As in the comment by @AndrewT., it appears that AT&T have locked the device against rooting (corrected the exploits used by these one-click root solutions back with Android 5.0).
From the link in the comment:
Emphasis and comments mine
...the fact remains that AT&T's Android 4.4.4, released in 2014, and Android 5.0 and 5.1.1, released in 2015, have all been unrootable and no-one in the worldwide rooting community has found a root method apart from downgrading to the AT&T, Android 4.4.2, NCE, firmware/kernel... [to get] an exploit that AT&T closed off almost year ago with Android 5.0. Make no mistake, this is not Google or Samsung preventing you from rooting... it is all down to your carriers, AT&T and Verizon.
This post appears to be from a reputed member of their community, so we can be reasonably assured that even the great people at XDA-Developers (AT&T SGS5 thread here) have not managed to root this particular device (see here).
Hence, we can say that as of now there is no way to root this particular device (AT&T Samsung Galaxy S5) on this particular Android version (5.1).