Integrand size = 21, antiderivative size = 20 \[ \int \frac {1}{\sqrt {2-5 x^2} \sqrt {1+x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{2}} x\right ),-\frac {2}{5}\right )}{\sqrt {5}} \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {430} \[ \int \frac {1}{\sqrt {2-5 x^2} \sqrt {1+x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{2}} x\right ),-\frac {2}{5}\right )}{\sqrt {5}} \]
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Rule 430
Rubi steps \begin{align*} \text {integral}& = \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {5}{2}} x\right )|-\frac {2}{5}\right )}{\sqrt {5}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {2-5 x^2} \sqrt {1+x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{2}} x\right ),-\frac {2}{5}\right )}{\sqrt {5}} \]
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Time = 3.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {F\left (\frac {\sqrt {10}\, x}{2}, \frac {i \sqrt {10}}{5}\right ) \sqrt {5}}{5}\) | \(19\) |
elliptic | \(\frac {\sqrt {-\left (5 x^{2}-2\right ) \left (x^{2}+1\right )}\, \sqrt {10}\, \sqrt {-10 x^{2}+4}\, F\left (\frac {\sqrt {10}\, x}{2}, \frac {i \sqrt {10}}{5}\right )}{10 \sqrt {-5 x^{2}+2}\, \sqrt {-5 x^{4}-3 x^{2}+2}}\) | \(67\) |
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none
Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {2-5 x^2} \sqrt {1+x^2}} \, dx=\frac {1}{5} \, \sqrt {5} F(\arcsin \left (\frac {1}{2} \, \sqrt {5} \sqrt {2} x\right )\,|\,-\frac {2}{5}) \]
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Time = 1.61 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {1}{\sqrt {2-5 x^2} \sqrt {1+x^2}} \, dx=\begin {cases} \frac {\sqrt {5} F\left (\operatorname {asin}{\left (\frac {\sqrt {10} x}{2} \right )}\middle | - \frac {2}{5}\right )}{5} & \text {for}\: x > - \frac {\sqrt {10}}{5} \wedge x < \frac {\sqrt {10}}{5} \end {cases} \]
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\[ \int \frac {1}{\sqrt {2-5 x^2} \sqrt {1+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} + 1} \sqrt {-5 \, x^{2} + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-5 x^2} \sqrt {1+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} + 1} \sqrt {-5 \, x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2-5 x^2} \sqrt {1+x^2}} \, dx=\int \frac {1}{\sqrt {x^2+1}\,\sqrt {2-5\,x^2}} \,d x \]
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