\[ \int \frac {1}{\left (a x^2+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {15 b^{2} \arctanh \left (\frac {x \sqrt {a}}{\sqrt {b \,x^{3}+a \,x^{2}}}\right )}{4 a^{\frac {7}{2}}}+\frac {2}{a x \sqrt {b \,x^{3}+a \,x^{2}}}-\frac {5 \sqrt {b \,x^{3}+a \,x^{2}}}{2 a^{2} x^{3}}+\frac {15 b \sqrt {b \,x^{3}+a \,x^{2}}}{4 a^{3} x^{2}} \]
command
integrate(1/(b*x^3+a*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {15 \, b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{3} \mathrm {sgn}\left (x\right )} + \frac {2 \, b^{2}}{\sqrt {b x + a} a^{3} \mathrm {sgn}\left (x\right )} + \frac {7 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2} - 9 \, \sqrt {b x + a} a b^{2}}{4 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________