Optimal. Leaf size=35 \[ -\frac {1}{3} \sqrt {2} E\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |6\right )+\frac {1}{3} \sqrt {2} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |6\right ) \]
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Rubi [A]
time = 0.02, 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 = {507, 435, 430}
\begin {gather*} \frac {1}{3} \sqrt {2} F\left (\left .\text {ArcSin}\left (\frac {x}{2}\right )\right |6\right )-\frac {1}{3} \sqrt {2} E\left (\left .\text {ArcSin}\left (\frac {x}{2}\right )\right |6\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 {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*}
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Mathematica [A]
time = 0.26, size = 38, normalized size = 1.09 \begin {gather*} -\frac {2 \left (E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {1}{6}\right )-F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {1}{6}\right )\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 33, normalized size = 0.94
method | result | size |
default | \(\frac {2 \sqrt {3}\, \left (\EllipticF \left (\frac {x \sqrt {6}}{2}, \frac {\sqrt {6}}{6}\right )-\EllipticE \left (\frac {x \sqrt {6}}{2}, \frac {\sqrt {6}}{6}\right )\right )}{3}\) | \(33\) |
elliptic | \(\frac {\sqrt {\left (3 x^{2}-2\right ) \left (x^{2}-4\right )}\, \sqrt {6}\, \sqrt {-6 x^{2}+4}\, \left (\EllipticF \left (\frac {x \sqrt {6}}{2}, \frac {\sqrt {6}}{6}\right )-\EllipticE \left (\frac {x \sqrt {6}}{2}, \frac {\sqrt {6}}{6}\right )\right )}{3 \sqrt {-3 x^{2}+2}\, \sqrt {3 x^{4}-14 x^{2}+8}}\) | \(80\) |
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 - 2\right ) \left (x + 2\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 {4-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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