Optimal. Leaf size=303 \[ \frac{2 a^4 \left (\frac{1}{a x}+1\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{4 a^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^4 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.320336, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6182, 6180, 88, 50, 63, 206} \[ \frac{2 a^4 \left (\frac{1}{a x}+1\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{4 a^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^4 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6180
Rule 88
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}}}{x^5} \, dx}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{x^3 \left (1+\frac{x}{a}\right )^{3/2}}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \left (-a^3 \left (1+\frac{x}{a}\right )^{3/2}+\frac{a^3 \left (1+\frac{x}{a}\right )^{3/2}}{1-\frac{x}{a}}+a^3 \left (1+\frac{x}{a}\right )^{5/2}-a^3 \left (1+\frac{x}{a}\right )^{7/2}\right ) \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (a^3 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (2 a^3 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 a^4 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 a^3 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 a^4 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 a^4 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 a^4 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^4 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.297134, size = 178, normalized size = 0.59 \[ \frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (788 a^4 x^4+236 a^3 x^3+138 a^2 x^2+95 a x+35\right ) \sqrt{c-\frac{c}{a x}}}{315 x^3 (a x-1)}-2 \sqrt{2} a^4 \sqrt{c} \log \left (2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (3 a^2 x^2-2 a x-1\right )\right )+2 \sqrt{2} a^4 \sqrt{c} \log \left ((a x-1)^2\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.195, size = 209, normalized size = 0.7 \begin{align*} -{\frac{2\,ax-2}{ \left ( 315\,ax+315 \right ){x}^{4}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 315\,{a}^{4}\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}^{5}-788\,{a}^{4}\sqrt{{a}^{-1}}{x}^{4}\sqrt{ \left ( ax+1 \right ) x}-236\,{a}^{3}\sqrt{{a}^{-1}}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}-138\,{a}^{2}\sqrt{{a}^{-1}}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}-95\,a\sqrt{{a}^{-1}}x\sqrt{ \left ( ax+1 \right ) x}-35\,\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\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{\sqrt{c - \frac{c}{a x}}}{x^{5} \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 = 1.94215, size = 926, normalized size = 3.06 \begin{align*} \left [\frac{315 \, \sqrt{2}{\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt{c} \log \left (-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt{2}{\left (3 \, a^{3} x^{3} + 4 \, 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^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \,{\left (788 \, a^{5} x^{5} + 1024 \, a^{4} x^{4} + 374 \, a^{3} x^{3} + 233 \, a^{2} x^{2} + 130 \, a x + 35\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{315 \,{\left (a x^{5} - x^{4}\right )}}, \frac{2 \,{\left (315 \, \sqrt{2}{\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt{-c} \arctan \left (\frac{2 \, \sqrt{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}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) +{\left (788 \, a^{5} x^{5} + 1024 \, a^{4} x^{4} + 374 \, a^{3} x^{3} + 233 \, a^{2} x^{2} + 130 \, a x + 35\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}\right )}}{315 \,{\left (a x^{5} - x^{4}\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{\sqrt{c - \frac{c}{a x}}}{x^{5} \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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