Optimal. Leaf size=96 \[ -\frac{2 x \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n+3}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+3}{2},\frac{3}{2},\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{(c-a c x)^{3/2}} \]
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Rubi [A] time = 0.200264, antiderivative size = 96, 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} \[ -\frac{2 x \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n+3}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \, _2F_1\left (\frac{1}{2},\frac{n+3}{2};\frac{3}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{(c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 132
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-\frac{3}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{\sqrt{x}} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac{2 \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{3+n}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x \, _2F_1\left (\frac{1}{2},\frac{3+n}{2};\frac{3}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{(c-a c x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0401119, size = 94, normalized size = 0.98 \[ \frac{2 \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{n/2} \left (\frac{a x-1}{a x+1}\right )^{\frac{n+1}{2}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+3}{2},\frac{3}{2},\frac{2}{a x+1}\right )}{a c \sqrt{c-a c x}} \]
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 )}} \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{acoth}{\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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