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Integrate the following

$$\int \frac{dx}{(3x^2-4x+2)^{3/2}}$$

Can this be integrated in terms of elementary functions? Could someone please give me some hint to proceed in this problem? I tried it by taking $3x^2-4x+2=t^2$ but it didn't help.

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$3x^2-4x+2$ is a polynomial in $\mathbb{R}[x]$ with a negative discriminant. It follows that, through a suitable substitution, your problem boils down to computing $$ \int \frac{dx}{(x^2+A^2)^{3/2}} =C+\frac{x}{A^2\sqrt{A^2+x^2}}$$ for some $A>0$. The last equality is easily proved by integration by parts.

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