Mathematics Question from JEE Main 2025 · 4 April, Shift 2

If $\int \frac{\left( \sqrt{1 + x^2} + x \right)^{10}}{\left( \sqrt{1 + x^2} - x \right)^9} \, dx = \frac{1}{m} \left( \left( \sqrt{1 + x^2} + x \right)^n \left( n\sqrt{1 + x^2} - x \right) \right) + C,$ where $m, n \in \mathbb{N}$ and $C$ is the constant of integration, then $m + n$ is equal to _____