Optimal. Leaf size=43 \[ -\frac {8 (2-e x)^{3/2}}{\sqrt {3} e}+\frac {2 \sqrt {3} (2-e x)^{5/2}}{5 e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 = {641, 45}
\begin {gather*} \frac {2 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {8 (2-e x)^{3/2}}{\sqrt {3} e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 641
Rubi steps
\begin {align*} \int \sqrt {2+e x} \sqrt {12-3 e^2 x^2} \, dx &=\int \sqrt {6-3 e x} (2+e x) \, dx\\ &=\int \left (4 \sqrt {6-3 e x}-\frac {1}{3} (6-3 e x)^{3/2}\right ) \, dx\\ &=-\frac {8 (2-e x)^{3/2}}{\sqrt {3} e}+\frac {2 \sqrt {3} (2-e x)^{5/2}}{5 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 42, normalized size = 0.98 \begin {gather*} \frac {2 (-2+e x) (14+3 e x) \sqrt {4-e^2 x^2}}{5 e \sqrt {6+3 e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.49, size = 36, normalized size = 0.84
method | result | size |
gosper | \(\frac {2 \left (e x -2\right ) \left (3 e x +14\right ) \sqrt {-3 e^{2} x^{2}+12}}{15 e \sqrt {e x +2}}\) | \(36\) |
default | \(\frac {2 \left (e x -2\right ) \left (3 e x +14\right ) \sqrt {-3 e^{2} x^{2}+12}}{15 e \sqrt {e x +2}}\) | \(36\) |
risch | \(-\frac {2 \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (3 e^{2} x^{2}+8 e x -28\right ) \left (e x -2\right )}{5 \sqrt {-3 e^{2} x^{2}+12}\, e \sqrt {-3 e x +6}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] Result contains complex when optimal does not.
time = 0.51, size = 51, normalized size = 1.19 \begin {gather*} -\frac {2 \, {\left (-3 i \, \sqrt {3} x^{2} e^{2} - 8 i \, \sqrt {3} x e + 28 i \, \sqrt {3}\right )} {\left (x e + 2\right )} \sqrt {x e - 2}}{15 \, {\left (x e^{2} + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.19, size = 46, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (3 \, x^{2} e^{2} + 8 \, x e - 28\right )} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{15 \, {\left (x e^{2} + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \sqrt {3} \int \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.49, size = 60, normalized size = 1.40 \begin {gather*} \frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {2\,x^2\,\sqrt {e\,x+2}}{5}-\frac {56\,\sqrt {e\,x+2}}{15\,e^2}+\frac {16\,x\,\sqrt {e\,x+2}}{15\,e}\right )}{x+\frac {2}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________