Optimal. Leaf size=78 \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (\frac{1}{2} (-n-1),-\frac{n}{2},\frac{1-n}{2},\frac{1}{2} (1-a x)\right )}{a c (n+1) \sqrt{c-a c x}} \]
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Rubi [A] time = 0.0682523, antiderivative size = 78, 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} \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (\frac{1}{2} (-n-1),-\frac{n}{2};\frac{1-n}{2};\frac{1}{2} (1-a x)\right )}{a c (n+1) \sqrt{c-a c x}} \]
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)}}{(c-a c x)^{3/2}} \, dx &=\int \frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{(c-a c x)^{3/2}} \, 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{3}{2}-\frac{n}{2}} \, dx\\ &=\frac{2^{1+\frac{n}{2}} (1-a x)^{-n/2} \, _2F_1\left (\frac{1}{2} (-1-n),-\frac{n}{2};\frac{1-n}{2};\frac{1}{2} (1-a x)\right )}{a c (1+n) \sqrt{c-a c x}}\\ \end{align*}
Mathematica [A] time = 0.0243085, size = 78, normalized size = 1. \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (-\frac{n}{2}-\frac{1}{2},-\frac{n}{2},\frac{1}{2}-\frac{n}{2},\frac{1}{2}-\frac{a x}{2}\right )}{a c (n+1) \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.212, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( -acx+c \right ) ^{-{\frac{3}{2}}}}\, 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}}{{\left (-a c x + c\right )}^{\frac{3}{2}}}\,{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^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}, 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 )}}}{\left (- c \left (a x - 1\right )\right )^{\frac{3}{2}}}\, 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}}{{\left (-a c x + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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