Optimal. Leaf size=34 \[ -\frac {x^2}{8}+\frac {1}{2} x^2 \log \left (\sqrt {x+2}\right )+\frac {x}{2}-\log (x+2) \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2395, 43} \[ -\frac {x^2}{8}+\frac {1}{2} x^2 \log \left (\sqrt {x+2}\right )+\frac {x}{2}-\log (x+2) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2395
Rubi steps
\begin {align*} \int x \log \left (\sqrt {2+x}\right ) \, dx &=\frac {1}{2} x^2 \log \left (\sqrt {2+x}\right )-\frac {1}{4} \int \frac {x^2}{2+x} \, dx\\ &=\frac {1}{2} x^2 \log \left (\sqrt {2+x}\right )-\frac {1}{4} \int \left (-2+x+\frac {4}{2+x}\right ) \, dx\\ &=\frac {x}{2}-\frac {x^2}{8}+\frac {1}{2} x^2 \log \left (\sqrt {2+x}\right )-\log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.88 \[ \frac {1}{2} \left (-\frac {x^2}{4}+\frac {1}{2} x^2 \log (x+2)+x-2 \log (x+2)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 20, normalized size = 0.59 \[ -\frac {1}{8} \, x^{2} + \frac {1}{4} \, {\left (x^{2} - 4\right )} \log \left (x + 2\right ) + \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 30, normalized size = 0.88 \[ \frac {1}{4} \, {\left (x + 2\right )}^{2} \log \left (x + 2\right ) - \frac {1}{8} \, {\left (x + 2\right )}^{2} - {\left (x + 2\right )} \log \left (x + 2\right ) + x + 2 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 31, normalized size = 0.91 \[ -\frac {x^{2}}{8}+\frac {x}{2}-\left (x +2\right ) \ln \left (x +2\right )+\frac {\left (x +2\right )^{2} \ln \left (x +2\right )}{4}+\frac {3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 24, normalized size = 0.71 \[ \frac {1}{4} \, x^{2} \log \left (x + 2\right ) - \frac {1}{8} \, x^{2} + \frac {1}{2} \, x - \log \left (x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 20, normalized size = 0.59 \[ \frac {x}{2}-\frac {x^2}{8}+\frac {\ln \left (x+2\right )\,\left (x^2-4\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 22, normalized size = 0.65 \[ \frac {x^{2} \log {\left (x + 2 \right )}}{4} - \frac {x^{2}}{8} + \frac {x}{2} - \log {\left (x + 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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