Optimal. Leaf size=190 \[ -\frac{47 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{2 x^2 \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{47 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.231778, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6176, 6181, 89, 80, 50, 54, 215} \[ -\frac{47 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{2 x^2 \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{47 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 89
Rule 80
Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx &=\frac{\sqrt{c-a c x} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}}}{x^{5/2}} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x} \left (1-\frac{x}{a}\right )^2}{\left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^2}+\frac{\left (2 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x} \left (\frac{11}{2 a^2}-\frac{x}{2 a^3}\right )}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^2}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x^2}+\frac{\left (47 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{4 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^2}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x^2}+\frac{47 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}-\frac{\left (47 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^2}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x^2}+\frac{47 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}-\frac{\left (47 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,\sqrt{\frac{1}{x}}\right )}{4 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^2}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x^2}+\frac{47 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}-\frac{47 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0828575, size = 90, normalized size = 0.47 \[ -\frac{\sqrt{c-a c x} \left (-47 a^2 x^2+\frac{47 a^{5/2} \sqrt{\frac{1}{a x}+1} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\left (\frac{1}{x}\right )^{5/2}}-13 a x+2\right )}{4 a x^3 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.158, size = 103, normalized size = 0.5 \begin{align*}{\frac{ax+1}{4\, \left ( ax-1 \right ) ^{2}{x}^{2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{-c \left ( ax-1 \right ) } \left ( 47\,\arctan \left ({\frac{\sqrt{-c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{2}{a}^{2}\sqrt{-c \left ( ax+1 \right ) }+47\,{x}^{2}{a}^{2}\sqrt{c}+13\,xa\sqrt{c}-2\,\sqrt{c} \right ){\frac{1}{\sqrt{c}}}} \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{-a c x + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71114, size = 587, normalized size = 3.09 \begin{align*} \left [\frac{47 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \,{\left (47 \, a^{2} x^{2} + 13 \, a x - 2\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{8 \,{\left (a x^{3} - x^{2}\right )}}, -\frac{47 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) -{\left (47 \, a^{2} x^{2} + 13 \, a x - 2\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{4 \,{\left (a x^{3} - x^{2}\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 [A] time = 1.23343, size = 113, normalized size = 0.59 \begin{align*} \frac{1}{4} \, a^{2} c{\left (\frac{47 \, \arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right )}{c^{\frac{3}{2}}} + \frac{32}{\sqrt{-a c x - c} c} + \frac{15 \,{\left (-a c x - c\right )}^{\frac{3}{2}} + 17 \, \sqrt{-a c x - c} c}{a^{2} c^{3} x^{2}}\right )}{\left | c \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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