<s>[INST] Show that $\int_{-\infty}^{\infty} \dfrac{x^2}{(x^2+1)^2(x^2+2x+2)} \ dx$ is convergent. 
Show that $$\int_{-\infty}^{\infty} \dfrac{x^2}{(x^2+1)^2(x^2+2x+2)} \ dx$$ is convergent.

$$\int_{0}^{\infty} \dfrac{x^2}{(x^2+1)^2(x^2+2x+2)} \ dx$$ is convergent by comparison test with $\int_{0}^{\infty} \dfrac{1}{x^4} dx$.
How do i show that $$\int_{-\infty}^{0} \dfrac{x^2}{(x^2+1)^2(x^2+2x+2)} \ dx$$ is convergent?
 [/INST] Another way:
$$
0\le\frac{x^2}{(x^2+1)^2(\underbrace{(x+1)^2+1)}_{\ge 1}}\le\frac{x^2}{(x^2+1)^2}\le\frac{x^2+1}{(x^2+1)^2}=\frac{1}{x^2+1} 
$$
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