Opinion...

#!/usr/bin/perl

use strict;
#use warnings;

for my $regexp (
  qr/^((a*)b|a*b?d)*c\2$/,
  qr/^(a*b?d|(a*)b)*c\2$/,

  qr/^((a*)b|a*b?d())*c\2$/, # NOTE these two fixed to work the way
  qr/^(?|a*b?d()|(a*)b)*c\1$/, # I think they should work
  )
  {
  for my $string (qw( aabadcaa aababdcaa ))
    {
    print "\n$string\n$regexp\n";
    print $string =~ /$regexp/ ? 'Yes' : 'No', "\n";
    }
  }

print <<END;

This is perl 5, version 38, subversion 0 (v5.38.0) built for x86_64-li
+nux-thread-multi

END
[download]

Outputs:

aabadcaa
(?^:^((a*)b|a*b?d)*c\2$)
No

aababdcaa
(?^:^((a*)b|a*b?d)*c\2$)
No

aabadcaa
(?^:^(a*b?d|(a*)b)*c\2$)
Yes

aababdcaa
(?^:^(a*b?d|(a*)b)*c\2$)
Yes

aabadcaa
(?^:^((a*)b|a*b?d())*c\2$)
No

aababdcaa
(?^:^((a*)b|a*b?d())*c\2$)
No

aabadcaa
(?^:^(?|a*b?d()|(a*)b)*c\1$)
No

aababdcaa
(?^:^(?|a*b?d()|(a*)b)*c\1$)
No

This is perl 5, version 38, subversion 0 (v5.38.0) built for x86_64-li
+nux-thread-multi
[download]