\[ \int \frac {(a+b \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (a^{2} \left (A -B \right )-b^{2} \left (A -B \right )-2 a b \left (A +B \right )\right ) \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{2 d}-\frac {\left (a^{2} \left (A -B \right )-b^{2} \left (A -B \right )-2 a b \left (A +B \right )\right ) \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{2 d}-\frac {\left (2 a b \left (A -B \right )+a^{2} \left (A +B \right )-b^{2} \left (A +B \right )\right ) \ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{4 d}+\frac {\left (2 a b \left (A -B \right )+a^{2} \left (A +B \right )-b^{2} \left (A +B \right )\right ) \ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{4 d}-\frac {2 a^{2} A}{d \sqrt {\tan \left (d x +c \right )}}+\frac {2 b^{2} B \left (\sqrt {\tan }\left (d x +c \right )\right )}{d} \]
command
integrate((a+b*tan(d*x+c))^2*(A+B*tan(d*x+c))/tan(d*x+c)^(3/2),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 \[ \text {Timed out} \]