x-5x-4-x-3-5x-1-dx-

Question Number 167598 by cortano1 last updated on 20/Mar/22

\int \frac{x}{{5 x}^{4} + x^{3} - 5 x - 1} dx = ?

Answered by MJS_new last updated on 20/Mar/22

\int \frac{x}{5 x^{4} - 5 x + x^{3} - 1} d x = \int \frac{x}{(5 x + 1) (x^{3} - 1)} d x =

= \int (\frac{25}{126 (5 x + 1)} + \frac{1}{18 (x - 1)} - \frac{2 x + 1}{21 (x^{2} + x + 1)} - \frac{2}{21 (x^{2} + x + 1)}) d x =

= \frac{5}{126} \ln ∣ 5 x + 1 ∣ + \frac{1}{18} \ln ∣ x - 1 ∣ - \frac{1}{21} \ln (x^{2} + x + 1) - \frac{4 \sqrt{3}}{63} \arctan \frac{\sqrt{3} (2 x + 1)}{3} + C

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