Optimal. Leaf size=156 \[ \frac{c^5 x \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}+\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a} \]
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Rubi [A] time = 0.268086, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {6177, 879, 865, 875, 208} \[ \frac{c^5 x \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}+\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 879
Rule 865
Rule 875
Rule 208
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{5/2} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{x^2 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{c^4 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{x \left (c-\frac{c x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{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{c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{c^3 \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{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{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}+\frac{c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{c^2 \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{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{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}+\frac{c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{c^4 \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{a^2}+\frac{c^2 x^2}{a^2}} \, dx,x,\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a^3}\\ &=-\frac{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{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}+\frac{c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}{\left (c-\frac{c}{a x}\right )^{5/2}}+\frac{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}\\ \end{align*}
Mathematica [A] time = 0.0658166, size = 89, normalized size = 0.57 \[ \frac{c^2 \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} \left (3 a^2 x^2+2 a x+2\right )+3 a x \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{3 a^2 x \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.18, size = 144, normalized size = 0.9 \begin{align*}{\frac{ \left ( ax-1 \right ){c}^{2}}{ \left ( 6\,ax+6 \right ) x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 6\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+3\,\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\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}+4\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{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}}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13802, size = 788, normalized size = 5.05 \begin{align*} \left [\frac{3 \,{\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} + 5 \, a^{2} c^{2} x^{2} + 4 \, 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{3 \,{\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} + 5 \, a^{2} c^{2} x^{2} + 4 \, 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}}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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