Optimal. Leaf size=69 \[ \frac{2 a^2 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}-\frac{2 a^2 \left (c-\frac{c}{a x}\right )^{3/2}}{c}+4 a^2 \sqrt{c-\frac{c}{a x}} \]
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Rubi [A] time = 0.332806, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6167, 6133, 25, 514, 446, 77} \[ \frac{2 a^2 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}-\frac{2 a^2 \left (c-\frac{c}{a x}\right )^{3/2}}{c}+4 a^2 \sqrt{c-\frac{c}{a x}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx\\ &=-\int \frac{\sqrt{c-\frac{c}{a x}} (1+a x)}{x^3 (1-a x)} \, dx\\ &=\frac{c \int \frac{1+a x}{\sqrt{c-\frac{c}{a x}} x^4} \, dx}{a}\\ &=\frac{c \int \frac{a+\frac{1}{x}}{\sqrt{c-\frac{c}{a x}} x^3} \, dx}{a}\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{x (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^2}{\sqrt{c-\frac{c x}{a}}}-\frac{3 a^2 \sqrt{c-\frac{c x}{a}}}{c}+\frac{a^2 \left (c-\frac{c x}{a}\right )^{3/2}}{c^2}\right ) \, dx,x,\frac{1}{x}\right )}{a}\\ &=4 a^2 \sqrt{c-\frac{c}{a x}}-\frac{2 a^2 \left (c-\frac{c}{a x}\right )^{3/2}}{c}+\frac{2 a^2 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}\\ \end{align*}
Mathematica [A] time = 0.0457306, size = 36, normalized size = 0.52 \[ \frac{2 \left (6 a^2 x^2+3 a x+1\right ) \sqrt{c-\frac{c}{a x}}}{5 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 35, normalized size = 0.5 \begin{align*}{\frac{12\,{a}^{2}{x}^{2}+6\,ax+2}{5\,{x}^{2}}\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 )} \sqrt{c - \frac{c}{a x}}}{{\left (a x - 1\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57498, size = 77, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (6 \, a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt{\frac{a c x - c}{a x}}}{5 \, x^{2}} \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^{3} \left (a x - 1\right )}\, 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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