\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {i \arctanh \left (\frac {\sqrt {2-3 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {3+2 \tan \left (d x +c \right )}}\right )}{d \sqrt {2-3 i}}-\frac {i \arctanh \left (\frac {\sqrt {2+3 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {3+2 \tan \left (d x +c \right )}}\right )}{d \sqrt {2+3 i}} \]
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 {1}{676} \, \sqrt {2} {\left (\frac {3 \, {\left (2 \, \sqrt {325 \, \sqrt {13} - 1118} \arctan \left (\frac {13 \, \left (\frac {4}{13}\right )^{\frac {3}{4}} {\left (\left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 1\right )}}{4 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right ) + 2 \, \sqrt {325 \, \sqrt {13} - 1118} \arctan \left (-\frac {13 \, \left (\frac {4}{13}\right )^{\frac {3}{4}} {\left (\left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} - 1\right )}}{4 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right ) - \sqrt {325 \, \sqrt {13} - 1118} \log \left (2 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 2 \, \sqrt {\frac {1}{13}} + 1\right ) + \sqrt {325 \, \sqrt {13} - 1118} \log \left (-2 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 2 \, \sqrt {\frac {1}{13}} + 1\right )\right )}}{d} - \frac {2 \, {\left (3 \, d^{2} \sqrt {325 \, \sqrt {13} - 1118} - 2 \, d \sqrt {325 \, \sqrt {13} - 1118} {\left | d \right |}\right )} \arctan \left (\frac {13 \, \left (\frac {4}{13}\right )^{\frac {3}{4}} {\left (\left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + \sqrt {-\frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1}\right )}}{4 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} - \frac {2 \, {\left (3 \, d^{2} \sqrt {325 \, \sqrt {13} - 1118} - 2 \, d \sqrt {325 \, \sqrt {13} - 1118} {\left | d \right |}\right )} \arctan \left (-\frac {13 \, \left (\frac {4}{13}\right )^{\frac {3}{4}} {\left (\left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} - \sqrt {-\frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1}\right )}}{4 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} + \frac {{\left (3 \, d^{2} \sqrt {325 \, \sqrt {13} - 1118} + 2 \, d \sqrt {325 \, \sqrt {13} - 1118} {\left | d \right |}\right )} \log \left (2 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} \sqrt {-\frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1} + 2 \, \sqrt {\frac {1}{13}} - \frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1\right )}{d^{3}} - \frac {{\left (3 \, d^{2} \sqrt {325 \, \sqrt {13} - 1118} + 2 \, d \sqrt {325 \, \sqrt {13} - 1118} {\left | d \right |}\right )} \log \left (-2 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} \sqrt {-\frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1} + 2 \, \sqrt {\frac {1}{13}} - \frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1\right )}{d^{3}}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________