Here is how to formulate x\log|I+ x(Y+xI)^{-1}|, x \geq 0,Y \in S^{+}, Y+xI \in S^{++}, which is jointly convex in x and Y.
x*log_det(eye(n)+x*eye(n)*inv(Y+x*eye(n)))
can be formulated as
quantum_rel_entr(x*eye(n)+Y,2*x*eye(n)+Y) + quantum_rel_entr(2*x*eye(n)+Y,x*eye(n)+Y)
which turns out to be the quantum (matrix) analog of the scalar formulation
x*log(1+x/(x+y)) = rel_entr(x+y,2*x+y) + rel_entr(2*x+y,x+y)
from Here's how to handle x*log(1+x/(x+y)) for x >= 0, y >= 0