Optimal. Leaf size=35 \[ \frac{1}{3} \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),6\right )-\frac{1}{3} \sqrt{2} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |6\right ) \]
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Rubi [A] time = 0.0316317, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {493, 424, 419} \[ \frac{1}{3} \sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |6\right )-\frac{1}{3} \sqrt{2} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |6\right ) \]
Antiderivative was successfully verified.
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Rule 493
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{4-x^2}} \, dx &=-\left (\frac{1}{3} \int \frac{\sqrt{2-3 x^2}}{\sqrt{4-x^2}} \, dx\right )+\frac{2}{3} \int \frac{1}{\sqrt{2-3 x^2} \sqrt{4-x^2}} \, dx\\ &=-\frac{1}{3} \sqrt{2} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |6\right )+\frac{1}{3} \sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |6\right )\\ \end{align*}
Mathematica [A] time = 0.0327616, size = 38, normalized size = 1.09 \[ -\frac{2 \left (E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{1}{6}\right )-\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),\frac{1}{6}\right )\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 33, normalized size = 0.9 \begin{align*}{\frac{2\,\sqrt{3}}{3} \left ({\it EllipticF} \left ({\frac{x\sqrt{6}}{2}},{\frac{\sqrt{6}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{6}}{2}},{\frac{\sqrt{6}}{6}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{-x^{2} + 4} \sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-x^{2} + 4} \sqrt{-3 \, x^{2} + 2} x^{2}}{3 \, x^{4} - 14 \, x^{2} + 8}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{- \left (x - 2\right ) \left (x + 2\right )} \sqrt{2 - 3 x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{-x^{2} + 4} \sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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