\[ \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx \]
Optimal antiderivative \[ \frac {2 \left (-1\right )^{\frac {1}{4}} a \left (i A +B \right ) \arctan \left (\left (-1\right )^{\frac {3}{4}} \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d}+\frac {2 a \left (i A +B \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{d}+\frac {2 i a B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3 d} \]
command
integrate(tan(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\left (i - 1\right ) \, \sqrt {2} {\left (A a - i \, B a\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, {\left (-i \, B a d^{2} \tan \left (d x + c\right )^{\frac {3}{2}} - 3 i \, A a d^{2} \sqrt {\tan \left (d x + c\right )} - 3 \, B a d^{2} \sqrt {\tan \left (d x + c\right )}\right )}}{3 \, d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )} \sqrt {\tan \left (d x + c\right )}\,{d x} \]________________________________________________________________________________________