Integrand size = 23, antiderivative size = 12 \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2+5 x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (x),-\frac {5}{2}\right )}{\sqrt {2}} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {430} \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2+5 x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (x),-\frac {5}{2}\right )}{\sqrt {2}} \]
[In]
[Out]
Rule 430
Rubi steps \begin{align*} \text {integral}& = \frac {F\left (\sin ^{-1}(x)|-\frac {5}{2}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2+5 x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (x),-\frac {5}{2}\right )}{\sqrt {2}} \]
[In]
[Out]
Time = 3.60 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17
method | result | size |
default | \(\frac {F\left (x , \frac {i \sqrt {10}}{2}\right ) \sqrt {2}}{2}\) | \(14\) |
elliptic | \(\frac {\sqrt {-\left (x^{2}-1\right ) \left (5 x^{2}+2\right )}\, \sqrt {10 x^{2}+4}\, F\left (x , \frac {i \sqrt {10}}{2}\right )}{2 \sqrt {5 x^{2}+2}\, \sqrt {-5 x^{4}+3 x^{2}+2}}\) | \(59\) |
[In]
[Out]
none
Time = 0.08 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2+5 x^2}} \, dx=\frac {1}{2} \, \sqrt {2} F(\arcsin \left (x\right )\,|\,-\frac {5}{2}) \]
[In]
[Out]
Time = 1.67 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2+5 x^2}} \, dx=\begin {cases} \frac {\sqrt {2} F\left (\operatorname {asin}{\left (x \right )}\middle | - \frac {5}{2}\right )}{2} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 2} \sqrt {-x^{2} + 1}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 2} \sqrt {-x^{2} + 1}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2+5 x^2}} \, dx=\int \frac {1}{\sqrt {1-x^2}\,\sqrt {5\,x^2+2}} \,d x \]
[In]
[Out]