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

Optimal. Leaf size=751 \[ -\frac{\log \left (-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}+\frac{\log \left (x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}-\frac{\log \left (-x \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left (x \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}\right )}{4 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}\right )}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}\right )}{4 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}\right )}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}} \]

[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]])

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Rubi [A]  time = 0.924504, antiderivative size = 751, normalized size of antiderivative = 1., 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} \[ -\frac{\log \left (-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}+\frac{\log \left (x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}-\frac{\log \left (-x \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left (x \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}\right )}{4 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}\right )}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}\right )}{4 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}\right )}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}} \]

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 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]]))

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 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 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 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 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]

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*}

Mathematica [C]  time = 0.0402587, size = 69, normalized size = 0.09 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8 e^2-\text{$\#$1}^4 f+d^2\& ,\frac{\text{$\#$1}^4 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 e^2-\text{$\#$1}^3 f}\& \right ] \]

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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Maple [C]  time = 0.034, size = 55, normalized size = 0.1 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({e}^{2}{{\it \_Z}}^{8}-f{{\it \_Z}}^{4}+{d}^{2} \right ) }{\frac{ \left ({{\it \_R}}^{4}e+d \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}{e}^{2}-{{\it \_R}}^{3}f}}} \end{align*}

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(x-_R),_R=RootOf(_Z^8*e^2-_Z^4*f+d^2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e x^{4} + d}{e^{2} x^{8} - f x^{4} + d^{2}}\,{d x} \end{align*}

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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Fricas [B]  time = 1.94836, size = 6278, normalized size = 8.36 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [A]  time = 5.33508, size = 136, normalized size = 0.18 \begin{align*} \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{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError