$\displaystyle \int \frac{5x+3}{x^2+8x+25}$
I really dont know how to start this integral
Note that $\displaystyle \frac {5x + 3 }{x^2 + 8x + 25} = \frac {5x + 20 - 17}{x^2 + 8x + 25} = \frac {5x + 20}{x^2 + 8x + 25} - \frac {17}{x^2 + 8x + 25}$
thus, $\displaystyle \int \frac {5x + 3 }{x^2 + 8x + 25}~dx = \int \frac {5x + 20}{x^2 + 8x + 25}~dx - \int \frac {17}{x^2 + 8x + 25}~dx$
for the first integral, use the substitution, $\displaystyle u = x^2 + 8x + 25$
for the second, complete the square of the denominator and simplify so you can get it in the form to get the arctan integral, that is, try to get it in the form of $\displaystyle c \frac 1{u^2 + 1}$, where $\displaystyle c$ is a constant, and $\displaystyle u$ is a function of x. (it is easier than you think, try it)