Optimal. Leaf size=87 \[ \frac{2 (2-e x)^{5/2}}{15 \sqrt{3} e}-\frac{8 (2-e x)^{3/2}}{3 \sqrt{3} e}+\frac{32 \sqrt{2-e x}}{\sqrt{3} e}+\frac{128}{3 \sqrt{3} e \sqrt{2-e x}} \]
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Rubi [A] time = 0.0245445, 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} \[ \frac{2 (2-e x)^{5/2}}{15 \sqrt{3} e}-\frac{8 (2-e x)^{3/2}}{3 \sqrt{3} e}+\frac{32 \sqrt{2-e x}}{\sqrt{3} e}+\frac{128}{3 \sqrt{3} e \sqrt{2-e x}} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(2+e x)^{9/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac{(2+e x)^3}{(6-3 e x)^{3/2}} \, dx\\ &=\int \left (\frac{64}{(6-3 e x)^{3/2}}-\frac{16}{\sqrt{6-3 e x}}+\frac{4}{3} \sqrt{6-3 e x}-\frac{1}{27} (6-3 e x)^{3/2}\right ) \, dx\\ &=\frac{128}{3 \sqrt{3} e \sqrt{2-e x}}+\frac{32 \sqrt{2-e x}}{\sqrt{3} e}-\frac{8 (2-e x)^{3/2}}{3 \sqrt{3} e}+\frac{2 (2-e x)^{5/2}}{15 \sqrt{3} e}\\ \end{align*}
Mathematica [A] time = 0.0805123, size = 51, normalized size = 0.59 \[ -\frac{2 \sqrt{e x+2} \left (e^3 x^3+14 e^2 x^2+172 e x-728\right )}{15 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 51, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ({e}^{3}{x}^{3}+14\,{e}^{2}{x}^{2}+172\,ex-728 \right ) }{5\,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 = 1.57932, size = 49, normalized size = 0.56 \begin{align*} \frac{2 i \, \sqrt{3}{\left (e^{3} x^{3} + 14 \, e^{2} x^{2} + 172 \, e x - 728\right )}}{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 = 1.75109, size = 131, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (e^{3} x^{3} + 14 \, e^{2} x^{2} + 172 \, e x - 728\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{45 \,{\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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