Optimal. Leaf size=116 \[ -\frac{16 c^2 \sqrt{c-a c x}}{a}+\frac{16 \sqrt{2} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a}-\frac{2 (c-a c x)^{7/2}}{7 a c}-\frac{4 (c-a c x)^{5/2}}{5 a}-\frac{8 c (c-a c x)^{3/2}}{3 a} \]
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Rubi [A] time = 0.131632, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6167, 6130, 21, 50, 63, 206} \[ -\frac{16 c^2 \sqrt{c-a c x}}{a}+\frac{16 \sqrt{2} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a}-\frac{2 (c-a c x)^{7/2}}{7 a c}-\frac{4 (c-a c x)^{5/2}}{5 a}-\frac{8 c (c-a c x)^{3/2}}{3 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^{5/2} \, dx\\ &=-\int \frac{(1-a x) (c-a c x)^{5/2}}{1+a x} \, dx\\ &=-\frac{\int \frac{(c-a c x)^{7/2}}{1+a x} \, dx}{c}\\ &=-\frac{2 (c-a c x)^{7/2}}{7 a c}-2 \int \frac{(c-a c x)^{5/2}}{1+a x} \, dx\\ &=-\frac{4 (c-a c x)^{5/2}}{5 a}-\frac{2 (c-a c x)^{7/2}}{7 a c}-(4 c) \int \frac{(c-a c x)^{3/2}}{1+a x} \, dx\\ &=-\frac{8 c (c-a c x)^{3/2}}{3 a}-\frac{4 (c-a c x)^{5/2}}{5 a}-\frac{2 (c-a c x)^{7/2}}{7 a c}-\left (8 c^2\right ) \int \frac{\sqrt{c-a c x}}{1+a x} \, dx\\ &=-\frac{16 c^2 \sqrt{c-a c x}}{a}-\frac{8 c (c-a c x)^{3/2}}{3 a}-\frac{4 (c-a c x)^{5/2}}{5 a}-\frac{2 (c-a c x)^{7/2}}{7 a c}-\left (16 c^3\right ) \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx\\ &=-\frac{16 c^2 \sqrt{c-a c x}}{a}-\frac{8 c (c-a c x)^{3/2}}{3 a}-\frac{4 (c-a c x)^{5/2}}{5 a}-\frac{2 (c-a c x)^{7/2}}{7 a c}+\frac{\left (32 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )}{a}\\ &=-\frac{16 c^2 \sqrt{c-a c x}}{a}-\frac{8 c (c-a c x)^{3/2}}{3 a}-\frac{4 (c-a c x)^{5/2}}{5 a}-\frac{2 (c-a c x)^{7/2}}{7 a c}+\frac{16 \sqrt{2} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0592854, size = 80, normalized size = 0.69 \[ \frac{2 c^2 \left (\left (15 a^3 x^3-87 a^2 x^2+269 a x-1037\right ) \sqrt{c-a c x}+840 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )\right )}{105 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 87, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{ac} \left ( 1/7\, \left ( -acx+c \right ) ^{7/2}+2/5\,c \left ( -acx+c \right ) ^{5/2}+4/3\, \left ( -acx+c \right ) ^{3/2}{c}^{2}+8\,\sqrt{-acx+c}{c}^{3}-8\,{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 = 1.77967, size = 462, normalized size = 3.98 \begin{align*} \left [\frac{2 \,{\left (420 \, \sqrt{2} c^{\frac{5}{2}} \log \left (\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) +{\left (15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right )} \sqrt{-a c x + c}\right )}}{105 \, a}, -\frac{2 \,{\left (840 \, \sqrt{2} \sqrt{-c} c^{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) -{\left (15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right )} \sqrt{-a c x + c}\right )}}{105 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 84.89, size = 110, normalized size = 0.95 \begin{align*} - \frac{16 \sqrt{2} c^{3} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{a \sqrt{- c}} - \frac{16 c^{2} \sqrt{- a c x + c}}{a} - \frac{8 c \left (- a c x + c\right )^{\frac{3}{2}}}{3 a} - \frac{4 \left (- a c x + c\right )^{\frac{5}{2}}}{5 a} - \frac{2 \left (- a c x + c\right )^{\frac{7}{2}}}{7 a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13706, size = 181, normalized size = 1.56 \begin{align*} -\frac{16 \, \sqrt{2} c^{3} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{a \sqrt{-c}} + \frac{2 \,{\left (15 \,{\left (a c x - c\right )}^{3} \sqrt{-a c x + c} a^{6} c^{6} - 42 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} a^{6} c^{7} - 140 \,{\left (-a c x + c\right )}^{\frac{3}{2}} a^{6} c^{8} - 840 \, \sqrt{-a c x + c} a^{6} c^{9}\right )}}{105 \, a^{7} c^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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