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

Let us write $$\sqrt{1+x^{2}}$$ as $$s$$ to shorten the steps.
Define $$t = s + x = \sqrt{1+x^{2}} + x$$.

Notice the important identity
$$(s + x)(s - x) = (1 + x^{2}) - x^{2} = 1 \; \Longrightarrow \; s - x = \frac{1}{t}.$$

With this, the integrand simplifies:

$$\frac{(\sqrt{1+x^{2}} + x)^{10}}{(\sqrt{1+x^{2}} - x)^{9}} = \frac{t^{10}}{\left(\dfrac{1}{t}\right)^{9}} = t^{10}\,t^{9} = t^{19}.$$

Differentiate the substitution to get $$dx$$ in terms of $$dt$$ :

$$\frac{dt}{dx} = \frac{x}{s} + 1 = \frac{x + s}{s} = \frac{t}{s}\; \Longrightarrow\; dx = \frac{s}{t}\,dt.$$

Replace $$s$$ by an expression involving only $$t$$. From $$t = s + x$$ and $$\dfrac{1}{t} = s - x$$, add the two equations:

$$t + \frac{1}{t} = 2s \;\Longrightarrow\; s = \frac{t + 1/t}{2}.$$

Hence

$$dx = \frac{s}{t}\,dt = \frac{\dfrac{t + 1/t}{2}}{t}\,dt = \frac{t^{2} + 1}{2t^{2}}\,dt.$$

Put everything in the integral:

$$I = \int t^{19}\,dx = \int t^{19}\left(\frac{t^{2} + 1}{2t^{2}}\right)dt = \frac12\int\left(t^{19}\cdot\frac{t^{2} + 1}{t^{2}}\right)dt = \frac12\int\left(t^{19} + t^{17}\right)dt.$$

Integrate term by term:

$$I = \frac12\left(\frac{t^{20}}{20} + \frac{t^{18}}{18}\right) + C = \frac{t^{20}}{40} + \frac{t^{18}}{36} + C.$$

Factor out $$t^{18}$$ :

$$I = t^{18}\left(\frac{t^{2}}{40} + \frac{1}{36}\right) + C = \frac{t^{18}\left(9t^{2} + 10\right)}{360} + C.$$

Now convert the bracket $$9t^{2} + 10$$ to the required form $$t\,(n s - x)$$. Using $$s=\dfrac{t + 1/t}{2}$$ and $$x=\dfrac{t - 1/t}{2}$$, compute

$$t\left(19s - x\right) = t\left[19\left(\frac{t + 1/t}{2}\right) - \left(\frac{t - 1/t}{2}\right)\right] = t\left(\frac{18t + 20/t}{2}\right) = 9t^{2} + 10.$$

Therefore

$$I = \frac{t^{18}\,t\,(19s - x)}{360} + C = \frac{t^{19}\left(19\sqrt{1+x^{2}} - x\right)}{360} + C.$$

The result is now in the demanded form $$\frac1m\Bigl((\sqrt{1+x^{2}} + x)^{\,n}\bigl(n\sqrt{1+x^{2}} - x\bigr)\Bigr) + C.$$

By comparison, $$m = 360$$ and $$n = 19$$. Hence

$$m + n = 360 + 19 = 379.$$

Answer: $$379$$