\(\int \frac {1}{(d+e x)^{3/2} (a+b x+c x^2)} \, dx\) [2293]

Optimal result

Integrand size = 22, antiderivative size = 310 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=-\frac {2 e}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}+\frac {\sqrt {2} \sqrt {c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \]

[Out]

-2*e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)-arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^
(1/2))*2^(1/2)*c^(1/2)*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b-(-4
*a*c+b^2)^(1/2)))^(1/2)+arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)*
c^(1/2)*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)
))^(1/2)

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {723, 840, 1180, 214} \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=-\frac {\sqrt {2} \sqrt {c} \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {2} \sqrt {c} \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac {2 e}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )} \]

[In]

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

[Out]

(-2*e)/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) - (Sqrt[2]*Sqrt[c]*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[
(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b -
Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) + (Sqrt[2]*Sqrt[c]*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*ArcTanh
[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2))

Rule 214

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

Rule 723

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

Rule 840

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

Rule 1180

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 e}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {\int \frac {c d-b e-c e x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{c d^2-b d e+a e^2} \\ & = -\frac {2 e}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {2 \text {Subst}\left (\int \frac {c d e+e (c d-b e)-c e x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c d^2-b d e+a e^2} \\ & = -\frac {2 e}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\left (c \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac {\left (c \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 e}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}+\frac {\sqrt {2} \sqrt {c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=-\frac {2 e}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {c} \left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e} \left (-c d^2+e (b d-a e)\right )}+\frac {\sqrt {2} \sqrt {c} \left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e} \left (-c d^2+e (b d-a e)\right )} \]

[In]

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

[Out]

(-2*e)/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) + (Sqrt[2]*Sqrt[c]*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[
(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b
- Sqrt[b^2 - 4*a*c])*e]*(-(c*d^2) + e*(b*d - a*e))) + (Sqrt[2]*Sqrt[c]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*A
rcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*
d + (b + Sqrt[b^2 - 4*a*c])*e]*(-(c*d^2) + e*(b*d - a*e)))

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(2 e \left (\frac {4 c \left (\frac {\left (b e -2 c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-b e +2 c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \,e^{2}-b d e +c \,d^{2}}-\frac {1}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}\right )\)	\(297\)
default	\(2 e \left (\frac {4 c \left (\frac {\left (b e -2 c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-b e +2 c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \,e^{2}-b d e +c \,d^{2}}-\frac {1}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}\right )\)	\(297\)
pseudoelliptic	\(-\frac {2 \left (-\frac {\sqrt {2}\, c \sqrt {e x +d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )}{2}+\left (\frac {c \sqrt {2}\, \sqrt {e x +d}\, \left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )}{2}+\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\right ) e}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {e x +d}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (a \,e^{2}-b d e +c \,d^{2}\right )}\)	\(365\)

[In]

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

[Out]

2*e*(4/(a*e^2-b*d*e+c*d^2)*c*(1/8*(b*e-2*c*d-(-e^2*(4*a*c-b^2))^(1/2))/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-
2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*
c)^(1/2))-1/8*(-b*e+2*c*d-(-e^2*(4*a*c-b^2))^(1/2))/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c
-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))-1/(a*
e^2-b*d*e+c*d^2)/(e*x+d)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11293 vs. \(2 (264) = 528\).

Time = 0.73 (sec) , antiderivative size = 11293, normalized size of antiderivative = 36.43 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1488 vs. \(2 (264) = 528\).

Time = 0.35 (sec) , antiderivative size = 1488, normalized size of antiderivative = 4.80 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-2*e/((c*d^2 - b*d*e + a*e^2)*sqrt(e*x + d)) - 1/4*((c*d^2*e - b*d*e^2 + a*e^3)^2*sqrt(-4*c^2*d + 2*(b*c - sqr
t(b^2 - 4*a*c)*c)*e)*(b^2 - 4*a*c)*e - 2*(2*sqrt(b^2 - 4*a*c)*c^2*d^3*e - 3*sqrt(b^2 - 4*a*c)*b*c*d^2*e^2 - sq
rt(b^2 - 4*a*c)*a*b*e^4 + (b^2 + 2*a*c)*sqrt(b^2 - 4*a*c)*d*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)
*e)*abs(c*d^2*e - b*d*e^2 + a*e^3) + (4*c^4*d^6*e - 12*b*c^3*d^5*e^2 + a^2*b^2*e^7 + (13*b^2*c^2 + 8*a*c^3)*d^
4*e^3 - 2*(3*b^3*c + 8*a*b*c^2)*d^3*e^4 + (b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2*e^5 - 2*(a*b^3 + 2*a^2*b*c)*d*e^6
)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c^2*d^3 - 3*b*c*
d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3 + sqrt((2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3)
^2 - 4*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*(c^2*d^2 - b*c*d*e + a*c*
e^2)))/(c^2*d^2 - b*c*d*e + a*c*e^2)))/((sqrt(b^2 - 4*a*c)*c^3*d^6 - 3*sqrt(b^2 - 4*a*c)*b*c^2*d^5*e - 3*sqrt(
b^2 - 4*a*c)*a^2*b*d*e^5 + sqrt(b^2 - 4*a*c)*a^3*e^6 + 3*(b^2*c + a*c^2)*sqrt(b^2 - 4*a*c)*d^4*e^2 - (b^3 + 6*
a*b*c)*sqrt(b^2 - 4*a*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*sqrt(b^2 - 4*a*c)*d^2*e^4)*abs(c*d^2*e - b*d*e^2 + a*e^3)
*abs(c)) + 1/4*((c*d^2*e - b*d*e^2 + a*e^3)^2*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(b^2 - 4*a*c)*e
+ 2*(2*sqrt(b^2 - 4*a*c)*c^2*d^3*e - 3*sqrt(b^2 - 4*a*c)*b*c*d^2*e^2 - sqrt(b^2 - 4*a*c)*a*b*e^4 + (b^2 + 2*a
*c)*sqrt(b^2 - 4*a*c)*d*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(c*d^2*e - b*d*e^2 + a*e^3) +
(4*c^4*d^6*e - 12*b*c^3*d^5*e^2 + a^2*b^2*e^7 + (13*b^2*c^2 + 8*a*c^3)*d^4*e^3 - 2*(3*b^3*c + 8*a*b*c^2)*d^3*
e^4 + (b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2*e^5 - 2*(a*b^3 + 2*a^2*b*c)*d*e^6)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2
- 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*
b*e^3 - sqrt((2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3)^2 - 4*(c^2*d^4 - 2*b*c*d^3*e + b^2*
d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*(c^2*d^2 - b*c*d*e + a*c*e^2)))/(c^2*d^2 - b*c*d*e + a*c*e^2)
))/((sqrt(b^2 - 4*a*c)*c^3*d^6 - 3*sqrt(b^2 - 4*a*c)*b*c^2*d^5*e - 3*sqrt(b^2 - 4*a*c)*a^2*b*d*e^5 + sqrt(b^2
- 4*a*c)*a^3*e^6 + 3*(b^2*c + a*c^2)*sqrt(b^2 - 4*a*c)*d^4*e^2 - (b^3 + 6*a*b*c)*sqrt(b^2 - 4*a*c)*d^3*e^3 + 3
*(a*b^2 + a^2*c)*sqrt(b^2 - 4*a*c)*d^2*e^4)*abs(c*d^2*e - b*d*e^2 + a*e^3)*abs(c))

Mupad [B] (verification not implemented)

Time = 14.60 (sec) , antiderivative size = 23975, normalized size of antiderivative = 77.34 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan((((d + e*x)^(1/2)*(16*c^8*d^8*e^2 - 16*a^4*c^4*e^10 + 32*a*c^7*d^6*e^4 - 64*b*c^7*d^7*e^3 + 8*a^3*b^2*c^3
*e^10 - 32*a^3*c^5*d^2*e^8 + 104*b^2*c^6*d^6*e^4 - 88*b^3*c^5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b^5*c^3*d^3*e^7
+ 24*a^2*b^2*c^4*d^2*e^8 - 96*a*b*c^6*d^5*e^5 + 32*a^3*b*c^4*d*e^9 + 120*a*b^2*c^5*d^4*e^6 - 80*a*b^3*c^4*d^3
*e^7 + 24*a*b^4*c^3*d^2*e^8 - 24*a^2*b^3*c^3*d*e^9) + (-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*
a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(
1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*
a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^
7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*
d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b
^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c
^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*((d + e*x)^(1/2)*(-(b^5
*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 +
3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*
b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6
+ a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*
e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b
^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48
*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*
a^3*b*c^3*d^3*e^3)))^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 3
20*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 +
88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8
*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5
*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*
d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4
- 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c
^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11
+ 144*a^5*b^2*c^3*d*e^12) - 32*a^5*b*c^3*e^12 + 64*a*c^8*d^9*e^3 + 64*a^5*c^4*d*e^11 + 8*a^4*b^3*c^2*e^12 + 25
6*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9 - 16*b^2*c^7*d^9*e^3 + 72*b^3*c^6*d^8*e^4 - 128*
b^4*c^5*d^7*e^5 + 112*b^5*c^4*d^6*e^6 - 48*b^6*c^3*d^5*e^7 + 8*b^7*c^2*d^4*e^8 + 1056*a^2*b^2*c^5*d^5*e^7 - 40
0*a^2*b^3*c^4*d^4*e^8 - 64*a^2*b^4*c^3*d^3*e^9 + 48*a^2*b^5*c^2*d^2*e^10 + 704*a^3*b^2*c^4*d^3*e^9 - 96*a^3*b^
3*c^3*d^2*e^10 - 288*a*b*c^7*d^8*e^4 + 448*a*b^2*c^6*d^7*e^5 - 224*a*b^3*c^5*d^6*e^6 - 96*a*b^4*c^4*d^5*e^7 +
128*a*b^5*c^3*d^4*e^8 - 32*a*b^6*c^2*d^3*e^9 - 896*a^2*b*c^6*d^6*e^6 - 960*a^3*b*c^5*d^4*e^8 - 32*a^3*b^4*c^2*
d*e^11 - 384*a^4*b*c^4*d^2*e^10 + 112*a^4*b^2*c^3*d*e^11))*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*
(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)
^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*
(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6
- b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b
^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*
a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^
4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*1i + ((d + e*x)^(1
/2)*(16*c^8*d^8*e^2 - 16*a^4*c^4*e^10 + 32*a*c^7*d^6*e^4 - 64*b*c^7*d^7*e^3 + 8*a^3*b^2*c^3*e^10 - 32*a^3*c^5*
d^2*e^8 + 104*b^2*c^6*d^6*e^4 - 88*b^3*c^5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b^5*c^3*d^3*e^7 + 24*a^2*b^2*c^4*d
^2*e^8 - 96*a*b*c^6*d^5*e^5 + 32*a^3*b*c^4*d*e^9 + 120*a*b^2*c^5*d^4*e^6 - 80*a*b^3*c^4*d^3*e^7 + 24*a*b^4*c^3
*d^2*e^8 - 24*a^2*b^3*c^3*d*e^9) + (-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2)
+ 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^
3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2)
+ 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2
*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^
4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24
*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4
*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*((d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3
- 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3
*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*
b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16
*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e
^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32
*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e +
24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3))
)^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4
+ 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3
- 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 +
2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a
^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4
*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^
8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a
^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d
*e^12) + 32*a^5*b*c^3*e^12 - 64*a*c^8*d^9*e^3 - 64*a^5*c^4*d*e^11 - 8*a^4*b^3*c^2*e^12 - 256*a^2*c^7*d^7*e^5 -
384*a^3*c^6*d^5*e^7 - 256*a^4*c^5*d^3*e^9 + 16*b^2*c^7*d^9*e^3 - 72*b^3*c^6*d^8*e^4 + 128*b^4*c^5*d^7*e^5 - 1
12*b^5*c^4*d^6*e^6 + 48*b^6*c^3*d^5*e^7 - 8*b^7*c^2*d^4*e^8 - 1056*a^2*b^2*c^5*d^5*e^7 + 400*a^2*b^3*c^4*d^4*e
^8 + 64*a^2*b^4*c^3*d^3*e^9 - 48*a^2*b^5*c^2*d^2*e^10 - 704*a^3*b^2*c^4*d^3*e^9 + 96*a^3*b^3*c^3*d^2*e^10 + 28
8*a*b*c^7*d^8*e^4 - 448*a*b^2*c^6*d^7*e^5 + 224*a*b^3*c^5*d^6*e^6 + 96*a*b^4*c^4*d^5*e^7 - 128*a*b^5*c^3*d^4*e
^8 + 32*a*b^6*c^2*d^3*e^9 + 896*a^2*b*c^6*d^6*e^6 + 960*a^3*b*c^5*d^4*e^8 + 32*a^3*b^4*c^2*d*e^11 + 384*a^4*b*
c^4*d^2*e^10 - 112*a^4*b^2*c^3*d*e^11))*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^
(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3
*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^
(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*
a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a
^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4
+ 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*
a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*1i)/(((d + e*x)^(1/2)*(16*c^8*d^8*e^2
- 16*a^4*c^4*e^10 + 32*a*c^7*d^6*e^4 - 64*b*c^7*d^7*e^3 + 8*a^3*b^2*c^3*e^10 - 32*a^3*c^5*d^2*e^8 + 104*b^2*c
^6*d^6*e^4 - 88*b^3*c^5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b^5*c^3*d^3*e^7 + 24*a^2*b^2*c^4*d^2*e^8 - 96*a*b*c^6
*d^5*e^5 + 32*a^3*b*c^4*d*e^9 + 120*a*b^2*c^5*d^4*e^6 - 80*a*b^3*c^4*d^3*e^7 + 24*a*b^4*c^3*d^2*e^8 - 24*a^2*b
^3*c^3*d*e^9) + (-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3
- 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*
c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e
^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^
2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*
c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e +
2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a
^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*((d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b
^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c
- b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c
*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*
c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*
e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^
3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5
- 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*(64*a*c^9*d
^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e
^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*
e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^
7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10
+ 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 -
200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c
^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 -
1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12) + 32*a^5*b*c
^3*e^12 - 64*a*c^8*d^9*e^3 - 64*a^5*c^4*d*e^11 - 8*a^4*b^3*c^2*e^12 - 256*a^2*c^7*d^7*e^5 - 384*a^3*c^6*d^5*e^
7 - 256*a^4*c^5*d^3*e^9 + 16*b^2*c^7*d^9*e^3 - 72*b^3*c^6*d^8*e^4 + 128*b^4*c^5*d^7*e^5 - 112*b^5*c^4*d^6*e^6
+ 48*b^6*c^3*d^5*e^7 - 8*b^7*c^2*d^4*e^8 - 1056*a^2*b^2*c^5*d^5*e^7 + 400*a^2*b^3*c^4*d^4*e^8 + 64*a^2*b^4*c^3
*d^3*e^9 - 48*a^2*b^5*c^2*d^2*e^10 - 704*a^3*b^2*c^4*d^3*e^9 + 96*a^3*b^3*c^3*d^2*e^10 + 288*a*b*c^7*d^8*e^4 -
448*a*b^2*c^6*d^7*e^5 + 224*a*b^3*c^5*d^6*e^6 + 96*a*b^4*c^4*d^5*e^7 - 128*a*b^5*c^3*d^4*e^8 + 32*a*b^6*c^2*d
^3*e^9 + 896*a^2*b*c^6*d^6*e^6 + 960*a^3*b*c^5*d^4*e^8 + 32*a^3*b^4*c^2*d*e^11 + 384*a^4*b*c^4*d^2*e^10 - 112*
a^4*b^2*c^3*d*e^11))*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^
2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-
(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^
2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a
^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48
*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5
*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 -
21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2) - ((d + e*x)^(1/2)*(16*c^8*d^8*e^2 - 16*a^4*c^4*e^10 + 3
2*a*c^7*d^6*e^4 - 64*b*c^7*d^7*e^3 + 8*a^3*b^2*c^3*e^10 - 32*a^3*c^5*d^2*e^8 + 104*b^2*c^6*d^6*e^4 - 88*b^3*c^
5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b^5*c^3*d^3*e^7 + 24*a^2*b^2*c^4*d^2*e^8 - 96*a*b*c^6*d^5*e^5 + 32*a^3*b*c^
4*d*e^9 + 120*a*b^2*c^5*d^4*e^6 - 80*a*b^3*c^4*d^3*e^7 + 24*a*b^4*c^3*d^2*e^8 - 24*a^2*b^3*c^3*d*e^9) + (-(b^5
*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 +
3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*
b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6
+ a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*
e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b
^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48
*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*
a^3*b*c^3*d^3*e^3)))^(1/2)*((d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)
^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a
*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)
^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3
- 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 +
48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2
*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 -
21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3
*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e
^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*
e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c
^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^
5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e
^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6
*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9
- 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12) - 32*a^5*b*c^3*e^12 + 64*a*c^8*d^9
*e^3 + 64*a^5*c^4*d*e^11 + 8*a^4*b^3*c^2*e^12 + 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^
9 - 16*b^2*c^7*d^9*e^3 + 72*b^3*c^6*d^8*e^4 - 128*b^4*c^5*d^7*e^5 + 112*b^5*c^4*d^6*e^6 - 48*b^6*c^3*d^5*e^7 +
8*b^7*c^2*d^4*e^8 + 1056*a^2*b^2*c^5*d^5*e^7 - 400*a^2*b^3*c^4*d^4*e^8 - 64*a^2*b^4*c^3*d^3*e^9 + 48*a^2*b^5*
c^2*d^2*e^10 + 704*a^3*b^2*c^4*d^3*e^9 - 96*a^3*b^3*c^3*d^2*e^10 - 288*a*b*c^7*d^8*e^4 + 448*a*b^2*c^6*d^7*e^5
- 224*a*b^3*c^5*d^6*e^6 - 96*a*b^4*c^4*d^5*e^7 + 128*a*b^5*c^3*d^4*e^8 - 32*a*b^6*c^2*d^3*e^9 - 896*a^2*b*c^6
*d^6*e^6 - 960*a^3*b*c^5*d^4*e^8 - 32*a^3*b^4*c^2*d*e^11 - 384*a^4*b*c^4*d^2*e^10 + 112*a^4*b^2*c^3*d*e^11))*(
-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e
^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2)
- 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^
5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6
*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*
a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3
- 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4
- 96*a^3*b*c^3*d^3*e^3)))^(1/2) + 16*a^3*c^4*e^9 + 16*c^7*d^6*e^3 + 48*a*c^6*d^4*e^5 - 48*b*c^6*d^5*e^4 + 48*a
^2*c^5*d^2*e^7 + 48*b^2*c^5*d^4*e^5 - 16*b^3*c^4*d^3*e^6 - 96*a*b*c^5*d^3*e^6 - 48*a^2*b*c^4*d*e^8 + 48*a*b^2*
c^4*d^2*e^7))*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 -
24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c
- b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2
)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*
c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^
3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*
a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2
*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*2i + atan((((d + e*x)^(1/2)*(16*c^8*d^8*e^2 - 16*a^4*c^4*e^10 +
32*a*c^7*d^6*e^4 - 64*b*c^7*d^7*e^3 + 8*a^3*b^2*c^3*e^10 - 32*a^3*c^5*d^2*e^8 + 104*b^2*c^6*d^6*e^4 - 88*b^3*
c^5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b^5*c^3*d^3*e^7 + 24*a^2*b^2*c^4*d^2*e^8 - 96*a*b*c^6*d^5*e^5 + 32*a^3*b*
c^4*d*e^9 + 120*a*b^2*c^5*d^4*e^6 - 80*a*b^3*c^4*d^3*e^7 + 24*a*b^4*c^3*d^2*e^8 - 24*a^2*b^3*c^3*d*e^9) + (-(b
^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2
+ 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) -
3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d
^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^
2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2
*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 -
48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 9
6*a^3*b*c^3*d^3*e^3)))^(1/2)*((d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^
2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7
*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^
2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^
3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2
+ 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d
^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5
- 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c
^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5
*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^
8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3
*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*
d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2
*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b
^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e
^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12) - 32*a^5*b*c^3*e^12 + 64*a*c^8*d
^9*e^3 + 64*a^5*c^4*d*e^11 + 8*a^4*b^3*c^2*e^12 + 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*
e^9 - 16*b^2*c^7*d^9*e^3 + 72*b^3*c^6*d^8*e^4 - 128*b^4*c^5*d^7*e^5 + 112*b^5*c^4*d^6*e^6 - 48*b^6*c^3*d^5*e^7
+ 8*b^7*c^2*d^4*e^8 + 1056*a^2*b^2*c^5*d^5*e^7 - 400*a^2*b^3*c^4*d^4*e^8 - 64*a^2*b^4*c^3*d^3*e^9 + 48*a^2*b^
5*c^2*d^2*e^10 + 704*a^3*b^2*c^4*d^3*e^9 - 96*a^3*b^3*c^3*d^2*e^10 - 288*a*b*c^7*d^8*e^4 + 448*a*b^2*c^6*d^7*e
^5 - 224*a*b^3*c^5*d^6*e^6 - 96*a*b^4*c^4*d^5*e^7 + 128*a*b^5*c^3*d^4*e^8 - 32*a*b^6*c^2*d^3*e^9 - 896*a^2*b*c
^6*d^6*e^6 - 960*a^3*b*c^5*d^4*e^8 - 32*a^3*b^4*c^2*d*e^11 - 384*a^4*b*c^4*d^2*e^10 + 112*a^4*b^2*c^3*d*e^11))
*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d
*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/
2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*
c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b
^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 2
4*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e
^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^
4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*1i + ((d + e*x)^(1/2)*(16*c^8*d^8*e^2 - 16*a^4*c^4*e^10 + 32*a*c^7*d^6*e^4 -
64*b*c^7*d^7*e^3 + 8*a^3*b^2*c^3*e^10 - 32*a^3*c^5*d^2*e^8 + 104*b^2*c^6*d^6*e^4 - 88*b^3*c^5*d^5*e^5 + 40*b^
4*c^4*d^4*e^6 - 8*b^5*c^3*d^3*e^7 + 24*a^2*b^2*c^4*d^2*e^8 - 96*a*b*c^6*d^5*e^5 + 32*a^3*b*c^4*d*e^9 + 120*a*b
^2*c^5*d^4*e^6 - 80*a*b^3*c^4*d^3*e^7 + 24*a*b^4*c^3*d^2*e^8 - 24*a^2*b^3*c^3*d*e^9) + (-(b^5*e^3 + 8*a*c^4*d^
3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e -
3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*
a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 +
16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d
*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 +
32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e
+ 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3
)))^(1/2)*((d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^
2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*
e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*
b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6
- 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^
2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c
^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4
*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^
4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*
d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4
*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*
a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*
b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8
*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40
*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2
*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12) + 32*a^5*b*c^3*e^12 - 64*a*c^8*d^9*e^3 - 64*a^5*c^4
*d*e^11 - 8*a^4*b^3*c^2*e^12 - 256*a^2*c^7*d^7*e^5 - 384*a^3*c^6*d^5*e^7 - 256*a^4*c^5*d^3*e^9 + 16*b^2*c^7*d^
9*e^3 - 72*b^3*c^6*d^8*e^4 + 128*b^4*c^5*d^7*e^5 - 112*b^5*c^4*d^6*e^6 + 48*b^6*c^3*d^5*e^7 - 8*b^7*c^2*d^4*e^
8 - 1056*a^2*b^2*c^5*d^5*e^7 + 400*a^2*b^3*c^4*d^4*e^8 + 64*a^2*b^4*c^3*d^3*e^9 - 48*a^2*b^5*c^2*d^2*e^10 - 70
4*a^3*b^2*c^4*d^3*e^9 + 96*a^3*b^3*c^3*d^2*e^10 + 288*a*b*c^7*d^8*e^4 - 448*a*b^2*c^6*d^7*e^5 + 224*a*b^3*c^5*
d^6*e^6 + 96*a*b^4*c^4*d^5*e^7 - 128*a*b^5*c^3*d^4*e^8 + 32*a*b^6*c^2*d^3*e^9 + 896*a^2*b*c^6*d^6*e^6 + 960*a^
3*b*c^5*d^4*e^8 + 32*a^3*b^4*c^2*d*e^11 + 384*a^4*b*c^4*d^2*e^10 - 112*a^4*b^2*c^3*d*e^11))*(-(b^5*e^3 + 8*a*c
^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^
2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2
- 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e
^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*
b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e
^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d
^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^
3*e^3)))^(1/2)*1i)/(((d + e*x)^(1/2)*(16*c^8*d^8*e^2 - 16*a^4*c^4*e^10 + 32*a*c^7*d^6*e^4 - 64*b*c^7*d^7*e^3 +
8*a^3*b^2*c^3*e^10 - 32*a^3*c^5*d^2*e^8 + 104*b^2*c^6*d^6*e^4 - 88*b^3*c^5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b
^5*c^3*d^3*e^7 + 24*a^2*b^2*c^4*d^2*e^8 - 96*a*b*c^6*d^5*e^5 + 32*a^3*b*c^4*d*e^9 + 120*a*b^2*c^5*d^4*e^6 - 80
*a*b^3*c^4*d^3*e^7 + 24*a*b^4*c^3*d^2*e^8 - 24*a^2*b^3*c^3*d*e^9) + (-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 -
b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a
*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b
*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^
4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^
5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*
e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^
5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*((d + e*x
)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^
2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)
^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(
16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6
+ 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*
e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*
c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c
*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^
3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c
^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c
^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8
+ 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 8
0*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b
^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9
- 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b
*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12) + 32*a^5*b*c^3*e^12 - 64*a*c^8*d^9*e^3 - 64*a^5*c^4*d*e^11 - 8*a^4*b^3
*c^2*e^12 - 256*a^2*c^7*d^7*e^5 - 384*a^3*c^6*d^5*e^7 - 256*a^4*c^5*d^3*e^9 + 16*b^2*c^7*d^9*e^3 - 72*b^3*c^6*
d^8*e^4 + 128*b^4*c^5*d^7*e^5 - 112*b^5*c^4*d^6*e^6 + 48*b^6*c^3*d^5*e^7 - 8*b^7*c^2*d^4*e^8 - 1056*a^2*b^2*c^
5*d^5*e^7 + 400*a^2*b^3*c^4*d^4*e^8 + 64*a^2*b^4*c^3*d^3*e^9 - 48*a^2*b^5*c^2*d^2*e^10 - 704*a^3*b^2*c^4*d^3*e
^9 + 96*a^3*b^3*c^3*d^2*e^10 + 288*a*b*c^7*d^8*e^4 - 448*a*b^2*c^6*d^7*e^5 + 224*a*b^3*c^5*d^6*e^6 + 96*a*b^4*
c^4*d^5*e^7 - 128*a*b^5*c^3*d^4*e^8 + 32*a*b^6*c^2*d^3*e^9 + 896*a^2*b*c^6*d^6*e^6 + 960*a^3*b*c^5*d^4*e^8 + 3
2*a^3*b^4*c^2*d*e^11 + 384*a^4*b*c^4*d^2*e^10 - 112*a^4*b^2*c^3*d*e^11))*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*
d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(
-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e
+ 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6
+ b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c
^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2
*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c
*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2) - ((
d + e*x)^(1/2)*(16*c^8*d^8*e^2 - 16*a^4*c^4*e^10 + 32*a*c^7*d^6*e^4 - 64*b*c^7*d^7*e^3 + 8*a^3*b^2*c^3*e^10 -
32*a^3*c^5*d^2*e^8 + 104*b^2*c^6*d^6*e^4 - 88*b^3*c^5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b^5*c^3*d^3*e^7 + 24*a^
2*b^2*c^4*d^2*e^8 - 96*a*b*c^6*d^5*e^5 + 32*a^3*b*c^4*d*e^9 + 120*a*b^2*c^5*d^4*e^6 - 80*a*b^3*c^4*d^3*e^7 + 2
4*a*b^4*c^3*d^2*e^8 - 24*a^2*b^3*c^3*d*e^9) + (-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^
2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7
*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^
2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^
3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2
+ 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d
^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5
- 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*((d + e*x)^(1/2)*(-(b^5*e^3 + 8
*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^
2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*
e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b
^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*
a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d
^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c
^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^
3*d^3*e^3)))^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c
^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c
^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2
*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4
*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11
+ 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a
*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e
^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^
5*b^2*c^3*d*e^12) - 32*a^5*b*c^3*e^12 + 64*a*c^8*d^9*e^3 + 64*a^5*c^4*d*e^11 + 8*a^4*b^3*c^2*e^12 + 256*a^2*c^
7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9 - 16*b^2*c^7*d^9*e^3 + 72*b^3*c^6*d^8*e^4 - 128*b^4*c^5*
d^7*e^5 + 112*b^5*c^4*d^6*e^6 - 48*b^6*c^3*d^5*e^7 + 8*b^7*c^2*d^4*e^8 + 1056*a^2*b^2*c^5*d^5*e^7 - 400*a^2*b^
3*c^4*d^4*e^8 - 64*a^2*b^4*c^3*d^3*e^9 + 48*a^2*b^5*c^2*d^2*e^10 + 704*a^3*b^2*c^4*d^3*e^9 - 96*a^3*b^3*c^3*d^
2*e^10 - 288*a*b*c^7*d^8*e^4 + 448*a*b^2*c^6*d^7*e^5 - 224*a*b^3*c^5*d^6*e^6 - 96*a*b^4*c^4*d^5*e^7 + 128*a*b^
5*c^3*d^4*e^8 - 32*a*b^6*c^2*d^3*e^9 - 896*a^2*b*c^6*d^6*e^6 - 960*a^3*b*c^5*d^4*e^8 - 32*a^3*b^4*c^2*d*e^11 -
384*a^4*b*c^4*d^2*e^10 + 112*a^4*b^2*c^3*d*e^11))*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c
- b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2
) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c
- b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*c^3*d^6 - b^7*d
^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4
*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^3 + 24*a^3*b^2*
c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*
d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2) + 16*a^3*c^4*e^9 + 16*c^7*
d^6*e^3 + 48*a*c^6*d^4*e^5 - 48*b*c^6*d^5*e^4 + 48*a^2*c^5*d^2*e^7 + 48*b^2*c^5*d^4*e^5 - 16*b^3*c^4*d^3*e^6 -
96*a*b*c^5*d^3*e^6 - 48*a^2*b*c^4*d*e^8 + 48*a*b^2*c^4*d^2*e^7))*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b
^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c
- b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c
*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5*d^6 + a^3*b^4*e^6 + 16*a^5*c^2*e^6 + b^4*
c^3*d^6 - b^7*d^3*e^3 - 8*a*b^2*c^4*d^6 - 8*a^4*b^2*c*e^6 + 3*a*b^6*d^2*e^4 - 3*a^2*b^5*d*e^5 - 3*b^5*c^2*d^5*
e + 3*b^6*c*d^4*e^2 + 48*a^3*c^4*d^4*e^2 + 48*a^4*c^3*d^2*e^4 + 24*a^2*b^2*c^3*d^4*e^2 + 32*a^2*b^3*c^2*d^3*e^
3 + 24*a^3*b^2*c^2*d^2*e^4 + 24*a*b^3*c^3*d^5*e + 2*a*b^5*c*d^3*e^3 - 48*a^2*b*c^4*d^5*e + 24*a^3*b^3*c*d*e^5
- 48*a^4*b*c^2*d*e^5 - 21*a*b^4*c^2*d^4*e^2 - 21*a^2*b^4*c*d^2*e^4 - 96*a^3*b*c^3*d^3*e^3)))^(1/2)*2i - (2*e)/
((d + e*x)^(1/2)*(a*e^2 + c*d^2 - b*d*e))