Optimal. Leaf size=174 \[ \frac{47 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2} \sqrt{1-a x}}+\frac{x^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{2 \sqrt{1-a x}}+\frac{8 x^2 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x} \sqrt{a x+1}}-\frac{47 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-a x}} \]
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Rubi [A] time = 0.198755, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {6134, 6129, 89, 80, 50, 54, 215} \[ \frac{47 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2} \sqrt{1-a x}}+\frac{x^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{2 \sqrt{1-a x}}+\frac{8 x^2 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x} \sqrt{a x+1}}-\frac{47 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 89
Rule 80
Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int e^{-3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int e^{-3 \tanh ^{-1}(a x)} \sqrt{x} \sqrt{1-a x} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{x} (1-a x)^2}{(1+a x)^{3/2}} \, dx}{\sqrt{1-a x}}\\ &=\frac{8 \sqrt{c-\frac{c}{a x}} x^2}{\sqrt{1-a x} \sqrt{1+a x}}-\frac{\left (2 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{x} \left (\frac{11 a^2}{2}-\frac{a^3 x}{2}\right )}{\sqrt{1+a x}} \, dx}{a^2 \sqrt{1-a x}}\\ &=\frac{8 \sqrt{c-\frac{c}{a x}} x^2}{\sqrt{1-a x} \sqrt{1+a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{2 \sqrt{1-a x}}-\frac{\left (47 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{x}}{\sqrt{1+a x}} \, dx}{4 \sqrt{1-a x}}\\ &=\frac{8 \sqrt{c-\frac{c}{a x}} x^2}{\sqrt{1-a x} \sqrt{1+a x}}-\frac{47 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{4 a \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{2 \sqrt{1-a x}}+\frac{\left (47 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{8 a \sqrt{1-a x}}\\ &=\frac{8 \sqrt{c-\frac{c}{a x}} x^2}{\sqrt{1-a x} \sqrt{1+a x}}-\frac{47 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{4 a \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{2 \sqrt{1-a x}}+\frac{\left (47 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{4 a \sqrt{1-a x}}\\ &=\frac{8 \sqrt{c-\frac{c}{a x}} x^2}{\sqrt{1-a x} \sqrt{1+a x}}-\frac{47 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{4 a \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{2 \sqrt{1-a x}}+\frac{47 \sqrt{c-\frac{c}{a x}} \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2} \sqrt{1-a x}}\\ \end{align*}
Mathematica [A] time = 0.0563745, size = 92, normalized size = 0.53 \[ \frac{\sqrt{x} \sqrt{c-\frac{c}{a x}} \left (\sqrt{a} \sqrt{x} \left (2 a^2 x^2-13 a x-47\right )+47 \sqrt{a x+1} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{4 a^{3/2} \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.151, size = 158, normalized size = 0.9 \begin{align*} -{\frac{x}{ \left ( 8\,ax+8 \right ) \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 4\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}-26\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}-47\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) xa-94\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}-47\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \right ) \sqrt{-{a}^{2}{x}^{2}+1}{a}^{-{\frac{3}{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 (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \sqrt{c - \frac{c}{a x}} x}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.54116, size = 640, normalized size = 3.68 \begin{align*} \left [\frac{47 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{-c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \,{\left (2 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \,{\left (2 \, a^{3} x^{3} - 13 \, a^{2} x^{2} - 47 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{16 \,{\left (a^{4} x^{2} - a^{2}\right )}}, -\frac{47 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (2 \, a^{3} x^{3} - 13 \, a^{2} x^{2} - 47 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{8 \,{\left (a^{4} x^{2} - a^{2}\right )}}\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{x \sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{\left (a x + 1\right )^{3}}\, dx \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 (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \sqrt{c - \frac{c}{a x}} x}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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