Optimal. Leaf size=155 \[ -\frac{x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x}}-\frac{5 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a} \sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{a} \sqrt{1-a x}} \]
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Rubi [A] time = 0.15448, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6134, 6129, 102, 157, 54, 215, 93, 206} \[ -\frac{x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-a x}}-\frac{5 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a} \sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{a} \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 102
Rule 157
Rule 54
Rule 215
Rule 93
Rule 206
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{1-a x}}{\sqrt{x}} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{(1+a x)^{3/2}}{\sqrt{x} (1-a x)} \, dx}{\sqrt{1-a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{\sqrt{1-a x}}-\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{-\frac{3 a}{2}-\frac{5 a^2 x}{2}}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{a \sqrt{1-a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{\sqrt{1-a x}}-\frac{\left (5 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 \sqrt{1-a x}}+\frac{\left (4 \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{\sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{\sqrt{1-a x}}-\frac{\left (5 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{\sqrt{1-a x}}+\frac{\left (8 \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{\sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{\sqrt{1-a x}}-\frac{5 \sqrt{c-\frac{c}{a x}} \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a} \sqrt{1-a x}}+\frac{4 \sqrt{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{a} \sqrt{1-a x}}\\ \end{align*}
Mathematica [A] time = 0.0742003, size = 105, normalized size = 0.68 \[ -\frac{\sqrt{x} \sqrt{c-\frac{c}{a x}} \left (\sqrt{a} \sqrt{x} \sqrt{a x+1}+5 \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )-4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )\right )}{\sqrt{a} \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.137, size = 165, normalized size = 1.1 \begin{align*}{\frac{x\sqrt{2}}{4\,ax-4}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}-5\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}+8\,\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 ){\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}{a}^{-{\frac{3}{2}}}{\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}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.57575, size = 975, normalized size = 6.29 \begin{align*} \left [\frac{4 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} + 4 \, \sqrt{2}{\left (a x - 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 ) + 5 \,{\left (a x - 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 )}{4 \,{\left (a^{2} x - a\right )}}, \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a x \sqrt{\frac{a c x - c}{a x}} - 4 \, \sqrt{2}{\left (a x - 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 ) + 5 \,{\left (a x - 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 )}{2 \,{\left (a^{2} x - a\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}}{\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}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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