Optimal. Leaf size=126 \[ \frac{\sqrt{\frac{1}{a x}+1} x^{m+1} \sqrt{c-\frac{c}{a x}}}{(m+1) \sqrt{1-\frac{1}{a x}}}-\frac{(4 m+3) x^m \sqrt{c-\frac{c}{a x}} \text{Hypergeometric2F1}\left (\frac{1}{2},-m,1-m,-\frac{1}{a x}\right )}{2 a m (m+1) \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.266844, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {6182, 6181, 79, 64} \[ \frac{\sqrt{\frac{1}{a x}+1} x^{m+1} \sqrt{c-\frac{c}{a x}}}{(m+1) \sqrt{1-\frac{1}{a x}}}-\frac{(4 m+3) x^m \sqrt{c-\frac{c}{a x}} \, _2F_1\left (\frac{1}{2},-m;1-m;-\frac{1}{a x}\right )}{2 a m (m+1) \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6181
Rule 79
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 \frac{x^{-2-m} \left (1-\frac{x}{a}\right )}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^{1+m}}{(1+m) \sqrt{1-\frac{1}{a x}}}+\frac{\left ((3+4 m) \sqrt{c-\frac{c}{a x}} \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1-m}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a (1+m) \sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^{1+m}}{(1+m) \sqrt{1-\frac{1}{a x}}}-\frac{(3+4 m) \sqrt{c-\frac{c}{a x}} x^m \, _2F_1\left (\frac{1}{2},-m;1-m;-\frac{1}{a x}\right )}{2 a m (1+m) \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0660519, size = 93, normalized size = 0.74 \[ \frac{x^m \sqrt{c-\frac{c}{a x}} \left (2 a m x \sqrt{\frac{1}{a x}+1}-(4 m+3) \text{Hypergeometric2F1}\left (\frac{1}{2},-m,1-m,-\frac{1}{a x}\right )\right )}{2 a m (m+1) \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.174, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{c-{\frac{c}{ax}}}\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 \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 (x^{m} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}, 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 \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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