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 = 0.09, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 21
Rule 43
Rule 6130
Rule 6167
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.50, size = 32, normalized size = 0.80 \[ -\frac {2 \, {\left (3 \, a^{2} c x^{2} + 4 \, a c x - 7 \, c\right )} \sqrt {-a c x + c}}{15 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 71, normalized size = 1.78 \[ \frac {2 \, {\left (15 \, \sqrt {-a c x + c} c - \frac {3 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} - 10 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {-a c x + c} c^{2}}{c}\right )}}{15 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 21, normalized size = 0.52 \[ \frac {2 \left (-a c x +c \right )^{\frac {3}{2}} \left (3 a x +7\right )}{15 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 32, normalized size = 0.80 \[ -\frac {2 \, {\left (3 \, {\left (-a c x + c\right )}^{\frac {5}{2}} - 10 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c\right )}}{15 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 32, normalized size = 0.80 \[ \frac {4\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a}-\frac {2\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.43, size = 61, normalized size = 1.52 \[ \begin {cases} \frac {- c \left (\begin {cases} 0 & \text {for}\: c = 0 \\- \frac {2 \left (- a c x + c\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) - \frac {2 \left (- \frac {c \left (- a c x + c\right )^{\frac {3}{2}}}{3} + \frac {\left (- a c x + c\right )^{\frac {5}{2}}}{5}\right )}{c}}{a} & \text {for}\: a \neq 0 \\- c^{\frac {3}{2}} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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