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