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I am struggling with the partial fraction decomposition of

$$\frac{(2l+1)(l+1)}{K+(l+1)(l+2)(l-1)},$$

where $K$ is a positive real number.

I would like to decompose it in a sum of simpler polynomial fractions.

I tried the Mathematica's function "Apart" but it is returning the fraction so I am thinking there is no simple partial fraction decomposition.

Could you help me?

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    The decomposition depends on how $K$ and $l$ are related. There is no general decomposition that will work for all $K.$ – 2017-01-24

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i think the simpliest form is this here $${\frac {2\,{l}^{2}+3\,l+1}{{l}^{3}+2\,{l}^{2}+K-l-2}}$$

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    Is it possible to have simpler than that? That's because I am eventually looking at series of legendre polynomials having that fraction as coefficient – 2017-01-23