I would like to solve the following indefinite integral
$$ \int \frac{1}{\sqrt{ax^4+bx^3+cx^2+gx+1}}dx $$
where a,b,c,g>0 (fyi: I applied g instead of d to the fourth coefficient in order no to confuse it with dx). Let us assume, if we had real values instead of the mentioned coefficients, that we cannot find any roots for the polynomial or rather we cannot simplify the argument of the integral (that would be an easy way to solve it as there are many examples on stack exchange). Actually, I can obtain an approximated expression for the argument by expanding it according to a Taylor series up to the first order.
$$ \frac{1}{\sqrt{u}}=1-\frac{u}{2} $$
I noticed though that the final solution is too much approximated especially if then, for instance, I integrate between zero and x>>1. It is not my target to approximate it. I am aware that it is not what it should be properly done. Your support with the integration procedure would be much appreciated. Thank you in advance. M.
Rewrite the integral as $$I=\frac 1{\sqrt a} \int \frac {dx}{\sqrt{(x-r_1)(x-r_2)(x-r_3)(x-r_4)}}$$ and have a look here.
If the link is broken, copy/paste for Wolfram Alpha
Integrate[1/Sqrt[(x - r[1])*(x - r[2])*(x - r[3])*(x - r[4])],x]