Optimal. Leaf size=65 \[ \frac{2 x^{m+1} \sqrt{c-a c x} \text{Hypergeometric2F1}\left (-\frac{1}{2},-m-\frac{3}{2},-m-\frac{1}{2},-\frac{1}{a x}\right )}{(2 m+3) \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.191909, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6176, 6181, 64} \[ \frac{2 x^{m+1} \sqrt{c-a c x} \, _2F_1\left (-\frac{1}{2},-m-\frac{3}{2};-m-\frac{1}{2};-\frac{1}{a x}\right )}{(2 m+3) \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 64
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} x^m \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^{\frac{1}{2}+m} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{\frac{1}{2}+m} x^m \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int x^{-\frac{5}{2}-m} \sqrt{1+\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 x^{1+m} \sqrt{c-a c x} \, _2F_1\left (-\frac{1}{2},-\frac{3}{2}-m;-\frac{1}{2}-m;-\frac{1}{a x}\right )}{(3+2 m) \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0266707, size = 67, normalized size = 1.03 \[ -\frac{x^{m+1} \sqrt{c-a c x} \text{Hypergeometric2F1}\left (-\frac{1}{2},-m-\frac{3}{2},-m-\frac{1}{2},-\frac{1}{a x}\right )}{\left (-m-\frac{3}{2}\right ) \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.388, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}\sqrt{-acx+c}{\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{-a c x + c} 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{\sqrt{-a c x + c}{\left (a x + 1\right )} x^{m} \sqrt{\frac{a x - 1}{a x + 1}}}{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{-a c x + c} 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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