Integrand size = 23, antiderivative size = 38 \[ \int \sqrt {3-6 x^2} \sqrt {2+4 x^2} \, dx=\sqrt {\frac {2}{3}} x \sqrt {1-4 x^4}+\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-1\right )}{\sqrt {3}} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {254, 201, 227} \[ \int \sqrt {3-6 x^2} \sqrt {2+4 x^2} \, dx=\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-1\right )}{\sqrt {3}}+\sqrt {\frac {2}{3}} \sqrt {1-4 x^4} x \]
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Rule 201
Rule 227
Rule 254
Rubi steps \begin{align*} \text {integral}& = \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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.58 \[ \int \sqrt {3-6 x^2} \sqrt {2+4 x^2} \, dx=\sqrt {6} x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},4 x^4\right ) \]
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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 = 2.75 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.97
method | result | size |
default | \(-\frac {\sqrt {-6 x^{2}+3}\, \sqrt {2}\, \sqrt {2 x^{2}+1}\, \left (\sqrt {2}\, \sqrt {3}\, \sqrt {-6 x^{2}+3}\, \sqrt {2 x^{2}+1}\, F\left (\sqrt {2}\, x , i\right )-12 x^{5}+3 x \right )}{9 \left (4 x^{4}-1\right )}\) | \(75\) |
elliptic | \(-\frac {\sqrt {-6 x^{2}+3}\, \sqrt {4 x^{2}+2}\, \sqrt {-24 x^{4}+6}\, \left (\frac {x \sqrt {-24 x^{4}+6}}{3}+\frac {2 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, F\left (\sqrt {2}\, x , i\right )}{\sqrt {-24 x^{4}+6}}\right )}{6 \left (4 x^{4}-1\right )}\) | \(92\) |
risch | \(-\frac {x \left (2 x^{2}-1\right ) \left (2 x^{2}+1\right ) \sqrt {\left (-6 x^{2}+3\right ) \left (4 x^{2}+2\right )}\, \sqrt {6}}{3 \sqrt {-\left (2 x^{2}-1\right ) \left (2 x^{2}+1\right )}\, \sqrt {-6 x^{2}+3}\, \sqrt {4 x^{2}+2}}+\frac {\sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, F\left (\sqrt {2}\, x , i\right ) \sqrt {\left (-6 x^{2}+3\right ) \left (4 x^{2}+2\right )}\, \sqrt {6}}{3 \sqrt {-4 x^{4}+1}\, \sqrt {-6 x^{2}+3}\, \sqrt {4 x^{2}+2}}\) | \(153\) |
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Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08 \[ \int \sqrt {3-6 x^2} \sqrt {2+4 x^2} \, dx=\frac {1}{3} \, \sqrt {4 \, x^{2} + 2} \sqrt {-6 \, x^{2} + 3} x + \frac {1}{3} \, \sqrt {2} \sqrt {-6} F(\arcsin \left (\frac {\sqrt {2}}{2 \, x}\right )\,|\,-1) \]
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\[ \int \sqrt {3-6 x^2} \sqrt {2+4 x^2} \, dx=\sqrt {6} \int \sqrt {1 - 2 x^{2}} \sqrt {2 x^{2} + 1}\, dx \]
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\[ \int \sqrt {3-6 x^2} \sqrt {2+4 x^2} \, dx=\int { \sqrt {4 \, x^{2} + 2} \sqrt {-6 \, x^{2} + 3} \,d x } \]
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\[ \int \sqrt {3-6 x^2} \sqrt {2+4 x^2} \, dx=\int { \sqrt {4 \, x^{2} + 2} \sqrt {-6 \, x^{2} + 3} \,d x } \]
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Timed out. \[ \int \sqrt {3-6 x^2} \sqrt {2+4 x^2} \, dx=\int \sqrt {4\,x^2+2}\,\sqrt {3-6\,x^2} \,d x \]
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