Optimal. Leaf size=22 \[ -\frac{6 \sqrt{3} (2-e x)^{5/2}}{5 e} \]
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Rubi [A] time = 0.0094595, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {627, 32} \[ -\frac{6 \sqrt{3} (2-e x)^{5/2}}{5 e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 32
Rubi steps
\begin{align*} \int \frac{\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{3/2}} \, dx &=\int (6-3 e x)^{3/2} \, dx\\ &=-\frac{6 \sqrt{3} (2-e x)^{5/2}}{5 e}\\ \end{align*}
Mathematica [A] time = 0.0447272, size = 37, normalized size = 1.68 \[ -\frac{6 (e x-2)^2 \sqrt{12-3 e^2 x^2}}{5 e \sqrt{e x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 30, normalized size = 1.4 \begin{align*}{\frac{2\,ex-4}{5\,e} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{{\frac{3}{2}}} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.78105, size = 49, normalized size = 2.23 \begin{align*} -\frac{{\left (6 i \, \sqrt{3} e^{2} x^{2} - 24 i \, \sqrt{3} e x + 24 i \, \sqrt{3}\right )} \sqrt{e x - 2}}{5 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7918, size = 105, normalized size = 4.77 \begin{align*} -\frac{6 \,{\left (e^{2} x^{2} - 4 \, e x + 4\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{5 \,{\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*} 3 \sqrt{3} \left (\int \frac{4 \sqrt{- e^{2} x^{2} + 4}}{e x \sqrt{e x + 2} + 2 \sqrt{e x + 2}}\, dx + \int - \frac{e^{2} x^{2} \sqrt{- e^{2} x^{2} + 4}}{e x \sqrt{e x + 2} + 2 \sqrt{e x + 2}}\, dx\right ) \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 (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}}}{{\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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