^5*d^8*e^21 + 820061*a^10*b^3*c^6*d^8*e^21 + 136004*a^11*b*c^7*d^8*e^21 - 2376*a^5*b^14*d^7*e^22 - 10032*
a^6*b^12*c*d^7*e^22 + 77088*a^7*b^10*c^2*d^7*e^22 + 201696*a^8*b^8*c^3*d^7*e^22 - 232232*a^9*b^6*c^4*d^7*e^22
- 604208*a^10*b^4*c^5*d^7*e^22 - 259136*a^11*b^2*c^6*d^7*e^22 - 13632*a^12*c^7*d^7*e^22 + 2772*a^6*b^13*d^6*e^
23 + 1320*a^7*b^11*c*d^6*e^23 - 71544*a^8*b^9*c^2*d^6*e^23 - 44880*a^9*b^7*c^3*d^6*e^23 + 228404*a^10*b^5*c^4*
d^6*e^23 + 244424*a^11*b^3*c^5*d^6*e^23 + 49472*a^12*b*c^6*d^6*e^23 - 2376*a^7*b^12*d^5*e^24 + 4950*a^8*b^10*c
*d^5*e^24 + 39710*a^9*b^8*c^2*d^5*e^24 - 22990*a^10*b^6*c^3*d^5*e^24 - 116540*a^11*b^4*c^4*d^5*e^24 - 65506*a^
12*b^2*c^5*d^5*e^24 - 4360*a^13*c^6*d^5*e^24 + 1485*a^8*b^11*d^4*e^25 - 5720*a^9*b^9*c*d^4*e^25 - 12287*a^10*b
^7*c^2*d^4*e^25 + 23528*a^11*b^5*c^3*d^4*e^25 + 38197*a^12*b^3*c^4*d^4*e^25 + 10804*a^13*b*c^5*d^4*e^25 - 660*
a^9*b^10*d^3*e^26 + 3344*a^10*b^8*c*d^3*e^26 + 992*a^11*b^6*c^2*d^3*e^26 - 9440*a^12*b^4*c^3*d^3*e^26 - 8344*a
^13*b^2*c^4*d^3*e^26 - 832*a^14*c^5*d^3*e^26 + 198*a^10*b^9*d^2*e^27 - 1164*a^11*b^7*c*d^2*e^27 + 676*a^12*b^5
*c^2*d^2*e^27 + 2152*a^13*b^3*c^3*d^2*e^27 + 1136*a^14*b*c^4*d^2*e^27 - 36*a^11*b^8*d*e^28 + 230*a^12*b^6*c*d*
e^28 - 238*a^13*b^4*c^2*d*e^28 - 290*a^14*b^2*c^3*d*e^28 - 72*a^15*c^4*d*e^28 + 3*a^12*b^7*e^29 - 20*a^13*b^5*
c*e^29 + 25*a^14*b^3*c^2*e^29 + 20*a^15*b*c^3*e^29)/(b^4*c^16*d^32 - 8*a*b^2*c^17*d^32 + 16*a^2*c^18*d^32 - 16
*b^5*c^15*d^31*e + 128*a*b^3*c^16*d^31*e - 256*a^2*b*c^17*d^31*e + 120*b^6*c^14*d^30*e^2 - 944*a*b^4*c^15*d^30
*e^2 + 1792*a^2*b^2*c^16*d^30*e^2 + 256*a^3*c^17*d^30*e^2 - 560*b^7*c^13*d^29*e^3 + 4240*a*b^5*c^14*d^29*e^3 -
7040*a^2*b^3*c^15*d^29*e^3 - 3840*a^3*b*c^16*d^29*e^3 + 1820*b^8*c^12*d^28*e^4 - 12880*a*b^6*c^13*d^28*e^4 +
15800*a^2*b^4*c^14*d^28*e^4 + 25920*a^3*b^2*c^15*d^28*e^4 + 1920*a^4*c^16*d^28*e^4 - 4368*b^9*c^11*d^27*e^5 +
27664*a*b^7*c^12*d^27*e^5 - 13328*a^2*b^5*c^13*d^27*e^5 - 103040*a^3*b^3*c^14*d^27*e^5 - 26880*a^4*b*c^15*d^27
*e^5 + 8008*b^10*c^10*d^26*e^6 - 42224*a*b^8*c^11*d^26*e^6 - 35672*a^2*b^6*c^12*d^26*e^6 + 262640*a^3*b^4*c^13
*d^26*e^6 + 170240*a^4*b^2*c^14*d^26*e^6 + 8960*a^5*c^15*d^26*e^6 - 11440*b^11*c^9*d^25*e^7 + 43472*a*b^9*c^10
*d^25*e^7 + 157664*a^2*b^7*c^11*d^25*e^7 - 426608*a^3*b^5*c^12*d^25*e^7 - 640640*a^4*b^3*c^13*d^25*e^7 - 11648
0*a^5*b*c^14*d^25*e^7 + 12870*b^12*c^8*d^24*e^8 - 22880*a*b^10*c^9*d^24*e^8 - 314600*a^2*b^8*c^10*d^24*e^8 + 3
64000*a^3*b^6*c^11*d^24*e^8 + 1574300*a^4*b^4*c^12*d^24*e^8 + 684320*a^5*b^2*c^13*d^24*e^8 + 29120*a^6*c^14*d^
24*e^8 - 11440*b^13*c^7*d^23*e^9 - 11440*a*b^11*c^8*d^23*e^9 + 400400*a^2*b^9*c^9*d^23*e^9 + 114400*a^3*b^7*c^
10*d^23*e^9 - 2584400*a^4*b^5*c^11*d^23*e^9 - 2387840*a^5*b^3*c^12*d^23*e^9 - 349440*a^6*b*c^13*d^23*e^9 + 800
8*b^14*c^6*d^22*e^10 + 38896*a*b^12*c^7*d^22*e^10 - 335192*a^2*b^10*c^8*d^22*e^10 - 835120*a^3*b^8*c^9*d^22*e^
10 + 2682680*a^4*b^6*c^10*d^22*e^10 + 5449808*a^5*b^4*c^11*d^22*e^10 + 1886976*a^6*b^2*c^12*d^22*e^10 + 69888*
a^7*c^13*d^22*e^10 - 4368*b^15*c^5*d^21*e^11 - 45136*a*b^13*c^6*d^21*e^11 + 158912*a^2*b^11*c^7*d^21*e^11 + 12
92720*a^3*b^9*c^8*d^21*e^11 - 1224080*a^4*b^7*c^9*d^21*e^11 - 8376368*a^5*b^5*c^10*d^21*e^11 - 6022016*a^6*b^3
*c^11*d^21*e^11 - 768768*a^7*b*c^12*d^21*e^11 + 1820*b^16*c^4*d^20*e^12 + 33488*a*b^14*c^5*d^20*e^12 + 5096*a^
2*b^12*c^6*d^20*e^12 - 1153152*a^3*b^10*c^7*d^20*e^12 - 1021020*a^4*b^8*c^8*d^20*e^12 + 8408400*a^5*b^6*c^9*d^
20*e^12 + 12500488*a^6*b^4*c^10*d^20*e^12 + 3779776*a^7*b^2*c^11*d^20*e^12 + 128128*a^8*c^12*d^20*e^12 - 560*b
^17*c^3*d^19*e^13 - 17360*a*b^15*c^4*d^19*e^13 - 74480*a^2*b^13*c^5*d^19*e^13 + 611520*a^3*b^11*c^6*d^19*e^13
+ 2402400*a^4*b^9*c^7*d^19*e^13 - 4564560*a^5*b^7*c^8*d^19*e^13 - 17377360*a^6*b^5*c^9*d^19*e^13 - 10890880*a^
7*b^3*c^10*d^19*e^13 - 1281280*a^8*b*c^11*d^19*e^13 + 120*b^18*c^2*d^18*e^14 + 6320*a*b^16*c^3*d^18*e^14 + 638
00*a^2*b^14*c^4*d^18*e^14 - 123760*a^3*b^12*c^5*d^18*e^14 - 2162160*a^4*b^10*c^6*d^18*e^14 - 480480*a^5*b^8*c^
7*d^18*e^14 + 15735720*a^6*b^6*c^8*d^18*e^14 + 20191600*a^7*b^4*c^9*d^18*e^14 + 5674240*a^8*b^2*c^10*d^18*e^14
+ 183040*a^9*c^11*d^18*e^14 - 16*b^19*c*d^17*e^15 - 1552*a*b^17*c^2*d^17*e^15 - 30496*a^2*b^15*c^3*d^17*e^15
- 77840*a^3*b^13*c^4*d^17*e^15 + 1062880*a^4*b^11*c^5*d^17*e^15 + 3107104*a^5*b^9*c^6*d^17*e^15 - 7879872*a^6*
b^7*c^7*d^17*e^15 - 24703536*a^7*b^5*c^8*d^17*e^15 - 14551680*a^8*b^3*c^9*d^17*e^15 - 1647360*a^9*b*c^10*d^17*
e^15 + b^20*d^16*e^16 + 232*a*b^18*c*d^16*e^16 + 9016*a^2*b^16*c^2*d^16*e^16 + 76640*a^3*b^14*c^3*d^16*e^16 -
205660*a^4*b^12*c^4*d^16*e^16 - 2626624*a^5*b^10*c^5*d^16*e^16 - 48048*a^6*b^8*c^6*d^16*e^16 + 19246656*a^7*b^
6*c^7*d^16*e^16 + 23625030*a^8*b^4*c^8*d^16*e^16 + 6486480*a^9*b^2*c^9*d^16*e^16 + 205920*a^10*c^10*d^16*e^16
- 16*a*b^19*d^15*e^17 - 1552*a^2*b^17*c*d^15*e^17 - 30496*a^3*b^15*c^2*d^15*e^17 - 77840*a^4*b^13*c^3*d^15*e^1
7 + 1062880*a^5*b^11*c^4*d^15*e^17 + 3107104*a^6*b^9*c^5*d^15*e^17 - 7879872*a^7*b^7*c^6*d^15*e^17 - 24703536*
a^8*b^5*c^7*d^15*e^17 - 14551680*a^9*b^3*c^8*d^15*e^17 - 1647360*a^10*b*c^9*d^15*e^17 + 120*a^2*b^18*d^14*e^18
+ 6320*a^3*b^16*c*d^14*e^18 + 63800*a^4*b^14*c^2*d^14*e^18 - 123760*a^5*b^12*c^3*d^14*e^18 - 2162160*a^6*b^10
*c^4*d^14*e^18 - 480480*a^7*b^8*c^5*d^14*e^18 + 15735720*a^8*b^6*c^6*d^14*e^18 + 20191600*a^9*b^4*c^7*d^14*e^1
8 + 5674240*a^10*b^2*c^8*d^14*e^18 + 183040*a^11*c^9*d^14*e^18 - 560*a^3*b^17*d^13*e^19 - 17360*a^4*b^15*c*d^1
3*e^19 - 74480*a^5*b^13*c^2*d^13*e^19 + 611520*a^6*b^11*c^3*d^13*e^19 + 2402400*a^7*b^9*c^4*d^13*e^19 - 456456
0*a^8*b^7*c^5*d^13*e^19 - 17377360*a^9*b^5*c^6*d^13*e^19 - 10890880*a^10*b^3*c^7*d^13*e^19 - 1281280*a^11*b*c^
8*d^13*e^19 + 1820*a^4*b^16*d^12*e^20 + 33488*a^5*b^14*c*d^12*e^20 + 5096*a^6*b^12*c^2*d^12*e^20 - 1153152*a^7
*b^10*c^3*d^12*e^20 - 1021020*a^8*b^8*c^4*d^12*e^20 + 8408400*a^9*b^6*c^5*d^12*e^20 + 12500488*a^10*b^4*c^6*d^
12*e^20 + 3779776*a^11*b^2*c^7*d^12*e^20 + 128128*a^12*c^8*d^12*e^20 - 4368*a^5*b^15*d^11*e^21 - 45136*a^6*b^1
3*c*d^11*e^21 + 158912*a^7*b^11*c^2*d^11*e^21 + 1292720*a^8*b^9*c^3*d^11*e^21 - 1224080*a^9*b^7*c^4*d^11*e^21
- 8376368*a^10*b^5*c^5*d^11*e^21 - 6022016*a^11*b^3*c^6*d^11*e^21 - 768768*a^12*b*c^7*d^11*e^21 + 8008*a^6*b^1
4*d^10*e^22 + 38896*a^7*b^12*c*d^10*e^22 - 335192*a^8*b^10*c^2*d^10*e^22 - 835120*a^9*b^8*c^3*d^10*e^22 + 2682
680*a^10*b^6*c^4*d^10*e^22 + 5449808*a^11*b^4*c^5*d^10*e^22 + 1886976*a^12*b^2*c^6*d^10*e^22 + 69888*a^13*c^7*
d^10*e^22 - 11440*a^7*b^13*d^9*e^23 - 11440*a^8*b^11*c*d^9*e^23 + 400400*a^9*b^9*c^2*d^9*e^23 + 114400*a^10*b^
7*c^3*d^9*e^23 - 2584400*a^11*b^5*c^4*d^9*e^23 - 2387840*a^12*b^3*c^5*d^9*e^23 - 349440*a^13*b*c^6*d^9*e^23 +
12870*a^8*b^12*d^8*e^24 - 22880*a^9*b^10*c*d^8*e^24 - 314600*a^10*b^8*c^2*d^8*e^24 + 364000*a^11*b^6*c^3*d^8*e
^24 + 1574300*a^12*b^4*c^4*d^8*e^24 + 684320*a^13*b^2*c^5*d^8*e^24 + 29120*a^14*c^6*d^8*e^24 - 11440*a^9*b^11*
d^7*e^25 + 43472*a^10*b^9*c*d^7*e^25 + 157664*a^11*b^7*c^2*d^7*e^25 - 426608*a^12*b^5*c^3*d^7*e^25 - 640640*a^
13*b^3*c^4*d^7*e^25 - 116480*a^14*b*c^5*d^7*e^25 + 8008*a^10*b^10*d^6*e^26 - 42224*a^11*b^8*c*d^6*e^26 - 35672
*a^12*b^6*c^2*d^6*e^26 + 262640*a^13*b^4*c^3*d^6*e^26 + 170240*a^14*b^2*c^4*d^6*e^26 + 8960*a^15*c^5*d^6*e^26
- 4368*a^11*b^9*d^5*e^27 + 27664*a^12*b^7*c*d^5*e^27 - 13328*a^13*b^5*c^2*d^5*e^27 - 103040*a^14*b^3*c^3*d^5*e
^27 - 26880*a^15*b*c^4*d^5*e^27 + 1820*a^12*b^8*d^4*e^28 - 12880*a^13*b^6*c*d^4*e^28 + 15800*a^14*b^4*c^2*d^4*
e^28 + 25920*a^15*b^2*c^3*d^4*e^28 + 1920*a^16*c^4*d^4*e^28 - 560*a^13*b^7*d^3*e^29 + 4240*a^14*b^5*c*d^3*e^29
- 7040*a^15*b^3*c^2*d^3*e^29 - 3840*a^16*b*c^3*d^3*e^29 + 120*a^14*b^6*d^2*e^30 - 944*a^15*b^4*c*d^2*e^30 + 1
792*a^16*b^2*c^2*d^2*e^30 + 256*a^17*c^3*d^2*e^30 - 16*a^15*b^5*d*e^31 + 128*a^16*b^3*c*d*e^31 - 256*a^17*b*c^
2*d*e^31 + a^16*b^4*e^32 - 8*a^17*b^2*c*e^32 + 16*a^18*c^2*e^32))*x - (b^3*c^16*d^29 - 12*a*b*c^17*d^29 - 16*b
^4*c^15*d^28*e + 186*a*b^2*c^16*d^28*e - 24*a^2*c^17*d^28*e + 120*b^5*c^14*d^27*e^2 - 1328*a*b^3*c^15*d^27*e^2
+ 144*a^2*b*c^16*d^27*e^2 - 560*b^6*c^13*d^26*e^3 + 5760*a*b^4*c^14*d^26*e^3 + 672*a^2*b^2*c^15*d^26*e^3 - 54
4*a^3*c^16*d^26*e^3 + 1820*b^7*c^12*d^25*e^4 - 16800*a*b^5*c^13*d^25*e^4 - 10875*a^2*b^3*c^14*d^25*e^4 + 5812*
a^3*b*c^15*d^25*e^4 - 4368*b^8*c^11*d^24*e^5 + 34216*a*b^6*c^12*d^24*e^5 + 58206*a^2*b^4*c^13*d^24*e^5 - 24558
*a^3*b^2*c^14*d^24*e^5 - 4600*a^4*c^15*d^24*e^5 + 8008*b^9*c^10*d^23*e^6 - 48048*a*b^7*c^11*d^23*e^6 - 186372*
a^2*b^5*c^12*d^23*e^6 + 39032*a^3*b^3*c^13*d^23*e^6 + 50688*a^4*b*c^14*d^23*e^6 - 11440*b^10*c^9*d^22*e^7 + 41
184*a*b^8*c^10*d^22*e^7 + 405648*a^2*b^6*c^11*d^22*e^7 + 73360*a^3*b^4*c^12*d^22*e^7 - 242448*a^4*b^2*c^13*d^2
2*e^7 - 21504*a^5*c^14*d^22*e^7 + 12870*b^11*c^8*d^21*e^8 - 5720*a*b^9*c^9*d^21*e^8 - 627033*a^2*b^7*c^10*d^21
*e^8 - 542784*a^3*b^5*c^11*d^21*e^8 + 633941*a^4*b^3*c^12*d^21*e^8 + 227172*a^5*b*c^13*d^21*e^8 - 11440*b^12*c
^7*d^20*e^9 - 42900*a*b^10*c^8*d^20*e^9 + 688710*a^2*b^8*c^9*d^20*e^9 + 1451186*a^3*b^6*c^10*d^20*e^9 - 866052
*a^4*b^4*c^11*d^20*e^9 - 1059102*a^5*b^2*c^12*d^20*e^9 - 64504*a^6*c^13*d^20*e^9 + 8008*b^13*c^6*d^19*e^10 + 7
5504*a*b^11*c^7*d^19*e^10 - 507342*a^2*b^9*c^8*d^19*e^10 - 2406404*a^3*b^7*c^9*d^19*e^10 + 102300*a^4*b^5*c^10
*d^19*e^10 + 2815032*a^5*b^3*c^11*d^19*e^10 + 635536*a^6*b*c^12*d^19*e^10 - 4368*b^14*c^5*d^18*e^11 - 75712*a*
b^12*c^6*d^18*e^11 + 188496*a^2*b^10*c^7*d^18*e^11 + 2714976*a^3*b^8*c^8*d^18*e^11 + 2038784*a^4*b^6*c^9*d^18*
e^11 - 4505952*a^5*b^4*c^10*d^18*e^11 - 2767248*a^6*b^2*c^11*d^18*e^11 - 133408*a^7*c^12*d^18*e^11 + 1820*b^15
*c^4*d^17*e^12 + 52416*a*b^13*c^5*d^17*e^12 + 66387*a^2*b^11*c^6*d^17*e^12 - 2086700*a^3*b^9*c^7*d^17*e^12 - 4
439259*a^4*b^7*c^8*d^17*e^12 + 3860208*a^5*b^5*c^9*d^17*e^12 + 6919209*a^6*b^3*c^10*d^17*e^12 + 1205028*a^7*b*
c^11*d^17*e^12 - 560*b^16*c^3*d^16*e^13 - 26040*a*b^14*c^4*d^16*e^13 - 151350*a^2*b^12*c^5*d^16*e^13 + 994070*
a^3*b^10*c^6*d^16*e^13 + 5238288*a^4*b^8*c^7*d^16*e^13 + 7326*a^5*b^6*c^8*d^16*e^13 - 10685070*a^6*b^4*c^9*d^1
6*e^13 - 4787046*a^7*b^2*c^10*d^16*e^13 - 197208*a^8*c^11*d^16*e^13 + 120*b^17*c^2*d^15*e^14 + 9200*a*b^15*c^3
*d^15*e^14 + 113640*a^2*b^13*c^4*d^15*e^14 - 155568*a^3*b^11*c^5*d^15*e^14 - 3907376*a^4*b^9*c^6*d^15*e^14 - 4
489056*a^5*b^7*c^7*d^15*e^14 + 9827928*a^6*b^5*c^8*d^15*e^14 + 10874160*a^7*b^3*c^9*d^15*e^14 + 1609344*a^8*b*
c^10*d^15*e^14 - 16*b^18*c*d^14*e^15 - 2208*a*b^16*c^2*d^14*e^15 - 51264*a^2*b^14*c^3*d^14*e^15 - 152288*a^3*b
^12*c^4*d^14*e^15 + 1774080*a^4*b^10*c^5*d^14*e^15 + 6025536*a^5*b^8*c^6*d^14*e^15 - 3729792*a^6*b^6*c^7*d^14*
e^15 - 15241248*a^7*b^4*c^8*d^14*e^15 - 5724576*a^8*b^2*c^9*d^14*e^15 - 211200*a^9*c^10*d^14*e^15 + b^19*d^13*
e^16 + 324*a*b^17*c*d^13*e^16 + 14625*a^2*b^15*c^2*d^13*e^16 + 131192*a^3*b^13*c^3*d^13*e^16 - 334497*a^4*b^11
*c^4*d^13*e^16 - 4234956*a^5*b^9*c^5*d^13*e^16 - 2639274*a^6*b^7*c^6*d^13*e^16 + 12912768*a^7*b^5*c^7*d^13*e^1
6 + 11515779*a^8*b^3*c^8*d^13*e^16 + 1534236*a^9*b*c^9*d^13*e^16 - 22*a*b^18*d^12*e^17 - 2454*a^2*b^16*c*d^12*
e^17 - 50010*a^3*b^14*c^2*d^12*e^17 - 116468*a^4*b^12*c^3*d^12*e^17 + 1652046*a^5*b^10*c^4*d^12*e^17 + 4616172
*a^6*b^8*c^5*d^12*e^17 - 5238156*a^7*b^6*c^6*d^12*e^17 - 14093640*a^8*b^4*c^7*d^12*e^17 - 4796946*a^9*b^2*c^8*
d^12*e^17 - 162888*a^10*c^9*d^12*e^17 + 186*a^2*b^17*d^11*e^18 + 10108*a^3*b^15*c*d^11*e^18 + 95796*a^4*b^13*c
^2*d^11*e^18 - 227448*a^5*b^11*c^3*d^11*e^18 - 2897180*a^6*b^9*c^4*d^11*e^18 - 1101672*a^7*b^7*c^5*d^11*e^18 +
10234224*a^8*b^5*c^6*d^11*e^18 + 8337472*a^9*b^3*c^7*d^11*e^18 + 1036464*a^10*b*c^8*d^11*e^18 - 880*a^3*b^16*
d^10*e^19 - 25872*a^4*b^14*c*d^10*e^19 - 86592*a^5*b^12*c^2*d^10*e^19 + 858176*a^6*b^10*c^3*d^10*e^19 + 253440
0*a^7*b^8*c^4*d^10*e^19 - 3470544*a^8*b^6*c^5*d^10*e^19 - 8593024*a^9*b^4*c^6*d^10*e^19 - 2792064*a^10*b^2*c^7
*d^10*e^19 - 87648*a^11*c^8*d^10*e^19 + 2695*a^4*b^15*d^9*e^20 + 43032*a^5*b^13*c*d^9*e^20 - 37719*a^6*b^11*c^
2*d^9*e^20 - 1258532*a^7*b^9*c^3*d^9*e^20 - 574794*a^8*b^7*c^4*d^9*e^20 + 5016000*a^9*b^5*c^5*d^9*e^20 + 40812
31*a^10*b^3*c^6*d^9*e^20 + 480348*a^11*b*c^7*d^9*e^20 - 5742*a^5*b^14*d^8*e^21 - 45078*a^6*b^12*c*d^8*e^21 + 2
20770*a^7*b^10*c^2*d^8*e^21 + 996336*a^8*b^8*c^3*d^8*e^21 - 1171676*a^9*b^6*c^4*d^8*e^21 - 3397086*a^10*b^4*c^
5*d^8*e^21 - 1092234*a^11*b^2*c^6*d^8*e^21 - 30184*a^12*c^7*d^8*e^21 + 8844*a^6*b^13*d^7*e^22 + 22968*a^7*b^11
*c*d^7*e^22 - 312840*a^8*b^9*c^2*d^7*e^22 - 327184*a^9*b^7*c^3*d^7*e^22 + 1465068*a^10*b^5*c^4*d^7*e^22 + 1303
512*a^11*b^3*c^5*d^7*e^22 + 141184*a^12*b*c^6*d^7*e^22 - 10032*a^7*b^12*d^6*e^23 + 9504*a^8*b^10*c*d^6*e^23 +
246048*a^9*b^8*c^2*d^6*e^23 - 151184*a^10*b^6*c^3*d^6*e^23 - 829680*a^11*b^4*c^4*d^6*e^23 - 268368*a^12*b^2*c^
5*d^6*e^23 - 4864*a^13*c^6*d^6*e^23 + 8415*a^8*b^11*d^5*e^24 - 28556*a^9*b^9*c*d^5*e^24 - 108141*a^10*b^7*c^2*
d^5*e^24 + 226368*a^11*b^5*c^3*d^5*e^24 + 255335*a^12*b^3*c^4*d^5*e^24 + 21132*a^13*b*c^5*d^5*e^24 - 5170*a^9*
b^10*d^4*e^25 + 26334*a^10*b^8*c*d^4*e^25 + 14202*a^11*b^6*c^2*d^4*e^25 - 114404*a^12*b^4*c^3*d^4*e^25 - 36618
*a^13*b^2*c^4*d^4*e^25 + 600*a^14*c^5*d^4*e^25 + 2266*a^10*b^9*d^3*e^26 - 14196*a^11*b^7*c*d^3*e^26 + 11580*a^
12*b^5*c^2*d^3*e^26 + 28472*a^13*b^3*c^3*d^3*e^26 + 48*a^14*b*c^4*d^3*e^26 - 672*a^11*b^8*d^2*e^27 + 4736*a^12
*b^6*c*d^2*e^27 - 7200*a^13*b^4*c^2*d^2*e^27 - 2448*a^14*b^2*c^3*d^2*e^27 + 416*a^15*c^4*d^2*e^27 + 121*a^12*b
^7*d*e^28 - 912*a^13*b^5*c*d*e^28 + 1731*a^14*b^3*c^2*d*e^28 - 308*a^15*b*c^3*d*e^28 - 10*a^13*b^6*e^29 + 78*a
^14*b^4*c*e^29 - 162*a^15*b^2*c^2*e^29 + 56*a^16*c^3*e^29)/(b^4*c^16*d^32 - 8*a*b^2*c^17*d^32 + 16*a^2*c^18*d^
32 - 16*b^5*c^15*d^31*e + 128*a*b^3*c^16*d^31*e - 256*a^2*b*c^17*d^31*e + 120*b^6*c^14*d^30*e^2 - 944*a*b^4*c^
15*d^30*e^2 + 1792*a^2*b^2*c^16*d^30*e^2 + 256*a^3*c^17*d^30*e^2 - 560*b^7*c^13*d^29*e^3 + 4240*a*b^5*c^14*d^2
9*e^3 - 7040*a^2*b^3*c^15*d^29*e^3 - 3840*a^3*b*c^16*d^29*e^3 + 1820*b^8*c^12*d^28*e^4 - 12880*a*b^6*c^13*d^28
*e^4 + 15800*a^2*b^4*c^14*d^28*e^4 + 25920*a^3*b^2*c^15*d^28*e^4 + 1920*a^4*c^16*d^28*e^4 - 4368*b^9*c^11*d^27
*e^5 + 27664*a*b^7*c^12*d^27*e^5 - 13328*a^2*b^5*c^13*d^27*e^5 - 103040*a^3*b^3*c^14*d^27*e^5 - 26880*a^4*b*c^
15*d^27*e^5 + 8008*b^10*c^10*d^26*e^6 - 42224*a*b^8*c^11*d^26*e^6 - 35672*a^2*b^6*c^12*d^26*e^6 + 262640*a^3*b
^4*c^13*d^26*e^6 + 170240*a^4*b^2*c^14*d^26*e^6 + 8960*a^5*c^15*d^26*e^6 - 11440*b^11*c^9*d^25*e^7 + 43472*a*b
^9*c^10*d^25*e^7 + 157664*a^2*b^7*c^11*d^25*e^7 - 426608*a^3*b^5*c^12*d^25*e^7 - 640640*a^4*b^3*c^13*d^25*e^7
- 116480*a^5*b*c^14*d^25*e^7 + 12870*b^12*c^8*d^24*e^8 - 22880*a*b^10*c^9*d^24*e^8 - 314600*a^2*b^8*c^10*d^24*
e^8 + 364000*a^3*b^6*c^11*d^24*e^8 + 1574300*a^4*b^4*c^12*d^24*e^8 + 684320*a^5*b^2*c^13*d^24*e^8 + 29120*a^6*
c^14*d^24*e^8 - 11440*b^13*c^7*d^23*e^9 - 11440*a*b^11*c^8*d^23*e^9 + 400400*a^2*b^9*c^9*d^23*e^9 + 114400*a^3
*b^7*c^10*d^23*e^9 - 2584400*a^4*b^5*c^11*d^23*e^9 - 2387840*a^5*b^3*c^12*d^23*e^9 - 349440*a^6*b*c^13*d^23*e^
9 + 8008*b^14*c^6*d^22*e^10 + 38896*a*b^12*c^7*d^22*e^10 - 335192*a^2*b^10*c^8*d^22*e^10 - 835120*a^3*b^8*c^9*
d^22*e^10 + 2682680*a^4*b^6*c^10*d^22*e^10 + 5449808*a^5*b^4*c^11*d^22*e^10 + 1886976*a^6*b^2*c^12*d^22*e^10 +
69888*a^7*c^13*d^22*e^10 - 4368*b^15*c^5*d^21*e^11 - 45136*a*b^13*c^6*d^21*e^11 + 158912*a^2*b^11*c^7*d^21*e^
11 + 1292720*a^3*b^9*c^8*d^21*e^11 - 1224080*a^4*b^7*c^9*d^21*e^11 - 8376368*a^5*b^5*c^10*d^21*e^11 - 6022016*
a^6*b^3*c^11*d^21*e^11 - 768768*a^7*b*c^12*d^21*e^11 + 1820*b^16*c^4*d^20*e^12 + 33488*a*b^14*c^5*d^20*e^12 +
5096*a^2*b^12*c^6*d^20*e^12 - 1153152*a^3*b^10*c^7*d^20*e^12 - 1021020*a^4*b^8*c^8*d^20*e^12 + 8408400*a^5*b^6
*c^9*d^20*e^12 + 12500488*a^6*b^4*c^10*d^20*e^12 + 3779776*a^7*b^2*c^11*d^20*e^12 + 128128*a^8*c^12*d^20*e^12
- 560*b^17*c^3*d^19*e^13 - 17360*a*b^15*c^4*d^19*e^13 - 74480*a^2*b^13*c^5*d^19*e^13 + 611520*a^3*b^11*c^6*d^1
9*e^13 + 2402400*a^4*b^9*c^7*d^19*e^13 - 4564560*a^5*b^7*c^8*d^19*e^13 - 17377360*a^6*b^5*c^9*d^19*e^13 - 1089
0880*a^7*b^3*c^10*d^19*e^13 - 1281280*a^8*b*c^11*d^19*e^13 + 120*b^18*c^2*d^18*e^14 + 6320*a*b^16*c^3*d^18*e^1
4 + 63800*a^2*b^14*c^4*d^18*e^14 - 123760*a^3*b^12*c^5*d^18*e^14 - 2162160*a^4*b^10*c^6*d^18*e^14 - 480480*a^5
*b^8*c^7*d^18*e^14 + 15735720*a^6*b^6*c^8*d^18*e^14 + 20191600*a^7*b^4*c^9*d^18*e^14 + 5674240*a^8*b^2*c^10*d^
18*e^14 + 183040*a^9*c^11*d^18*e^14 - 16*b^19*c*d^17*e^15 - 1552*a*b^17*c^2*d^17*e^15 - 30496*a^2*b^15*c^3*d^1
7*e^15 - 77840*a^3*b^13*c^4*d^17*e^15 + 1062880*a^4*b^11*c^5*d^17*e^15 + 3107104*a^5*b^9*c^6*d^17*e^15 - 78798
72*a^6*b^7*c^7*d^17*e^15 - 24703536*a^7*b^5*c^8*d^17*e^15 - 14551680*a^8*b^3*c^9*d^17*e^15 - 1647360*a^9*b*c^1
0*d^17*e^15 + b^20*d^16*e^16 + 232*a*b^18*c*d^16*e^16 + 9016*a^2*b^16*c^2*d^16*e^16 + 76640*a^3*b^14*c^3*d^16*
e^16 - 205660*a^4*b^12*c^4*d^16*e^16 - 2626624*a^5*b^10*c^5*d^16*e^16 - 48048*a^6*b^8*c^6*d^16*e^16 + 19246656
*a^7*b^6*c^7*d^16*e^16 + 23625030*a^8*b^4*c^8*d^16*e^16 + 6486480*a^9*b^2*c^9*d^16*e^16 + 205920*a^10*c^10*d^1
6*e^16 - 16*a*b^19*d^15*e^17 - 1552*a^2*b^17*c*d^15*e^17 - 30496*a^3*b^15*c^2*d^15*e^17 - 77840*a^4*b^13*c^3*d
^15*e^17 + 1062880*a^5*b^11*c^4*d^15*e^17 + 3107104*a^6*b^9*c^5*d^15*e^17 - 7879872*a^7*b^7*c^6*d^15*e^17 - 24
703536*a^8*b^5*c^7*d^15*e^17 - 14551680*a^9*b^3*c^8*d^15*e^17 - 1647360*a^10*b*c^9*d^15*e^17 + 120*a^2*b^18*d^
14*e^18 + 6320*a^3*b^16*c*d^14*e^18 + 63800*a^4*b^14*c^2*d^14*e^18 - 123760*a^5*b^12*c^3*d^14*e^18 - 2162160*a
^6*b^10*c^4*d^14*e^18 - 480480*a^7*b^8*c^5*d^14*e^18 + 15735720*a^8*b^6*c^6*d^14*e^18 + 20191600*a^9*b^4*c^7*d
^14*e^18 + 5674240*a^10*b^2*c^8*d^14*e^18 + 183040*a^11*c^9*d^14*e^18 - 560*a^3*b^17*d^13*e^19 - 17360*a^4*b^1
5*c*d^13*e^19 - 74480*a^5*b^13*c^2*d^13*e^19 + 611520*a^6*b^11*c^3*d^13*e^19 + 2402400*a^7*b^9*c^4*d^13*e^19 -
4564560*a^8*b^7*c^5*d^13*e^19 - 17377360*a^9*b^5*c^6*d^13*e^19 - 10890880*a^10*b^3*c^7*d^13*e^19 - 1281280*a^
11*b*c^8*d^13*e^19 + 1820*a^4*b^16*d^12*e^20 + 33488*a^5*b^14*c*d^12*e^20 + 5096*a^6*b^12*c^2*d^12*e^20 - 1153
152*a^7*b^10*c^3*d^12*e^20 - 1021020*a^8*b^8*c^4*d^12*e^20 + 8408400*a^9*b^6*c^5*d^12*e^20 + 12500488*a^10*b^4
*c^6*d^12*e^20 + 3779776*a^11*b^2*c^7*d^12*e^20 + 128128*a^12*c^8*d^12*e^20 - 4368*a^5*b^15*d^11*e^21 - 45136*
a^6*b^13*c*d^11*e^21 + 158912*a^7*b^11*c^2*d^11*e^21 + 1292720*a^8*b^9*c^3*d^11*e^21 - 1224080*a^9*b^7*c^4*d^1
1*e^21 - 8376368*a^10*b^5*c^5*d^11*e^21 - 6022016*a^11*b^3*c^6*d^11*e^21 - 768768*a^12*b*c^7*d^11*e^21 + 8008*
a^6*b^14*d^10*e^22 + 38896*a^7*b^12*c*d^10*e^22 - 335192*a^8*b^10*c^2*d^10*e^22 - 835120*a^9*b^8*c^3*d^10*e^22
+ 2682680*a^10*b^6*c^4*d^10*e^22 + 5449808*a^11*b^4*c^5*d^10*e^22 + 1886976*a^12*b^2*c^6*d^10*e^22 + 69888*a^
13*c^7*d^10*e^22 - 11440*a^7*b^13*d^9*e^23 - 11440*a^8*b^11*c*d^9*e^23 + 400400*a^9*b^9*c^2*d^9*e^23 + 114400*
a^10*b^7*c^3*d^9*e^23 - 2584400*a^11*b^5*c^4*d^9*e^23 - 2387840*a^12*b^3*c^5*d^9*e^23 - 349440*a^13*b*c^6*d^9*
e^23 + 12870*a^8*b^12*d^8*e^24 - 22880*a^9*b^10*c*d^8*e^24 - 314600*a^10*b^8*c^2*d^8*e^24 + 364000*a^11*b^6*c^
3*d^8*e^24 + 1574300*a^12*b^4*c^4*d^8*e^24 + 684320*a^13*b^2*c^5*d^8*e^24 + 29120*a^14*c^6*d^8*e^24 - 11440*a^
9*b^11*d^7*e^25 + 43472*a^10*b^9*c*d^7*e^25 + 157664*a^11*b^7*c^2*d^7*e^25 - 426608*a^12*b^5*c^3*d^7*e^25 - 64
0640*a^13*b^3*c^4*d^7*e^25 - 116480*a^14*b*c^5*d^7*e^25 + 8008*a^10*b^10*d^6*e^26 - 42224*a^11*b^8*c*d^6*e^26
- 35672*a^12*b^6*c^2*d^6*e^26 + 262640*a^13*b^4*c^3*d^6*e^26 + 170240*a^14*b^2*c^4*d^6*e^26 + 8960*a^15*c^5*d^
6*e^26 - 4368*a^11*b^9*d^5*e^27 + 27664*a^12*b^7*c*d^5*e^27 - 13328*a^13*b^5*c^2*d^5*e^27 - 103040*a^14*b^3*c^
3*d^5*e^27 - 26880*a^15*b*c^4*d^5*e^27 + 1820*a^12*b^8*d^4*e^28 - 12880*a^13*b^6*c*d^4*e^28 + 15800*a^14*b^4*c
^2*d^4*e^28 + 25920*a^15*b^2*c^3*d^4*e^28 + 1920*a^16*c^4*d^4*e^28 - 560*a^13*b^7*d^3*e^29 + 4240*a^14*b^5*c*d
^3*e^29 - 7040*a^15*b^3*c^2*d^3*e^29 - 3840*a^16*b*c^3*d^3*e^29 + 120*a^14*b^6*d^2*e^30 - 944*a^15*b^4*c*d^2*e
^30 + 1792*a^16*b^2*c^2*d^2*e^30 + 256*a^17*c^3*d^2*e^30 - 16*a^15*b^5*d*e^31 + 128*a^16*b^3*c*d*e^31 - 256*a^
17*b*c^2*d*e^31 + a^16*b^4*e^32 - 8*a^17*b^2*c*e^32 + 16*a^18*c^2*e^32))/(c*x^2 + b*x + a)^(3/2) - 1/4*(40*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*d^2*e^5 - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*d*e^6 + 11*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*e^7 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*e^7 + 88*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^2*c^(5/2)*d^3*e^4 - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2)*d^2*e^5 + 17*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*sqrt(c)*d*e^6 - 44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2)*d*e^6
+ 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*sqrt(c)*e^7 + 88*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*d^3
*e^4 - 68*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c*d^2*e^5 - 136*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*c^2*d^
2*e^5 + 13*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*d*e^6 + 92*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c*d*e^6
- 13*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*e^7 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*e^7 + 22*b^2*
c^(3/2)*d^3*e^4 - 11*b^3*sqrt(c)*d^2*e^5 - 68*a*b*c^(3/2)*d^2*e^5 + 35*a*b^2*sqrt(c)*d*e^6 + 44*a^2*c^(3/2)*d*
e^6 - 24*a^2*b*sqrt(c)*e^7)/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3
- 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3
*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e
+ 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)*d + b*d - a*e)^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]

[In]

int(1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)),x)

[Out]

int(1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)), x)