Find the indefinite integral without using substitution. \[ \int 8 x^{7}\left(x^{8}-1\right) d x=\square \]

Step 1 ：Given the integral \(\int 8 x^{7}(x^{8}-1) dx\)

Step 2 ：We can simplify the integral by expanding the expression inside the integral. The integral of a sum of terms can be found by integrating each term separately. The integral of a constant times a function is equal to the constant times the integral of the function. The power rule for integration states that the integral of x^n dx is \((1/(n+1))x^{(n+1)}\).

Step 3 ：Applying these rules, we get \(\int 8 x^{7}(x^{8}-1) dx = \int 8 x^{15} dx - \int 8 x^{7} dx\)

Step 4 ：Integrating each term separately, we get \(\frac{x^{16}}{2} - x^{8}\)

Step 5 ：Adding the constant of integration, we get the final answer: \(\boxed{\frac{x^{16}}{2} - x^{8} + C}\)