<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] For any integral of the forms, you have mentioned, 
transform the integrand into the sum or difference of two perfect square expressions and the resulting expression will look like  $\sqrt{(x\pm a)^2+(b)^2}$ ,$\sqrt{(x\pm a)^2-(b)^2}$  or $\sqrt{(b)^2-(x\pm a)^2}$.
And then use the following formulae which can be deduced with the help of integration by parts:


*

*$$\sqrt{(x\pm a)^2+(b)^2}dx=\frac{(x\pm a)\sqrt{(x\pm a)^2+(b)^2}}{2}+\frac{b^2}{2}\ln |(x\pm a)+\sqrt{(x\pm a)^2+(b)^2}|$$

*$$\sqrt{(x\pm a)^2-(b)^2}dx=\frac{(x\pm a)\sqrt{(x\pm a)^2-(b)^2}}{2}-\frac{b^2}{2}\ln |(x\pm a)+\sqrt{(x\pm a)^2-(b)^2}|$$

*$$\sqrt{(b)^2-(x\pm a)^2}dx=\frac{(x\pm a)\sqrt{(b)^2-(x\pm a)^2}}{2}+\frac{b^2}{2}\sin \frac{x\pm a}{b}$$


Hope you can apply this to your problem and solve it now.
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