Optimal. Leaf size=71 \[ -\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{c \sqrt{c-\frac{c}{a x}}}{a}-x \left (c-\frac{c}{a x}\right )^{3/2} \]
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Rubi [A] time = 0.120584, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6133, 25, 514, 375, 78, 50, 63, 208} \[ -\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{c \sqrt{c-\frac{c}{a x}}}{a}-x \left (c-\frac{c}{a x}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 6133
Rule 25
Rule 514
Rule 375
Rule 78
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{3/2} \, dx &=\int \frac{\left (c-\frac{c}{a x}\right )^{3/2} (1+a x)}{1-a x} \, dx\\ &=-\frac{c \int \frac{\sqrt{c-\frac{c}{a x}} (1+a x)}{x} \, dx}{a}\\ &=-\frac{c \int \left (a+\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}} \, dx}{a}\\ &=\frac{c \operatorname{Subst}\left (\int \frac{(a+x) \sqrt{c-\frac{c x}{a}}}{x^2} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\left (c-\frac{c}{a x}\right )^{3/2} x+\frac{c \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{c \sqrt{c-\frac{c}{a x}}}{a}-\left (c-\frac{c}{a x}\right )^{3/2} x+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{c \sqrt{c-\frac{c}{a x}}}{a}-\left (c-\frac{c}{a x}\right )^{3/2} x-c \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )\\ &=\frac{c \sqrt{c-\frac{c}{a x}}}{a}-\left (c-\frac{c}{a x}\right )^{3/2} x-\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0470816, size = 57, normalized size = 0.8 \[ \frac{c (2-a x) \sqrt{c-\frac{c}{a x}}-c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.121, size = 103, normalized size = 1.5 \begin{align*} -{\frac{c}{2\,x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -2\,\sqrt{a{x}^{2}-x}{a}^{3/2}{x}^{2}+4\, \left ( a{x}^{2}-x \right ) ^{3/2}\sqrt{a}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1 \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}a \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{a}^{-{\frac{3}{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 (a x + 1\right )}^{2}{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15055, size = 297, normalized size = 4.18 \begin{align*} \left [\frac{c^{\frac{3}{2}} \log \left (-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right ) - 2 \,{\left (a c x - 2 \, c\right )} \sqrt{\frac{a c x - c}{a x}}}{2 \, a}, \frac{\sqrt{-c} c \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right ) -{\left (a c x - 2 \, c\right )} \sqrt{\frac{a c x - c}{a x}}}{a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 21.4053, size = 163, normalized size = 2.3 \begin{align*} - c \left (\begin{cases} \frac{\sqrt{a} \sqrt{c} x^{\frac{3}{2}}}{\sqrt{a x - 1}} - \frac{\sqrt{c} \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{a} - \frac{\sqrt{c} \sqrt{x}}{\sqrt{a} \sqrt{a x - 1}} & \text{for}\: \left |{a x}\right | > 1 \\\frac{i \sqrt{c} \operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{a} + \frac{i \sqrt{c} \sqrt{x} \sqrt{- a x + 1}}{\sqrt{a}} & \text{otherwise} \end{cases}\right ) + \frac{2 c^{2} \operatorname{atan}{\left (\frac{\sqrt{c - \frac{c}{a x}}}{\sqrt{- c}} \right )}}{a \sqrt{- c}} + \frac{2 c \sqrt{c - \frac{c}{a x}}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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