Optimal. Leaf size=19 \[ \frac {2}{5} (-x)^{5/2}-\frac {x^2}{2} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14}
\begin {gather*} \frac {2}{5} (-x)^{5/2}-\frac {x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int \sqrt {-x} \left (\sqrt {-x}+x\right ) \, dx &=\int \left (-(-x)^{3/2}-x\right ) \, dx\\ &=\frac {2}{5} (-x)^{5/2}-\frac {x^2}{2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.95 \begin {gather*} \frac {1}{10} \left (-5+4 \sqrt {-x}\right ) x^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 14, normalized size = 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 | \(-\frac {\left (1+x \right ) \left (-1+x \right )}{2}+\frac {2 x^{2} \sqrt {-x}}{5}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{5} \, \left (-x\right )^{\frac {5}{2}} - \frac {1}{2} \, x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 16, normalized size = 0.84 \begin {gather*} \frac {2}{5} \, \sqrt {-x} x^{2} - \frac {1}{2} \, x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 x^{2} \sqrt {- x}}{5} - \frac {x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.16, size = 16, normalized size = 0.84 \begin {gather*} \frac {2}{5} \, \sqrt {-x} x^{2} - \frac {1}{2} \, x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 13, normalized size = 0.68 \begin {gather*} \frac {2\,{\left (-x\right )}^{5/2}}{5}-\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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