Answer
$$\frac{49}{12 \sqrt{6}}\left[\frac{\sqrt{6 x^{2}-49}}{7} \cdot \frac{\sqrt{6} x}{7}+\ln \left|\frac{\sqrt{6} x}{7}+\frac{\sqrt{6 x^{2}-49}}{7}\right|\right]+C$$
Work Step by Step
Given $$\int \frac{x^{2} d x}{\left(6 x^{2}-49\right)^{1 / 2}}$$
Let
$$\sqrt{6}x=7\sec u\ \ \ \ \ \ \to \ \ \ \ \ \sqrt{6}dx=7\sec u\tan udu $$
Then
\begin{align*}
\int \frac{x^{2} d x}{\left(6 x^{2}-49\right)^{1 / 2}}&=\frac{7}{\sqrt{6}}\frac{49}{6}\int \frac{\sec^3 u\tan udu}{\left(49\sec^2u-49\right)^{1 / 2}}\\
&=\frac{49}{6\sqrt{6}}\int \frac{\sec^3 u\tan udu}{7\tan u}\\
&= \frac{49}{6\sqrt{6}}\int \sec^3 udu
\end{align*}
Use
$$\int \sec ^{n} u d u=\frac{1}{n-1} \tan u \sec ^{n-2} u+\frac{n-2}{n-1} \int \sec ^{n-2} u d u$$
Then
$$\int \sec ^{3} u d u=\frac{1}{2} \sec u \tan u+\frac{1}{2} \ln |\sec u+\tan u|+C $$
Hence
\begin{align*}
\int \frac{x^{2} d x}{\left(6 x^{2}-49\right)^{1 / 2}} &= \frac{49}{6\sqrt{6}}\int \sec^3 udu\\
&= \frac{49}{6\sqrt{6}}\left(\frac{1}{2} \sec u \tan u+\frac{1}{2} \ln |\sec u+\tan u|+C\right) \\
&=\frac{49}{12 \sqrt{6}}\left[\frac{\sqrt{6 x^{2}-49}}{7} \cdot \frac{\sqrt{6} x}{7}+\ln \left|\frac{\sqrt{6} x}{7}+\frac{\sqrt{6 x^{2}-49}}{7}\right|\right]+C
\end{align*}
