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Let $I=\int \sqrt{1-4 x-x^{2}} d x$

$=\int \sqrt{1-\left(x^{2}+4 x+4-4\right)} d x$

$=\int \sqrt{1+4-(x+2)^{2}} d x$

$=\int \sqrt{(\sqrt{5})^{2}-(x+2)^{2}} d x$

It is known that, $\int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+\mathrm{C}$

$\therefore I=\frac{(x+2)}{2} \sqrt{1-4 x-x^{2}}+\frac{5}{2} \sin ^{-1}\left(\frac{x+2}{\sqrt{5}}\right)+\mathrm{C}$