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Let $I=\int_{0}^{1} \frac{d x}{\sqrt{1+x}-\sqrt{x}}$

$I=\int_{0}^{1} \frac{1}{(\sqrt{1+x}-\sqrt{x})} \times \frac{(\sqrt{1+x}+\sqrt{x})}{(\sqrt{1+x}+\sqrt{x})} d x$

$=\int_{0}^{1} \frac{\sqrt{1+x}+\sqrt{x}}{1+x-x} d x$

$=\int_{0}^{1} \sqrt{1+x} d x+\int_{0}^{1} \sqrt{x} d x$

$=\left[\frac{2}{3}(1+x)^{\frac{3}{2}}\right]_{0}^{1}+\left[\frac{2}{3}(x)^{\frac{3}{2}}\right]_{0}^{1}$

$=\frac{2}{3}\left[(2)^{\frac{3}{2}}-1\right]+\frac{2}{3}[1]$

$=\frac{2}{3}(2)^{\frac{3}{2}}$

$=\frac{2 \cdot 2 \sqrt{2}}{3}$

$=\frac{4 \sqrt{2}}{3}$