\int _0^3\frac{x^2}{\sqrt{81-x^4}}dx
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Definite integral
\[\int_{0}^{3}{\frac{{x}^{2}}{\sqrt{81 - {x}^{4}}}}dx = \frac{3 \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle| {1} \right)}}{4 \Gamma\left(\frac{7}{4}\right)} \approx 1.79721035210338831115988373842049\]
\(\Gamma\) is the gamma function, which is an extension of the factorial function to complex number arguments. \(\Gamma\) is defined for any positive integer \(n\) with the formula \(\Gamma(n) = (n-1)!\) and for complex numbers with a positive real part with the formula \(\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\).
\({}_{p}F_{q}\) is the generalized hypergeometric function, defined by a series for which the ratio of consecutive terms is a rational function of the summation index, also known as a hypergeometric series. \({{}_{p}F_{q}\left(\begin{smallmatrix} a_1, ..., a_p \\ b_1,...,b_p \end{smallmatrix}\middle| {z} \right)} = \sum_{k=0}^{\infty} \frac{(a_1)_k \cdots (a_p)_k}{(b_1)_k \cdots (b_q)_k} \frac{z^k}{k!}\) where \((a)_k\) denotes the rising factorial. The function can also be written in the forms \({{}_{p}F_{q}\left(\begin{smallmatrix} a \\ b \end{smallmatrix}\middle| {z} \right)}\) or \({{}_{p}F_{q}\left(a; b; z\right)}\).