My friends are preparing project about a Solow model. The asked me to calculate such integral:
$ s(1-a)\int e^{(1-a)(b+c)t} \cdot (d-ge^{ft})^{1-a}dt$
where: $b,c,s,d,g>0$ and $a∈(0,1)$. $[a,b,c,d,g,f$-constans] They recived a hint to use hypergeometric series.
Unfortunately, I do not know anything about hypergeometric series.



Difficul integral- Solow model
It is long continuation of a comment above, only on the rights of comment (Thanks to user @alpoge, please check and type your own full answer): 


If I did not mistake somewhere in arithmetic after change of variable $t=\ln z$ 
we have $$s(1-a)\int e^{(1-a)(b+c)t}(d-e^{ft})^{(1-a)}dt =s(1-a)\int z^{(1-a)(b+c)-1}(d-gz^f)^{(1-a)}dz$$


then set $u=\frac{g}{d}z^f$ 


the integral then has following form 


$$\frac{s}{f}(\frac{d^{2f+2}}{g^{f+2}})^{\frac{1}{f}}(1-a)\int u^{\frac{(1-a)(b+c)}{f}}(1-u)^{(1-a)}du$$


which is [incomplete beta function](http://mathworld.wolfram.com/IncompleteBetaFunction.html) $ B(u,1+\frac{(1-a)(b+c)}{f},2-a)$ multiplied by $\frac{s(1-a)}{f}(\frac{d^{2f+2}}{g^{f+2}})^{\frac{1}{f}}$


Then using $B(u,\alpha,\beta)=\frac{u^{\alpha}}{\alpha} ~\_2F\_1(\alpha,1-\beta,\alpha+1,u)$ we can give it in terms of hypergeometric functions.


