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Your input: perform the partial fraction decomposition of $$$\frac{1}{x^{4} - 5}$$$

Factor the denominator: $$$\frac{1}{x^{4} - 5}=\frac{1}{\left(x - \sqrt[4]{5}\right) \left(x + \sqrt[4]{5}\right) \left(x^{2} + \sqrt{5}\right)}$$$

The form of the partial fraction decomposition is

$$\frac{1}{\left(x - \sqrt[4]{5}\right) \left(x + \sqrt[4]{5}\right) \left(x^{2} + \sqrt{5}\right)}=\frac{A}{x + \sqrt[4]{5}}+\frac{B}{x - \sqrt[4]{5}}+\frac{C x + D}{x^{2} + \sqrt{5}}$$

Write the right-hand side as a single fraction:

$$\frac{1}{\left(x - \sqrt[4]{5}\right) \left(x + \sqrt[4]{5}\right) \left(x^{2} + \sqrt{5}\right)}=\frac{\left(x - \sqrt[4]{5}\right) \left(x + \sqrt[4]{5}\right) \left(C x + D\right) + \left(x - \sqrt[4]{5}\right) \left(x^{2} + \sqrt{5}\right) A + \left(x + \sqrt[4]{5}\right) \left(x^{2} + \sqrt{5}\right) B}{\left(x - \sqrt[4]{5}\right) \left(x + \sqrt[4]{5}\right) \left(x^{2} + \sqrt{5}\right)}$$

The denominators are equal, so we require the equality of the numerators:

$$1=\left(x - \sqrt[4]{5}\right) \left(x + \sqrt[4]{5}\right) \left(C x + D\right) + \left(x - \sqrt[4]{5}\right) \left(x^{2} + \sqrt{5}\right) A + \left(x + \sqrt[4]{5}\right) \left(x^{2} + \sqrt{5}\right) B$$

Expand the right-hand side:

$$1=x^{3} A + x^{3} B + x^{3} C - \sqrt[4]{5} x^{2} A + \sqrt[4]{5} x^{2} B + x^{2} D + \sqrt{5} x A + \sqrt{5} x B - \sqrt{5} x C - 5^{\frac{3}{4}} A + 5^{\frac{3}{4}} B - \sqrt{5} D$$

Collect up the like terms:

$$1=x^{3} \left(A + B + C\right) + x^{2} \left(- \sqrt[4]{5} A + \sqrt[4]{5} B + D\right) + x \left(\sqrt{5} A + \sqrt{5} B - \sqrt{5} C\right) - 5^{\frac{3}{4}} A + 5^{\frac{3}{4}} B - \sqrt{5} D$$

The coefficients near the like terms should be equal, so the following system is obtained:

$$\begin{cases} A + B + C = 0\\- \sqrt[4]{5} A + \sqrt[4]{5} B + D = 0\\\sqrt{5} A + \sqrt{5} B - \sqrt{5} C = 0\\- 5^{\frac{3}{4}} A + 5^{\frac{3}{4}} B - \sqrt{5} D = 1 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=- \frac{\sqrt[4]{5}}{20}$$$, $$$B=\frac{\sqrt[4]{5}}{20}$$$, $$$C=0$$$, $$$D=- \frac{\sqrt{5}}{10}$$$

Therefore,

$$\frac{1}{\left(x - \sqrt[4]{5}\right) \left(x + \sqrt[4]{5}\right) \left(x^{2} + \sqrt{5}\right)}=\frac{- \frac{\sqrt[4]{5}}{20}}{x + \sqrt[4]{5}}+\frac{\frac{\sqrt[4]{5}}{20}}{x - \sqrt[4]{5}}+\frac{- \frac{\sqrt{5}}{10}}{x^{2} + \sqrt{5}}$$

Answer: $$$\frac{1}{x^{4} - 5}=\frac{- \frac{\sqrt[4]{5}}{20}}{x + \sqrt[4]{5}}+\frac{\frac{\sqrt[4]{5}}{20}}{x - \sqrt[4]{5}}+\frac{- \frac{\sqrt{5}}{10}}{x^{2} + \sqrt{5}}$$$