\(\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx\) [5]
Optimal result
Integrand size = 26, antiderivative size = 791 \[
\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\arctan \left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}+\frac {\arctan \left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}
\]
[Out]
-1/4*arctan((-2*x*e^(1/2)+(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^(1/2))/(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2)
)/d^(1/2)/(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2)+1/4*arctan((2*x*e^(1/2)+(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2)
)^(1/2))/(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2)-1/8*ln(d
^(1/2)+x^2*e^(1/2)-x*(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1
/2)+1/8*ln(d^(1/2)+x^2*e^(1/2)+x*(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/2)-(2*d*e-
b)^(1/2))^(1/2)-1/4*arctan((-2*x*e^(1/2)+(2*d^(1/2)*e^(1/2)-(2*d*e-b)^(1/2))^(1/2))/(2*d^(1/2)*e^(1/2)+(2*d*e-
b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^(1/2)+1/4*arctan((2*x*e^(1/2)+(2*d^(1/2)*e^(1/2)-
(2*d*e-b)^(1/2))^(1/2))/(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))
^(1/2)-1/8*ln(d^(1/2)+x^2*e^(1/2)-x*(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*e^(1/2)+(2*d
*e-b)^(1/2))^(1/2)+1/8*ln(d^(1/2)+x^2*e^(1/2)+x*(2*d^(1/2)*e^(1/2)+(2*d*e-b)^(1/2))^(1/2))/d^(1/2)/(2*d^(1/2)*
e^(1/2)+(2*d*e-b)^(1/2))^(1/2)
Rubi [A] (verified)
Time = 0.59 (sec) , antiderivative size = 791, normalized size of antiderivative = 1.00, number of
steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1433, 1108, 648, 632, 210,
642} \[
\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}-2 \sqrt {e} x}{\sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}\right )}{4 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}-\frac {\arctan \left (\frac {\sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {\arctan \left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}+2 \sqrt {e} x}{\sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}\right )}{4 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}+\frac {\arctan \left (\frac {\sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}-\frac {\log \left (-x \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {\log \left (x \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}-\frac {\log \left (-x \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}+\frac {\log \left (x \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}
\]
[In]
Int[(d + e*x^4)/(d^2 + b*x^4 + e^2*x^8),x]
[Out]
-1/4*ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*
e]]]/(Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) - ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]
- 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*
e]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*
d*e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d
*e]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b +
2*d*e]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]
*Sqrt[e] - Sqrt[-b + 2*d*e]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*S
qrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]*
x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e]
+ Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]])
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 632
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 642
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 648
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 1108
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Rule 1433
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[2*(d/e) -
b/c, 2]}, Dist[e/(2*c), Int[1/Simp[d/e + q*x^(n/2) + x^n, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x^(n/2
) + x^n, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2,
0] && IGtQ[n/2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d, e*Rt[a/c, 2]]))
Rubi steps \begin{align*}
\text {integral}& = \frac {\int \frac {1}{\frac {d}{e}-\frac {\sqrt {-b+2 d e} x^2}{e}+x^4} \, dx}{2 e}+\frac {\int \frac {1}{\frac {d}{e}+\frac {\sqrt {-b+2 d e} x^2}{e}+x^4} \, dx}{2 e} \\ & = \frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}{\sqrt {e}}-x}{\frac {\sqrt {d}}{\sqrt {e}}-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}{\sqrt {e}}+x}{\frac {\sqrt {d}}{\sqrt {e}}+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}{\sqrt {e}}-x}{\frac {\sqrt {d}}{\sqrt {e}}-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}+\frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}{\sqrt {e}}+x}{\frac {\sqrt {d}}{\sqrt {e}}+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}} \\ & = \frac {\int \frac {1}{\frac {\sqrt {d}}{\sqrt {e}}-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {e}}+\frac {\int \frac {1}{\frac {\sqrt {d}}{\sqrt {e}}+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {e}}+\frac {\int \frac {1}{\frac {\sqrt {d}}{\sqrt {e}}-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {e}}+\frac {\int \frac {1}{\frac {\sqrt {d}}{\sqrt {e}}+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {e}}-\frac {\int \frac {-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}{\sqrt {e}}+2 x}{\frac {\sqrt {d}}{\sqrt {e}}-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}{\sqrt {e}}+2 x}{\frac {\sqrt {d}}{\sqrt {e}}+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}-\frac {\int \frac {-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}{\sqrt {e}}+2 x}{\frac {\sqrt {d}}{\sqrt {e}}-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}+\frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}{\sqrt {e}}+2 x}{\frac {\sqrt {d}}{\sqrt {e}}+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}} \\ & = -\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}{e}-x^2} \, dx,x,-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}{\sqrt {e}}+2 x\right )}{4 \sqrt {d} \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}{e}-x^2} \, dx,x,\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}{\sqrt {e}}+2 x\right )}{4 \sqrt {d} \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}{e}-x^2} \, dx,x,-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}{\sqrt {e}}+2 x\right )}{4 \sqrt {d} \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}{e}-x^2} \, dx,x,\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}{\sqrt {e}}+2 x\right )}{4 \sqrt {d} \sqrt {e}} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}} \\
\end{align*}
Mathematica [C] (verified)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.08
\[
\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\frac {1}{4} \text {RootSum}\left [d^2+b \text {$\#$1}^4+e^2 \text {$\#$1}^8\&,\frac {d \log (x-\text {$\#$1})+e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}^3+2 e^2 \text {$\#$1}^7}\&\right ]
\]
[In]
Integrate[(d + e*x^4)/(d^2 + b*x^4 + e^2*x^8),x]
[Out]
RootSum[d^2 + b*#1^4 + e^2*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/(b*#1^3 + 2*e^2*#1^7) & ]/4
Maple [C] (verified)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.07
| | |
method | result | size |
| | |
default |
\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (e^{2} \textit {\_Z}^{8}+\textit {\_Z}^{4} b +d^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} e^{2}+\textit {\_R}^{3} b}\right )}{4}\) |
\(53\) |
risch |
\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (e^{2} \textit {\_Z}^{8}+\textit {\_Z}^{4} b +d^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} e^{2}+\textit {\_R}^{3} b}\right )}{4}\) |
\(53\) |
| | |
|
|
|
[In]
int((e*x^4+d)/(e^2*x^8+b*x^4+d^2),x,method=_RETURNVERBOSE)
[Out]
1/4*sum((_R^4*e+d)/(2*_R^7*e^2+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*e^2+_Z^4*b+d^2))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2461 vs. \(2 (581) = 1162\).
Time = 0.32 (sec) , antiderivative size = 2461, normalized size of antiderivative = 3.11
\[
\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\text {Too large to display}
\]
[In]
integrate((e*x^4+d)/(e^2*x^8+b*x^4+d^2),x, algorithm="fricas")
[Out]
1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^
2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*log(e*x + 1/2*(2*d*e - (4*d^4*e^2 + 4*b*d^3*e + b
^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)*sqrt(sqrt(1/2)*sqrt(-((4*d^
4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4
*e^2 + 4*b*d^3*e + b^2*d^2)))) - 1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)
/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*log(e*x - 1/2*(2
*d*e - (4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4))
+ b)*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*
b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))) + 1/4*sqrt(-sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*
d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d
^3*e + b^2*d^2)))*log(e*x + 1/2*(2*d*e - (4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b
*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)*sqrt(-sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e
- b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))) - 1/4*sqrt(
-sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e
+ b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*log(e*x - 1/2*(2*d*e - (4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*
sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)*sqrt(-sqrt(1/2)*sqrt(-((4*d^4*e^2 +
4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 +
4*b*d^3*e + b^2*d^2)))) + 1/4*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*
e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*log(e*x + 1/2*(2*d*e + (
4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)*sq
rt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e
+ b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))) - 1/4*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2
*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*
d^2)))*log(e*x - 1/2*(2*d*e + (4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 +
6*b^2*d^5*e + b^3*d^4)) + b)*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e
^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))) + 1/4*sqrt(-sqrt(1/2)*sq
rt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) -
b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*log(e*x + 1/2*(2*d*e + (4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e -
b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)*sqrt(-sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^
2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2
*d^2)))) - 1/4*sqrt(-sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6
*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*log(e*x - 1/2*(2*d*e + (4*d^4*e^2 + 4*
b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)*sqrt(-sqrt(1/2)*
sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4))
- b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2))))
Sympy [A] (verification not implemented)
Time = 19.01 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.17
\[
\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\operatorname {RootSum} {\left (t^{8} \cdot \left (65536 b^{4} d^{2} + 524288 b^{3} d^{3} e + 1572864 b^{2} d^{4} e^{2} + 2097152 b d^{5} e^{3} + 1048576 d^{6} e^{4}\right ) + t^{4} \cdot \left (256 b^{3} + 1024 b^{2} d e + 1024 b d^{2} e^{2}\right ) + e^{2}, \left ( t \mapsto t \log {\left (x + \frac {1024 t^{5} b^{2} d^{2} + 4096 t^{5} b d^{3} e + 4096 t^{5} d^{4} e^{2} + 4 t b + 4 t d e}{e} \right )} \right )\right )}
\]
[In]
integrate((e*x**4+d)/(e**2*x**8+b*x**4+d**2),x)
[Out]
RootSum(_t**8*(65536*b**4*d**2 + 524288*b**3*d**3*e + 1572864*b**2*d**4*e**2 + 2097152*b*d**5*e**3 + 1048576*d
**6*e**4) + _t**4*(256*b**3 + 1024*b**2*d*e + 1024*b*d**2*e**2) + e**2, Lambda(_t, _t*log(x + (1024*_t**5*b**2
*d**2 + 4096*_t**5*b*d**3*e + 4096*_t**5*d**4*e**2 + 4*_t*b + 4*_t*d*e)/e)))
Maxima [F]
\[
\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\int { \frac {e x^{4} + d}{e^{2} x^{8} + b x^{4} + d^{2}} \,d x }
\]
[In]
integrate((e*x^4+d)/(e^2*x^8+b*x^4+d^2),x, algorithm="maxima")
[Out]
integrate((e*x^4 + d)/(e^2*x^8 + b*x^4 + d^2), x)
Giac [F]
\[
\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\int { \frac {e x^{4} + d}{e^{2} x^{8} + b x^{4} + d^{2}} \,d x }
\]
[In]
integrate((e*x^4+d)/(e^2*x^8+b*x^4+d^2),x, algorithm="giac")
[Out]
integrate((e*x^4 + d)/(e^2*x^8 + b*x^4 + d^2), x)
Mupad [B] (verification not implemented)
Time = 9.72 (sec) , antiderivative size = 10409, normalized size of antiderivative = 13.16
\[
\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx=\text {Too large to display}
\]
[In]
int((d + e*x^4)/(b*x^4 + d^2 + e^2*x^8),x)
[Out]
2*atan(((x*(32*b*d^5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) + (-(b^3 + ((b - 2*d*e)*(b + 2
*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4
*e^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^1
0 + 20480*b^4*d^5*e^11 + 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) - (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2)
+ 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*
(262144*d^10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*b^4*d
^6*e^11 + 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i)*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*
e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i - 256*d^
7*e^14 + 256*b*d^6*e^13 + 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11)*1i)*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) +
4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4) + (
x*(32*b*d^5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) + (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^
(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(
1/4)*((x*(65536*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480
*b^4*d^5*e^11 + 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) + (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^
2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d
^10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 +
196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i)*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b
^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i + 256*d^7*e^14 -
256*b*d^6*e^13 - 16*b^4*d^3*e^10 + 64*b^3*d^4*e^11)*1i)*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e
^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4))/((x*(32*b*d
^5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) + (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4
*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((x*
(65536*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*b^4*d^5*
e^11 + 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) - (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4
*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15
- 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 + 196608*b
^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i)*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(
512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i - 256*d^7*e^14 + 256*b*d^6
*e^13 + 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11)*1i)*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^
2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i - (x*(32*b*d^5*e^1
3 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) + (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2
*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((x*(65536
*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 +
32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) + (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d
*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 2621
44*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 + 196608*b^3*d^7
*e^12 - 196608*b^2*d^8*e^13)*1i)*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b
^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i + 256*d^7*e^14 - 256*b*d^6*e^13
- 16*b^4*d^3*e^10 + 64*b^3*d^4*e^11)*1i)*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)
/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i))*(-(b^3 + ((b - 2*d*e)*
(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b
^2*d^4*e^2)))^(1/4) - atan(((x*(32*b*d^5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) - (-(b^3 +
((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*
d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9
- 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 + 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) + (-(b^3 + ((b - 2*d*e)*
(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b
^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49152*b^5*d^
5*e^10 + 49152*b^4*d^6*e^11 + 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13))*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)
^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^
(3/4) - 256*d^7*e^14 + 256*b*d^6*e^13 + 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11))*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^
5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2))
)^(1/4)*1i + (x*(32*b*d^5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) - (-(b^3 + ((b - 2*d*e)*(
b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^
2*d^4*e^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^
4*e^10 + 20480*b^4*d^5*e^11 + 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) - (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^
(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(
1/4)*(262144*d^10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*
b^4*d^6*e^11 + 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13))*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^
2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) + 256*d^7
*e^14 - 256*b*d^6*e^13 - 16*b^4*d^3*e^10 + 64*b^3*d^4*e^11))*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*
d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i)/((x
*(32*b*d^5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) - (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(
1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1
/4)*((x*(65536*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*
b^4*d^5*e^11 + 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) + (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2
*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^
10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 +
196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13))*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d
*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) - 256*d^7*e^14 + 256*b*d
^6*e^13 + 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11))*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2
*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4) - (x*(32*b*d^5*e^13 -
4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) - (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2
+ 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((x*(65536*d^9
*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 + 327
68*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) - (-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/
(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 262144*b
*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 + 196608*b^3*d^7*e^1
2 - 196608*b^2*d^8*e^13))*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2
+ 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) + 256*d^7*e^14 - 256*b*d^6*e^13 - 16*b^4*d
^3*e^10 + 64*b^3*d^4*e^11))*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^
2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)))*(-(b^3 + ((b - 2*d*e)*(b + 2*d*e)^5)^(
1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1
/4)*2i + atan(((x*(32*b*d^5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) + (-(b^3 - ((b - 2*d*e)
*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*
b^2*d^4*e^2)))^(1/4)*(((-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 1
6*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 262144*b*d^9*e^14 + 4096*
b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 + 196608*b^3*d^7*e^12 - 196608*b^2*d^
8*e^13) - x*(65536*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20
480*b^4*d^5*e^11 + 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*
d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) - 256*d
^7*e^14 + 256*b*d^6*e^13 + 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*
b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i +
(x*(32*b*d^5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) - (-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)
^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^
(1/4)*(((-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b
^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4
096*b^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 + 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13) + x*(65
536*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^1
1 + 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2
*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) - 256*d^7*e^14 + 256*b
*d^6*e^13 + 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b
^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i)/((x*(32*b*d^5*e^
13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) + (-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^
2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(((-(b^3
- ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b
*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9
- 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 + 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13) - x*(65536*d^9*e^15 -
32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 + 32768*b^3*d
^6*e^12 - 65536*b^2*d^7*e^13))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4
*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) - 256*d^7*e^14 + 256*b*d^6*e^13 + 16*
b^4*d^3*e^10 - 64*b^3*d^4*e^11))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b
^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4) - (x*(32*b*d^5*e^13 - 4*b^4*d^2*e^1
0 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) - (-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)
/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(((-(b^3 - ((b - 2*d*e)*(b
+ 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*
d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49152*b^5*d^5*e
^10 + 49152*b^4*d^6*e^11 + 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13) + x*(65536*d^9*e^15 - 32768*b*d^8*e^14 +
1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 + 32768*b^3*d^6*e^12 - 65536*b^
2*d^7*e^13))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4
+ 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) - 256*d^7*e^14 + 256*b*d^6*e^13 + 16*b^4*d^3*e^10 - 64*
b^3*d^4*e^11))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^
4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^
2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*2i - 2*at
an(((x*(32*b*d^5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) + (-(b^3 - ((b - 2*d*e)*(b + 2*d*e
)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2
)))^(1/4)*(((-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 +
8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8
- 4096*b^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 + 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i
+ x*(65536*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*b^4*
d^5*e^11 + 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2
+ 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i + 256*d^7*e^
14 - 256*b*d^6*e^13 - 16*b^4*d^3*e^10 + 64*b^3*d^4*e^11)*1i)*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*
d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4) + (x*(3
2*b*d^5*e^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) - (-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2
) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)
*(((-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^
3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b
^6*d^4*e^9 - 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 + 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i - x*(6553
6*d^9*e^15 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11
+ 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d
*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i + 256*d^7*e^14 - 256*
b*d^6*e^13 - 16*b^4*d^3*e^10 + 64*b^3*d^4*e^11)*1i)*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 +
4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4))/((x*(32*b*d^5*e
^13 - 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) + (-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d
^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(((-(b^3
- ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*
b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^
9 - 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 + 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i + x*(65536*d^9*e^1
5 - 32768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 + 32768*b
^3*d^6*e^12 - 65536*b^2*d^7*e^13))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*
(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i + 256*d^7*e^14 - 256*b*d^6*e^1
3 - 16*b^4*d^3*e^10 + 64*b^3*d^4*e^11)*1i)*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*
e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i - (x*(32*b*d^5*e^13 -
4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 - 48*b^2*d^4*e^12) - (-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2
+ 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(((-(b^3 - ((b
- 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*
e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 - 262144*b*d^9*e^14 + 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 - 49
152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 + 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i - x*(65536*d^9*e^15 - 32
768*b*d^8*e^14 + 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 - 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 + 32768*b^3*d^6
*e^12 - 65536*b^2*d^7*e^13))*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d
^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i + 256*d^7*e^14 - 256*b*d^6*e^13 - 16
*b^4*d^3*e^10 + 64*b^3*d^4*e^11)*1i)*(-(b^3 - ((b - 2*d*e)*(b + 2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(51
2*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i))*(-(b^3 - ((b - 2*d*e)*(b +
2*d*e)^5)^(1/2) + 4*b*d^2*e^2 + 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 + 8*b^3*d^3*e + 32*b*d^5*e^3 + 24*b^2*d
^4*e^2)))^(1/4)