Does this spoof integral from a meme have a coherent solution $\int\frac {3x^3 -x^2 + 2x-4}{\sqrt{x^2-3x+2}}dx$

Question

I ask because I have what I think is a solution, but it doesn't fit the spirit of the joke, which is to say, it doesn't produce anything like a pin.

The solution I get is the equation below:

$$-\frac{135\sqrt2\ln(3-2\sqrt2)+404}{16\sqrt2}$$

However, this solves to a transcendental number, beginning:

$$-2.98126694400553644032103778411344302709190188721887186739\cdots$$

So, I'm appealing for better or more informed solutions if I have made a mistake.

Does the equation solve to something nice and friendly, or is the whole thing just a wheeze?