Optimal. Leaf size=31 \[ \frac {1}{3} \sqrt {2} F\left (\sin ^{-1}(x)|\frac {3}{2}\right )-\frac {1}{3} \sqrt {2} E\left (\sin ^{-1}(x)|\frac {3}{2}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{3} \sqrt {2} F\left (\sin ^{-1}(x)|\frac {3}{2}\right )-\frac {1}{3} \sqrt {2} E\left (\sin ^{-1}(x)|\frac {3}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 493
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx &=-\left (\frac {1}{3} \int \frac {\sqrt {2-3 x^2}}{\sqrt {1-x^2}} \, dx\right )+\frac {2}{3} \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx\\ &=-\frac {1}{3} \sqrt {2} E\left (\sin ^{-1}(x)|\frac {3}{2}\right )+\frac {1}{3} \sqrt {2} F\left (\sin ^{-1}(x)|\frac {3}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 37, normalized size = 1.19 \[ \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2} x^{2}}{3 \, x^{4} - 5 \, x^{2} + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 23, normalized size = 0.74 \[ \frac {\sqrt {2}\, \left (-\EllipticE \left (x , \frac {\sqrt {6}}{2}\right )+\EllipticF \left (x , \frac {\sqrt {6}}{2}\right )\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^2}{\sqrt {1-x^2}\,\sqrt {2-3\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {2 - 3 x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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