Optimal. Leaf size=45 \[ \frac{6 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{24 \sqrt{3} (2-e x)^{5/2}}{5 e} \]
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Rubi [A] time = 0.0156231, antiderivative size = 45, 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} \[ \frac{6 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{24 \sqrt{3} (2-e x)^{5/2}}{5 e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (12-3 e^2 x^2\right )^{3/2}}{\sqrt{2+e x}} \, dx &=\int (6-3 e x)^{3/2} (2+e x) \, dx\\ &=\int \left (4 (6-3 e x)^{3/2}-\frac{1}{3} (6-3 e x)^{5/2}\right ) \, dx\\ &=-\frac{24 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{6 \sqrt{3} (2-e x)^{7/2}}{7 e}\\ \end{align*}
Mathematica [A] time = 0.0545767, size = 43, normalized size = 0.96 \[ -\frac{6 (e x-2)^2 (5 e x+18) \sqrt{12-3 e^2 x^2}}{35 e \sqrt{e x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 36, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 5\,ex+18 \right ) }{35\,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 = 1.78863, size = 63, normalized size = 1.4 \begin{align*} -\frac{{\left (30 i \, \sqrt{3} e^{3} x^{3} - 12 i \, \sqrt{3} e^{2} x^{2} - 312 i \, \sqrt{3} e x + 432 i \, \sqrt{3}\right )} \sqrt{e x - 2}}{35 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77618, size = 128, normalized size = 2.84 \begin{align*} -\frac{6 \,{\left (5 \, e^{3} x^{3} - 2 \, e^{2} x^{2} - 52 \, e x + 72\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{35 \,{\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 \frac{4 \sqrt{- e^{2} x^{2} + 4}}{\sqrt{e x + 2}}\, dx + \int - \frac{e^{2} x^{2} \sqrt{- e^{2} x^{2} + 4}}{\sqrt{e x + 2}}\, 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 \frac{{\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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