Generating function for cardinality of Bruhat intervals

Question

For a finite Coxeter system $(W, S)$, the Poincare polynomial counts the number of elements of a certain length: $$ P_W(t) = \sum_{x \in W} t^{l(x)}. $$ The polynomials for types $A$, $B$, and $D$ are relatively simple to write down, for example we have in type $A_n$ that $$ P_W(t) = \prod_{i = 1}^{n}(1 + t + t^2 + \cdots + t^i). $$ For two elements $x, y \in W$ such that $x \leq y$ in the Bruhat order, let $[x, y]$ denote the interval $\{z \in W \mid x \leq z \leq y\}$. I am most interested in the intervals $[e, x]$ from the identity up to some $x \in W$. Let $Q_W(t)$ be the generating function for the cardinality of these intervals: $$ Q_W(t) := \sum_{x \in W} t^{\#[\operatorname{id}, x]}.$$ Question: Is there a closed form for the polynomials $Q_W(t)$, at least in finite types $A$, $B$, $D$?