1. Sustitución inteligente:
Sea $u = x + \sqrt{x^{2}+1}$.
Entonces, despejamos $x$:
$u - x = \sqrt{x^{2}+1} \implies (u-x)^{2} = x^{2}+1$
$u^{2} - 2ux + x^{2} = x^{2} + 1 \implies x = \frac{u^{2}-1}{2u}$

2. Diferencial:
$dx = \frac{1}{2} \left( \frac{2u(2u) - 2(u^{2}-1)}{4u^{2}} \right) du = \frac{u^{2}+1}{2u^{2}} du$

3. Integración:
$$ \begin{aligned} \int u^{10} \left( \frac{u^{2}+1}{2u^{2}} \right) du &= \frac{1}{2} \int (u^{10} + u^{8}) du \\ &= \frac{1}{2} \left( \frac{u^{11}}{11} + \frac{u^{9}}{9} \right) + C \end{aligned} $$

4. Resultado final:
$$ \boxed{\frac{(x + \sqrt{x^{2}+1})^{11}}{22} + \frac{(x + \sqrt{x^{2}+1})^{9}}{18} + C} $$