These are *not* datetime. These are strings and integers, don't mistake the two as the DateTime is an object type, not a scalar type.
This is your order of evaluation of relative versus non-relative formats:
$constructed = strtotime("2013-02-06 17:59:19");
$date = date('Y:m:d H:i:s', $constructed);
$final=date('Y:m:d H:i:s',strtotime("$date + 5 weekdays"));
print $final . PHP_EOL;// 2013:02:13 00:00:00 (which is correct for the criteria)
$final = date('Y:m:d H:i:s', strtotime("+5 weekdays $date"));
print $final . PHP_EOL; // 2013:02:13 17:59:19 (which is correct for the criteria)
In hindsight, this may be a bug. The relative should always apply *after* the absolute, unless its 'yesterday', 'midnight', 'today', 'noon', or 'tomorrow'. So that says that the above should work in either order which it certainly does not appear to.
I'd simply use the DateTime myself:
Which gives me 'February 13 2013, 17:59:19' as the result.
$dt = new DateTime('2013-02-06 17:59:19');
print $dt->format('F j Y H:i:s');