Following the work of Andreas Wurfl i am trying to implement the Risch algorithm on $\int{\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}dx}$ following his method for extensions that are purely logarithmic, we let $\theta=\log(x)$ then the integrand becomes $\dfrac{\theta+2}{x^{2}\theta^{3}} \in \mathbb{Q}(x,\theta)$, let $p$ be the numerator and $q$ be the denominator of our integrant, first to show that this has an elementary integral we need to show that the roots (solutions for $z$) of the resultant (determinant of the sylvester matrix) of the polynomials $p-zq'$ and $q$ after some calculation this boils down to finding the resultant of $\theta+2-2xz\theta^{3}-3xz\theta^{2}$ with $x^{2}\theta^{3}$ i.e. the determinant of the matrix
$ S =\begin{pmatrix} -2xz & -3xz& 1 &2 &0& 0 \\ 0&-2xz&-3xz&1&2&0 \\ 0&0&-2xz&-3xz&1&2 \\ x^{2}&0&0&0&0&0 \\ 0&x^{2}&0&0&0&0 \\ 0&0&x^{2}&0&0&0 \end{pmatrix}$ (i think although my resultant knowledge is a bit rusty so don't quote me on this)
Which i get as 0 which causes problems as the next bit involves writing it in the form where we need to multiply a quotient by the roots of our resultant (roots for z) but this will just be 0 so we get a problem (see theorem 4 of the article.) Any help would be apprecaited