Optimal. Leaf size=20 \[ -\frac {2 \sqrt {2-e x}}{\sqrt {3} e} \]
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Rubi [A] time = 0.01, antiderivative size = 20, 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} \[ -\frac {2 \sqrt {2-e x}}{\sqrt {3} e} \]
Antiderivative was successfully verified.
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Rule 32
Rule 627
Rubi steps
\begin {align*} \int \frac {\sqrt {2+e x}}{\sqrt {12-3 e^2 x^2}} \, dx &=\int \frac {1}{\sqrt {6-3 e x}} \, dx\\ &=-\frac {2 \sqrt {2-e x}}{\sqrt {3} e}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 1.65 \[ \frac {2 (e x-2) \sqrt {e x+2}}{e \sqrt {12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 32, normalized size = 1.60 \[ -\frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{3 \, {\left (e^{2} x + 2 \, e\right )}} \]
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 \[ \int \frac {\sqrt {e x + 2}}{\sqrt {-3 \, e^{2} x^{2} + 12}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 1.50 \[ \frac {2 \left (e x -2\right ) \sqrt {e x +2}}{\sqrt {-3 e^{2} x^{2}+12}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.98, size = 25, normalized size = 1.25 \[ -\frac {2 i \, \sqrt {3} e x - 4 i \, \sqrt {3}}{3 \, \sqrt {e x - 2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 24, normalized size = 1.20 \[ -\frac {2\,\sqrt {12-3\,e^2\,x^2}}{3\,e\,\sqrt {e\,x+2}} \]
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 \[ \frac {\sqrt {3} \int \frac {\sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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