3.7 \(\int \frac{d+e x^4}{d^2-b x^4+e^2 x^8} \, dx\)
Optimal. Leaf size=349 \[ -\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}} \]
[Out]
-((Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[
Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]])) - (Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d
*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]) - (Sqrt[e]*ArcTanh[(Sqrt[2]*Sqrt[e]*x
)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]])
- (Sqrt[e]*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt
[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]])
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Rubi [A] time = 0.421594, antiderivative size = 349, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used =
{1419, 1093, 207, 203} \[ -\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x^4)/(d^2 - b*x^4 + e^2*x^8),x]
[Out]
-((Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[
Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]])) - (Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d
*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]) - (Sqrt[e]*ArcTanh[(Sqrt[2]*Sqrt[e]*x
)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]])
- (Sqrt[e]*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt
[Sqrt[b - 2*d*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 1093
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]
Rule 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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{1}{-\frac{\sqrt{b-2 d e}}{2 e}-\frac{\sqrt{b+2 d e}}{2 e}+x^2} \, dx}{2 \sqrt{b-2 d e}}-\frac{\int \frac{1}{\frac{\sqrt{b-2 d e}}{2 e}-\frac{\sqrt{b+2 d e}}{2 e}+x^2} \, dx}{2 \sqrt{b-2 d e}}+\frac{\int \frac{1}{-\frac{\sqrt{b-2 d e}}{2 e}+\frac{\sqrt{b+2 d e}}{2 e}+x^2} \, dx}{2 \sqrt{b-2 d e}}-\frac{\int \frac{1}{\frac{\sqrt{b-2 d e}}{2 e}+\frac{\sqrt{b+2 d e}}{2 e}+x^2} \, dx}{2 \sqrt{b-2 d e}}\\ &=-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\\ \end{align*}
Mathematica [C] time = 0.0449291, size = 69, normalized size = 0.2 \[ \frac{1}{4} \text{RootSum}\left [-\text{$\#$1}^4 b+\text{$\#$1}^8 e^2+d^2\& ,\frac{\text{$\#$1}^4 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 e^2-\text{$\#$1}^3 b}\& \right ] \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x^4)/(d^2 - b*x^4 + e^2*x^8),x]
[Out]
RootSum[d^2 - b*#1^4 + e^2*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/(-(b*#1^3) + 2*e^2*#1^7) & ]/4
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Maple [C] time = 0.031, size = 55, normalized size = 0.2 \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 = 1.89971, size = 6279, normalized size = 17.99 \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)*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)*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)*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 = 5.86887, size = 136, normalized size = 0.39 \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