Optimal. Leaf size=50 \[ \frac{1}{6 \sqrt{3} e \sqrt{2-e x}}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{12 \sqrt{3} e} \]
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Rubi [A] time = 0.0209086, antiderivative size = 50, 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} \[ \frac{1}{6 \sqrt{3} e \sqrt{2-e x}}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{12 \sqrt{3} e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{2+e x}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac{1}{(6-3 e x)^{3/2} (2+e x)} \, dx\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x}}+\frac{1}{12} \int \frac{1}{\sqrt{6-3 e x} (2+e x)} \, dx\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x}}-\frac{\operatorname{Subst}\left (\int \frac{1}{4-\frac{x^2}{3}} \, dx,x,\sqrt{6-3 e x}\right )}{18 e}\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x}}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{12 \sqrt{3} e}\\ \end{align*}
Mathematica [C] time = 0.0423811, size = 48, normalized size = 0.96 \[ \frac{\sqrt{e x+2} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{1}{2}-\frac{e x}{4}\right )}{6 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 60, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ( 108\,ex-216 \right ) e}\sqrt{-3\,{e}^{2}{x}^{2}+12} \left ( \sqrt{3}{\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ) \sqrt{-3\,ex+6}-6 \right ){\frac{1}{\sqrt{ex+2}}}} \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{e x + 2}}{{\left (-3 \, e^{2} x^{2} + 12\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 = 1.84519, size = 254, normalized size = 5.08 \begin{align*} \frac{\sqrt{3}{\left (e^{2} x^{2} - 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}}{72 \,{\left (e^{3} x^{2} - 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{\sqrt{e x + 2}}{- e^{2} x^{2} \sqrt{- e^{2} x^{2} + 4} + 4 \sqrt{- e^{2} x^{2} + 4}}\, dx}{9} \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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