If
$$\int \frac{x^{-7/6}-x^{5/6}}{x^{1/3}(x^2+x+1)^{1/2}-x^{1/2}(x^2+x+1)^{1/3}} \, dx = -\lambda \left(\frac{z^3}{3}+\frac{z^p}{2}+\frac{z^q}{r}+\ln|z-1|\right) + \text{constant}$$
Find the value of $\lambda,p,q,r$.
After doing algebraic manipulation, I got $$-\int \frac 1 {z^{1/2}-z^{1/3}} \, dz$$
where $z=x+1+1/x$
How to proceed from here to get desired values?