Optimal. Leaf size=35 \[ \frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.0451377, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6127, 649} \[ \frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 649
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} \sqrt{c-a c x} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}} \, dx\\ &=\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.012753, size = 37, normalized size = 1.06 \[ \frac{2 (a x+1)^{3/2} \sqrt{c-a c x}}{3 a \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 34, normalized size = 1. \begin{align*}{\frac{2\, \left ( ax+1 \right ) ^{2}}{3\,a}\sqrt{-acx+c}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01662, size = 78, normalized size = 2.23 \begin{align*} \frac{2 \,{\left (a^{2} \sqrt{c} x^{2} - a \sqrt{c} x - 2 \, \sqrt{c}\right )}}{3 \, \sqrt{a x + 1} a} + \frac{2 \,{\left (a \sqrt{c} x + \sqrt{c}\right )}}{\sqrt{a x + 1} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78012, size = 86, normalized size = 2.46 \begin{align*} -\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x + 1\right )}}{3 \,{\left (a^{2} x - a\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 )}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35827, size = 43, normalized size = 1.23 \begin{align*} -\frac{2 \,{\left (2 \, \sqrt{2} \sqrt{c} - \frac{{\left (a c x + c\right )}^{\frac{3}{2}}}{c}\right )} c}{3 \, a{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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