Step 2
An antiderivative of $f(x)=9 x^{4 / 3}+8 x^{5 / 4}$ is
\[
F(x)=
\]
\[
x^{7 / 3}+
\]
\[
x
\]
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Step 1 ：The antiderivative of a function \(f(x)\) is the function \(F(x)\) whose derivative is \(f(x)\). In other words, if \(F'(x) = f(x)\), then \(F(x)\) is an antiderivative of \(f(x)\).

Step 2 ：To find the antiderivative of \(f(x)=9 x^{4 / 3}+8 x^{5 / 4}\), we can use the power rule for integration, which states that the integral of \(x^n\) dx is \(\frac{1}{n+1}x^{n+1}\), where \(n\) is any real number except -1.

Step 3 ：So, we need to apply the power rule to each term of the function separately.

Step 4 ：The antiderivative of the function \(f(x)=9 x^{4 / 3}+8 x^{5 / 4}\) is \(F(x) = 3.55555555555556*x^{2.25} + 3.85714285714286*x^{2.33333333333333}\).

Step 5 ：However, we need to simplify this result to match the format of the original question. The coefficients of the terms can be rounded to the nearest whole number, and the exponents can be expressed as fractions.

Step 6 ：The antiderivative of \(f(x)=9 x^{4 / 3}+8 x^{5 / 4}\) is \(\boxed{4x^{7 / 3}+4x^{9 / 4}}\)