\[ \int \frac {1}{x^2 \left (a x^2+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {315 b^{4} \arctanh \left (\frac {x \sqrt {a}}{\sqrt {b \,x^{3}+a \,x^{2}}}\right )}{64 a^{\frac {11}{2}}}+\frac {2}{a \,x^{3} \sqrt {b \,x^{3}+a \,x^{2}}}-\frac {9 \sqrt {b \,x^{3}+a \,x^{2}}}{4 a^{2} x^{5}}+\frac {21 b \sqrt {b \,x^{3}+a \,x^{2}}}{8 a^{3} x^{4}}-\frac {105 b^{2} \sqrt {b \,x^{3}+a \,x^{2}}}{32 a^{4} x^{3}}+\frac {315 b^{3} \sqrt {b \,x^{3}+a \,x^{2}}}{64 a^{5} x^{2}} \]
command
integrate(1/x^2/(b*x^3+a*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {315 \, b^{4} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a^{5} \mathrm {sgn}\left (x\right )} + \frac {2 \, b^{4}}{\sqrt {b x + a} a^{5} \mathrm {sgn}\left (x\right )} + \frac {187 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{4} - 643 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{4} + 765 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{4} - 325 \, \sqrt {b x + a} a^{3} b^{4}}{64 \, a^{5} b^{4} x^{4} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]________________________________________________________________________________________