Optimal. Leaf size=163 \[ \frac{2 a^4 \left (c-\frac{c}{a x}\right )^{9/2}}{9 c^4}-\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}+4 a^4 \sqrt{c-\frac{c}{a x}}-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right ) \]
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Rubi [A] time = 0.458131, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6167, 6133, 25, 514, 446, 88, 50, 63, 208} \[ \frac{2 a^4 \left (c-\frac{c}{a x}\right )^{9/2}}{9 c^4}-\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}+4 a^4 \sqrt{c-\frac{c}{a x}}-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 446
Rule 88
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx\\ &=-\int \frac{\sqrt{c-\frac{c}{a x}} (1-a x)}{x^5 (1+a x)} \, dx\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{3/2}}{x^4 (1+a x)} \, dx}{c}\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{3/2}}{\left (a+\frac{1}{x}\right ) x^5} \, dx}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{x^3 \left (c-\frac{c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \left (a^2 \left (c-\frac{c x}{a}\right )^{3/2}-\frac{a^3 \left (c-\frac{c x}{a}\right )^{3/2}}{a+x}-\frac{a^2 \left (c-\frac{c x}{a}\right )^{5/2}}{c}+\frac{a^2 \left (c-\frac{c x}{a}\right )^{7/2}}{c^2}\right ) \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}-\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{9/2}}{9 c^4}+\frac{a^4 \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}-\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{9/2}}{9 c^4}+\left (2 a^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{a+x} \, dx,x,\frac{1}{x}\right )\\ &=4 a^4 \sqrt{c-\frac{c}{a x}}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}-\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{9/2}}{9 c^4}+\left (4 a^4 c\right ) \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=4 a^4 \sqrt{c-\frac{c}{a x}}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}-\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{9/2}}{9 c^4}-\left (8 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )\\ &=4 a^4 \sqrt{c-\frac{c}{a x}}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}-\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}+\frac{2 a^4 \left (c-\frac{c}{a x}\right )^{9/2}}{9 c^4}-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.139376, size = 95, normalized size = 0.58 \[ \frac{2 \left (788 a^4 x^4-236 a^3 x^3+138 a^2 x^2-95 a x+35\right ) \sqrt{c-\frac{c}{a x}}}{315 x^4}-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.186, size = 326, normalized size = 2. \begin{align*} -{\frac{1}{315\,{x}^{5}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -1890\,\sqrt{a{x}^{2}-x}{a}^{11/2}\sqrt{{a}^{-1}}{x}^{6}+630\,{a}^{11/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}{x}^{6}+1260\, \left ( a{x}^{2}-x \right ) ^{3/2}{a}^{9/2}\sqrt{{a}^{-1}}{x}^{4}+945\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{x}^{6}{a}^{5}-630\,{a}^{9/2}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1}{ax+1}} \right ){x}^{6}-945\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{x}^{6}{a}^{5}-316\, \left ( a{x}^{2}-x \right ) ^{3/2}{a}^{7/2}\sqrt{{a}^{-1}}{x}^{3}+156\,{a}^{5/2}\sqrt{{a}^{-1}} \left ( a{x}^{2}-x \right ) ^{3/2}{x}^{2}-120\,{a}^{3/2} \left ( a{x}^{2}-x \right ) ^{3/2}x\sqrt{{a}^{-1}}+70\, \left ( a{x}^{2}-x \right ) ^{3/2}\sqrt{a}\sqrt{{a}^{-1}} \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \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 )} \sqrt{c - \frac{c}{a x}}}{{\left (a x + 1\right )} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65622, size = 517, normalized size = 3.17 \begin{align*} \left [\frac{2 \,{\left (315 \, \sqrt{2} a^{4} \sqrt{c} x^{4} \log \left (\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) +{\left (788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right )} \sqrt{\frac{a c x - c}{a x}}\right )}}{315 \, x^{4}}, \frac{2 \,{\left (630 \, \sqrt{2} a^{4} \sqrt{-c} x^{4} \arctan \left (\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right ) +{\left (788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right )} \sqrt{\frac{a c x - c}{a x}}\right )}}{315 \, x^{4}}\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{\sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x - 1\right )}{x^{5} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.70279, size = 586, normalized size = 3.6 \begin{align*} \frac{4 \, \sqrt{2} a^{5} c \arctan \left (\frac{\sqrt{2}{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )} a + \sqrt{c}{\left | a \right |}\right )}}{2 \, a \sqrt{-c}}\right )}{\sqrt{-c}{\left | a \right |} \mathrm{sgn}\left (x\right )} + \frac{2 \,{\left (1260 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{8} a^{9} c - 1260 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{7} a^{8} c^{\frac{3}{2}}{\left | a \right |} + 2100 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{6} a^{9} c^{2} - 3150 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{5} a^{8} c^{\frac{5}{2}}{\left | a \right |} + 3528 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{4} a^{9} c^{3} - 2625 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{3} a^{8} c^{\frac{7}{2}}{\left | a \right |} + 1215 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{2} a^{9} c^{4} - 315 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )} a^{8} c^{\frac{9}{2}}{\left | a \right |} + 35 \, a^{9} c^{5}\right )}}{315 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{9} a^{4}{\left | a \right |} \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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