∫x^3/(9+x^2)dx
=1/2∫x^2/(9+x^2)dx^2 (x^2=t)
=1/2∫t/(9+t)dt
=1/2∫(t+9-9)/(9+t)dt
=1/2∫[1-9/(9+t)]dt
=1/2t-9/2ln(9+t)+C
=1/2x^2-9/2ln(9+x^2)+C