calculus - U substitution of indefinite integrals like $int frac{5+3x}{1+x^2} dx$.

I've spent the better part of today trying to understand conceptually how to solve indefinite integrals using the "u-substitution" method.

I am able to solve relatively easy indefinite integrals using u-substituion, but when it comes to more complicated ones I struggle and never end up with the correct answer which means i do not fully understand what is going on and am simply memorizing basic procedure for solving basic indefinite integrals.

For example:

$\int \frac{5+3x}{1+x^2} dx$

I did the following:

$\frac{5}{1+x^2} + \frac{3x}{1+x^2} dx$

$u = 1+x^2$

$du = 2x dx$

$\frac{1}{2x} du = dx$

And from there I am stuck and to be honest I don't even know if my approach is correct.

I know I am asking a lot, but is there anyone that can solve this and explain why they did what they did?