Optimal. Leaf size=147 \[ \frac{4 \sqrt{2} a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{1-a x}}-\frac{4 a \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x}}-\frac{2 (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x}} \]
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Rubi [A] time = 0.245573, antiderivative size = 147, 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 = {6134, 6129, 94, 93, 206} \[ \frac{4 \sqrt{2} a^{3/2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{1-a x}}-\frac{4 a \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x}}-\frac{2 (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{3 x \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 94
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{3 \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^{3 \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} (1-a x)} \, dx}{\sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}} (1+a x)^{3/2}}{3 x \sqrt{1-a x}}+\frac{\left (2 a \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{1+a x}}{x^{3/2} (1-a x)} \, dx}{\sqrt{1-a x}}\\ &=-\frac{4 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{\sqrt{1-a x}}-\frac{2 \sqrt{c-\frac{c}{a x}} (1+a x)^{3/2}}{3 x \sqrt{1-a x}}+\frac{\left (4 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{\sqrt{1-a x}}\\ &=-\frac{4 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{\sqrt{1-a x}}-\frac{2 \sqrt{c-\frac{c}{a x}} (1+a x)^{3/2}}{3 x \sqrt{1-a x}}+\frac{\left (8 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{\sqrt{1-a x}}\\ &=-\frac{4 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{\sqrt{1-a x}}-\frac{2 \sqrt{c-\frac{c}{a x}} (1+a x)^{3/2}}{3 x \sqrt{1-a x}}+\frac{4 \sqrt{2} a^{3/2} \sqrt{c-\frac{c}{a x}} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{\sqrt{1-a x}}\\ \end{align*}
Mathematica [A] time = 0.0538605, size = 93, normalized size = 0.63 \[ \frac{2 \sqrt{c-\frac{c}{a x}} \left (6 \sqrt{2} a^{3/2} x^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )-\sqrt{a x+1} (7 a x+1)\right )}{3 x \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.154, size = 150, normalized size = 1. \begin{align*}{\frac{\sqrt{2}}{3\, \left ( ax-1 \right ) x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 7\,x\sqrt{- \left ( ax+1 \right ) x}a\sqrt{2}\sqrt{-{a}^{-1}}+6\,a\ln \left ({\frac{1}{ax-1} \left ( 2\,\sqrt{2}\sqrt{-{a}^{-1}}\sqrt{- \left ( ax+1 \right ) x}a-3\,ax-1 \right ) } \right ){x}^{2}+\sqrt{- \left ( ax+1 \right ) x}\sqrt{2}\sqrt{-{a}^{-1}} \right ){\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{-{a}^{-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 )}^{3} \sqrt{c - \frac{c}{a x}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.59657, size = 668, normalized size = 4.54 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left (a^{2} x^{2} - a x\right )} \sqrt{-c} \log \left (-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2}{\left (3 \, 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^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, \sqrt{-a^{2} x^{2} + 1}{\left (7 \, a x + 1\right )} \sqrt{\frac{a c x - c}{a x}}}{3 \,{\left (a x^{2} - x\right )}}, -\frac{2 \,{\left (3 \, \sqrt{2}{\left (a^{2} x^{2} - a x\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left (7 \, a x + 1\right )} \sqrt{\frac{a c x - c}{a x}}\right )}}{3 \,{\left (a x^{2} - x\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{\sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x + 1\right )^{3}}{x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, 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 )}^{3} \sqrt{c - \frac{c}{a x}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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