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.

The term

is giving hyperbolic arcsin. That shouldn't happen...the next two terms are integrable as per our syllabus but I don't think this one is...if there is any other way to do it I'd love to know.