Find $\int\left(x^{4}-\frac{4}{x}-\frac{5}{x^{3}}+6 \sqrt{x}\right) d x$

Step 1 ：Given the function \(f(x) = x^{4}-\frac{4}{x}-\frac{5}{x^{3}}+6 \sqrt{x}\), we are asked to find its integral.

Step 2 ：We can integrate each term separately. The integral of a sum/difference of functions is the sum/difference of their integrals.

Step 3 ：For the first term, the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), so the integral of \(x^4\) is \(\frac{x^5}{5}\).

Step 4 ：For the second term, the integral of \(\frac{1}{x}\) is \(\ln|x|\), so the integral of \(\frac{4}{x}\) is \(4\ln|x|\).

Step 5 ：For the third term, the integral of \(\frac{1}{x^n}\) is \(\frac{-1}{(n-1)x^{n-1}}\), so the integral of \(\frac{5}{x^3}\) is \(\frac{-5}{2x^2}\).

Step 6 ：For the fourth term, the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), so the integral of \(6\sqrt{x}\) is \(6\frac{2}{3}x^{3/2}\).

Step 7 ：Finally, we need to add a constant of integration, which we'll denote as \(C\).

Step 8 ：Combining all these, the integral of the given function is \(\frac{x^{5}}{5}-4 \ln |x|+\frac{5}{2 x^{2}}+4 x^{\frac{3}{2}}+C\).

Step 9 ：\(\boxed{\frac{x^{5}}{5}-4 \ln |x|+\frac{5}{2 x^{2}}+4 x^{\frac{3}{2}}+C}\) is the final answer.