How should I sample from a mixture distribution?

Let’s say we have a mixture distribution, defined by density $f(x)= w_1 p_1(x) + w_2 p_2(x)$, where $w_i$ is a scalar weight. Furthermore, we have efficient methods to evaluate the pdf and cdf/icdf for the distribution $D_i$ corresponding to density $p_i$. I would like to sample from such a distribution.

The method I currently employ is an implementation of rejection sampling. I construct a proposal function $M*u(x)$, where $u sim text{Unif}(lb,ub)$ ($lb,ub$ are constructed such that at least 99% of the cdf of each $D_i$ is contained within, using icdf) and $M$ is constructed by finding the $max_{x in [lb,ub],i} p_i(x)$. Because such proposal function envelopes $f$, I am able to sample $f$ by choosing $x in X sim text{Unif}(lb,ub) times text{Unif}(0,M)$ and rejecting if $pi_2 x > f(x)$ [$pi_2 x$ being the second coordinate of $x$].

However, doing this is quite slow. Not only is the proposal function inopportune (it is scaled uniform, which likely will lead to many rejections), but the construction of $M$ is very slow, as maximizing a function is a non-trivial task. Is there a more efficient way to sample such a distribution? I had considered icdf sampling, but constructing the icdf for $f$ seems non-trivially difficult. Is this impression incorrect? Or perhaps is there some other effective method? If it is helpful, I am implementing this in python and am currently using the scipy and pytorch libraries.