Integrand size = 26, antiderivative size = 31 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=-\frac {1}{3} \sqrt {2} E\left (\arcsin (x)\left |\frac {3}{2}\right .\right )+\frac {1}{3} \sqrt {2} \operatorname {EllipticF}\left (\arcsin (x),\frac {3}{2}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 31, 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} \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\frac {1}{3} \sqrt {2} \operatorname {EllipticF}\left (\arcsin (x),\frac {3}{2}\right )-\frac {1}{3} \sqrt {2} E\left (\arcsin (x)\left |\frac {3}{2}\right .\right ) \]
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Rule 430
Rule 435
Rule 507
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\frac {-E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )+\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{\sqrt {3}} \]
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Time = 3.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {\sqrt {2}\, \left (F\left (x , \frac {\sqrt {6}}{2}\right )-E\left (x , \frac {\sqrt {6}}{2}\right )\right )}{3}\) | \(23\) |
elliptic | \(\frac {\sqrt {\left (3 x^{2}-2\right ) \left (x^{2}-1\right )}\, \sqrt {-6 x^{2}+4}\, \left (F\left (x , \frac {\sqrt {6}}{2}\right )-E\left (x , \frac {\sqrt {6}}{2}\right )\right )}{3 \sqrt {-3 x^{2}+2}\, \sqrt {3 x^{4}-5 x^{2}+2}}\) | \(67\) |
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Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\frac {\sqrt {3} x E(\arcsin \left (\frac {1}{x}\right )\,|\,\frac {2}{3}) - \sqrt {3} x F(\arcsin \left (\frac {1}{x}\right )\,|\,\frac {2}{3}) + \sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}{3 \, x} \]
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\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int \frac {x^{2}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {2 - 3 x^{2}}}\, dx \]
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\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int \frac {x^2}{\sqrt {1-x^2}\,\sqrt {2-3\,x^2}} \,d x \]
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