Find the following indefinite integral:
$$\int \sqrt{x}(5x+3{)}^{2}dx$$
$\frac{175}{2}{x}^{\frac{7}{2}}+75x^{\frac{5}{2}}+\frac{27}{2}{x}^{\frac{3}{2}}+c$
$\frac{50}{7}{x}^{\frac{7}{2}}+12x^{\frac{5}{2}}+6x^{\frac{3}{2}}+c$
$\frac{50}{7}{x}^{\frac{7}{2}}+6x^{\frac{3}{2}}+c$
$\frac{175}{2}{x}^{\frac{7}{2}}+\frac{27}{2}{x}^{\frac{3}{2}}+c$

Step 1 ：First, we recognize that this is an integral of the form $\int udv$, where $u=\sqrt{x}=x^{\frac{1}{2}}$ and $dv=(5x+3{)}^{2}dx$.

Step 2 ：We need to find $du$ and $v$. The derivative of $u$ with respect to $x$ is $\frac{1}{2}{x}^{-\frac{1}{2}}$, so $du=\frac{1}{2}{x}^{-\frac{1}{2}}dx$.

Step 3 ：To find $v$, we need to integrate $dv$. The integral of $(5x+3{)}^2$ with respect to $x$ is $\frac{5}{3}{x}^3+3x^2+C$, so $v=\frac{5}{3}{x}^3+3x^2+C$.

Step 4 ：Now we can use the integration by parts formula, $\int udv=uv-\int vdu$. Substituting our expressions for $u$, $v$, and $du$, we get $\int \sqrt{x}(5x+3{)}^{2}dx=x^{\frac{1}{2}}(\frac{5}{3}{x}^3+3x^2+C)-\int (\frac{5}{3}{x}^3+3x^2+C)\frac{1}{2}{x}^{-\frac{1}{2}}dx$.

Step 5 ：Simplify the integral on the right to get $\int \sqrt{x}(5x+3{)}^{2}dx=x^{\frac{1}{2}}(\frac{5}{3}{x}^3+3x^2+C)-\frac{5}{6}{x}^2-\frac{3}{2}x+C$.

Step 6 ：Finally, simplify the expression to get the final answer: $\boxed{{\displaystyle \frac{175}{2}{x}^{\frac{7}{2}}+75x^{\frac{5}{2}}+\frac{27}{2}{x}^{\frac{3}{2}}+c}}$.