Optimal. Leaf size=24 \[ \frac {x}{\sqrt {3-x^2}}-\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 24, 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 = {288, 216} \[ \frac {x}{\sqrt {3-x^2}}-\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 288
Rubi steps
\begin {align*} \int \frac {x^2}{\left (3-x^2\right )^{3/2}} \, dx &=\frac {x}{\sqrt {3-x^2}}-\int \frac {1}{\sqrt {3-x^2}} \, dx\\ &=\frac {x}{\sqrt {3-x^2}}-\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 24, normalized size = 1.00 \[ \frac {x}{\sqrt {3-x^2}}-\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.06, size = 44, normalized size = 1.83 \[ -\frac {\sqrt {3-x^2} x}{x^2-3}-i \log \left (\sqrt {3-x^2}-i x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 41, normalized size = 1.71 \[ \frac {{\left (x^{2} - 3\right )} \arctan \left (\frac {\sqrt {-x^{2} + 3}}{x}\right ) - \sqrt {-x^{2} + 3} x}{x^{2} - 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 29, normalized size = 1.21 \[ -\frac {\sqrt {-x^{2} + 3} x}{x^{2} - 3} - \arcsin \left (\frac {1}{3} \, \sqrt {3} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 22, normalized size = 0.92
method | result | size |
default | \(-\arcsin \left (\frac {x \sqrt {3}}{3}\right )+\frac {x}{\sqrt {-x^{2}+3}}\) | \(22\) |
risch | \(-\arcsin \left (\frac {x \sqrt {3}}{3}\right )+\frac {x}{\sqrt {-x^{2}+3}}\) | \(22\) |
meijerg | \(\frac {i \left (-\frac {i \sqrt {\pi }\, x \sqrt {3}}{3 \sqrt {-\frac {x^{2}}{3}+1}}+i \sqrt {\pi }\, \arcsin \left (\frac {x \sqrt {3}}{3}\right )\right )}{\sqrt {\pi }}\) | \(40\) |
trager | \(-\frac {x \sqrt {-x^{2}+3}}{x^{2}-3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+3}+x \right )\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 21, normalized size = 0.88 \[ \frac {x}{\sqrt {-x^{2} + 3}} - \arcsin \left (\frac {1}{3} \, \sqrt {3} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 54, normalized size = 2.25 \[ -\mathrm {asin}\left (\frac {\sqrt {3}\,x}{3}\right )-\frac {\sqrt {3-x^2}}{2\,\left (x-\sqrt {3}\right )}-\frac {\sqrt {3-x^2}}{2\,\left (x+\sqrt {3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.53, size = 49, normalized size = 2.04 \[ - \frac {x^{2} \operatorname {asin}{\left (\frac {\sqrt {3} x}{3} \right )}}{x^{2} - 3} - \frac {x \sqrt {3 - x^{2}}}{x^{2} - 3} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {3} x}{3} \right )}}{x^{2} - 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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