\(\int \frac {(d+e x)^{3/2}}{x^2 (a+b x+c x^2)} \, dx\) [539]

Optimal result

Integrand size = 25, antiderivative size = 403 \[ \int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx=-\frac {d \sqrt {d+e x}}{a x}+\frac {\sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a}+\frac {2 \sqrt {d} (b d-2 a e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a^2}-\frac {\sqrt {2} \sqrt {c} \left (b^2 d^2+b d \left (\sqrt {b^2-4 a c} d-2 a e\right )-2 a \left (c d^2+e \left (\sqrt {b^2-4 a c} d-a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (b^2 d^2-b d \left (\sqrt {b^2-4 a c} d+2 a e\right )-2 a \left (c d^2-e \left (\sqrt {b^2-4 a c} d+a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

e*arctanh((e*x+d)^(1/2)/d^(1/2))*d^(1/2)/a+2*(-2*a*e+b*d)*arctanh((e*x+d)^(1/2)/d^(1/2))*d^(1/2)/a^2-d*(e*x+d)
^(1/2)/a/x-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)*(b^2*
d^2+b*d*(-2*a*e+d*(-4*a*c+b^2)^(1/2))-2*a*(c*d^2+e*(-a*e+d*(-4*a*c+b^2)^(1/2))))/a^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)*(b^2*d^2-b*d*(2*a*e+d*(-4*a*c+b^2)^(1/2))-2*a*(c*d^2-e*(a*e+d*(-4*a*c+b^2)^(1/2))))/a^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 = 2.01 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {911, 1301, 205, 212, 1180, 214} \[ \int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx=-\frac {\sqrt {2} \sqrt {c} \left (b d \left (d \sqrt {b^2-4 a c}-2 a e\right )-2 a e \left (d \sqrt {b^2-4 a c}-a e\right )-2 a c d^2+b^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \sqrt {c} \left (-b d \left (d \sqrt {b^2-4 a c}+2 a e\right )+2 a e \left (d \sqrt {b^2-4 a c}+a e\right )-2 a c d^2+b^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 \sqrt {d} (b d-2 a e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a^2}+\frac {\sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a}-\frac {d \sqrt {d+e x}}{a x} \]

[In]

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

[Out]

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

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 212

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

Rule 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 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

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]

Rule 1301

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.59 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx=\frac {-\frac {a d \sqrt {d+e x}}{x}+\frac {\sqrt {2} \sqrt {c} \left (-i b^2 d^2+b d \left (\sqrt {-b^2+4 a c} d+2 i a e\right )-2 i a \left (-c d^2+e \left (-i \sqrt {-b^2+4 a c} d+a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (i b^2 d^2+b d \left (\sqrt {-b^2+4 a c} d-2 i a e\right )+2 i a \left (-c d^2+e \left (i \sqrt {-b^2+4 a c} d+a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}+\sqrt {d} (2 b d-3 a e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a^2} \]

[In]

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

[Out]

(-((a*d*Sqrt[d + e*x])/x) + (Sqrt[2]*Sqrt[c]*((-I)*b^2*d^2 + b*d*(Sqrt[-b^2 + 4*a*c]*d + (2*I)*a*e) - (2*I)*a*
(-(c*d^2) + e*((-I)*Sqrt[-b^2 + 4*a*c]*d + a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I
*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (Sqrt[2]*Sqrt[c]*(
I*b^2*d^2 + b*d*(Sqrt[-b^2 + 4*a*c]*d - (2*I)*a*e) + (2*I)*a*(-(c*d^2) + e*(I*Sqrt[-b^2 + 4*a*c]*d + a*e)))*Ar
cTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*Sqrt[-2
*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]) + Sqrt[d]*(2*b*d - 3*a*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/a^2

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.01

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {4 c \left (-\frac {\left (2 a^{2} e^{3}-2 a b d \,e^{2}-2 a c \,d^{2} e +b^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} e^{3}+2 a b d \,e^{2}+2 a c \,d^{2} e -b^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a^{2} e^{3}}-\frac {d \left (\frac {a \sqrt {e x +d}}{2 x}+\frac {\left (3 a e -2 b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{a^{2} e^{3}}\right )\)	\(409\)
default	\(2 e^{3} \left (\frac {4 c \left (-\frac {\left (2 a^{2} e^{3}-2 a b d \,e^{2}-2 a c \,d^{2} e +b^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} e^{3}+2 a b d \,e^{2}+2 a c \,d^{2} e -b^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a^{2} e^{3}}-\frac {d \left (\frac {a \sqrt {e x +d}}{2 x}+\frac {\left (3 a e -2 b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{a^{2} e^{3}}\right )\)	\(409\)
risch	\(-\frac {d \sqrt {e x +d}}{a x}-\frac {e \left (\frac {\sqrt {d}\, \left (3 a e -2 b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{e a}+\frac {8 c \left (-\frac {\left (-2 a^{2} e^{3}+2 a b d \,e^{2}+2 a c \,d^{2} e -b^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (2 a^{2} e^{3}-2 a b d \,e^{2}-2 a c \,d^{2} e +b^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a e}\right )}{a}\)	\(411\)
pseudoelliptic	\(-\frac {2 \left (\left (\left (-d^{\frac {3}{2}} e a +\frac {b \,d^{\frac {5}{2}}}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (\left (-a c +\frac {b^{2}}{2}\right ) d^{\frac {5}{2}}+a e \left (\sqrt {d}\, e a -b \,d^{\frac {3}{2}}\right )\right )\right ) \sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, x c \,\operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\left (\left (d^{\frac {3}{2}} e a -\frac {b \,d^{\frac {5}{2}}}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (\left (-a c +\frac {b^{2}}{2}\right ) d^{\frac {5}{2}}+a e \left (\sqrt {d}\, e a -b \,d^{\frac {3}{2}}\right )\right )\right ) \sqrt {2}\, x c \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\frac {\left (\left (3 a d e -2 b \,d^{2}\right ) x \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\sqrt {e x +d}\, d^{\frac {3}{2}} a \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}{2}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {d}\, a^{2} x}\)	\(454\)

[In]

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

[Out]

2*e^3*(4/a^2/e^3*c*(-1/8*(2*a^2*e^3-2*a*b*d*e^2-2*a*c*d^2*e+b^2*d^2*e-2*(-e^2*(4*a*c-b^2))^(1/2)*a*d*e+(-e^2*(
4*a*c-b^2))^(1/2)*b*d^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)*arct
anh(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*(-2*a^2*e^3+2*a*b*d*e^2+2*a*c
*d^2*e-b^2*d^2*e-2*(-e^2*(4*a*c-b^2))^(1/2)*a*d*e+(-e^2*(4*a*c-b^2))^(1/2)*b*d^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)))-d/a^2/e^3*(1/2*a*(e*x+d)^(1/2)/x+1/2*(3*a*e-2*b*d)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)
)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4324 vs. \(2 (344) = 688\).

Time = 24.31 (sec) , antiderivative size = 8653, normalized size of antiderivative = 21.47 \[ \int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 890 vs. \(2 (344) = 688\).

Time = 0.36 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.21 \[ \int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx=-\frac {\sqrt {e x + d} d}{a x} - \frac {{\left (2 \, b d^{2} - 3 \, a d e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{a^{2} \sqrt {-d}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{3} - 4 \, a b c\right )} d^{2} - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d e\right )} e^{2} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} b c d^{3} + 2 \, \sqrt {b^{2} - 4 \, a c} a b d e^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} e^{3} - {\left (b^{2} + a c\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | e \right |} + {\left (2 \, a^{2} b e^{4} - 2 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d^{3} e + {\left (b^{3} + 2 \, a b c\right )} d^{2} e^{2} - 2 \, {\left (a b^{2} + 2 \, a^{2} c\right )} d e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, a^{2} c d - a^{2} b e + \sqrt {-4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} a^{2} c + {\left (2 \, a^{2} c d - a^{2} b e\right )}^{2}}}{a^{2} c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{2} c d^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} b d e + \sqrt {b^{2} - 4 \, a c} a^{3} e^{2}\right )} {\left | c \right |} {\left | e \right |}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{3} - 4 \, a b c\right )} d^{2} - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d e\right )} e^{2} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} b c d^{3} + 2 \, \sqrt {b^{2} - 4 \, a c} a b d e^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} e^{3} - {\left (b^{2} + a c\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | e \right |} + {\left (2 \, a^{2} b e^{4} - 2 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d^{3} e + {\left (b^{3} + 2 \, a b c\right )} d^{2} e^{2} - 2 \, {\left (a b^{2} + 2 \, a^{2} c\right )} d e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, a^{2} c d - a^{2} b e - \sqrt {-4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} a^{2} c + {\left (2 \, a^{2} c d - a^{2} b e\right )}^{2}}}{a^{2} c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{2} c d^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} b d e + \sqrt {b^{2} - 4 \, a c} a^{3} e^{2}\right )} {\left | c \right |} {\left | e \right |}} \]

[In]

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

[Out]

-sqrt(e*x + d)*d/(a*x) - (2*b*d^2 - 3*a*d*e)*arctan(sqrt(e*x + d)/sqrt(-d))/(a^2*sqrt(-d)) + 1/4*(sqrt(-4*c^2*
d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^3 - 4*a*b*c)*d^2 - 2*(a*b^2 - 4*a^2*c)*d*e)*e^2 - 2*(sqrt(b^2 - 4*a*c
)*b*c*d^3 + 2*sqrt(b^2 - 4*a*c)*a*b*d*e^2 - sqrt(b^2 - 4*a*c)*a^2*e^3 - (b^2 + a*c)*sqrt(b^2 - 4*a*c)*d^2*e)*s
qrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(e) + (2*a^2*b*e^4 - 2*(b^2*c - 2*a*c^2)*d^3*e + (b^3 + 2*a
*b*c)*d^2*e^2 - 2*(a*b^2 + 2*a^2*c)*d*e^3)*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*a^2*c*d - a^2*b*e + sqrt(-4*(a^2*c*d^2 - a^2*b*d*e + a^3*e^2)*a^2*c + (2*a^2*c*d - a^
2*b*e)^2))/(a^2*c)))/((sqrt(b^2 - 4*a*c)*a^2*c*d^2 - sqrt(b^2 - 4*a*c)*a^2*b*d*e + sqrt(b^2 - 4*a*c)*a^3*e^2)*
abs(c)*abs(e)) - 1/4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*((b^3 - 4*a*b*c)*d^2 - 2*(a*b^2 - 4*a^2
*c)*d*e)*e^2 + 2*(sqrt(b^2 - 4*a*c)*b*c*d^3 + 2*sqrt(b^2 - 4*a*c)*a*b*d*e^2 - sqrt(b^2 - 4*a*c)*a^2*e^3 - (b^2
+ a*c)*sqrt(b^2 - 4*a*c)*d^2*e)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(e) + (2*a^2*b*e^4 - 2*(b
^2*c - 2*a*c^2)*d^3*e + (b^3 + 2*a*b*c)*d^2*e^2 - 2*(a*b^2 + 2*a^2*c)*d*e^3)*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*a^2*c*d - a^2*b*e - sqrt(-4*(a^2*c*d^2 - a^2*b*d*e
+ a^3*e^2)*a^2*c + (2*a^2*c*d - a^2*b*e)^2))/(a^2*c)))/((sqrt(b^2 - 4*a*c)*a^2*c*d^2 - sqrt(b^2 - 4*a*c)*a^2*b
*d*e + sqrt(b^2 - 4*a*c)*a^3*e^2)*abs(c)*abs(e))

Mupad [B] (verification not implemented)

Time = 17.20 (sec) , antiderivative size = 29890, normalized size of antiderivative = 74.17 \[ \int \frac {(d+e x)^{3/2}}{x^2 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(d^(1/2)*atan(((d^(1/2)*((8*(d + e*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4
*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 +
8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*
b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*
a^4*b*c^4*d^3*e^13))/a^4 - (d^(1/2)*((8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*
a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3
*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^
3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 +
6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*
e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4 + (d^(1/2)*((8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^
6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 5
6*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^
3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^
3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/
a^4 + (d^(1/2)*(3*a*e - 2*b*d)*((8*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4
*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8
+ 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 - (4*d^(1/2)*(3*a*e - 2*b*d)*(d
+ e*x)^(1/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3
*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^6))/(
2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b*d)*1i)/(2*a^2) + (d^(1/2)*((8*(d + e*
x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14
+ 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2
*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8
- 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4 + (d^(1/2)
*((8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^
3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4
*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^1
3 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d
^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4 -
(d^(1/2)*((8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^8
+ 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*
c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^3
*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9
+ 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4 - (d^(1/2)*(3*a*e - 2*b*d)*((8*
(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^
10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b
*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 + (4*d^(1/2)*(3*a*e - 2*b*d)*(d + e*x)^(1/2)*(64*a^9*c^4*e^10 + 4*
a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 -
112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^6))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*(
3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b*d)*1i)/(2*a^2))/((16*(6*a*c^7*d^9*e^9 + 6*a^5*c^3*d*e^17 - 4*b*c^7*d^10*
e^8 + 6*a^2*c^6*d^7*e^11 + 6*a^4*c^4*d^3*e^15 + 8*b^2*c^6*d^9*e^9 - 4*b^3*c^5*d^8*e^10 + 4*a^2*b^2*c^4*d^5*e^1
3 - 11*a^2*b^3*c^3*d^4*e^14 + 22*a^3*b^2*c^3*d^3*e^15 - 16*a*b*c^6*d^8*e^10 + 8*a*b^2*c^5*d^7*e^11 + 2*a*b^4*c
^3*d^5*e^13 - 3*a^2*b*c^5*d^6*e^12 - 10*a^3*b*c^4*d^4*e^14 - 19*a^4*b*c^3*d^2*e^16))/a^4 - (d^(1/2)*((8*(d + e
*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14
+ 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^
2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8
- 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4 - (d^(1/2
)*((8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c
^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^
4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^
13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*
d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4
+ (d^(1/2)*((8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^
8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5
*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^
3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9
+ 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4 + (d^(1/2)*(3*a*e - 2*b*d)*((8
*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e
^10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*
b*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 - (4*d^(1/2)*(3*a*e - 2*b*d)*(d + e*x)^(1/2)*(64*a^9*c^4*e^10 + 4
*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8
- 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^6))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*
(3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2) + (d^(1/2)*((8*(d + e*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*
d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5
*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^
13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*
d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4 + (d^(1/2)*((8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5
*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11
- 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237
*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6
*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a
^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4 - (d^(1/2)*((8*(d + e*x)^(1/2)*(16*a^7
*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c
^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10
+ 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234
*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c
^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4 - (d^(1/2)*(3*a*e - 2*b*d)*((8*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e
^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b
^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11
))/a^4 + (4*d^(1/2)*(3*a*e - 2*b*d)*(d + e*x)^(1/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^1
0 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*
d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^6))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b*d))/(2*a^2))*(3*a*e - 2*b
*d))/(2*a^2)))*(3*a*e - 2*b*d)*1i)/a^2 - atan(((((((8*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*
d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 3
2*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 - (8*(d + e*
x)^(1/2)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^
3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d
^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c
- b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)
^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*
c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b
^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3
)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4
*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^
3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e
^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(d + e*x)^(1
/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 +
8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c
^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2
*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 -
348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a
*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^
3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*
a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^
3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(
56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*
e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11
+ 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 11
1*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^1
0 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4)*((b^6*
d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^
2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*
d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2)
+ 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(
a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - (8*(d + e*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c
^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^
2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c
^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3
*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^
2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a
*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b
^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c
*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*1i - (((((8*(
80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^1
0 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*
c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 + (8*(d + e*x)^(1/2)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*
e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*
b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d
^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2
*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1
/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 -
56*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^4)*((b^6*d^3 -
a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*
d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-
(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*
a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^
4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - (8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*
c^2*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*
b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*
d^4*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*
e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4)*(
(b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) +
3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a
*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^
(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))
/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*
d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^
9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 -
161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*
c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*
a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c -
b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8
*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c -
b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2
*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(d + e
*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14
+ 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^
2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8
- 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4)*((b^6*d^
3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*
b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^
3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) +
21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^
4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*1i)/((((((8*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*
d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 3
2*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 - (8*(d + e*
x)^(1/2)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^
3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d
^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c
- b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)
^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*
c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b
^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3
)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4
*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^
3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e
^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(d + e*x)^(1
/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 +
8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c
^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2
*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 -
348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a
*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^
3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*
a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^
3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(
56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*
e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11
+ 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 11
1*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^1
0 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4)*((b^6*
d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^
2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*
d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2)
+ 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(
a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - (8*(d + e*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c
^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^
2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c
^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3
*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^
2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a
*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b
^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c
*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (((((8*(80*
a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 +
4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4
*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 + (8*(d + e*x)^(1/2)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3
*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2
*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*
e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e
- 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)
*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56
*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^4)*((b^6*d^3 - a^3
*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e
^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*
a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2
*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 +
16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - (8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2
*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4
*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4
*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^1
1 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4)*((b^
6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*
a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*
c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/
2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2
*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2
*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 +
96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161
*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4
*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5
*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2
)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*
b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^
2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*
d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(d + e*x)
^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 +
4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c
^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 -
28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4)*((b^6*d^3 -
a^3*b^3*e^3 - 8*a^3*c^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4
*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(
-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21
*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b
^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (16*(6*a*c^7*d^9*e^9 + 6*a^5*c^3*d*e^17 - 4*b*c^7*d^10*e^8 + 6*a^2*c^
6*d^7*e^11 + 6*a^4*c^4*d^3*e^15 + 8*b^2*c^6*d^9*e^9 - 4*b^3*c^5*d^8*e^10 + 4*a^2*b^2*c^4*d^5*e^13 - 11*a^2*b^3
*c^3*d^4*e^14 + 22*a^3*b^2*c^3*d^3*e^15 - 16*a*b*c^6*d^8*e^10 + 8*a*b^2*c^5*d^7*e^11 + 2*a*b^4*c^3*d^5*e^13 -
3*a^2*b*c^5*d^6*e^12 - 10*a^3*b*c^4*d^4*e^14 - 19*a^4*b*c^3*d^2*e^16))/a^4))*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c
^3*d^3 - a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*
e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e - 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2
) - 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*
a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 + 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^
5*b^2*c)))^(1/2)*2i - atan(((((((8*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4
*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8
+ 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 - (8*(d + e*x)^(1/2)*((b^6*d^3
- a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^
4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*
(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2
1*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*
b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c
^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a
^7*b^3*c^3*d*e^9))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(
-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*
e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*
d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*
(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e
^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e
^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a
^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2
*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e
^11 - 84*a^6*b^2*c^3*d*e^12))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2)
- b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 +
4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2)
- 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a
^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(56*a^4*c^6*d^6*e^9
- 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c
^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^
3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e
^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^
7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4)*((b^6*d^3 - a^3*b^3*e^3 -
8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^
4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)
^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2
*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^
2 - 8*a^5*b^2*c)))^(1/2) - (8*(d + e*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a
^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11
+ 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^
5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 6
0*a^4*b*c^4*d^3*e^13))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d
^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*
b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^
2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^
2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*1i - (((((8*(80*a^8*c^4*d*e^11 +
80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 + 4*a^6*b^2*c^4*d
^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4*d^2*e^10 - 28*a
^7*b^2*c^3*d*e^11))/a^4 + (8*(d + e*x)^(1/2)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)
^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b
^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2
)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d
*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(64*a^9*c^4*e^1
0 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2
*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3
*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*
d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1
/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 3
6*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*
a^5*b^2*c)))^(1/2) - (8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c
^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 3
6*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b
^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^
5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4)*((b^6*d^3 - a^3*b^3*
e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 +
24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c -
b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*
c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a
^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b
^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*
d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3
*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b
^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12
+ 6*a^5*b^3*c^2*d*e^14))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3
*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^
4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*
a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*
d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(d + e*x)^(1/2)*(4*a^6*c^
3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^
8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28
*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7
*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8
*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*
c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3
)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e
- 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2
- 8*a^5*b^2*c)))^(1/2)*1i)/((((((8*(80*a^8*c^4*d*e^11 + 80*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4
*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 + 4*a^6*b^2*c^4*d^3*e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8
+ 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4*d^2*e^10 - 28*a^7*b^2*c^3*d*e^11))/a^4 - (8*(d + e*x)^(1/2)*((b^6*d^3
- a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^
4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*
(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2
1*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*
b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c
^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a
^7*b^3*c^3*d*e^9))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(
-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*
e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*
d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*
(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e
^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e
^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a
^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2
*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d^4*e^9 + 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e
^11 - 84*a^6*b^2*c^3*d*e^12))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2)
- b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 +
4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2)
- 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a
^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(56*a^4*c^6*d^6*e^9
- 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c
^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^
3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e
^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^
7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14))/a^4)*((b^6*d^3 - a^3*b^3*e^3 -
8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^
4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)
^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2
*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^
2 - 8*a^5*b^2*c)))^(1/2) - (8*(d + e*x)^(1/2)*(4*a^6*c^3*e^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a
^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11
+ 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^
5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 6
0*a^4*b*c^4*d^3*e^13))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d
^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*
b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^
2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^
2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (((((8*(80*a^8*c^4*d*e^11 + 80
*a^7*c^5*d^3*e^9 + 8*a^5*b^3*c^4*d^4*e^8 - 6*a^5*b^4*c^3*d^3*e^9 - 2*a^5*b^5*c^2*d^2*e^10 + 4*a^6*b^2*c^4*d^3*
e^9 + 36*a^6*b^3*c^3*d^2*e^10 - 32*a^6*b*c^5*d^4*e^8 + 2*a^6*b^4*c^2*d*e^11 - 112*a^7*b*c^4*d^2*e^10 - 28*a^7*
b^2*c^3*d*e^11))/a^4 + (8*(d + e*x)^(1/2)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)
^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*
c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3
)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^
2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(64*a^9*c^4*e^10 +
4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^
8 - 112*a^8*b*c^4*d*e^9 - 8*a^6*b^5*c^2*d*e^9 + 60*a^7*b^3*c^3*d*e^9))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^
3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e
^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2)
+ 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a
^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5
*b^2*c)))^(1/2) - (8*(d + e*x)^(1/2)*(16*a^7*b*c^3*e^13 + 88*a^7*c^4*d*e^12 - 4*a^6*b^3*c^2*e^13 - 40*a^5*c^6*
d^5*e^8 + 184*a^6*c^5*d^3*e^10 + 8*a^2*b^6*c^3*d^5*e^8 - 8*a^2*b^7*c^2*d^4*e^9 - 56*a^3*b^4*c^4*d^5*e^8 + 36*a
^3*b^5*c^3*d^4*e^9 + 28*a^3*b^6*c^2*d^3*e^10 + 108*a^4*b^2*c^5*d^5*e^8 + 36*a^4*b^3*c^4*d^4*e^9 - 179*a^4*b^4*
c^3*d^3*e^10 - 33*a^4*b^5*c^2*d^2*e^11 + 234*a^5*b^2*c^4*d^3*e^10 + 215*a^5*b^3*c^3*d^2*e^11 - 224*a^5*b*c^5*d
^4*e^9 + 16*a^5*b^4*c^2*d*e^12 - 348*a^6*b*c^4*d^2*e^11 - 84*a^6*b^2*c^3*d*e^12))/a^4)*((b^6*d^3 - a^3*b^3*e^3
- 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*
a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^
2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d
^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*
c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*
c^5*d^7*e^8 - 39*a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5
*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^
12 - 19*a^4*b^4*c^2*d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d^7*e^8 + 6*a*b^6*
c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6
*a^5*b^3*c^2*d*e^14))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^
3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b
*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2
*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2
*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (8*(d + e*x)^(1/2)*(4*a^6*c^3*e
^16 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5*c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 +
129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^
3*b^3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^
9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^5*e^11 - 60*a^4*b*c^4*d^3*e^13))/a^4)*((b^6*d^3 - a^3*b^3*e^3 - 8*a^
3*c^3*d^3 + a^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2
*d*e^2 + 18*a^2*b^2*c^2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(
1/2) + 3*a*b^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e -
36*a^3*b*c^2*d^2*e - 18*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8
*a^5*b^2*c)))^(1/2) + (16*(6*a*c^7*d^9*e^9 + 6*a^5*c^3*d*e^17 - 4*b*c^7*d^10*e^8 + 6*a^2*c^6*d^7*e^11 + 6*a^4*
c^4*d^3*e^15 + 8*b^2*c^6*d^9*e^9 - 4*b^3*c^5*d^8*e^10 + 4*a^2*b^2*c^4*d^5*e^13 - 11*a^2*b^3*c^3*d^4*e^14 + 22*
a^3*b^2*c^3*d^3*e^15 - 16*a*b*c^6*d^8*e^10 + 8*a*b^2*c^5*d^7*e^11 + 2*a*b^4*c^3*d^5*e^13 - 3*a^2*b*c^5*d^6*e^1
2 - 10*a^3*b*c^4*d^4*e^14 - 19*a^4*b*c^3*d^2*e^16))/a^4))*((b^6*d^3 - a^3*b^3*e^3 - 8*a^3*c^3*d^3 + a^3*e^3*(-
(4*a*c - b^2)^3)^(1/2) - b^3*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^4*d*e^2 + 24*a^4*c^2*d*e^2 + 18*a^2*b^2*c^
2*d^3 - 8*a*b^4*c*d^3 + 4*a^4*b*c*e^3 - 3*a*b^5*d^2*e + 2*a*b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^2*d^2*e*(
-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 21*a^2*b^3*c*d^2*e - 36*a^3*b*c^2*d^2*e - 1
8*a^3*b^2*c*d*e^2 - 3*a^2*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*2i
- (d*(d + e*x)^(1/2))/(a*x)