Optimal. Leaf size=46 \[ \frac{2 \sqrt{3} \sqrt{2-e x}}{e}-\frac{4 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]
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Rubi [A] time = 0.0202946, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {627, 50, 63, 206} \[ \frac{2 \sqrt{3} \sqrt{2-e x}}{e}-\frac{4 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx &=\int \frac{\sqrt{6-3 e x}}{2+e x} \, dx\\ &=\frac{2 \sqrt{3} \sqrt{2-e x}}{e}+12 \int \frac{1}{\sqrt{6-3 e x} (2+e x)} \, dx\\ &=\frac{2 \sqrt{3} \sqrt{2-e x}}{e}-\frac{8 \operatorname{Subst}\left (\int \frac{1}{4-\frac{x^2}{3}} \, dx,x,\sqrt{6-3 e x}\right )}{e}\\ &=\frac{2 \sqrt{3} \sqrt{2-e x}}{e}-\frac{4 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.0602468, size = 63, normalized size = 1.37 \[ \frac{2 \sqrt{12-3 e^2 x^2} \left (\sqrt{e x-2}-2 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )\right )}{e \sqrt{e x-2} \sqrt{e x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.143, size = 66, normalized size = 1.4 \begin{align*} -2\,{\frac{\sqrt{-{e}^{2}{x}^{2}+4} \left ( 2\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -\sqrt{-3\,ex+6} \right ) \sqrt{3}}{\sqrt{ex+2}\sqrt{-3\,ex+6}e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-3 \, e^{2} x^{2} + 12}}{{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13564, size = 239, normalized size = 5.2 \begin{align*} \frac{2 \,{\left (\sqrt{3}{\left (e x + 2\right )} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) + \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}\right )}}{e^{2} x + 2 \, e} \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 \frac{\sqrt{- e^{2} x^{2} + 4}}{e x \sqrt{e x + 2} + 2 \sqrt{e x + 2}}\, 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 \frac{\sqrt{-3 \, e^{2} x^{2} + 12}}{{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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