Optimal. Leaf size=79 \[ \frac{1}{16 \sqrt{3} e \sqrt{2-e x}}-\frac{1}{12 \sqrt{3} e \sqrt{2-e x} (e x+2)}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{32 \sqrt{3} e} \]
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Rubi [A] time = 0.0303714, antiderivative size = 86, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {627, 51, 63, 206} \[ -\frac{\sqrt{2-e x}}{16 \sqrt{3} e (e x+2)}+\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (e x+2)}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{32 \sqrt{3} e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\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)^2} \, dx\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (2+e x)}+\frac{1}{4} \int \frac{1}{\sqrt{6-3 e x} (2+e x)^2} \, dx\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (2+e x)}-\frac{\sqrt{2-e x}}{16 \sqrt{3} e (2+e x)}+\frac{1}{32} \int \frac{1}{\sqrt{6-3 e x} (2+e x)} \, dx\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (2+e x)}-\frac{\sqrt{2-e x}}{16 \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 )}{48 e}\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (2+e x)}-\frac{\sqrt{2-e x}}{16 \sqrt{3} e (2+e x)}-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{32 \sqrt{3} e}\\ \end{align*}
Mathematica [C] time = 0.0496762, size = 48, normalized size = 0.61 \[ \frac{\sqrt{e x+2} \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{1}{2}-\frac{e x}{4}\right )}{24 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 93, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ( 288\,ex-576 \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}xe+2\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{-3\,ex+6}-6\,ex-4 \right ) \left ( ex+2 \right ) ^{-{\frac{3}{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{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}} \sqrt{e x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85329, size = 331, normalized size = 4.19 \begin{align*} \frac{3 \, \sqrt{3}{\left (e^{3} x^{3} + 2 \, e^{2} x^{2} - 4 \, e x - 8\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}{\left (3 \, e x + 2\right )} \sqrt{e x + 2}}{576 \,{\left (e^{4} x^{3} + 2 \, e^{3} x^{2} - 4 \, e^{2} x - 8 \, 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^{2} x^{2} \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4} + 4 \sqrt{e x + 2} \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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