Optimal. Leaf size=81 \[ -\frac{2^{\frac{n}{2}+1} \sqrt{c-a c x} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (\frac{1-n}{2},-\frac{n}{2},\frac{3-n}{2},\frac{1}{2} (1-a x)\right )}{a c (1-n)} \]
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Rubi [A] time = 0.0648541, antiderivative size = 81, 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 = {6130, 23, 69} \[ -\frac{2^{\frac{n}{2}+1} \sqrt{c-a c x} (1-a x)^{-n/2} \, _2F_1\left (\frac{1-n}{2},-\frac{n}{2};\frac{3-n}{2};\frac{1}{2} (1-a x)\right )}{a c (1-n)} \]
Antiderivative was successfully verified.
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Rule 6130
Rule 23
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=\int \frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{\sqrt{c-a c x}} \, dx\\ &=\left ((1-a x)^{-n/2} (c-a c x)^{n/2}\right ) \int (1+a x)^{n/2} (c-a c x)^{-\frac{1}{2}-\frac{n}{2}} \, dx\\ &=-\frac{2^{1+\frac{n}{2}} (1-a x)^{-n/2} \sqrt{c-a c x} \, _2F_1\left (\frac{1-n}{2},-\frac{n}{2};\frac{3-n}{2};\frac{1}{2} (1-a x)\right )}{a c (1-n)}\\ \end{align*}
Mathematica [A] time = 0.0256864, size = 78, normalized size = 0.96 \[ \frac{2^{\frac{n}{2}+1} \sqrt{c-a c x} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (\frac{1}{2}-\frac{n}{2},-\frac{n}{2},\frac{3}{2}-\frac{n}{2},\frac{1}{2}-\frac{a x}{2}\right )}{a c (n-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.231, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}{\frac{1}{\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 \frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{\sqrt{-a c x + c}}\,{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 (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a c x - c}, 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*} \int \frac{e^{n \operatorname{atanh}{\left (a x \right )}}}{\sqrt{- c \left (a x - 1\right )}}\, dx \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{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{\sqrt{-a c x + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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