Optimal. Leaf size=20 \[ -\frac {2 (2-e x)^{3/2}}{\sqrt {3} e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {627, 32} \[ -\frac {2 (2-e x)^{3/2}}{\sqrt {3} e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 627
Rubi steps
\begin {align*} \int \frac {\sqrt {12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx &=\int \sqrt {6-3 e x} \, dx\\ &=-\frac {2 (2-e x)^{3/2}}{\sqrt {3} e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 34, normalized size = 1.70 \[ \frac {2 (e x-2) \sqrt {4-e^2 x^2}}{e \sqrt {3 e x+6}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.02, size = 37, normalized size = 1.85 \[ \frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} {\left (e x - 2\right )}}{3 \, {\left (e^{2} x + 2 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-3 \, e^{2} x^{2} + 12}}{\sqrt {e x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 30, normalized size = 1.50 \[ \frac {2 \left (e x -2\right ) \sqrt {-3 e^{2} x^{2}+12}}{3 \sqrt {e x +2}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 3.08, size = 25, normalized size = 1.25 \[ \frac {{\left (2 i \, \sqrt {3} e x - 4 i \, \sqrt {3}\right )} \sqrt {e x - 2}}{3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.11, size = 29, normalized size = 1.45 \[ \frac {\left (\frac {2\,x}{3}-\frac {4}{3\,e}\right )\,\sqrt {12-3\,e^2\,x^2}}{\sqrt {e\,x+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \sqrt {3} \int \frac {\sqrt {- e^{2} x^{2} + 4}}{\sqrt {e x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________