\[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx \]
Optimal antiderivative \[ -\frac {\left (a^{2} \left (A -B \right )-b^{2} \left (A -B \right )-2 a b \left (A +B \right )\right ) \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{2 d}-\frac {\left (a^{2} \left (A -B \right )-b^{2} \left (A -B \right )-2 a b \left (A +B \right )\right ) \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{2 d}-\frac {\left (2 a b \left (A -B \right )+a^{2} \left (A +B \right )-b^{2} \left (A +B \right )\right ) \ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{4 d}+\frac {\left (2 a b \left (A -B \right )+a^{2} \left (A +B \right )-b^{2} \left (A +B \right )\right ) \ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{4 d}-\frac {2 \left (2 A a b +B \,a^{2}-b^{2} B \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{d}+\frac {2 \left (a^{2} A -A \,b^{2}-2 a b B \right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3 d}+\frac {2 \left (2 A a b +B \,a^{2}-b^{2} B \right ) \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5 d}+\frac {2 b \left (9 A b +11 B a \right ) \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{63 d}+\frac {2 b B \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )}{9 d} \]
command
integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]