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Let a and b be positive real numbers. How to find a closed form answer to the integral

$$\int_0^t \left(-a t + \big(1+ \dfrac{2bt}{3}\big)^{-3/2}\right)^{5/3} dt$$

If it is not possible to find a closed form answer. How can I find an approximate answer through some quadrature?

Thank you. I appreciate your help.

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    $\begingroup$ you want an indefinite integral? (the upper limit as it is written now makes no sense); in any case, an exact answer will not be forthcoming, and for an approximation you'll want to tell us something about these numbers $a$ and $b$. $\endgroup$
    – Carlo Beenakker
    Commented Aug 31, 2015 at 20:21
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    $\begingroup$ Just to be more specific about the reason why the integral is unlikely to have a nice "closed" form, it is an integral of an algebraic function defined by an irreducible polynomial or genus 3 (at least so Maple tells me). Such "Abelian integrals" are in general not elementary. For comparison, if the integrand came from a genus 1 polynomial, it would be expressible in "incomplete elliptic integrals". So, the answer you would get here would be two complexity steps beyond that. $\endgroup$
    – Igor Khavkine
    Commented Sep 1, 2015 at 7:15

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