Optimal. Leaf size=65 \[ -\frac{2 (2-e x)^{5/2}}{5 \sqrt{3} e}+\frac{16 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{32 \sqrt{2-e x}}{\sqrt{3} e} \]
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Rubi [A] time = 0.0199924, 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} \[ -\frac{2 (2-e x)^{5/2}}{5 \sqrt{3} e}+\frac{16 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{32 \sqrt{2-e x}}{\sqrt{3} e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(2+e x)^{5/2}}{\sqrt{12-3 e^2 x^2}} \, dx &=\int \frac{(2+e x)^2}{\sqrt{6-3 e x}} \, dx\\ &=\int \left (\frac{16}{\sqrt{6-3 e x}}-\frac{8}{3} \sqrt{6-3 e x}+\frac{1}{9} (6-3 e x)^{3/2}\right ) \, dx\\ &=-\frac{32 \sqrt{2-e x}}{\sqrt{3} e}+\frac{16 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{2 (2-e x)^{5/2}}{5 \sqrt{3} e}\\ \end{align*}
Mathematica [A] time = 0.0591318, size = 49, normalized size = 0.75 \[ \frac{2 (e x-2) \sqrt{e x+2} \left (3 e^2 x^2+28 e x+172\right )}{15 e \sqrt{12-3 e^2 x^2}} \]
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 ( 3\,{e}^{2}{x}^{2}+28\,ex+172 \right ) }{15\,e}\sqrt{ex+2}{\frac{1}{\sqrt{-3\,{e}^{2}{x}^{2}+12}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.88108, size = 63, normalized size = 0.97 \begin{align*} -\frac{6 i \, \sqrt{3} e^{3} x^{3} + 44 i \, \sqrt{3} e^{2} x^{2} + 232 i \, \sqrt{3} e x - 688 i \, \sqrt{3}}{45 \, \sqrt{e x - 2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82014, size = 113, normalized size = 1.74 \begin{align*} -\frac{2 \,{\left (3 \, e^{2} x^{2} + 28 \, e x + 172\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{45 \,{\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 \frac{{\left (e x + 2\right )}^{\frac{5}{2}}}{\sqrt{-3 \, e^{2} x^{2} + 12}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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