\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {i \arctan \left (\frac {\sqrt {2-3 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {-3-2 \tan \left (d x +c \right )}}\right )}{d \sqrt {2-3 i}}-\frac {i \arctan \left (\frac {\sqrt {2+3 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {-3-2 \tan \left (d x +c \right )}}\right )}{d \sqrt {2+3 i}} \]
command
integrate(tan(d*x+c)^(1/2)/(-3-2*tan(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (-i \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{4} - 6 i \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - 9 i\right )} \log \left ({\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{8} + 12 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{6} + 118 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{4} + 108 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 81\right )}{27 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________