Optimal. Leaf size=133 \[ \frac{p \sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} x^{m-p+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m-p+1}{2 p},\frac{m+p+1}{2 p},a^2 x^{2 p}\right )}{a (m+1) (m-p+1)}+\frac{p x^{m-p+1}}{a (m+1) (m-p+1)}+\frac{x^{m+1} e^{\text{sech}^{-1}\left (a x^p\right )}}{m+1} \]
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Rubi [A] time = 0.0834416, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6335, 30, 259, 364} \[ \frac{p \sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} x^{m-p+1} \, _2F_1\left (\frac{1}{2},\frac{m-p+1}{2 p};\frac{m+p+1}{2 p};a^2 x^{2 p}\right )}{a (m+1) (m-p+1)}+\frac{p x^{m-p+1}}{a (m+1) (m-p+1)}+\frac{x^{m+1} e^{\text{sech}^{-1}\left (a x^p\right )}}{m+1} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 259
Rule 364
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}\left (a x^p\right )} x^m \, dx &=\frac{e^{\text{sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac{p \int x^{m-p} \, dx}{a (1+m)}+\frac{\left (p \sqrt{\frac{1}{1+a x^p}} \sqrt{1+a x^p}\right ) \int \frac{x^{m-p}}{\sqrt{1-a x^p} \sqrt{1+a x^p}} \, dx}{a (1+m)}\\ &=\frac{e^{\text{sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac{p x^{1+m-p}}{a (1+m) (1+m-p)}+\frac{\left (p \sqrt{\frac{1}{1+a x^p}} \sqrt{1+a x^p}\right ) \int \frac{x^{m-p}}{\sqrt{1-a^2 x^{2 p}}} \, dx}{a (1+m)}\\ &=\frac{e^{\text{sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac{p x^{1+m-p}}{a (1+m) (1+m-p)}+\frac{p x^{1+m-p} \sqrt{\frac{1}{1+a x^p}} \sqrt{1+a x^p} \, _2F_1\left (\frac{1}{2},\frac{1+m-p}{2 p};\frac{1+m+p}{2 p};a^2 x^{2 p}\right )}{a (1+m) (1+m-p)}\\ \end{align*}
Mathematica [A] time = 4.95324, size = 186, normalized size = 1.4 \[ \frac{2^{\frac{m+1}{p}} x^{m+1} \left (a x^p\right )^{-\frac{m+1}{p}} e^{\text{sech}^{-1}\left (a x^p\right )} \left (\frac{e^{\text{sech}^{-1}\left (a x^p\right )}}{e^{2 \text{sech}^{-1}\left (a x^p\right )}+1}\right )^{\frac{m+1}{p}} \left ((m+3 p+1) \text{Hypergeometric2F1}\left (1,1-\frac{m+p+1}{2 p},\frac{m+3 p+1}{2 p},-e^{2 \text{sech}^{-1}\left (a x^p\right )}\right )-(m+p+1) e^{2 \text{sech}^{-1}\left (a x^p\right )} \text{Hypergeometric2F1}\left (1,-\frac{m-3 p+1}{2 p},\frac{m+5 p+1}{2 p},-e^{2 \text{sech}^{-1}\left (a x^p\right )}\right )\right )}{(m+p+1) (m+3 p+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.224, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{1}{a{x}^{p}}}+\sqrt{{\frac{1}{a{x}^{p}}}-1}\sqrt{{\frac{1}{a{x}^{p}}}+1} \right ){x}^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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*} \frac{\int x^{m} x^{- p}\, dx + \int a x^{m} \sqrt{-1 + \frac{x^{- p}}{a}} \sqrt{1 + \frac{x^{- p}}{a}}\, dx}{a} \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 x^{m}{\left (\sqrt{\frac{1}{a x^{p}} + 1} \sqrt{\frac{1}{a x^{p}} - 1} + \frac{1}{a x^{p}}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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