\[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x) \sqrt {3+4 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \arctan \left (\frac {2 i c x +d \sqrt {3}}{\sqrt {-2 i x^{2}+\sqrt {3}}\, \sqrt {2 i c^{2}-d^{2} \sqrt {3}}}\right )}{\sqrt {2 i c^{2}-d^{2} \sqrt {3}}}+\frac {\left (-\frac {1}{2}-\frac {i}{2}\right ) \arctanh \left (\frac {-2 i c x +d \sqrt {3}}{\sqrt {2 i x^{2}+\sqrt {3}}\, \sqrt {2 i c^{2}+d^{2} \sqrt {3}}}\right )}{\sqrt {2 i c^{2}+d^{2} \sqrt {3}}} \]
command
Integrate[Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]/((c + d*x)*Sqrt[3 + 4*x^4]),x]
Mathematica 13.1 output
\[ \frac {-\sqrt {-2 c^2-\sqrt {4 c^4+3 d^4}} \tan ^{-1}\left (\frac {d \sqrt {2 x^2+\sqrt {3+4 x^4}}}{\sqrt {-2 c^2-\sqrt {4 c^4+3 d^4}}}\right )+\sqrt {-2 c^2+\sqrt {4 c^4+3 d^4}} \tan ^{-1}\left (\frac {d \sqrt {2 x^2+\sqrt {3+4 x^4}}}{\sqrt {-2 c^2+\sqrt {4 c^4+3 d^4}}}\right )+2 i c \sqrt {12 c^4+9 d^4} \text {RootSum}\left [12 d^2+16 i \sqrt {3} c^2 \text {$\#$1}+24 d^2 \text {$\#$1}+24 i \sqrt {3} c^2 \text {$\#$1}^2+24 d^2 \text {$\#$1}^2+8 i \sqrt {3} c^2 \text {$\#$1}^3+12 d^2 \text {$\#$1}^3+3 d^2 \text {$\#$1}^4\&,\frac {\log \left (2 x^2+\sqrt {3+4 x^4}\right )-\log \left (i \sqrt {3}-2 x^2+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-2 x^2 \text {$\#$1}-\sqrt {3+4 x^4} (1+\text {$\#$1})\right )+\log \left (2 x^2+\sqrt {3+4 x^4}\right ) \text {$\#$1}-\log \left (i \sqrt {3}-2 x^2+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-2 x^2 \text {$\#$1}-\sqrt {3+4 x^4} (1+\text {$\#$1})\right ) \text {$\#$1}}{4 i \sqrt {3} c^2+6 d^2+12 i \sqrt {3} c^2 \text {$\#$1}+12 d^2 \text {$\#$1}+6 i \sqrt {3} c^2 \text {$\#$1}^2+9 d^2 \text {$\#$1}^2+3 d^2 \text {$\#$1}^3}\&\right ]}{\sqrt {4 c^4+3 d^4}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x) \sqrt {3+4 x^4}} \, dx \]________________________________________________________________________________________