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]
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Rubi [A] time = 0.153278, 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]
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Rubi in Sympy [A] time = 13.6053, size = 104, normalized size = 0.89 \[ \frac{\sqrt{5} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{x^{4}}{- \frac{\sqrt{5}}{2} + \frac{3}{2}}} \right )}}{5 \left (- \frac{\sqrt{5}}{2} + \frac{3}{2}\right ) \left (m + 1\right )} - \frac{\sqrt{5} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{x^{4}}{\frac{\sqrt{5}}{2} + \frac{3}{2}}} \right )}}{5 \left (\frac{\sqrt{5}}{2} + \frac{3}{2}\right ) \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m/(x**8+3*x**4+1),x)
[Out]
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Mathematica [C] time = 0.0731782, size = 79, normalized size = 0.68 \[ \frac{x^m \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\&\right ]}{4 m} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x^m/(1 + 3*x^4 + x^8),x]
[Out]
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Maple [F] time = 0.026, 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]
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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]
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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]
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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]
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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]