Initially, the differential equation $\displaystyle dx= bx\left(1- \left(\frac{x}{F}\right)^m\right)dt$ (I have dropped the subscript "i" since it is not relevant to this question). As suggested, dividing both sides by F gives $\displaystyle \frac{dx}{F}= b \frac{x}{F} \left ( 1- \frac{x}{F} \right )^m dt$ and then, letting $\displaystyle y= \frac{dx}{F}$, $\displaystyle dy= by(1- y)dt$.
Now, let $\displaystyle z= y^{-m}$. Then $\displaystyle y= z^m$ so that $\displaystyle dy= mz^{m-1}dz= bz^m(1- z)dt$ which reduces to $\displaystyle mdz= bz(1- z)dt$ or $\displaystyle \frac{mdz}{z(1- z)}= dt$. On the left use "partial fractions": Find numbers A and B such that $\displaystyle \frac{1}{z(1- z)}= \frac{A}{z}+ \frac{B}{1- z}$ for all z. Multiply on both sides by z(1- z) to get $\displaystyle 1= A(1- z)+ Bz$. When z= 1, 1= B. When z= 0, 1= A. So $\displaystyle \frac{m}{z}dz + \frac{m}{1- z}dz= dt$. Integrating now, $\displaystyle mln(z)- mlog(1- z)= m ln \left ( \frac{z}{1-z} \right )= t+ C$