\[ \int \frac {(a+b \text {ArcTan}(c x))^3}{(d+e x)^2} \, dx \]
Optimal antiderivative \[ \frac {\mathrm {I} c \left (a +b \arctan \! \left (c x \right )\right )^{3}}{c^{2} d^{2}+e^{2}}+\frac {c^{2} d \left (a +b \arctan \! \left (c x \right )\right )^{3}}{e \left (c^{2} d^{2}+e^{2}\right )}-\frac {\left (a +b \arctan \! \left (c x \right )\right )^{3}}{e \left (e x +d \right )}-\frac {3 b c \left (a +b \arctan \! \left (c x \right )\right )^{2} \ln \! \left (\frac {2}{1-\mathrm {I} c x}\right )}{c^{2} d^{2}+e^{2}}+\frac {3 b c \left (a +b \arctan \! \left (c x \right )\right )^{2} \ln \! \left (\frac {2}{1+\mathrm {I} c x}\right )}{c^{2} d^{2}+e^{2}}+\frac {3 b c \left (a +b \arctan \! \left (c x \right )\right )^{2} \ln \! \left (\frac {2 c \left (e x +d \right )}{\left (c d +\mathrm {I} e \right ) \left (1-\mathrm {I} c x \right )}\right )}{c^{2} d^{2}+e^{2}}+\frac {3 \,\mathrm {I} b^{2} c \left (a +b \arctan \! \left (c x \right )\right ) \polylog \! \left (2, 1-\frac {2}{1-\mathrm {I} c x}\right )}{c^{2} d^{2}+e^{2}}+\frac {3 \,\mathrm {I} b^{2} c \left (a +b \arctan \! \left (c x \right )\right ) \polylog \! \left (2, 1-\frac {2}{1+\mathrm {I} c x}\right )}{c^{2} d^{2}+e^{2}}-\frac {3 \,\mathrm {I} b^{2} c \left (a +b \arctan \! \left (c x \right )\right ) \polylog \! \left (2, 1-\frac {2 c \left (e x +d \right )}{\left (c d +\mathrm {I} e \right ) \left (1-\mathrm {I} c x \right )}\right )}{c^{2} d^{2}+e^{2}}-\frac {3 b^{3} c \polylog \! \left (3, 1-\frac {2}{1-\mathrm {I} c x}\right )}{2 \left (c^{2} d^{2}+e^{2}\right )}+\frac {3 b^{3} c \polylog \! \left (3, 1-\frac {2}{1+\mathrm {I} c x}\right )}{2 \left (c^{2} d^{2}+e^{2}\right )}+\frac {3 b^{3} c \polylog \! \left (3, 1-\frac {2 c \left (e x +d \right )}{\left (c d +\mathrm {I} e \right ) \left (1-\mathrm {I} c x \right )}\right )}{2 \left (c^{2} d^{2}+e^{2}\right )} \]
command
Integrate[(a + b*ArcTan[c*x])^3/(d + e*x)^2,x]
Mathematica 13.1 output
\[ \int \frac {(a+b \text {ArcTan}(c x))^3}{(d+e x)^2} \, dx \]
Mathematica 12.3 output
\[ -\frac {a^3}{e (d+e x)}+\frac {3 a^2 b c^2 d \tan ^{-1}(c x)}{c^2 d^2 e+e^3}-\frac {3 a^2 b c \log \left (c^2 x^2+1\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 a^2 b c \log (d+e x)}{c^2 d^2+e^2}-\frac {3 a^2 b \tan ^{-1}(c x)}{e (d+e x)}+\frac {3 a b^2 \left (-\frac {c d \left (-\frac {1}{2} \pi \log \left (c^2 x^2+1\right )+i \text {Li}_2\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-i \tan ^{-1}(c x) \left (\pi -2 \tan ^{-1}\left (\frac {c d}{e}\right )\right )-2 \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right ) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+2 \tan ^{-1}\left (\frac {c d}{e}\right ) \log \left (\sin \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )\right )-\pi \log \left (1+e^{-2 i \tan ^{-1}(c x)}\right )\right )}{c^2 d^2+e^2}-\frac {\tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left (\frac {c d}{e}\right )}}{e \sqrt {\frac {c^2 d^2}{e^2}+1}}+\frac {x \tan ^{-1}(c x)^2}{d+e x}\right )}{d}+\frac {b^3 \left (\frac {2 x \tan ^{-1}(c x)^3}{d+e x}+\frac {2 \tan ^{-1}(c x) \left (\tan ^{-1}(c x)^2 \left (-2 e \sqrt {\frac {c^2 d^2}{e^2}+1} e^{i \tan ^{-1}\left (\frac {c d}{e}\right )}+i c d+e\right )+3 c d \left (2 \tan ^{-1}\left (\frac {c d}{e}\right ) \left (\log \left (\frac {e^{-i \tan ^{-1}\left (\frac {c d}{e}\right )} \left ((c x-i) e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )}+c x+i\right )}{2 \sqrt {c^2 x^2+1}}\right )+\log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-\log \left (\sin \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )\right )-\log \left (-i e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \sin \left (2 \tan ^{-1}(c x)\right )-e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \cos \left (2 \tan ^{-1}(c x)\right )+1\right )\right )+\pi \left (\log \left (1+e^{-2 i \tan ^{-1}(c x)}\right )-\log \left (-\frac {2 i}{c x-i}\right )\right )\right )+3 c d \tan ^{-1}(c x) \left (2 \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-\log \left (-i e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \sin \left (2 \tan ^{-1}(c x)\right )-e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \cos \left (2 \tan ^{-1}(c x)\right )+1\right )\right )\right )-6 i c d \tan ^{-1}(c x) \text {Li}_2\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+3 c d \text {Li}_3\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )}{c^2 d^2+e^2}\right )}{2 d} \]