Integrand size = 24, antiderivative size = 20 \[ \int \frac {\sqrt {2+e x}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 \sqrt {2-e x}}{\sqrt {3} e} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 20, 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 = {641, 32} \[ \int \frac {\sqrt {2+e x}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 \sqrt {2-e x}}{\sqrt {3} e} \]
[In]
[Out]
Rule 32
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {6-3 e x}} \, dx \\ & = -\frac {2 \sqrt {2-e x}}{\sqrt {3} e} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {2+e x}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 \sqrt {4-e^2 x^2}}{e \sqrt {6+3 e x}} \]
[In]
[Out]
Time = 2.55 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25
method | result | size |
default | \(-\frac {2 \sqrt {-3 x^{2} e^{2}+12}}{3 \sqrt {e x +2}\, e}\) | \(25\) |
gosper | \(\frac {2 \left (e x -2\right ) \sqrt {e x +2}}{e \sqrt {-3 x^{2} e^{2}+12}}\) | \(30\) |
risch | \(\frac {2 \left (e x -2\right ) \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}\) | \(58\) |
[In]
[Out]
none
Time = 0.41 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {\sqrt {2+e x}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{3 \, {\left (e^{2} x + 2 \, e\right )}} \]
[In]
[Out]
\[ \int \frac {\sqrt {2+e x}}{\sqrt {12-3 e^2 x^2}} \, dx=\frac {\sqrt {3} \int \frac {\sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx}{3} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {2+e x}}{\sqrt {12-3 e^2 x^2}} \, dx=\frac {2 \, {\left (-i \, \sqrt {3} e x + 2 i \, \sqrt {3}\right )}}{3 \, \sqrt {e x - 2} e} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {2+e x}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 \, \sqrt {3} {\left (\sqrt {-e x + 2} - 2\right )}}{3 \, e} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {2+e x}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2\,\sqrt {12-3\,e^2\,x^2}}{3\,e\,\sqrt {e\,x+2}} \]
[In]
[Out]