\[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {3}\, \arctan \left (\frac {5 \sqrt {3}\, \left (x^{5}+9 x^{4}+28 x^{3}+36 x^{2}+27 x +27\right )^{\frac {1}{3}}}{12 \,10^{\frac {1}{3}}-2 \,10^{\frac {1}{3}} x -2 \,10^{\frac {1}{3}} x^{2}+5 \left (x^{5}+9 x^{4}+28 x^{3}+36 x^{2}+27 x +27\right )^{\frac {1}{3}}}\right ) 10^{\frac {2}{3}}}{50}+\frac {\ln \left (-6 \,10^{\frac {1}{3}}+10^{\frac {1}{3}} x +10^{\frac {1}{3}} x^{2}+5 \left (x^{5}+9 x^{4}+28 x^{3}+36 x^{2}+27 x +27\right )^{\frac {1}{3}}\right ) 10^{\frac {2}{3}}}{50}-\frac {\ln \left (36 \,10^{\frac {2}{3}}-12 \,10^{\frac {2}{3}} x -11 \,10^{\frac {2}{3}} x^{2}+2 \,10^{\frac {2}{3}} x^{3}+10^{\frac {2}{3}} x^{4}+\left (30 \,10^{\frac {1}{3}}-5 \,10^{\frac {1}{3}} x -5 \,10^{\frac {1}{3}} x^{2}\right ) \left (x^{5}+9 x^{4}+28 x^{3}+36 x^{2}+27 x +27\right )^{\frac {1}{3}}+25 \left (x^{5}+9 x^{4}+28 x^{3}+36 x^{2}+27 x +27\right )^{\frac {2}{3}}\right ) 10^{\frac {2}{3}}}{100} \]
command
Integrate[(1 + x)/((1 + 2*x)*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3)),x]
Mathematica 13.1 output
\[ -\frac {(3+x) \sqrt [3]{1+x^2} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {4 \sqrt [3]{10}-2 \sqrt [3]{10} x+5 \sqrt [3]{1+x^2}}{5 \sqrt {3} \sqrt [3]{1+x^2}}\right )-2 \log \left (-2 \sqrt [3]{10}+\sqrt [3]{10} x+5 \sqrt [3]{1+x^2}\right )+\log \left (4\ 10^{2/3}-4\ 10^{2/3} x+10^{2/3} x^2-5 \sqrt [3]{10} (-2+x) \sqrt [3]{1+x^2}+25 \left (1+x^2\right )^{2/3}\right )\right )}{10 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \]
Mathematica 12.3 output
\[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx \]________________________________________________________________________________________