Optimal. Leaf size=35 \[ \frac {E\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}}-\frac {F\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 35, 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} \[ \frac {E\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}}-\frac {F\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 493
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx &=\frac {1}{3} \int \frac {\sqrt {2+3 x^2}}{\sqrt {1-4 x^2}} \, dx-\frac {2}{3} \int \frac {1}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx\\ &=\frac {E\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}}-\frac {F\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 28, normalized size = 0.80 \[ \frac {E\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )-F\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 2} \sqrt {-4 \, x^{2} + 1} x^{2}}{12 \, x^{4} + 5 \, x^{2} - 2}, x\right ) \]
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 \[ \int \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {-4 \, x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 0.83 \[ -\frac {\left (-\EllipticE \left (2 x , \frac {i \sqrt {6}}{4}\right )+\EllipticF \left (2 x , \frac {i \sqrt {6}}{4}\right )\right ) \sqrt {2}}{6} \]
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 \[ \int \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {-4 \, x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^2}{\sqrt {3\,x^2+2}\,\sqrt {1-4\,x^2}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {- \left (2 x - 1\right ) \left (2 x + 1\right )} \sqrt {3 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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