∫ d+ex4 ____________________

3.5 $\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx$

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

[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 = 0.86, 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 = {1419, 1094, 634, 618, 204, 628} \[ -\frac {\log \left (-x \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {\log \left (x \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}-\frac {\log \left (-x \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}+\frac {\log \left (x \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}-2 \sqrt {e} x}{\sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}\right )}{4 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}+2 \sqrt {e} x}{\sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}\right )}{4 \sqrt {d} \sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2 d e-b}+2 \sqrt {d} \sqrt {e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-b}}} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x^4)/(d^2 + b*x^4 + e^2*x^8),x]

[Out]

-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]]) + 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]*S

qrt[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*Sqr

t[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 204

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

; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

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 628

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 634

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 1094

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 1419

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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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] time = 0.04, size = 67, normalized size = 0.08 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8 e^2+\text {$\#$1}^4 b+d^2\& ,\frac {\text {$\#$1}^4 e \log (x-\text {$\#$1})+d \log (x-\text {$\#$1})}{2 \text {$\#$1}^7 e^2+\text {$\#$1}^3 b}\& \right ] \]

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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fricas [B] time = 1.33, size = 3059, normalized size = 3.87 \[ \text {result too large to display} \]

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(-1/4*(2*sqrt(1/2)*((8*d^5*e^3 + 12*b*d^4*e^2

+ 6*b^2*d^3*e + b^3*d^2)*x*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)) - (4*d^2*e^2

+ 4*b*d*e + b^2)*x)*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)) + (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(-

((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((2*e^2*x^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)))/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)))/e) + sqrt(sqrt(1/2)*sqr

t(((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(-1/4*(2*sqrt(1/2)*((8*d^5*e^3 + 12*b*d^4*e^2 + 6*b^2*d^3*e + b^3*

d^2)*x*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)) + (4*d^2*e^2 + 4*b*d*e + b^2)*x)*

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)) - (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(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((2*e^2*x^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)

))/e^2))/e) + 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)*sqr

t(-(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)))*lo

g(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)*sqr

t(-(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))))

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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [C] time = 0.05, size = 53, normalized size = 0.07 \[ \frac {\left (\RootOf \left (e^{2} \textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+d^{2}\right )^{4} e +d \right ) \ln \left (-\RootOf \left (e^{2} \textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+d^{2}\right )+x \right )}{8 \RootOf \left (e^{2} \textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+d^{2}\right )^{7} e^{2}+4 \RootOf \left (e^{2} \textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+d^{2}\right )^{3} b} \]

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)

[Out]

1/4*sum((_R^4*e+d)/(2*_R^7*e^2+_R^3*b)*ln(-_R+x),_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 \[ \int \frac {e x^{4} + d}{e^{2} x^{8} + b x^{4} + d^{2}}\,{d x} \]

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

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mupad [B] time = 3.83, size = 10409, normalized size = 13.16 \[ \text {result too large to display} \]

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)/(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)

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sympy [A] time = 8.50, size = 136, normalized size = 0.17 \[ \operatorname {RootSum} {\left (t^{8} \left (65536 b^{4} d^{2} + 524288 b^{3} d^{3} e + 1572864 b^{2} d^{4} e^{2} + 2097152 b d^{5} e^{3} + 1048576 d^{6} e^{4}\right ) + t^{4} \left (256 b^{3} + 1024 b^{2} d e + 1024 b d^{2} e^{2}\right ) + e^{2}, \left (t \mapsto t \log {\left (x + \frac {1024 t^{5} b^{2} d^{2} + 4096 t^{5} b d^{3} e + 4096 t^{5} d^{4} e^{2} + 4 t b + 4 t d e}{e} \right )} \right )\right )} \]

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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3.5 \(\int \frac {d+e x^4}{d^2+b x^4+e^2 x^8} \, dx\)