$\sqrt {1 - 4x - {x^2}}$

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

$= \int {\sqrt {5 - {{\left( {x + 2} \right)}^2}} dx} = \int {\sqrt {{{\left( 5 \right)}^2} - {{\left( {x + 2} \right)}^2}} dx}$

$= \cfrac{{x + 2}}{2}\sqrt {5 - {{\left( {x + 2} \right)}^2}} + \cfrac{5}{2}{\sin ^{ - 1}}\left( {\cfrac{{x + 2}}{{\sqrt 5 }}} \right) + C$

$= \cfrac{{x + 2}}{2}\sqrt {{x^2} + 4x + 1} - \cfrac{3}{2}\log \left| {\left( {x + 2} \right) + \sqrt {{x^2} + 4x + 1} } \right| + C$

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