To put the above in the proper context, I am trying to solve a Bernoulli equation of the second order:
$$\frac{dy}{dt} = -\frac{A}{p-q}(e^{-pt}-e^{-qt})y-Be^{-rt}y^2$$
where constants $A, B, p, q, r$ are real, positive and generally non-integer. Also, $p$ and $q$ assume different values. Applying the standard procedure, i.e., $$w=\frac{1}{y}$$ substitution and multiplying the resulting equation by integrating factor:
$$M(t) = e^{-\frac{A}{p-q} \int e^{-pt}-e^{-qt}dt}$$
ultimately leads to:
$$\frac{d(wM)}{dt} = Be^{\frac{A}{p(p-q)} e^{-pt}}e^{-\frac{A}{q(p-q)}e^{-qt}}e^{-rt}$$
which can be solved by the integral mentioned in title. Thank you very much for any useful tip.