The following integral cannot be expressed in elementary terms:
$\displaystyle\int_{0}^t \frac{1}{\sqrt{0.3(1+x)^3+0.7}}\,dx$
with $t>0$ real.
What are possible techniques for approaching this kind of integral, and how do they apply to this concrete case?
The integral you posted cannot be evaluated in terms of elementary functions. You will need either elliptic integrals, hypergeometric series, or incomplete beta functions to express it.
Hypergeometric series: It is quite simple, really, but there are a few details that we have to pay attention to: Basically, it all relies upon expanding the integrand into its binomial series, then switching the order of summation and integration. The trick is that, in order for that to happen, the new variable has to be lesser than $1$ in absolute value. Notice that we can rewrite the integrand as $~\sqrt{\dfrac{10}7}\cdot\bigg(1+\dfrac37u^3\bigg)^{-1/2}~$ where u is $x+1$, and the new limits of integration are $1$ and $t+1$. Or $\sqrt{\dfrac{10}3}~u^{-3/2}\bigg(1+\dfrac73u^{-3}\bigg)^{-1/2}$ with the same limits in place. The former expression works for the case $|u|\le\sqrt[3]{\dfrac73}$ , and the latter when $|u|\ge\sqrt[3]{\dfrac73}$ . The whole idea is to place the relation of $\sqrt[3]{\dfrac73}>1$ with respect to $t+1$, and to break up the interval of integration accordingly $($if necessary$)$, then to employ the appropriate formula on each such subinterval.
Incomplete beta functions:
First, factor the free term outside the radical sign, then let $u=x+1$, and $v=\dfrac37u^3.~$ Finally, let $w=\dfrac1{v+1}$ . You will see the expression of the $($incomplete$)$ beta function slowly emerging from the mist of successive substitutions.