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