Solving a non-integer polynomial where the exponent is greater than one I'm trying to solve an equation of the form:
$ax + bx^{1+c} + d = 0$, where $0 < c < 1$, and the reciprocal of $c$ is not necessarily an integer either.
Mathematica protests to me that it is not up to the task of solving this, and I'd like a general solution rather than FindRoot at a particular value.
I've poked around and there are a couple work-arounds - usually involving some substitution - for solving a non-integer polynomials but (as far as I can tell - because I'm having trouble applying them to this problem) they all only really seem to work if the non-integer polynomial is less than one and this is definitely more than one.
I've come to the point of getting a derivative w.r.t. to one of the parameters of interest with the implicit function theorem, which is fine for my purposes, but obviously getting a general solution would be preferable.
Is there a general solution for this?

If $|b/a|$ and $|d/a|$ are small, you can solve it in series.  For convenience, take $a=1$.  Then
$$x = -d + \sum_{n=0}^\infty \frac{\Gamma(nc+n+1)}{n! \Gamma(nc+2)} (-d)^{nc+1} (-b)^n$$
