Optimal. Leaf size=98 \[ \frac{2}{3} x \sqrt{c-a c x} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{n-1}{2},-\frac{1}{2},\frac{2}{x \left (a+\frac{1}{x}\right )}\right ) \]
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Rubi [A] time = 0.177348, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6176, 6181, 132} \[ \frac{2}{3} x \sqrt{c-a c x} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \, _2F_1\left (-\frac{3}{2},\frac{n-1}{2};-\frac{1}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right ) \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 132
Rubi steps
\begin{align*} \int e^{n \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{n \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} \sqrt{x} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{x^{5/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2}{3} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (-1+n)} \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x \sqrt{c-a c x} \, _2F_1\left (-\frac{3}{2},\frac{1}{2} (-1+n);-\frac{1}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )\\ \end{align*}
Mathematica [A] time = 0.0567682, size = 98, normalized size = 1. \[ \frac{2 (a x+1) \sqrt{c-a c x} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{n/2} \left (\frac{a x-1}{a x+1}\right )^{\frac{n-1}{2}} \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{n-1}{2},-\frac{1}{2},\frac{2}{a x+1}\right )}{3 a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.367, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}}\sqrt{-acx+c}\, 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{-a c x + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{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 (\sqrt{-a c x + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}, 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{-a c x + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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