I'm not sure what you mean. Let Ax mean rotating x by A (which is a rotation matrix, quaternion, rotor, whatever). Then, the composition of Ax + b and Cx + d is

    C(Ax + b) + d = (CA)x + (Cb + d).

yes but what about non-uniform scaling. Is it possible to handle it in this case?

Not with just a rotor (quaternion) and an offset, or with motors (dual quaternions). If you want nonuniform scaling, rotation and translation, then matrices are probably the nicest way to do that.

You work with 4x4 (in 3D space) matrices, with a specific form (a row/columb of 1’s). The affine part of a projective transformation.

All this GA stuff is OK but what is the problem with matrices of affine transformations?

a selection:

- no closed form exponential/logarithm (i.e. cant interpolate)

- no numerical stability for subgroups (i.e. need Gramm-Shmidt, SVD)

- no covariant transformations (i.e. need adjugate to transform axial vectors)

- no geometric construction. (e.g. in GA product of two elements is (square of) versor between them)

- costly inverses (just some sign swaps in GA)