Optimal. Leaf size=40 \[ \frac{4 (c-a c x)^{5/2}}{5 a}-\frac{2 (c-a c x)^{7/2}}{7 a c} \]
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Rubi [A] time = 0.0876669, 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} \[ \frac{4 (c-a c x)^{5/2}}{5 a}-\frac{2 (c-a c x)^{7/2}}{7 a c} \]
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)^{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\\ &=-\left (c \int (1+a x) (c-a c x)^{3/2} \, dx\right )\\ &=-\left (c \int \left (2 (c-a c x)^{3/2}-\frac{(c-a c x)^{5/2}}{c}\right ) \, dx\right )\\ &=\frac{4 (c-a c x)^{5/2}}{5 a}-\frac{2 (c-a c x)^{7/2}}{7 a c}\\ \end{align*}
Mathematica [A] time = 0.039523, size = 34, normalized size = 0.85 \[ \frac{2 c^2 (a x-1)^2 (5 a x+9) \sqrt{c-a c x}}{35 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 21, normalized size = 0.5 \begin{align*}{\frac{10\,ax+18}{35\,a} \left ( -acx+c \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00252, size = 43, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (5 \,{\left (-a c x + c\right )}^{\frac{7}{2}} - 14 \,{\left (-a c x + c\right )}^{\frac{5}{2}} c\right )}}{35 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60886, size = 103, normalized size = 2.58 \begin{align*} \frac{2 \,{\left (5 \, a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} - 13 \, a c^{2} x + 9 \, c^{2}\right )} \sqrt{-a c x + c}}{35 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.5798, size = 80, normalized size = 2. \begin{align*} \begin{cases} \frac{- c^{2} \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^{2} \left (- a c x + c\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (- a c x + c\right )^{\frac{5}{2}}}{5} + \frac{\left (- a c x + c\right )^{\frac{7}{2}}}{7}\right )}{c}}{a} & \text{for}\: a \neq 0 \\- c^{\frac{5}{2}} x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12242, size = 108, normalized size = 2.7 \begin{align*} \frac{2 \,{\left (35 \,{\left (-a c x + c\right )}^{\frac{3}{2}} c + \frac{15 \,{\left (a c x - c\right )}^{3} \sqrt{-a c x + c} + 42 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} c - 35 \,{\left (-a c x + c\right )}^{\frac{3}{2}} c^{2}}{c}\right )}}{105 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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