\[ \int \frac {a+b \text {ArcTan}\left (c x^2\right )}{(d+e x)^2} \, dx \]
Optimal antiderivative \[ \frac {b \,c^{2} d^{3} \arctan \! \left (c \,x^{2}\right )}{e \left (c^{2} d^{4}+e^{4}\right )}+\frac {-a -b \arctan \! \left (c \,x^{2}\right )}{e \left (e x +d \right )}-\frac {2 b c d e \ln \! \left (e x +d \right )}{c^{2} d^{4}+e^{4}}+\frac {b c d e \ln \! \left (c^{2} x^{4}+1\right )}{2 c^{2} d^{4}+2 e^{4}}-\frac {b \left (c \,d^{2}-e^{2}\right ) \arctan \! \left (-1+x \sqrt {2}\, \sqrt {c}\right ) \sqrt {c}\, \sqrt {2}}{2 \left (c^{2} d^{4}+e^{4}\right )}-\frac {b \left (c \,d^{2}-e^{2}\right ) \arctan \! \left (1+x \sqrt {2}\, \sqrt {c}\right ) \sqrt {c}\, \sqrt {2}}{2 \left (c^{2} d^{4}+e^{4}\right )}-\frac {b \left (c \,d^{2}+e^{2}\right ) \ln \! \left (1+c \,x^{2}-x \sqrt {2}\, \sqrt {c}\right ) \sqrt {c}\, \sqrt {2}}{4 \left (c^{2} d^{4}+e^{4}\right )}+\frac {b \left (c \,d^{2}+e^{2}\right ) \ln \! \left (1+c \,x^{2}+x \sqrt {2}\, \sqrt {c}\right ) \sqrt {c}\, \sqrt {2}}{4 c^{2} d^{4}+4 e^{4}} \]
command
integrate((a+b*arctan(c*x^2))/(e*x+d)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {output too large to display} \]