Optimal. Leaf size=43 \[ \frac {2 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {8 (2-e x)^{3/2}}{\sqrt {3} e} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {627, 43} \begin {gather*} \frac {2 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {8 (2-e x)^{3/2}}{\sqrt {3} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int \sqrt {2+e x} \sqrt {12-3 e^2 x^2} \, dx &=\int \sqrt {6-3 e x} (2+e x) \, dx\\ &=\int \left (4 \sqrt {6-3 e x}-\frac {1}{3} (6-3 e x)^{3/2}\right ) \, dx\\ &=-\frac {8 (2-e x)^{3/2}}{\sqrt {3} e}+\frac {2 \sqrt {3} (2-e x)^{5/2}}{5 e}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 0.98 \begin {gather*} \frac {2 (e x-2) (3 e x+14) \sqrt {4-e^2 x^2}}{5 e \sqrt {3 e x+6}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 51, normalized size = 1.19 \begin {gather*} -\frac {2 (3 (e x+2)+8) \left (4 (e x+2)-(e x+2)^2\right )^{3/2}}{5 \sqrt {3} e (e x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 46, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (3 \, e^{2} x^{2} + 8 \, e x - 28\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{15 \, {\left (e^{2} x + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 0.84 \begin {gather*} \frac {2 \left (e x -2\right ) \left (3 e x +14\right ) \sqrt {-3 e^{2} x^{2}+12}}{15 \sqrt {e x +2}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 3.12, size = 49, normalized size = 1.14 \begin {gather*} \frac {{\left (6 i \, \sqrt {3} e^{2} x^{2} + 16 i \, \sqrt {3} e x - 56 i \, \sqrt {3}\right )} {\left (e x + 2\right )} \sqrt {e x - 2}}{15 \, {\left (e^{2} x + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 60, normalized size = 1.40 \begin {gather*} \frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {2\,x^2\,\sqrt {e\,x+2}}{5}-\frac {56\,\sqrt {e\,x+2}}{15\,e^2}+\frac {16\,x\,\sqrt {e\,x+2}}{15\,e}\right )}{x+\frac {2}{e}} \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 {3} \int \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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