\[ \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (-a \,x^{6}+b \right )^{\frac {1}{3}}}{x}+\frac {c^{\frac {1}{3}} \arctan \! \left (\frac {\sqrt {3}\, c^{\frac {1}{3}} x}{c^{\frac {1}{3}} x +2 \left (-a \,x^{6}+b \right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}+\frac {c^{\frac {1}{3}} \ln \! \left (-c^{\frac {1}{3}} x +\left (-a \,x^{6}+b \right )^{\frac {1}{3}}\right )}{3}-\frac {c^{\frac {1}{3}} \ln \! \left (c^{\frac {2}{3}} x^{2}+c^{\frac {1}{3}} x \left (-a \,x^{6}+b \right )^{\frac {1}{3}}+\left (-a \,x^{6}+b \right )^{\frac {2}{3}}\right )}{6} \]
command
Int[((b - a*x^6)^(1/3)*(b + a*x^6))/(x^2*(-b + c*x^3 + a*x^6)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {2 a^2 c \sqrt [3]{b-a x^6} \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^5}{5 \sqrt {c^2+4 a b} \left (c^2-\sqrt {c^2+4 a b} c+2 a b\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (1-\frac {c}{\sqrt {c^2+4 a b}}\right ) \sqrt [3]{b-a x^6} \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^5}{5 \left (2 a b+c \left (c-\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (\frac {c}{\sqrt {c^2+4 a b}}+1\right ) \sqrt [3]{b-a x^6} \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^5}{5 \left (2 a b+c \left (c+\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {2 a^2 c \sqrt [3]{b-a x^6} \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^5}{5 \sqrt {c^2+4 a b} \left (2 a b+c \left (c+\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a \sqrt [3]{b-a x^6} \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^2}{\sqrt {c^2+4 a b} \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a c \left (1-\frac {c}{\sqrt {c^2+4 a b}}\right ) \sqrt [3]{b-a x^6} \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^2}{2 \left (2 a b+c \left (c-\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {a \sqrt [3]{b-a x^6} \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^2}{\sqrt {c^2+4 a b} \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a c \left (\frac {c}{\sqrt {c^2+4 a b}}+1\right ) \sqrt [3]{b-a x^6} \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^2}{2 \left (2 a b+c \left (c+\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\sqrt [3]{b-a x^6} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},\frac {a x^6}{b}\right )}{\sqrt [3]{1-\frac {a x^6}{b}} x} \]