(x4+x3)/(x2-x-56)=x3(x+1)/[(x-8)(x+7)].
Let (x+1)/[(x-8)(x+7)]=A/(x-8)+B/(x+7),
x+1=Ax+7A+Bx-8B.
Equating coefficients:
constant: 7A-8B=1
x: A+B=1, B=1-A, so 7A-8(1-A)=1, 7A-8+8A=1, 15A=9, A=9/15=⅗⇒B=⅖.
(x+1)/[(x-8)(x+7)]=3/(5(x-8))+2/(5(x+7)).
I=∫{(x4+x3)/(x2-x-56)}dx=⅕∫{x3[3/(x-8)+2/(x+7)]}dx.
Synthetic division:
x3/(x-8)
8 | 1 0 0 0
1 8 64 | 512
1 8 64 | 512 = x2+8x+64+[512/(x-8)]
x3/(x+7)
-7 | 1 0 0 0
1 -7 49 | -343
1 -7 49 | -343 = x2-7x+49-[343/(x+7)]
I=⅗∫{x2+8x+64+[512/(x-8)]}dx+⅖∫{x2-7x+49-[343/(x+7)]}dx,
I=⅗(x3/3+4x2+64x+512ln|x-8|)+⅖(x3/3-7x2/2+49x-343ln|x+7|),
I=x3/3+x2+58x+307.2ln|x-8|-137.2ln|x+7|+C where C is integration constant.