Semidefinite program with objective function containing a square root

Question

By using a software package that is compatible with YALMIP, I would like to solve an optimization program of the form \begin{align*} \begin{array}{cr} \max\limits_{X\in \mathbb{S}^n}& \frac{x_1+\sqrt{x_1^2+c\left(\sum \operatorname{tr}(X_k \cdot B_k)-x_1\right)}}{x_2}\\ \text{s.t.}&\operatorname{tr}(A_k,X)=b_k\\ &X\succeq 0 \end{array} \end{align*} where $A_k, B_k, c$ and $b_k$ are given matrices and scalars, respectively, and $x_1,x_2,X$ are the optimization variables. Currently, I don't see a theoretical way to remove the square root function in the objective function. So, is there a practical way to solve this optimization program? Any solver that can handle this kind of programs, or any relaxation method that I could use?