Optimal. Leaf size=98 \[ \frac{2}{7} x (c-a c x)^{5/2} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n-5}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \text{Hypergeometric2F1}\left (-\frac{7}{2},\frac{n-5}{2},-\frac{5}{2},\frac{2}{x \left (a+\frac{1}{x}\right )}\right ) \]
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Rubi [A] time = 0.199031, 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} \[ \frac{2}{7} x (c-a c x)^{5/2} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{n-5}{2}} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \, _2F_1\left (-\frac{7}{2},\frac{n-5}{2};-\frac{5}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right ) \]
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)^{5/2} \, dx &=\frac{(c-a c x)^{5/2} \int e^{n \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{5/2} x^{5/2} \, dx}{\left (1-\frac{1}{a x}\right )^{5/2} x^{5/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{\frac{5}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{x^{9/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=\frac{2}{7} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (-5+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)^{5/2} \, _2F_1\left (-\frac{7}{2},\frac{1}{2} (-5+n);-\frac{5}{2};\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )\\ \end{align*}
Mathematica [A] time = 0.096458, size = 103, normalized size = 1.05 \[ \frac{2 c^2 (a x+1)^3 \sqrt{c-a c x} \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{n/2} \left (\frac{a x-1}{a x+1}\right )^{\frac{n-1}{2}} \text{Hypergeometric2F1}\left (-\frac{7}{2},\frac{n-5}{2},-\frac{5}{2},\frac{2}{a x+1}\right )}{7 a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.404, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -acx+c \right ) ^{{\frac{5}{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{5}{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^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\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{5}{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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