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