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

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

[Out]

-ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]]
/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) - ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]] -
2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]
]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*
e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e
]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*
d*e]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*S
qrt[e] - Sqrt[-b + 2*d*e]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqr
t[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]*x
+ Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] +
Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]])

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Rubi [A]  time = 0.862735, antiderivative size = 791, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1419, 1094, 634, 618, 204, 628} \[ -\frac{\log \left (-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}+\frac{\log \left (x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}-\frac{\log \left (-x \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left (x \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right )}{8 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}\right )}{4 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}\right )}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}\right )}{4 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}\right )}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]]
/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) - ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]] -
2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]
]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*
e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e
]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*
d*e]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*S
qrt[e] - Sqrt[-b + 2*d*e]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqr
t[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]*x
+ Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] +
Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]])

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

Mathematica [C]  time = 0.0449239, size = 67, normalized size = 0.08 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^4 b+\text{$\#$1}^8 e^2+d^2\& ,\frac{\text{$\#$1}^4 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{\text{$\#$1}^3 b+2 \text{$\#$1}^7 e^2}\& \right ] \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.059, size = 53, normalized size = 0.1 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({e}^{2}{{\it \_Z}}^{8}+b{{\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}b}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sum((_R^4*e+d)/(2*_R^7*e^2+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*e^2+_Z^4*b+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} + b 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+b*x^4+d^2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.94518, size = 6279, normalized size = 7.94 \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+b*x^4+d^2),x, algorithm="fricas")

[Out]

-sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d
^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*arctan(-1/4*(2*sqrt(1/2)*((8*d^5*e^3 + 12*b*d^4*e^2
+ 6*b^2*d^3*e + b^3*d^2)*x*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - (4*d^2*e^2
+ 4*b*d*e + b^2)*x)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b
^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)) + (4*d^2*e^2 + 4*b*d*e + b^2 - (8*d^5*e^3 + 12*b*
d^4*e^2 + 6*b^2*d^3*e + b^3*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)))*sqrt(-
((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/
(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2))*sqrt((2*e^2*x^2 + sqrt(1/2)*(2*b*d*e + b^2 - (8*d^5*e^3 + 12*b*d^4*e^2 + 6*
b^2*d^3*e + b^3*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)))*sqrt(-((4*d^4*e^2
+ 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 +
4*b*d^3*e + b^2*d^2)))/e^2))*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7
*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))/e) + sqrt(sqrt(1/2)*sqr
t(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b
)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*arctan(-1/4*(2*sqrt(1/2)*((8*d^5*e^3 + 12*b*d^4*e^2 + 6*b^2*d^3*e + b^3*
d^2)*x*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + (4*d^2*e^2 + 4*b*d*e + b^2)*x)*
sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5
*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e -
b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)) - (4*d^2*e^2 +
4*b*d*e + b^2 + (8*d^5*e^3 + 12*b*d^4*e^2 + 6*b^2*d^3*e + b^3*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^
2 + 6*b^2*d^5*e + b^3*d^4)))*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e
^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*sqrt(((4*d^4*e^2 + 4*b*d^
3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3
*e + b^2*d^2))*sqrt((2*e^2*x^2 + sqrt(1/2)*(2*b*d*e + b^2 + (8*d^5*e^3 + 12*b*d^4*e^2 + 6*b^2*d^3*e + b^3*d^2)
*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)))*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2
)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)
))/e^2))/e) + 1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*
d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*log(e*x + 1/2*(2*d*e - (4*d^4*e^2 +
4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)*sqrt(sqrt(1/2
)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4
)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))) - 1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqr
t(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*lo
g(e*x - 1/2*(2*d*e - (4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5
*e + b^3*d^4)) + b)*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*
b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))) + 1/4*sqrt(sqrt(1/2)*sqrt(((4*d^
4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4
*e^2 + 4*b*d^3*e + b^2*d^2)))*log(e*x + 1/2*(2*d*e + (4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^
7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqr
t(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))) -
1/4*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^
2*d^5*e + b^3*d^4)) - b)/(4*d^4*e^2 + 4*b*d^3*e + b^2*d^2)))*log(e*x - 1/2*(2*d*e + (4*d^4*e^2 + 4*b*d^3*e + b
^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) + b)*sqrt(sqrt(1/2)*sqrt(((4*d^4
*e^2 + 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e - b)/(8*d^7*e^3 + 12*b*d^6*e^2 + 6*b^2*d^5*e + b^3*d^4)) - b)/(4*d^4*
e^2 + 4*b*d^3*e + b^2*d^2))))

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Sympy [A]  time = 5.78472, size = 136, normalized size = 0.17 \begin{align*} \operatorname{RootSum}{\left (t^{8} \left (65536 b^{4} d^{2} + 524288 b^{3} d^{3} e + 1572864 b^{2} d^{4} e^{2} + 2097152 b d^{5} e^{3} + 1048576 d^{6} e^{4}\right ) + t^{4} \left (256 b^{3} + 1024 b^{2} d e + 1024 b d^{2} e^{2}\right ) + e^{2}, \left ( t \mapsto t \log{\left (x + \frac{1024 t^{5} b^{2} d^{2} + 4096 t^{5} b d^{3} e + 4096 t^{5} d^{4} e^{2} + 4 t b + 4 t d e}{e} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**8*(65536*b**4*d**2 + 524288*b**3*d**3*e + 1572864*b**2*d**4*e**2 + 2097152*b*d**5*e**3 + 1048576*d
**6*e**4) + _t**4*(256*b**3 + 1024*b**2*d*e + 1024*b*d**2*e**2) + e**2, Lambda(_t, _t*log(x + (1024*_t**5*b**2
*d**2 + 4096*_t**5*b*d**3*e + 4096*_t**5*d**4*e**2 + 4*_t*b + 4*_t*d*e)/e)))

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Giac [F(-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+b*x^4+d^2),x, algorithm="giac")

[Out]

Exception raised: TypeError