3.5.6.3 Algorithm description to obtain the above solutions

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

$$s_i=b{s}_i+a$$

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.