Optimal. Leaf size=40 \[ \frac{4 (c-a c x)^{3/2}}{3 a}-\frac{2 (c-a c x)^{5/2}}{5 a c} \]
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Rubi [A] time = 0.0891647, antiderivative size = 40, 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} \[ \frac{4 (c-a c x)^{3/2}}{3 a}-\frac{2 (c-a c x)^{5/2}}{5 a c} \]
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)} (c-a c x)^{3/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^{3/2} \, dx\\ &=-\int \frac{(1+a x) (c-a c x)^{3/2}}{1-a x} \, dx\\ &=-\left (c \int (1+a x) \sqrt{c-a c x} \, dx\right )\\ &=-\left (c \int \left (2 \sqrt{c-a c x}-\frac{(c-a c x)^{3/2}}{c}\right ) \, dx\right )\\ &=\frac{4 (c-a c x)^{3/2}}{3 a}-\frac{2 (c-a c x)^{5/2}}{5 a c}\\ \end{align*}
Mathematica [A] time = 0.0301593, size = 30, normalized size = 0.75 \[ -\frac{2 c (a x-1) (3 a x+7) \sqrt{c-a c x}}{15 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 21, normalized size = 0.5 \begin{align*}{\frac{6\,ax+14}{15\,a} \left ( -acx+c \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01969, size = 43, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (3 \,{\left (-a c x + c\right )}^{\frac{5}{2}} - 10 \,{\left (-a c x + c\right )}^{\frac{3}{2}} c\right )}}{15 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50783, size = 76, normalized size = 1.9 \begin{align*} -\frac{2 \,{\left (3 \, a^{2} c x^{2} + 4 \, a c x - 7 \, c\right )} \sqrt{-a c x + c}}{15 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.8728, size = 61, normalized size = 1.52 \begin{align*} \begin{cases} - \frac{c \left (\begin{cases} 0 & \text{for}\: c = 0 \\- \frac{2 \left (- a c x + c\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + \frac{2 \left (- \frac{c \left (- a c x + c\right )^{\frac{3}{2}}}{3} + \frac{\left (- a c x + c\right )^{\frac{5}{2}}}{5}\right )}{c}}{a} & \text{for}\: a \neq 0 \\- c^{\frac{3}{2}} x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10256, size = 76, normalized size = 1.9 \begin{align*} \frac{2 \,{\left (5 \,{\left (-a c x + c\right )}^{\frac{3}{2}} - \frac{3 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} - 5 \,{\left (-a c x + c\right )}^{\frac{3}{2}} c}{c}\right )}}{15 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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