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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)

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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.

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