Split the numerator and complete the square under a radical

The source guidance is to split the numerator: $x+2=-\frac18(-8x-8)+1$. Thus the integral is $-\frac18\int\dfrac{-8x-8}{\sqrt{1-8x-4x^2}}dx+\int\dfrac1{\sqrt{5-4(x+1)^2}}dx$. The first term is $-\frac14\sqrt{1-8x-4x^2}$. The second is $\frac12\sin^{-1}(2(x+1)/\sqrt5)$. Hence the answer is $-\frac14\sqrt{1-8x-4x^2}+\frac12\sin^{-1}(2(x+1)/\sqrt5)+C$, equivalent to the printed argument $2\sqrt5(x+1)/5$.