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