Optimal. Leaf size=38 \[ \sqrt {\frac {2}{3}} x \sqrt {1-4 x^4}+\frac {2 F\left (\left .\sin ^{-1}\left (\sqrt {2} x\right )\right |-1\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {254, 201, 227}
\begin {gather*} \frac {2 F\left (\left .\text {ArcSin}\left (\sqrt {2} x\right )\right |-1\right )}{\sqrt {3}}+\sqrt {\frac {2}{3}} \sqrt {1-4 x^4} x \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 227
Rule 254
Rubi steps
\begin {align*} \int \sqrt {3-6 x^2} \sqrt {2+4 x^2} \, dx &=\int \sqrt {6-24 x^4} \, dx\\ &=\sqrt {\frac {2}{3}} x \sqrt {1-4 x^4}+4 \int \frac {1}{\sqrt {6-24 x^4}} \, dx\\ &=\sqrt {\frac {2}{3}} x \sqrt {1-4 x^4}+\frac {2 F\left (\left .\sin ^{-1}\left (\sqrt {2} x\right )\right |-1\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.08, size = 22, normalized size = 0.58 \begin {gather*} \sqrt {6} x \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};4 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 74 vs. \(2 (29 ) = 58\).
time = 0.14, size = 75, normalized size = 1.97
method | result | size |
default | \(-\frac {\sqrt {-6 x^{2}+3}\, \sqrt {2}\, \sqrt {2 x^{2}+1}\, \left (\sqrt {2}\, \sqrt {3}\, \sqrt {-6 x^{2}+3}\, \sqrt {2 x^{2}+1}\, \EllipticF \left (x \sqrt {2}, i\right )-12 x^{5}+3 x \right )}{9 \left (4 x^{4}-1\right )}\) | \(75\) |
elliptic | \(-\frac {\sqrt {-6 x^{2}+3}\, \sqrt {4 x^{2}+2}\, \sqrt {-24 x^{4}+6}\, \left (\frac {x \sqrt {-24 x^{4}+6}}{3}+\frac {2 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticF \left (x \sqrt {2}, i\right )}{\sqrt {-24 x^{4}+6}}\right )}{6 \left (4 x^{4}-1\right )}\) | \(92\) |
risch | \(-\frac {x \left (2 x^{2}-1\right ) \left (2 x^{2}+1\right ) \sqrt {\left (-6 x^{2}+3\right ) \left (4 x^{2}+2\right )}\, \sqrt {6}}{3 \sqrt {-\left (2 x^{2}-1\right ) \left (2 x^{2}+1\right )}\, \sqrt {-6 x^{2}+3}\, \sqrt {4 x^{2}+2}}+\frac {\sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticF \left (x \sqrt {2}, i\right ) \sqrt {\left (-6 x^{2}+3\right ) \left (4 x^{2}+2\right )}\, \sqrt {6}}{3 \sqrt {-4 x^{4}+1}\, \sqrt {-6 x^{2}+3}\, \sqrt {4 x^{2}+2}}\) | \(153\) |
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 [A]
time = 0.24, size = 21, normalized size = 0.55 \begin {gather*} \frac {1}{3} \, \sqrt {4 \, x^{2} + 2} \sqrt {-6 \, x^{2} + 3} x \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*} \sqrt {6} \int \sqrt {1 - 2 x^{2}} \sqrt {2 x^{2} + 1}\, 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 \sqrt {4\,x^2+2}\,\sqrt {3-6\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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