Optimal. Leaf size=84 \[ \frac{10 a \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x}}-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-\frac{c}{a x}}}{3 x (1-a x)} \]
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Rubi [A] time = 0.259863, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6134, 6128, 879, 848, 37} \[ \frac{10 a \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x}}-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-\frac{c}{a x}}}{3 x (1-a x)} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6128
Rule 879
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{(1-a x)^{3/2}}{x^{5/2} \sqrt{1-a^2 x^2}} \, dx}{\sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1-a^2 x^2}}{3 x (1-a x)}-\frac{\left (5 a \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{1-a x}}{x^{3/2} \sqrt{1-a^2 x^2}} \, dx}{3 \sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1-a^2 x^2}}{3 x (1-a x)}-\frac{\left (5 a \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{x^{3/2} \sqrt{1+a x}} \, dx}{3 \sqrt{1-a x}}\\ &=\frac{10 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{3 \sqrt{1-a x}}-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1-a^2 x^2}}{3 x (1-a x)}\\ \end{align*}
Mathematica [A] time = 0.0306416, size = 47, normalized size = 0.56 \[ \frac{2 \sqrt{a x+1} (5 a x-1) \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.093, size = 46, normalized size = 0.6 \begin{align*} -{\frac{10\,ax-2}{ \left ( 3\,ax-3 \right ) x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\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{\sqrt{-a^{2} x^{2} + 1} \sqrt{c - \frac{c}{a x}}}{{\left (a x + 1\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33845, size = 99, normalized size = 1.18 \begin{align*} -\frac{2 \, \sqrt{-a^{2} x^{2} + 1}{\left (5 \, 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 )} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{2} \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{\sqrt{-a^{2} x^{2} + 1} \sqrt{c - \frac{c}{a x}}}{{\left (a x + 1\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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