Optimal. Leaf size=57 \[ -\frac{\sqrt{2-e x}}{4 \sqrt{3} e (e x+2)}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{8 \sqrt{3} e} \]
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Rubi [A] time = 0.0207186, antiderivative size = 57, 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, 51, 63, 206} \[ -\frac{\sqrt{2-e x}}{4 \sqrt{3} e (e x+2)}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{8 \sqrt{3} e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(2+e x)^{3/2} \sqrt{12-3 e^2 x^2}} \, dx &=\int \frac{1}{\sqrt{6-3 e x} (2+e x)^2} \, dx\\ &=-\frac{\sqrt{2-e x}}{4 \sqrt{3} e (2+e x)}+\frac{1}{8} \int \frac{1}{\sqrt{6-3 e x} (2+e x)} \, dx\\ &=-\frac{\sqrt{2-e x}}{4 \sqrt{3} e (2+e x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{4-\frac{x^2}{3}} \, dx,x,\sqrt{6-3 e x}\right )}{12 e}\\ &=-\frac{\sqrt{2-e x}}{4 \sqrt{3} e (2+e x)}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{8 \sqrt{3} e}\\ \end{align*}
Mathematica [A] time = 0.0614101, size = 54, normalized size = 0.95 \[ \frac{-2 \sqrt{2-e x}-(e x+2) \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{8 \sqrt{3} e (e x+2)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.127, size = 88, normalized size = 1.5 \begin{align*} -{\frac{\sqrt{3}}{24\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ( \sqrt{3}{\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ) xe+2\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) +2\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{3}}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \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{1}{\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 = 1.78991, size = 278, normalized size = 4.88 \begin{align*} \frac{\sqrt{3}{\left (e^{2} x^{2} + 4 \, e x + 4\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 ) - 4 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{48 \,{\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, 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{1}{e x \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4} + 2 \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{1}{\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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