Optimal. Leaf size=43 \[ \frac {2 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {8 \sqrt {2-e x}}{\sqrt {3} e} \]
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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 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {8 \sqrt {2-e x}}{\sqrt {3} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx &=\int \frac {2+e x}{\sqrt {6-3 e x}} \, dx\\ &=\int \left (\frac {4}{\sqrt {6-3 e x}}-\frac {1}{3} \sqrt {6-3 e x}\right ) \, dx\\ &=-\frac {8 \sqrt {2-e x}}{\sqrt {3} e}+\frac {2 (2-e x)^{3/2}}{3 \sqrt {3} e}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 40, normalized size = 0.93 \begin {gather*} \frac {2 (e x-2) \sqrt {e x+2} (e x+10)}{3 e \sqrt {12-3 e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 56, normalized size = 1.30 \begin {gather*} -\frac {2 \left (\sqrt {3} (e x+2)+8 \sqrt {3}\right ) \sqrt {4 (e x+2)-(e x+2)^2}}{9 e \sqrt {e x+2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 37, normalized size = 0.86 \begin {gather*} -\frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 10\right )} \sqrt {e x + 2}}{9 \, {\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + 2\right )}^{\frac {3}{2}}}{\sqrt {-3 \, e^{2} x^{2} + 12}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 0.81 \begin {gather*} \frac {2 \left (e x -2\right ) \left (e x +10\right ) \sqrt {e x +2}}{3 \sqrt {-3 e^{2} x^{2}+12}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 3.09, size = 28, normalized size = 0.65 \begin {gather*} -\frac {2 i \, \sqrt {3} {\left (e^{2} x^{2} + 8 \, e x - 20\right )}}{9 \, \sqrt {e x - 2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 49, normalized size = 1.14 \begin {gather*} -\frac {\left (\frac {20\,\sqrt {e\,x+2}}{9\,e^2}+\frac {2\,x\,\sqrt {e\,x+2}}{9\,e}\right )\,\sqrt {12-3\,e^2\,x^2}}{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*} \frac {\sqrt {3} \left (\int \frac {2 \sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {e x \sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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