Starting with \[ \left ( y^{\prime }\right ) ^{\frac {n}{m}}=ax+by+c \] Find the solution \(z\) of equation \[ z^{\frac {n}{m}}=u \] Where \(u\) now is a symbol. Lets say we found \(s_{1},s_{2},\cdots \) solutions (depending on what \(n,m\) are). Then for each solution \(s_{i}\) change it to be
\[ s_{i}=bs_{i}+a \]
Then write
\[ \int \frac {du}{s_{i}}=x+c_{1}\] Then replace each with letter \(u\) in each \(s_{i}\) by new letter say \(z\) (the integration variable). Now the solution becomes\[ \int ^{ax+by+c}\frac {dz}{s_{i}}=x+c_{1}\] This is basically what was done in the above examples. There is no need to find an explicit solution for the integral. But this can be done if needed afterwords.