\[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (\left (4+i\right ) A +\left (1+6 i\right ) B \right ) \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{a d}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (\left (4+i\right ) A +\left (1+6 i\right ) B \right ) \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{a d}+\frac {\left (-\frac {1}{16}-\frac {i}{16}\right ) \left (\left (1+4 i\right ) A +\left (-6-i\right ) B \right ) \ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{a d}-\frac {\left (\left (3-5 i\right ) A +\left (5+7 i\right ) B \right ) \ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{16 a d}-\frac {5 \left (i A -B \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{2 a d}-\frac {\left (7 i B +3 A \right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{6 a d}+\frac {\left (i A -B \right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{2 d \left (a +i a \tan \left (d x +c \right )\right )} \]
command
integrate(tan(d*x+c)^(5/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\left (i - 1\right ) \, \sqrt {2} {\left (-2 i \, A + 3 \, B\right )} \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{2 \, a d} + \frac {\left (i + 1\right ) \, \sqrt {2} {\left (-i \, A - B\right )} \arctan \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{4 \, a d} - \frac {A \sqrt {\tan \left (d x + c\right )} + i \, B \sqrt {\tan \left (d x + c\right )}}{2 \, a d {\left (\tan \left (d x + c\right ) - i\right )}} - \frac {2 \, {\left (i \, B a^{2} d^{2} \tan \left (d x + c\right )^{\frac {3}{2}} + 3 i \, A a^{2} d^{2} \sqrt {\tan \left (d x + c\right )} - 3 \, B a^{2} d^{2} \sqrt {\tan \left (d x + c\right )}\right )}}{3 \, a^{3} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{\frac {5}{2}}}{i \, a \tan \left (d x + c\right ) + a}\,{d x} \]________________________________________________________________________________________