3.1.5 \(\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx\) [5]

Optimal. Leaf size=791 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {-b+2 d e}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {-b+2 d e}}} \]

[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)

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Rubi [A]
time = 0.58, antiderivative size = 791, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1433, 1108, 648, 632, 210, 642} \begin {gather*} -\frac {\text {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 {\text {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 {\text {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 {\text {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}}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx &=\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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.03, size = 67, normalized size = 0.08 \begin {gather*} \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 ] \end {gather*}

Antiderivative was successfully verified.

[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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.06, size = 53, normalized size = 0.07

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (e^{2} \textit {\_Z}^{8}+\textit {\_Z}^{4} b +d^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} e^{2}+\textit {\_R}^{3} b}\right )}{4}\)	\(53\)
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (e^{2} \textit {\_Z}^{8}+\textit {\_Z}^{4} b +d^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} e^{2}+\textit {\_R}^{3} b}\right )}{4}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^4+d)/(e^2*x^8+b*x^4+d^2),x, algorithm="maxima")

[Out]

integrate((x^4*e + d)/(x^8*e^2 + b*x^4 + d^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3072 vs. \(2 (573) = 1146\).
time = 0.51, size = 3072, normalized size = 3.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^4+d)/(e^2*x^8+b*x^4+d^2),x, algorithm="fricas")

[Out]

-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)))*arctan(-1/2*(sqrt(1/2)*(4*d^2*e^2 + 4*b*d*e + b^2 -
(8*d^5*e^3 + 12*b*d^4*e^2 + 6*b^2*d^3*e + b^3*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)))*sqrt(x^2*e^2 + 1/2*sqrt(1/2)*(2*b*d*e + b^2 - (8*d^5*e^3 + 12*b*d^4*e^2 + 6*b^2*d^3*e + b^3*d^2)*s
qrt(-(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)))*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))
)*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)) - sqrt(1/2)*(4*d^2*x*e^3 + 4*b*d*x*e^2 + b^2*x*e - (8*d^5*x*e^4 + 1
2*b*d^4*x*e^3 + 6*b^2*d^3*x*e^2 + b^3*d^2*x*e)*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)))*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)))*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)))*e^(
-2)) + 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)))*arctan(1/2*(sqrt(1/2)*(4*d^2*e^2 + 4*b*d*e + b^
2 + (8*d^5*e^3 + 12*b*d^4*e^2 + 6*b^2*d^3*e + b^3*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)))*sqrt(x^2*e^2 + 1/2*sqrt(1/2)*(2*b*d*e + b^2 + (8*d^5*e^3 + 12*b*d^4*e^2 + 6*b^2*d^3*e + b^3*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)))*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)))*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)))*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)) - sqrt(1/
2)*(4*d^2*x*e^3 + 4*b*d*x*e^2 + b^2*x*e + (8*d^5*x*e^4 + 12*b*d^4*x*e^3 + 6*b^2*d^3*x*e^2 + b^3*d^2*x*e)*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)))*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)))*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)))*e^(-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(x*e + 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(x*e - 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 + 1
2*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(x*e + 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)*s
qrt(-(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(x*e - 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))))

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Sympy [A]
time = 18.75, size = 136, normalized size = 0.17 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^4+d)/(e^2*x^8+b*x^4+d^2),x, algorithm="giac")

[Out]

integrate((x^4*e + d)/(x^8*e^2 + b*x^4 + d^2), x)

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Mupad [B]
time = 3.83, size = 2500, normalized size = 3.16 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)...

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