Optimal. Leaf size=22 \[ \frac{2}{3 \sqrt{3} e \sqrt{2-e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0096814, antiderivative size = 22, 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} \[ \frac{2}{3 \sqrt{3} e \sqrt{2-e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 627
Rule 32
Rubi steps
\begin{align*} \int \frac{(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac{1}{(6-3 e x)^{3/2}} \, dx\\ &=\frac{2}{3 \sqrt{3} e \sqrt{2-e x}}\\ \end{align*}
Mathematica [A] time = 0.0455999, size = 30, normalized size = 1.36 \[ \frac{2 \sqrt{e x+2}}{3 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 30, normalized size = 1.4 \begin{align*} -2\,{\frac{ \left ( ex-2 \right ) \left ( ex+2 \right ) ^{3/2}}{e \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.71926, size = 20, normalized size = 0.91 \begin{align*} -\frac{2 i \, \sqrt{3}}{9 \, \sqrt{e x - 2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.67795, size = 78, normalized size = 3.55 \begin{align*} -\frac{2 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{9 \,{\left (e^{3} x^{2} - 4 \, 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*} \frac{\sqrt{3} \left (\int \frac{2 \sqrt{e x + 2}}{- e^{2} x^{2} \sqrt{- e^{2} x^{2} + 4} + 4 \sqrt{- e^{2} x^{2} + 4}}\, dx + \int \frac{e x \sqrt{e x + 2}}{- e^{2} x^{2} \sqrt{- e^{2} x^{2} + 4} + 4 \sqrt{- e^{2} x^{2} + 4}}\, dx\right )}{9} \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]