\(\int \frac {(d+e x^2)^3}{(a+b x^2+c x^4)^2} \, dx\) [270]

Optimal result

Integrand size = 24, antiderivative size = 563 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac {a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (a b^3 e^3+6 a c \left (2 c d+\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt {b^2-4 a c} e^3\right )-b c \left (a e^2 \left (3 \sqrt {b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt {b^2-4 a c} d+12 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (a b^3 e^3+6 a c \left (2 c d-\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2-a \sqrt {b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt {b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt {b^2-4 a c} d-8 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]

[Out]

1/2*x*(c*(b^2*d^3-2*a*d*(-3*a*e^2+c*d^2)-a*b*e*(a*e^2+3*c*d^2)/c)-(a*b^2*e^3+2*a*c*e*(-a*e^2+3*c*d^2)-b*c*d*(3
*a*e^2+c*d^2))*x^2)/a/c/(-4*a*c+b^2)/(c*x^4+b*x^2+a)-1/4*arctan(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)
)*(a*b^3*e^3+6*a*c*(a*e^2+c*d^2)*(2*c*d+e*(-4*a*c+b^2)^(1/2))-b^2*(c^2*d^3-3*a*c*d*e^2+a*e^3*(-4*a*c+b^2)^(1/2
))-b*c*(c*d^2*(12*a*e+d*(-4*a*c+b^2)^(1/2))+a*e^2*(8*a*e+3*d*(-4*a*c+b^2)^(1/2))))/a/c^(3/2)/(-4*a*c+b^2)^(3/2
)*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/4*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(a*b^3*e^3+6
*a*c*(a*e^2+c*d^2)*(2*c*d-e*(-4*a*c+b^2)^(1/2))-b^2*(c^2*d^3-3*a*c*d*e^2-a*e^3*(-4*a*c+b^2)^(1/2))+b*c*(c*d^2*
(-12*a*e+d*(-4*a*c+b^2)^(1/2))+a*e^2*(-8*a*e+3*d*(-4*a*c+b^2)^(1/2))))/a/c^(3/2)/(-4*a*c+b^2)^(3/2)*2^(1/2)/(b
+(-4*a*c+b^2)^(1/2))^(1/2)

Rubi [A] (verified)

Time = 2.37 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1219, 1180, 211} \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (a b^3 e^3-b^2 \left (a e^3 \sqrt {b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (e \sqrt {b^2-4 a c}+2 c d\right )-b c \left (c d^2 \left (d \sqrt {b^2-4 a c}+12 a e\right )+a e^2 \left (3 d \sqrt {b^2-4 a c}+8 a e\right )\right )\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (a b^3 e^3-b^2 \left (-a e^3 \sqrt {b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (2 c d-e \sqrt {b^2-4 a c}\right )+b c \left (c d^2 \left (d \sqrt {b^2-4 a c}-12 a e\right )+a e^2 \left (3 d \sqrt {b^2-4 a c}-8 a e\right )\right )\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {x \left (c \left (-\frac {a b e \left (a e^2+3 c d^2\right )}{c}-2 a d \left (c d^2-3 a e^2\right )+b^2 d^3\right )-x^2 \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[In]

Int[(d + e*x^2)^3/(a + b*x^2 + c*x^4)^2,x]

[Out]

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

Rule 211

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

Rule 1180

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

Rule 1219

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x
^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2
*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS
um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c
*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], 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] && IGtQ[q, 1] && LtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac {a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {-b^2 d^3+6 a d \left (c d^2+a e^2\right )-\frac {a b e \left (3 c d^2+a e^2\right )}{c}+\left (-\frac {a b^2 e^3}{c}+6 a e \left (c d^2+a e^2\right )-b \left (c d^3+3 a d e^2\right )\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )} \\ & = \frac {x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac {a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (a b^3 e^3+6 a c \left (2 c d-\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2-a \sqrt {b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt {b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt {b^2-4 a c} d-8 a e\right )\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{4 a c \left (b^2-4 a c\right )^{3/2}}-\frac {\left (a b^3 e^3+6 a c \left (2 c d+\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt {b^2-4 a c} e^3\right )-b c \left (a e^2 \left (3 \sqrt {b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt {b^2-4 a c} d+12 a e\right )\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{4 a c \left (b^2-4 a c\right )^{3/2}} \\ & = \frac {x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac {a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (a b^3 e^3+6 a c \left (2 c d+\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt {b^2-4 a c} e^3\right )-b c \left (a e^2 \left (3 \sqrt {b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt {b^2-4 a c} d+12 a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (a b^3 e^3+6 a c \left (2 c d-\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2-a \sqrt {b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt {b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt {b^2-4 a c} d-8 a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 540, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {\frac {2 \sqrt {c} x \left (b^2 \left (c d^3-a e^3 x^2\right )+b \left (-a^2 e^3+c^2 d^3 x^2-3 a c d e \left (d-e x^2\right )\right )+2 a c \left (a e^2 \left (3 d+e x^2\right )-c d^2 \left (d+3 e x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \left (-a b^3 e^3-6 a c \left (2 c d+\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )+b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt {b^2-4 a c} e^3\right )+b c \left (a e^2 \left (3 \sqrt {b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt {b^2-4 a c} d+12 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (a b^3 e^3+6 a c \left (2 c d-\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )+b^2 \left (-c^2 d^3+3 a c d e^2+a \sqrt {b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt {b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt {b^2-4 a c} d-8 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{4 a c^{3/2}} \]

[In]

Integrate[(d + e*x^2)^3/(a + b*x^2 + c*x^4)^2,x]

[Out]

((2*Sqrt[c]*x*(b^2*(c*d^3 - a*e^3*x^2) + b*(-(a^2*e^3) + c^2*d^3*x^2 - 3*a*c*d*e*(d - e*x^2)) + 2*a*c*(a*e^2*(
3*d + e*x^2) - c*d^2*(d + 3*e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(a*b^3*e^3) - 6*a*c*(2*
c*d + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) + b^2*(c^2*d^3 - 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(a*e^
2*(3*Sqrt[b^2 - 4*a*c]*d + 8*a*e) + c*d^2*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b -
Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(a*b^3*e^3 + 6*a*c*(2*c*d -
Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) + b^2*(-(c^2*d^3) + 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(c*d^2*
(Sqrt[b^2 - 4*a*c]*d - 12*a*e) + a*e^2*(3*Sqrt[b^2 - 4*a*c]*d - 8*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + S
qrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a*c^(3/2))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.26 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.57

method	result	size
risch	\(\frac {-\frac {\left (2 a^{2} c \,e^{3}-a \,b^{2} e^{3}+3 a b c d \,e^{2}-6 a \,c^{2} d^{2} e +b \,c^{2} d^{3}\right ) x^{3}}{2 a c \left (4 a c -b^{2}\right )}+\frac {\left (a^{2} b \,e^{3}-6 a^{2} c d \,e^{2}+3 a b c \,d^{2} e +2 a \,c^{2} d^{3}-b^{2} c \,d^{3}\right ) x}{2 c \left (4 a c -b^{2}\right ) a}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {\left (6 a^{2} c \,e^{3}-a \,b^{2} e^{3}-3 a b c d \,e^{2}+6 a \,c^{2} d^{2} e -b \,c^{2} d^{3}\right ) \textit {\_R}^{2}}{4 a c -b^{2}}-\frac {a^{2} b \,e^{3}-6 a^{2} c d \,e^{2}+3 a b c \,d^{2} e -6 a \,c^{2} d^{3}+b^{2} c \,d^{3}}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{4 a c}\)	\(320\)
default	\(\frac {-\frac {\left (2 a^{2} c \,e^{3}-a \,b^{2} e^{3}+3 a b c d \,e^{2}-6 a \,c^{2} d^{2} e +b \,c^{2} d^{3}\right ) x^{3}}{2 a c \left (4 a c -b^{2}\right )}+\frac {\left (a^{2} b \,e^{3}-6 a^{2} c d \,e^{2}+3 a b c \,d^{2} e +2 a \,c^{2} d^{3}-b^{2} c \,d^{3}\right ) x}{2 c \left (4 a c -b^{2}\right ) a}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (6 a^{2} c \,e^{3} \sqrt {-4 a c +b^{2}}-a \,b^{2} e^{3} \sqrt {-4 a c +b^{2}}-3 a b c d \,e^{2} \sqrt {-4 a c +b^{2}}+6 a \,c^{2} d^{2} e \sqrt {-4 a c +b^{2}}-b \,c^{2} d^{3} \sqrt {-4 a c +b^{2}}+8 a^{2} b \,e^{3} c -12 a^{2} c^{2} d \,e^{2}-a \,b^{3} e^{3}-3 a \,b^{2} c d \,e^{2}+12 a b \,c^{2} d^{2} e -12 a \,c^{3} d^{3}+b^{2} c^{2} d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (6 a^{2} c \,e^{3} \sqrt {-4 a c +b^{2}}-a \,b^{2} e^{3} \sqrt {-4 a c +b^{2}}-3 a b c d \,e^{2} \sqrt {-4 a c +b^{2}}+6 a \,c^{2} d^{2} e \sqrt {-4 a c +b^{2}}-b \,c^{2} d^{3} \sqrt {-4 a c +b^{2}}-8 a^{2} b \,e^{3} c +12 a^{2} c^{2} d \,e^{2}+a \,b^{3} e^{3}+3 a \,b^{2} c d \,e^{2}-12 a b \,c^{2} d^{2} e +12 a \,c^{3} d^{3}-b^{2} c^{2} d^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{a \left (4 a c -b^{2}\right )}\)	\(628\)

[In]

int((e*x^2+d)^3/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

(-1/2*(2*a^2*c*e^3-a*b^2*e^3+3*a*b*c*d*e^2-6*a*c^2*d^2*e+b*c^2*d^3)/a/c/(4*a*c-b^2)*x^3+1/2/c*(a^2*b*e^3-6*a^2
*c*d*e^2+3*a*b*c*d^2*e+2*a*c^2*d^3-b^2*c*d^3)/(4*a*c-b^2)/a*x)/(c*x^4+b*x^2+a)+1/4/a/c*sum(((6*a^2*c*e^3-a*b^2
*e^3-3*a*b*c*d*e^2+6*a*c^2*d^2*e-b*c^2*d^3)/(4*a*c-b^2)*_R^2-(a^2*b*e^3-6*a^2*c*d*e^2+3*a*b*c*d^2*e-6*a*c^2*d^
3+b^2*c*d^3)/(4*a*c-b^2))/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*b+a))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 12117 vs. \(2 (507) = 1014\).

Time = 107.28 (sec) , antiderivative size = 12117, normalized size of antiderivative = 21.52 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x^2+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((e*x**2+d)**3/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \]

[In]

integrate((e*x^2+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8992 vs. \(2 (507) = 1014\).

Time = 1.77 (sec) , antiderivative size = 8992, normalized size of antiderivative = 15.97 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x^2+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

1/2*(b*c^2*d^3*x^3 - 6*a*c^2*d^2*e*x^3 + 3*a*b*c*d*e^2*x^3 - a*b^2*e^3*x^3 + 2*a^2*c*e^3*x^3 + b^2*c*d^3*x - 2
*a*c^2*d^3*x - 3*a*b*c*d^2*e*x + 6*a^2*c*d*e^2*x - a^2*b*e^3*x)/((c*x^4 + b*x^2 + a)*(a*b^2*c - 4*a^2*c^2)) +
1/16*((2*b^3*c^4 - 8*a*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4
*a*c)*c)*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^4 - 2*(b^2 - 4*a*c)*b*c^4)*(a
*b^2*c - 4*a^2*c^2)^2*d^3 - 6*(2*a*b^2*c^4 - 8*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c
)*c)*a*b^2*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*
a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^4
- 2*(b^2 - 4*a*c)*a*c^4)*(a*b^2*c - 4*a^2*c^2)^2*d^2*e + 3*(2*a*b^3*c^3 - 8*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*
a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2
*b*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*a*b*c^3)*(a*b^2*c - 4*a^2*c^2)^2*d*e^2 + (2*a*b^4*c^2 -
20*a^2*b^2*c^3 + 48*a^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4 + 10*sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a*b^3*c - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^2 - 12*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c
)*a*b^2*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 - 2*(b^2 - 4*a*c)*a*b^2*c^2
+ 12*(b^2 - 4*a*c)*a^2*c^3)*(a*b^2*c - 4*a^2*c^2)^2*e^3 + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c^3
- 14*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^4 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^
4 - 2*a*b^6*c^4 + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^5 + 20*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*
a*c)*c)*a^2*b^3*c^5 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^5 + 28*a^2*b^4*c^5 - 96*sqrt(2)*sqrt(b*c
+ sqrt(b^2 - 4*a*c)*c)*a^4*c^6 - 48*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^6 - 10*sqrt(2)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^6 - 128*a^3*b^2*c^6 + 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^7 + 192
*a^4*c^7 + 2*(b^2 - 4*a*c)*a*b^4*c^4 - 20*(b^2 - 4*a*c)*a^2*b^2*c^5 + 48*(b^2 - 4*a*c)*a^3*c^6)*d^3*abs(a*b^2*
c - 4*a^2*c^2) + 6*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^3 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*
c)*c)*a^3*b^3*c^4 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^4 - 2*a^2*b^5*c^4 + 16*sqrt(2)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^5 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^5 + sqrt(2)*sqrt(b*c
+ sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^5 + 16*a^3*b^3*c^5 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^6 - 32
*a^4*b*c^6 + 2*(b^2 - 4*a*c)*a^2*b^3*c^4 - 8*(b^2 - 4*a*c)*a^3*b*c^5)*d^2*e*abs(a*b^2*c - 4*a^2*c^2) - 12*(sqr
t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^3 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^4 - 2*s
qrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^4 - 2*a^3*b^4*c^4 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c
)*a^5*c^5 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^5 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*
b^2*c^5 + 16*a^4*b^2*c^5 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*c^6 - 32*a^5*c^6 + 2*(b^2 - 4*a*c)*a^
3*b^2*c^4 - 8*(b^2 - 4*a*c)*a^4*c^5)*d*e^2*abs(a*b^2*c - 4*a^2*c^2) + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*a^3*b^5*c^2 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^3 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c
)*c)*a^3*b^4*c^3 - 2*a^3*b^5*c^3 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b*c^4 + 8*sqrt(2)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^4 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^4 + 16*a^4*b^3*c^4 - 4*s
qrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^5 - 32*a^5*b*c^5 + 2*(b^2 - 4*a*c)*a^3*b^3*c^3 - 8*(b^2 - 4*a*c
)*a^4*b*c^4)*e^3*abs(a*b^2*c - 4*a^2*c^2) + (2*a^2*b^7*c^6 - 40*a^3*b^5*c^7 + 224*a^4*b^3*c^8 - 384*a^5*b*c^9
- sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^7*c^4 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^5 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^6*c^5
- 112*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^6 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*
c^6 + 192*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b*c^7 + 96*sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^7 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*
b^3*c^7 - 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^8 - 2*(b^2 - 4*a*c)*a^2*b^5*c^6
+ 32*(b^2 - 4*a*c)*a^3*b^3*c^7 - 96*(b^2 - 4*a*c)*a^4*b*c^8)*d^3 + 12*(2*a^3*b^6*c^6 - 16*a^4*b^4*c^7 + 32*a^
5*b^2*c^8 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c
)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^4*c^5 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^
3*b^5*c^5 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^6 - 8*sqrt(2)*sqrt(b^2 - 4*
a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
^3*b^4*c^6 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^7 - 2*(b^2 - 4*a*c)*a^3*b^4
*c^6 + 8*(b^2 - 4*a*c)*a^4*b^2*c^7)*d^2*e - 3*(2*a^3*b^7*c^5 - 8*a^4*b^5*c^6 - 32*a^5*b^3*c^7 + 128*a^6*b*c^8
- sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^7*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
+ sqrt(b^2 - 4*a*c)*c)*a^4*b^5*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^4
+ 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^5 - 64*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b*c^6
- 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^6 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt
(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b*c^7 - 2*(b^2 - 4*a*c)*a^3*b^5*c^5 + 32*(b^2 - 4*a*c)*a^5*b*c^7)*d*e^2 - (2*a
^3*b^8*c^4 - 32*a^4*b^6*c^5 + 160*a^5*b^4*c^6 - 256*a^6*b^2*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*a^3*b^8*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c^3 + 2*sqrt(
2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^7*c^3 - 80*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr
t(b^2 - 4*a*c)*c)*a^5*b^4*c^4 - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^5*c^4 - sqr
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^4 + 128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^5 + 64*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^3*c^5 +
12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^4*c^5 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b
*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^6 - 2*(b^2 - 4*a*c)*a^3*b^6*c^4 + 24*(b^2 - 4*a*c)*a^4*b^4*c^5 - 64*(b^2 -
4*a*c)*a^5*b^2*c^6)*e^3)*arctan(2*sqrt(1/2)*x/sqrt((a*b^3*c - 4*a^2*b*c^2 + sqrt((a*b^3*c - 4*a^2*b*c^2)^2 -
4*(a^2*b^2*c - 4*a^3*c^2)*(a*b^2*c^2 - 4*a^2*c^3)))/(a*b^2*c^2 - 4*a^2*c^3)))/((a^3*b^6*c^3 - 12*a^4*b^4*c^4 -
2*a^3*b^5*c^4 + 48*a^5*b^2*c^5 + 16*a^4*b^3*c^5 + a^3*b^4*c^5 - 64*a^6*c^6 - 32*a^5*b*c^6 - 8*a^4*b^2*c^6 + 1
6*a^5*c^7)*abs(a*b^2*c - 4*a^2*c^2)*abs(c)) - 1/16*((2*b^3*c^4 - 8*a*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + 2*sqr
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*b*c^4 - 2*(b^2 - 4*a*c)*b*c^4)*(a*b^2*c - 4*a^2*c^2)^2*d^3 - 6*(2*a*b^2*c^4 - 8*a^2*c^5 - sqrt(2)
*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*a^2*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^2 - 4*a*c)*a*c^4)*(a*b^2*c - 4*a^2*c^2)^2*d^2*e + 3*(2*a
*b^3*c^3 - 8*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 4*sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c
)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*a*b*c^3)*
(a*b^2*c - 4*a^2*c^2)^2*d*e^2 + (2*a*b^4*c^2 - 20*a^2*b^2*c^3 + 48*a^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*a*b^4 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c + 2*sq
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt
(b^2 - 4*a*c)*c)*a^3*c^2 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 -
4*a*c)*c)*a^2*c^3 - 2*(b^2 - 4*a*c)*a*b^2*c^2 + 12*(b^2 - 4*a*c)*a^2*c^3)*(a*b^2*c - 4*a^2*c^2)^2*e^3 - 2*(sqr
t(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^6*c^3 - 14*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^4 - 2*sq
rt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*c^4 + 2*a*b^6*c^4 + 64*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3
*b^2*c^5 + 20*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^5 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*
b^4*c^5 - 28*a^2*b^4*c^5 - 96*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*c^6 - 48*sqrt(2)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*a^3*b*c^6 - 10*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^6 + 128*a^3*b^2*c^6 + 24*sqrt(2)
*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^7 - 192*a^4*c^7 - 2*(b^2 - 4*a*c)*a*b^4*c^4 + 20*(b^2 - 4*a*c)*a^2*b^2*
c^5 - 48*(b^2 - 4*a*c)*a^3*c^6)*d^3*abs(a*b^2*c - 4*a^2*c^2) - 6*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*
b^5*c^3 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^4 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^
2*b^4*c^4 + 2*a^2*b^5*c^4 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b*c^5 + 8*sqrt(2)*sqrt(b*c - sqrt(b
^2 - 4*a*c)*c)*a^3*b^2*c^5 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^5 - 16*a^3*b^3*c^5 - 4*sqrt(2)*
sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^6 + 32*a^4*b*c^6 - 2*(b^2 - 4*a*c)*a^2*b^3*c^4 + 8*(b^2 - 4*a*c)*a^3*b
*c^5)*d^2*e*abs(a*b^2*c - 4*a^2*c^2) + 12*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^3 - 8*sqrt(2)*sqr
t(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^4 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^4 + 2*a^3*b^4*c
^4 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*c^5 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b*c^5
+ sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^5 - 16*a^4*b^2*c^5 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c
)*c)*a^4*c^6 + 32*a^5*c^6 - 2*(b^2 - 4*a*c)*a^3*b^2*c^4 + 8*(b^2 - 4*a*c)*a^4*c^5)*d*e^2*abs(a*b^2*c - 4*a^2*c
^2) - 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^2 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b
^3*c^3 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^3 + 2*a^3*b^5*c^3 + 16*sqrt(2)*sqrt(b*c - sqrt(b^
2 - 4*a*c)*c)*a^5*b*c^4 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^4 + sqrt(2)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*a^3*b^3*c^4 - 16*a^4*b^3*c^4 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b*c^5 + 32*a^5*b*c^5
- 2*(b^2 - 4*a*c)*a^3*b^3*c^3 + 8*(b^2 - 4*a*c)*a^4*b*c^4)*e^3*abs(a*b^2*c - 4*a^2*c^2) + (2*a^2*b^7*c^6 - 40*
a^3*b^5*c^7 + 224*a^4*b^3*c^8 - 384*a^5*b*c^9 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*
b^7*c^4 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^5 + 2*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^6*c^5 - 112*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)
*a^4*b^3*c^6 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^6 - sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^6 + 192*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)
*c)*a^5*b*c^7 + 96*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^7 + 16*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^7 - 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*a^4*b*c^8 - 2*(b^2 - 4*a*c)*a^2*b^5*c^6 + 32*(b^2 - 4*a*c)*a^3*b^3*c^7 - 96*(b^2 - 4*a*c)*a^4*b*c^8)*d^
3 + 12*(2*a^3*b^6*c^6 - 16*a^4*b^4*c^7 + 32*a^5*b^2*c^8 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*
c)*c)*a^3*b^6*c^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^4*c^5 + 2*sqrt(2)*sqrt(b
^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^5 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4
*a*c)*c)*a^5*b^2*c^6 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^6 - sqrt(2)*sqrt(
b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^6 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4
*a*c)*c)*a^4*b^2*c^7 - 2*(b^2 - 4*a*c)*a^3*b^4*c^6 + 8*(b^2 - 4*a*c)*a^4*b^2*c^7)*d^2*e - 3*(2*a^3*b^7*c^5 - 8
*a^4*b^5*c^6 - 32*a^5*b^3*c^7 + 128*a^6*b*c^8 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*
b^7*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^5*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c
)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^4 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a
^5*b^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^5 - 64*sqrt(2)*sqrt(b^2 - 4*a
*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b*c^6 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a
^5*b^2*c^6 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b*c^7 - 2*(b^2 - 4*a*c)*a^3*b^5*
c^5 + 32*(b^2 - 4*a*c)*a^5*b*c^7)*d*e^2 - (2*a^3*b^8*c^4 - 32*a^4*b^6*c^5 + 160*a^5*b^4*c^6 - 256*a^6*b^2*c^7
- sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^8*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^7*c^3
- 80*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^4 - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^5*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c
^4 + 128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^5 + 64*sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^3*c^5 + 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4
*b^4*c^5 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^6 - 2*(b^2 - 4*a*c)*a^3*b^6*
c^4 + 24*(b^2 - 4*a*c)*a^4*b^4*c^5 - 64*(b^2 - 4*a*c)*a^5*b^2*c^6)*e^3)*arctan(2*sqrt(1/2)*x/sqrt((a*b^3*c - 4
*a^2*b*c^2 - sqrt((a*b^3*c - 4*a^2*b*c^2)^2 - 4*(a^2*b^2*c - 4*a^3*c^2)*(a*b^2*c^2 - 4*a^2*c^3)))/(a*b^2*c^2 -
4*a^2*c^3)))/((a^3*b^6*c^3 - 12*a^4*b^4*c^4 - 2*a^3*b^5*c^4 + 48*a^5*b^2*c^5 + 16*a^4*b^3*c^5 + a^3*b^4*c^5 -
64*a^6*c^6 - 32*a^5*b*c^6 - 8*a^4*b^2*c^6 + 16*a^5*c^7)*abs(a*b^2*c - 4*a^2*c^2)*abs(c))

Mupad [B] (verification not implemented)

Time = 10.27 (sec) , antiderivative size = 29030, normalized size of antiderivative = 51.56 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

int((d + e*x^2)^3/(a + b*x^2 + c*x^4)^2,x)

[Out]

- ((x^3*(b*c^2*d^3 - a*b^2*e^3 + 2*a^2*c*e^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2))/(2*a*c*(4*a*c - b^2)) - (x*(2*a
*c^2*d^3 + a^2*b*e^3 - b^2*c*d^3 - 6*a^2*c*d*e^2 + 3*a*b*c*d^2*e))/(2*a*c*(4*a*c - b^2)))/(a + b*x^2 + c*x^4)
- atan(((((6144*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*b*c^5*e^3 + 6144*a^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3
+ 1920*a^3*b^4*c^5*d^3 - 5632*a^4*b^2*c^6*d^3 + 16*a^3*b^7*c^2*e^3 - 192*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^
3 - 3072*a^5*b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*b^5*c^4*d^2*e - 96*a^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*
c^5*d^2*e + 1152*a^4*b^4*c^4*d*e^2 - 4608*a^5*b^2*c^5*d*e^2)/(8*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*
a^4*b^2*c^3)) - (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*
c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^
8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6
*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-
(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*
c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2
*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e
^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2
)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4
*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*
e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^
2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 2
40*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*(1024*a^5*b*c^6 - 16*a^2*b^7*
c^3 + 192*a^3*b^5*c^4 - 768*a^4*b^3*c^5))/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b
^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 38
40*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^
7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c
^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^
3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 374
4*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^
5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^
3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b
^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 +
4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b
*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c -
b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*
a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2) - (x*(72*a^5*c^3*e^6 - 72*a^2*c^6*d^6 - a^2*b^6*e^6 - b^4*c^4*d^6 + 14
*a*b^2*c^5*d^6 + 16*a^3*b^4*c*e^6 - 74*a^4*b^2*c^2*e^6 - 72*a^3*c^5*d^4*e^2 + 72*a^4*c^4*d^2*e^4 - 102*a^2*b^2
*c^4*d^4*e^2 + 44*a^2*b^3*c^3*d^3*e^3 + 9*a^2*b^4*c^2*d^2*e^4 - 174*a^3*b^2*c^3*d^2*e^4 - 6*a*b^3*c^4*d^5*e +
120*a^2*b*c^5*d^5*e - 6*a^2*b^5*c*d*e^5 + 24*a^4*b*c^3*d*e^5 + 144*a^3*b*c^4*d^3*e^3 + 42*a^3*b^3*c^2*d*e^5))/
(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^
8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c -
b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^
4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b
^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^
7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^
3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e
^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3
*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*
a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d
*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*
a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^1
2*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*1i
- (((6144*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*b*c^5*e^3 + 6144*a^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3 + 19
20*a^3*b^4*c^5*d^3 - 5632*a^4*b^2*c^6*d^3 + 16*a^3*b^7*c^2*e^3 - 192*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^3 - 3
072*a^5*b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*b^5*c^4*d^2*e - 96*a^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*c^5*d
^2*e + 1152*a^4*b^4*c^4*d*e^2 - 4608*a^5*b^2*c^5*d*e^2)/(8*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b
^2*c^3)) + (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b
^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5
*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4
*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*
c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d
^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4
- 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 -
13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^
(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*
c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 +
384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)
^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^
5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*(1024*a^5*b*c^6 - 16*a^2*b^7*c^3 +
192*a^3*b^5*c^4 - 768*a^4*b^3*c^5))/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c
^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^
8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5
*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^
6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3
+ 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4
*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4
*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4
*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c
*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*
a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*
d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^
9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b
^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2) + (x*(72*a^5*c^3*e^6 - 72*a^2*c^6*d^6 - a^2*b^6*e^6 - b^4*c^4*d^6 + 14*a*b^
2*c^5*d^6 + 16*a^3*b^4*c*e^6 - 74*a^4*b^2*c^2*e^6 - 72*a^3*c^5*d^4*e^2 + 72*a^4*c^4*d^2*e^4 - 102*a^2*b^2*c^4*
d^4*e^2 + 44*a^2*b^3*c^3*d^3*e^3 + 9*a^2*b^4*c^2*d^2*e^4 - 174*a^3*b^2*c^3*d^2*e^4 - 6*a*b^3*c^4*d^5*e + 120*a
^2*b*c^5*d^5*e - 6*a^2*b^5*c*d*e^5 + 24*a^4*b*c^3*d*e^5 + 144*a^3*b*c^4*d^3*e^3 + 42*a^3*b^3*c^2*d*e^5))/(2*(1
6*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6
- 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^
9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3
*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^
4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4
*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3
- 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 +
8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*
d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b
^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5
+ 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c
^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3
- 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*1i)/((5
*a^4*b^4*e^9 + 216*a^6*c^2*e^9 + 5*b^3*c^5*d^9 - 66*a^5*b^2*c*e^9 + a*b^7*d^3*e^6 - 9*a^3*b^5*d*e^8 + 216*a^2*
c^6*d^8*e - 9*b^4*c^4*d^8*e + 3*a^2*b^6*d^2*e^7 + 864*a^3*c^5*d^6*e^3 + 1296*a^4*c^4*d^4*e^5 + 864*a^5*c^3*d^2
*e^7 + 3*b^5*c^3*d^7*e^2 + b^6*c^2*d^6*e^3 - 36*a*b*c^6*d^9 + 624*a^2*b^2*c^4*d^6*e^3 - 6*a^2*b^3*c^3*d^5*e^4
- 108*a^2*b^4*c^2*d^4*e^5 + 1020*a^3*b^2*c^3*d^4*e^5 + 128*a^3*b^3*c^2*d^3*e^6 + 384*a^4*b^2*c^2*d^2*e^7 + 54*
a*b^2*c^5*d^8*e + 6*a*b^6*c*d^4*e^5 + 153*a^4*b^3*c*d*e^8 - 612*a^5*b*c^2*d*e^8 + 24*a*b^3*c^4*d^7*e^2 - 46*a*
b^4*c^3*d^6*e^3 - 3*a*b^5*c^2*d^5*e^4 - 720*a^2*b*c^5*d^7*e^2 - 3*a^2*b^5*c*d^3*e^6 - 1944*a^3*b*c^4*d^5*e^4 -
90*a^3*b^4*c*d^2*e^7 - 1872*a^4*b*c^3*d^3*e^6)/(4*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b^2*c^3))
+ (((6144*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*b*c^5*e^3 + 6144*a^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3 + 19
20*a^3*b^4*c^5*d^3 - 5632*a^4*b^2*c^6*d^3 + 16*a^3*b^7*c^2*e^3 - 192*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^3 - 3
072*a^5*b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*b^5*c^4*d^2*e - 96*a^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*c^5*d
^2*e + 1152*a^4*b^4*c^4*d*e^2 - 4608*a^5*b^2*c^5*d*e^2)/(8*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b
^2*c^3)) - (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b
^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5
*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4
*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*
c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d
^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4
- 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 -
13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^
(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*
c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 +
384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)
^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^
5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*(1024*a^5*b*c^6 - 16*a^2*b^7*c^3 +
192*a^3*b^5*c^4 - 768*a^4*b^3*c^5))/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c
^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^
8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5
*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^
6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3
+ 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4
*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4
*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4
*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c
*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*
a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*
d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^
9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b
^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2) - (x*(72*a^5*c^3*e^6 - 72*a^2*c^6*d^6 - a^2*b^6*e^6 - b^4*c^4*d^6 + 14*a*b^
2*c^5*d^6 + 16*a^3*b^4*c*e^6 - 74*a^4*b^2*c^2*e^6 - 72*a^3*c^5*d^4*e^2 + 72*a^4*c^4*d^2*e^4 - 102*a^2*b^2*c^4*
d^4*e^2 + 44*a^2*b^3*c^3*d^3*e^3 + 9*a^2*b^4*c^2*d^2*e^4 - 174*a^3*b^2*c^3*d^2*e^4 - 6*a*b^3*c^4*d^5*e + 120*a
^2*b*c^5*d^5*e - 6*a^2*b^5*c*d*e^5 + 24*a^4*b*c^3*d*e^5 + 144*a^3*b*c^4*d^3*e^3 + 42*a^3*b^3*c^2*d*e^5))/(2*(1
6*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6
- 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^
9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3
*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^
4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4
*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3
- 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 +
8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*
d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b
^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5
+ 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c
^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3
- 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2) + (((61
44*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*b*c^5*e^3 + 6144*a^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3 + 1920*a^3*b
^4*c^5*d^3 - 5632*a^4*b^2*c^6*d^3 + 16*a^3*b^7*c^2*e^3 - 192*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^3 - 3072*a^5*
b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*b^5*c^4*d^2*e - 96*a^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*c^5*d^2*e + 1
152*a^4*b^4*c^4*d*e^2 - 4608*a^5*b^2*c^5*d*e^2)/(8*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b^2*c^3))
+ (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(
1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 921
6*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b
^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)
^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 +
9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*
a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^
6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) -
6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*
e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6
*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) -
6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^
5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*(1024*a^5*b*c^6 - 16*a^2*b^7*c^3 + 192*a^3
*b^5*c^4 - 768*a^4*b^3*c^5))/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 -
a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*
e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1
504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504
*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2
*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5
*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3
*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(
4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 +
108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*
c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4
+ 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2)
)/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 -
6144*a^8*b^2*c^8)))^(1/2) + (x*(72*a^5*c^3*e^6 - 72*a^2*c^6*d^6 - a^2*b^6*e^6 - b^4*c^4*d^6 + 14*a*b^2*c^5*d^
6 + 16*a^3*b^4*c*e^6 - 74*a^4*b^2*c^2*e^6 - 72*a^3*c^5*d^4*e^2 + 72*a^4*c^4*d^2*e^4 - 102*a^2*b^2*c^4*d^4*e^2
+ 44*a^2*b^3*c^3*d^3*e^3 + 9*a^2*b^4*c^2*d^2*e^4 - 174*a^3*b^2*c^3*d^2*e^4 - 6*a*b^3*c^4*d^5*e + 120*a^2*b*c^5
*d^5*e - 6*a^2*b^5*c*d*e^5 + 24*a^4*b*c^3*d*e^5 + 144*a^3*b*c^4*d^3*e^3 + 42*a^3*b^3*c^2*d*e^5))/(2*(16*a^4*c^
3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c
^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2)
- 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6
- a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 +
b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2
+ 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a
^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6
*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*
(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d
^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*
a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e
*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^
4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)))*((27*a*b^9*c^
4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c
*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 2
88*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*
a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a
^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*
e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2
+ 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 -
9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e
- 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2
*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17
664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-
(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^
6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*2i - atan(((((6144*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*
b*c^5*e^3 + 6144*a^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3 + 1920*a^3*b^4*c^5*d^3 - 5632*a^4*b^2*c^6*d^3 + 16*a^3*b^
7*c^2*e^3 - 192*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^3 - 3072*a^5*b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*
b^5*c^4*d^2*e - 96*a^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*c^5*d^2*e + 1152*a^4*b^4*c^4*d*e^2 - 4608*a^5*b^2*c^5*d*e^
2)/(8*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b^2*c^3)) - (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3
*b^11*e^6 + 3840*a^5*b*c^8*d^6 + 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6
- 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*
a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 + a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6
*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 - b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9
*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4
*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3
+ 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 + 9*a^2*c^3*d^4*e^2*(-(4*a*
c - b^2)^9)^(1/2) - 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108
*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*
d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 46
08*a^7*b^2*c^5*d*e^5 - 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(3
2*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 614
4*a^8*b^2*c^8)))^(1/2)*(1024*a^5*b*c^6 - 16*a^2*b^7*c^3 + 192*a^3*b^5*c^4 - 768*a^4*b^3*c^5))/(2*(16*a^4*c^3 +
a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 + 9*a*c^4*
d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 - 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) -
9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 +
a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 - b^2
*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 +
88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*
b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^
2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 + 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) - 9*a^3*c^2*d^2*e^4*(-(
4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*
e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6
*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 - 6*a*b*c^3*d^5*e*(-
(4*a*c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b
^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2) - (x*(72*a^5*c^3*e
^6 - 72*a^2*c^6*d^6 - a^2*b^6*e^6 - b^4*c^4*d^6 + 14*a*b^2*c^5*d^6 + 16*a^3*b^4*c*e^6 - 74*a^4*b^2*c^2*e^6 - 7
2*a^3*c^5*d^4*e^2 + 72*a^4*c^4*d^2*e^4 - 102*a^2*b^2*c^4*d^4*e^2 + 44*a^2*b^3*c^3*d^3*e^3 + 9*a^2*b^4*c^2*d^2*
e^4 - 174*a^3*b^2*c^3*d^2*e^4 - 6*a*b^3*c^4*d^5*e + 120*a^2*b*c^5*d^5*e - 6*a^2*b^5*c*d*e^5 + 24*a^4*b*c^3*d*e
^5 + 144*a^3*b*c^4*d^3*e^3 + 42*a^3*b^3*c^2*d*e^5))/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c
^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 + 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*
c*e^6 + 3840*a^8*b*c^5*e^6 - 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 -
288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 + a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288
*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 - b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*
a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2
*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^
2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4
+ 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) - 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e
- 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^
2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 1
7664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 - 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(
-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c
^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*1i - (((6144*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*b*c^5
*e^3 + 6144*a^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3 + 1920*a^3*b^4*c^5*d^3 - 5632*a^4*b^2*c^6*d^3 + 16*a^3*b^7*c^2
*e^3 - 192*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^3 - 3072*a^5*b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*b^5*c
^4*d^2*e - 96*a^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*c^5*d^2*e + 1152*a^4*b^4*c^4*d*e^2 - 4608*a^5*b^2*c^5*d*e^2)/(8
*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b^2*c^3)) + (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11
*e^6 + 3840*a^5*b*c^8*d^6 + 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 - 9*a
^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b
^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 + a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*
c^3*e^6 - 3840*a^7*b^3*c^4*e^6 - b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*
d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2
- 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 37
44*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 + 9*a^2*c^3*d^4*e^2*(-(4*a*c - b
^2)^9)^(1/2) - 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*
b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e
- 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^
7*b^2*c^5*d*e^5 - 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(40
96*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8
*b^2*c^8)))^(1/2)*(1024*a^5*b*c^6 - 16*a^2*b^7*c^3 + 192*a^3*b^5*c^4 - 768*a^4*b^3*c^5))/(2*(16*a^4*c^3 + a^2*
b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 + 9*a*c^4*d^6*(
-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 - 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*
a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 + a^3*b
^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 - b^2*c^3*
d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^
3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c
^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6
*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 + 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) - 9*a^3*c^2*d^2*e^4*(-(4*a*c
- b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 3
84*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^
7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 - 6*a*b*c^3*d^5*e*(-(4*a*
c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c
^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2) + (x*(72*a^5*c^3*e^6 -
72*a^2*c^6*d^6 - a^2*b^6*e^6 - b^4*c^4*d^6 + 14*a*b^2*c^5*d^6 + 16*a^3*b^4*c*e^6 - 74*a^4*b^2*c^2*e^6 - 72*a^3
*c^5*d^4*e^2 + 72*a^4*c^4*d^2*e^4 - 102*a^2*b^2*c^4*d^4*e^2 + 44*a^2*b^3*c^3*d^3*e^3 + 9*a^2*b^4*c^2*d^2*e^4 -
174*a^3*b^2*c^3*d^2*e^4 - 6*a*b^3*c^4*d^5*e + 120*a^2*b*c^5*d^5*e - 6*a^2*b^5*c*d*e^5 + 24*a^4*b*c^3*d*e^5 +
144*a^3*b*c^4*d^3*e^3 + 42*a^3*b^3*c^2*d*e^5))/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^
6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 + 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6
+ 3840*a^8*b*c^5*e^6 - 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a
^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 + a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*
b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 - b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c
^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4
+ 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 7
68*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 + 9*a
^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) - 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*
a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e
^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*
a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 - 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a
*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 +
3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*1i)/((5*a^4*b^4*e^9 + 216*a^6*c^2*e^9 + 5*b^3*c^5*d^9 - 66*a^5*b^
2*c*e^9 + a*b^7*d^3*e^6 - 9*a^3*b^5*d*e^8 + 216*a^2*c^6*d^8*e - 9*b^4*c^4*d^8*e + 3*a^2*b^6*d^2*e^7 + 864*a^3*
c^5*d^6*e^3 + 1296*a^4*c^4*d^4*e^5 + 864*a^5*c^3*d^2*e^7 + 3*b^5*c^3*d^7*e^2 + b^6*c^2*d^6*e^3 - 36*a*b*c^6*d^
9 + 624*a^2*b^2*c^4*d^6*e^3 - 6*a^2*b^3*c^3*d^5*e^4 - 108*a^2*b^4*c^2*d^4*e^5 + 1020*a^3*b^2*c^3*d^4*e^5 + 128
*a^3*b^3*c^2*d^3*e^6 + 384*a^4*b^2*c^2*d^2*e^7 + 54*a*b^2*c^5*d^8*e + 6*a*b^6*c*d^4*e^5 + 153*a^4*b^3*c*d*e^8
- 612*a^5*b*c^2*d*e^8 + 24*a*b^3*c^4*d^7*e^2 - 46*a*b^4*c^3*d^6*e^3 - 3*a*b^5*c^2*d^5*e^4 - 720*a^2*b*c^5*d^7*
e^2 - 3*a^2*b^5*c*d^3*e^6 - 1944*a^3*b*c^4*d^5*e^4 - 90*a^3*b^4*c*d^2*e^7 - 1872*a^4*b*c^3*d^3*e^6)/(4*(64*a^5
*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b^2*c^3)) + (((6144*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*b*c^5
*e^3 + 6144*a^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3 + 1920*a^3*b^4*c^5*d^3 - 5632*a^4*b^2*c^6*d^3 + 16*a^3*b^7*c^2
*e^3 - 192*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^3 - 3072*a^5*b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*b^5*c
^4*d^2*e - 96*a^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*c^5*d^2*e + 1152*a^4*b^4*c^4*d*e^2 - 4608*a^5*b^2*c^5*d*e^2)/(8
*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b^2*c^3)) - (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11
*e^6 + 3840*a^5*b*c^8*d^6 + 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 - 9*a
^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b
^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 + a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*
c^3*e^6 - 3840*a^7*b^3*c^4*e^6 - b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*
d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2
- 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 37
44*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 + 9*a^2*c^3*d^4*e^2*(-(4*a*c - b
^2)^9)^(1/2) - 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*
b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e
- 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^
7*b^2*c^5*d*e^5 - 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(40
96*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8
*b^2*c^8)))^(1/2)*(1024*a^5*b*c^6 - 16*a^2*b^7*c^3 + 192*a^3*b^5*c^4 - 768*a^4*b^3*c^5))/(2*(16*a^4*c^3 + a^2*
b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 + 9*a*c^4*d^6*(
-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 - 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*
a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 + a^3*b
^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 - b^2*c^3*
d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^
3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c
^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6
*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 + 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) - 9*a^3*c^2*d^2*e^4*(-(4*a*c
- b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 3
84*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^
7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 - 6*a*b*c^3*d^5*e*(-(4*a*
c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c
^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2) - (x*(72*a^5*c^3*e^6 -
72*a^2*c^6*d^6 - a^2*b^6*e^6 - b^4*c^4*d^6 + 14*a*b^2*c^5*d^6 + 16*a^3*b^4*c*e^6 - 74*a^4*b^2*c^2*e^6 - 72*a^3
*c^5*d^4*e^2 + 72*a^4*c^4*d^2*e^4 - 102*a^2*b^2*c^4*d^4*e^2 + 44*a^2*b^3*c^3*d^3*e^3 + 9*a^2*b^4*c^2*d^2*e^4 -
174*a^3*b^2*c^3*d^2*e^4 - 6*a*b^3*c^4*d^5*e + 120*a^2*b*c^5*d^5*e - 6*a^2*b^5*c*d*e^5 + 24*a^4*b*c^3*d*e^5 +
144*a^3*b*c^4*d^3*e^3 + 42*a^3*b^3*c^2*d*e^5))/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^
6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 + 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6
+ 3840*a^8*b*c^5*e^6 - 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a
^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 + a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*
b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 - b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c
^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4
+ 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 7
68*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 + 9*a
^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) - 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*
a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e
^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*
a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 - 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a
*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 +
3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2) + (((6144*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*b*c^5*e^3 + 6
144*a^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3 + 1920*a^3*b^4*c^5*d^3 - 5632*a^4*b^2*c^6*d^3 + 16*a^3*b^7*c^2*e^3 - 1
92*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^3 - 3072*a^5*b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*b^5*c^4*d^2*e
- 96*a^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*c^5*d^2*e + 1152*a^4*b^4*c^4*d*e^2 - 4608*a^5*b^2*c^5*d*e^2)/(8*(64*a^5
*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b^2*c^3)) + (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3
840*a^5*b*c^8*d^6 + 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 - 9*a^4*c*e^6
*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d
^6 - 3840*a^4*b^3*c^7*d^6 + a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6
- 3840*a^7*b^3*c^4*e^6 - b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2
- 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^
4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b
^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 + 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(
1/2) - 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*
d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a
^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^
5*d*e^5 - 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c
^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8
)))^(1/2)*(1024*a^5*b*c^6 - 16*a^2*b^7*c^3 + 192*a^3*b^5*c^4 - 768*a^4*b^3*c^5))/(2*(16*a^4*c^3 + a^2*b^4*c -
8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 + 9*a*c^4*d^6*(-(4*a*c
- b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 - 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*
d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 + a^3*b^2*e^6*(
-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 - b^2*c^3*d^6*(-(4
*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^
3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e
^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3
- 13824*a^6*b^3*c^5*d^2*e^4 + 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) - 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^
9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b
^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^
2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 - 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)
^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240
*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2) + (x*(72*a^5*c^3*e^6 - 72*a^2*c
^6*d^6 - a^2*b^6*e^6 - b^4*c^4*d^6 + 14*a*b^2*c^5*d^6 + 16*a^3*b^4*c*e^6 - 74*a^4*b^2*c^2*e^6 - 72*a^3*c^5*d^4
*e^2 + 72*a^4*c^4*d^2*e^4 - 102*a^2*b^2*c^4*d^4*e^2 + 44*a^2*b^3*c^3*d^3*e^3 + 9*a^2*b^4*c^2*d^2*e^4 - 174*a^3
*b^2*c^3*d^2*e^4 - 6*a*b^3*c^4*d^5*e + 120*a^2*b*c^5*d^5*e - 6*a^2*b^5*c*d*e^5 + 24*a^4*b*c^3*d*e^5 + 144*a^3*
b*c^4*d^3*e^3 + 42*a^3*b^3*c^2*d*e^5))/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11
*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 + 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*
a^8*b*c^5*e^6 - 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c
^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 + a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*
e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 - b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e
^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a
^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b
^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 + 9*a^2*c^3*d
^4*e^2*(-(4*a*c - b^2)^9)^(1/2) - 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10
*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 460
8*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^
6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 - 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a*c - b^2
)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7
*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 +
9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 - 9*a^4*c*e^6*(-(4*a*c - b^2)^9)
^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c
^7*d^6 + a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*
e^6 - b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d
^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 -
384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 81
92*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 + 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) - 9*a^3*c^2*d^
2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6
*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 +
17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 - 6*a*b*c^3
*d^5*e*(-(4*a*c - b^2)^9)^(1/2) + 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 -
24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*2i