Integrand size = 24, antiderivative size = 80 \[ \int \frac {x^2}{\sqrt {1+x^2} \sqrt {2+3 x^2}} \, dx=\frac {x \sqrt {2+3 x^2}}{3 \sqrt {1+x^2}}-\frac {\sqrt {2} \sqrt {2+3 x^2} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {1+x^2} \sqrt {\frac {2+3 x^2}{1+x^2}}} \]
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Time = 0.02 (sec) , antiderivative size = 80, 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} \[ \int \frac {x^2}{\sqrt {1+x^2} \sqrt {2+3 x^2}} \, dx=\frac {x \sqrt {3 x^2+2}}{3 \sqrt {x^2+1}}-\frac {\sqrt {2} \sqrt {3 x^2+2} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {x^2+1} \sqrt {\frac {3 x^2+2}{x^2+1}}} \]
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Rule 422
Rule 506
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {2+3 x^2}}{3 \sqrt {1+x^2}}-\frac {1}{3} \int \frac {\sqrt {2+3 x^2}}{\left (1+x^2\right )^{3/2}} \, dx \\ & = \frac {x \sqrt {2+3 x^2}}{3 \sqrt {1+x^2}}-\frac {\sqrt {2} \sqrt {2+3 x^2} E\left (\tan ^{-1}(x)|-\frac {1}{2}\right )}{3 \sqrt {1+x^2} \sqrt {\frac {2+3 x^2}{1+x^2}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.60 \[ \int \frac {x^2}{\sqrt {1+x^2} \sqrt {2+3 x^2}} \, dx=-\frac {i \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )\right )}{\sqrt {3}} \]
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Time = 3.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38
method | result | size |
default | \(\frac {i \left (F\left (i x , \frac {\sqrt {6}}{2}\right )-E\left (i x , \frac {\sqrt {6}}{2}\right )\right ) \sqrt {2}}{3}\) | \(30\) |
elliptic | \(\frac {i \sqrt {\left (3 x^{2}+2\right ) \left (x^{2}+1\right )}\, \sqrt {6 x^{2}+4}\, \left (F\left (i x , \frac {\sqrt {6}}{2}\right )-E\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{3 \sqrt {3 x^{2}+2}\, \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(74\) |
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none
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {1+x^2} \sqrt {2+3 x^2}} \, dx=-\frac {2 \, \sqrt {-2} x E(\arcsin \left (\frac {\sqrt {3} \sqrt {-2}}{3 \, x}\right )\,|\,\frac {3}{2}) - 2 \, \sqrt {-2} x F(\arcsin \left (\frac {\sqrt {3} \sqrt {-2}}{3 \, x}\right )\,|\,\frac {3}{2}) - 3 \, \sqrt {3 \, x^{2} + 2} \sqrt {x^{2} + 1}}{9 \, x} \]
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\[ \int \frac {x^2}{\sqrt {1+x^2} \sqrt {2+3 x^2}} \, dx=\int \frac {x^{2}}{\sqrt {x^{2} + 1} \sqrt {3 x^{2} + 2}}\, dx \]
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\[ \int \frac {x^2}{\sqrt {1+x^2} \sqrt {2+3 x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {x^{2} + 1}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {1+x^2} \sqrt {2+3 x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {3 \, x^{2} + 2} \sqrt {x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt {1+x^2} \sqrt {2+3 x^2}} \, dx=\int \frac {x^2}{\sqrt {x^2+1}\,\sqrt {3\,x^2+2}} \,d x \]
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