Optimal. Leaf size=29 \[ \frac{2 (a x+1) \sqrt{c-a c x} e^{\coth ^{-1}(a x)}}{3 a} \]
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Rubi [A] time = 0.0341808, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {6174} \[ \frac{2 (a x+1) \sqrt{c-a c x} e^{\coth ^{-1}(a x)}}{3 a} \]
Antiderivative was successfully verified.
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Rule 6174
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \sqrt{c-a c x} \, dx &=\frac{2 e^{\coth ^{-1}(a x)} (1+a x) \sqrt{c-a c x}}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0198929, size = 43, normalized size = 1.48 \[ \frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.04, size = 35, normalized size = 1.2 \begin{align*}{\frac{2\,ax+2}{3\,a}\sqrt{-acx+c}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06179, size = 35, normalized size = 1.21 \begin{align*} \frac{2 \,{\left (a \sqrt{-c} x + \sqrt{-c}\right )} \sqrt{a x + 1}}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51021, size = 111, normalized size = 3.83 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{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 )}}{\sqrt{\frac{a x - 1}{a x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16605, size = 66, normalized size = 2.28 \begin{align*} \frac{2 \,{\left (\frac{2 \, \sqrt{2} \sqrt{-c} c}{\mathrm{sgn}\left (c\right )} - \frac{{\left (-a c x - c\right )}^{\frac{3}{2}}}{\mathrm{sgn}\left (-a c x - c\right )}\right )}}{3 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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