Integrand size = 24, antiderivative size = 82 \[ \int \frac {x^2}{\sqrt {4+x^2} \sqrt {2+3 x^2}} \, dx=\frac {x \sqrt {2+3 x^2}}{3 \sqrt {4+x^2}}-\frac {\sqrt {2} \sqrt {2+3 x^2} E\left (\left .\arctan \left (\frac {x}{2}\right )\right |-5\right )}{3 \sqrt {4+x^2} \sqrt {\frac {2+3 x^2}{4+x^2}}} \]
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Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {506, 422} \[ \int \frac {x^2}{\sqrt {4+x^2} \sqrt {2+3 x^2}} \, dx=\frac {x \sqrt {3 x^2+2}}{3 \sqrt {x^2+4}}-\frac {\sqrt {2} \sqrt {3 x^2+2} E\left (\left .\arctan \left (\frac {x}{2}\right )\right |-5\right )}{3 \sqrt {x^2+4} \sqrt {\frac {3 x^2+2}{x^2+4}}} \]
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Rule 422
Rule 506
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {2+3 x^2}}{3 \sqrt {4+x^2}}-\frac {4}{3} \int \frac {\sqrt {2+3 x^2}}{\left (4+x^2\right )^{3/2}} \, dx \\ & = \frac {x \sqrt {2+3 x^2}}{3 \sqrt {4+x^2}}-\frac {\sqrt {2} \sqrt {2+3 x^2} E\left (\left .\tan ^{-1}\left (\frac {x}{2}\right )\right |-5\right )}{3 \sqrt {4+x^2} \sqrt {\frac {2+3 x^2}{4+x^2}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.59 \[ \int \frac {x^2}{\sqrt {4+x^2} \sqrt {2+3 x^2}} \, dx=-\frac {2 i \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {1}{6}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{6}\right )\right )}{\sqrt {3}} \]
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Time = 3.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.32
method | result | size |
default | \(\frac {i \left (F\left (\frac {i x}{2}, \sqrt {6}\right )-E\left (\frac {i x}{2}, \sqrt {6}\right )\right ) \sqrt {2}}{3}\) | \(26\) |
elliptic | \(\frac {i \sqrt {\left (3 x^{2}+2\right ) \left (x^{2}+4\right )}\, \sqrt {6 x^{2}+4}\, \left (F\left (\frac {i x}{2}, \sqrt {6}\right )-E\left (\frac {i x}{2}, \sqrt {6}\right )\right )}{3 \sqrt {3 x^{2}+2}\, \sqrt {3 x^{4}+14 x^{2}+8}}\) | \(70\) |
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none
Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{\sqrt {4+x^2} \sqrt {2+3 x^2}} \, dx=-\frac {2 \, \sqrt {-2} x E(\arcsin \left (\frac {\sqrt {3} \sqrt {-2}}{3 \, x}\right )\,|\,6) - 2 \, \sqrt {-2} x F(\arcsin \left (\frac {\sqrt {3} \sqrt {-2}}{3 \, x}\right )\,|\,6) - 3 \, \sqrt {3 \, x^{2} + 2} \sqrt {x^{2} + 4}}{9 \, x} \]
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\[ \int \frac {x^2}{\sqrt {4+x^2} \sqrt {2+3 x^2}} \, dx=\int \frac {x^{2}}{\sqrt {x^{2} + 4} \sqrt {3 x^{2} + 2}}\, dx \]
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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. \[ \int \frac {x^2}{\sqrt {4+x^2} \sqrt {2+3 x^2}} \, dx=\int \frac {x^2}{\sqrt {x^2+4}\,\sqrt {3\,x^2+2}} \,d x \]
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