Integrand size = 17, antiderivative size = 19 \[ \int \sqrt {-x} \left (\sqrt {-x}+x\right ) \, dx=\frac {2}{5} (-x)^{5/2}-\frac {x^2}{2} \]
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Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14} \[ \int \sqrt {-x} \left (\sqrt {-x}+x\right ) \, dx=\frac {2}{5} (-x)^{5/2}-\frac {x^2}{2} \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (-(-x)^{3/2}-x\right ) \, dx \\ & = \frac {2}{5} (-x)^{5/2}-\frac {x^2}{2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \sqrt {-x} \left (\sqrt {-x}+x\right ) \, dx=\frac {1}{10} \left (-5+4 \sqrt {-x}\right ) x^2 \]
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Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {2 \left (-x \right )^{\frac {5}{2}}}{5}-\frac {x^{2}}{2}\) | \(14\) |
default | \(\frac {2 \left (-x \right )^{\frac {5}{2}}}{5}-\frac {x^{2}}{2}\) | \(14\) |
trager | \(-\frac {\left (x -1\right ) \left (x +1\right )}{2}+\frac {2 x^{2} \sqrt {-x}}{5}\) | \(20\) |
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none
Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \sqrt {-x} \left (\sqrt {-x}+x\right ) \, dx=\frac {2}{5} \, \sqrt {-x} x^{2} - \frac {1}{2} \, x^{2} \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \sqrt {-x} \left (\sqrt {-x}+x\right ) \, dx=\frac {2 x^{2} \sqrt {- x}}{5} - \frac {x^{2}}{2} \]
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none
Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \sqrt {-x} \left (\sqrt {-x}+x\right ) \, dx=\frac {2}{5} \, \left (-x\right )^{\frac {5}{2}} - \frac {1}{2} \, x^{2} \]
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none
Time = 0.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \sqrt {-x} \left (\sqrt {-x}+x\right ) \, dx=\frac {2}{5} \, \sqrt {-x} x^{2} - \frac {1}{2} \, x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \sqrt {-x} \left (\sqrt {-x}+x\right ) \, dx=\frac {2\,{\left (-x\right )}^{5/2}}{5}-\frac {x^2}{2} \]
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