Optimal. Leaf size=38 \[ \frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c} \]
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Rubi [A] time = 0.08, antiderivative size = 38, 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} \[ \frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c} \]
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)} \sqrt {c-a c x} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt {c-a c x} \, dx\\ &=-\int \frac {(1+a x) \sqrt {c-a c x}}{1-a x} \, dx\\ &=-\left (c \int \frac {1+a x}{\sqrt {c-a c x}} \, dx\right )\\ &=-\left (c \int \left (\frac {2}{\sqrt {c-a c x}}-\frac {\sqrt {c-a c x}}{c}\right ) \, dx\right )\\ &=\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 23, normalized size = 0.61 \[ \frac {2 (a x+5) \sqrt {c-a c x}}{3 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 19, normalized size = 0.50 \[ \frac {2 \, \sqrt {-a c x + c} {\left (a x + 5\right )}}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 44, normalized size = 1.16 \[ \frac {2 \, {\left (3 \, \sqrt {-a c x + c} - \frac {{\left (-a c x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {-a c x + c} c}{c}\right )}}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 20, normalized size = 0.53 \[ \frac {2 \sqrt {-a c x +c}\, \left (a x +5\right )}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 30, normalized size = 0.79 \[ -\frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} - 6 \, \sqrt {-a c x + c} c\right )}}{3 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 32, normalized size = 0.84 \[ \frac {4\,\sqrt {c-a\,c\,x}}{a}-\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.89, size = 31, normalized size = 0.82 \[ - \frac {2 \left (- 2 c \sqrt {- a c x + c} + \frac {\left (- a c x + c\right )^{\frac {3}{2}}}{3}\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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