Optimal. Leaf size=27 \[ -\frac {1}{8} x \sqrt {9-4 x^2}+\frac {9}{16} \sin ^{-1}\left (\frac {2 x}{3}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 27, 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 {9}{16} \text {ArcSin}\left (\frac {2 x}{3}\right )-\frac {1}{8} x \sqrt {9-4 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 {9-4 x^2}} \, dx &=-\frac {1}{8} x \sqrt {9-4 x^2}+\frac {9}{8} \int \frac {1}{\sqrt {9-4 x^2}} \, dx\\ &=-\frac {1}{8} x \sqrt {9-4 x^2}+\frac {9}{16} \sin ^{-1}\left (\frac {2 x}{3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 40, normalized size = 1.48 \begin {gather*} -\frac {1}{8} x \sqrt {9-4 x^2}+\frac {9}{8} \tan ^{-1}\left (\frac {2 x}{-3+\sqrt {9-4 x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 20, normalized size = 0.74
method | result | size |
default | \(\frac {9 \arcsin \left (\frac {2 x}{3}\right )}{16}-\frac {x \sqrt {-4 x^{2}+9}}{8}\) | \(20\) |
risch | \(\frac {x \left (4 x^{2}-9\right )}{8 \sqrt {-4 x^{2}+9}}+\frac {9 \arcsin \left (\frac {2 x}{3}\right )}{16}\) | \(27\) |
meijerg | \(\frac {9 i \left (\frac {2 i \sqrt {\pi }\, x \sqrt {1-\frac {4 x^{2}}{9}}}{3}-i \sqrt {\pi }\, \arcsin \left (\frac {2 x}{3}\right )\right )}{16 \sqrt {\pi }}\) | \(34\) |
trager | \(-\frac {x \sqrt {-4 x^{2}+9}}{8}+\frac {9 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-4 x^{2}+9}+2 x \right )}{16}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 19, normalized size = 0.70 \begin {gather*} -\frac {1}{8} \, \sqrt {-4 \, x^{2} + 9} x + \frac {9}{16} \, \arcsin \left (\frac {2}{3} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.11, size = 32, normalized size = 1.19 \begin {gather*} -\frac {1}{8} \, \sqrt {-4 \, x^{2} + 9} x - \frac {9}{8} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{2 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 22, normalized size = 0.81 \begin {gather*} - \frac {x \sqrt {9 - 4 x^{2}}}{8} + \frac {9 \operatorname {asin}{\left (\frac {2 x}{3} \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 19, normalized size = 0.70 \begin {gather*} -\frac {1}{8} \, \sqrt {-4 \, x^{2} + 9} x + \frac {9}{16} \, \arcsin \left (\frac {2}{3} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 19, normalized size = 0.70 \begin {gather*} \frac {9\,\mathrm {asin}\left (\frac {2\,x}{3}\right )}{16}-\frac {x\,\sqrt {\frac {9}{4}-x^2}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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