\[ \int \frac {1}{x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 b}{3 a^{3} \sqrt {\left (b \,x^{3}+a \right )^{2}}}-\frac {b}{6 a^{2} \left (b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {-b \,x^{3}-a}{3 a^{3} x^{3} \sqrt {\left (b \,x^{3}+a \right )^{2}}}-\frac {3 b \left (b \,x^{3}+a \right ) \ln \left (x \right )}{a^{4} \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {b \left (b \,x^{3}+a \right ) \ln \left (b \,x^{3}+a \right )}{a^{4} \sqrt {\left (b \,x^{3}+a \right )^{2}}} \]
command
integrate(1/x^4/(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
\[ \frac {b \log \left ({\left | b x^{3} + a \right |}\right )}{a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {3 \, b \log \left ({\left | x \right |}\right )}{a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {9 \, b^{3} x^{6} + 22 \, a b^{2} x^{3} + 14 \, a^{2} b}{6 \, {\left (b x^{3} + a\right )}^{2} a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {3 \, b x^{3} - a}{3 \, a^{4} x^{3} \mathrm {sgn}\left (b x^{3} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________