Optimal. Leaf size=119 \[ -\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}-\frac{4 c \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a} \]
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Rubi [A] time = 0.114404, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6127, 665, 661, 208} \[ -\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}-\frac{4 c \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 665
Rule 661
Rule 208
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \sqrt{c-a c x} \, dx &=c^3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{5/2}} \, dx\\ &=-\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}+\left (2 c^2\right ) \int \frac{\sqrt{1-a^2 x^2}}{(c-a c x)^{3/2}} \, dx\\ &=-\frac{4 c \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}-\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}+(4 c) \int \frac{1}{\sqrt{c-a c x} \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{4 c \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}-\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}-\left (8 a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )\\ &=-\frac{4 c \sqrt{1-a^2 x^2}}{a \sqrt{c-a c x}}-\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0316262, size = 67, normalized size = 0.56 \[ -\frac{2 \sqrt{c-a c x} \left (\sqrt{a x+1} (a x+7)-6 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )\right )}{3 a \sqrt{1-a x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.108, size = 95, normalized size = 0.8 \begin{align*} -{\frac{2}{ \left ( 3\,ax-3 \right ) a}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( 6\,\sqrt{c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -xa\sqrt{c \left ( ax+1 \right ) }-7\,\sqrt{c \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}} \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{\sqrt{-a c x + c}{\left (a x + 1\right )}^{3}}{{\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 = 1.78455, size = 518, normalized size = 4.35 \begin{align*} \left [\frac{2 \,{\left (3 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x + 7\right )}\right )}}{3 \,{\left (a^{2} x - a\right )}}, \frac{2 \,{\left (6 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x + 7\right )}\right )}}{3 \,{\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 (a x - 1\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 [A] time = 1.18129, size = 142, normalized size = 1.19 \begin{align*} -\frac{2 \,{\left (\frac{6 \, \sqrt{2} c^{2} \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} +{\left (a c x + c\right )}^{\frac{3}{2}} + 6 \, \sqrt{a c x + c} c\right )}}{3 \, a{\left | c \right |}} + \frac{4 \, \sqrt{2}{\left (3 \, c^{2} \arctan \left (\frac{\sqrt{c}}{\sqrt{-c}}\right ) + 4 \, \sqrt{-c} c^{\frac{3}{2}}\right )}}{3 \, a \sqrt{-c}{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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