Find the general antiderivative of $f(x)=-3 x^{3}+\frac{6}{x^{5}}-6 \sqrt{x}$

Step 1 ：We are given the function \(f(x)=-3 x^{3}+\frac{6}{x^{5}}-6 \sqrt{x}\) and we are asked to find its general antiderivative.

Step 2 ：The antiderivative of a function \(f(x)\) is a function \(F(x)\) whose derivative is \(f(x)\). The antiderivative is also known as the indefinite integral of the function.

Step 3 ：To find the antiderivative of the given function, we need to apply the power rule for integration, which states that the integral of \(x^n dx\) is \((1/(n+1))x^(n+1)\), and the integral of \(1/x^n dx\) is \(-(1/(n-1))x^(n-1)\) for \(n ≠ -1, 0\).

Step 4 ：For the given function, we have three terms: \(-3x^3\), \(6/x^5\), and \(-6\sqrt{x}\).

Step 5 ：The antiderivative of \(-3x^3\) is \((-3/4)x^4\).

Step 6 ：The antiderivative of \(6/x^5\) is \(-6/(4)x^(-4)\).

Step 7 ：The antiderivative of \(-6\sqrt{x}\) is \(-6*(2/3)x^(3/2)\).

Step 8 ：We can calculate these separately and then add them together to get the final answer.

Step 9 ：The antiderivative of the function \(f(x)=-3 x^{3}+\frac{6}{x^{5}}-6 \sqrt{x}\) is \(F(x)=-4x^{3/2} - \frac{3}{4}x^{4} - \frac{3}{2x^{4}}\).

Step 10 ：Final Answer: The general antiderivative of \(f(x)=-3 x^{3}+\frac{6}{x^{5}}-6 \sqrt{x}\) is \(\boxed{F(x)=-4x^{3/2} - \frac{3}{4}x^{4} - \frac{3}{2x^{4}}\)