\[ \int \frac {1}{(-b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \! \left (\frac {2^{\frac {3}{4}} a^{\frac {1}{4}} b^{\frac {1}{4}} \left (a^{2} x^{3}+b^{2} x \right )^{\frac {1}{4}}}{a x +b}\right ) 2^{\frac {3}{4}}}{4 a^{\frac {3}{4}} b^{\frac {3}{4}}}-\frac {\operatorname {arctanh}\! \left (\frac {2^{\frac {3}{4}} a^{\frac {1}{4}} b^{\frac {1}{4}} \left (a^{2} x^{3}+b^{2} x \right )^{\frac {1}{4}}}{a x +b}\right ) 2^{\frac {3}{4}}}{4 a^{\frac {3}{4}} b^{\frac {3}{4}}} \]
command
Int[1/((-b + a*x)*(b^2*x + a^2*x^3)^(1/4)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {1}{(-b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {4 a x^2 \sqrt [4]{\frac {a^2 x^2}{b^2}+1} \operatorname {AppellF1}\left (\frac {7}{8},1,\frac {1}{4},\frac {15}{8},\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{7 b^2 \sqrt [4]{a^2 x^3+b^2 x}}-\frac {4 x \sqrt [4]{\frac {a^2 x^2}{b^2}+1} \operatorname {AppellF1}\left (\frac {3}{8},1,\frac {1}{4},\frac {11}{8},\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{3 b \sqrt [4]{a^2 x^3+b^2 x}} \]