\[ \int \frac {b+2 a x}{\sqrt [4]{c+b x+a x^2} \left (5 c+4 b x+4 a x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (1-\frac {2 \left (a \,x^{2}+b x +c \right )^{\frac {1}{4}}}{c^{\frac {1}{4}}}\right )}{2 c^{\frac {1}{4}}}+\frac {\arctan \left (1+\frac {2 \left (a \,x^{2}+b x +c \right )^{\frac {1}{4}}}{c^{\frac {1}{4}}}\right )}{2 c^{\frac {1}{4}}}-\frac {\arctanh \left (\frac {\frac {c^{\frac {1}{4}}}{2}+\frac {\sqrt {a \,x^{2}+b x +c}}{c^{\frac {1}{4}}}}{\left (a \,x^{2}+b x +c \right )^{\frac {1}{4}}}\right )}{2 c^{\frac {1}{4}}} \]
command
Integrate[(b + 2*a*x)/((c + b*x + a*x^2)^(1/4)*(5*c + 4*b*x + 4*a*x^2)),x]
Mathematica 13.1 output
\[ -\frac {\text {ArcTan}\left (1-\frac {2 \sqrt [4]{c+x (b+a x)}}{\sqrt [4]{c}}\right )-\text {ArcTan}\left (1+\frac {2 \sqrt [4]{c+x (b+a x)}}{\sqrt [4]{c}}\right )+\tanh ^{-1}\left (\frac {\sqrt {c}+2 \sqrt {c+x (b+a x)}}{2 \sqrt [4]{c} \sqrt [4]{c+x (b+a x)}}\right )}{2 \sqrt [4]{c}} \]
Mathematica 12.3 output
\[ \int \frac {b+2 a x}{\sqrt [4]{c+b x+a x^2} \left (5 c+4 b x+4 a x^2\right )} \, dx \]________________________________________________________________________________________