MFB low-pass filter transfer function derivation

The low-pass MFB filter can be a little tricky to analyse so, if you are only interested in the result skip to the end. The MFB low-pass filter: -

The first thing to do is convert impedances to functional blocks using circuit theorems: -

The stages above should all be fairly clear. In the centre stage the integrating op-amp, R3 and C2 are converted to a functional block but, R3 still acts as impedance to ground because of the op-amp's virtual ground. Stage 3 mops up things a bit more leaving two remaining impedances. We are trying to get rid of impedances because they simultaneously have a loading effect and, shape functional blocks. We just want to make functional blocks.

We can convert the two remaining impedances to functional blocks using superposition theorem: -

From this point onwards it can be a math problem but, I choose to solve this by keeping an image of the overall functional transfer function so that it's clear what's happening: -

We can now put the above in standard form: -

That's it and remember that some derivations swap the positions of R2 and R3 so, watch out for that if you are double checking my post.