Optimal. Leaf size=95 \[ \frac{c^2 \sqrt{c-\frac{c}{a x}}}{a}-\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{c \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+x \left (c-\frac{c}{a x}\right )^{5/2} \]
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Rubi [A] time = 0.20145, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{c^2 \sqrt{c-\frac{c}{a x}}}{a}-\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{c \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+x \left (c-\frac{c}{a x}\right )^{5/2} \]
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 )^{5/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{5/2} \, dx\\ &=-\int \frac{\left (c-\frac{c}{a x}\right )^{5/2} (1+a x)}{1-a x} \, dx\\ &=\frac{c \int \frac{\left (c-\frac{c}{a x}\right )^{3/2} (1+a x)}{x} \, dx}{a}\\ &=\frac{c \int \left (a+\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2} \, dx}{a}\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{(a+x) \left (c-\frac{c x}{a}\right )^{3/2}}{x^2} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\left (c-\frac{c}{a x}\right )^{5/2} x+\frac{c \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{x} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{c \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac{c}{a x}\right )^{5/2} x+\frac{c^2 \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{c^2 \sqrt{c-\frac{c}{a x}}}{a}+\frac{c \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac{c}{a x}\right )^{5/2} x+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{c^2 \sqrt{c-\frac{c}{a x}}}{a}+\frac{c \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac{c}{a x}\right )^{5/2} x-c^2 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )\\ &=\frac{c^2 \sqrt{c-\frac{c}{a x}}}{a}+\frac{c \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac{c}{a x}\right )^{5/2} x-\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0648904, size = 75, normalized size = 0.79 \[ \frac{c^2 \left (3 a^2 x^2-2 a x+2\right ) \sqrt{c-\frac{c}{a x}}-3 a c^{5/2} x \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{3 a^2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.164, size = 108, normalized size = 1.1 \begin{align*} -{\frac{{c}^{2}}{6\,{x}^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -6\,\sqrt{a{x}^{2}-x}{a}^{5/2}{x}^{3}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{3}{a}^{2}+4\, \left ( a{x}^{2}-x \right ) ^{3/2}\sqrt{a} \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{a}^{-{\frac{5}{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{5}{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 = 1.86239, size = 397, normalized size = 4.18 \begin{align*} \left [\frac{3 \, a c^{\frac{5}{2}} x \log \left (-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right ) + 2 \,{\left (3 \, a^{2} c^{2} x^{2} - 2 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt{\frac{a c x - c}{a x}}}{6 \, a^{2} x}, \frac{3 \, a \sqrt{-c} c^{2} x \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right ) +{\left (3 \, a^{2} c^{2} x^{2} - 2 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} x}\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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