Optimal. Leaf size=118 \[ -\frac{9}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{9 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{5/2}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{3}{a c \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{9}{5 a \left (c-\frac{c}{a x}\right )^{5/2}} \]
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Rubi [A] time = 0.215616, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 11, 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, 51, 63, 208} \[ -\frac{9}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{9 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{5/2}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{3}{a c \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{9}{5 a \left (c-\frac{c}{a x}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 375
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{5/2}} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{5/2}} \, dx\\ &=-\int \frac{1+a x}{\left (c-\frac{c}{a x}\right )^{5/2} (1-a x)} \, dx\\ &=\frac{c \int \frac{1+a x}{\left (c-\frac{c}{a x}\right )^{7/2} x} \, dx}{a}\\ &=\frac{c \int \frac{a+\frac{1}{x}}{\left (c-\frac{c}{a x}\right )^{7/2}} \, dx}{a}\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{a+x}{x^2 \left (c-\frac{c x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{x}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{(9 c) \operatorname{Subst}\left (\int \frac{1}{x \left (c-\frac{c x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{9}{5 a \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{x \left (c-\frac{c x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{9}{5 a \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{3}{a c \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{x \left (c-\frac{c x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{2 a c}\\ &=-\frac{9}{5 a \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{3}{a c \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{9}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a c^2}\\ &=-\frac{9}{5 a \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{3}{a c \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{9}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{5/2}}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{c^3}\\ &=-\frac{9}{5 a \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{3}{a c \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{9}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{5/2}}+\frac{9 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0353326, size = 58, normalized size = 0.49 \[ \frac{x}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{9 \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},1-\frac{1}{a x}\right )}{5 a \left (c-\frac{c}{a x}\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.173, size = 328, normalized size = 2.8 \begin{align*} -{\frac{x}{10\,{c}^{3} \left ( ax-1 \right ) ^{4}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -90\,\sqrt{ \left ( ax-1 \right ) x}{a}^{9/2}{x}^{4}+80\, \left ( \left ( ax-1 \right ) x \right ) ^{3/2}{a}^{7/2}{x}^{2}-45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{4}{a}^{4}+360\,\sqrt{ \left ( ax-1 \right ) x}{a}^{7/2}{x}^{3}-132\, \left ( \left ( ax-1 \right ) x \right ) ^{3/2}{a}^{5/2}x+180\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{3}{a}^{3}-540\,{a}^{5/2}\sqrt{ \left ( ax-1 \right ) x}{x}^{2}+60\,{a}^{3/2} \left ( \left ( ax-1 \right ) x \right ) ^{3/2}-270\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{2}{a}^{2}+360\,{a}^{3/2}\sqrt{ \left ( ax-1 \right ) x}x+180\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) xa-90\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}-45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{\frac{1}{\sqrt{a}}}} \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{a x + 1}{{\left (a x - 1\right )}{\left (c - \frac{c}{a x}\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 = 1.93625, size = 629, normalized size = 5.33 \begin{align*} \left [\frac{45 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{c} \log \left (-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right ) + 2 \,{\left (5 \, a^{4} x^{4} - 69 \, a^{3} x^{3} + 105 \, a^{2} x^{2} - 45 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{10 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}, -\frac{45 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right ) -{\left (5 \, a^{4} x^{4} - 69 \, a^{3} x^{3} + 105 \, a^{2} x^{2} - 45 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{5 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x + 1}{\left (- c \left (-1 + \frac{1}{a x}\right )\right )^{\frac{5}{2}} \left (a x - 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25614, size = 220, normalized size = 1.86 \begin{align*} -\frac{1}{5} \, a c{\left (\frac{2 \,{\left (2 \, c^{2} + \frac{5 \,{\left (a c x - c\right )} c}{a x} + \frac{20 \,{\left (a c x - c\right )}^{2}}{a^{2} x^{2}}\right )} x^{2}}{{\left (a c x - c\right )}^{2} c^{3} \sqrt{\frac{a c x - c}{a x}}} + \frac{45 \, \arctan \left (\frac{\sqrt{\frac{a c x - c}{a x}}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{3}} - \frac{5 \, \sqrt{\frac{a c x - c}{a x}}}{a^{2}{\left (c - \frac{a c x - c}{a x}\right )} c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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