Optimal. Leaf size=60 \[ \frac{x^{m+1} \sqrt{c-\frac{c}{a x}} \text{Hypergeometric2F1}\left (-\frac{1}{2},-m-1,-m,-\frac{1}{a x}\right )}{(m+1) \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.221682, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6182, 6181, 64} \[ \frac{x^{m+1} \sqrt{c-\frac{c}{a x}} \, _2F_1\left (-\frac{1}{2},-m-1;-m;-\frac{1}{a x}\right )}{(m+1) \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6181
Rule 64
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^m \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int e^{\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^m \, dx}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\left (\sqrt{c-\frac{c}{a x}} \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \sqrt{1+\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x^{1+m} \, _2F_1\left (-\frac{1}{2},-1-m;-m;-\frac{1}{a x}\right )}{(1+m) \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0311424, size = 60, normalized size = 1. \[ \frac{x^{m+1} \sqrt{c-\frac{c}{a x}} \text{Hypergeometric2F1}\left (-\frac{1}{2},-m-1,-m,-\frac{1}{a x}\right )}{(m+1) \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.17, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}\sqrt{c-{\frac{c}{ax}}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+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{\sqrt{c - \frac{c}{a x}} x^{m}}{\sqrt{\frac{a x - 1}{a x + 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{{\left (a x + 1\right )} x^{m} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\sqrt{c - \frac{c}{a x}} x^{m}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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