Optimal. Leaf size=117 \[ \frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{3-\sqrt{5}}\right )}{\sqrt{5} \left (3-\sqrt{5}\right ) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{3+\sqrt{5}}\right )}{\sqrt{5} \left (3+\sqrt{5}\right ) (m+1)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.128404, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{3-\sqrt{5}}\right )}{\sqrt{5} \left (3-\sqrt{5}\right ) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{3+\sqrt{5}}\right )}{\sqrt{5} \left (3+\sqrt{5}\right ) (m+1)} \]
Antiderivative was successfully verified.
[In] Int[x^m/(1 - 3*x^4 + x^8),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.9538, size = 105, normalized size = 0.9 \[ - \frac{\sqrt{5} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{x^{4}}{- \frac{3}{2} - \frac{\sqrt{5}}{2}}} \right )}}{5 \left (\frac{\sqrt{5}}{2} + \frac{3}{2}\right ) \left (m + 1\right )} + \frac{\sqrt{5} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{x^{4}}{- \frac{3}{2} + \frac{\sqrt{5}}{2}}} \right )}}{5 \left (- \frac{\sqrt{5}}{2} + \frac{3}{2}\right ) \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m/(x**8-3*x**4+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 1.0223, size = 575, normalized size = 4.91 \[ \frac{x^m \left (\frac{\text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2-1\&,\frac{\text{$\#$1}^2 m^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+3 \text{$\#$1}^2 m \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+2 \text{$\#$1}^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+\text{$\#$1}^2 m \left (\frac{x}{\text{$\#$1}}\right )^{-m}+\text{$\#$1} m^2 x+2 \text{$\#$1} m x+m^2 x^2+m x^2}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right ]-\text{RootSum}\left [\text{$\#$1}^4+\text{$\#$1}^2-1\&,\frac{\text{$\#$1}^2 m^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+3 \text{$\#$1}^2 m \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+2 \text{$\#$1}^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+\text{$\#$1}^2 m \left (\frac{x}{\text{$\#$1}}\right )^{-m}+\text{$\#$1} m^2 x+2 \text{$\#$1} m x+m^2 x^2+m x^2}{2 \text{$\#$1}^3+\text{$\#$1}}\&\right ]-\left (m^2+3 m+2\right ) \text{RootSum}\left [\text{$\#$1}^4+\text{$\#$1}^2-1\&,\frac{\left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )}{2 \text{$\#$1}^3+\text{$\#$1}}\&\right ]}{m^2+3 m+2}-\text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2-1\&,\frac{\left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right ]\right )}{4 m} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x^m/(1 - 3*x^4 + x^8),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.027, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{{x}^{8}-3\,{x}^{4}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m/(x^8-3*x^4+1),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(x^8 - 3*x^4 + 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{x^{8} - 3 \, x^{4} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(x^8 - 3*x^4 + 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m/(x**8-3*x**4+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(x^8 - 3*x^4 + 1),x, algorithm="giac")
[Out]