\[ \int \frac {1+x}{(3+x) (1+2 x) \sqrt [3]{1+x^2}} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\frac {4 \,2^{\frac {1}{3}} \sqrt {3}\, 5^{\frac {1}{3}}}{15}-\frac {2 \,2^{\frac {1}{3}} x \sqrt {3}\, 5^{\frac {1}{3}}}{15}+\frac {\left (x^{2}+1\right )^{\frac {1}{3}} \sqrt {3}}{3}}{\left (x^{2}+1\right )^{\frac {1}{3}}}\right ) 10^{\frac {2}{3}}}{50}+\frac {\ln \left (-2 \,10^{\frac {1}{3}}+10^{\frac {1}{3}} x +5 \left (x^{2}+1\right )^{\frac {1}{3}}\right ) 10^{\frac {2}{3}}}{50}-\frac {\ln \left (4 \,10^{\frac {2}{3}}-4 \,10^{\frac {2}{3}} x +10^{\frac {2}{3}} x^{2}+\left (10 \,10^{\frac {1}{3}}-5 \,10^{\frac {1}{3}} x \right ) \left (x^{2}+1\right )^{\frac {1}{3}}+25 \left (x^{2}+1\right )^{\frac {2}{3}}\right ) 10^{\frac {2}{3}}}{100} \]
command
Integrate[(1 + x)/((3 + x)*(1 + 2*x)*(1 + x^2)^(1/3)),x]
Mathematica 13.1 output
\[ -\frac {2 \sqrt {3} \text {ArcTan}\left (\frac {4 \sqrt [3]{10}-2 \sqrt [3]{10} x+5 \sqrt [3]{1+x^2}}{5 \sqrt {3} \sqrt [3]{1+x^2}}\right )-2 \log \left (-2 \sqrt [3]{10}+\sqrt [3]{10} x+5 \sqrt [3]{1+x^2}\right )+\log \left (4\ 10^{2/3}-4\ 10^{2/3} x+10^{2/3} x^2-5 \sqrt [3]{10} (-2+x) \sqrt [3]{1+x^2}+25 \left (1+x^2\right )^{2/3}\right )}{10 \sqrt [3]{10}} \]
Mathematica 12.3 output
\[ \int \frac {1+x}{(3+x) (1+2 x) \sqrt [3]{1+x^2}} \, dx \]________________________________________________________________________________________