Optimal. Leaf size=65 \[ -\frac{2 (2-e x)^{9/2}}{\sqrt{3} e}+\frac{48 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e} \]
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Rubi [A] time = 0.0196335, antiderivative size = 65, normalized size of antiderivative = 1., 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} \[ -\frac{2 (2-e x)^{9/2}}{\sqrt{3} e}+\frac{48 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \sqrt{2+e x} \left (12-3 e^2 x^2\right )^{3/2} \, dx &=\int (6-3 e x)^{3/2} (2+e x)^2 \, dx\\ &=\int \left (16 (6-3 e x)^{3/2}-\frac{8}{3} (6-3 e x)^{5/2}+\frac{1}{9} (6-3 e x)^{7/2}\right ) \, dx\\ &=-\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{48 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{2 (2-e x)^{9/2}}{\sqrt{3} e}\\ \end{align*}
Mathematica [A] time = 0.0447866, size = 52, normalized size = 0.8 \[ -\frac{2 (e x-2)^2 \sqrt{4-e^2 x^2} \left (35 e^2 x^2+220 e x+428\right )}{35 e \sqrt{3 e x+6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 44, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 35\,{e}^{2}{x}^{2}+220\,ex+428 \right ) }{315\,e} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{{\frac{3}{2}}} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.99547, size = 96, normalized size = 1.48 \begin{align*} -\frac{{\left (70 i \, \sqrt{3} e^{4} x^{4} + 160 i \, \sqrt{3} e^{3} x^{3} - 624 i \, \sqrt{3} e^{2} x^{2} - 1664 i \, \sqrt{3} e x + 3424 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{105 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86763, size = 155, normalized size = 2.38 \begin{align*} -\frac{2 \,{\left (35 \, e^{4} x^{4} + 80 \, e^{3} x^{3} - 312 \, e^{2} x^{2} - 832 \, e x + 1712\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{105 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} 3 \sqrt{3} \left (\int 4 \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}\, dx + \int - e^{2} x^{2} \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}} \sqrt{e x + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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