I am trying to integrate
$$ \int x^a f'(x)\lambda \exp(f(x)\lambda) dx$$
I'm not that great at these things, but I noticed that $f'(x)\lambda \exp(f(x)\lambda = \frac{d}{dx} \exp(f(x)\lambda)$. I then tried several substitutions, but they all failed. Is there a way to integrate this expression?
Just for completeness:
\begin{align*} f(x) &= \frac{xk_1}{x-c} + \frac{1}{\tilde \lambda}(1 + W_{-1}(-e^z))\\ z &= -\tilde\lambda k_2\frac{r-x}{x-c} - 1 \end{align*}
where $k_1, k_2, r, c$ are constants. $W_{-1}(\cdot)$ is the $-1$ branch of Lambert's W. In general, $\tilde \lambda$ is not equal to $\lambda$ - but the special case $\lambda = \tilde \lambda$ is useful, too.