Optimal. Leaf size=159 \[ -\frac{16 a^4 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{315 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{4 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 x^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 x^3 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{16 a^4 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 \sqrt{c-\frac{c}{a x}}} \]
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Rubi [A] time = 0.306008, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6178, 871, 795, 649} \[ -\frac{16 a^4 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{315 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{4 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 x^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 x^3 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{16 a^4 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6178
Rule 871
Rule 795
Rule 649
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{x^3 \sqrt{1-\frac{x^2}{a^2}}}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 \left (c-\frac{c}{a x}\right )^{3/2} x^3}+\frac{1}{3} (2 a c) \operatorname{Subst}\left (\int \frac{x^2 \sqrt{1-\frac{x^2}{a^2}}}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 \left (c-\frac{c}{a x}\right )^{3/2} x^3}+\frac{4 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 \left (c-\frac{c}{a x}\right )^{3/2} x^2}-\frac{1}{21} \left (8 a^2 c\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{1-\frac{x^2}{a^2}}}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{16 a^4 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 \sqrt{c-\frac{c}{a x}}}-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 \left (c-\frac{c}{a x}\right )^{3/2} x^3}+\frac{4 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 \left (c-\frac{c}{a x}\right )^{3/2} x^2}-\frac{1}{105} \left (8 a^3 c\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{16 a^4 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{315 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{16 a^4 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 \sqrt{c-\frac{c}{a x}}}-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 \left (c-\frac{c}{a x}\right )^{3/2} x^3}+\frac{4 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 \left (c-\frac{c}{a x}\right )^{3/2} x^2}\\ \end{align*}
Mathematica [A] time = 0.116209, size = 74, normalized size = 0.47 \[ \frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a^4 x^4-8 a^3 x^3+6 a^2 x^2-5 a x-35\right ) \sqrt{c-\frac{c}{a x}}}{315 x^3 (a x-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.117, size = 63, normalized size = 0.4 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 16\,{x}^{3}{a}^{3}-24\,{a}^{2}{x}^{2}+30\,ax-35 \right ) }{315\,{x}^{4}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+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} \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 = 1.56027, size = 178, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (16 \, a^{5} x^{5} + 8 \, a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2} - 40 \, 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 )}} \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} \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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