Evaluation of a Subclass <span class="math-tex">\( K_{p,r} =\int x^p \left ( 1+ x^2\right )^r dx \) of the Binomial Integral

<jats:p>Employing the fundamental techniques of substitution (change of variables) and integration by parts, the discussion in this letter focuses on obtaining explicit formulas of the integral\( K_{p,r} =\int x^p \left ( 1+ x^2\right )^r dx \) mainly for \( p \) an integer and \( r \) an integral multiple of \( \frac{1}{2} \). As the case where \( r \) is a non-negative integer is rather simple and obvious, we concentrate mainly on the cases where \( r \) is a negative integer or an odd multiple of \( \frac{1}{2} \) except when consideration of other cases is appropriate.</jats:p>