For sufficiently large values of C, say 100, how do I go about evaluating this sort of integral:
$$\int\frac{(2x+3)^{100}}{\sqrt x}dx$$
Of course, one could, in theory, just expand the expression with the binomial theorem, but I'm looking for a faster method. I tried substitution but I don't think it's even possible in this case.
Your case is not the most general one, due to the expression in the brackets, but we can mind as follows.
Set
$$(2x +3)^C = z$$
Hence reversing the substitution:
$$dx = \frac{ z^{1/C - 1}}{2C}\ dz ~~~~~~~~~~~ x = \frac{z^{1/C} - 3}{2}$$
Hence your integral becomes
$$\frac{\sqrt{2}}{2C}\int \frac{z^{1/C}}{(z^{1/C}-3)^{1/2}} dz$$
Which is known to be an integral in terms of the HyperGeometric Function.
The final result is
$$3\sqrt{2}\frac{ z \left(\sqrt{9-3 z^{1/C}} \, _2F_1\left(\frac{1}{2},C;C+1;\frac{z^{1/C}}{3}\right)+z^{1/C}-3\right)}{(2 C+1) \sqrt{z^{1/C}-3}}$$
Where $_2F_1$ stands for the HyperGeometric function.
More on HGF
https://en.wikipedia.org/wiki/Hypergeometric_function
http://mathworld.wolfram.com/HypergeometricFunction.html