Optimal. Leaf size=152 \[ \frac{x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{5 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.133691, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6182, 6179, 98, 156, 63, 208, 206} \[ \frac{x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{5 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 98
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} \, dx}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^2 \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{\sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{-\frac{5}{2 a}-\frac{3 x}{2 a^2}}{x \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{\sqrt{1-\frac{1}{a x}}}-\frac{\left (4 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a x}}}-\frac{\left (5 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{\sqrt{1-\frac{1}{a x}}}-\frac{\left (5 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\sqrt{1-\frac{1}{a x}}}-\frac{\left (8 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{a \sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{\sqrt{1-\frac{1}{a x}}}+\frac{5 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{a \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0726192, size = 93, normalized size = 0.61 \[ \frac{\sqrt{c-\frac{c}{a x}} \left (a x \sqrt{\frac{1}{a x}+1}+5 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )-4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )\right )}{a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.18, size = 160, normalized size = 1.1 \begin{align*}{\frac{ \left ( ax-1 \right ) x}{2\,ax+2}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}-4\,\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ) \sqrt{a}+5\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \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{\sqrt{c - \frac{c}{a x}}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\sqrt{c - \frac{c}{a x}}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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