First, just look at #(x-1)/(x+3)^3#
#(x-1)/(x+3)^3=a/(x+3)+b/(x+3)^2+c/(x+3)^3|* (x+3)^3#
#x-1=a* (x+3)^2+b* (x+3)+c#
#color(red)(0)x^2+color(blue)(1)xcolor(orange)(-1)=(a)x^2+(6a+b)x+9a+3b+c#
#|(a=color(red)(0)), (6a+b=color(blue)(1)), (9a+3b+c=color(orange)(-1))|#
#|(a=0), (b=1), (c=-4)|#
#(x-1)/(x+3)^3=0/(x+3)+1/(x+3)^2-4/(x+3)^3#
#(x-1)/(x+3)^3=1/(x+3)^2-4/(x+3)^3#
#int(x-1)/(x+3)^3 dx=int(1/(x+3)^2-4/(x+3)^3)dx#
#int(1/(x+3)^2-4/(x+3)^3)dx=-1/(x+3)+2/(x+3)^2+c#
