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