Optimal. Leaf size=42 \[ \frac{2 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-4 a \sqrt{c-\frac{c}{a x}} \]
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Rubi [A] time = 0.215732, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6133, 25, 514, 444, 43} \[ \frac{2 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-4 a \sqrt{c-\frac{c}{a x}} \]
Antiderivative was successfully verified.
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Rule 6133
Rule 25
Rule 514
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx &=\int \frac{\sqrt{c-\frac{c}{a x}} (1+a x)}{x^2 (1-a x)} \, dx\\ &=-\frac{c \int \frac{1+a x}{\sqrt{c-\frac{c}{a x}} x^3} \, dx}{a}\\ &=-\frac{c \int \frac{a+\frac{1}{x}}{\sqrt{c-\frac{c}{a x}} x^2} \, dx}{a}\\ &=\frac{c \operatorname{Subst}\left (\int \frac{a+x}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{c \operatorname{Subst}\left (\int \left (\frac{2 a}{\sqrt{c-\frac{c x}{a}}}-\frac{a \sqrt{c-\frac{c x}{a}}}{c}\right ) \, dx,x,\frac{1}{x}\right )}{a}\\ &=-4 a \sqrt{c-\frac{c}{a x}}+\frac{2 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}\\ \end{align*}
Mathematica [A] time = 0.0336075, size = 28, normalized size = 0.67 \[ -\frac{2 (5 a x+1) \sqrt{c-\frac{c}{a x}}}{3 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 27, normalized size = 0.6 \begin{align*} -{\frac{10\,ax+2}{3\,x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}} \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 )}^{2} \sqrt{c - \frac{c}{a x}}}{{\left (a^{2} x^{2} - 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 = 1.75613, size = 59, normalized size = 1.4 \begin{align*} -\frac{2 \,{\left (5 \, a x + 1\right )} \sqrt{\frac{a c x - c}{a x}}}{3 \, x} \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 - \frac{c}{a x}}}{a x^{3} - x^{2}}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x^{3} - x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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