Optimal. Leaf size=30 \[ \frac {5}{16} \sin ^{-1}\left (\frac {2 x}{\sqrt {5}}\right )-\frac {1}{8} x \sqrt {5-4 x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 30, 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 = {321, 216} \[ \frac {5}{16} \sin ^{-1}\left (\frac {2 x}{\sqrt {5}}\right )-\frac {1}{8} x \sqrt {5-4 x^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {5-4 x^2}} \, dx &=-\frac {1}{8} x \sqrt {5-4 x^2}+\frac {5}{8} \int \frac {1}{\sqrt {5-4 x^2}} \, dx\\ &=-\frac {1}{8} x \sqrt {5-4 x^2}+\frac {5}{16} \sin ^{-1}\left (\frac {2 x}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ \frac {5}{16} \sin ^{-1}\left (\frac {2 x}{\sqrt {5}}\right )-\frac {1}{8} x \sqrt {5-4 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 30, normalized size = 1.00 \[ -\frac {1}{8} \, \sqrt {-4 \, x^{2} + 5} x - \frac {5}{16} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 5}}{2 \, x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 22, normalized size = 0.73 \[ -\frac {1}{8} \, \sqrt {-4 \, x^{2} + 5} x + \frac {5}{16} \, \arcsin \left (\frac {2}{5} \, \sqrt {5} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.77 \[ -\frac {\sqrt {-4 x^{2}+5}\, x}{8}+\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, x}{5}\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 22, normalized size = 0.73 \[ -\frac {1}{8} \, \sqrt {-4 \, x^{2} + 5} x + \frac {5}{16} \, \arcsin \left (\frac {2}{5} \, \sqrt {5} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 22, normalized size = 0.73 \[ \frac {5\,\mathrm {asin}\left (\frac {2\,\sqrt {5}\,x}{5}\right )}{16}-\frac {x\,\sqrt {\frac {5}{4}-x^2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 27, normalized size = 0.90 \[ - \frac {x \sqrt {5 - 4 x^{2}}}{8} + \frac {5 \operatorname {asin}{\left (\frac {2 \sqrt {5} x}{5} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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