Optimal. Leaf size=120 \[ \frac{a x^2 \left (1-\frac{1}{a x}\right )^{5/2}}{\sqrt{\frac{1}{a x}+1} (c-a c x)^{5/2}}-\frac{a^{3/2} \left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{2} \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}} \]
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Rubi [A] time = 0.179631, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6176, 6181, 94, 93, 206} \[ \frac{a x^2 \left (1-\frac{1}{a x}\right )^{5/2}}{\sqrt{\frac{1}{a x}+1} (c-a c x)^{5/2}}-\frac{a^{3/2} \left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{2} \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{5/2} x^{5/2}} \, dx}{(c-a c x)^{5/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (1-\frac{x}{a}\right ) \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=\frac{a \left (1-\frac{1}{a x}\right )^{5/2} x^2}{\sqrt{1+\frac{1}{a x}} (c-a c x)^{5/2}}-\frac{\left (a \left (1-\frac{1}{a x}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=\frac{a \left (1-\frac{1}{a x}\right )^{5/2} x^2}{\sqrt{1+\frac{1}{a x}} (c-a c x)^{5/2}}-\frac{\left (a \left (1-\frac{1}{a x}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{\left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=\frac{a \left (1-\frac{1}{a x}\right )^{5/2} x^2}{\sqrt{1+\frac{1}{a x}} (c-a c x)^{5/2}}-\frac{a^{3/2} \left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{\sqrt{2} \left (\frac{1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0779689, size = 122, normalized size = 1.02 \[ \frac{\sqrt{1-\frac{1}{a x}} \left (2 \sqrt{\frac{1}{x}}-\sqrt{2} \sqrt{a} \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{2 a c^2 \sqrt{\frac{1}{x}} \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.149, size = 85, normalized size = 0.7 \begin{align*} -{\frac{ax+1}{2\, \left ( ax-1 \right ) ^{2}a} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{-c \left ( ax-1 \right ) } \left ( \arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) \sqrt{2}\sqrt{-c \left ( ax+1 \right ) }+2\,\sqrt{c} \right ){c}^{-{\frac{7}{2}}}} \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{3}{2}}}{{\left (-a c x + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6623, size = 560, normalized size = 4.67 \begin{align*} \left [-\frac{\sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{4 \,{\left (a^{2} c^{3} x - a c^{3}\right )}}, \frac{\sqrt{2}{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) - 2 \, \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \,{\left (a^{2} c^{3} x - a c^{3}\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 [A] time = 1.15681, size = 134, normalized size = 1.12 \begin{align*} -\frac{{\left (\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{a c^{\frac{3}{2}}} - \frac{\sqrt{2}{\left (\sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{c}}\right ) + \sqrt{c}\right )}}{a \sqrt{-c} c^{\frac{3}{2}}} + \frac{2}{\sqrt{-a c x - c} a c}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{2 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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