Optimal. Leaf size=56 \[ \frac{2 x (1-a x)^{3/2} F_1\left (\frac{5}{2};\frac{n+3}{2},-\frac{n}{2};\frac{7}{2};a x,-a x\right )}{5 \left (c-\frac{c}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.170338, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6134, 6129, 133} \[ \frac{2 x (1-a x)^{3/2} F_1\left (\frac{5}{2};\frac{n+3}{2},-\frac{n}{2};\frac{7}{2};a x,-a x\right )}{5 \left (c-\frac{c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 133
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{3/2}} \, dx &=\frac{(1-a x)^{3/2} \int \frac{e^{n \tanh ^{-1}(a x)} x^{3/2}}{(1-a x)^{3/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{(1-a x)^{3/2} \int x^{3/2} (1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{n/2} \, dx}{\left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac{2 x (1-a x)^{3/2} F_1\left (\frac{5}{2};\frac{3+n}{2},-\frac{n}{2};\frac{7}{2};a x,-a x\right )}{5 \left (c-\frac{c}{a x}\right )^{3/2}}\\ \end{align*}
Mathematica [F] time = 180.005, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 0.122, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( c-{\frac{c}{ax}} \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 (c - \frac{c}{a x}\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{a^{2} x^{2} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n} \sqrt{\frac{a c x - c}{a x}}}{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(-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{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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