\(\int \frac {(d+e x)^{5/2}}{(a+b x+c x^2)^2} \, dx\) [2296]

Optimal result

Integrand size = 22, antiderivative size = 504 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+3 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d-4 a b e-3 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-(e*x+d)^(3/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)+e*(-b*e+2*c*d)*(e*x+d)^(1/2)/c/(-4*a*c+b^
2)+1/2*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*(8*c^3*d^3+b^2*e^3*(b-(-4
*a*c+b^2)^(1/2))-2*c^2*d*e*(6*b*d-8*a*e-d*(-4*a*c+b^2)^(1/2))+2*c*e^2*(b^2*d+3*a*e*(-4*a*c+b^2)^(1/2)-b*(4*a*e
+d*(-4*a*c+b^2)^(1/2))))/c^(3/2)/(-4*a*c+b^2)^(3/2)*2^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)-1/2*arctanh
(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*(8*c^3*d^3+b^2*e^3*(b+(-4*a*c+b^2)^(1/2
))-2*c^2*d*e*(6*b*d-8*a*e+d*(-4*a*c+b^2)^(1/2))+2*c*e^2*(b^2*d-4*a*b*e+b*d*(-4*a*c+b^2)^(1/2)-3*a*e*(-4*a*c+b^
2)^(1/2)))/c^(3/2)/(-4*a*c+b^2)^(3/2)*2^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 2.84 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {752, 838, 840, 1180, 214} \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\left (-2 c^2 d e \left (-d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (-b \left (d \sqrt {b^2-4 a c}+4 a e\right )+3 a e \sqrt {b^2-4 a c}+b^2 d\right )+b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+8 c^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-2 c^2 d e \left (d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt {b^2-4 a c}-3 a e \sqrt {b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+8 c^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e \sqrt {d+e x} (2 c d-b e)}{c \left (b^2-4 a c\right )} \]

[In]

Int[(d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x]

[Out]

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

Rule 214

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

Rule 752

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

Rule 838

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*
((d + e*x)^m/(c*m)), x] + Dist[1/c, Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/
(a + b*x + c*x^2)), 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] && FractionQ[m] && GtQ[m, 0]

Rule 840

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

Rule 1180

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (4 c d^2-5 b d e+6 a e^2\right )-\frac {1}{2} e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{-b^2+4 a c} \\ & = \frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (4 c^2 d^3-a b e^3-c d e (5 b d-8 a e)\right )+\frac {1}{2} e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{c \left (b^2-4 a c\right )} \\ & = \frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} e \left (4 c^2 d^3-a b e^3-c d e (5 b d-8 a e)\right )-\frac {1}{2} d e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )+\frac {1}{2} e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c \left (b^2-4 a c\right )} \\ & = \frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d-4 a b e-3 a \sqrt {b^2-4 a c} e\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c \left (b^2-4 a c\right )^{3/2}}-\frac {\left (8 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+3 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c \left (b^2-4 a c\right )^{3/2}} \\ & = \frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+3 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+4 a e\right )\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 )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d-4 a b e-3 a \sqrt {b^2-4 a c} 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 )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.09 (sec) , antiderivative size = 796, normalized size of antiderivative = 1.58 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {c} \sqrt {d+e x} \left (a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 \sqrt {2} e^2 \left (6 c d+\left (-3 b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 i c^3 d^3-b^2 \left (5 i b+\sqrt {-b^2+4 a c}\right ) e^3-2 c^2 d e \left (6 i b d+\sqrt {-b^2+4 a c} d+16 i a e\right )+2 c e^2 \left (7 i b^2 d+b \sqrt {-b^2+4 a c} d+8 i a b e+a \sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\left (-b^2+4 a c\right )^{3/2} \sqrt {-c d+\frac {1}{2} \left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {\left (-8 i c^3 d^3-b^2 \left (-5 i b+\sqrt {-b^2+4 a c}\right ) e^3+2 c^2 d e \left (6 i b d-\sqrt {-b^2+4 a c} d+16 i a e\right )+2 c e^2 \left (-7 i b^2 d+b \sqrt {-b^2+4 a c} d-8 i a b e+a \sqrt {-b^2+4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\left (-b^2+4 a c\right )^{3/2} \sqrt {-c d+\frac {1}{2} \left (b+i \sqrt {-b^2+4 a c}\right ) e}}+\frac {2 \sqrt {2} e^2 \left (-6 c d+\left (3 b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{2 c^{3/2}} \]

[In]

Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x]

[Out]

((-2*Sqrt[c]*Sqrt[d + e*x]*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x)))/((b^
2 - 4*a*c)*(a + x*(b + c*x))) + (2*Sqrt[2]*e^2*(6*c*d + (-3*b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*
Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a
*c])*e]) - (((8*I)*c^3*d^3 - b^2*((5*I)*b + Sqrt[-b^2 + 4*a*c])*e^3 - 2*c^2*d*e*((6*I)*b*d + Sqrt[-b^2 + 4*a*c
]*d + (16*I)*a*e) + 2*c*e^2*((7*I)*b^2*d + b*Sqrt[-b^2 + 4*a*c]*d + (8*I)*a*b*e + a*Sqrt[-b^2 + 4*a*c]*e))*Arc
Tan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/((-b^2 + 4*a*c)^(3/2)*Sqrt[-
(c*d) + ((b - I*Sqrt[-b^2 + 4*a*c])*e)/2]) - (((-8*I)*c^3*d^3 - b^2*((-5*I)*b + Sqrt[-b^2 + 4*a*c])*e^3 + 2*c^
2*d*e*((6*I)*b*d - Sqrt[-b^2 + 4*a*c]*d + (16*I)*a*e) + 2*c*e^2*((-7*I)*b^2*d + b*Sqrt[-b^2 + 4*a*c]*d - (8*I)
*a*b*e + a*Sqrt[-b^2 + 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a
*c]*e]])/((-b^2 + 4*a*c)^(3/2)*Sqrt[-(c*d) + ((b + I*Sqrt[-b^2 + 4*a*c])*e)/2]) + (2*Sqrt[2]*e^2*(-6*c*d + (3*
b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]])/(S
qrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]))/(2*c^(3/2))

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(\frac {-3 \sqrt {2}\, \left (c \,x^{2}+b x +a \right ) e \left (\left (\frac {c^{2} d^{2}}{3}+e \left (a e -\frac {b d}{3}\right ) c -\frac {b^{2} e^{2}}{6}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}-\frac {4 \left (b e -2 c d \right ) \left (\frac {c^{2} d^{2}}{2}+e \left (a e -\frac {b d}{2}\right ) c -\frac {b^{2} e^{2}}{8}\right )}{3}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (3 \sqrt {2}\, \left (c \,x^{2}+b x +a \right ) \left (\left (\frac {c^{2} d^{2}}{3}+e \left (a e -\frac {b d}{3}\right ) c -\frac {b^{2} e^{2}}{6}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+\frac {4 \left (b e -2 c d \right ) \left (\frac {c^{2} d^{2}}{2}+e \left (a e -\frac {b d}{2}\right ) c -\frac {b^{2} e^{2}}{8}\right )}{3}\right ) e \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )+\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \left (2 c^{2} d^{2} x +\left (-2 a \,e^{2} x -4 \left (\frac {b x}{2}+a \right ) d e +b \,d^{2}\right ) c +b \,e^{2} \left (b x +a \right )\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {e x +d}\right )}{4 \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (a c -\frac {b^{2}}{4}\right ) \left (c \,x^{2}+b x +a \right ) c}\)	\(542\)
derivativedivides	\(2 e^{3} \left (\frac {-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}+\frac {\left (a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+3 b c \,d^{2} e -2 c^{2} d^{3}\right ) \sqrt {e x +d}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {\frac {\left (8 a b c \,e^{3}-16 a \,c^{2} d \,e^{2}-b^{3} e^{3}-2 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-8 a b c \,e^{3}+16 a \,c^{2} d \,e^{2}+b^{3} e^{3}+2 b^{2} c d \,e^{2}-12 b \,c^{2} d^{2} e +8 c^{3} d^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(692\)
default	\(2 e^{3} \left (\frac {-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}+\frac {\left (a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+3 b c \,d^{2} e -2 c^{2} d^{3}\right ) \sqrt {e x +d}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {\frac {\left (8 a b c \,e^{3}-16 a \,c^{2} d \,e^{2}-b^{3} e^{3}-2 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-8 a b c \,e^{3}+16 a \,c^{2} d \,e^{2}+b^{3} e^{3}+2 b^{2} c d \,e^{2}-12 b \,c^{2} d^{2} e +8 c^{3} d^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(692\)

[In]

int((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/4/(-4*(a*c-1/4*b^2)*e^2)^(1/2)*(-3*2^(1/2)*(c*x^2+b*x+a)*e*((1/3*c^2*d^2+e*(a*e-1/3*b*d)*c-1/6*b^2*e^2)*(-4*
(a*c-1/4*b^2)*e^2)^(1/2)-4/3*(b*e-2*c*d)*(1/2*c^2*d^2+e*(a*e-1/2*b*d)*c-1/8*b^2*e^2))*((b*e-2*c*d+(-4*(a*c-1/4
*b^2)*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2)
)+((-b*e+2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2)*(3*2^(1/2)*(c*x^2+b*x+a)*((1/3*c^2*d^2+e*(a*e-1/3*b*d)*c
-1/6*b^2*e^2)*(-4*(a*c-1/4*b^2)*e^2)^(1/2)+4/3*(b*e-2*c*d)*(1/2*c^2*d^2+e*(a*e-1/2*b*d)*c-1/8*b^2*e^2))*e*arct
an(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2))+(-4*(a*c-1/4*b^2)*e^2)^(1/2)*(2
*c^2*d^2*x+(-2*a*e^2*x-4*(1/2*b*x+a)*d*e+b*d^2)*c+b*e^2*(b*x+a))*((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^
(1/2)*(e*x+d)^(1/2)))/((-b*e+2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(1/2))*c)^(1/2)/((b*e-2*c*d+(-4*(a*c-1/4*b^2)*e^2)^(
1/2))*c)^(1/2)/(a*c-1/4*b^2)/(c*x^2+b*x+a)/c

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5256 vs. \(2 (445) = 890\).

Time = 1.21 (sec) , antiderivative size = 5256, normalized size of antiderivative = 10.43 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1398 vs. \(2 (445) = 890\).

Time = 0.80 (sec) , antiderivative size = 1398, normalized size of antiderivative = 2.77 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

-(2*(e*x + d)^(3/2)*c^2*d^2*e - 2*sqrt(e*x + d)*c^2*d^3*e - 2*(e*x + d)^(3/2)*b*c*d*e^2 + 3*sqrt(e*x + d)*b*c*
d^2*e^2 + (e*x + d)^(3/2)*b^2*e^3 - 2*(e*x + d)^(3/2)*a*c*e^3 - sqrt(e*x + d)*b^2*d*e^3 - 2*sqrt(e*x + d)*a*c*
d*e^3 + sqrt(e*x + d)*a*b*e^4)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e + a*e^2)*(b^2
*c - 4*a*c^2)) - (16*(b^2*c^6 - 4*a*c^7)*d^4*e - 32*(b^3*c^5 - 4*a*b*c^6)*d^3*e^2 + 16*(b^4*c^4 - 2*a*b^2*c^5
- 8*a^2*c^6)*d^2*e^3 - 32*(a*b^3*c^4 - 4*a^2*b*c^5)*d*e^4 - (b^6*c^2 - 12*a*b^4*c^3 + 32*a^2*b^2*c^4)*e^5 - (2
*c^2*d^2*e - 2*b*c*d*e^2 - (b^2 - 6*a*c)*e^3)*(b^2*c*e - 4*a*c^2*e)^2 - 2*(2*sqrt(b^2 - 4*a*c)*c^4*d^3*e - 3*s
qrt(b^2 - 4*a*c)*b*c^3*d^2*e^2 - sqrt(b^2 - 4*a*c)*a*b*c^2*e^4 + (b^2*c^2 + 2*a*c^3)*sqrt(b^2 - 4*a*c)*d*e^3)*
abs(-b^2*c*e + 4*a*c^2*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*b*c
^2*e + sqrt((2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*b*c^2*e)^2 - 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 - 4*a*c^3)))/(b^2*c^2 - 4*a*c^3)))/(sqrt(-4*c^2*d + 2*(b*
c + sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^2*c^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3 - (
b^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*c))*e)*abs(-b^2*c*e + 4*a*c^2*e)*abs(c)) + (16*(b^2*c^6 - 4*a*c^7)*d^4*e - 3
2*(b^3*c^5 - 4*a*b*c^6)*d^3*e^2 + 16*(b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d^2*e^3 - 32*(a*b^3*c^4 - 4*a^2*b*c^5
)*d*e^4 - (b^6*c^2 - 12*a*b^4*c^3 + 32*a^2*b^2*c^4)*e^5 - (2*c^2*d^2*e - 2*b*c*d*e^2 - (b^2 - 6*a*c)*e^3)*(b^2
*c*e - 4*a*c^2*e)^2 + 2*(2*sqrt(b^2 - 4*a*c)*c^4*d^3*e - 3*sqrt(b^2 - 4*a*c)*b*c^3*d^2*e^2 - sqrt(b^2 - 4*a*c)
*a*b*c^2*e^4 + (b^2*c^2 + 2*a*c^3)*sqrt(b^2 - 4*a*c)*d*e^3)*abs(-b^2*c*e + 4*a*c^2*e))*arctan(2*sqrt(1/2)*sqrt
(e*x + d)/sqrt(-(2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*b*c^2*e - sqrt((2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4
*a*b*c^2*e)^2 - 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 - 4*a*c^3)))/(b^2*c^2 - 4*a*c^3)))/(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^2*c^2 - 4*a*c^3)
*sqrt(b^2 - 4*a*c)*d - (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3 + (b^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*c))*e)*abs(-b^2*
c*e + 4*a*c^2*e)*abs(c))

Mupad [B] (verification not implemented)

Time = 13.52 (sec) , antiderivative size = 21160, normalized size of antiderivative = 41.98 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*
a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576
*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^
5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) - (2*(
d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c
^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e
^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e
^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d
*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^
2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 4
80*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^
4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*
a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^
3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a
^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^
5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e +
1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^
2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e
^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2
+ 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*
b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e
- 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a
^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8))
)^(1/2) - (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3
+ 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16
*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d
^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c
- b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b
^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8
*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*
c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2
*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*
e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*
b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b
^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*1i
- (((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3
*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^
2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 +
768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) + (2*(d +
e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*
d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5
+ 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3
- 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^
4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c
^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*
a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d
*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2
*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 -
256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*
b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(
-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 153
6*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 +
50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*
(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2
400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8
*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 4
80*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*
c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(
1/2) + (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 +
74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*
b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*
e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c -
b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*
c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^
3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c -
b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^
5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4
+ 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6
*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12
*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*1i)/(
(((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3*b^
3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^2*b
^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 + 76
8*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) - (2*(d + e*
x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5
+ 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1
504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5
*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 +
5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*
d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b
^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^
4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^
8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 - 25
6*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*b^2
*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4
*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a
^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50
*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(
4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400
*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^
2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*
a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9
+ b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2
) - (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 + 74*
a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*b^4
*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*e^6
))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2
)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7
*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d
^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^
2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c
^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 +
5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^
3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^
3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) - (2*(5*
a^2*b^4*e^11 + 216*a^4*c^2*e^11 + 5*b^6*d^2*e^9 + 32*c^6*d^8*e^3 - 66*a^3*b^2*c*e^11 + 232*a*c^5*d^6*e^5 - 128
*b*c^5*d^7*e^4 + 16*b^5*c*d^3*e^8 + 584*a^2*c^4*d^4*e^7 + 600*a^3*c^3*d^2*e^9 + 166*b^2*c^4*d^6*e^5 - 50*b^3*c
^3*d^5*e^6 - 41*b^4*c^2*d^4*e^7 - 10*a*b^5*d*e^10 + 426*a^2*b^2*c^2*d^2*e^9 - 696*a*b*c^4*d^5*e^6 - 108*a*b^4*
c*d^2*e^9 + 158*a^2*b^3*c*d*e^10 - 600*a^3*b*c^2*d*e^10 + 578*a*b^2*c^3*d^4*e^7 + 4*a*b^3*c^2*d^3*e^8 - 1168*a
^2*b*c^3*d^3*e^8))/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) + (((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e
^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*
b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5
- 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^
5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) + (2*(d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3
*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6
*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c
^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2
) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2
) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 +
960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e
^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a
^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*
b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2
- 8*b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8
*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6
*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5
+ 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3
- 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e
^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*
c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480
*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*
d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^
2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) + (2*(d + e*x)^(1/2)*(b^6*e^8 - 72
*a^3*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 + 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6
+ 90*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4
*d^3*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2
))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 384
0*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^
3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d
^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c
*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^
2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4
*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11
520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5
- 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11
*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7
*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a
^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*
e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4
*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*
d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b
^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^
4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*
a^5*b^2*c^8)))^(1/2)*2i - (((d + e*x)^(1/2)*(b^2*d*e^3 + 2*c^2*d^3*e - a*b*e^4 + 2*a*c*d*e^3 - 3*b*c*d^2*e^2))
/(c*(4*a*c - b^2)) - (e*(d + e*x)^(3/2)*(b^2*e^2 + 2*c^2*d^2 - 2*a*c*e^2 - 2*b*c*d*e))/(c*(4*a*c - b^2)))/((b*
e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + a*e^2 + c*d^2 - b*d*e) + atan(((((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 -
512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*
c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96
*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*
e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) - (2*(d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8
*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e
^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e
^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) +
27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) +
960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960
*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 -
3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b
^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*
c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*
b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b
^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5
+ 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1
504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5
*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 -
5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*
d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b
^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^
4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^
8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) - (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3
*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 + 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 9
0*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3
*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*(
(32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^
5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^
5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e
^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e
^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 -
8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3
*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*
a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 12
80*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*1i - (((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 51
2*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4
*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*
b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5
)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) + (2*(d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^
5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4
- 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5
- 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*
a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960
*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*
b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 38
40*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*
c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7
- 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6
*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*
c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 +
3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504
*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^
2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*
b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3
*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*
c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 +
11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c
^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) + (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^
3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 + 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b
^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^
5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32
*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b
*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c
^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*
(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*
(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 896
0*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^
2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4
*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*
a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*1i)/((((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a
^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^
3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4
*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(
b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) - (2*(d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 -
b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 8
0*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7
680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b
^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^
2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5
*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*
a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5
*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 -
6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6*c^
4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2
))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 384
0*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^
3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d
^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c
*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^
2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4
*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11
520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5
- 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) - (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^3*e
^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3 + 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b^2*
c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^5 -
20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^
6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^
5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*
e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(
4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(
4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a
^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 -
240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*
c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3
*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) - (2*(5*a^2*b^4*e^11 + 216*a^4*c^2*e^11 + 5*b^6*d^2*e^
9 + 32*c^6*d^8*e^3 - 66*a^3*b^2*c*e^11 + 232*a*c^5*d^6*e^5 - 128*b*c^5*d^7*e^4 + 16*b^5*c*d^3*e^8 + 584*a^2*c^
4*d^4*e^7 + 600*a^3*c^3*d^2*e^9 + 166*b^2*c^4*d^6*e^5 - 50*b^3*c^3*d^5*e^6 - 41*b^4*c^2*d^4*e^7 - 10*a*b^5*d*e
^10 + 426*a^2*b^2*c^2*d^2*e^9 - 696*a*b*c^4*d^5*e^6 - 108*a*b^4*c*d^2*e^9 + 158*a^2*b^3*c*d*e^10 - 600*a^3*b*c
^2*d*e^10 + 578*a*b^2*c^3*d^4*e^7 + 4*a*b^3*c^2*d^3*e^8 - 1168*a^2*b*c^3*d^3*e^8))/(b^6*c - 64*a^3*c^4 - 12*a*
b^4*c^2 + 48*a^2*b^2*c^3) + (((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*
a^2*b^5*c^3*e^6 - 192*a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2
*b^2*c^6*d^3*e^3 - 576*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4
+ 96*a^2*b^4*c^4*d*e^5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 +
48*a^2*b^2*c^3) + (2*(d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^
9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d
^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3
*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)
^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4
*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 51
20*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*
d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3
- 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3
*e^3 - 48*a*b^5*c^4*e^3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a
*b^4*c^5*d*e^2 - 384*a^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d
^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4
- 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5
- 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27
*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 96
0*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a
*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3
840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2
*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^
7 - 6144*a^5*b^2*c^8)))^(1/2) + (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*
e^4 - 96*b*c^5*d^5*e^3 + 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 1
0*b^4*c^2*d^2*e^6 - 16*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d
*e^7 + 140*a*b^2*c^3*d^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*
e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*
c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^
4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e
^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*
c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d
^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^
3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4
)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a
^5*b^2*c^8)))^(1/2)))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 - b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*
a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^
7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^
2*d^2*e^3 + 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 + 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b
^10*c*d*e^4 - 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560
*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^
4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3
*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4
+ 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*2i