Optimal. Leaf size=23 \[ \frac {x^2}{2}-\frac {1}{3} \left (1-x^2\right )^{3/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {14, 267}
\begin {gather*} \frac {x^2}{2}-\frac {1}{3} \left (1-x^2\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 267
Rubi steps
\begin {align*} \int x \left (1+\sqrt {1-x^2}\right ) \, dx &=\int \left (x+x \sqrt {1-x^2}\right ) \, dx\\ &=\frac {x^2}{2}+\int x \sqrt {1-x^2} \, dx\\ &=\frac {x^2}{2}-\frac {1}{3} \left (1-x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{6} \left (-1+x^2\right ) \left (3+2 \sqrt {1-x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 18, normalized size = 0.78
method | result | size |
default | \(\frac {x^{2}}{2}-\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{3}\) | \(18\) |
derivativedivides | \(\frac {x^{2}}{2}-\frac {1}{2}-\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{3}\) | \(19\) |
trager | \(\frac {x^{2}}{2}+\left (\frac {x^{2}}{3}-\frac {1}{3}\right ) \sqrt {-x^{2}+1}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 22, normalized size = 0.96 \begin {gather*} \frac {1}{2} \, x^{2} + \frac {1}{3} \, {\left (x^{2} - 1\right )} \sqrt {-x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 27, normalized size = 1.17 \begin {gather*} \frac {x^{2} \sqrt {1 - x^{2}}}{3} + \frac {x^{2}}{2} - \frac {\sqrt {1 - x^{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.77, size = 18, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} - \frac {1}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 23, normalized size = 1.00 \begin {gather*} \frac {x^2}{2}+\sqrt {1-x^2}\,\left (\frac {x^2}{3}-\frac {1}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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