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 = 0.0062046, antiderivative size = 30, normalized size of antiderivative = 1., 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 321
Rule 216
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*}
Mathematica [A] time = 0.008429, size = 30, normalized size = 1. \[ \frac{5}{16} \sin ^{-1}\left (\frac{2 x}{\sqrt{5}}\right )-\frac{1}{8} x \sqrt{5-4 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 23, normalized size = 0.8 \begin{align*}{\frac{5}{16}\arcsin \left ({\frac{2\,x\sqrt{5}}{5}} \right ) }-{\frac{x}{8}\sqrt{-4\,{x}^{2}+5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44639, size = 30, normalized size = 1. \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{2} + 5} x + \frac{5}{16} \, \arcsin \left (\frac{2}{5} \, \sqrt{5} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9948, size = 85, normalized size = 2.83 \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{2} + 5} x - \frac{5}{16} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} + 5}}{2 \, x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.206622, size = 27, normalized size = 0.9 \begin{align*} - \frac{x \sqrt{5 - 4 x^{2}}}{8} + \frac{5 \operatorname{asin}{\left (\frac{2 \sqrt{5} x}{5} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07222, size = 30, normalized size = 1. \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{2} + 5} x + \frac{5}{16} \, \arcsin \left (\frac{2}{5} \, \sqrt{5} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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