\[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(b*x^3 + a*x^4)^(1/4)/(-2*b + a*x + 2*x^2),x]
Mathematica 13.1 output
\[ -\frac {x^{9/4} (b+a x)^{3/4} \left (8 \sqrt [4]{a} \left (\text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )+\text {RootSum}\left [3 a^2-2 b-5 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {3 a^2 \log (x)-2 b \log (x)-12 a^2 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+8 b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-2 a \log (x) \text {$\#$1}^4+8 a \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{5 a \text {$\#$1}^3-4 \text {$\#$1}^7}\&\right ]\right )}{8 \left (x^3 (b+a x)\right )^{3/4}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx \]________________________________________________________________________________________