Optimal. Leaf size=43 \[ \frac{2 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{8 (2-e x)^{3/2}}{\sqrt{3} e} \]
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Rubi [A] time = 0.0153062, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {627, 43} \[ \frac{2 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{8 (2-e x)^{3/2}}{\sqrt{3} e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \sqrt{2+e x} \sqrt{12-3 e^2 x^2} \, dx &=\int \sqrt{6-3 e x} (2+e x) \, dx\\ &=\int \left (4 \sqrt{6-3 e x}-\frac{1}{3} (6-3 e x)^{3/2}\right ) \, dx\\ &=-\frac{8 (2-e x)^{3/2}}{\sqrt{3} e}+\frac{2 \sqrt{3} (2-e x)^{5/2}}{5 e}\\ \end{align*}
Mathematica [A] time = 0.0439205, size = 42, normalized size = 0.98 \[ \frac{2 (e x-2) (3 e x+14) \sqrt{4-e^2 x^2}}{5 e \sqrt{3 e x+6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 36, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 3\,ex+14 \right ) }{15\,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.
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Maxima [C] time = 2.1625, size = 66, normalized size = 1.53 \begin{align*} \frac{{\left (6 i \, \sqrt{3} e^{2} x^{2} + 16 i \, \sqrt{3} e x - 56 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{15 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11975, size = 109, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (3 \, e^{2} x^{2} + 8 \, e x - 28\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{15 \,{\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*} \sqrt{3} \int \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}\, dx \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 \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.
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