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 = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{5 e (e x+2)^{5/2}} \end {gather*}
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*}
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Mathematica [A] time = 0.04, size = 35, normalized size = 1.00 \begin {gather*} \frac {(e x-2) \sqrt [4]{12-3 e^2 x^2}}{5 e (e x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 42, normalized size = 1.20 \begin {gather*} -\frac {\sqrt [4]{3} \left (4 (e x+2)-(e x+2)^2\right )^{5/4}}{5 e (e x+2)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 45, normalized size = 1.29 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 46, normalized size = 1.31 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 30, normalized size = 0.86 \begin {gather*} \frac {\left (e x -2\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}{5 \left (e x +2\right )^{\frac {3}{2}} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}}{{\left (e x + 2\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 49, normalized size = 1.40 \begin {gather*} \frac {\left (\frac {x}{5\,e}-\frac {2}{5\,e^2}\right )\,{\left (12-3\,e^2\,x^2\right )}^{1/4}}{\frac {2\,\sqrt {e\,x+2}}{e}+x\,\sqrt {e\,x+2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \sqrt [4]{3} \int \frac {\sqrt [4]{- e^{2} x^{2} + 4}}{e^{2} x^{2} \sqrt {e x + 2} + 4 e x \sqrt {e x + 2} + 4 \sqrt {e x + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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