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