\[ \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx \]
Optimal antiderivative \[ \frac {4 \left (-1\right )^{\frac {1}{4}} a^{2} \left (i A +B \right ) \arctan \left (\left (-1\right )^{\frac {3}{4}} \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d}+\frac {4 a^{2} \left (i A +B \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{d}-\frac {2 a^{2} \left (-7 i B +5 A \right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{15 d}+\frac {2 i B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a^{2}+i a^{2} \tan \left (d x +c \right )\right )}{5 d} \]
command
integrate(tan(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^2*(A+B*tan(d*x+c)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\left (2 i - 2\right ) \, \sqrt {2} {\left (A a^{2} - i \, B a^{2}\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, {\left (3 \, B a^{2} d^{4} \tan \left (d x + c\right )^{\frac {5}{2}} + 5 \, A a^{2} d^{4} \tan \left (d x + c\right )^{\frac {3}{2}} - 10 i \, B a^{2} d^{4} \tan \left (d x + c\right )^{\frac {3}{2}} - 30 i \, A a^{2} d^{4} \sqrt {\tan \left (d x + c\right )} - 30 \, B a^{2} d^{4} \sqrt {\tan \left (d x + c\right )}\right )}}{15 \, d^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \sqrt {\tan \left (d x + c\right )}\,{d x} \]________________________________________________________________________________________