Integrand size = 16, antiderivative size = 92 \[ \int \frac {1}{\sqrt {3-5 x^2+2 x^4}} \, dx=\frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3-5 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {2}{3}} x\right ),\frac {1}{24} \left (12+5 \sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3-5 x^2+2 x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1110} \[ \int \frac {1}{\sqrt {3-5 x^2+2 x^4}} \, dx=\frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4-5 x^2+3}{\left (\sqrt {6} x^2+3\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {2}{3}} x\right ),\frac {1}{24} \left (12+5 \sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {2 x^4-5 x^2+3}} \]
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Rule 1110
Rubi steps \begin{align*} \text {integral}& = \frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3-5 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{24} \left (12+5 \sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3-5 x^2+2 x^4}} \\ \end{align*}
Time = 10.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\sqrt {3-5 x^2+2 x^4}} \, dx=\frac {\sqrt {3-2 x^2} \sqrt {1-x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{3}} x\right ),\frac {3}{2}\right )}{\sqrt {6-10 x^2+4 x^4}} \]
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Time = 0.60 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.46
method | result | size |
default | \(\frac {\sqrt {-x^{2}+1}\, \sqrt {-6 x^{2}+9}\, F\left (x , \frac {\sqrt {6}}{3}\right )}{3 \sqrt {2 x^{4}-5 x^{2}+3}}\) | \(42\) |
elliptic | \(\frac {\sqrt {-x^{2}+1}\, \sqrt {-6 x^{2}+9}\, F\left (x , \frac {\sqrt {6}}{3}\right )}{3 \sqrt {2 x^{4}-5 x^{2}+3}}\) | \(42\) |
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Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.10 \[ \int \frac {1}{\sqrt {3-5 x^2+2 x^4}} \, dx=\frac {1}{3} \, \sqrt {3} F(\arcsin \left (x\right )\,|\,\frac {2}{3}) \]
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\[ \int \frac {1}{\sqrt {3-5 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2 x^{4} - 5 x^{2} + 3}}\, dx \]
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\[ \int \frac {1}{\sqrt {3-5 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} - 5 \, x^{2} + 3}} \,d x } \]
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\[ \int \frac {1}{\sqrt {3-5 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} - 5 \, x^{2} + 3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {3-5 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2\,x^4-5\,x^2+3}} \,d x \]
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