Simplifying expressions with dot or non-commutative product (without the help of NCAlgebra)

Question

I have two expressions (with dot and non-commutative product):

$f=J^T.x+J.x$

$g=J^T**x+J**x$

I want to simplify them and get the following expressions:

$f=(J^T+J).x$

$g=(J^T+J)**x$

In an attempt to solve this problem, I used some tricks from: Symbolic Tensors

$Assumptions = {x \[Element] Arrays[{3, 1}, Reals], J \[Element] Matrices[{3, 3}, Reals]};

expr = Transpose[J].x + J.x // TensorReduce

But the expression doesn't change. What needs to be done so that Mathematica can simplify expressions with scalar and non-commutative products?

Some EDIT:

Clear["Derivative"]; ClearAll["Global`*"];

Unprotect[Dot, Plus];

(*Dot rules*)
Dot[A_, x_] + Dot[B_, x_] := Dot[A + B, x];
Protect[Dot, Plus];

(***)

J.x + Transpose[J].x // TraditionalForm;

Clear["Derivative"]; ClearAll["Global`*"];

Unprotect[NonCommutativeMultiply, Plus];

(*NonCommutativeMultiply rules*)
NonCommutativeMultiply[A_, x_] + NonCommutativeMultiply[B_, x_] := 
  NonCommutativeMultiply[A + B, x];
Protect[NonCommutativeMultiply, Plus];

(***)

J ** x + Transpose[J] ** x // TraditionalForm;

Answer 1

Try:

f = Transpose[J] . x + J . x
f /. c1_ . x + c2_ . x -> (c1 + c2) . x

(* J + Transpose[J]) . x *)

Note: take care to put a space between "" and "." because "." represents an argument that can be omitted and the above does not work:

f /. c1_. x + c2_. x -> (c1 + c2) . x
(* J . x + Transpose[J] . x *)