\[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {x \left (b \,x^{3}+a \right )}{12 a \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {5}{2}}}+\frac {11 x \left (b \,x^{3}+a \right )^{2}}{108 a^{2} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {5}{2}}}+\frac {11 x \left (b \,x^{3}+a \right )^{3}}{81 a^{3} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {5}{2}}}+\frac {55 x \left (b \,x^{3}+a \right )^{4}}{243 a^{4} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {5}{2}}}+\frac {110 \left (b \,x^{3}+a \right )^{5} \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{729 a^{\frac {14}{3}} b^{\frac {1}{3}} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {5}{2}}}-\frac {55 \left (b \,x^{3}+a \right )^{5} \ln \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )}{729 a^{\frac {14}{3}} b^{\frac {1}{3}} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {5}{2}}}-\frac {110 \left (b \,x^{3}+a \right )^{5} \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x \right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{729 a^{\frac {14}{3}} b^{\frac {1}{3}} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{\frac {5}{2}}} \]
command
integrate(1/(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
\[ -\frac {110 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {110 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{5} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {55 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{5} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {220 \, b^{3} x^{10} + 792 \, a b^{2} x^{7} + 1023 \, a^{2} b x^{4} + 532 \, a^{3} x}{972 \, {\left (b x^{3} + a\right )}^{4} a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________