\[ \int \tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx \]
Optimal antiderivative \[ -\frac {4 \left (-1\right )^{\frac {1}{4}} a^{2} \left (i A +B \right ) \arctan \left (\left (-1\right )^{\frac {3}{4}} \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d}-\frac {4 a^{2} \left (i A +B \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{d}+\frac {4 a^{2} \left (-i B +A \right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3 d}+\frac {4 a^{2} \left (i A +B \right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5 d}-\frac {2 a^{2} \left (-11 i B +9 A \right ) \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{63 d}+\frac {2 i B \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) \left (a^{2}+i a^{2} \tan \left (d x +c \right )\right )}{9 d} \]
command
integrate(tan(d*x+c)^(5/2)*(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
\[ \frac {\left (2 i - 2\right ) \, \sqrt {2} {\left (A a^{2} - i \, B a^{2}\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, {\left (35 \, B a^{2} d^{8} \tan \left (d x + c\right )^{\frac {9}{2}} + 45 \, A a^{2} d^{8} \tan \left (d x + c\right )^{\frac {7}{2}} - 90 i \, B a^{2} d^{8} \tan \left (d x + c\right )^{\frac {7}{2}} - 126 i \, A a^{2} d^{8} \tan \left (d x + c\right )^{\frac {5}{2}} - 126 \, B a^{2} d^{8} \tan \left (d x + c\right )^{\frac {5}{2}} - 210 \, A a^{2} d^{8} \tan \left (d x + c\right )^{\frac {3}{2}} + 210 i \, B a^{2} d^{8} \tan \left (d x + c\right )^{\frac {3}{2}} + 630 i \, A a^{2} d^{8} \sqrt {\tan \left (d x + c\right )} + 630 \, B a^{2} d^{8} \sqrt {\tan \left (d x + c\right )}\right )}}{315 \, d^{9}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \tan \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]________________________________________________________________________________________