Let $A,B \in \mathcal{M}_n(\mathbb{K}) $ (where $\mathbb{K} \in \{\mathbb{R},\mathbb{C} \} $) invertible matrices such that $A + B = I_n$ (where $I_n$ is the identity matrix of order $n$). Solve the following equation for $X$.
$$ A^2XB^T + (BX^TB^TA^T)^T = BAB^T $$
I started by doing the following
$ A^2XB^T + (BX^TB^TA^T)^T = BAB^T \iff A^2XB^T + ABXB^T = BAB^T \iff$
$ \iff (A^2X + ABX)B^T = BAB^T \iff A^2X + ABX = BA \iff$
$ \iff (A^2 + AB)X = BA \iff X = (A^2+AB)^{-1}BA$
But I sense that this is not the simplest answer. Any hints on how to simplify this?
Best regards