Integrand size = 13, antiderivative size = 41 \[ \int \frac {x^6}{2+3 x^2} \, dx=\frac {4 x}{27}-\frac {2 x^3}{27}+\frac {x^5}{15}-\frac {4}{27} \sqrt {\frac {2}{3}} \arctan \left (\sqrt {\frac {3}{2}} x\right ) \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {308, 209} \[ \int \frac {x^6}{2+3 x^2} \, dx=-\frac {4}{27} \sqrt {\frac {2}{3}} \arctan \left (\sqrt {\frac {3}{2}} x\right )+\frac {x^5}{15}-\frac {2 x^3}{27}+\frac {4 x}{27} \]
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Rule 209
Rule 308
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4}{27}-\frac {2 x^2}{9}+\frac {x^4}{3}-\frac {8}{27 \left (2+3 x^2\right )}\right ) \, dx \\ & = \frac {4 x}{27}-\frac {2 x^3}{27}+\frac {x^5}{15}-\frac {8}{27} \int \frac {1}{2+3 x^2} \, dx \\ & = \frac {4 x}{27}-\frac {2 x^3}{27}+\frac {x^5}{15}-\frac {4}{27} \sqrt {\frac {2}{3}} \arctan \left (\sqrt {\frac {3}{2}} x\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \frac {x^6}{2+3 x^2} \, dx=\frac {1}{405} \left (60 x-30 x^3+27 x^5-20 \sqrt {6} \arctan \left (\sqrt {\frac {3}{2}} x\right )\right ) \]
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Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66
method | result | size |
default | \(\frac {4 x}{27}-\frac {2 x^{3}}{27}+\frac {x^{5}}{15}-\frac {4 \arctan \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {6}}{81}\) | \(27\) |
risch | \(\frac {4 x}{27}-\frac {2 x^{3}}{27}+\frac {x^{5}}{15}-\frac {4 \arctan \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {6}}{81}\) | \(27\) |
meijerg | \(\frac {2 \sqrt {2}\, \sqrt {3}\, \left (\frac {x \sqrt {2}\, \sqrt {3}\, \left (\frac {189}{4} x^{4}-\frac {105}{2} x^{2}+105\right )}{105}-2 \arctan \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{81}\) | \(43\) |
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Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.78 \[ \int \frac {x^6}{2+3 x^2} \, dx=\frac {1}{15} \, x^{5} - \frac {2}{27} \, x^{3} - \frac {4}{81} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {3} \sqrt {2} x\right ) + \frac {4}{27} \, x \]
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Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.83 \[ \int \frac {x^6}{2+3 x^2} \, dx=\frac {x^{5}}{15} - \frac {2 x^{3}}{27} + \frac {4 x}{27} - \frac {4 \sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} x}{2} \right )}}{81} \]
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Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int \frac {x^6}{2+3 x^2} \, dx=\frac {1}{15} \, x^{5} - \frac {2}{27} \, x^{3} - \frac {4}{81} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {4}{27} \, x \]
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int \frac {x^6}{2+3 x^2} \, dx=\frac {1}{15} \, x^{5} - \frac {2}{27} \, x^{3} - \frac {4}{81} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {4}{27} \, x \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.78 \[ \int \frac {x^6}{2+3 x^2} \, dx=\frac {4\,x}{27}-\frac {2\,x^3}{27}+\frac {x^5}{15}-\frac {4\,\sqrt {2}\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{81} \]
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