Optimal. Leaf size=67 \[ -\frac{2 (2-e x)^{3/2}}{9 \sqrt{3} e}+\frac{16 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{32}{3 \sqrt{3} e \sqrt{2-e x}} \]
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Rubi [A] time = 0.021156, antiderivative size = 67, 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} \[ -\frac{2 (2-e x)^{3/2}}{9 \sqrt{3} e}+\frac{16 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{32}{3 \sqrt{3} e \sqrt{2-e x}} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac{(2+e x)^2}{(6-3 e x)^{3/2}} \, dx\\ &=\int \left (\frac{16}{(6-3 e x)^{3/2}}-\frac{8}{3 \sqrt{6-3 e x}}+\frac{1}{9} \sqrt{6-3 e x}\right ) \, dx\\ &=\frac{32}{3 \sqrt{3} e \sqrt{2-e x}}+\frac{16 \sqrt{2-e x}}{3 \sqrt{3} e}-\frac{2 (2-e x)^{3/2}}{9 \sqrt{3} e}\\ \end{align*}
Mathematica [A] time = 0.0643718, size = 43, normalized size = 0.64 \[ -\frac{2 \sqrt{e x+2} \left (e^2 x^2+20 e x-92\right )}{9 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 43, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ({e}^{2}{x}^{2}+20\,ex-92 \right ) }{3\,e} \left ( ex+2 \right ) ^{{\frac{3}{2}}} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.68838, size = 49, normalized size = 0.73 \begin{align*} \frac{2 i \, \sqrt{3} e^{2} x^{2} + 40 i \, \sqrt{3} e x - 184 i \, \sqrt{3}}{27 \, \sqrt{e x - 2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85742, size = 111, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (e^{2} x^{2} + 20 \, e x - 92\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{27 \,{\left (e^{3} x^{2} - 4 \, 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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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