Optimal. Leaf size=29 \[ \frac{5}{2} \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right )-\frac{1}{2} x \sqrt{5-x^2} \]
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Rubi [A] time = 0.0061221, antiderivative size = 29, 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} \[ \frac{5}{2} \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right )-\frac{1}{2} x \sqrt{5-x^2} \]
Antiderivative was successfully verified.
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Rule 321
Rule 216
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*}
Mathematica [A] time = 0.0089389, size = 29, normalized size = 1. \[ \frac{5}{2} \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right )-\frac{1}{2} x \sqrt{5-x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 23, normalized size = 0.8 \begin{align*}{\frac{5}{2}\arcsin \left ({\frac{x\sqrt{5}}{5}} \right ) }-{\frac{x}{2}\sqrt{-{x}^{2}+5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40043, size = 30, normalized size = 1.03 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 5} x + \frac{5}{2} \, \arcsin \left (\frac{1}{5} \, \sqrt{5} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27144, size = 73, normalized size = 2.52 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 5} x - \frac{5}{2} \, \arctan \left (\frac{\sqrt{-x^{2} + 5}}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.202197, size = 24, normalized size = 0.83 \begin{align*} - \frac{x \sqrt{5 - x^{2}}}{2} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{5} x}{5} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08117, size = 30, normalized size = 1.03 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 5} x + \frac{5}{2} \, \arcsin \left (\frac{1}{5} \, \sqrt{5} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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