Optimal. Leaf size=27 \[ \frac{2}{27} (2-3 x)^{3/2}-\frac{4}{9} \sqrt{2-3 x} \]
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Rubi [A] time = 0.0046574, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{2}{27} (2-3 x)^{3/2}-\frac{4}{9} \sqrt{2-3 x} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{2-3 x}} \, dx &=\int \left (\frac{2}{3 \sqrt{2-3 x}}-\frac{1}{3} \sqrt{2-3 x}\right ) \, dx\\ &=-\frac{4}{9} \sqrt{2-3 x}+\frac{2}{27} (2-3 x)^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0049493, size = 18, normalized size = 0.67 \[ -\frac{2}{27} \sqrt{2-3 x} (3 x+4) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 15, normalized size = 0.6 \begin{align*} -{\frac{6\,x+8}{27}\sqrt{2-3\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947613, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{27} \,{\left (-3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{9} \, \sqrt{-3 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.416598, size = 43, normalized size = 1.59 \begin{align*} -\frac{2}{27} \,{\left (3 \, x + 4\right )} \sqrt{-3 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.82036, size = 61, normalized size = 2.26 \begin{align*} \begin{cases} - \frac{2 i x \sqrt{3 x - 2}}{9} - \frac{8 i \sqrt{3 x - 2}}{27} & \text{for}\: \frac{3 \left |{x}\right |}{2} > 1 \\- \frac{2 x \sqrt{2 - 3 x}}{9} - \frac{8 \sqrt{2 - 3 x}}{27} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06409, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{27} \,{\left (-3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{9} \, \sqrt{-3 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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