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What is the integral or maybe a suggested technique to find $$\int_0^x \left(1+\left(\frac{t}{b}\right)^a\right)^{-q} d t,$$ $a,b,q\in \mathbb{R}:a\cdot q>1,b>0.$ ?

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Note that there will not, in general, be an elementary antiderivative. For example, if $a=3$, $b=1$, $q=1/2$, you are asking for $$\int_0^x{dt\over\sqrt{1+t^3}}$$ which is an example of an elliptic integral, a so-called "special function", and not expressible in closed form in terms of rational functions, exponentials, logarithms, trig and inverse trig functions. For other values of the parameters you probably get functions that don't even have names, much less closed form expressions in terms of familiar functions.

There are techniques of numerical integration for finding approximate values of definite integrals, e.g., the trapezoid rule, and Simpson's rule. There are also computer algebra packages which have implemented these approximation techniques --- Maple, Mathematica, and others. It might even be possible to get Wolfram Alpha to do it online for you.

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