\(\int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx\) [535]

Optimal result

Integrand size = 25, antiderivative size = 441 \[ \int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=-\frac {2 \left (b c d-b^2 e+a c e\right ) \sqrt {d+e x}}{c^3}-\frac {2 b (d+e x)^{3/2}}{3 c^2}+\frac {2 (d+e x)^{5/2}}{5 c e}+\frac {\sqrt {2} \left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )+\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{7/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )-\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{7/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-2/3*b*(e*x+d)^(3/2)/c^2+2/5*(e*x+d)^(5/2)/c/e-2*(a*c*e-b^2*e+b*c*d)*(e*x+d)^(1/2)/c^3+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)*((-b*e+c*d)*(2*a*c*e-b^2*e+b*c*d)+(2*b^3*c*d*e-
6*a*b*c^2*d*e-b^4*e^2-b^2*c*(-4*a*e^2+c*d^2)+2*a*c^2*(-a*e^2+c*d^2))/(-4*a*c+b^2)^(1/2))/c^(7/2)/(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
)*((-b*e+c*d)*(2*a*c*e-b^2*e+b*c*d)+(-2*b^3*c*d*e+6*a*b*c^2*d*e+b^4*e^2+b^2*c*(-4*a*e^2+c*d^2)-2*a*c^2*(-a*e^2
+c*d^2))/(-4*a*c+b^2)^(1/2))/c^(7/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {911, 1301, 1180, 214} \[ \int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {\sqrt {2} \left ((c d-b e) \left (2 a c e+b^2 (-e)+b c d\right )+\frac {-b^2 c \left (c d^2-4 a e^2\right )-6 a b c^2 d e+2 a c^2 \left (c d^2-a e^2\right )+b^4 \left (-e^2\right )+2 b^3 c d e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{c^{7/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \left ((c d-b e) \left (2 a c e+b^2 (-e)+b c d\right )-\frac {-b^2 c \left (c d^2-4 a e^2\right )-6 a b c^2 d e+2 a c^2 \left (c d^2-a e^2\right )+b^4 \left (-e^2\right )+2 b^3 c d e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{7/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {2 \sqrt {d+e x} \left (a c e+b^2 (-e)+b c d\right )}{c^3}-\frac {2 b (d+e x)^{3/2}}{3 c^2}+\frac {2 (d+e x)^{5/2}}{5 c e} \]

[In]

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

[Out]

(-2*(b*c*d - b^2*e + a*c*e)*Sqrt[d + e*x])/c^3 - (2*b*(d + e*x)^(3/2))/(3*c^2) + (2*(d + e*x)^(5/2))/(5*c*e) +
(Sqrt[2]*((c*d - b*e)*(b*c*d - b^2*e + 2*a*c*e) + (2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a
*e^2) + 2*a*c^2*(c*d^2 - a*e^2))/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b -
Sqrt[b^2 - 4*a*c])*e]])/(c^(7/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*((c*d - b*e)*(b*c*d - b^2
*e + 2*a*c*e) - (2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2*a*c^2*(c*d^2 - a*e^2))/Sq
rt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(7/2)*Sq
rt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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}{\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}} \, dx,x,\sqrt {d+e x}\right )}{e} \\ & = \frac {2 \text {Subst}\left (\int \left (-\frac {e \left (b c d-b^2 e+a c e\right )}{c^3}-\frac {b e x^2}{c^2}+\frac {x^4}{c}+\frac {\left (b c d-b^2 e+a c e\right ) \left (c d^2-b d e+a e^2\right )-(c d-b e) \left (b c d-b^2 e+2 a c e\right ) x^2}{c^3 e \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 )}\right ) \, dx,x,\sqrt {d+e x}\right )}{e} \\ & = -\frac {2 \left (b c d-b^2 e+a c e\right ) \sqrt {d+e x}}{c^3}-\frac {2 b (d+e x)^{3/2}}{3 c^2}+\frac {2 (d+e x)^{5/2}}{5 c e}+\frac {2 \text {Subst}\left (\int \frac {\left (b c d-b^2 e+a c e\right ) \left (c d^2-b d e+a e^2\right )-(c d-b e) \left (b c d-b^2 e+2 a c e\right ) x^2}{\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}} \, dx,x,\sqrt {d+e x}\right )}{c^3 e^2} \\ & = -\frac {2 \left (b c d-b^2 e+a c e\right ) \sqrt {d+e x}}{c^3}-\frac {2 b (d+e x)^{3/2}}{3 c^2}+\frac {2 (d+e x)^{5/2}}{5 c e}-\frac {\left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )-\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b^2-4 a c}}{2 e}-\frac {2 c d-b e}{2 e^2}+\frac {c x^2}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{c^3 e^2}-\frac {\left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )+\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {\sqrt {b^2-4 a c}}{2 e}-\frac {2 c d-b e}{2 e^2}+\frac {c x^2}{e^2}} \, dx,x,\sqrt {d+e x}\right )}{c^3 e^2} \\ & = -\frac {2 \left (b c d-b^2 e+a c e\right ) \sqrt {d+e x}}{c^3}-\frac {2 b (d+e x)^{3/2}}{3 c^2}+\frac {2 (d+e x)^{5/2}}{5 c e}+\frac {\sqrt {2} \left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )+\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\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 )}{c^{7/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left ((c d-b e) \left (b c d-b^2 e+2 a c e\right )-\frac {2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )}{\sqrt {b^2-4 a c}}\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 )}{c^{7/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.72 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.22 \[ \int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {d+e x} \left (15 b^2 e^2+3 c^2 (d+e x)^2-5 c e (4 b d+3 a e+b e x)\right )}{e}-\frac {15 \sqrt {2} \left (-b^4 e^2+b^3 e \left (2 c d+\sqrt {b^2-4 a c} e\right )+b c \left (-2 a \sqrt {b^2-4 a c} e^2+c d \left (\sqrt {b^2-4 a c} d-6 a e\right )\right )-b^2 c \left (c d^2+2 e \left (\sqrt {b^2-4 a c} d-2 a e\right )\right )+2 a c^2 \left (c d^2+e \left (\sqrt {b^2-4 a c} d-a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {15 \sqrt {2} \left (b^4 e^2+b^3 e \left (-2 c d+\sqrt {b^2-4 a c} e\right )+2 a c^2 \left (-c d^2+e \left (\sqrt {b^2-4 a c} d+a e\right )\right )+b^2 c \left (c d^2-2 e \left (\sqrt {b^2-4 a c} d+2 a e\right )\right )+b c \left (-2 a \sqrt {b^2-4 a c} e^2+c d \left (\sqrt {b^2-4 a c} d+6 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{15 c^{7/2}} \]

[In]

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

[Out]

((2*Sqrt[c]*Sqrt[d + e*x]*(15*b^2*e^2 + 3*c^2*(d + e*x)^2 - 5*c*e*(4*b*d + 3*a*e + b*e*x)))/e - (15*Sqrt[2]*(-
(b^4*e^2) + b^3*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e) + b*c*(-2*a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d -
6*a*e)) - b^2*c*(c*d^2 + 2*e*(Sqrt[b^2 - 4*a*c]*d - 2*a*e)) + 2*a*c^2*(c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - a*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]) - (15*Sqrt[2]*(b^4*e^2 + b^3*e*(-2*c*d + Sqrt[b^2 - 4*a*c]*e) + 2*a*c^2*(-(
c*d^2) + e*(Sqrt[b^2 - 4*a*c]*d + a*e)) + b^2*c*(c*d^2 - 2*e*(Sqrt[b^2 - 4*a*c]*d + 2*a*e)) + b*c*(-2*a*Sqrt[b
^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d + 6*a*e)))*ArcTan[(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]))/(15*c^(7/2))

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.23

method	result	size
pseudoelliptic	\(-\frac {2 \left (\sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, e \left (\left (\left (a c -\frac {b^{2}}{2}\right ) e +\frac {b c d}{2}\right ) \left (b e -c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (\left (a^{2} c^{2}-2 a \,b^{2} c +\frac {1}{2} b^{4}\right ) e^{2}+d \left (3 b a \,c^{2}-b^{3} c \right ) e -d^{2} \left (a c -\frac {b^{2}}{2}\right ) c^{2}\right )\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\sqrt {2}\, e \left (-\left (\left (a c -\frac {b^{2}}{2}\right ) e +\frac {b c d}{2}\right ) \left (b e -c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (\left (a^{2} c^{2}-2 a \,b^{2} c +\frac {1}{2} b^{4}\right ) e^{2}+d \left (3 b a \,c^{2}-b^{3} c \right ) e -d^{2} \left (a c -\frac {b^{2}}{2}\right ) c^{2}\right )\right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (\left (-\frac {c^{2} x^{2}}{5}+\left (\frac {b x}{3}+a \right ) c -b^{2}\right ) e^{2}+\frac {4 d \left (-\frac {3 c x}{10}+b \right ) c e}{3}-\frac {c^{2} d^{2}}{5}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {e x +d}\right )\right )}{\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, e \,c^{3}}\)	\(542\)
risch	\(-\frac {2 \left (-3 c^{2} x^{2} e^{2}+5 b c \,e^{2} x -6 c^{2} d e x +15 a c \,e^{2}-15 b^{2} e^{2}+20 b c d e -3 c^{2} d^{2}\right ) \sqrt {e x +d}}{15 e \,c^{3}}+\frac {-\frac {\left (2 a^{2} c^{2} e^{3}-4 a \,b^{2} c \,e^{3}+6 a b \,c^{2} d \,e^{2}-2 a \,c^{3} d^{2} e +b^{4} e^{3}-2 b^{3} c d \,e^{2}+b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{3}+4 a \,b^{2} c \,e^{3}-6 a b \,c^{2} d \,e^{2}+2 a \,c^{3} d^{2} e -b^{4} e^{3}+2 b^{3} c d \,e^{2}-b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c^{2}}\)	\(670\)
derivativedivides	\(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}+a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}+b c d e \sqrt {e x +d}\right )}{c^{3}}+\frac {8 e \left (-\frac {\left (2 a^{2} c^{2} e^{3}-4 a \,b^{2} c \,e^{3}+6 a b \,c^{2} d \,e^{2}-2 a \,c^{3} d^{2} e +b^{4} e^{3}-2 b^{3} c d \,e^{2}+b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{3}+4 a \,b^{2} c \,e^{3}-6 a b \,c^{2} d \,e^{2}+2 a \,c^{3} d^{2} e -b^{4} e^{3}+2 b^{3} c d \,e^{2}-b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{2}}}{e}\)	\(674\)
default	\(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}+a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}+b c d e \sqrt {e x +d}\right )}{c^{3}}+\frac {8 e \left (-\frac {\left (2 a^{2} c^{2} e^{3}-4 a \,b^{2} c \,e^{3}+6 a b \,c^{2} d \,e^{2}-2 a \,c^{3} d^{2} e +b^{4} e^{3}-2 b^{3} c d \,e^{2}+b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{3}+4 a \,b^{2} c \,e^{3}-6 a b \,c^{2} d \,e^{2}+2 a \,c^{3} d^{2} e -b^{4} e^{3}+2 b^{3} c d \,e^{2}-b^{2} c^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{2}}}{e}\)	\(674\)

[In]

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

[Out]

-2/(-4*e^2*(a*c-1/4*b^2))^(1/2)*(2^(1/2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*e*(((a*c-1/2*b^2)*
e+1/2*b*c*d)*(b*e-c*d)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+e*((a^2*c^2-2*a*b^2*c+1/2*b^4)*e^2+d*(3*a*b*c^2-b^3*c)*e-d
^2*(a*c-1/2*b^2)*c^2))*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+((
-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(2^(1/2)*e*(-((a*c-1/2*b^2)*e+1/2*b*c*d)*(b*e-c*d)*(-4*e^2*(
a*c-1/4*b^2))^(1/2)+e*((a^2*c^2-2*a*b^2*c+1/2*b^4)*e^2+d*(3*a*b*c^2-b^3*c)*e-d^2*(a*c-1/2*b^2)*c^2))*arctan(c*
(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+(-4*e^2*(a*c-1/4*b^2))^(1/2)*((-1/5*
c^2*x^2+(1/3*b*x+a)*c-b^2)*e^2+4/3*d*(-3/10*c*x+b)*c*e-1/5*c^2*d^2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*
c)^(1/2)*(e*x+d)^(1/2)))/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2)
)^(1/2))*c)^(1/2)/e/c^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8530 vs. \(2 (391) = 782\).

Time = 3.55 (sec) , antiderivative size = 8530, normalized size of antiderivative = 19.34 \[ \int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1195 vs. \(2 (391) = 782\).

Time = 0.37 (sec) , antiderivative size = 1195, normalized size of antiderivative = 2.71 \[ \int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^3*c^2 - 4*a*b*c^3)*d^2 - 2*(b^4*c - 5*a*b^2*c^2 + 4
*a^2*c^3)*d*e + (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*e^2)*c^2*e^2 - 2*(sqrt(b^2 - 4*a*c)*b*c^4*d^3 + sqrt(b^2 - 4*a
*c)*b^3*c^2*d*e^2 - (2*b^2*c^3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2*e - (a*b^2*c^2 - a^2*c^3)*sqrt(b^2 - 4*a*c)*e^3)
*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) + (2*(b^2*c^5 - 2*a*c^6)*d^3*e - (5*b^3*c^4 -
14*a*b*c^5)*d^2*e^2 + 2*(2*b^4*c^3 - 7*a*b^2*c^4 + 2*a^2*c^5)*d*e^3 - (b^5*c^2 - 4*a*b^3*c^3 + 2*a^2*b*c^4)*e^
4)*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^6*d*e^6 - b*c
^5*e^7 + sqrt(-4*(c^6*d^2*e^6 - b*c^5*d*e^7 + a*c^5*e^8)*c^6*e^6 + (2*c^6*d*e^6 - b*c^5*e^7)^2))/(c^6*e^6)))/(
(sqrt(b^2 - 4*a*c)*c^6*d^2 - sqrt(b^2 - 4*a*c)*b*c^5*d*e + sqrt(b^2 - 4*a*c)*a*c^5*e^2)*c^2*abs(e)) + 1/4*(sqr
t(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*((b^3*c^2 - 4*a*b*c^3)*d^2 - 2*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)
*d*e + (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*e^2)*c^2*e^2 + 2*(sqrt(b^2 - 4*a*c)*b*c^4*d^3 + sqrt(b^2 - 4*a*c)*b^3*c
^2*d*e^2 - (2*b^2*c^3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2*e - (a*b^2*c^2 - a^2*c^3)*sqrt(b^2 - 4*a*c)*e^3)*sqrt(-4*
c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) + (2*(b^2*c^5 - 2*a*c^6)*d^3*e - (5*b^3*c^4 - 14*a*b*c^
5)*d^2*e^2 + 2*(2*b^4*c^3 - 7*a*b^2*c^4 + 2*a^2*c^5)*d*e^3 - (b^5*c^2 - 4*a*b^3*c^3 + 2*a^2*b*c^4)*e^4)*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^6*d*e^6 - b*c^5*e^7 -
sqrt(-4*(c^6*d^2*e^6 - b*c^5*d*e^7 + a*c^5*e^8)*c^6*e^6 + (2*c^6*d*e^6 - b*c^5*e^7)^2))/(c^6*e^6)))/((sqrt(b^2
- 4*a*c)*c^6*d^2 - sqrt(b^2 - 4*a*c)*b*c^5*d*e + sqrt(b^2 - 4*a*c)*a*c^5*e^2)*c^2*abs(e)) + 2/15*(3*(e*x + d)
^(5/2)*c^4*e^4 - 5*(e*x + d)^(3/2)*b*c^3*e^5 - 15*sqrt(e*x + d)*b*c^3*d*e^5 + 15*sqrt(e*x + d)*b^2*c^2*e^6 - 1
5*sqrt(e*x + d)*a*c^3*e^6)/(c^5*e^5)

Mupad [B] (verification not implemented)

Time = 14.90 (sec) , antiderivative size = 19465, normalized size of antiderivative = 44.14 \[ \int \frac {x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((((8*(4*a^3*c^6*e^5 + a*b^4*c^4*e^5 - b^5*c^4*d*e^4 - 5*a^2*b^2*c^5*e^5 + 4*a^2*c^7*d^2*e^3 - b^3*c^6*d^
3*e^2 + 2*b^4*c^5*d^2*e^3 + 4*a*b*c^7*d^3*e^2 + 4*a*b^3*c^5*d*e^4 - 9*a*b^2*c^6*d^2*e^3))/c^5 - (8*(d + e*x)^(
1/2)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*
c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 +
a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 -
6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c
- b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)
^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b
^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^
3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 -
8*a*b^2*c^8)))^(1/2)*(b^3*c^7*e^3 - 2*b^2*c^8*d*e^2 - 4*a*b*c^8*e^3 + 8*a*c^9*d*e^2))/c^5)*(-(b^9*e^3 + 8*a^3*
c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e
^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b
^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*
a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b
^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c
^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c^2
*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b
^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (
8*(d + e*x)^(1/2)*(b^8*e^6 + 2*a^4*c^4*e^6 + 20*a^2*b^4*c^2*e^6 - 16*a^3*b^2*c^3*e^6 + 2*a^2*c^6*d^4*e^2 - 12*
a^3*c^5*d^2*e^4 + b^4*c^4*d^4*e^2 - 4*b^5*c^3*d^3*e^3 + 6*b^6*c^2*d^2*e^4 - 8*a*b^6*c*e^6 - 4*b^7*c*d*e^5 + 54
*a^2*b^2*c^4*d^2*e^4 + 28*a*b^5*c^2*d*e^5 + 28*a^3*b*c^4*d*e^5 - 4*a*b^2*c^5*d^4*e^2 + 20*a*b^3*c^4*d^3*e^3 -
36*a*b^4*c^3*d^2*e^4 - 20*a^2*b*c^5*d^3*e^3 - 56*a^2*b^3*c^3*d*e^5))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3
*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^
2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^
3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/
2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30
*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2
*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c -
b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*
a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*1i - (((8*(4*a^3*c^6
*e^5 + a*b^4*c^4*e^5 - b^5*c^4*d*e^4 - 5*a^2*b^2*c^5*e^5 + 4*a^2*c^7*d^2*e^3 - b^3*c^6*d^3*e^2 + 2*b^4*c^5*d^2
*e^3 + 4*a*b*c^7*d^3*e^2 + 4*a*b^3*c^5*d*e^4 - 9*a*b^2*c^6*d^2*e^3))/c^5 + (8*(d + e*x)^(1/2)*(-(b^9*e^3 + 8*a
^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*
d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c
- b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-
(4*a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*
a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^
3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*
c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c
- b^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*
(b^3*c^7*e^3 - 2*b^2*c^8*d*e^2 - 4*a*b*c^8*e^3 + 8*a*c^9*d*e^2))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3
- b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e
- 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^
3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) -
2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b
^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4
*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c - b^2
)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*
b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) + (8*(d + e*x)^(1/2)*(b^
8*e^6 + 2*a^4*c^4*e^6 + 20*a^2*b^4*c^2*e^6 - 16*a^3*b^2*c^3*e^6 + 2*a^2*c^6*d^4*e^2 - 12*a^3*c^5*d^2*e^4 + b^4
*c^4*d^4*e^2 - 4*b^5*c^3*d^3*e^3 + 6*b^6*c^2*d^2*e^4 - 8*a*b^6*c*e^6 - 4*b^7*c*d*e^5 + 54*a^2*b^2*c^4*d^2*e^4
+ 28*a*b^5*c^2*d*e^5 + 28*a^3*b*c^4*d*e^5 - 4*a*b^2*c^5*d^4*e^2 + 20*a*b^3*c^4*d^3*e^3 - 36*a*b^4*c^3*d^2*e^4
- 20*a^2*b*c^5*d^3*e^3 - 56*a^2*b^3*c^3*d*e^5))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a
*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*
d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c -
b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-
(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60
*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*
a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*
b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*
a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*1i)/((((8*(4*a^3*c^6*e^5 + a*b^4*c^4*e^5
- b^5*c^4*d*e^4 - 5*a^2*b^2*c^5*e^5 + 4*a^2*c^7*d^2*e^3 - b^3*c^6*d^3*e^2 + 2*b^4*c^5*d^2*e^3 + 4*a*b*c^7*d^3*
e^2 + 4*a*b^3*c^5*d*e^4 - 9*a*b^2*c^6*d^2*e^3))/c^5 - (8*(d + e*x)^(1/2)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*
d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2
*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3
*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2
) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*
a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*
b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c -
b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a
^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*(b^3*c^7*e^3 - 2*b^2*
c^8*d*e^2 - 4*a*b*c^8*e^3 + 8*a*c^9*d*e^2))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c -
b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3
+ 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)
^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a
*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3
*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*
b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*
c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c
- b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (8*(d + e*x)^(1/2)*(b^8*e^6 + 2*a^4*c^4*e^6
+ 20*a^2*b^4*c^2*e^6 - 16*a^3*b^2*c^3*e^6 + 2*a^2*c^6*d^4*e^2 - 12*a^3*c^5*d^2*e^4 + b^4*c^4*d^4*e^2 - 4*b^5*
c^3*d^3*e^3 + 6*b^6*c^2*d^2*e^4 - 8*a*b^6*c*e^6 - 4*b^7*c*d*e^5 + 54*a^2*b^2*c^4*d^2*e^4 + 28*a*b^5*c^2*d*e^5
+ 28*a^3*b*c^4*d*e^5 - 4*a*b^2*c^5*d^4*e^2 + 20*a*b^3*c^4*d^3*e^3 - 36*a*b^4*c^3*d^2*e^4 - 20*a^2*b*c^5*d^3*e^
3 - 56*a^2*b^3*c^3*d*e^5))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) +
8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*
e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*
b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2
) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*
b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3
*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*
c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/
(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) + (((8*(4*a^3*c^6*e^5 + a*b^4*c^4*e^5 - b^5*c^4*d*e^4 - 5*a^2*
b^2*c^5*e^5 + 4*a^2*c^7*d^2*e^3 - b^3*c^6*d^3*e^2 + 2*b^4*c^5*d^2*e^3 + 4*a*b*c^7*d^3*e^2 + 4*a*b^3*c^5*d*e^4
- 9*a*b^2*c^6*d^2*e^3))/c^5 + (8*(d + e*x)^(1/2)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c -
b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3
+ 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)
^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a
*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3
*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*
b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*
c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c
- b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*(b^3*c^7*e^3 - 2*b^2*c^8*d*e^2 - 4*a*b*c^8*e^
3 + 8*a*c^9*d*e^2))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4
*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 6
3*a^3*b^3*c^3*e^3 + a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e
^3 - 3*b^8*c*d*e^2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a
*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d
*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^
4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2
)^3)^(1/2) - 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*
a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) + (8*(d + e*x)^(1/2)*(b^8*e^6 + 2*a^4*c^4*e^6 + 20*a^2*b^4*c^2*e^6 -
16*a^3*b^2*c^3*e^6 + 2*a^2*c^6*d^4*e^2 - 12*a^3*c^5*d^2*e^4 + b^4*c^4*d^4*e^2 - 4*b^5*c^3*d^3*e^3 + 6*b^6*c^2*
d^2*e^4 - 8*a*b^6*c*e^6 - 4*b^7*c*d*e^5 + 54*a^2*b^2*c^4*d^2*e^4 + 28*a*b^5*c^2*d*e^5 + 28*a^3*b*c^4*d*e^5 - 4
*a*b^2*c^5*d^4*e^2 + 20*a*b^3*c^4*d^3*e^3 - 36*a*b^4*c^3*d^2*e^4 - 20*a^2*b*c^5*d^3*e^3 - 56*a^2*b^3*c^3*d*e^5
))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4
*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3
+ a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^
2 - 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a
*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b
^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c
- b^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a
*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7
- 8*a*b^2*c^8)))^(1/2) - (16*(a^4*b^3*e^8 - 2*a^3*b^4*d*e^7 + 2*a^5*c^2*d*e^7 + a^2*b^5*d^2*e^6 + 2*a^3*c^4*d
^5*e^3 + 4*a^4*c^3*d^3*e^5 - 2*a^5*b*c*e^8 - 4*a^2*b^2*c^3*d^5*e^3 + 6*a^2*b^3*c^2*d^4*e^4 + 2*a^4*b^2*c*d*e^7
+ a^2*b*c^4*d^6*e^2 - 4*a^2*b^4*c*d^3*e^5 - 4*a^3*b*c^3*d^4*e^4 + 4*a^3*b^3*c*d^2*e^6 - 7*a^4*b*c^2*d^2*e^6))
/c^5))*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 - b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*
b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3
+ a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) + b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2
- 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e^3*(-(4*a*
c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e + 3*b^5*c*d*e^2*(-(4*a*c - b^
2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 - 3*a^2*c^4*d^2*e*(-(4*a*c -
b^2)^3)^(1/2) - 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 12*a*
b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7
- 8*a*b^2*c^8)))^(1/2)*2i + atan(((((8*(4*a^3*c^6*e^5 + a*b^4*c^4*e^5 - b^5*c^4*d*e^4 - 5*a^2*b^2*c^5*e^5 + 4*
a^2*c^7*d^2*e^3 - b^3*c^6*d^3*e^2 + 2*b^4*c^5*d^2*e^3 + 4*a*b*c^7*d^3*e^2 + 4*a*b^3*c^5*d*e^4 - 9*a*b^2*c^6*d^
2*e^3))/c^5 - (8*(d + e*x)^(1/2)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 + b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) +
8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2
*e^3 - 63*a^3*b^3*c^3*e^3 - a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a
*b^7*c*e^3 - 3*b^8*c*d*e^2 + 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/
2) - 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e - 3
*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 +
3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^2*c^3*d^2*e*(-(4*a
*c - b^2)^3)^(1/2) + 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))
/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*(b^3*c^7*e^3 - 2*b^2*c^8*d*e^2 - 4*a*b*c^8*e^3 + 8*a*c^9*d*e^
2))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 + b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^
4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^
3 - a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e
^2 + 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*e^3*(-(4*
a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e - 3*b^5*c*d*e^2*(-(4*a*c -
b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 + 3*a^2*c^4*d^2*e*(-(4*a*c
- b^2)^3)^(1/2) + 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*
a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^
7 - 8*a*b^2*c^8)))^(1/2) - (8*(d + e*x)^(1/2)*(b^8*e^6 + 2*a^4*c^4*e^6 + 20*a^2*b^4*c^2*e^6 - 16*a^3*b^2*c^3*e
^6 + 2*a^2*c^6*d^4*e^2 - 12*a^3*c^5*d^2*e^4 + b^4*c^4*d^4*e^2 - 4*b^5*c^3*d^3*e^3 + 6*b^6*c^2*d^2*e^4 - 8*a*b^
6*c*e^6 - 4*b^7*c*d*e^5 + 54*a^2*b^2*c^4*d^2*e^4 + 28*a*b^5*c^2*d*e^5 + 28*a^3*b*c^4*d*e^5 - 4*a*b^2*c^5*d^4*e
^2 + 20*a*b^3*c^4*d^3*e^3 - 36*a*b^4*c^3*d^2*e^4 - 20*a^2*b*c^5*d^3*e^3 - 56*a^2*b^3*c^3*d*e^5))/c^5)*(-(b^9*e
^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 + b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*
a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 - a^3*c^3*e^3*(
-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 + 6*a^2*b^2*c^
2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/
2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e - 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 7
5*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 + 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2)
+ 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^3*c^2*d*e^2*(
-(4*a*c - b^2)^3)^(1/2) - 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8))
)^(1/2)*1i - (((8*(4*a^3*c^6*e^5 + a*b^4*c^4*e^5 - b^5*c^4*d*e^4 - 5*a^2*b^2*c^5*e^5 + 4*a^2*c^7*d^2*e^3 - b^3
*c^6*d^3*e^2 + 2*b^4*c^5*d^2*e^3 + 4*a*b*c^7*d^3*e^2 + 4*a*b^3*c^5*d*e^4 - 9*a*b^2*c^6*d^2*e^3))/c^5 + (8*(d +
e*x)^(1/2)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 + b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28
*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3
*e^3 - a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*
d*e^2 + 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*e^3*(-
(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e - 3*b^5*c*d*e^2*(-(4*a*c
- b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 + 3*a^2*c^4*d^2*e*(-(4*
a*c - b^2)^3)^(1/2) + 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) +
12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4
*c^7 - 8*a*b^2*c^8)))^(1/2)*(b^3*c^7*e^3 - 2*b^2*c^8*d*e^2 - 4*a*b*c^8*e^3 + 8*a*c^9*d*e^2))/c^5)*(-(b^9*e^3 +
8*a^3*c^6*d^3 - b^6*c^3*d^3 + b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*
c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 - a^3*c^3*e^3*(-(4*
a*c - b^2)^3)^(1/2) - b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 + 6*a^2*b^2*c^2*e^
3*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) -
27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e - 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^
2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 + 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*
b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^3*c^2*d*e^2*(-(4*
a*c - b^2)^3)^(1/2) - 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1
/2) + (8*(d + e*x)^(1/2)*(b^8*e^6 + 2*a^4*c^4*e^6 + 20*a^2*b^4*c^2*e^6 - 16*a^3*b^2*c^3*e^6 + 2*a^2*c^6*d^4*e^
2 - 12*a^3*c^5*d^2*e^4 + b^4*c^4*d^4*e^2 - 4*b^5*c^3*d^3*e^3 + 6*b^6*c^2*d^2*e^4 - 8*a*b^6*c*e^6 - 4*b^7*c*d*e
^5 + 54*a^2*b^2*c^4*d^2*e^4 + 28*a*b^5*c^2*d*e^5 + 28*a^3*b*c^4*d*e^5 - 4*a*b^2*c^5*d^4*e^2 + 20*a*b^3*c^4*d^3
*e^3 - 36*a*b^4*c^3*d^2*e^4 - 20*a^2*b*c^5*d^3*e^3 - 56*a^2*b^3*c^3*d*e^5))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 -
b^6*c^3*d^3 + b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7
*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 - a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/
2) - b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 + 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)
^3)^(1/2) + 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2
*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e - 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e -
99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 + 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^4*c^2*d^2*e*(-(
4*a*c - b^2)^3)^(1/2) - 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/
2) - 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*1i)/((((8*(4*
a^3*c^6*e^5 + a*b^4*c^4*e^5 - b^5*c^4*d*e^4 - 5*a^2*b^2*c^5*e^5 + 4*a^2*c^7*d^2*e^3 - b^3*c^6*d^3*e^2 + 2*b^4*
c^5*d^2*e^3 + 4*a*b*c^7*d^3*e^2 + 4*a*b^3*c^5*d*e^4 - 9*a*b^2*c^6*d^2*e^3))/c^5 - (8*(d + e*x)^(1/2)*(-(b^9*e^
3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 + b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a
^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 - a^3*c^3*e^3*(-
(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 + 6*a^2*b^2*c^2
*e^3*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2
) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e - 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75
*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 + 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) +
3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^3*c^2*d*e^2*(-
(4*a*c - b^2)^3)^(1/2) - 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))
^(1/2)*(b^3*c^7*e^3 - 2*b^2*c^8*d*e^2 - 4*a*b*c^8*e^3 + 8*a*c^9*d*e^2))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*
c^3*d^3 + b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2
*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 - a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) -
b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 + 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^
(1/2) + 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e +
30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e - 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*
a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 + 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^4*c^2*d^2*e*(-(4*a*
c - b^2)^3)^(1/2) - 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) -
9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (8*(d + e*x)^(1
/2)*(b^8*e^6 + 2*a^4*c^4*e^6 + 20*a^2*b^4*c^2*e^6 - 16*a^3*b^2*c^3*e^6 + 2*a^2*c^6*d^4*e^2 - 12*a^3*c^5*d^2*e^
4 + b^4*c^4*d^4*e^2 - 4*b^5*c^3*d^3*e^3 + 6*b^6*c^2*d^2*e^4 - 8*a*b^6*c*e^6 - 4*b^7*c*d*e^5 + 54*a^2*b^2*c^4*d
^2*e^4 + 28*a*b^5*c^2*d*e^5 + 28*a^3*b*c^4*d*e^5 - 4*a*b^2*c^5*d^4*e^2 + 20*a*b^3*c^4*d^3*e^3 - 36*a*b^4*c^3*d
^2*e^4 - 20*a^2*b*c^5*d^3*e^3 - 56*a^2*b^3*c^3*d*e^5))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 + b^6*e^3
*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b
^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 - a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^3*(-(4
*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 + 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^4
*d^3*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e
^2 - 60*a^3*b*c^5*d^2*e - 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2
+ 114*a^3*b^2*c^4*d*e^2 + 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2)
- 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a^2*b*c^3*d*e^
2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) + (((8*(4*a^3*c^6*e^5 + a*b^4*c^4*
e^5 - b^5*c^4*d*e^4 - 5*a^2*b^2*c^5*e^5 + 4*a^2*c^7*d^2*e^3 - b^3*c^6*d^3*e^2 + 2*b^4*c^5*d^2*e^3 + 4*a*b*c^7*
d^3*e^2 + 4*a*b^3*c^5*d*e^4 - 9*a*b^2*c^6*d^2*e^3))/c^5 + (8*(d + e*x)^(1/2)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*
c^3*d^3 + b^6*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2
*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 - a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) -
b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 + 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^
(1/2) + 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e +
30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e - 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*
a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2 + 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^4*c^2*d^2*e*(-(4*a*
c - b^2)^3)^(1/2) - 9*a*b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) -
9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*(b^3*c^7*e^3 - 2*
b^2*c^8*d*e^2 - 4*a*b*c^8*e^3 + 8*a*c^9*d*e^2))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 + b^6*e^3*(-(4*a
*c - b^2)^3)^(1/2) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*
d^3 + 42*a^2*b^5*c^2*e^3 - 63*a^3*b^3*c^3*e^3 - a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^3*(-(4*a*c -
b^2)^3)^(1/2) - 11*a*b^7*c*e^3 - 3*b^8*c*d*e^2 + 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^4*d^3*(-
(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60
*a^3*b*c^5*d^2*e - 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*
a^3*b^2*c^4*d*e^2 + 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*
b^2*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a^2*b*c^3*d*e^2*(-(4*
a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) + (8*(d + e*x)^(1/2)*(b^8*e^6 + 2*a^4*c^4
*e^6 + 20*a^2*b^4*c^2*e^6 - 16*a^3*b^2*c^3*e^6 + 2*a^2*c^6*d^4*e^2 - 12*a^3*c^5*d^2*e^4 + b^4*c^4*d^4*e^2 - 4*
b^5*c^3*d^3*e^3 + 6*b^6*c^2*d^2*e^4 - 8*a*b^6*c*e^6 - 4*b^7*c*d*e^5 + 54*a^2*b^2*c^4*d^2*e^4 + 28*a*b^5*c^2*d*
e^5 + 28*a^3*b*c^4*d*e^5 - 4*a*b^2*c^5*d^4*e^2 + 20*a*b^3*c^4*d^3*e^3 - 36*a*b^4*c^3*d^2*e^4 - 20*a^2*b*c^5*d^
3*e^3 - 56*a^2*b^3*c^3*d*e^5))/c^5)*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 + b^6*e^3*(-(4*a*c - b^2)^3)^(1/2
) + 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*
c^2*e^3 - 63*a^3*b^3*c^3*e^3 - a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 1
1*a*b^7*c*e^3 - 3*b^8*c*d*e^2 + 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^
(1/2) - 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e
- 3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2
+ 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^2*c^3*d^2*e*(-(
4*a*c - b^2)^3)^(1/2) + 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/
2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (16*(a^4*b^3*e^8 - 2*a^3*b^4*d*e^7 + 2*a^5*c^2*d*e^7 + a
^2*b^5*d^2*e^6 + 2*a^3*c^4*d^5*e^3 + 4*a^4*c^3*d^3*e^5 - 2*a^5*b*c*e^8 - 4*a^2*b^2*c^3*d^5*e^3 + 6*a^2*b^3*c^2
*d^4*e^4 + 2*a^4*b^2*c*d*e^7 + a^2*b*c^4*d^6*e^2 - 4*a^2*b^4*c*d^3*e^5 - 4*a^3*b*c^3*d^4*e^4 + 4*a^3*b^3*c*d^2
*e^6 - 7*a^4*b*c^2*d^2*e^6))/c^5))*(-(b^9*e^3 + 8*a^3*c^6*d^3 - b^6*c^3*d^3 + b^6*e^3*(-(4*a*c - b^2)^3)^(1/2)
+ 8*a*b^4*c^4*d^3 + 28*a^4*b*c^4*e^3 - 24*a^4*c^5*d*e^2 + 3*b^7*c^2*d^2*e - 18*a^2*b^2*c^5*d^3 + 42*a^2*b^5*c
^2*e^3 - 63*a^3*b^3*c^3*e^3 - a^3*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11
*a*b^7*c*e^3 - 3*b^8*c*d*e^2 + 6*a^2*b^2*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^4*d^3*(-(4*a*c - b^2)^3)^(
1/2) - 5*a*b^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^5*c^3*d^2*e + 30*a*b^6*c^2*d*e^2 - 60*a^3*b*c^5*d^2*e -
3*b^5*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 75*a^2*b^3*c^4*d^2*e - 99*a^2*b^4*c^3*d*e^2 + 114*a^3*b^2*c^4*d*e^2
+ 3*a^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^4*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^2*c^3*d^2*e*(-(4
*a*c - b^2)^3)^(1/2) + 12*a*b^3*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a^2*b*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2
))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2)*2i + (d + e*x)^(1/2)*((2*d^2)/(c*e) - (2*(a*e^3 - b*d*e^2 +
c*d^2*e))/(c^2*e^2) + (((4*d)/(c*e) + (2*(b*e^2 - 2*c*d*e))/(c^2*e^2))*(b*e^2 - 2*c*d*e))/(c*e)) - ((4*d)/(3*
c*e) + (2*(b*e^2 - 2*c*d*e))/(3*c^2*e^2))*(d + e*x)^(3/2) + (2*(d + e*x)^(5/2))/(5*c*e)