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I get this in a very old version of Sage's Maxima, the current Sage Maxima, and the most recent one I have.

Maxima 5.27.0 http://maxima.sourceforge.net
using Lisp SBCL 1.0.24
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1)  residue(( x/((4*x - 1)*(t^2 - t + x)) ),t,(1-sqrt(1-4*x))/2);
(%o1)                                  0

What/when was the answer you got before? I can't get this to do anything else in any version of Sage, either, not just the Maxima in it.

This is now Maxima artifact 3541292; I've also put it at

I get this in a very old version of Sage's Maxima, the current Sage Maxima, and the most recent one I have.

Maxima 5.27.0 http://maxima.sourceforge.net
using Lisp SBCL 1.0.24
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1)  residue(( x/((4*x - 1)*(t^2 - t + x)) ),t,(1-sqrt(1-4*x))/2);
(%o1)                                  0

What/when was the answer you got before? I can't get this to do anything else in any version of Sage, either, not just the Maxima in it.

This is now Maxima artifact 3541292; I've also put it at the relevant Sage ticket.