how to integrate ?

write

x^2 + 2x + 3 as

m ( x^2 + x + 1) + n \frac{d\left(x^2 + x + 1 \right)}{dx} + u

evaluate m , n, u

u get

m = 1

n=1/2

u = 3/2

integral reduces as

{\color{blue} \int\left[ \left(x^2 + x +1 \right) + \frac{1}{2} ( 2x + 1) + \frac{3}{2}\right]\sqrt{x^2 + x +1} . dx }

convert these into perfect squares and use formula for

now use the formula

{\color{blue} \int \sqrt{x^2 + a^2} . dx }

and done

kaise please elaborate.
perfect squares kaise karoon. full soln .please.