∫(x^5+x^4-8)dx/(x^3-x)=∫(x^5-x^3+x^4-x^2+x^3-x+x^2+x-8)dx/(x^3-x)=∫(x^2+x+1)dx+∫(x^2+x-8)dx/(x^3-x)=x^3/3+x^2/2+x+∫(x^2+x)dx/(x^3-x)-∫8dx/(x^3-x)=x^3/3+x^2/2+x+∫dx/(x-1)-8∫dx/[x(x-1)(x+1)]
=x^3/3+x^2/2+x+ln(x-1)+4∫dx/x(x-1)]-4∫dx/[x(x+1)]=x^3/3+x^2/2+x+ln(x-1)+4∫dx/(x-1)-4∫dx/x-4∫dx/x+4∫dx/(x+1)=x^3/3+x^2/2+x+5ln(x-1)-5lnx+4ln(x+1)+C