Optimal. Leaf size=43 \[ \frac{2 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{8 \sqrt{2-e x}}{\sqrt{3} e} \]
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Rubi [A] time = 0.0167946, 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} \[ \frac{2 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{8 \sqrt{2-e x}}{\sqrt{3} e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(2+e x)^{3/2}}{\sqrt{12-3 e^2 x^2}} \, dx &=\int \frac{2+e x}{\sqrt{6-3 e x}} \, dx\\ &=\int \left (\frac{4}{\sqrt{6-3 e x}}-\frac{1}{3} \sqrt{6-3 e x}\right ) \, dx\\ &=-\frac{8 \sqrt{2-e x}}{\sqrt{3} e}+\frac{2 (2-e x)^{3/2}}{3 \sqrt{3} e}\\ \end{align*}
Mathematica [A] time = 0.0506008, size = 40, normalized size = 0.93 \[ \frac{2 (e x-2) \sqrt{e x+2} (e x+10)}{3 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 35, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( ex+10 \right ) }{3\,e}\sqrt{ex+2}{\frac{1}{\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 = 1.96245, size = 38, normalized size = 0.88 \begin{align*} -\frac{2 i \, \sqrt{3}{\left (e^{2} x^{2} + 8 \, e x - 20\right )}}{9 \, \sqrt{e x - 2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72781, size = 90, normalized size = 2.09 \begin{align*} -\frac{2 \, \sqrt{-3 \, e^{2} x^{2} + 12}{\left (e x + 10\right )} \sqrt{e x + 2}}{9 \,{\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} \left (\int \frac{2 \sqrt{e x + 2}}{\sqrt{- e^{2} x^{2} + 4}}\, dx + \int \frac{e x \sqrt{e x + 2}}{\sqrt{- e^{2} x^{2} + 4}}\, dx\right )}{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{{\left (e x + 2\right )}^{\frac{3}{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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