Optimal. Leaf size=29 \[ -\frac {1}{2} x \sqrt {5-x^2}+\frac {5}{2} \sin ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {327, 222}
\begin {gather*} \frac {5}{2} \text {ArcSin}\left (\frac {x}{\sqrt {5}}\right )-\frac {1}{2} x \sqrt {5-x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 327
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {5-x^2}} \, dx &=-\frac {1}{2} x \sqrt {5-x^2}+\frac {5}{2} \int \frac {1}{\sqrt {5-x^2}} \, dx\\ &=-\frac {1}{2} x \sqrt {5-x^2}+\frac {5}{2} \sin ^{-1}\left (\frac {x}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 35, normalized size = 1.21 \begin {gather*} -\frac {1}{2} x \sqrt {5-x^2}+\frac {5}{2} \tan ^{-1}\left (\frac {x}{\sqrt {5-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 23, normalized size = 0.79
method | result | size |
default | \(\frac {5 \arcsin \left (\frac {x \sqrt {5}}{5}\right )}{2}-\frac {x \sqrt {-x^{2}+5}}{2}\) | \(23\) |
risch | \(\frac {x \left (x^{2}-5\right )}{2 \sqrt {-x^{2}+5}}+\frac {5 \arcsin \left (\frac {x \sqrt {5}}{5}\right )}{2}\) | \(28\) |
meijerg | \(\frac {5 i \left (\frac {i \sqrt {\pi }\, x \sqrt {5}\, \sqrt {-\frac {x^{2}}{5}+1}}{5}-i \sqrt {\pi }\, \arcsin \left (\frac {x \sqrt {5}}{5}\right )\right )}{2 \sqrt {\pi }}\) | \(40\) |
trager | \(-\frac {x \sqrt {-x^{2}+5}}{2}+\frac {5 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+5}+x \right )}{2}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.61, size = 22, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 5} x + \frac {5}{2} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 29, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 5} x - \frac {5}{2} \, \arctan \left (\frac {\sqrt {-x^{2} + 5}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 24, normalized size = 0.83 \begin {gather*} - \frac {x \sqrt {5 - x^{2}}}{2} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {5} x}{5} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 22, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 5} x + \frac {5}{2} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 22, normalized size = 0.76 \begin {gather*} \frac {5\,\mathrm {asin}\left (\frac {\sqrt {5}\,x}{5}\right )}{2}-\frac {x\,\sqrt {5-x^2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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