Optimal. Leaf size=35 \[ -\frac{\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (e x+2)^{3/2}} \]
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Rubi [A] time = 0.0141082, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {651} \[ -\frac{\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (e x+2)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin{align*} \int \frac{1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx &=-\frac{\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (2+e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0460585, size = 35, normalized size = 1. \[ \frac{e x-2}{3 e \sqrt{e x+2} \sqrt [4]{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 30, normalized size = 0.9 \begin{align*}{\frac{ex-2}{3\,e}{\frac{1}{\sqrt{ex+2}}}{\frac{1}{\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}}}} \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{1}{4}}{\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 [A] time = 1.85474, size = 95, normalized size = 2.71 \begin{align*} -\frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{4}} \sqrt{e x + 2}}{9 \,{\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{3^{\frac{3}{4}} \int \frac{1}{e x \sqrt{e x + 2} \sqrt [4]{- e^{2} x^{2} + 4} + 2 \sqrt{e x + 2} \sqrt [4]{- 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}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\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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