\[ \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx \]
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
Integrate[((b - a*x^6)^(1/3)*(b + a*x^6))/(x^2*(-b + c*x^3 + a*x^6)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{b-a x^6}}{x}+\frac {\sqrt [3]{c} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{c} x}{\sqrt [3]{c} x+2 \sqrt [3]{b-a x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{c} \log \left (-\sqrt [3]{c} x+\sqrt [3]{b-a x^6}\right )-\frac {1}{6} \sqrt [3]{c} \log \left (c^{2/3} x^2+\sqrt [3]{c} x \sqrt [3]{b-a x^6}+\left (b-a x^6\right )^{2/3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx \]________________________________________________________________________________________