Integral of polynomial to some power

[Wiemster]

Does anybody know in general how (if) one can perform the integral of a general polynomial to some, not necessarily integer, power? I.e.

$$\int \left(\Sigma_{i=0} ^n c_i x^i \right)^a dx$$

with $c_i$ and $a$arbitrary (real) numbers,

$$\int \left(1+x + x^2 + 2x^5 \right)^{1.7} dx$$.

Maybe what I'm looking for is some generalization of Newton's binomium?

 

[HallsofIvy]

There is no simple way to do that.

 

[Wiemster]

I already thought it would be difficult, if not impossible. Too bad. Thanks anyway.

 

[g_edgar]

$\int\sqrt{\text{fourth degree}}$ = elliptic integral

$\int_0^1 [x(1-x)]^a\,dx$ = Beta function