Optimal. Leaf size=133 \[ -\frac{a x^2 (23-a x) \left (c-\frac{c}{a x}\right )^{3/2}}{(1-a x)^{3/2} \sqrt{a x+1}}+\frac{9 \sqrt{a} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{3/2}}-\frac{2 x \sqrt{1-a x} \left (c-\frac{c}{a x}\right )^{3/2}}{\sqrt{a x+1}} \]
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Rubi [A] time = 0.175303, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6134, 6129, 98, 143, 54, 215} \[ -\frac{a x^2 (23-a x) \left (c-\frac{c}{a x}\right )^{3/2}}{(1-a x)^{3/2} \sqrt{a x+1}}+\frac{9 \sqrt{a} x^{3/2} \left (c-\frac{c}{a x}\right )^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{3/2}}-\frac{2 x \sqrt{1-a x} \left (c-\frac{c}{a x}\right )^{3/2}}{\sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 98
Rule 143
Rule 54
Rule 215
Rubi steps
\begin{align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{3/2} \, dx &=\frac{\left (\left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac{e^{-3 \tanh ^{-1}(a x)} (1-a x)^{3/2}}{x^{3/2}} \, dx}{(1-a x)^{3/2}}\\ &=\frac{\left (\left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac{(1-a x)^3}{x^{3/2} (1+a x)^{3/2}} \, dx}{(1-a x)^{3/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{3/2} x \sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\left (2 \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac{(1-a x) \left (\frac{7 a}{2}+\frac{a^2 x}{2}\right )}{\sqrt{x} (1+a x)^{3/2}} \, dx}{(1-a x)^{3/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{3/2} x \sqrt{1-a x}}{\sqrt{1+a x}}-\frac{a \left (c-\frac{c}{a x}\right )^{3/2} x^2 (23-a x)}{(1-a x)^{3/2} \sqrt{1+a x}}+\frac{\left (9 a \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 (1-a x)^{3/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{3/2} x \sqrt{1-a x}}{\sqrt{1+a x}}-\frac{a \left (c-\frac{c}{a x}\right )^{3/2} x^2 (23-a x)}{(1-a x)^{3/2} \sqrt{1+a x}}+\frac{\left (9 a \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{(1-a x)^{3/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{3/2} x \sqrt{1-a x}}{\sqrt{1+a x}}-\frac{a \left (c-\frac{c}{a x}\right )^{3/2} x^2 (23-a x)}{(1-a x)^{3/2} \sqrt{1+a x}}+\frac{9 \sqrt{a} \left (c-\frac{c}{a x}\right )^{3/2} x^{3/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0512834, size = 80, normalized size = 0.6 \[ \frac{c \sqrt{c-\frac{c}{a x}} \left (a^2 x^2+19 a x-9 \sqrt{a} \sqrt{x} \sqrt{a x+1} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )+2\right )}{a \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.157, size = 164, normalized size = 1.2 \begin{align*} -{\frac{c}{ \left ( 2\,ax+2 \right ) \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}+9\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ){x}^{2}{a}^{2}+38\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}+9\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) xa+4\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x} \right ) \sqrt{-{a}^{2}{x}^{2}+1}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \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^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12749, size = 621, normalized size = 4.67 \begin{align*} \left [\frac{9 \,{\left (a^{2} c x^{2} - c\right )} \sqrt{-c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, 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 x - 1}\right ) - 4 \,{\left (a^{2} c x^{2} + 19 \, a c x + 2 \, c\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \,{\left (a^{3} x^{2} - a\right )}}, \frac{9 \,{\left (a^{2} c x^{2} - c\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \,{\left (a^{2} c x^{2} + 19 \, a c x + 2 \, c\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a^{3} x^{2} - a\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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