3.15.34 \(\int \frac {(b+2 c x) \sqrt {d+e x}}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=463 \[ \frac {\sqrt {c} e \left (-2 c e \left (-d \sqrt {b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (\sqrt {b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {c} e \left (-2 c e \left (d \sqrt {b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (b-\sqrt {b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac {e \sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x}}{2 \left (a+b x+c x^2\right )^2} \]

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Rubi [A]  time = 1.89, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {768, 740, 826, 1166, 208} \begin {gather*} \frac {\sqrt {c} e \left (-2 c e \left (-d \sqrt {b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (\sqrt {b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\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 )}{4 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {c} e \left (-2 c e \left (d \sqrt {b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (b-\sqrt {b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\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 )}{4 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac {e \sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x}}{2 \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[d + e*x]/(2*(a + b*x + c*x^2)^2) - (e*Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(4*(b
^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) + (Sqrt[c]*e*(8*c^2*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*e^2
- 2*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - 6*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt
[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*
e^2)) - (Sqrt[c]*e*(8*c^2*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 6*a*e))*A
rcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(3/2
)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2))

Rule 208

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

Rule 740

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

Rule 768

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

Rule 826

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

Rule 1166

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

Rubi steps

\begin {align*} \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {\sqrt {d+e x}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{4} e \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {\sqrt {d+e x}}{2 \left (a+b x+c x^2\right )^2}-\frac {e \sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {e \int \frac {\frac {1}{2} \left (4 c^2 d^2-b^2 e^2-3 c e (b d-2 a e)\right )+\frac {1}{2} c e (2 c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x}}{2 \left (a+b x+c x^2\right )^2}-\frac {e \sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {e \operatorname {Subst}\left (\int \frac {-\frac {1}{2} c d e (2 c d-b e)+\frac {1}{2} e \left (4 c^2 d^2-b^2 e^2-3 c e (b d-2 a e)\right )+\frac {1}{2} c e (2 c d-b 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 )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x}}{2 \left (a+b x+c x^2\right )^2}-\frac {e \sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\left (c e \left (8 c^2 d^2-b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-6 a e\right )\right )\right ) \operatorname {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 )}{8 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )}+\frac {\left (c e \left (8 c^2 d^2-b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-6 a e\right )\right )\right ) \operatorname {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 )}{8 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x}}{2 \left (a+b x+c x^2\right )^2}-\frac {e \sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {c} e \left (8 c^2 d^2-b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-6 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 )}{4 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {c} e \left (8 c^2 d^2-b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-6 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 )}{4 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}\\ \end {align*}

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Mathematica [B]  time = 6.41, size = 1080, normalized size = 2.33 \begin {gather*} -\frac {\left (-2 a c (2 c d-b e)+b \left (-e b^2+c d b+2 a c e\right )+c (b (2 c d-b e)-2 c (b d-2 a e)) x\right ) (d+e x)^{3/2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {-\frac {\left (\frac {3}{2} a c \left (b^2-4 a c\right ) (2 c d-b e) e^2-\frac {1}{2} \left (b^2-4 a c\right ) (c d+b e) \left (-e b^2+c d b+2 a c e\right ) e+c \left (\frac {3}{2} c \left (b^2-4 a c\right ) e^2 (b d-2 a e)-\frac {1}{2} \left (b^2-4 a c\right ) e (2 c d-b e) (c d+b e)\right ) x\right ) (d+e x)^{3/2}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {\frac {1}{2} \left (b^2-4 a c\right ) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {d+e x} e^2+\frac {4 \left (\frac {\sqrt {2 c d-b e-\sqrt {b^2-4 a c} e} \left (-\frac {1}{8} c^2 \left (b^2-4 a c\right ) (2 c d-b e) \left (c d^2-e (b d-a e)\right ) e^2-\frac {\frac {1}{8} c^2 \left (b^2-4 a c\right ) (2 c d-b e) (b e-2 c d) \left (c d^2-e (b d-a e)\right ) e^2+2 c \left (\frac {1}{8} c^2 \left (b^2-4 a c\right ) d e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )-\frac {1}{8} c \left (b^2-4 a c\right ) e^2 \left (c d^2-b e d+a e^2\right ) \left (4 c^2 d^2-3 b c e d-b^2 e^2+6 a c e^2\right )\right )}{\sqrt {b^2-4 a c} e}\right ) \tanh ^{-1}\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 {2} \sqrt {c} \left (-2 c d+b e+\sqrt {b^2-4 a c} e\right )}+\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} \left (\frac {\frac {1}{8} c^2 \left (b^2-4 a c\right ) (2 c d-b e) (b e-2 c d) \left (c d^2-e (b d-a e)\right ) e^2+2 c \left (\frac {1}{8} c^2 \left (b^2-4 a c\right ) d e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )-\frac {1}{8} c \left (b^2-4 a c\right ) e^2 \left (c d^2-b e d+a e^2\right ) \left (4 c^2 d^2-3 b c e d-b^2 e^2+6 a c e^2\right )\right )}{\sqrt {b^2-4 a c} e}-\frac {1}{8} c^2 \left (b^2-4 a c\right ) e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )\right ) \tanh ^{-1}\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 {2} \sqrt {c} \left (-2 c d+b e-\sqrt {b^2-4 a c} e\right )}\right )}{c}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((d + e*x)^(3/2)*(-2*a*c*(2*c*d - b*e) + b*(b*c*d - b^2*e + 2*a*c*e) + c*(-2*c*(b*d - 2*a*e) + b*(2*c*d -
b*e))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (-(((d + e*x)^(3/2)*((3*a*c*(b^2 - 4*
a*c)*e^2*(2*c*d - b*e))/2 - ((b^2 - 4*a*c)*e*(c*d + b*e)*(b*c*d - b^2*e + 2*a*c*e))/2 + c*((3*c*(b^2 - 4*a*c)*
e^2*(b*d - 2*a*e))/2 - ((b^2 - 4*a*c)*e*(2*c*d - b*e)*(c*d + b*e))/2)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^
2)*(a + b*x + c*x^2))) - (((b^2 - 4*a*c)*e^2*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*Sqrt[d + e*x])/2 + (4
*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]*(-1/8*(c^2*(b^2 - 4*a*c)*e^2*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e)))
- ((c^2*(b^2 - 4*a*c)*e^2*(2*c*d - b*e)*(-2*c*d + b*e)*(c*d^2 - e*(b*d - a*e)))/8 + 2*c*(-1/8*(c*(b^2 - 4*a*c
)*e^2*(c*d^2 - b*d*e + a*e^2)*(4*c^2*d^2 - 3*b*c*d*e - b^2*e^2 + 6*a*c*e^2)) + (c^2*(b^2 - 4*a*c)*d*e^2*(2*c*d
- b*e)*(c*d^2 - e*(b*d - a*e)))/8))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d
- b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e)) + (Sqrt[2*c*d - b*e + S
qrt[b^2 - 4*a*c]*e]*(-1/8*(c^2*(b^2 - 4*a*c)*e^2*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))) + ((c^2*(b^2 - 4*a*c)*
e^2*(2*c*d - b*e)*(-2*c*d + b*e)*(c*d^2 - e*(b*d - a*e)))/8 + 2*c*(-1/8*(c*(b^2 - 4*a*c)*e^2*(c*d^2 - b*d*e +
a*e^2)*(4*c^2*d^2 - 3*b*c*d*e - b^2*e^2 + 6*a*c*e^2)) + (c^2*(b^2 - 4*a*c)*d*e^2*(2*c*d - b*e)*(c*d^2 - e*(b*d
- a*e)))/8))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a
*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e))))/c)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/
(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))

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IntegrateAlgebraic [C]  time = 17.62, size = 873, normalized size = 1.89 \begin {gather*} -\frac {\sqrt {d+e x} \left (2 c^3 d^4-4 b c^2 e d^3-6 c^3 (d+e x) d^3+8 a c^2 e^2 d^2+b^2 c e^2 d^2+6 c^3 (d+e x)^2 d^2+9 b c^2 e (d+e x) d^2+b^3 e^3 d-8 a b c e^3 d-2 c^3 (d+e x)^3 d-6 b c^2 e (d+e x)^2 d+2 a c^2 e^2 (d+e x) d-5 b^2 c e^2 (d+e x) d-a b^2 e^4+6 a^2 c e^4+b c^2 e (d+e x)^3-2 a c^2 e^2 (d+e x)^2+2 b^2 c e^2 (d+e x)^2+b^3 e^3 (d+e x)-a b c e^3 (d+e x)\right ) e^2}{4 \left (b^2-4 a c\right ) \left (-c d^2+b e d-a e^2\right ) \left (-c d^2+b e d+2 c (d+e x) d-a e^2-c (d+e x)^2-b e (d+e x)\right )^2}+\frac {\left (-12 i \sqrt {2} a c^{3/2} e^3+i \sqrt {2} b^2 \sqrt {c} e^3-\sqrt {2} b \sqrt {c} \sqrt {4 a c-b^2} e^3+8 i \sqrt {2} b c^{3/2} d e^2+2 \sqrt {2} c^{3/2} \sqrt {4 a c-b^2} d e^2-8 i \sqrt {2} c^{5/2} d^2 e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {4 a c-b^2} e}}\right )}{8 \left (b^2-4 a c\right ) \sqrt {4 a c-b^2} \sqrt {-2 c d+b e-i \sqrt {4 a c-b^2} e} \left (-c d^2+b e d-a e^2\right )}+\frac {\left (12 i \sqrt {2} a c^{3/2} e^3-i \sqrt {2} b^2 \sqrt {c} e^3-\sqrt {2} b \sqrt {c} \sqrt {4 a c-b^2} e^3-8 i \sqrt {2} b c^{3/2} d e^2+2 \sqrt {2} c^{3/2} \sqrt {4 a c-b^2} d e^2+8 i \sqrt {2} c^{5/2} d^2 e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {4 a c-b^2} e}}\right )}{8 \left (b^2-4 a c\right ) \sqrt {4 a c-b^2} \sqrt {-2 c d+b e+i \sqrt {4 a c-b^2} e} \left (-c d^2+b e d-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(e^2*Sqrt[d + e*x]*(2*c^3*d^4 - 4*b*c^2*d^3*e + b^2*c*d^2*e^2 + 8*a*c^2*d^2*e^2 + b^3*d*e^3 - 8*a*b*c*d*e
^3 - a*b^2*e^4 + 6*a^2*c*e^4 - 6*c^3*d^3*(d + e*x) + 9*b*c^2*d^2*e*(d + e*x) - 5*b^2*c*d*e^2*(d + e*x) + 2*a*c
^2*d*e^2*(d + e*x) + b^3*e^3*(d + e*x) - a*b*c*e^3*(d + e*x) + 6*c^3*d^2*(d + e*x)^2 - 6*b*c^2*d*e*(d + e*x)^2
+ 2*b^2*c*e^2*(d + e*x)^2 - 2*a*c^2*e^2*(d + e*x)^2 - 2*c^3*d*(d + e*x)^3 + b*c^2*e*(d + e*x)^3))/((b^2 - 4*a
*c)*(-(c*d^2) + b*d*e - a*e^2)*(-(c*d^2) + b*d*e - a*e^2 + 2*c*d*(d + e*x) - b*e*(d + e*x) - c*(d + e*x)^2)^2)
+ (((-8*I)*Sqrt[2]*c^(5/2)*d^2*e + (8*I)*Sqrt[2]*b*c^(3/2)*d*e^2 + 2*Sqrt[2]*c^(3/2)*Sqrt[-b^2 + 4*a*c]*d*e^2
+ I*Sqrt[2]*b^2*Sqrt[c]*e^3 - (12*I)*Sqrt[2]*a*c^(3/2)*e^3 - Sqrt[2]*b*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*e^3)*ArcTan
[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(8*(b^2 - 4*a*c)*Sqrt[-b^2 + 4*
a*c]*Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]*(-(c*d^2) + b*d*e - a*e^2)) + (((8*I)*Sqrt[2]*c^(5/2)*d^2*e -
(8*I)*Sqrt[2]*b*c^(3/2)*d*e^2 + 2*Sqrt[2]*c^(3/2)*Sqrt[-b^2 + 4*a*c]*d*e^2 - I*Sqrt[2]*b^2*Sqrt[c]*e^3 + (12*
I)*Sqrt[2]*a*c^(3/2)*e^3 - Sqrt[2]*b*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*e^3)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sq
rt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(8*(b^2 - 4*a*c)*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e + I*Sqrt[-b^
2 + 4*a*c]*e]*(-(c*d^2) + b*d*e - a*e^2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [B]  time = 5.53, size = 3071, normalized size = 6.63

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*((b^2*c*d^2*e - 4*a*c^2*d^2*e - b^3*d*e^2 + 4*a*b*c*d*e^2 + a*b^2*e^3 - 4*a^2*c*e^3)^2*sqrt(-4*c^2*d + 2
*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*c*d*e^2 - b*e^3) + 2*(2*sqrt(b^2 - 4*a*c)*c^3*d^4*e^2 - 4*sqrt(b^2 - 4*a*c)
*b*c^2*d^3*e^3 + (b^2*c + 8*a*c^2)*sqrt(b^2 - 4*a*c)*d^2*e^4 + (b^3 - 8*a*b*c)*sqrt(b^2 - 4*a*c)*d*e^5 - (a*b^
2 - 6*a^2*c)*sqrt(b^2 - 4*a*c)*e^6)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(-b^2*c*d^2*e + 4*a*c^
2*d^2*e + b^3*d*e^2 - 4*a*b*c*d*e^2 - a*b^2*e^3 + 4*a^2*c*e^3) - (16*(b^2*c^5 - 4*a*c^6)*d^7*e^2 - 56*(b^3*c^4
- 4*a*b*c^5)*d^6*e^3 + 14*(5*b^4*c^3 - 16*a*b^2*c^4 - 16*a^2*c^5)*d^5*e^4 - 35*(b^5*c^2 - 16*a^2*b*c^4)*d^4*e
^5 + 4*(b^6*c + 23*a*b^4*c^2 - 92*a^2*b^2*c^3 - 64*a^3*c^4)*d^3*e^6 + (b^7 - 26*a*b^5*c - 8*a^2*b^3*c^2 + 384*
a^3*b*c^3)*d^2*e^7 - 2*(a*b^6 - 19*a^2*b^4*c + 48*a^3*b^2*c^2 + 48*a^4*c^3)*d*e^8 + (a^2*b^5 - 16*a^3*b^3*c +
48*a^4*b*c^2)*e^9)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2
*b^2*c^2*d^3 - 8*a*c^3*d^3 - 3*b^3*c*d^2*e + 12*a*b*c^2*d^2*e + b^4*d*e^2 - 2*a*b^2*c*d*e^2 - 8*a^2*c^2*d*e^2
- a*b^3*e^3 + 4*a^2*b*c*e^3 + sqrt((2*b^2*c^2*d^3 - 8*a*c^3*d^3 - 3*b^3*c*d^2*e + 12*a*b*c^2*d^2*e + b^4*d*e^2
- 2*a*b^2*c*d*e^2 - 8*a^2*c^2*d*e^2 - a*b^3*e^3 + 4*a^2*b*c*e^3)^2 - 4*(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d
^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3
+ a^2*b^2*e^4 - 4*a^3*c*e^4)*(b^2*c^2*d^2 - 4*a*c^3*d^2 - b^3*c*d*e + 4*a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2
*e^2)))/(b^2*c^2*d^2 - 4*a*c^3*d^2 - b^3*c*d*e + 4*a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2*e^2)))/(((b^2*c^3 - 4
*a*c^4)*sqrt(b^2 - 4*a*c)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*sqrt(b^2 - 4*a*c)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a
^2*c^3)*sqrt(b^2 - 4*a*c)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*sqrt(b^2 - 4*a*c)*d^3*e^3 + 3*(a*b^4 - 3*
a^2*b^2*c - 4*a^3*c^2)*sqrt(b^2 - 4*a*c)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*sqrt(b^2 - 4*a*c)*d*e^5 + (a^3*b^2
- 4*a^4*c)*sqrt(b^2 - 4*a*c)*e^6)*abs(-b^2*c*d^2*e + 4*a*c^2*d^2*e + b^3*d*e^2 - 4*a*b*c*d*e^2 - a*b^2*e^3 + 4
*a^2*c*e^3)*abs(c)) + 1/32*((b^2*c*d^2*e - 4*a*c^2*d^2*e - b^3*d*e^2 + 4*a*b*c*d*e^2 + a*b^2*e^3 - 4*a^2*c*e^3
)^2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*c*d*e^2 - b*e^3) - 2*(2*sqrt(b^2 - 4*a*c)*c^3*d^4*e^2
- 4*sqrt(b^2 - 4*a*c)*b*c^2*d^3*e^3 + (b^2*c + 8*a*c^2)*sqrt(b^2 - 4*a*c)*d^2*e^4 + (b^3 - 8*a*b*c)*sqrt(b^2 -
4*a*c)*d*e^5 - (a*b^2 - 6*a^2*c)*sqrt(b^2 - 4*a*c)*e^6)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(
-b^2*c*d^2*e + 4*a*c^2*d^2*e + b^3*d*e^2 - 4*a*b*c*d*e^2 - a*b^2*e^3 + 4*a^2*c*e^3) - (16*(b^2*c^5 - 4*a*c^6)*
d^7*e^2 - 56*(b^3*c^4 - 4*a*b*c^5)*d^6*e^3 + 14*(5*b^4*c^3 - 16*a*b^2*c^4 - 16*a^2*c^5)*d^5*e^4 - 35*(b^5*c^2
- 16*a^2*b*c^4)*d^4*e^5 + 4*(b^6*c + 23*a*b^4*c^2 - 92*a^2*b^2*c^3 - 64*a^3*c^4)*d^3*e^6 + (b^7 - 26*a*b^5*c -
8*a^2*b^3*c^2 + 384*a^3*b*c^3)*d^2*e^7 - 2*(a*b^6 - 19*a^2*b^4*c + 48*a^3*b^2*c^2 + 48*a^4*c^3)*d*e^8 + (a^2*
b^5 - 16*a^3*b^3*c + 48*a^4*b*c^2)*e^9)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*s
qrt(x*e + d)/sqrt(-(2*b^2*c^2*d^3 - 8*a*c^3*d^3 - 3*b^3*c*d^2*e + 12*a*b*c^2*d^2*e + b^4*d*e^2 - 2*a*b^2*c*d*e
^2 - 8*a^2*c^2*d*e^2 - a*b^3*e^3 + 4*a^2*b*c*e^3 - sqrt((2*b^2*c^2*d^3 - 8*a*c^3*d^3 - 3*b^3*c*d^2*e + 12*a*b*
c^2*d^2*e + b^4*d*e^2 - 2*a*b^2*c*d*e^2 - 8*a^2*c^2*d*e^2 - a*b^3*e^3 + 4*a^2*b*c*e^3)^2 - 4*(b^2*c^2*d^4 - 4*
a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*
e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*e^4)*(b^2*c^2*d^2 - 4*a*c^3*d^2 - b^3*c*d*e + 4*a*b*c^2*d*e + a*
b^2*c*e^2 - 4*a^2*c^2*e^2)))/(b^2*c^2*d^2 - 4*a*c^3*d^2 - b^3*c*d*e + 4*a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2*
e^2)))/(((b^2*c^3 - 4*a*c^4)*sqrt(b^2 - 4*a*c)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*sqrt(b^2 - 4*a*c)*d^5*e + 3*(b^4*
c - 3*a*b^2*c^2 - 4*a^2*c^3)*sqrt(b^2 - 4*a*c)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*sqrt(b^2 - 4*a*c)*d^
3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*sqrt(b^2 - 4*a*c)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*sqrt(b^2 - 4*a
*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*sqrt(b^2 - 4*a*c)*e^6)*abs(-b^2*c*d^2*e + 4*a*c^2*d^2*e + b^3*d*e^2 - 4*a*b*c*
d*e^2 - a*b^2*e^3 + 4*a^2*c*e^3)*abs(c)) - 1/4*(2*(x*e + d)^(7/2)*c^3*d*e^2 - 6*(x*e + d)^(5/2)*c^3*d^2*e^2 +
6*(x*e + d)^(3/2)*c^3*d^3*e^2 - 2*sqrt(x*e + d)*c^3*d^4*e^2 - (x*e + d)^(7/2)*b*c^2*e^3 + 6*(x*e + d)^(5/2)*b*
c^2*d*e^3 - 9*(x*e + d)^(3/2)*b*c^2*d^2*e^3 + 4*sqrt(x*e + d)*b*c^2*d^3*e^3 - 2*(x*e + d)^(5/2)*b^2*c*e^4 + 2*
(x*e + d)^(5/2)*a*c^2*e^4 + 5*(x*e + d)^(3/2)*b^2*c*d*e^4 - 2*(x*e + d)^(3/2)*a*c^2*d*e^4 - sqrt(x*e + d)*b^2*
c*d^2*e^4 - 8*sqrt(x*e + d)*a*c^2*d^2*e^4 - (x*e + d)^(3/2)*b^3*e^5 + (x*e + d)^(3/2)*a*b*c*e^5 - sqrt(x*e + d
)*b^3*d*e^5 + 8*sqrt(x*e + d)*a*b*c*d*e^5 + sqrt(x*e + d)*a*b^2*e^6 - 6*sqrt(x*e + d)*a^2*c*e^6)/((b^2*c*d^2 -
4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 - 4*a^2*c*e^2)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*
e + d)*b*e - b*d*e + a*e^2)^2)

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maple [B]  time = 0.12, size = 3056, normalized size = 6.60 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*e^4/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/(
(b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2
))*c)^(1/2)*c)*b^2-e^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^3/(-(4*a*c-b^2)*e^2
)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a
*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d^2-3/2*e^4/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*
c^2/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(
1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a-1/4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a^2*c*e^2-a*b
^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(3/2)*b^3+e^3/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a
*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^2/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/
2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*d-3/2*e^4/(4*a^2*c*e^2-a
*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^2/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-
b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a+e^3
/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^2/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-
2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)
^(1/2)*c)*b*d+1/8*e^4/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c/(-(4*a*c-b^2)*e^2)^(
1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a
*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2-e^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^3/
(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)
/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d^2+1/4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d
)^(1/2)*b^2-3/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^3/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2
*c*d^2)*(e*x+d)^(5/2)*d^2+3/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b
^3*d*e-b^2*c*d^2)*(e*x+d)^(3/2)*c^3*d^3-1/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*c^2*d^2-
1/4*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x
+d)^(7/2)*b+1/2*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2
*c*d^2)*(e*x+d)^(5/2)*a-1/2*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b
^3*d*e-b^2*c*d^2)*(e*x+d)^(5/2)*b^2-3/2*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*a*c+1/2*e^2/
(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^3/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(7/2
)*d+1/2*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*b*c*d+3/2*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^
2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(5/2)*b*d+1/4*e^2/(4*a^2*c*e^2-a*b
^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^2*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*a
rctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d-1/2*e^4/(c*e^2*x^2+b*e^2*x+a*e
^2)^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(3/2)*c^2*a*d-1/8*e^3/(4*a^2*c
*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(
1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b+1/8*e^3/(4*a^2*c*e^2-a*b
^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*ar
ctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b+5/4*e^4/(c*e^2*x^2+b*e^2*x+a*
e^2)^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(3/2)*b^2*c*d-9/4*e^3/(c*e^2*
x^2+b*e^2*x+a*e^2)^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(3/2)*b*c^2*d^2
-1/4*e^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^2*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b
^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d+1/
4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(3
/2)*a*b*c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, c x + b\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 9.03, size = 46559, normalized size = 100.56

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((d + e*x)^(1/2)*(b^2*e^4 - 2*c^2*d^2*e^2 - 6*a*c*e^4 + 2*b*c*d*e^3))/(4*(4*a*c - b^2)) + ((d + e*x)^(5/2)*(a
*c^2*e^4 - b^2*c*e^4 - 3*c^3*d^2*e^2 + 3*b*c^2*d*e^3))/(2*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)) - ((d + e*x)^
(3/2)*(b^3*e^5 - 6*c^3*d^3*e^2 + 9*b*c^2*d^2*e^3 - a*b*c*e^5 + 2*a*c^2*d*e^4 - 5*b^2*c*d*e^4))/(4*(4*a*c - b^2
)*(a*e^2 + c*d^2 - b*d*e)) + (c*(2*c^2*d*e^2 - b*c*e^3)*(d + e*x)^(7/2))/(4*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d
*e)))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 2*a*b*e^3 + 4*a*c*d*e^2 - 6*b*c*d^2*e) - (4*c^2*
d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 + 2*a*c*e^2 - 6*b*c*d*e) + a^2*e^4 + c^2*d^4 + b^2
*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2) - atan(((((24576*a^5*c^6*e^8 + 64*a*b^8*c^2*e^8 - 64*b^9
*c^2*d*e^7 - 1152*a^2*b^6*c^3*e^8 + 7680*a^3*b^4*c^4*e^8 - 22528*a^4*b^2*c^5*e^8 + 8192*a^3*c^8*d^4*e^4 + 3276
8*a^4*c^7*d^2*e^6 - 128*b^6*c^5*d^4*e^4 + 256*b^7*c^4*d^3*e^5 - 64*b^8*c^3*d^2*e^6 - 6144*a^2*b^2*c^7*d^4*e^4
+ 12288*a^2*b^3*c^6*d^3*e^5 + 3072*a^2*b^4*c^5*d^2*e^6 - 20480*a^3*b^2*c^6*d^2*e^6 + 1280*a*b^7*c^3*d*e^7 - 32
768*a^4*b*c^6*d*e^7 + 1536*a*b^4*c^6*d^4*e^4 - 3072*a*b^5*c^5*d^3*e^5 + 256*a*b^6*c^4*d^2*e^6 - 9216*a^2*b^5*c
^4*d*e^7 - 16384*a^3*b*c^7*d^3*e^5 + 28672*a^3*b^3*c^5*d*e^7)/(64*(a^2*b^6*e^4 - 64*a^3*c^5*d^4 - 64*a^5*c^3*e
^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 - 12*a^3*b^4*c*e^4 + 48*a^2*b^2*c^4*d^4 + 48*a^4*b^2*c^2*e^4
- 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3*e + 24*a^2*b^4*c^2*d^2*e^2 + 32*a^3*b^2*c^3*d^2*e^2 + 24*
a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*d*e^3 + 128*a^3*b*c^4*d^3*e + 128*a^4*b*c^3*d*e^3 - 96*a^2
*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3)) - ((d + e*x)^(1/2)*(-(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 38
40*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 20
48*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b
^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2)
- 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3
+ 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 9
0*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*
e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*
a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24
*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^
2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2
*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*
c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*
d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^
2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4
*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e -
18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4
*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*
a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c
^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*(8192*a^5*c^6*d*e^6 - 4096*a^5*b*c^5*e^7 + 64*a^2*b^7*c^2*e^7 -
768*a^3*b^5*c^3*e^7 + 3072*a^4*b^3*c^4*e^7 + 8192*a^3*c^8*d^5*e^2 + 16384*a^4*c^7*d^3*e^4 - 128*b^6*c^5*d^5*e^
2 + 320*b^7*c^4*d^4*e^3 - 256*b^8*c^3*d^3*e^4 + 64*b^9*c^2*d^2*e^5 - 6144*a^2*b^2*c^7*d^5*e^2 + 15360*a^2*b^3*
c^6*d^4*e^3 - 9216*a^2*b^4*c^5*d^3*e^4 - 1536*a^2*b^5*c^4*d^2*e^5 + 4096*a^3*b^2*c^6*d^3*e^4 + 14336*a^3*b^3*c
^5*d^2*e^5 - 128*a*b^8*c^2*d*e^6 + 1536*a*b^4*c^6*d^5*e^2 - 3840*a*b^5*c^5*d^4*e^3 + 2816*a*b^6*c^4*d^3*e^4 -
384*a*b^7*c^3*d^2*e^5 + 1408*a^2*b^6*c^3*d*e^6 - 20480*a^3*b*c^7*d^4*e^3 - 4608*a^3*b^4*c^4*d*e^6 - 24576*a^4*
b*c^6*d^2*e^5 + 2048*a^4*b^2*c^5*d*e^6))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6
*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c
^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)))*(-(b^2*e^7*(-(4
*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5
*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^
4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 -
9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7
*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^
3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*
b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e
^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^
6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5
+ 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 61
44*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6
- 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^
3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b
^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^
6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2
*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288
*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^
5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 115
20*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2) - ((d + e*x)^(1/2)*(72*a^2*c
^5*e^8 + b^4*c^3*e^8 + 32*c^7*d^4*e^4 - 14*a*b^2*c^4*e^8 + 88*a*c^6*d^2*e^6 - 64*b*c^6*d^3*e^5 + 6*b^3*c^4*d*e
^7 + 26*b^2*c^5*d^2*e^6 - 88*a*b*c^5*d*e^7))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 +
b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b
^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)))*(-(b^2*e^7*
(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3
*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^
7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e
^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2
*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^
6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 48
0*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4
*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^
6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*
e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6
+ 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*
e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 312
0*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a
^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^
5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7
*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 1
2288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^
5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 +
11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*1i - (((24576*a^5*c^6*e^
8 + 64*a*b^8*c^2*e^8 - 64*b^9*c^2*d*e^7 - 1152*a^2*b^6*c^3*e^8 + 7680*a^3*b^4*c^4*e^8 - 22528*a^4*b^2*c^5*e^8
+ 8192*a^3*c^8*d^4*e^4 + 32768*a^4*c^7*d^2*e^6 - 128*b^6*c^5*d^4*e^4 + 256*b^7*c^4*d^3*e^5 - 64*b^8*c^3*d^2*e^
6 - 6144*a^2*b^2*c^7*d^4*e^4 + 12288*a^2*b^3*c^6*d^3*e^5 + 3072*a^2*b^4*c^5*d^2*e^6 - 20480*a^3*b^2*c^6*d^2*e^
6 + 1280*a*b^7*c^3*d*e^7 - 32768*a^4*b*c^6*d*e^7 + 1536*a*b^4*c^6*d^4*e^4 - 3072*a*b^5*c^5*d^3*e^5 + 256*a*b^6
*c^4*d^2*e^6 - 9216*a^2*b^5*c^4*d*e^7 - 16384*a^3*b*c^7*d^3*e^5 + 28672*a^3*b^3*c^5*d*e^7)/(64*(a^2*b^6*e^4 -
64*a^3*c^5*d^4 - 64*a^5*c^3*e^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 - 12*a^3*b^4*c*e^4 + 48*a^2*b^2
*c^4*d^4 + 48*a^4*b^2*c^2*e^4 - 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3*e + 24*a^2*b^4*c^2*d^2*e^2 +
32*a^3*b^2*c^3*d^2*e^2 + 24*a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*d*e^3 + 128*a^3*b*c^4*d^3*e +
128*a^4*b*c^3*d*e^3 - 96*a^2*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3)) + ((d + e*x)^(1/2)*(-(b^2*e^7*(-(4*a*c -
b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^
7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e
^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*
e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^
2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 -
8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4
*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11
520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^1
2*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^1
3*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*
b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288
*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c
^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*
d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d
^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^
2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*
c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 1
1520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*
b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*(8192*a^5*c^6*d*e^6 - 4096*a^5*b*c^
5*e^7 + 64*a^2*b^7*c^2*e^7 - 768*a^3*b^5*c^3*e^7 + 3072*a^4*b^3*c^4*e^7 + 8192*a^3*c^8*d^5*e^2 + 16384*a^4*c^7
*d^3*e^4 - 128*b^6*c^5*d^5*e^2 + 320*b^7*c^4*d^4*e^3 - 256*b^8*c^3*d^3*e^4 + 64*b^9*c^2*d^2*e^5 - 6144*a^2*b^2
*c^7*d^5*e^2 + 15360*a^2*b^3*c^6*d^4*e^3 - 9216*a^2*b^4*c^5*d^3*e^4 - 1536*a^2*b^5*c^4*d^2*e^5 + 4096*a^3*b^2*
c^6*d^3*e^4 + 14336*a^3*b^3*c^5*d^2*e^5 - 128*a*b^8*c^2*d*e^6 + 1536*a*b^4*c^6*d^5*e^2 - 3840*a*b^5*c^5*d^4*e^
3 + 2816*a*b^6*c^4*d^3*e^4 - 384*a*b^7*c^3*d^2*e^5 + 1408*a^2*b^6*c^3*d*e^6 - 20480*a^3*b*c^7*d^4*e^3 - 4608*a
^3*b^4*c^4*d*e^6 - 24576*a^4*b*c^6*d^2*e^5 + 2048*a^4*b^2*c^5*d*e^6))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^
4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3
- 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b
*c^2*d*e^3)))*(-(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 288*a
^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 3
2*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c - b^2
)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c - b^2
)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*
d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 9
60*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*
d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*
a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^1
4*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^
6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^
4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a
^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4
*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*
b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b
^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 +
12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*
d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e -
3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)
+ ((d + e*x)^(1/2)*(72*a^2*c^5*e^8 + b^4*c^3*e^8 + 32*c^7*d^4*e^4 - 14*a*b^2*c^4*e^8 + 88*a*c^6*d^2*e^6 - 64*
b*c^6*d^3*e^5 + 6*b^3*c^4*d*e^7 + 26*b^2*c^5*d^2*e^6 - 88*a*b*c^5*d*e^7))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 1
6*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d
*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a
^3*b*c^2*d*e^3)))*(-(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 2
88*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4
+ 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c -
b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c -
b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*
c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2
+ 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*
c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4
096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a
*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^
6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^
4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 +
96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680
*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*
a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a
^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^
5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^1
2*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5
*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(
1/2)*1i)/((5*b^3*c^4*e^9 + 32*c^7*d^3*e^6 - 48*b*c^6*d^2*e^7 + 6*b^2*c^5*d*e^8 - 36*a*b*c^5*e^9 + 72*a*c^6*d*e
^8)/(32*(a^2*b^6*e^4 - 64*a^3*c^5*d^4 - 64*a^5*c^3*e^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 - 12*a^3
*b^4*c*e^4 + 48*a^2*b^2*c^4*d^4 + 48*a^4*b^2*c^2*e^4 - 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3*e + 2
4*a^2*b^4*c^2*d^2*e^2 + 32*a^3*b^2*c^3*d^2*e^2 + 24*a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*d*e^3
+ 128*a^3*b*c^4*d^3*e + 128*a^4*b*c^3*d*e^3 - 96*a^2*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3)) + (((24576*a^5*c^6
*e^8 + 64*a*b^8*c^2*e^8 - 64*b^9*c^2*d*e^7 - 1152*a^2*b^6*c^3*e^8 + 7680*a^3*b^4*c^4*e^8 - 22528*a^4*b^2*c^5*e
^8 + 8192*a^3*c^8*d^4*e^4 + 32768*a^4*c^7*d^2*e^6 - 128*b^6*c^5*d^4*e^4 + 256*b^7*c^4*d^3*e^5 - 64*b^8*c^3*d^2
*e^6 - 6144*a^2*b^2*c^7*d^4*e^4 + 12288*a^2*b^3*c^6*d^3*e^5 + 3072*a^2*b^4*c^5*d^2*e^6 - 20480*a^3*b^2*c^6*d^2
*e^6 + 1280*a*b^7*c^3*d*e^7 - 32768*a^4*b*c^6*d*e^7 + 1536*a*b^4*c^6*d^4*e^4 - 3072*a*b^5*c^5*d^3*e^5 + 256*a*
b^6*c^4*d^2*e^6 - 9216*a^2*b^5*c^4*d*e^7 - 16384*a^3*b*c^7*d^3*e^5 + 28672*a^3*b^3*c^5*d*e^7)/(64*(a^2*b^6*e^4
- 64*a^3*c^5*d^4 - 64*a^5*c^3*e^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 - 12*a^3*b^4*c*e^4 + 48*a^2*
b^2*c^4*d^4 + 48*a^4*b^2*c^2*e^4 - 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3*e + 24*a^2*b^4*c^2*d^2*e^
2 + 32*a^3*b^2*c^3*d^2*e^2 + 24*a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*d*e^3 + 128*a^3*b*c^4*d^3*
e + 128*a^4*b*c^3*d*e^3 - 96*a^2*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3)) - ((d + e*x)^(1/2)*(-(b^2*e^7*(-(4*a*c
- b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3
*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^
4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a
*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5
*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^
4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*
c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 +
11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 -
b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*
b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a
^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12
288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^
8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c
^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^
4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6
*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8
*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e
+ 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a
^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*(8192*a^5*c^6*d*e^6 - 4096*a^5*b
*c^5*e^7 + 64*a^2*b^7*c^2*e^7 - 768*a^3*b^5*c^3*e^7 + 3072*a^4*b^3*c^4*e^7 + 8192*a^3*c^8*d^5*e^2 + 16384*a^4*
c^7*d^3*e^4 - 128*b^6*c^5*d^5*e^2 + 320*b^7*c^4*d^4*e^3 - 256*b^8*c^3*d^3*e^4 + 64*b^9*c^2*d^2*e^5 - 6144*a^2*
b^2*c^7*d^5*e^2 + 15360*a^2*b^3*c^6*d^4*e^3 - 9216*a^2*b^4*c^5*d^3*e^4 - 1536*a^2*b^5*c^4*d^2*e^5 + 4096*a^3*b
^2*c^6*d^3*e^4 + 14336*a^3*b^3*c^5*d^2*e^5 - 128*a*b^8*c^2*d*e^6 + 1536*a*b^4*c^6*d^5*e^2 - 3840*a*b^5*c^5*d^4
*e^3 + 2816*a*b^6*c^4*d^3*e^4 - 384*a*b^7*c^3*d^2*e^5 + 1408*a^2*b^6*c^3*d*e^6 - 20480*a^3*b*c^7*d^4*e^3 - 460
8*a^3*b^4*c^4*d*e^6 - 24576*a^4*b*c^6*d^2*e^5 + 2048*a^4*b^2*c^5*d*e^6))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16
*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*
e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^
3*b*c^2*d*e^3)))*(-(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 28
8*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4
+ 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c -
b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c -
b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c
^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2
+ 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c
^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 40
96*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*
b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6
*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4
*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 9
6*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*
a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a
^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^
6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5
+ 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12
*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*
e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1
/2) - ((d + e*x)^(1/2)*(72*a^2*c^5*e^8 + b^4*c^3*e^8 + 32*c^7*d^4*e^4 - 14*a*b^2*c^4*e^8 + 88*a*c^6*d^2*e^6 -
64*b*c^6*d^3*e^5 + 6*b^3*c^4*d*e^7 + 26*b^2*c^5*d^2*e^6 - 88*a*b*c^5*d*e^7))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4
+ 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^
5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 3
2*a^3*b*c^2*d*e^3)))*(-(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6
- 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*
e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*
c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*
c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b
^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*
e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3
*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3
- 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 -
3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6
*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7
*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2
+ 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7
680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 168
96*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 691
2*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d
*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*
b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*
d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5))
)^(1/2) + (((24576*a^5*c^6*e^8 + 64*a*b^8*c^2*e^8 - 64*b^9*c^2*d*e^7 - 1152*a^2*b^6*c^3*e^8 + 7680*a^3*b^4*c^4
*e^8 - 22528*a^4*b^2*c^5*e^8 + 8192*a^3*c^8*d^4*e^4 + 32768*a^4*c^7*d^2*e^6 - 128*b^6*c^5*d^4*e^4 + 256*b^7*c^
4*d^3*e^5 - 64*b^8*c^3*d^2*e^6 - 6144*a^2*b^2*c^7*d^4*e^4 + 12288*a^2*b^3*c^6*d^3*e^5 + 3072*a^2*b^4*c^5*d^2*e
^6 - 20480*a^3*b^2*c^6*d^2*e^6 + 1280*a*b^7*c^3*d*e^7 - 32768*a^4*b*c^6*d*e^7 + 1536*a*b^4*c^6*d^4*e^4 - 3072*
a*b^5*c^5*d^3*e^5 + 256*a*b^6*c^4*d^2*e^6 - 9216*a^2*b^5*c^4*d*e^7 - 16384*a^3*b*c^7*d^3*e^5 + 28672*a^3*b^3*c
^5*d*e^7)/(64*(a^2*b^6*e^4 - 64*a^3*c^5*d^4 - 64*a^5*c^3*e^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 -
12*a^3*b^4*c*e^4 + 48*a^2*b^2*c^4*d^4 + 48*a^4*b^2*c^2*e^4 - 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3
*e + 24*a^2*b^4*c^2*d^2*e^2 + 32*a^3*b^2*c^3*d^2*e^2 + 24*a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*
d*e^3 + 128*a^3*b*c^4*d^3*e + 128*a^4*b*c^3*d*e^3 - 96*a^2*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3)) + ((d + e*x)
^(1/2)*(-(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*
c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c
^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1
/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1
/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5
+ 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^
5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3
+ 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9
*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e
^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 384
0*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 +
6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11
*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^
5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5
*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*
d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a
^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4
- 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a
^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*(8192*
a^5*c^6*d*e^6 - 4096*a^5*b*c^5*e^7 + 64*a^2*b^7*c^2*e^7 - 768*a^3*b^5*c^3*e^7 + 3072*a^4*b^3*c^4*e^7 + 8192*a^
3*c^8*d^5*e^2 + 16384*a^4*c^7*d^3*e^4 - 128*b^6*c^5*d^5*e^2 + 320*b^7*c^4*d^4*e^3 - 256*b^8*c^3*d^3*e^4 + 64*b
^9*c^2*d^2*e^5 - 6144*a^2*b^2*c^7*d^5*e^2 + 15360*a^2*b^3*c^6*d^4*e^3 - 9216*a^2*b^4*c^5*d^3*e^4 - 1536*a^2*b^
5*c^4*d^2*e^5 + 4096*a^3*b^2*c^6*d^3*e^4 + 14336*a^3*b^3*c^5*d^2*e^5 - 128*a*b^8*c^2*d*e^6 + 1536*a*b^4*c^6*d^
5*e^2 - 3840*a*b^5*c^5*d^4*e^3 + 2816*a*b^6*c^4*d^3*e^4 - 384*a*b^7*c^3*d^2*e^5 + 1408*a^2*b^6*c^3*d*e^6 - 204
80*a^3*b*c^7*d^4*e^3 - 4608*a^3*b^4*c^4*d*e^6 - 24576*a^4*b*c^6*d^2*e^5 + 2048*a^4*b^2*c^5*d*e^6))/(8*(a^2*b^4
*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^
3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e +
16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)))*(-(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7
- 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e
^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 -
5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6
+ 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^
5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^
6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^
6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^
6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6
+ 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*
c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*
b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 64
8*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^
3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5
*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6
*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3
*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b
^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^
5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 184
32*a^7*b^3*c^5*d*e^5)))^(1/2) + ((d + e*x)^(1/2)*(72*a^2*c^5*e^8 + b^4*c^3*e^8 + 32*c^7*d^4*e^4 - 14*a*b^2*c^4
*e^8 + 88*a*c^6*d^2*e^6 - 64*b*c^6*d^3*e^5 + 6*b^3*c^4*d*e^7 + 26*b^2*c^5*d^2*e^6 - 88*a*b*c^5*d*e^7))/(8*(a^2
*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 3
2*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*
e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)))*(-(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5
*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d
^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e
^5 - 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d
*e^6 + 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^
4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*
d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^
2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*
d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*
d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*
b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*
a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4
- 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 64
8*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912
*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768
*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c
*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a
^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*
d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 -
18432*a^7*b^3*c^5*d*e^5)))^(1/2)))*(-(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680
*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 76
80*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*
d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*
c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e
^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384
*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d
*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128
*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^
4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6
+ 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3
*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b
^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*
c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^
6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^
6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 -
72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*
d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 184
32*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*
b^3*c^5*d*e^5)))^(1/2)*2i - atan(((((24576*a^5*c^6*e^8 + 64*a*b^8*c^2*e^8 - 64*b^9*c^2*d*e^7 - 1152*a^2*b^6*c^
3*e^8 + 7680*a^3*b^4*c^4*e^8 - 22528*a^4*b^2*c^5*e^8 + 8192*a^3*c^8*d^4*e^4 + 32768*a^4*c^7*d^2*e^6 - 128*b^6*
c^5*d^4*e^4 + 256*b^7*c^4*d^3*e^5 - 64*b^8*c^3*d^2*e^6 - 6144*a^2*b^2*c^7*d^4*e^4 + 12288*a^2*b^3*c^6*d^3*e^5
+ 3072*a^2*b^4*c^5*d^2*e^6 - 20480*a^3*b^2*c^6*d^2*e^6 + 1280*a*b^7*c^3*d*e^7 - 32768*a^4*b*c^6*d*e^7 + 1536*a
*b^4*c^6*d^4*e^4 - 3072*a*b^5*c^5*d^3*e^5 + 256*a*b^6*c^4*d^2*e^6 - 9216*a^2*b^5*c^4*d*e^7 - 16384*a^3*b*c^7*d
^3*e^5 + 28672*a^3*b^3*c^5*d*e^7)/(64*(a^2*b^6*e^4 - 64*a^3*c^5*d^4 - 64*a^5*c^3*e^4 + b^6*c^2*d^4 + b^8*d^2*e
^2 - 12*a*b^4*c^3*d^4 - 12*a^3*b^4*c*e^4 + 48*a^2*b^2*c^4*d^4 + 48*a^4*b^2*c^2*e^4 - 128*a^4*c^4*d^2*e^2 - 2*a
*b^7*d*e^3 - 2*b^7*c*d^3*e + 24*a^2*b^4*c^2*d^2*e^2 + 32*a^3*b^2*c^3*d^2*e^2 + 24*a*b^5*c^2*d^3*e - 10*a*b^6*c
*d^2*e^2 + 24*a^2*b^5*c*d*e^3 + 128*a^3*b*c^4*d^3*e + 128*a^4*b*c^3*d*e^3 - 96*a^2*b^3*c^3*d^3*e - 96*a^3*b^3*
c^2*d*e^3)) - ((d + e*x)^(1/2)*(-(3840*a^5*b*c^5*e^7 - b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 - 7680*a^5*
c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^
4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e
^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 + 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e
^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 +
2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^
4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6
+ 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^1
5*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^1
0*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 12
80*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6
- 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c
^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d
^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4
*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3
*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^
3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e
+ 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^
5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c
^5*d*e^5)))^(1/2)*(8192*a^5*c^6*d*e^6 - 4096*a^5*b*c^5*e^7 + 64*a^2*b^7*c^2*e^7 - 768*a^3*b^5*c^3*e^7 + 3072*a
^4*b^3*c^4*e^7 + 8192*a^3*c^8*d^5*e^2 + 16384*a^4*c^7*d^3*e^4 - 128*b^6*c^5*d^5*e^2 + 320*b^7*c^4*d^4*e^3 - 25
6*b^8*c^3*d^3*e^4 + 64*b^9*c^2*d^2*e^5 - 6144*a^2*b^2*c^7*d^5*e^2 + 15360*a^2*b^3*c^6*d^4*e^3 - 9216*a^2*b^4*c
^5*d^3*e^4 - 1536*a^2*b^5*c^4*d^2*e^5 + 4096*a^3*b^2*c^6*d^3*e^4 + 14336*a^3*b^3*c^5*d^2*e^5 - 128*a*b^8*c^2*d
*e^6 + 1536*a*b^4*c^6*d^5*e^2 - 3840*a*b^5*c^5*d^4*e^3 + 2816*a*b^6*c^4*d^3*e^4 - 384*a*b^7*c^3*d^2*e^5 + 1408
*a^2*b^6*c^3*d*e^6 - 20480*a^3*b*c^7*d^4*e^3 - 4608*a^3*b^4*c^4*d*e^6 - 24576*a^4*b*c^6*d^2*e^5 + 2048*a^4*b^2
*c^5*d*e^6))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 -
8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2
- 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)))*(-(3840*a^5*b*c^5*e^7 - b^2*e^7*(-(4*a*c -
b^2)^9)^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*
e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e
^4 + 5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 + 9*a*c*e^7*(-(4*a*c - b^2)^9
)^(1/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*
d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2
*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*
c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5
+ 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*
e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*
c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5
*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12
288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*
b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^
8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2
*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3
*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12
*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*
e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576
*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2) - ((d + e*x)^(1/2)*(72*a^2*c^5*e^8 + b^4*c^3*e^8 + 32*c^
7*d^4*e^4 - 14*a*b^2*c^4*e^8 + 88*a*c^6*d^2*e^6 - 64*b*c^6*d^3*e^5 + 6*b^3*c^4*d*e^7 + 26*b^2*c^5*d^2*e^6 - 88
*a*b*c^5*d*e^7))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d
^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2
*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)))*(-(3840*a^5*b*c^5*e^7 - b^2*e^7*(-(4*a*
c - b^2)^9)^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*
c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d
^3*e^4 + 5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 + 9*a*c*e^7*(-(4*a*c - b^
2)^9)^(1/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*
c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5
*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*
b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2
*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b
^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b
^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240
*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2
- 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*
a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^
4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6
*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11
*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*
b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*
d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 2
4576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*1i - (((24576*a^5*c^6*e^8 + 64*a*b^8*c^2*e^8 - 64*b^
9*c^2*d*e^7 - 1152*a^2*b^6*c^3*e^8 + 7680*a^3*b^4*c^4*e^8 - 22528*a^4*b^2*c^5*e^8 + 8192*a^3*c^8*d^4*e^4 + 327
68*a^4*c^7*d^2*e^6 - 128*b^6*c^5*d^4*e^4 + 256*b^7*c^4*d^3*e^5 - 64*b^8*c^3*d^2*e^6 - 6144*a^2*b^2*c^7*d^4*e^4
+ 12288*a^2*b^3*c^6*d^3*e^5 + 3072*a^2*b^4*c^5*d^2*e^6 - 20480*a^3*b^2*c^6*d^2*e^6 + 1280*a*b^7*c^3*d*e^7 - 3
2768*a^4*b*c^6*d*e^7 + 1536*a*b^4*c^6*d^4*e^4 - 3072*a*b^5*c^5*d^3*e^5 + 256*a*b^6*c^4*d^2*e^6 - 9216*a^2*b^5*
c^4*d*e^7 - 16384*a^3*b*c^7*d^3*e^5 + 28672*a^3*b^3*c^5*d*e^7)/(64*(a^2*b^6*e^4 - 64*a^3*c^5*d^4 - 64*a^5*c^3*
e^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 - 12*a^3*b^4*c*e^4 + 48*a^2*b^2*c^4*d^4 + 48*a^4*b^2*c^2*e^
4 - 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3*e + 24*a^2*b^4*c^2*d^2*e^2 + 32*a^3*b^2*c^3*d^2*e^2 + 24
*a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*d*e^3 + 128*a^3*b*c^4*d^3*e + 128*a^4*b*c^3*d*e^3 - 96*a^
2*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3)) + ((d + e*x)^(1/2)*(-(3840*a^5*b*c^5*e^7 - b^2*e^7*(-(4*a*c - b^2)^9)
^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2
048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*
b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 + 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2)
- 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3
+ 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 +
90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2
*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840
*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 2
4*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e
^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^
2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8
*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3
*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d
^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^
4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e
- 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^
4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720
*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*
c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*(8192*a^5*c^6*d*e^6 - 4096*a^5*b*c^5*e^7 + 64*a^2*b^7*c^2*e^7 -
768*a^3*b^5*c^3*e^7 + 3072*a^4*b^3*c^4*e^7 + 8192*a^3*c^8*d^5*e^2 + 16384*a^4*c^7*d^3*e^4 - 128*b^6*c^5*d^5*e
^2 + 320*b^7*c^4*d^4*e^3 - 256*b^8*c^3*d^3*e^4 + 64*b^9*c^2*d^2*e^5 - 6144*a^2*b^2*c^7*d^5*e^2 + 15360*a^2*b^3
*c^6*d^4*e^3 - 9216*a^2*b^4*c^5*d^3*e^4 - 1536*a^2*b^5*c^4*d^2*e^5 + 4096*a^3*b^2*c^6*d^3*e^4 + 14336*a^3*b^3*
c^5*d^2*e^5 - 128*a*b^8*c^2*d*e^6 + 1536*a*b^4*c^6*d^5*e^2 - 3840*a*b^5*c^5*d^4*e^3 + 2816*a*b^6*c^4*d^3*e^4 -
384*a*b^7*c^3*d^2*e^5 + 1408*a^2*b^6*c^3*d*e^6 - 20480*a^3*b*c^7*d^4*e^3 - 4608*a^3*b^4*c^4*d*e^6 - 24576*a^4
*b*c^6*d^2*e^5 + 2048*a^4*b^2*c^5*d*e^6))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^
6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*
c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)))*(-(3840*a^5*b*
c^5*e^7 - b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^
5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c
^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7
+ 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^
7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d
^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a
*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*
e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e
^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5
+ 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6
144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6
- 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a
^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*
b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b
^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^
2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 1228
8*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d
^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11
520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2) + ((d + e*x)^(1/2)*(72*a^2*
c^5*e^8 + b^4*c^3*e^8 + 32*c^7*d^4*e^4 - 14*a*b^2*c^4*e^8 + 88*a*c^6*d^2*e^6 - 64*b*c^6*d^3*e^5 + 6*b^3*c^4*d*
e^7 + 26*b^2*c^5*d^2*e^6 - 88*a*b*c^5*d*e^7))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4
+ b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*
b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)))*(-(3840*a^
5*b*c^5*e^7 - b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^
3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b
^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*
e^7 + 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^
2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c
^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 4
80*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^
4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c
^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d
*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6
+ 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5
*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 31
20*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*
a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a
^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^
7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e +
12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c
^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5
+ 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*1i)/((5*b^3*c^4*e^9 + 3
2*c^7*d^3*e^6 - 48*b*c^6*d^2*e^7 + 6*b^2*c^5*d*e^8 - 36*a*b*c^5*e^9 + 72*a*c^6*d*e^8)/(32*(a^2*b^6*e^4 - 64*a^
3*c^5*d^4 - 64*a^5*c^3*e^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 - 12*a^3*b^4*c*e^4 + 48*a^2*b^2*c^4*
d^4 + 48*a^4*b^2*c^2*e^4 - 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3*e + 24*a^2*b^4*c^2*d^2*e^2 + 32*a
^3*b^2*c^3*d^2*e^2 + 24*a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*d*e^3 + 128*a^3*b*c^4*d^3*e + 128*
a^4*b*c^3*d*e^3 - 96*a^2*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3)) + (((24576*a^5*c^6*e^8 + 64*a*b^8*c^2*e^8 - 64
*b^9*c^2*d*e^7 - 1152*a^2*b^6*c^3*e^8 + 7680*a^3*b^4*c^4*e^8 - 22528*a^4*b^2*c^5*e^8 + 8192*a^3*c^8*d^4*e^4 +
32768*a^4*c^7*d^2*e^6 - 128*b^6*c^5*d^4*e^4 + 256*b^7*c^4*d^3*e^5 - 64*b^8*c^3*d^2*e^6 - 6144*a^2*b^2*c^7*d^4*
e^4 + 12288*a^2*b^3*c^6*d^3*e^5 + 3072*a^2*b^4*c^5*d^2*e^6 - 20480*a^3*b^2*c^6*d^2*e^6 + 1280*a*b^7*c^3*d*e^7
- 32768*a^4*b*c^6*d*e^7 + 1536*a*b^4*c^6*d^4*e^4 - 3072*a*b^5*c^5*d^3*e^5 + 256*a*b^6*c^4*d^2*e^6 - 9216*a^2*b
^5*c^4*d*e^7 - 16384*a^3*b*c^7*d^3*e^5 + 28672*a^3*b^3*c^5*d*e^7)/(64*(a^2*b^6*e^4 - 64*a^3*c^5*d^4 - 64*a^5*c
^3*e^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 - 12*a^3*b^4*c*e^4 + 48*a^2*b^2*c^4*d^4 + 48*a^4*b^2*c^2
*e^4 - 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3*e + 24*a^2*b^4*c^2*d^2*e^2 + 32*a^3*b^2*c^3*d^2*e^2 +
24*a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*d*e^3 + 128*a^3*b*c^4*d^3*e + 128*a^4*b*c^3*d*e^3 - 96
*a^2*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3)) - ((d + e*x)^(1/2)*(-(3840*a^5*b*c^5*e^7 - b^2*e^7*(-(4*a*c - b^2)
^9)^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7
- 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 +
5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 + 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1
/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*
e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5
+ 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*
d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3
840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6
+ 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^
4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8
*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*
a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*
c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^
3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7
*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5
*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2
*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e +
720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7
*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*(8192*a^5*c^6*d*e^6 - 4096*a^5*b*c^5*e^7 + 64*a^2*b^7*c^2*e^
7 - 768*a^3*b^5*c^3*e^7 + 3072*a^4*b^3*c^4*e^7 + 8192*a^3*c^8*d^5*e^2 + 16384*a^4*c^7*d^3*e^4 - 128*b^6*c^5*d^
5*e^2 + 320*b^7*c^4*d^4*e^3 - 256*b^8*c^3*d^3*e^4 + 64*b^9*c^2*d^2*e^5 - 6144*a^2*b^2*c^7*d^5*e^2 + 15360*a^2*
b^3*c^6*d^4*e^3 - 9216*a^2*b^4*c^5*d^3*e^4 - 1536*a^2*b^5*c^4*d^2*e^5 + 4096*a^3*b^2*c^6*d^3*e^4 + 14336*a^3*b
^3*c^5*d^2*e^5 - 128*a*b^8*c^2*d*e^6 + 1536*a*b^4*c^6*d^5*e^2 - 3840*a*b^5*c^5*d^4*e^3 + 2816*a*b^6*c^4*d^3*e^
4 - 384*a*b^7*c^3*d^2*e^5 + 1408*a^2*b^6*c^3*d*e^6 - 20480*a^3*b*c^7*d^4*e^3 - 4608*a^3*b^4*c^4*d*e^6 - 24576*
a^4*b*c^6*d^2*e^5 + 2048*a^4*b^2*c^5*d*e^6))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 +
b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b
^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)))*(-(3840*a^5
*b*c^5*e^7 - b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3
*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^
7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e
^7 + 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2
*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^
6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 48
0*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4
*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^
6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*
e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6
+ 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*
e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 312
0*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a
^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^
5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7
*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 1
2288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^
5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 +
11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2) - ((d + e*x)^(1/2)*(72*a
^2*c^5*e^8 + b^4*c^3*e^8 + 32*c^7*d^4*e^4 - 14*a*b^2*c^4*e^8 + 88*a*c^6*d^2*e^6 - 64*b*c^6*d^3*e^5 + 6*b^3*c^4
*d*e^7 + 26*b^2*c^5*d^2*e^6 - 88*a*b*c^5*d*e^7))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d
^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16
*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)))*(-(3840
*a^5*b*c^5*e^7 - b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504
*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 8
0*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9
*c*e^7 + 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2
*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^
2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3
- 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4
*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^
9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^1
3*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*
d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*
c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 +
3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 38
40*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 768
0*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144
*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e
+ 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^
7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e
^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2) + (((24576*a^5*c^6*e
^8 + 64*a*b^8*c^2*e^8 - 64*b^9*c^2*d*e^7 - 1152*a^2*b^6*c^3*e^8 + 7680*a^3*b^4*c^4*e^8 - 22528*a^4*b^2*c^5*e^8
+ 8192*a^3*c^8*d^4*e^4 + 32768*a^4*c^7*d^2*e^6 - 128*b^6*c^5*d^4*e^4 + 256*b^7*c^4*d^3*e^5 - 64*b^8*c^3*d^2*e
^6 - 6144*a^2*b^2*c^7*d^4*e^4 + 12288*a^2*b^3*c^6*d^3*e^5 + 3072*a^2*b^4*c^5*d^2*e^6 - 20480*a^3*b^2*c^6*d^2*e
^6 + 1280*a*b^7*c^3*d*e^7 - 32768*a^4*b*c^6*d*e^7 + 1536*a*b^4*c^6*d^4*e^4 - 3072*a*b^5*c^5*d^3*e^5 + 256*a*b^
6*c^4*d^2*e^6 - 9216*a^2*b^5*c^4*d*e^7 - 16384*a^3*b*c^7*d^3*e^5 + 28672*a^3*b^3*c^5*d*e^7)/(64*(a^2*b^6*e^4 -
64*a^3*c^5*d^4 - 64*a^5*c^3*e^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^4 - 12*a^3*b^4*c*e^4 + 48*a^2*b^
2*c^4*d^4 + 48*a^4*b^2*c^2*e^4 - 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c*d^3*e + 24*a^2*b^4*c^2*d^2*e^2
+ 32*a^3*b^2*c^3*d^2*e^2 + 24*a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^5*c*d*e^3 + 128*a^3*b*c^4*d^3*e
+ 128*a^4*b*c^3*d*e^3 - 96*a^2*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3)) + ((d + e*x)^(1/2)*(-(3840*a^5*b*c^5*e^7
- b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e
^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*
e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 + 9*a*c
*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e
^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4
- 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^
4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 1
1520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^
12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^
13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5
*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 1228
8*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*
c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4
*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*
d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d
^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b
*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e +
11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6
*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*(8192*a^5*c^6*d*e^6 - 4096*a^5*b*c
^5*e^7 + 64*a^2*b^7*c^2*e^7 - 768*a^3*b^5*c^3*e^7 + 3072*a^4*b^3*c^4*e^7 + 8192*a^3*c^8*d^5*e^2 + 16384*a^4*c^
7*d^3*e^4 - 128*b^6*c^5*d^5*e^2 + 320*b^7*c^4*d^4*e^3 - 256*b^8*c^3*d^3*e^4 + 64*b^9*c^2*d^2*e^5 - 6144*a^2*b^
2*c^7*d^5*e^2 + 15360*a^2*b^3*c^6*d^4*e^3 - 9216*a^2*b^4*c^5*d^3*e^4 - 1536*a^2*b^5*c^4*d^2*e^5 + 4096*a^3*b^2
*c^6*d^3*e^4 + 14336*a^3*b^3*c^5*d^2*e^5 - 128*a*b^8*c^2*d*e^6 + 1536*a*b^4*c^6*d^5*e^2 - 3840*a*b^5*c^5*d^4*e
^3 + 2816*a*b^6*c^4*d^3*e^4 - 384*a*b^7*c^3*d^2*e^5 + 1408*a^2*b^6*c^3*d*e^6 - 20480*a^3*b*c^7*d^4*e^3 - 4608*
a^3*b^4*c^4*d*e^6 - 24576*a^4*b*c^6*d^2*e^5 + 2048*a^4*b^2*c^5*d*e^6))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a
^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^
3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*
b*c^2*d*e^3)))*(-(3840*a^5*b*c^5*e^7 - b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 - 288*
a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 +
32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c - b^
2)^9)^(1/2) + 27*a*b^9*c*e^7 + 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c - b^
2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4
*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 +
960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7
*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096
*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^
14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d
^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c
^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*
a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^
4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5
*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*
b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 +
12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c
*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e
- 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2
) + ((d + e*x)^(1/2)*(72*a^2*c^5*e^8 + b^4*c^3*e^8 + 32*c^7*d^4*e^4 - 14*a*b^2*c^4*e^8 + 88*a*c^6*d^2*e^6 - 64
*b*c^6*d^3*e^5 + 6*b^3*c^4*d*e^7 + 26*b^2*c^5*d^2*e^6 - 88*a*b*c^5*d*e^7))/(8*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 +
16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*
d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*
a^3*b*c^2*d*e^3)))*(-(3840*a^5*b*c^5*e^7 - b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 -
288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^
4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c
- b^2)^9)^(1/2) + 27*a*b^9*c*e^7 + 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c
- b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5
*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^
2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b
*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 -
4096*a^6*c^9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*
a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c
^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b
^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 +
96*a^2*b^11*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 768
0*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896
*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*
a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e
^5 + 12288*a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^
12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^
5*e - 3840*a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^
(1/2)))*(-(3840*a^5*b*c^5*e^7 - b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7
*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*
c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 + 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(
1/2) + 27*a*b^9*c*e^7 + 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 - 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(
1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^
5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b
^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^
3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6)/(128*(b^15*d^3*e^3 - 4096*a^6*c^
9*d^6 - 4096*a^9*c^6*e^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*
e^4 + 3*a^2*b^13*d*e^5 + 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 38
40*a^4*b^4*c^7*d^6 + 6144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6
+ 6144*a^8*b^2*c^5*e^6 - 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^1
1*c^2*d^3*e^3 + 3120*a^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c
^5*d^4*e^2 - 3840*a^4*b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^
5*d^3*e^3 - 7680*a^5*b^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5
*d^2*e^4 + 6144*a^7*b^2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*
a^6*b*c^8*d^5*e + 12288*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^
4 - 3840*a^3*b^7*c^5*d^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*
a^5*b^7*c^3*d*e^5 + 11520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*2i

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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