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