Optimal. Leaf size=199 \[ -\frac{2 \sqrt{a x+1} (1-a x)^{5/2}}{a^3 x^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{(1-a x)^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{(1-a x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{2} a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{(1-a x)^{5/2}}{a^2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{5/2}} \]
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Rubi [A] time = 0.205231, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6134, 6129, 98, 154, 157, 54, 215, 93, 206} \[ -\frac{2 \sqrt{a x+1} (1-a x)^{5/2}}{a^3 x^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{(1-a x)^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{(1-a x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{2} a^{7/2} x^{5/2} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{(1-a x)^{5/2}}{a^2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 98
Rule 154
Rule 157
Rule 54
Rule 215
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{5/2}} \, dx &=\frac{(1-a x)^{5/2} \int \frac{e^{-3 \tanh ^{-1}(a x)} x^{5/2}}{(1-a x)^{5/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{5/2} \int \frac{x^{5/2}}{(1-a x) (1+a x)^{3/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{5/2}}{a^2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}-\frac{(1-a x)^{5/2} \int \frac{\sqrt{x} \left (\frac{3}{2}-2 a x\right )}{(1-a x) \sqrt{1+a x}} \, dx}{a^2 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{5/2}}{a^2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}-\frac{2 (1-a x)^{5/2} \sqrt{1+a x}}{a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{(1-a x)^{5/2} \int \frac{a-\frac{a^2 x}{2}}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{a^4 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{5/2}}{a^2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}-\frac{2 (1-a x)^{5/2} \sqrt{1+a x}}{a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{(1-a x)^{5/2} \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}+\frac{(1-a x)^{5/2} \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{2 a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{5/2}}{a^2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}-\frac{2 (1-a x)^{5/2} \sqrt{1+a x}}{a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{(1-a x)^{5/2} \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}+\frac{(1-a x)^{5/2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac{(1-a x)^{5/2}}{a^2 \left (c-\frac{c}{a x}\right )^{5/2} x \sqrt{1+a x}}-\frac{2 (1-a x)^{5/2} \sqrt{1+a x}}{a^3 \left (c-\frac{c}{a x}\right )^{5/2} x^2}+\frac{(1-a x)^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{a^{7/2} \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}+\frac{(1-a x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{\sqrt{2} a^{7/2} \left (c-\frac{c}{a x}\right )^{5/2} x^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.153795, size = 162, normalized size = 0.81 \[ \frac{\sqrt{1-a x} \left (5 \left (2 \sqrt{a} \sqrt{x}-4 \sqrt{a x+1} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )+\sqrt{2 a x+2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )\right )-4 a^{5/2} x^{5/2} \sqrt{a x+1} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{5}{2},\frac{7}{2},-a x\right )\right )}{10 a^{3/2} c^2 \sqrt{x} \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.147, size = 279, normalized size = 1.4 \begin{align*} -{\frac{x\sqrt{2}}{4\,{c}^{3} \left ( ax+1 \right ) \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,\sqrt{- \left ( ax+1 \right ) x}{a}^{5/2}\sqrt{2}\sqrt{-{a}^{-1}}x+{a}^{2}\arctan \left ({\frac{2\,ax+1}{2}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \right ) \sqrt{2}\sqrt{-{a}^{-1}}x+4\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}+\arctan \left ({\frac{2\,ax+1}{2}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}+{a}^{{\frac{3}{2}}}\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+\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 ) \sqrt{a} \right ) \sqrt{-{a}^{2}{x}^{2}+1}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-{a}^{-1}}}}{\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 (a x + 1\right )}^{3}{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.65381, size = 1057, normalized size = 5.31 \begin{align*} \left [-\frac{\sqrt{2}{\left (a^{2} x^{2} - 1\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 \,{\left (a^{2} x^{2} - 1\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 ) + 8 \,{\left (a^{2} x^{2} + 2 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{8 \,{\left (a^{3} c^{3} x^{2} - a c^{3}\right )}}, \frac{\sqrt{2}{\left (a^{2} x^{2} - 1\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 ) + 2 \,{\left (a^{2} x^{2} - 1\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 ) - 4 \,{\left (a^{2} x^{2} + 2 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{4 \,{\left (a^{3} c^{3} x^{2} - a c^{3}\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 (a x + 1\right )}^{3}{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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