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 = 0.0837868, 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, 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 1375
Rule 364
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*}
Mathematica [C] time = 0.0413588, size = 54, normalized size = 0.46 \[ \frac{x^{m+1} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\& ,\frac{\, _2F_1\left (1,m+1;m+2;\frac{x}{\text{$\#$1}}\right )}{3 \text{$\#$1}^4+2}\& \right ]}{4 (m+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{{x}^{8}+3\,{x}^{4}+1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m}}{x^{8} + 3 \, x^{4} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{x^{8} + 3 x^{4} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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