Since I figured I can't use the integration by parts method right away, I tried the simple u-substitution method first, with $u=x+x^2$ and $dy/dx=2x+1$.
What I usually did for other problems like this was manipulate u in some way to cancel out the dy/dx, which leads to a form of a product between two functions, and then use the integration by parts method. But here I'm not sure how to cancel out the dy/dx.
HINT: $$\ln(x^2+x)=\ln(x+1)+\ln(x)$$
$\int\ln x dx$? Can be get by integration by parts with $f(x)=x$ ; $g(x)=\ln(x)$.