Optimal. Leaf size=138 \[ -\frac{x^2 \sqrt{1-a^2 x^2} \sqrt{c-\frac{c}{a x}}}{2 (1-a x)}-\frac{7 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2} \sqrt{1-a x}}+\frac{7 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-a x}} \]
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Rubi [A] time = 0.230064, antiderivative size = 138, 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, 6128, 881, 848, 50, 54, 215} \[ -\frac{x^2 \sqrt{1-a^2 x^2} \sqrt{c-\frac{c}{a x}}}{2 (1-a x)}-\frac{7 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2} \sqrt{1-a x}}+\frac{7 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6128
Rule 881
Rule 848
Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int e^{-\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^{-\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)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{1-a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x^2 \sqrt{1-a^2 x^2}}{2 (1-a x)}+\frac{\left (7 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{x} \sqrt{1-a x}}{\sqrt{1-a^2 x^2}} \, dx}{4 \sqrt{1-a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x^2 \sqrt{1-a^2 x^2}}{2 (1-a x)}+\frac{\left (7 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{x}}{\sqrt{1+a x}} \, dx}{4 \sqrt{1-a x}}\\ &=\frac{7 \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^2 x^2}}{2 (1-a x)}-\frac{\left (7 \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{7 \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^2 x^2}}{2 (1-a x)}-\frac{\left (7 \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{7 \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^2 x^2}}{2 (1-a x)}-\frac{7 \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 = 0.0508709, size = 80, normalized size = 0.58 \[ -\frac{\sqrt{x} \sqrt{c-\frac{c}{a x}} \left (\sqrt{a} \sqrt{x} \sqrt{a x+1} (2 a x-7)+7 \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{4 a^{3/2} \sqrt{1-a x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.123, size = 107, normalized size = 0.8 \begin{align*}{\frac{x}{8\,ax-8}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}-14\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}-7\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \right ){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{\sqrt{-a^{2} x^{2} + 1} \sqrt{c - \frac{c}{a x}} x}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.00311, size = 582, normalized size = 4.22 \begin{align*} \left [\frac{7 \,{\left (a x - 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^{2} x^{2} - 7 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{16 \,{\left (a^{3} x - a^{2}\right )}}, \frac{7 \,{\left (a x - 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^{2} x^{2} - 7 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{8 \,{\left (a^{3} x - 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 )} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, 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{\sqrt{-a^{2} x^{2} + 1} \sqrt{c - \frac{c}{a x}} x}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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