$\int_0^z a^x b^{x^{m+1}} dx$ Let $a$ and $b$  be  positive constants, $m$ be any non-negative constant , how do we evaluate the following integral: $$\int_0^z a^x b^{x^{m+1}} dx$$ (at least a good approximation would do) 

Well, you can define:
$$ F(z)= \int_{0}^{z} a^x b^{x^{m+1}} dx$$
Take derivative:
$$\frac{\partial F}{\partial z} = a^z b^{z^{m+1}}$$
Now you can approximate this by using power series around $z=0$ on $G(z)=\frac{\partial F}{\partial z}$, just taking more derivatives of the RHS.
Hope it helped.
