We can decompose the integrand as follows:
#x/((x+7)(x+8)(x+9))=A/(x+7)+B/(x+8)+C/(x+9)#
Add up the right side:
#x/((x+7)(x+8)(x+9))=(A(x+8)(x+9))/((x+7)(x+8)(x+9))+((B)(x+7)(x+9))/((x+7)(x+8)(x+9))+((C)(x+7)(x+8))/((x+7)(x+8)(x+9))#
Set numerators equal:
#x=A(x+8)(x+9)+B(x+7)(x+9)+C(x+7)(x+8)#
We need to find #A,B,C.# Fortunately, it looks like we can do this by choosing particular values of #x# which will eliminate some terms on the right side.
#x=-8:#
#-8=B(-1)(1), B=8#
#x=-7:#
#-7=A(-1)(2), A=7/2#
#x=-9:#
#-9=C(-2)(-1), C=-9/2#
So, with the decomposed fraction, the integral becomes
#7/2intdx/(x+7)+8intdx/(x+8)-9/2intdx/(x+9)#
Note that all constants were factored out.
These are all simple integrals.
In general, #intdx/(x+-a)=ln|x+-a|+C#
Integrating yields
#intx/((x+7)(x+8)(x+9))dx=7/2ln|x+7|+8ln|x+8|-9/2ln|x+9|+C#