integration by exponential proofs

[stoic canopy]

So I have created a formula that could integrate any function that is in the format (a^m1x+b1)/(v^m2x+b2). It works completely fine for most functions, but there are some that are not integratable I plug the functions that are not integratable into the formula and I get a solution. It is extremely confusing why these functions are considered non-integrable.

For example: (3^2x+1)/(5^4x-6) is valid and I get a correct answer but integrating (1^x+2)/(6^2x-3) is apparently not possible but I get a solution for this answer anyways

The valid work is done in red and the invalid work is done in black