Optimal. Leaf size=38 \[ \frac{4 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{3/2}}{3 a c} \]
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Rubi [A] time = 0.0813062, antiderivative size = 38, 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 = {6167, 6130, 21, 43} \[ \frac{4 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{3/2}}{3 a c} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 43
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt{c-a c x} \, dx\\ &=-\int \frac{(1+a x) \sqrt{c-a c x}}{1-a x} \, dx\\ &=-\left (c \int \frac{1+a x}{\sqrt{c-a c x}} \, dx\right )\\ &=-\left (c \int \left (\frac{2}{\sqrt{c-a c x}}-\frac{\sqrt{c-a c x}}{c}\right ) \, dx\right )\\ &=\frac{4 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{3/2}}{3 a c}\\ \end{align*}
Mathematica [A] time = 0.0242868, size = 23, normalized size = 0.61 \[ \frac{2 (a x+5) \sqrt{c-a c x}}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 20, normalized size = 0.5 \begin{align*}{\frac{2\,ax+10}{3\,a}\sqrt{-acx+c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00853, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left ({\left (-a c x + c\right )}^{\frac{3}{2}} - 6 \, \sqrt{-a c x + c} c\right )}}{3 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52457, size = 46, normalized size = 1.21 \begin{align*} \frac{2 \, \sqrt{-a c x + c}{\left (a x + 5\right )}}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.19414, size = 31, normalized size = 0.82 \begin{align*} - \frac{2 \left (- 2 c \sqrt{- a c x + c} + \frac{\left (- a c x + c\right )^{\frac{3}{2}}}{3}\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11735, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left ({\left (-a c x + c\right )}^{\frac{3}{2}} - 6 \, \sqrt{-a c x + c} c\right )}}{3 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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