Optimal. Leaf size=31 \[ \frac {1}{3} \sqrt {2} E\left (\sin ^{-1}(x)|-\frac {3}{2}\right )-\frac {1}{3} \sqrt {2} F\left (\sin ^{-1}(x)|-\frac {3}{2}\right ) \]
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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} \[ \frac {1}{3} \sqrt {2} E\left (\sin ^{-1}(x)|-\frac {3}{2}\right )-\frac {1}{3} \sqrt {2} F\left (\sin ^{-1}(x)|-\frac {3}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 493
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {1-x^2} \sqrt {2+3 x^2}} \, dx &=\frac {1}{3} \int \frac {\sqrt {2+3 x^2}}{\sqrt {1-x^2}} \, dx-\frac {2}{3} \int \frac {1}{\sqrt {1-x^2} \sqrt {2+3 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 = 0.03, size = 24, normalized size = 0.77 \[ \frac {1}{3} \sqrt {2} \left (E\left (\sin ^{-1}(x)|-\frac {3}{2}\right )-F\left (\sin ^{-1}(x)|-\frac {3}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 1} x^{2}}{3 \, x^{4} - x^{2} - 2}, x\right ) \]
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 \[ \int \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 25, normalized size = 0.81 \[ -\frac {\left (-\EllipticE \left (x , \frac {i \sqrt {6}}{2}\right )+\EllipticF \left (x , \frac {i \sqrt {6}}{2}\right )\right ) \sqrt {2}}{3} \]
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 \[ \int \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^2}{\sqrt {1-x^2}\,\sqrt {3\,x^2+2}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {3 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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