Optimal. Leaf size=157 \[ \frac{c^4 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{3 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}-\frac{3 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a} \]
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Rubi [A] time = 0.132817, antiderivative size = 189, normalized size of antiderivative = 1.2, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6182, 6179, 89, 80, 50, 63, 208} \[ -\frac{2 \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2}}{3 a \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{x \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2}}{\left (1-\frac{1}{a x}\right )^{5/2}}+\frac{3 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{5/2}}{a \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{3 \left (c-\frac{c}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Rule 6182
Rule 6179
Rule 89
Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{5/2} \, dx &=\frac{\left (c-\frac{c}{a x}\right )^{5/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{5/2} \, dx}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{\left (c-\frac{c}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2 \sqrt{1+\frac{x}{a}}}{x^2} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=\frac{\left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2} x}{\left (1-\frac{1}{a x}\right )^{5/2}}-\frac{\left (c-\frac{c}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{\left (-\frac{3}{2 a}+\frac{x}{a^2}\right ) \sqrt{1+\frac{x}{a}}}{x} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=-\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2}}{3 a \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2} x}{\left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (3 \left (c-\frac{c}{a x}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x} \, dx,x,\frac{1}{x}\right )}{2 a \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=\frac{3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}{a \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2}}{3 a \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2} x}{\left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (3 \left (c-\frac{c}{a x}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \left (1-\frac{1}{a x}\right )^{5/2}}\\ &=\frac{3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}{a \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2}}{3 a \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2} x}{\left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (3 \left (c-\frac{c}{a x}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\left (1-\frac{1}{a x}\right )^{5/2}}\\ &=\frac{3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}{a \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2}}{3 a \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{\left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2} x}{\left (1-\frac{1}{a x}\right )^{5/2}}-\frac{3 \left (c-\frac{c}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \left (1-\frac{1}{a x}\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0736338, size = 89, normalized size = 0.57 \[ \frac{c^2 \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} \left (3 a^2 x^2+10 a x-2\right )-9 a x \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{3 a^2 x \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.18, size = 132, normalized size = 0.8 \begin{align*}{\frac{{c}^{2}}{6\,x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 6\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+20\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{2}{a}^{2}-4\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \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 (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18703, size = 790, normalized size = 5.03 \begin{align*} \left [\frac{9 \,{\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (3 \, a^{3} c^{2} x^{3} + 13 \, a^{2} c^{2} x^{2} + 8 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{12 \,{\left (a^{3} x^{2} - a^{2} x\right )}}, \frac{9 \,{\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (3 \, a^{3} c^{2} x^{3} + 13 \, a^{2} c^{2} x^{2} + 8 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{6 \,{\left (a^{3} x^{2} - a^{2} x\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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