\[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {7}{18 a^{2} x \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {1}{6 a x \left (b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}-\frac {14 \left (b \,x^{3}+a \right )}{9 a^{3} x \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {14 b^{\frac {1}{3}} \left (b \,x^{3}+a \right ) \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{27 a^{\frac {10}{3}} \sqrt {\left (b \,x^{3}+a \right )^{2}}}-\frac {7 b^{\frac {1}{3}} \left (b \,x^{3}+a \right ) \ln \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{27 a^{\frac {10}{3}} \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {14 b^{\frac {1}{3}} \left (b \,x^{3}+a \right ) \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{27 a^{\frac {10}{3}} \sqrt {\left (b \,x^{3}+a \right )^{2}}} \]
command
integrate(1/x^2/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {14 \, b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {14 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {7 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{4} b \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {10 \, b^{2} x^{5} + 13 \, a b x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {1}{a^{3} x \mathrm {sgn}\left (b x^{3} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________