<s>[INST] Partial fraction decomposition,how? I need to decompose this fraction:
$${x^2+1\over (x-1)^3\cdot(x+3)}$$
I tried to write it up like this:  $${A\over (x-1)}+{B\over (x-1)^2}+{C\over (x-1)^3}+{D\over (x+3)}$$ 
But now i get $$A\cdot(x-1)^5\cdot (x+3) +B\cdot (x-1)^4\cdot (x+3) +C\cdot(x-1)^3\cdot (x+3) +D\cdot (x-1)^6 =x^2+1$$  which is not correct i think.
What should i do?
 [/INST] Try this:
$
 \frac{x^2+1}{(x-1)^3(x+3)} =
 \frac{(x-1)^2+2x}{(x-1)^3(x+3)} =
 \frac{(x-1)^2}{(x-1)^3(x+3)}+\frac{2x}{(x-1)^3(x+3)} =
 \frac{1}{(x-1)(x+3)}+\frac{2x}{(x-1)^3(x+3)}
$
</s>