<s>[INST] integate $\int\sqrt{x^2+3x+3} dx$ Is there any universal method how to solve integrals like this?
$$\int\sqrt{x^2+3x+3} dx$$
Or this?
$$\int\sqrt{-x^2+3x+3} dx$$
I tried use first Euler subs, but it was not good idea. 
$$\sqrt{ax^2+bx+c} = t\pm\sqrt{ax}$$
 [/INST] By a linear transform of the argument, you can normalize the integrand to one of
$$\sqrt{1+x^2}\text{ or }\sqrt{1-x^2}.$$
Then by parts,
$$\int\sqrt{1\pm x^2}\,dx=x\sqrt{1\pm x^2}\mp\int\frac{x^2}{\sqrt{1\pm x^2}}dx=x\sqrt{1\pm x^2}-\int\frac{(1\pm x^2)-1}{\sqrt{1\pm x^2}}dx.$$
Move the first part of the integral to the LHS and remains to integrate
$$\int\frac{dx}{\sqrt{1\pm x^2}},$$ which is elementary.
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