\[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {x \left (b \,x^{3}+a \right )}{6 a \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {3}{2}}}+\frac {5 x \left (b \,x^{3}+a \right )^{2}}{18 a^{2} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {3}{2}}}+\frac {5 \left (b \,x^{3}+a \right )^{3} \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{27 a^{\frac {8}{3}} b^{\frac {1}{3}} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {3}{2}}}-\frac {5 \left (b \,x^{3}+a \right )^{3} \ln \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{54 a^{\frac {8}{3}} b^{\frac {1}{3}} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {3}{2}}}-\frac {5 \left (b \,x^{3}+a \right )^{3} \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{27 a^{\frac {8}{3}} b^{\frac {1}{3}} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {3}{2}}} \]
command
integrate(1/(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
\[ -\frac {5 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {5 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {5 \, b x^{4} + 8 \, a x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} \mathrm {sgn}\left (b x^{3} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________