\[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {154}{243 a^{4} x^{2} \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {1}{12 a \,x^{2} \left (b \,x^{3}+a \right )^{3} \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {7}{54 a^{2} x^{2} \left (b \,x^{3}+a \right )^{2} \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {77}{324 a^{3} x^{2} \left (b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}-\frac {385 \left (b \,x^{3}+a \right )}{243 a^{5} x^{2} \sqrt {\left (b \,x^{3}+a \right )^{2}}}-\frac {770 b^{\frac {2}{3}} \left (b \,x^{3}+a \right ) \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{729 a^{\frac {17}{3}} \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {385 b^{\frac {2}{3}} \left (b \,x^{3}+a \right ) \ln \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{729 a^{\frac {17}{3}} \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {770 b^{\frac {2}{3}} \left (b \,x^{3}+a \right ) \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{729 a^{\frac {17}{3}} \sqrt {\left (b \,x^{3}+a \right )^{2}}} \]
command
integrate(1/x^3/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {770 \, b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{6} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {770 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{6} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {385 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{6} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {1}{2 \, a^{5} x^{2} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {1054 \, b^{4} x^{10} + 3600 \, a b^{3} x^{7} + 4245 \, a^{2} b^{2} x^{4} + 1780 \, a^{3} b x}{972 \, {\left (b x^{3} + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x^{3} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________