Optimal. Leaf size=70 \[ \frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}-\frac{c \sqrt{c-\frac{c}{a x}}}{a}+x \left (c-\frac{c}{a x}\right )^{3/2} \]
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Rubi [A] time = 0.177587, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6167, 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 6167
Rule 6133
Rule 25
Rule 514
Rule 375
Rule 78
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{3/2} \, dx &=-\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 = 0.047223, size = 55, normalized size = 0.79 \[ \frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )+c (a x-2) \sqrt{c-\frac{c}{a x}}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.163, 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 )}{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}{a x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80755, size = 298, normalized size = 4.26 \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 [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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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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