\[ \int \frac {1}{x^3 \sqrt {a x^2+b x^3}} \, dx \]
Optimal antiderivative \[ \frac {5 b^{3} \arctanh \left (\frac {x \sqrt {a}}{\sqrt {b \,x^{3}+a \,x^{2}}}\right )}{8 a^{\frac {7}{2}}}-\frac {\sqrt {b \,x^{3}+a \,x^{2}}}{3 a \,x^{4}}+\frac {5 b \sqrt {b \,x^{3}+a \,x^{2}}}{12 a^{2} x^{3}}-\frac {5 b^{2} \sqrt {b \,x^{3}+a \,x^{2}}}{8 a^{3} x^{2}} \]
command
integrate(1/x^3/(b*x^3+a*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {15 \, b^{4} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {15 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{4} - 40 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{4} + 33 \, \sqrt {b x + a} a^{2} b^{4}}{a^{3} b^{3} x^{3}}}{24 \, b \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________