\[ \int \frac {1}{(b+a x) \sqrt [3]{-b^3+a^3 x^3}} \, dx \]
Optimal antiderivative \[ \frac {\left (-1\right )^{\frac {5}{6}} \sqrt {3}\, \arctanh \left (\frac {\frac {\left (i b +b \sqrt {3}\right ) 2^{\frac {1}{3}} \sqrt {3}}{6}+\frac {\left (-i a -\sqrt {3}\, a \right ) x 2^{\frac {1}{3}} \sqrt {3}}{6}+\frac {i \left (a^{3} x^{3}-b^{3}\right )^{\frac {1}{3}} \sqrt {3}}{3}}{\left (a^{3} x^{3}-b^{3}\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}}}{4 a b}+\frac {\left (-1\right )^{\frac {1}{3}} 2^{\frac {2}{3}} \ln \left (-\left (-1\right )^{\frac {2}{3}} \sqrt {a}\, b^{\frac {3}{2}}+\left (-1\right )^{\frac {2}{3}} a^{\frac {3}{2}} \sqrt {b}\, x -2^{\frac {2}{3}} \sqrt {a}\, \sqrt {b}\, \left (a^{3} x^{3}-b^{3}\right )^{\frac {1}{3}}\right )}{4 a b}-\frac {\left (-1\right )^{\frac {1}{3}} 2^{\frac {2}{3}} \ln \left (\left (-1\right )^{\frac {1}{3}} a \,b^{3}-2 \left (-1\right )^{\frac {1}{3}} a^{2} b^{2} x +\left (-1\right )^{\frac {1}{3}} a^{3} b \,x^{2}+\left (\left (-2\right )^{\frac {2}{3}} a \,b^{2}-\left (-2\right )^{\frac {2}{3}} a^{2} b x \right ) \left (a^{3} x^{3}-b^{3}\right )^{\frac {1}{3}}-2 \,2^{\frac {1}{3}} a b \left (a^{3} x^{3}-b^{3}\right )^{\frac {2}{3}}\right )}{8 a b} \]
command
Integrate[1/((b + a*x)*(-b^3 + a^3*x^3)^(1/3)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{-\frac {1}{2}} \left (2 i \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt [3]{2} \left (3+i \sqrt {3}\right ) b+\sqrt [3]{2} \left (-3-i \sqrt {3}\right ) a x+2 i \sqrt {3} \sqrt [3]{-b^3+a^3 x^3}}{6 \sqrt [3]{-b^3+a^3 x^3}}\right )+2 \log \left (\sqrt {a} \sqrt {b} \left (-b+i \sqrt {3} b+a x-i \sqrt {3} a x+2\ 2^{2/3} \sqrt [3]{-b^3+a^3 x^3}\right )\right )-\log \left (a b \left (\left (-1-i \sqrt {3}\right ) b^2+\left (-1-i \sqrt {3}\right ) a^2 x^2+2 a b \left (x+i \sqrt {3} x\right )-2 (-2)^{2/3} b \sqrt [3]{-b^3+a^3 x^3}+2 (-2)^{2/3} a x \sqrt [3]{-b^3+a^3 x^3}+4 \sqrt [3]{2} \left (-b^3+a^3 x^3\right )^{2/3}\right )\right )\right )}{4 a b} \]
Mathematica 12.3 output
\[ \int \frac {1}{(b+a x) \sqrt [3]{-b^3+a^3 x^3}} \, dx \]________________________________________________________________________________________