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 = {507, 435, 430}
\begin {gather*} \frac {1}{3} \sqrt {2} F\left (\text {ArcSin}(x)\left |\frac {3}{2}\right .\right )-\frac {1}{3} \sqrt {2} E\left (\text {ArcSin}(x)\left |\frac {3}{2}\right .\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 507
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 = 0.27, size = 37, normalized size = 1.19 \begin {gather*} \frac {-E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )+F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 23, normalized size = 0.74
method | result | size |
default | \(\frac {\sqrt {2}\, \left (\EllipticF \left (x , \frac {\sqrt {6}}{2}\right )-\EllipticE \left (x , \frac {\sqrt {6}}{2}\right )\right )}{3}\) | \(23\) |
elliptic | \(\frac {\sqrt {\left (3 x^{2}-2\right ) \left (x^{2}-1\right )}\, \sqrt {-6 x^{2}+4}\, \left (\EllipticF \left (x , \frac {\sqrt {6}}{2}\right )-\EllipticE \left (x , \frac {\sqrt {6}}{2}\right )\right )}{3 \sqrt {-3 x^{2}+2}\, \sqrt {3 x^{4}-5 x^{2}+2}}\) | \(67\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {2 - 3 x^{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2}{\sqrt {1-x^2}\,\sqrt {2-3\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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