What technique would I use in order to reach the correct answer?

Question

∫(3x²+4x+4)/(x³+x)dx

I tried doing integration of rational functions by partial fractions but I end with ∫4/x dx + ∫(4x-1)/(x²+1)dx and I try doing integration of rational functions for ∫(4x+1)/(x²+x)dx but get the same ∫(4x+1)/(x²+x)dx as the answer. I don't know if I am doing it incorrectly or if I was suppose to use another technique.

Answer 1

desmos.com/calculator/r80g0hz2ax

Answer 2

For the answer you got, the first integral is 4ln lxl, The second integral can be written as 4x/(x²+1) - 1/(x²+1).

Integrating, we get 2ln(x²+1) - Arctanx