Integrand size = 26, antiderivative size = 35 \[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {2+3 x^2}} \, dx=\frac {1}{3} \sqrt {2} E\left (\left .\arcsin \left (\frac {x}{2}\right )\right |-6\right )-\frac {1}{3} \sqrt {2} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-6\right ) \]
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Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {507, 435, 430} \[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {2+3 x^2}} \, dx=\frac {1}{3} \sqrt {2} E\left (\left .\arcsin \left (\frac {x}{2}\right )\right |-6\right )-\frac {1}{3} \sqrt {2} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-6\right ) \]
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Rule 430
Rule 435
Rule 507
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {\sqrt {2+3 x^2}}{\sqrt {4-x^2}} \, dx-\frac {2}{3} \int \frac {1}{\sqrt {4-x^2} \sqrt {2+3 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*}
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37 \[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {2+3 x^2}} \, dx=\frac {2 i \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),-\frac {1}{6}\right )\right )}{\sqrt {3}} \]
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Time = 3.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {\left (F\left (\frac {x}{2}, i \sqrt {6}\right )-E\left (\frac {x}{2}, i \sqrt {6}\right )\right ) \sqrt {2}}{3}\) | \(29\) |
elliptic | \(-\frac {\sqrt {-\left (3 x^{2}+2\right ) \left (x^{2}-4\right )}\, \sqrt {6 x^{2}+4}\, \left (F\left (\frac {x}{2}, i \sqrt {6}\right )-E\left (\frac {x}{2}, i \sqrt {6}\right )\right )}{3 \sqrt {3 x^{2}+2}\, \sqrt {-3 x^{4}+10 x^{2}+8}}\) | \(74\) |
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none
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {2+3 x^2}} \, dx=-\frac {8 \, \sqrt {-3} x E(\arcsin \left (\frac {2}{x}\right )\,|\,-\frac {1}{6}) - 8 \, \sqrt {-3} x F(\arcsin \left (\frac {2}{x}\right )\,|\,-\frac {1}{6}) + \sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 4}}{3 \, x} \]
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\[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {2+3 x^2}} \, dx=\int \frac {x^{2}}{\sqrt {- \left (x - 2\right ) \left (x + 2\right )} \sqrt {3 x^{2} + 2}}\, dx \]
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\[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {2+3 x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 4}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {2+3 x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 4}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {2+3 x^2}} \, dx=\int \frac {x^2}{\sqrt {4-x^2}\,\sqrt {3\,x^2+2}} \,d x \]
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