Optimal. Leaf size=20 \[ -\frac{2 (2-e x)^{3/2}}{\sqrt{3} e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0090526, antiderivative size = 20, normalized size of antiderivative = 1., 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 627
Rule 32
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 = 0.0380533, size = 34, normalized size = 1.7 \[ \frac{2 (e x-2) \sqrt{4-e^2 x^2}}{e \sqrt{3 e x+6}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 30, normalized size = 1.5 \begin{align*}{\frac{2\,ex-4}{3\,e}\sqrt{-3\,{e}^{2}{x}^{2}+12}{\frac{1}{\sqrt{ex+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.87714, size = 34, normalized size = 1.7 \begin{align*} \frac{{\left (2 i \, \sqrt{3} e x - 4 i \, \sqrt{3}\right )} \sqrt{e x - 2}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.04526, size = 88, normalized size = 4.4 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt{3} \int \frac{\sqrt{- e^{2} x^{2} + 4}}{\sqrt{e x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-3 \, e^{2} x^{2} + 12}}{\sqrt{e x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]