3.2399 \(\int \frac{1}{(d+e x)^3 (a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=621 \[ \frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-16 c^2 e^2 \left (81 a^2 e^2+28 a b d e+b^2 d^2\right )+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right )}{12 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^4}+\frac{e \sqrt{a+b x+c x^2} \left (8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right )}{6 \left (b^2-4 a c\right )^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac{2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-9 a e)-7 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (20 a c e^2-7 b^2 e^2+8 c^2 d^2\right )+8 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac{5 e^4 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 1.04109, antiderivative size = 621, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {740, 822, 834, 806, 724, 206} \[ \frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-16 c^2 e^2 \left (81 a^2 e^2+28 a b d e+b^2 d^2\right )+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right )}{12 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^4}+\frac{e \sqrt{a+b x+c x^2} \left (8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right )}{6 \left (b^2-4 a c\right )^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac{2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-9 a e)-7 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (20 a c e^2-7 b^2 e^2+8 c^2 d^2\right )+8 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac{5 e^4 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[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 740

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 822

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 834

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])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((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[Sim
plify[m + 2*p + 3], 0]

Rule 724

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 206

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

Rubi steps

\begin{align*} \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 \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 ) \operatorname{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]  time = 3.53391, size = 626, normalized size = 1.01 \[ \frac{2 \left (\frac{-8 c^2 \left (5 a^2 e^3+a c d e (9 e x-2 d)+2 c^2 d^3 x\right )+2 b^2 c e \left (21 a e^2+c d (4 d+3 e x)\right )-4 b c^2 \left (a e^2 (13 d-9 e x)+2 c d^2 (d-3 e x)\right )+7 b^3 c e^2 (d-e x)-7 b^4 e^3}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt{a+x (b+c x)} \left (e (b d-a e)-c d^2\right )}+\frac{e \left (\frac{2 \sqrt{a+x (b+c x)} (2 c d-b e) \left (-16 c^2 e^2 \left (81 a^2 e^2+28 a b d e+b^2 d^2\right )+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right )}{(d+e x) \left (e (a e-b d)+c d^2\right )}+\frac{4 \sqrt{a+x (b+c x)} \left (8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right )}{(d+e x)^2}-\frac{15 e^3 \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}\right )}{16 \left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )^2}+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{(d+e x)^2 (a+x (b+c x))^{3/2}}\right )}{3 \left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[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]  time = 0.241, size = 4942, normalized size = 8. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35*e/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*c^
3*b*d^2-280*e/(a*e^2-b*d*e+c*d^2)^3*c^4/(4*a*c-b^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e
^2)^(1/2)*x*d^2*b+140*e^2/(a*e^2-b*d*e+c*d^2)^3*c^3/(4*a*c-b^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*
d*e+c*d^2)/e^2)^(1/2)*x*b^2*d-35/8*e^5/(a*e^2-b*d*e+c*d^2)^4/((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*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2
-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^2+35/6*e/(a*e^2-b*d*e+c*d^2)^3/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-
b*d*e+c*d^2)/e^2)^(3/2)*c^2*d^2+7/4*e/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-
b*d*e+c*d^2)/e^2)^(3/2)*b-35/24*e^3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^
2-b*d*e+c*d^2)/e^2)^(3/2)*b^4+5/2*e^3*c/(a*e^2-b*d*e+c*d^2)^3/((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*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^
2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))+35/2*e^3/(a*e^2-b*d*e+c*d^2)^4/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b
*d*e+c*d^2)/e^2)^(1/2)*c^2*d^2-35/8*e^5/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^4-7/2/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-
b*d*e+c*d^2)/e^2)^(3/2)*c*d+35/2*e^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e
^2-b*d*e+c*d^2)/e^2)^(3/2)*x*c^2*b^2*d+105/2*e^4/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*
(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^2*c^2*d-105*e^3/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((d/e+x)^2*c+(b*e
-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c^3*d^2+35/24*e^3/(a*e^2-b*d*e+c*d^2)^3/((d/e+x)^2*c+(b*e
-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2+35/8*e^5/(a*e^2-b*d*e+c*d^2)^4/((d/e+x)^2*c+(b*e-2*c*d)/e
*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-5/6*e*c/(a*e^2-b*d*e+c*d^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(
a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-5/2*e^3*c/(a*e^2-b*d*e+c*d^2)^3/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+
c*d^2)/e^2)^(1/2)-1/2/e/(a*e^2-b*d*e+c*d^2)/(d/e+x)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e
^2)^(3/2)+35/2*e^4/(a*e^2-b*d*e+c*d^2)^4/((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*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(
1/2))/(d/e+x))*b*c*d-35/12*e^3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d
*e+c*d^2)/e^2)^(3/2)*x*c*b^3-35/4*e^5/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*
e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^3*c+70*e^2/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x
)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^4*d^3+11*e/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)
/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*b+105/4*e^4/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*
c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*c*d-105/2*e^3/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((d/e+x)^2*c
+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*c^2*d^2+35*e^2/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((d
/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c^3*d^3-35/2*e/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b
^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2*c^2*d^2-70/3*e^3/(a*e^2-b*d*e+c*d^2)
^3*c^2/(4*a*c-b^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^3+5*e^3*c^2/(a*e^2-
b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b-10*e^2*c^3/(a
*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d-5*e^2*c^
2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d+70*e
^2/(a*e^2-b*d*e+c*d^2)^3*c^2/(4*a*c-b^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b
^3*d-140*e/(a*e^2-b*d*e+c*d^2)^3*c^3/(4*a*c-b^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*b^2*d^2+88*e/(a*e^2-b*d*e+c*d^2)^2*c^3/(4*a*c-b^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*
d^2)/e^2)^(1/2)*x*b+35/4*e^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e
+c*d^2)/e^2)^(3/2)*b^3*c*d+560/3/(a*e^2-b*d*e+c*d^2)^3*c^5/(4*a*c-b^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d^3-22/(a*e^2-b*d*e+c*d^2)^2*c^3/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)
+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*d-11/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x
)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*d-176/(a*e^2-b*d*e+c*d^2)^2*c^4/(4*a*c-b^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d
/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d-88/(a*e^2-b*d*e+c*d^2)^2*c^3/(4*a*c-b^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e
*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d+11/2*e/(a*e^2-b*d*e+c*d^2)^2*c/(4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d
)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2+44*e/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^2)^2/((d/e+x)^2*c+(b*e-
2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-35/3*e^3/(a*e^2-b*d*e+c*d^2)^3*c/(4*a*c-b^2)^2/((d/e+x)^2*
c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^4-35/2*e^3/(a*e^2-b*d*e+c*d^2)^4/((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*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c
+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*c^2*d^2-35/2*e^4/(a*e^2-b*d*e+c*d^2)^4/((d/e+x
)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d*b+5/2*e^3*c/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/(
(d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2+70/3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((
d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*c^4*d^3+280/3/(a*e^2-b*d*e+c*d^2)^3*c^4/(4*a
*c-b^2)^2/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d^3+35/3/(a*e^2-b*d*e+c*d^2)^3/(
4*a*c-b^2)/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*c^3*d^3-35/6*e^2/(a*e^2-b*d*e+c
*d^2)^3/((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c*d*b

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 17.856, size = 16020, normalized size = 25.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)
) + 1/3*((((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^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 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^1
0*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 + 16390*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 - 50
4944*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 - 152504*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 - 1601
556*a^2*b^8*c^9*d^18*e^11 - 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 + 3675936*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^12*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 - 2728968*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 - 3413784*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 + 24128*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^1
7 - 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 + 48576*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 + 78886*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 + 692956*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 + 432344*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^2
2 - 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 - 93698*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 + 2
970*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^2
7 - 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^28 + 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^2 - 8*a*b^2*c^3 + 16*a^2*c
^4))*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 + 13330*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 + 1144*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 + 1503
26*a*b^8*c^10*d^21*e^8 + 494186*a^2*b^6*c^11*d^21*e^8 + 426052*a^3*b^4*c^12*d^21*e^8 + 131450*a^4*b^2*c^13*d^2
1*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^2
0*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 + 859056*a^2*b^8*c^9*d^19*e^10 + 1418912*a^3*b^6*c^10*d^19*e^10 + 74
0960*a^4*b^4*c^11*d^19*e^10 + 159544*a^5*b^2*c^12*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 - 4425
8*a*b^12*c^6*d^17*e^12 + 468886*a^2*b^10*c^7*d^17*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 - 159463*a^2*b^11*c^6*d^16*e^13 - 1786884*a^3*b^9*c^7*d^16*e^13 - 2
793945*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 - 2231
328*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 + 2569116*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 + 4
96*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 - 2751936*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 + 6820*a^3*b^15*c*d^10*e^19 + 34540*a^4*b^13*c^2*d^10*e^19 - 1896
40*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 + 243760*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 + 352594*a^6*b^10*c^3*d^9*e^20 + 635976*a^7*b^8*c^4*d^9*e^20 - 797
148*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 - 4
4880*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^2 -
8*a*b^2*c^3 + 16*a^2*c^4))*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 - 544*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 + 41184*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^22*e^7 - 21504*a^5*c^14*d^22*e^7 + 1287
0*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 + 75504*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 - 4439259*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^1
3 - 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^16*e^13 - 4787046*a^7*b^2*c^10*d^16*e^1
3 - 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 - 4489056*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^16 + 11515779*a^8*b^3*c^8*d^13*e^16 + 1
534236*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^1
1*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 + 8337
472*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 + 2534400*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 + 4081231*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 + 220770*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 + 1303512*a^11*b^3*c^5*d^7*e^22 + 141184*a^1
2*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^2
4 + 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 + 17
31*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^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2) - 1/4*(88*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^2*c^(5/2)*d^3*e^4 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*d^2*e^5 + 88*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*b*c^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 + 22*b^2*c^(
3/2)*d^3*e^4 - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*d*e^6 - 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 + 17*(sqrt(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 - 11*b^3*sqrt(c)*d^2*e^5 - 68*a*b
*c^(3/2)*d^2*e^5 + 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 + 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 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*sqrt(c)*e^7 + 35*a*b^2*sqrt(c)*d*e^6 + 44*a^2*c^(3/2)*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 - 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)