Optimal. Leaf size=35 \[ -\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{5 e (e x+2)^{5/2}} \]
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Rubi [A] time = 0.0100162, 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} \[ -\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{5 e (e x+2)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx &=-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{5 e (2+e x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0533383, size = 35, normalized size = 1. \[ \frac{(e x-2) \sqrt [4]{12-3 e^2 x^2}}{5 e (e x+2)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 30, normalized size = 0.9 \begin{align*}{\frac{ex-2}{5\,e}\sqrt [4]{-3\,{e}^{2}{x}^{2}+12} \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{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}}{{\left (e x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7725, size = 107, normalized size = 3.06 \begin{align*} \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}{\left (e x - 2\right )}}{5 \,{\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25996, size = 62, normalized size = 1.77 \begin{align*} -\frac{3^{\frac{1}{4}}{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}{\left (\frac{4}{x e + 2} - 1\right )} e^{\left (-1\right )}}{5 \, \sqrt{x e + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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