Optimal. Leaf size=140 \[ \frac{x \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{3/2}}{\left (1-\frac{1}{a x}\right )^{3/2}}-\frac{2 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{3/2}}{a \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{5 \left (c-\frac{c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.125543, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6182, 6179, 89, 80, 63, 208} \[ \frac{x \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{3/2}}{\left (1-\frac{1}{a x}\right )^{3/2}}-\frac{2 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{3/2}}{a \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{5 \left (c-\frac{c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 89
Rule 80
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{3/2} \, dx &=\frac{\left (c-\frac{c}{a x}\right )^{3/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{3/2} \, dx}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{\left (c-\frac{c}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^2 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=\frac{\sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{3/2} x}{\left (1-\frac{1}{a x}\right )^{3/2}}-\frac{\left (c-\frac{c}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{-\frac{5}{2 a}+\frac{x}{a^2}}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{3/2}}{a \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{\sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{3/2} x}{\left (1-\frac{1}{a x}\right )^{3/2}}+\frac{\left (5 \left (c-\frac{c}{a x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{3/2}}{a \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{\sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{3/2} x}{\left (1-\frac{1}{a x}\right )^{3/2}}+\frac{\left (5 \left (c-\frac{c}{a x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{3/2}}{a \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{\sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{3/2} x}{\left (1-\frac{1}{a x}\right )^{3/2}}-\frac{5 \left (c-\frac{c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \left (1-\frac{1}{a x}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0514815, size = 70, normalized size = 0.5 \[ \frac{c \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} (a x-2)-5 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 118, normalized size = 0.8 \begin{align*}{\frac{c \left ( ax+1 \right ) }{2\,ax-2}\sqrt{{\frac{ax-1}{ax+1}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-5\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) xa-4\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \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 (c - \frac{c}{a x}\right )}^{\frac{3}{2}} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.64371, size = 667, normalized size = 4.76 \begin{align*} \left [\frac{5 \,{\left (a c x - c\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (a^{2} c x^{2} - a c x - 2 \, c\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{4 \,{\left (a^{2} x - a\right )}}, \frac{5 \,{\left (a c x - c\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (a^{2} c x^{2} - a c x - 2 \, c\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a^{2} x - a\right )}}\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 (c - \frac{c}{a x}\right )}^{\frac{3}{2}} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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