∫ d+ex4 ____________________

3.1.6 $\int \frac {d+e x^4}{d^2+f x^4+e^2 x^8} \, dx$

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

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Rubi [A] time = 0.81, 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} \begin {gather*} -\frac {\log \left (-x \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}+\frac {\log \left (x \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}-\frac {\log \left (-x \sqrt {\sqrt {2 d e-f}+2 \sqrt {d} \sqrt {e}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {\sqrt {2 d e-f}+2 \sqrt {d} \sqrt {e}}}+\frac {\log \left (x \sqrt {\sqrt {2 d e-f}+2 \sqrt {d} \sqrt {e}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {\sqrt {2 d e-f}+2 \sqrt {d} \sqrt {e}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}-2 \sqrt {e} x}{\sqrt {\sqrt {2 d e-f}+2 \sqrt {d} \sqrt {e}}}\right )}{4 \sqrt {d} \sqrt {\sqrt {2 d e-f}+2 \sqrt {d} \sqrt {e}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2 d e-f}+2 \sqrt {d} \sqrt {e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}+2 \sqrt {e} x}{\sqrt {\sqrt {2 d e-f}+2 \sqrt {d} \sqrt {e}}}\right )}{4 \sqrt {d} \sqrt {\sqrt {2 d e-f}+2 \sqrt {d} \sqrt {e}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2 d e-f}+2 \sqrt {d} \sqrt {e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]]/(

4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]) - ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]] - 2*S

qrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]) + A

rcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]]/(4*

Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]] + 2*Sqr

t[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]) - Log

[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt

[2*d*e - f]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqr

t[d]*Sqrt[e] - Sqrt[2*d*e - f]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]*x + Sqrt[e]*x^2]/(8

*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]*

x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]])

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+f x^4+e^2 x^8} \, dx &=\frac {\int \frac {1}{\frac {d}{e}-\frac {\sqrt {2 d e-f} x^2}{e}+x^4} \, dx}{2 e}+\frac {\int \frac {1}{\frac {d}{e}+\frac {\sqrt {2 d e-f} x^2}{e}+x^4} \, dx}{2 e}\\ &=\frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}{\sqrt {e}}-x}{\frac {\sqrt {d}}{\sqrt {e}}-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}} x}{\sqrt {e}}+x^2} \, dx}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}+\frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}{\sqrt {e}}+x}{\frac {\sqrt {d}}{\sqrt {e}}+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}} x}{\sqrt {e}}+x^2} \, dx}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}+\frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}{\sqrt {e}}-x}{\frac {\sqrt {d}}{\sqrt {e}}-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}} x}{\sqrt {e}}+x^2} \, dx}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}+\frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}{\sqrt {e}}+x}{\frac {\sqrt {d}}{\sqrt {e}}+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}} x}{\sqrt {e}}+x^2} \, dx}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}\\ &=\frac {\int \frac {1}{\frac {\sqrt {d}}{\sqrt {e}}-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}} 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 {2 d e-f}} 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 {2 d e-f}} 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 {2 d e-f}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {e}}-\frac {\int \frac {-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}{\sqrt {e}}+2 x}{\frac {\sqrt {d}}{\sqrt {e}}-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}+\frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}{\sqrt {e}}+2 x}{\frac {\sqrt {d}}{\sqrt {e}}+\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}-\frac {\int \frac {-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}{\sqrt {e}}+2 x}{\frac {\sqrt {d}}{\sqrt {e}}-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}+\frac {\int \frac {\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}{\sqrt {e}}+2 x}{\frac {\sqrt {d}}{\sqrt {e}}+\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}} x}{\sqrt {e}}+x^2} \, dx}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}\\ &=-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}{e}-x^2} \, dx,x,-\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}{\sqrt {e}}+2 x\right )}{4 \sqrt {d} \sqrt {e}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}{e}-x^2} \, dx,x,\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}{\sqrt {e}}+2 x\right )}{4 \sqrt {d} \sqrt {e}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}{e}-x^2} \, dx,x,-\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}{\sqrt {e}}+2 x\right )}{4 \sqrt {d} \sqrt {e}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}{e}-x^2} \, dx,x,\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}{\sqrt {e}}+2 x\right )}{4 \sqrt {d} \sqrt {e}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e-f}}}-\frac {\log \left (\sqrt {d}-\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}+\frac {\log \left (\sqrt {d}+\sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}} x+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}+\sqrt {2 d e-f}}}\\ \end {align*}

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

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x^4)/(d^2 + f*x^4 + e^2*x^8),x]

[Out]

RootSum[d^2 + f*#1^4 + e^2*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/(f*#1^3 + 2*e^2*#1^7) & ]/4

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x^4}{d^2+f x^4+e^2 x^8} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x^4)/(d^2 + f*x^4 + e^2*x^8),x]

[Out]

IntegrateAlgebraic[(d + e*x^4)/(d^2 + f*x^4 + e^2*x^8), x]

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fricas [B] time = 1.80, size = 3059, normalized size = 3.87

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+f*x^4+d^2),x, algorithm="fricas")

[Out]

-sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e

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

+ 6*d^3*e*f^2 + d^2*f^3)*x*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - (4*d^2*e^2

+ 4*d*e*f + f^2)*x)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d

^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)) + (4*d^2*e^2 + 4*d*e*f + f^2 - (8*d^5*e^3 + 12*d^

4*e^2*f + 6*d^3*e*f^2 + d^2*f^3)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)))*sqrt(-

((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/

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

d^3*e*f^2 + d^2*f^3)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)))*sqrt(-((4*d^4*e^2

+ 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 +

4*d^3*e*f + d^2*f^2)))/e^2))*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7

*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))/e) + sqrt(sqrt(1/2)*sqr

t(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f

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

f^3)*x*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + (4*d^2*e^2 + 4*d*e*f + f^2)*x)*

sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f

^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e -

f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)) - (4*d^2*e^2 +

4*d*e*f + f^2 + (8*d^5*e^3 + 12*d^4*e^2*f + 6*d^3*e*f^2 + d^2*f^3)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*

f + 6*d^5*e*f^2 + d^4*f^3)))*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e

^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))*sqrt(((4*d^4*e^2 + 4*d^3*

e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e

*f + d^2*f^2))*sqrt((2*e^2*x^2 + sqrt(1/2)*(2*d*e*f + f^2 + (8*d^5*e^3 + 12*d^4*e^2*f + 6*d^3*e*f^2 + d^2*f^3)

*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)))*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2

)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)

))/e^2))/e) + 1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^

6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))*log(e*x + 1/2*(2*d*e - (4*d^4*e^2 +

4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)*sqrt(sqrt(1/2

)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3

)) + f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))) - 1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqr

t(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))*lo

g(e*x - 1/2*(2*d*e - (4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f

^2 + d^4*f^3)) + f)*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*

d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))) + 1/4*sqrt(sqrt(1/2)*sqrt(((4*d^

4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4

*e^2 + 4*d^3*e*f + d^2*f^2)))*log(e*x + 1/2*(2*d*e + (4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^

7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqr

t(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))) -

1/4*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^

5*e*f^2 + d^4*f^3)) - f)/(4*d^4*e^2 + 4*d^3*e*f + d^2*f^2)))*log(e*x - 1/2*(2*d*e + (4*d^4*e^2 + 4*d^3*e*f + d

^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) + f)*sqrt(sqrt(1/2)*sqrt(((4*d^4

*e^2 + 4*d^3*e*f + d^2*f^2)*sqrt(-(2*d*e - f)/(8*d^7*e^3 + 12*d^6*e^2*f + 6*d^5*e*f^2 + d^4*f^3)) - f)/(4*d^4*

e^2 + 4*d^3*e*f + d^2*f^2))))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^4+d)/(e^2*x^8+f*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 \begin {gather*} \frac {\left (\RootOf \left (e^{2} \textit {\_Z}^{8}+f \,\textit {\_Z}^{4}+d^{2}\right )^{4} e +d \right ) \ln \left (-\RootOf \left (e^{2} \textit {\_Z}^{8}+f \,\textit {\_Z}^{4}+d^{2}\right )+x \right )}{8 \RootOf \left (e^{2} \textit {\_Z}^{8}+f \,\textit {\_Z}^{4}+d^{2}\right )^{7} e^{2}+4 \RootOf \left (e^{2} \textit {\_Z}^{8}+f \,\textit {\_Z}^{4}+d^{2}\right )^{3} f} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sum((_R^4*e+d)/(2*_R^7*e^2+_R^3*f)*ln(-_R+x),_R=RootOf(_Z^8*e^2+_Z^4*f+d^2))

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e x^{4} + d}{e^{2} x^{8} + f x^{4} + d^{2}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B] time = 4.03, size = 10411, normalized size = 13.16

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan((((-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*

d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 -

2048*d^3*e^9*f^6 - 10240*d^4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2) - (-(f^

3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32

*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f + 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f

^6 - 49152*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 196608*d^8*e^13*f^2)*1i)*(-(f^3 + ((f - 2

*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f

+ 24*d^4*e^2*f^2)))^(3/4)*1i - 256*d^7*e^14 + 256*d^6*e^13*f + 16*d^3*e^10*f^4 - 64*d^4*e^11*f^3)*1i + x*(32*d

^5*e^13*f - 4*d^2*e^10*f^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) +

4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4) + (

(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3

+ 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3

*e^9*f^6 - 10240*d^4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2) + (-(f^3 + ((f -

2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*

f + 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f + 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 4915

2*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 196608*d^8*e^13*f^2)*1i)*(-(f^3 + ((f - 2*d*e)*(f

+ 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*

e^2*f^2)))^(3/4)*1i + 256*d^7*e^14 - 256*d^6*e^13*f - 16*d^3*e^10*f^4 + 64*d^4*e^11*f^3)*1i + x*(32*d^5*e^13*f

- 4*d^2*e^10*f^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2

*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4))/(((-(f^3 +

((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5

*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3*e^9*f^6

- 10240*d^4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2) - (-(f^3 + ((f - 2*d*e)*(

f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^

4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f + 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 49152*d^5*e^1

0*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 196608*d^8*e^13*f^2)*1i)*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^

5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2))

)^(3/4)*1i - 256*d^7*e^14 + 256*d^6*e^13*f + 16*d^3*e^10*f^4 - 64*d^4*e^11*f^3)*1i + x*(32*d^5*e^13*f - 4*d^2*

e^10*f^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*

e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*1i - ((-(f^3 + ((f -

2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f

+ 24*d^4*e^2*f^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3*e^9*f^6 - 1024

0*d^4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2) + (-(f^3 + ((f - 2*d*e)*(f + 2*

d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*

f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f + 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 49152*d^5*e^10*f^5

+ 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 196608*d^8*e^13*f^2)*1i)*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/

2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(3/4

)*1i + 256*d^7*e^14 - 256*d^6*e^13*f - 16*d^3*e^10*f^4 + 64*d^4*e^11*f^3)*1i + x*(32*d^5*e^13*f - 4*d^2*e^10*f

^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)

/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*1i))*(-(f^3 + ((f - 2*d*e)*

(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d

^4*e^2*f^2)))^(1/4) - atan((((-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^

6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*d^8*e^14*f

+ 1024*d^2*e^8*f^7 - 2048*d^3*e^9*f^6 - 10240*d^4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d

^7*e^13*f^2) + (-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^

4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f + 4096*d^3*e^8*

f^7 - 4096*d^4*e^9*f^6 - 49152*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 196608*d^8*e^13*f^2))

*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^

3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(3/4) - 256*d^7*e^14 + 256*d^6*e^13*f + 16*d^3*e^10*f^4 - 64*d^4*e^11*f^3

) - x*(32*d^5*e^13*f - 4*d^2*e^10*f^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^

5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2))

)^(1/4)*1i + ((-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4

+ 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*

f^7 - 2048*d^3*e^9*f^6 - 10240*d^4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2) -

(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3

+ 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f + 4096*d^3*e^8*f^7 - 4096*d^4*

e^9*f^6 - 49152*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 196608*d^8*e^13*f^2))*(-(f^3 + ((f -

2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*

f + 24*d^4*e^2*f^2)))^(3/4) + 256*d^7*e^14 - 256*d^6*e^13*f - 16*d^3*e^10*f^4 + 64*d^4*e^11*f^3) - x*(32*d^5*e

^13*f - 4*d^2*e^10*f^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^

2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*1i)/(((

-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3

+ 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3*

e^9*f^6 - 10240*d^4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2) + (-(f^3 + ((f -

2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f

+ 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f + 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 49152

*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 196608*d^8*e^13*f^2))*(-(f^3 + ((f - 2*d*e)*(f + 2*

d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*

f^2)))^(3/4) - 256*d^7*e^14 + 256*d^6*e^13*f + 16*d^3*e^10*f^4 - 64*d^4*e^11*f^3) - x*(32*d^5*e^13*f - 4*d^2*e

^10*f^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e

*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4) - ((-(f^3 + ((f - 2*d*

e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 2

4*d^4*e^2*f^2)))^(1/4)*((x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3*e^9*f^6 - 10240*d^

4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2) - (-(f^3 + ((f - 2*d*e)*(f + 2*d*e)

^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)

))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f + 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 49152*d^5*e^10*f^5 + 49

152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 196608*d^8*e^13*f^2))*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*

d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(3/4) + 256

*d^7*e^14 - 256*d^6*e^13*f - 16*d^3*e^10*f^4 + 64*d^4*e^11*f^3) - x*(32*d^5*e^13*f - 4*d^2*e^10*f^4 + 24*d^3*e

^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6

*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)))*(-(f^3 + ((f - 2*d*e)*(f + 2*d*e)^5)^(

1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1

/4)*2i - atan((((-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f

^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(((-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2

*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(262144*

d^10*e^15 - 262144*d^9*e^14*f + 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 49152*d^5*e^10*f^5 + 49152*d^6*e^11*f^4

+ 196608*d^7*e^12*f^3 - 196608*d^8*e^13*f^2) + x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*

d^3*e^9*f^6 - 10240*d^4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2))*(-(f^3 - ((f

- 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^

3*f + 24*d^4*e^2*f^2)))^(3/4) - 256*d^7*e^14 + 256*d^6*e^13*f + 16*d^3*e^10*f^4 - 64*d^4*e^11*f^3) - x*(32*d^5

*e^13*f - 4*d^2*e^10*f^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*

d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*1i -

((-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^

3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(((-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*

f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262

144*d^9*e^14*f + 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 49152*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^

12*f^3 - 196608*d^8*e^13*f^2) - x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3*e^9*f^6 - 1

0240*d^4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2))*(-(f^3 - ((f - 2*d*e)*(f +

2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^

2*f^2)))^(3/4) - 256*d^7*e^14 + 256*d^6*e^13*f + 16*d^3*e^10*f^4 - 64*d^4*e^11*f^3) + x*(32*d^5*e^13*f - 4*d^2

*e^10*f^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d

*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*1i)/(((-(f^3 - ((f -

2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*

f + 24*d^4*e^2*f^2)))^(1/4)*(((-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d

^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f

+ 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 49152*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 196608

*d^8*e^13*f^2) + x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3*e^9*f^6 - 10240*d^4*e^10*f

^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2)

+ 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(3/4)

- 256*d^7*e^14 + 256*d^6*e^13*f + 16*d^3*e^10*f^4 - 64*d^4*e^11*f^3) - x*(32*d^5*e^13*f - 4*d^2*e^10*f^4 + 24*

d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(1

6*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4) + ((-(f^3 - ((f - 2*d*e)*(f + 2*d*e

)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2

)))^(1/4)*(((-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 +

8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f + 4096*d^3*e^8*f^7

- 4096*d^4*e^9*f^6 - 49152*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 196608*d^8*e^13*f^2) - x

*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3*e^9*f^6 - 10240*d^4*e^10*f^5 + 20480*d^5*e^1

1*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4

*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(3/4) - 256*d^7*e^14 + 2

56*d^6*e^13*f + 16*d^3*e^10*f^4 - 64*d^4*e^11*f^3) + x*(32*d^5*e^13*f - 4*d^2*e^10*f^4 + 24*d^3*e^11*f^3 - 48*

d^4*e^12*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^

4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*

e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*2i - 2*at

an((((-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*

e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(((-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*

d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 -

262144*d^9*e^14*f + 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 49152*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^

7*e^12*f^3 - 196608*d^8*e^13*f^2)*1i + x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3*e^9*

f^6 - 10240*d^4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2))*(-(f^3 - ((f - 2*d*e

)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24

*d^4*e^2*f^2)))^(3/4)*1i + 256*d^7*e^14 - 256*d^6*e^13*f - 16*d^3*e^10*f^4 + 64*d^4*e^11*f^3)*1i + x*(32*d^5*e

^13*f - 4*d^2*e^10*f^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^

2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4) - ((-(f

^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 3

2*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(((-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/

(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d

^9*e^14*f + 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 49152*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^

3 - 196608*d^8*e^13*f^2)*1i - x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3*e^9*f^6 - 102

40*d^4*e^10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*

d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*

f^2)))^(3/4)*1i + 256*d^7*e^14 - 256*d^6*e^13*f - 16*d^3*e^10*f^4 + 64*d^4*e^11*f^3)*1i - x*(32*d^5*e^13*f - 4

*d^2*e^10*f^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f +

4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4))/(((-(f^3 - ((f

- 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3

*f + 24*d^4*e^2*f^2)))^(1/4)*(((-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*

d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f

+ 4096*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 49152*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 19660

8*d^8*e^13*f^2)*1i + x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3*e^9*f^6 - 10240*d^4*e^

10*f^5 + 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(

1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(3

/4)*1i + 256*d^7*e^14 - 256*d^6*e^13*f - 16*d^3*e^10*f^4 + 64*d^4*e^11*f^3)*1i + x*(32*d^5*e^13*f - 4*d^2*e^10

*f^4 + 24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^

2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*1i + ((-(f^3 - ((f - 2*d*

e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 2

4*d^4*e^2*f^2)))^(1/4)*(((-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^

4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*(262144*d^10*e^15 - 262144*d^9*e^14*f + 409

6*d^3*e^8*f^7 - 4096*d^4*e^9*f^6 - 49152*d^5*e^10*f^5 + 49152*d^6*e^11*f^4 + 196608*d^7*e^12*f^3 - 196608*d^8*

e^13*f^2)*1i - x*(65536*d^9*e^15 - 32768*d^8*e^14*f + 1024*d^2*e^8*f^7 - 2048*d^3*e^9*f^6 - 10240*d^4*e^10*f^5

+ 20480*d^5*e^11*f^4 + 32768*d^6*e^12*f^3 - 65536*d^7*e^13*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) +

4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(3/4)*1i

+ 256*d^7*e^14 - 256*d^6*e^13*f - 16*d^3*e^10*f^4 + 64*d^4*e^11*f^3)*1i - x*(32*d^5*e^13*f - 4*d^2*e^10*f^4 +

24*d^3*e^11*f^3 - 48*d^4*e^12*f^2))*(-(f^3 - ((f - 2*d*e)*(f + 2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(51

2*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e^2*f^2)))^(1/4)*1i))*(-(f^3 - ((f - 2*d*e)*(f +

2*d*e)^5)^(1/2) + 4*d^2*e^2*f + 4*d*e*f^2)/(512*(16*d^6*e^4 + d^2*f^4 + 8*d^3*e*f^3 + 32*d^5*e^3*f + 24*d^4*e

^2*f^2)))^(1/4)

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sympy [A] time = 7.14, size = 136, normalized size = 0.17 \begin {gather*} \operatorname {RootSum} {\left (t^{8} \left (1048576 d^{6} e^{4} + 2097152 d^{5} e^{3} f + 1572864 d^{4} e^{2} f^{2} + 524288 d^{3} e f^{3} + 65536 d^{2} f^{4}\right ) + t^{4} \left (1024 d^{2} e^{2} f + 1024 d e f^{2} + 256 f^{3}\right ) + e^{2}, \left (t \mapsto t \log {\left (x + \frac {4096 t^{5} d^{4} e^{2} + 4096 t^{5} d^{3} e f + 1024 t^{5} d^{2} f^{2} + 4 t d e + 4 t f}{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+f*x**4+d**2),x)

[Out]

RootSum(_t**8*(1048576*d**6*e**4 + 2097152*d**5*e**3*f + 1572864*d**4*e**2*f**2 + 524288*d**3*e*f**3 + 65536*d

**2*f**4) + _t**4*(1024*d**2*e**2*f + 1024*d*e*f**2 + 256*f**3) + e**2, Lambda(_t, _t*log(x + (4096*_t**5*d**4

*e**2 + 4096*_t**5*d**3*e*f + 1024*_t**5*d**2*f**2 + 4*_t*d*e + 4*_t*f)/e)))

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