Optimal. Leaf size=109 \[ -\frac{\sqrt{\frac{1}{\frac{a}{x}+1}} \sqrt{\frac{a}{x}+1} x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (-m-2),-\frac{m}{2},\frac{a^2}{x^2}\right )}{a \left (m^2+3 m+2\right )}-\frac{x^{m+2}}{a \left (m^2+3 m+2\right )}+\frac{x^{m+1} e^{\text{sech}^{-1}\left (\frac{a}{x}\right )}}{m+1} \]
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Rubi [A] time = 0.0751257, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6335, 30, 259, 339, 364} \[ -\frac{\sqrt{\frac{1}{\frac{a}{x}+1}} \sqrt{\frac{a}{x}+1} x^{m+2} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-m-2);-\frac{m}{2};\frac{a^2}{x^2}\right )}{a \left (m^2+3 m+2\right )}-\frac{x^{m+2}}{a \left (m^2+3 m+2\right )}+\frac{x^{m+1} e^{\text{sech}^{-1}\left (\frac{a}{x}\right )}}{m+1} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 259
Rule 339
Rule 364
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}\left (\frac{a}{x}\right )} x^m \, dx &=\frac{e^{\text{sech}^{-1}\left (\frac{a}{x}\right )} x^{1+m}}{1+m}-\frac{\int x^{1+m} \, dx}{a (1+m)}-\frac{\left (\sqrt{\frac{1}{1+\frac{a}{x}}} \sqrt{1+\frac{a}{x}}\right ) \int \frac{x^{1+m}}{\sqrt{1-\frac{a}{x}} \sqrt{1+\frac{a}{x}}} \, dx}{a (1+m)}\\ &=\frac{e^{\text{sech}^{-1}\left (\frac{a}{x}\right )} x^{1+m}}{1+m}-\frac{x^{2+m}}{a \left (2+3 m+m^2\right )}-\frac{\left (\sqrt{\frac{1}{1+\frac{a}{x}}} \sqrt{1+\frac{a}{x}}\right ) \int \frac{x^{1+m}}{\sqrt{1-\frac{a^2}{x^2}}} \, dx}{a (1+m)}\\ &=\frac{e^{\text{sech}^{-1}\left (\frac{a}{x}\right )} x^{1+m}}{1+m}-\frac{x^{2+m}}{a \left (2+3 m+m^2\right )}+\frac{\left (\sqrt{\frac{1}{1+\frac{a}{x}}} \sqrt{1+\frac{a}{x}} \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-3-m}}{\sqrt{1-a^2 x^2}} \, dx,x,\frac{1}{x}\right )}{a (1+m)}\\ &=\frac{e^{\text{sech}^{-1}\left (\frac{a}{x}\right )} x^{1+m}}{1+m}-\frac{x^{2+m}}{a \left (2+3 m+m^2\right )}-\frac{\sqrt{\frac{1}{1+\frac{a}{x}}} \sqrt{1+\frac{a}{x}} x^{2+m} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-2-m);-\frac{m}{2};\frac{a^2}{x^2}\right )}{a \left (2+3 m+m^2\right )}\\ \end{align*}
Mathematica [A] time = 0.825083, size = 139, normalized size = 1.28 \[ -\frac{a 2^{-m-1} x^m \left (\frac{a}{x}\right )^m e^{\text{sech}^{-1}\left (\frac{a}{x}\right )} \left (\frac{e^{\text{sech}^{-1}\left (\frac{a}{x}\right )}}{e^{2 \text{sech}^{-1}\left (\frac{a}{x}\right )}+1}\right )^{-m-1} \left (m e^{2 \text{sech}^{-1}\left (\frac{a}{x}\right )} \text{Hypergeometric2F1}\left (1,\frac{m}{2}+2,2-\frac{m}{2},-e^{2 \text{sech}^{-1}\left (\frac{a}{x}\right )}\right )-(m-2) \text{Hypergeometric2F1}\left (1,\frac{m}{2}+1,1-\frac{m}{2},-e^{2 \text{sech}^{-1}\left (\frac{a}{x}\right )}\right )\right )}{(m-2) m} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.643, size = 0, normalized size = 0. \begin{align*} \int \left ({\frac{x}{a}}+\sqrt{-1+{\frac{x}{a}}}\sqrt{1+{\frac{x}{a}}} \right ){x}^{m}\, 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*} \frac{x^{2} x^{m}}{a{\left (m + 2\right )}} + \frac{\int \sqrt{a + x} \sqrt{-a + x} x^{m}\,{d x}}{a} \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{a x^{m} \sqrt{\frac{a + x}{a}} \sqrt{-\frac{a - x}{a}} + x x^{m}}{a}, 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*} \frac{\int x x^{m}\, dx + \int a x^{m} \sqrt{-1 + \frac{x}{a}} \sqrt{1 + \frac{x}{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{x}{a} + 1} \sqrt{\frac{x}{a} - 1} + \frac{x}{a}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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