\[ \int \frac {1}{x^2 \sqrt {c x^2} (a+b x)^2} \, dx \]
Optimal antiderivative \[ \frac {2 b}{a^{3} \sqrt {c \,x^{2}}}-\frac {1}{2 a^{2} x \sqrt {c \,x^{2}}}+\frac {b^{2} x}{a^{3} \left (b x +a \right ) \sqrt {c \,x^{2}}}+\frac {3 b^{2} x \ln \! \left (x \right )}{a^{4} \sqrt {c \,x^{2}}}-\frac {3 b^{2} x \ln \! \left (b x +a \right )}{a^{4} \sqrt {c \,x^{2}}} \]
command
integrate(1/x^2/(b*x+a)^2/(c*x^2)^(1/2),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
\[ -\frac {\frac {6 \, b^{2} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{4} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} + \frac {2 \, b^{2}}{{\left (b x + a\right )} a^{3} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} - \frac {\frac {6 \, a b^{2}}{b x + a} - 5 \, b^{2}}{a^{4} {\left (\frac {a}{b x + a} - 1\right )}^{2} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}}{2 \, \sqrt {c}} \]