Optimal. Leaf size=22 \[ \frac {2}{3 \sqrt {3} e \sqrt {2-e x}} \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \[ \frac {2}{3 \sqrt {3} e \sqrt {2-e x}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 627
Rubi steps
\begin {align*} \int \frac {(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{(6-3 e x)^{3/2}} \, dx\\ &=\frac {2}{3 \sqrt {3} e \sqrt {2-e x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 30, normalized size = 1.36 \[ \frac {2 \sqrt {e x+2}}{3 e \sqrt {12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 34, normalized size = 1.55 \[ -\frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{9 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 30, normalized size = 1.36 \[ -\frac {2 \left (e x -2\right ) \left (e x +2\right )^{\frac {3}{2}}}{\left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 3.02, size = 15, normalized size = 0.68 \[ -\frac {2 i \, \sqrt {3}}{9 \, \sqrt {e x - 2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 24, normalized size = 1.09 \[ \frac {2\,\sqrt {e\,x+2}}{3\,e\,\sqrt {12-3\,e^2\,x^2}} \]
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 \[ \frac {\sqrt {3} \left (\int \frac {2 \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {e x \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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