$\int {\sqrt {5 - 2x + {x^2}} } dx$

Let $I = \int {\sqrt {5 - 2x + {x^2}} } dx = \int {\sqrt {{x^2} - 2x + 1 + 4} } dx$

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

$= \frac{{x - 1}}{2}\sqrt {{2^2} + {{(x - 1)}^2}} + 2\log \left| {x - 1 + \sqrt {{2^2} + {{(x - 1)}^2}} } \right| + C$

$= \frac{{x - 1}}{2}\sqrt {5 - 2x + {x^2}} + 2\log \left| {x - 1 + \sqrt {5 - 2x + {x^2}} } \right| + C$