Final Answer
$\frac{x^{3}}{3}-4x^2+66x+543\ln\left(\frac{3\sqrt{2}}{\sqrt{-18+\left(x+4\right)^2}}\right)-271.646855\ln\left(8.242641+x\right)+271.646855\ln\left(-\frac{33}{136}+x\right)+C_0$
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Divide $x^4+x+1$ by $x^2+8x-2$
$\begin{array}{l}\phantom{\phantom{;}x^{2}+8x\phantom{;}-2;}{\phantom{;}x^{2}-8x\phantom{;}+66\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+8x\phantom{;}-2\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}\phantom{-;x^n}+x\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+8x\phantom{;}-2;}\underline{-x^{4}-8x^{3}+2x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{4}-8x^{3}+2x^{2};}-8x^{3}+2x^{2}+x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+8x\phantom{;}-2-;x^n;}\underline{\phantom{;}8x^{3}+64x^{2}-16x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}8x^{3}+64x^{2}-16x\phantom{;}-;x^n;}\phantom{;}66x^{2}-15x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+8x\phantom{;}-2-;x^n-;x^n;}\underline{-66x^{2}-528x\phantom{;}+132\phantom{;}\phantom{;}}\\\phantom{;;-66x^{2}-528x\phantom{;}+132\phantom{;}\phantom{;}-;x^n-;x^n;}-543x\phantom{;}+133\phantom{;}\phantom{;}\\\end{array}$
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$\begin{array}{l}\phantom{\phantom{;}x^{2}+8x\phantom{;}-2;}{\phantom{;}x^{2}-8x\phantom{;}+66\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+8x\phantom{;}-2\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}\phantom{-;x^n}+x\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+8x\phantom{;}-2;}\underline{-x^{4}-8x^{3}+2x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{4}-8x^{3}+2x^{2};}-8x^{3}+2x^{2}+x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+8x\phantom{;}-2-;x^n;}\underline{\phantom{;}8x^{3}+64x^{2}-16x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}8x^{3}+64x^{2}-16x\phantom{;}-;x^n;}\phantom{;}66x^{2}-15x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+8x\phantom{;}-2-;x^n-;x^n;}\underline{-66x^{2}-528x\phantom{;}+132\phantom{;}\phantom{;}}\\\phantom{;;-66x^{2}-528x\phantom{;}+132\phantom{;}\phantom{;}-;x^n-;x^n;}-543x\phantom{;}+133\phantom{;}\phantom{;}\\\end{array}$
Learn how to solve problems step by step online. Find the integral int((x^4+x+1)/(x^2+8x+-2))dx. Divide x^4+x+1 by x^2+8x-2. Resulting polynomial. Expand the integral \int\left(x^{2}-8x+66+\frac{-543x+133}{x^2+8x-2}\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int x^{2}dx results in: \frac{x^{3}}{3}.
Final Answer
$\frac{x^{3}}{3}-4x^2+66x+543\ln\left(\frac{3\sqrt{2}}{\sqrt{-18+\left(x+4\right)^2}}\right)-271.646855\ln\left(8.242641+x\right)+271.646855\ln\left(-\frac{33}{136}+x\right)+C_0$