3.1.8 \(\int \frac {d+e x^4}{d^2-f x^4+e^2 x^8} \, dx\)
Optimal. Leaf size=751 \[ -\frac {\log \left (-x \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e+f}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e+f}}}+\frac {\log \left (x \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e+f}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e+f}}}-\frac {\log \left (-x \sqrt {\sqrt {2 d e+f}+2 \sqrt {d} \sqrt {e}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {\sqrt {2 d e+f}+2 \sqrt {d} \sqrt {e}}}+\frac {\log \left (x \sqrt {\sqrt {2 d e+f}+2 \sqrt {d} \sqrt {e}}+\sqrt {d}+\sqrt {e} x^2\right )}{8 \sqrt {d} \sqrt {\sqrt {2 d e+f}+2 \sqrt {d} \sqrt {e}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e+f}}-2 \sqrt {e} x}{\sqrt {\sqrt {2 d e+f}+2 \sqrt {d} \sqrt {e}}}\right )}{4 \sqrt {d} \sqrt {\sqrt {2 d e+f}+2 \sqrt {d} \sqrt {e}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2 d e+f}+2 \sqrt {d} \sqrt {e}}-2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e+f}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e+f}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e+f}}+2 \sqrt {e} x}{\sqrt {\sqrt {2 d e+f}+2 \sqrt {d} \sqrt {e}}}\right )}{4 \sqrt {d} \sqrt {\sqrt {2 d e+f}+2 \sqrt {d} \sqrt {e}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2 d e+f}+2 \sqrt {d} \sqrt {e}}+2 \sqrt {e} x}{\sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e+f}}}\right )}{4 \sqrt {d} \sqrt {2 \sqrt {d} \sqrt {e}-\sqrt {2 d e+f}}} \]
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Rubi [A] time = 0.92, antiderivative size = 751, normalized size of antiderivative = 1.00,
number of steps used = 19, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, 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 = 69, normalized size = 0.09 \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 = 1.62, size = 3051, normalized size = 4.06
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)*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(s
qrt(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)*s
qrt(-(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)*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)*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)*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 = 0.03, size = 55, 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 = 4.20, size = 10343, normalized size = 13.77
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x^4)/(d^2 - f*x^4 + e^2*x^8),x)
[Out]
2*atan(((((f^3 + ((f - 2*d*e)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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
)^5*(f + 2*d*e))^(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)^5*(f + 2*d
*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*
e))^(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)*1i)*((f^3 + ((f - 2*d*e)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f +
2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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^1
1*f^3 + 48*d^4*e^12*f^2))*((f^3 + ((f - 2*d*e)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)*(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)^5*(f + 2*d*e))^(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)))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5
*(f + 2*d*e))^(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))*((f^3 + ((f - 2*d*e)^5*(f + 2*d*e))^(
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)^5*(f + 2*d*e))^(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)^5*(f
+ 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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*(65
536*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)^5*(f + 2*d*e))^(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 + 26
2144*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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(1/2) + 4*d^2*e^2*f - 4*d*e*f^2)/(5
12*(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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 + 2
0480*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)^5*(f + 2*d*e))^(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^1
1*f^3 + 48*d^4*e^12*f^2))*((f^3 - ((f - 2*d*e)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(
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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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) + 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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))
^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e)
)^(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)^5*(f + 2*d*e))^(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*atan(((((f^3 - ((f - 2*d*e)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)*(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)*((((f^3 - ((f - 2*d*e)^5*(f + 2
*d*e))^(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 - 1
024*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)^5*(f + 2*d*e))^(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
)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2*d*e))^(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)^5*(f + 2
*d*e))^(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)^5*(f + 2*d*e))^(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 = 7.25, size = 136, normalized size = 0.18 \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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