Optimal. Leaf size=97 \[ -\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac{2 (c-a c x)^{3/2}}{3 a^3 c}-\frac{4 \sqrt{c-a c x}}{a^3}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a^3} \]
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Rubi [A] time = 0.248465, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6167, 6130, 21, 88, 50, 63, 206} \[ -\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac{2 (c-a c x)^{3/2}}{3 a^3 c}-\frac{4 \sqrt{c-a c x}}{a^3}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a^3} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 88
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx\\ &=-\int \frac{x^2 (1-a x) \sqrt{c-a c x}}{1+a x} \, dx\\ &=-\frac{\int \frac{x^2 (c-a c x)^{3/2}}{1+a x} \, dx}{c}\\ &=-\frac{\int \left (\frac{(c-a c x)^{3/2}}{a^2 (1+a x)}-\frac{(c-a c x)^{5/2}}{a^2 c}\right ) \, dx}{c}\\ &=-\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac{\int \frac{(c-a c x)^{3/2}}{1+a x} \, dx}{a^2 c}\\ &=-\frac{2 (c-a c x)^{3/2}}{3 a^3 c}-\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac{2 \int \frac{\sqrt{c-a c x}}{1+a x} \, dx}{a^2}\\ &=-\frac{4 \sqrt{c-a c x}}{a^3}-\frac{2 (c-a c x)^{3/2}}{3 a^3 c}-\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac{(4 c) \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx}{a^2}\\ &=-\frac{4 \sqrt{c-a c x}}{a^3}-\frac{2 (c-a c x)^{3/2}}{3 a^3 c}-\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac{8 \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )}{a^3}\\ &=-\frac{4 \sqrt{c-a c x}}{a^3}-\frac{2 (c-a c x)^{3/2}}{3 a^3 c}-\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0899396, size = 78, normalized size = 0.8 \[ \frac{2 \left (3 a^3 x^3-9 a^2 x^2+16 a x-52\right ) \sqrt{c-a c x}+84 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{21 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 75, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{{c}^{3}{a}^{3}} \left ( 1/7\, \left ( -acx+c \right ) ^{7/2}+1/3\, \left ( -acx+c \right ) ^{3/2}{c}^{2}+2\,\sqrt{-acx+c}{c}^{3}-2\,{c}^{7/2}\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 = 1.7128, size = 400, normalized size = 4.12 \begin{align*} \left [\frac{2 \,{\left (21 \, \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^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right )} \sqrt{-a c x + c}\right )}}{21 \, a^{3}}, -\frac{2 \,{\left (42 \, \sqrt{2} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) -{\left (3 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 16 \, a x - 52\right )} \sqrt{-a c x + c}\right )}}{21 \, a^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.76929, size = 95, normalized size = 0.98 \begin{align*} - \frac{2 \left (\frac{2 \sqrt{2} c^{4} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{\sqrt{- c}} + 2 c^{3} \sqrt{- a c x + c} + \frac{c^{2} \left (- a c x + c\right )^{\frac{3}{2}}}{3} + \frac{\left (- a c x + c\right )^{\frac{7}{2}}}{7}\right )}{a^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14079, 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^{3} \sqrt{-c}} + \frac{2 \,{\left (3 \,{\left (a c x - c\right )}^{3} \sqrt{-a c x + c} a^{18} c^{18} - 7 \,{\left (-a c x + c\right )}^{\frac{3}{2}} a^{18} c^{20} - 42 \, \sqrt{-a c x + c} a^{18} c^{21}\right )}}{21 \, a^{21} c^{21}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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