Optimal. Leaf size=87 \[ \frac{2 (2-e x)^{9/2}}{3 \sqrt{3} e}-\frac{24 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{128 (2-e x)^{3/2}}{\sqrt{3} e} \]
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Rubi [A] time = 0.0267335, antiderivative size = 87, 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} \[ \frac{2 (2-e x)^{9/2}}{3 \sqrt{3} e}-\frac{24 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{128 (2-e x)^{3/2}}{\sqrt{3} e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int (2+e x)^{5/2} \sqrt{12-3 e^2 x^2} \, dx &=\int \sqrt{6-3 e x} (2+e x)^3 \, dx\\ &=\int \left (64 \sqrt{6-3 e x}-16 (6-3 e x)^{3/2}+\frac{4}{3} (6-3 e x)^{5/2}-\frac{1}{27} (6-3 e x)^{7/2}\right ) \, dx\\ &=-\frac{128 (2-e x)^{3/2}}{\sqrt{3} e}+\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{24 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{2 (2-e x)^{9/2}}{3 \sqrt{3} e}\\ \end{align*}
Mathematica [A] time = 0.0752209, size = 58, normalized size = 0.67 \[ \frac{2 (e x-2) \sqrt{4-e^2 x^2} \left (35 e^3 x^3+330 e^2 x^2+1284 e x+2552\right )}{105 e \sqrt{3 e x+6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 52, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 35\,{e}^{3}{x}^{3}+330\,{e}^{2}{x}^{2}+1284\,ex+2552 \right ) }{315\,e}\sqrt{-3\,{e}^{2}{x}^{2}+12}{\frac{1}{\sqrt{ex+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.71542, size = 96, normalized size = 1.1 \begin{align*} \frac{{\left (70 i \, \sqrt{3} e^{4} x^{4} + 520 i \, \sqrt{3} e^{3} x^{3} + 1248 i \, \sqrt{3} e^{2} x^{2} - 32 i \, \sqrt{3} e x - 10208 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{315 \,{\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 = 2.11747, size = 154, normalized size = 1.77 \begin{align*} \frac{2 \,{\left (35 \, e^{4} x^{4} + 260 \, e^{3} x^{3} + 624 \, e^{2} x^{2} - 16 \, e x - 5104\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{315 \,{\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \sqrt{-3 \, e^{2} x^{2} + 12}{\left (e x + 2\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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