Intuition of Liouville's Theorem (differential algebra) Proof

Question

At the end of my abstract algebra class this spring, we were given an overview of differential algebra and some differential Galois theory. We went too fast to prove anything nontrivial, but I found Liouville's theorem on elementary antiderivatives really interesting and have been trying to get a handle on the proof recently. The theorem is as follows:

Theorem (Liouville) Let $\mathbb{F}$ be a differential field. For $a\in \mathbb{F}$, there exists an elementary differential extension $\mathbb{E}$ containing an antiderivative of $a$ if and only if there exist $u_1,...,u_n,v\in\mathbb{F}$, and constants $c_1,...,c_n\in\mathbb{F}$ such that

$$ a = v' + \sum_{i=1}^nc_i\frac{u_i'}{u_i}.$$

The only proof I've found online is that due to Maxwell Rosenlicht (see theorem 4.3 here). Its logic is reasonably easy to follow but it's long enough that I can't piece together an intuition from it. Does anyone have / know of an intuitive account for why Liouville's differential algebra theorem is true?