Optimal. Leaf size=117 \[ \frac {2 x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};\frac {2 x^4}{3-\sqrt {5}}\right )}{\sqrt {5} \left (3-\sqrt {5}\right ) (m+1)}-\frac {2 x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};\frac {2 x^4}{3+\sqrt {5}}\right )}{\sqrt {5} \left (3+\sqrt {5}\right ) (m+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1375, 364} \[ \frac {2 x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};\frac {2 x^4}{3-\sqrt {5}}\right )}{\sqrt {5} \left (3-\sqrt {5}\right ) (m+1)}-\frac {2 x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};\frac {2 x^4}{3+\sqrt {5}}\right )}{\sqrt {5} \left (3+\sqrt {5}\right ) (m+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 1375
Rubi steps
\begin {align*} \int \frac {x^m}{1-3 x^4+x^8} \, dx &=\frac {\int \frac {x^m}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}}-\frac {\int \frac {x^m}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}}\\ &=\frac {2 x^{1+m} \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};\frac {2 x^4}{3-\sqrt {5}}\right )}{\sqrt {5} \left (3-\sqrt {5}\right ) (1+m)}-\frac {2 x^{1+m} \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};\frac {2 x^4}{3+\sqrt {5}}\right )}{\sqrt {5} \left (3+\sqrt {5}\right ) (1+m)}\\ \end {align*}
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Mathematica [C] time = 0.57, 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.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{x^{8} - 3 \, x^{4} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{x^{8}-3 x^{4}+1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m}{x^8-3\,x^4+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (x^{4} - x^{2} - 1\right ) \left (x^{4} + x^{2} - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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