Optimal. Leaf size=27 \[ -\frac {4}{9} \sqrt {2-3 x}+\frac {2}{27} (2-3 x)^{3/2} \]
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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {45}
\begin {gather*} \frac {2}{27} (2-3 x)^{3/2}-\frac {4}{9} \sqrt {2-3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
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*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} -\frac {2}{27} \sqrt {2-3 x} (4+3 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 20, normalized size = 0.74
method | result | size |
trager | \(\left (-\frac {2 x}{9}-\frac {8}{27}\right ) \sqrt {2-3 x}\) | \(14\) |
gosper | \(-\frac {2 \left (3 x +4\right ) \sqrt {2-3 x}}{27}\) | \(15\) |
derivativedivides | \(\frac {2 \left (2-3 x \right )^{\frac {3}{2}}}{27}-\frac {4 \sqrt {2-3 x}}{9}\) | \(20\) |
default | \(\frac {2 \left (2-3 x \right )^{\frac {3}{2}}}{27}-\frac {4 \sqrt {2-3 x}}{9}\) | \(20\) |
risch | \(\frac {2 \left (-2+3 x \right ) \left (3 x +4\right )}{27 \sqrt {2-3 x}}\) | \(20\) |
meijerg | \(\frac {2 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (6 x +8\right ) \sqrt {1-\frac {3 x}{2}}}{6}\right )}{9 \sqrt {\pi }}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.82, size = 19, normalized size = 0.70 \begin {gather*} \frac {2}{27} \, {\left (-3 \, x + 2\right )}^{\frac {3}{2}} - \frac {4}{9} \, \sqrt {-3 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 14, normalized size = 0.52 \begin {gather*} -\frac {2}{27} \, {\left (3 \, x + 4\right )} \sqrt {-3 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.47, size = 60, normalized size = 2.22 \begin {gather*} \begin {cases} - \frac {2 i x \sqrt {3 x - 2}}{9} - \frac {8 i \sqrt {3 x - 2}}{27} & \text {for}\: \left |{x}\right | > \frac {2}{3} \\- \frac {2 x \sqrt {2 - 3 x}}{9} - \frac {8 \sqrt {2 - 3 x}}{27} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.72, size = 19, normalized size = 0.70 \begin {gather*} \frac {2}{27} \, {\left (-3 \, x + 2\right )}^{\frac {3}{2}} - \frac {4}{9} \, \sqrt {-3 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 14, normalized size = 0.52 \begin {gather*} -\frac {2\,\sqrt {2-3\,x}\,\left (3\,x+4\right )}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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