Optimal. Leaf size=65 \[ -\frac{2 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{16 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{32 (2-e x)^{3/2}}{\sqrt{3} e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0202403, 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} \[ -\frac{2 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{16 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{32 (2-e x)^{3/2}}{\sqrt{3} e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 627
Rule 43
Rubi steps
\begin{align*} \int (2+e x)^{3/2} \sqrt{12-3 e^2 x^2} \, dx &=\int \sqrt{6-3 e x} (2+e x)^2 \, dx\\ &=\int \left (16 \sqrt{6-3 e x}-\frac{8}{3} (6-3 e x)^{3/2}+\frac{1}{9} (6-3 e x)^{5/2}\right ) \, dx\\ &=-\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}+\frac{16 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{2 \sqrt{3} (2-e x)^{7/2}}{7 e}\\ \end{align*}
Mathematica [A] time = 0.0554336, size = 50, normalized size = 0.77 \[ \frac{2 (e x-2) \sqrt{4-e^2 x^2} \left (15 e^2 x^2+108 e x+284\right )}{35 e \sqrt{3 e x+6}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.042, size = 44, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 15\,{e}^{2}{x}^{2}+108\,ex+284 \right ) }{105\,e}\sqrt{-3\,{e}^{2}{x}^{2}+12}{\frac{1}{\sqrt{ex+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.71175, size = 81, normalized size = 1.25 \begin{align*} \frac{{\left (30 i \, \sqrt{3} e^{3} x^{3} + 156 i \, \sqrt{3} e^{2} x^{2} + 136 i \, \sqrt{3} e x - 1136 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{105 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.09546, size = 132, normalized size = 2.03 \begin{align*} \frac{2 \,{\left (15 \, e^{3} x^{3} + 78 \, e^{2} x^{2} + 68 \, e x - 568\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{105 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt{3} \left (\int 2 \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}\, dx + \int e x \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}\, dx\right ) \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*} \int \sqrt{-3 \, e^{2} x^{2} + 12}{\left (e x + 2\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]