\[ \int \frac {1}{x \left (a x^2+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {35 b^{3} \arctanh \left (\frac {x \sqrt {a}}{\sqrt {b \,x^{3}+a \,x^{2}}}\right )}{8 a^{\frac {9}{2}}}+\frac {2}{a \,x^{2} \sqrt {b \,x^{3}+a \,x^{2}}}-\frac {7 \sqrt {b \,x^{3}+a \,x^{2}}}{3 a^{2} x^{4}}+\frac {35 b \sqrt {b \,x^{3}+a \,x^{2}}}{12 a^{3} x^{3}}-\frac {35 b^{2} \sqrt {b \,x^{3}+a \,x^{2}}}{8 a^{4} x^{2}} \]
command
integrate(1/x/(b*x^3+a*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {35 \, b^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{4} \mathrm {sgn}\left (x\right )} - \frac {2 \, b^{3}}{\sqrt {b x + a} a^{4} \mathrm {sgn}\left (x\right )} - \frac {57 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{3} - 136 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{3} + 87 \, \sqrt {b x + a} a^{2} b^{3}}{24 \, a^{4} b^{3} x^{3} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}} x}\,{d x} \]________________________________________________________________________________________