Optimal. Leaf size=20 \[ -\frac{2 \sqrt{2-e x}}{\sqrt{3} e} \]
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Rubi [A] time = 0.0085675, 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} \[ -\frac{2 \sqrt{2-e x}}{\sqrt{3} e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 32
Rubi steps
\begin{align*} \int \frac{\sqrt{2+e x}}{\sqrt{12-3 e^2 x^2}} \, dx &=\int \frac{1}{\sqrt{6-3 e x}} \, dx\\ &=-\frac{2 \sqrt{2-e x}}{\sqrt{3} e}\\ \end{align*}
Mathematica [A] time = 0.0347937, size = 33, normalized size = 1.65 \[ \frac{2 (e x-2) \sqrt{e x+2}}{e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 30, normalized size = 1.5 \begin{align*} 2\,{\frac{ \left ( ex-2 \right ) \sqrt{ex+2}}{e\sqrt{-3\,{e}^{2}{x}^{2}+12}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.73152, size = 34, normalized size = 1.7 \begin{align*} -\frac{2 i \, \sqrt{3} e x - 4 i \, \sqrt{3}}{3 \, \sqrt{e x - 2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75109, size = 76, normalized size = 3.8 \begin{align*} -\frac{2 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{3 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{3} \int \frac{\sqrt{e x + 2}}{\sqrt{- e^{2} x^{2} + 4}}\, dx}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + 2}}{\sqrt{-3 \, e^{2} x^{2} + 12}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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