Optimal. Leaf size=98 \[ \frac{2}{5} x (c-a c x)^{3/2} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \text{Hypergeometric2F1}\left (-\frac{5}{2},\frac{n-3}{2},-\frac{3}{2},\frac{2}{x \left (a+\frac{1}{x}\right )}\right ) \]
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Rubi [A] time = 0.199218, 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} \[ \frac{2}{5} x (c-a c x)^{3/2} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \, _2F_1\left (-\frac{5}{2},\frac{n-3}{2};-\frac{3}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right ) \]
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)} (c-a c x)^{3/2} \, dx &=\frac{(c-a c x)^{3/2} \int e^{n \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \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}}{x^{7/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=\frac{2}{5} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (-3+n)} \left (1-\frac{1}{a x}\right )^{-n/2} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x (c-a c x)^{3/2} \, _2F_1\left (-\frac{5}{2},\frac{1}{2} (-3+n);-\frac{3}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )\\ \end{align*}
Mathematica [A] time = 0.0705067, size = 101, normalized size = 1.03 \[ -\frac{2 c (a x+1)^2 \sqrt{c-a c x} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{n/2} \left (\frac{a x-1}{a x+1}\right )^{\frac{n-1}{2}} \text{Hypergeometric2F1}\left (-\frac{5}{2},\frac{n-3}{2},-\frac{3}{2},\frac{2}{a x+1}\right )}{5 a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.381, 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{\left (-a c x + c\right )}^{\frac{3}{2}} \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 (-{\left (a c x - c\right )} \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{\left (-a c x + c\right )}^{\frac{3}{2}} \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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