How to solve:
$$\int \frac{x}{\sqrt{4x^2 + 8x + 5}} dx$$
This question is from a list and it's in the category of problems that involving $\sqrt{x^2\pm a^2}$ and $\sqrt{a^2\pm x^2}$ (triangle rules). I tried some methods, but I didn't get anywhere. Where this type of problem fits?
First note that $$ \frac{d}{dx} (4x^2+8x+5) = 8(x+1), $$ so add and subtract $1$ in the numerator: $$ \int \frac{x}{\sqrt{4x^2 + 8x + 5}} \, dx = \int \frac{x+1}{\sqrt{4x^2 + 8x + 5}} \, dx - \int \frac{1}{\sqrt{4x^2 + 8x + 5}} \, dx $$
The first integral is then in the form $(f)^{-1/2}f'$, which is easy to integrate.
For the other, complete the square on the denominator: $$ 4x^2+8x+5 = 4(x+1)^2 + 1 $$ This transforms the integral into one of the form $$ \int \frac{dy}{\sqrt{a^2y^2+1}}, $$ (with here $y=x+1$, $a=2$) which you can then turn into one of the forms you know how to do.