Optimal. Leaf size=97 \[ \frac{2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac{2 (c-a c x)^{3/2}}{3 a^2 c}+\frac{4 \sqrt{c-a c x}}{a^2}-\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a^2} \]
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Rubi [A] time = 0.170344, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6167, 6130, 21, 80, 50, 63, 206} \[ \frac{2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac{2 (c-a c x)^{3/2}}{3 a^2 c}+\frac{4 \sqrt{c-a c x}}{a^2}-\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 80
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} x \sqrt{c-a c x} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} x \sqrt{c-a c x} \, dx\\ &=-\int \frac{x (1-a x) \sqrt{c-a c x}}{1+a x} \, dx\\ &=-\frac{\int \frac{x (c-a c x)^{3/2}}{1+a x} \, dx}{c}\\ &=\frac{2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac{\int \frac{(c-a c x)^{3/2}}{1+a x} \, dx}{a c}\\ &=\frac{2 (c-a c x)^{3/2}}{3 a^2 c}+\frac{2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac{2 \int \frac{\sqrt{c-a c x}}{1+a x} \, dx}{a}\\ &=\frac{4 \sqrt{c-a c x}}{a^2}+\frac{2 (c-a c x)^{3/2}}{3 a^2 c}+\frac{2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac{(4 c) \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx}{a}\\ &=\frac{4 \sqrt{c-a c x}}{a^2}+\frac{2 (c-a c x)^{3/2}}{3 a^2 c}+\frac{2 (c-a c x)^{5/2}}{5 a^2 c^2}-\frac{8 \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )}{a^2}\\ &=\frac{4 \sqrt{c-a c x}}{a^2}+\frac{2 (c-a c x)^{3/2}}{3 a^2 c}+\frac{2 (c-a c x)^{5/2}}{5 a^2 c^2}-\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0818633, size = 70, normalized size = 0.72 \[ \frac{2 \left (3 a^2 x^2-11 a x+38\right ) \sqrt{c-a c x}-60 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{15 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 73, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{{a}^{2}{c}^{2}} \left ( 1/5\, \left ( -acx+c \right ) ^{5/2}+1/3\,c \left ( -acx+c \right ) ^{3/2}+2\,\sqrt{-acx+c}{c}^{2}-2\,{c}^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-acx+c}\sqrt{2}}{\sqrt{c}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64144, size = 366, normalized size = 3.77 \begin{align*} \left [\frac{2 \,{\left (15 \, \sqrt{2} \sqrt{c} \log \left (\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) +{\left (3 \, a^{2} x^{2} - 11 \, a x + 38\right )} \sqrt{-a c x + c}\right )}}{15 \, a^{2}}, \frac{2 \,{\left (30 \, \sqrt{2} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) +{\left (3 \, a^{2} x^{2} - 11 \, a x + 38\right )} \sqrt{-a c x + c}\right )}}{15 \, a^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.81473, size = 92, normalized size = 0.95 \begin{align*} \frac{2 \left (\frac{2 \sqrt{2} c^{3} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{\sqrt{- c}} + 2 c^{2} \sqrt{- a c x + c} + \frac{c \left (- a c x + c\right )^{\frac{3}{2}}}{3} + \frac{\left (- a c x + c\right )^{\frac{5}{2}}}{5}\right )}{a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12735, size = 142, normalized size = 1.46 \begin{align*} \frac{4 \, \sqrt{2} c \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{a^{2} \sqrt{-c}} + \frac{2 \,{\left (3 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} a^{8} c^{8} + 5 \,{\left (-a c x + c\right )}^{\frac{3}{2}} a^{8} c^{9} + 30 \, \sqrt{-a c x + c} a^{8} c^{10}\right )}}{15 \, a^{10} c^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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