Consider two vector fields:
$f_1(x)=\begin{bmatrix} x_1+3x_2^2x_3\\ x_3-x_1\\ x_2+x_3x_1 \end{bmatrix}$
$f_2(x)=\begin{bmatrix} x_3+\frac{x_1^2}{x_2}\\ x_1\\ 0 \end{bmatrix}$
Does it exists a quick way to check if these vector fields are independent with respect to each other?
I don't think you need a quick way. Usually you would check if there are any $\alpha, \beta$, that are not all equal to $0$ and
$$\alpha f_1 + \beta f_2 = 0.$$
The third row shows that $\alpha = 0$ for any $\beta$. Then the first two rows imply $\beta = 0$. $\alpha = \beta = 0$ is sufficient for the independence of $f_1$ and $f_2$.
If you only have two vectors, as it is in your example, you can also check the existence of $\alpha$, so that $\alpha f_1 = f_2$.
Note that $\alpha, \beta$ are not dependent on $x_1,x_2,x_3$.