Answer
$$\ln |\sqrt{x^{2}+1}+x|-\frac{x}{\sqrt{x^{2}+1}}+C$$
Work Step by Step
Given $$\int \frac{x^{2} d x}{\left(x^{2}+1\right)^{3 / 2}} $$ Let $$x=\tan u \ \ \ \ \to \ \ \ dx=\sec^2udu $$ Then \begin{align*} \int \frac{x^{2} d x}{\left(x^{2}+1\right)^{3 / 2}}&= \int \frac{\tan^{2}u\sec^2udu}{\left(\tan^{2}+1\right)^{3 / 2}} \\ &= \int \frac{\tan^{2}u\sec^2udu}{\sec^3u} \\ &= \int \frac{\tan^{2}u du}{\sec u} \\ &= \int \frac{(\sec^{2}u-1) du}{\sec u} \\ &= \int (\sec u-\cos u)du\\ &= \ln|\sec u+\tan u| -sin u+C\\ &= \ln |\sqrt{x^{2}+1}+x|-\frac{x}{\sqrt{x^{2}+1}}+C \end{align*}
