We first expand the given expression into partial fractions:
#(x+1)/((x+5)(x+6)(x-1)) -= A/(x+5)+B/(x+6)+C/(x-1)#
#(x+1)/((x+5)(x+6)(x-1)) -= (A(x+6)(x-1) + B(x+5)(x-1) + C(x+5)(x+6))/((x+5)(x+6)(x-1))#
And so. #(x+1) -= A(x+6)(x-1) + B(x+5)(x-1) + C(x+5)(x+6))#
Put #x=-5=>-5+1=A(-5+6)(-5-1)+0+0#
#:. (1)(-6)A=-4=>A=2/3#
Put #x=-6=>-6+1=0+B(-6+5)(-6-1)+0#
#:. (-1)(-7)A=-5=>B=-5/7#
Put #x=1=>1+1=0+0+C(1+5)(1+6)#
#:. (6)(7)C=2=>C=1/21#
So the partial fraction decomposition is:
#(x+1)/((x+5)(x+6)(x-1)) -= 2/(3(x+5))-5/(7(x+6))+1/(21(x-1))#
We now want to integrate; so
#int(x+1)/((x+5)(x+6)(x-1))dx = int(2/(3(x+5))-5/(7(x+6))+1/(21(x-1)))dx#
#int(x+1)/((x+5)(x+6)(x-1))dx = 2/3int1/(x+5)dx - 5/7 int1/(x+6)dx+1/21 int 1/(x-1)dx#
#int(x+1)/((x+5)(x+6)(x-1))dx = 2/3ln|x+5| - 5/7 ln|x+6|+1/21 ln|x-1| + c#
