Optimal. Leaf size=45 \[ \frac{2 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{8}{3 \sqrt{3} e \sqrt{2-e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0171074, 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} \[ \frac{2 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{8}{3 \sqrt{3} e \sqrt{2-e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(2+e x)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac{2+e x}{(6-3 e x)^{3/2}} \, dx\\ &=\int \left (\frac{4}{(6-3 e x)^{3/2}}-\frac{1}{3 \sqrt{6-3 e x}}\right ) \, dx\\ &=\frac{8}{3 \sqrt{3} e \sqrt{2-e x}}+\frac{2 \sqrt{2-e x}}{3 \sqrt{3} e}\\ \end{align*}
Mathematica [A] time = 0.0566121, size = 35, normalized size = 0.78 \[ -\frac{2 (e x-6) \sqrt{e x+2}}{3 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 35, normalized size = 0.8 \begin{align*} 2\,{\frac{ \left ( ex-2 \right ) \left ( ex-6 \right ) \left ( ex+2 \right ) ^{3/2}}{e \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.55537, size = 27, normalized size = 0.6 \begin{align*} \frac{2 i \, \sqrt{3}{\left (e x - 6\right )}}{9 \, \sqrt{e x - 2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.82831, size = 90, normalized size = 2. \begin{align*} \frac{2 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}{\left (e x - 6\right )}}{9 \,{\left (e^{3} x^{2} - 4 \, e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]