Locally reduce space between equation and qed / placement of qed

Question

If (using amsthm) I have a proof which ends on an equation, and after that some text goes on, I have something like:

text / (bit of space) / equation / (bit of space which looks huge because there is nothing in the line with qed) / empty line with the qed / (bit of space) / further text.

Here's a MWE ...

\documentclass{scrartcl}
\usepackage[UKenglish]{babel}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{amsthm}

\begin{document}

\begin{proof}
Here is a bit of text which explains how the last step will work. The last step will only be some easy equation or some property with no need to write text between this and the q.e.d.
\begin{equation*}
\int\limits_1^\infty f(t) g(t) dt = \Biggl[ \frac{a^b}{c^d} \Biggr] = \infty 
\end{equation*}
\end{proof}
And then, some other text goes on...

\end{document}

...and here's a picture of it:

What I would like to have is the qed to move to the blue position (that is the "lower border" of the equation line).

I tried inserting \vspace{-...} which - no matter if I put it in the equation, between the equation or the proof - will just move the text of after the proof upwards (yellow arrow) while leaving the rest unchanged. I also tried \qedhere in the equation's line, which is almost what I would like to have but then the qed is centered in the equation line (indicated in red).