$u=x+1 \Rightarrow x=u-1\\ du=dx\\ \displaystyle\int\dfrac{x^2-2x-5}{(x+1)^9} \, dx\\ =\displaystyle\int\dfrac{(u-1)^2-2(u-1)-5}{u^9} \, du\\ =\displaystyle\int\dfrac{u^2-4u-2}{u^9} \, du\\ =\displaystyle\int\left(\dfrac{1}{u^7} -\dfrac{4u}{u^8}-\dfrac{2}{u^9}\right)\, du\\ =-\dfrac{1}{6u^6}+\dfrac{4}{7u^7}+\dfrac{2}{8u^8}+C\\ =-\dfrac{1}{6(x+1)^6}+\dfrac{4}{7(x+1)^7}+\dfrac{1}{4(x+1)^8}+C$
