\[ \int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {3}\, \arctan \left (\frac {5 \sqrt {3}\, \left (8 x^{8}+84 x^{7}+338 x^{6}+679 x^{5}+825 x^{4}+784 x^{3}+522 x^{2}+189 x +27\right )^{\frac {1}{3}}}{12 \,10^{\frac {1}{3}}+22 \,10^{\frac {1}{3}} x -6 \,10^{\frac {1}{3}} x^{2}-4 \,10^{\frac {1}{3}} x^{3}+5 \left (8 x^{8}+84 x^{7}+338 x^{6}+679 x^{5}+825 x^{4}+784 x^{3}+522 x^{2}+189 x +27\right )^{\frac {1}{3}}}\right ) 10^{\frac {2}{3}}}{50}-\frac {\ln \left (2 x^{2}+7 x +3\right ) 10^{\frac {2}{3}}}{50}+\frac {\ln \left (4 x^{4}+28 x^{3}+61 x^{2}+42 x +9\right ) 10^{\frac {2}{3}}}{100}+\frac {\ln \left (-6 \,10^{\frac {1}{3}}-11 \,10^{\frac {1}{3}} x +3 \,10^{\frac {1}{3}} x^{2}+2 \,10^{\frac {1}{3}} x^{3}+5 \left (8 x^{8}+84 x^{7}+338 x^{6}+679 x^{5}+825 x^{4}+784 x^{3}+522 x^{2}+189 x +27\right )^{\frac {1}{3}}\right ) 10^{\frac {2}{3}}}{50}-\frac {\ln \left (36 \,10^{\frac {2}{3}}+132 \,10^{\frac {2}{3}} x +85 \,10^{\frac {2}{3}} x^{2}-90 \,10^{\frac {2}{3}} x^{3}-35 \,10^{\frac {2}{3}} x^{4}+12 \,10^{\frac {2}{3}} x^{5}+4 \,10^{\frac {2}{3}} x^{6}+\left (30 \,10^{\frac {1}{3}}+55 \,10^{\frac {1}{3}} x -15 \,10^{\frac {1}{3}} x^{2}-10 \,10^{\frac {1}{3}} x^{3}\right ) \left (8 x^{8}+84 x^{7}+338 x^{6}+679 x^{5}+825 x^{4}+784 x^{3}+522 x^{2}+189 x +27\right )^{\frac {1}{3}}+25 \left (8 x^{8}+84 x^{7}+338 x^{6}+679 x^{5}+825 x^{4}+784 x^{3}+522 x^{2}+189 x +27\right )^{\frac {2}{3}}\right ) 10^{\frac {2}{3}}}{100} \]
command
Integrate[(1 + x)/(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3),x]
Mathematica 13.1 output
\[ -\frac {\sqrt [3]{1+x^2} \left (3+7 x+2 x^2\right ) \left (2 \sqrt {3} \text {ArcTan}\left (\frac {4 \sqrt [3]{10}-2 \sqrt [3]{10} x+5 \sqrt [3]{1+x^2}}{5 \sqrt {3} \sqrt [3]{1+x^2}}\right )-2 \log \left (-2 \sqrt [3]{10}+\sqrt [3]{10} x+5 \sqrt [3]{1+x^2}\right )+\log \left (4\ 10^{2/3}-4\ 10^{2/3} x+10^{2/3} x^2-5 \sqrt [3]{10} (-2+x) \sqrt [3]{1+x^2}+25 \left (1+x^2\right )^{2/3}\right )\right )}{10 \sqrt [3]{10} \sqrt [3]{\left (1+x^2\right ) \left (3+7 x+2 x^2\right )^3}} \]
Mathematica 12.3 output
\[ \int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx \]________________________________________________________________________________________