Suppose $x(t)$ satisfies the following conditions: \[ \frac{d x}{d t}=\frac{5 \sqrt{t^{3}}-7 t}{\sqrt{t^{3}}}, \quad x(9)=8 \] Find the particular antiderivative, $x(t)$. \[ x(t)= \]

Step 1 ：Given the first order ordinary differential equation \(\frac{d x}{d t} = \frac{5 \sqrt{t^{3}}-7 t}{\sqrt{t^{3}}}, \quad x(9)=8\)

Step 2 ：Integrate the right hand side of the equation with respect to t to find the antiderivative.

Step 3 ：The antiderivative is \(-14t^{2/\sqrt{3}} + 5t\)

Step 4 ：Use the initial condition x(9)=8 to find the constant of integration.

Step 5 ：The constant of integration is 5.

Step 6 ：Substitute the constant of integration into the antiderivative to find the particular solution.

Step 7 ：The particular solution is \(x(t) = -14t^{2/\sqrt{3}} + 5t + 5\)

Step 8 ：\(\boxed{x(t) = -14t^{2/\sqrt{3}} + 5t + 5}\) is the final answer.