Optimal. Leaf size=41 \[ -\frac{2 (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x}} \]
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Rubi [A] time = 0.226816, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6134, 6128, 848, 37} \[ -\frac{2 (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6128
Rule 848
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{e^{\tanh ^{-1}(a x)} \sqrt{1-a x}}{x^{5/2}} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{1-a^2 x^2}}{x^{5/2} \sqrt{1-a x}} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{1+a x}}{x^{5/2}} \, dx}{\sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}} (1+a x)^{3/2}}{3 x \sqrt{1-a x}}\\ \end{align*}
Mathematica [A] time = 0.0212255, size = 41, normalized size = 1. \[ -\frac{2 (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.083, size = 40, normalized size = 1. \begin{align*} -{\frac{2\, \left ( ax+1 \right ) ^{2}}{3\,x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+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{{\left (a x + 1\right )} \sqrt{c - \frac{c}{a x}}}{\sqrt{-a^{2} x^{2} + 1} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90373, size = 95, normalized size = 2.32 \begin{align*} \frac{2 \, \sqrt{-a^{2} x^{2} + 1}{\left (a x + 1\right )} \sqrt{\frac{a c x - c}{a x}}}{3 \,{\left (a x^{2} - x\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{\sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x + 1\right )}{x^{2} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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 x + 1\right )} \sqrt{c - \frac{c}{a x}}}{\sqrt{-a^{2} x^{2} + 1} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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