Optimal. Leaf size=46 \[ \frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1-m}{5},\frac{6-m}{5},-\frac{x^5}{a^5}\right )}{a^5 (1-m)} \]
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Rubi [A] time = 0.0095126, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {364} \[ \frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1-m}{5},\frac{6-m}{5},-\frac{x^5}{a^5}\right )}{a^5 (1-m)} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin{align*} \int \frac{x^{-m}}{a^5+x^5} \, dx &=\frac{x^{1-m} \, _2F_1\left (1,\frac{1-m}{5};\frac{6-m}{5};-\frac{x^5}{a^5}\right )}{a^5 (1-m)}\\ \end{align*}
Mathematica [A] time = 0.0093539, size = 45, normalized size = 0.98 \[ -\frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1}{5}-\frac{m}{5},\frac{6}{5}-\frac{m}{5},-\frac{x^5}{a^5}\right )}{a^5 (m-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{m} \left ({a}^{5}+{x}^{5} \right ) }}\, 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{1}{{\left (a^{5} + x^{5}\right )} x^{m}}\,{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{1}{{\left (a^{5} + x^{5}\right )} x^{m}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 34.013, size = 92, normalized size = 2. \begin{align*} - \frac{m x x^{- m} \Phi \left (\frac{x^{5} e^{i \pi }}{a^{5}}, 1, \frac{1}{5} - \frac{m}{5}\right ) \Gamma \left (\frac{1}{5} - \frac{m}{5}\right )}{25 a^{5} \Gamma \left (\frac{6}{5} - \frac{m}{5}\right )} + \frac{x x^{- m} \Phi \left (\frac{x^{5} e^{i \pi }}{a^{5}}, 1, \frac{1}{5} - \frac{m}{5}\right ) \Gamma \left (\frac{1}{5} - \frac{m}{5}\right )}{25 a^{5} \Gamma \left (\frac{6}{5} - \frac{m}{5}\right )} \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{1}{{\left (a^{5} + x^{5}\right )} x^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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