Integrand size = 8, antiderivative size = 21 \[ \int \sqrt {x} \log (x) \, dx=-\frac {4 x^{3/2}}{9}+\frac {2}{3} x^{3/2} \log (x) \]
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Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2341} \[ \int \sqrt {x} \log (x) \, dx=\frac {2}{3} x^{3/2} \log (x)-\frac {4 x^{3/2}}{9} \]
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Rule 2341
Rubi steps \begin{align*} \text {integral}& = -\frac {4 x^{3/2}}{9}+\frac {2}{3} x^{3/2} \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \sqrt {x} \log (x) \, dx=\frac {2}{9} x^{3/2} (-2+3 \log (x)) \]
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Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(-\frac {4 x^{\frac {3}{2}}}{9}+\frac {2 x^{\frac {3}{2}} \ln \left (x \right )}{3}\) | \(14\) |
default | \(-\frac {4 x^{\frac {3}{2}}}{9}+\frac {2 x^{\frac {3}{2}} \ln \left (x \right )}{3}\) | \(14\) |
risch | \(-\frac {4 x^{\frac {3}{2}}}{9}+\frac {2 x^{\frac {3}{2}} \ln \left (x \right )}{3}\) | \(14\) |
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \sqrt {x} \log (x) \, dx=\frac {2}{9} \, {\left (3 \, x \log \left (x\right ) - 2 \, x\right )} \sqrt {x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (19) = 38\).
Time = 0.93 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.00 \[ \int \sqrt {x} \log (x) \, dx=\begin {cases} - \frac {2 x^{\frac {3}{2}} \log {\left (\frac {1}{x} \right )}}{3} + \frac {2 x^{\frac {3}{2}} \log {\left (x \right )}}{3} - \frac {8 x^{\frac {3}{2}}}{9} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\frac {2 x^{\frac {3}{2}} \log {\left (x \right )}}{3} - \frac {4 x^{\frac {3}{2}}}{9} & \text {for}\: \left |{x}\right | < 1 \\- \frac {2 x^{\frac {3}{2}} \log {\left (\frac {1}{x} \right )}}{3} - \frac {4 x^{\frac {3}{2}}}{9} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{3, 3}^{2, 1}\left (\begin {matrix} 1 & \frac {5}{2}, \frac {5}{2} \\\frac {3}{2}, \frac {3}{2} & 0 \end {matrix} \middle | {x} \right )} + {G_{3, 3}^{0, 3}\left (\begin {matrix} \frac {5}{2}, \frac {5}{2}, 1 & \\ & \frac {3}{2}, \frac {3}{2}, 0 \end {matrix} \middle | {x} \right )} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt {x} \log (x) \, dx=\frac {2}{3} \, x^{\frac {3}{2}} \log \left (x\right ) - \frac {4}{9} \, x^{\frac {3}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt {x} \log (x) \, dx=\frac {2}{3} \, x^{\frac {3}{2}} \log \left (x\right ) - \frac {4}{9} \, x^{\frac {3}{2}} \]
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Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.43 \[ \int \sqrt {x} \log (x) \, dx=\frac {2\,x^{3/2}\,\left (\ln \left (x\right )-\frac {2}{3}\right )}{3} \]
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