\[ \int \frac {x^6}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {5 x}{486 a^{2} b^{2} \sqrt {\left (b \,x^{3}+a \right )^{2}}}-\frac {x^{4}}{12 b \left (b \,x^{3}+a \right )^{3} \sqrt {\left (b \,x^{3}+a \right )^{2}}}-\frac {x}{27 b^{2} \left (b \,x^{3}+a \right )^{2} \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {x}{162 a \,b^{2} \left (b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}+\frac {5 \left (b \,x^{3}+a \right ) \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{729 a^{\frac {8}{3}} b^{\frac {7}{3}} \sqrt {\left (b \,x^{3}+a \right )^{2}}}-\frac {5 \left (b \,x^{3}+a \right ) \ln \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{1458 a^{\frac {8}{3}} b^{\frac {7}{3}} \sqrt {\left (b \,x^{3}+a \right )^{2}}}-\frac {5 \left (b \,x^{3}+a \right ) \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{729 a^{\frac {8}{3}} b^{\frac {7}{3}} \sqrt {\left (b \,x^{3}+a \right )^{2}}} \]
command
integrate(x^6/(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
\[ -\frac {5 \, \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {5 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{3} b^{2} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{3} b^{3} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {10 \, b^{3} x^{10} + 36 \, a b^{2} x^{7} - 75 \, a^{2} b x^{4} - 20 \, a^{3} x}{972 \, {\left (b x^{3} + a\right )}^{4} a^{2} b^{2} \mathrm {sgn}\left (b x^{3} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________