\[ \int \frac {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((b*x + a*x^2)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2),x]
Mathematica 13.1 output
\[ \frac {x^{9/4} (b+a x)^{3/4} \left (-16 a^{11/4} x^{3/4} \sqrt [4]{b+a x}+18 a^{3/4} b x^{3/4} \sqrt [4]{b+a x}+8 a^{7/4} x^{7/4} \sqrt [4]{b+a x}-32 a^4 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+8 a^2 b \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-5 b^2 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+\left (32 a^4-8 a^2 b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-4 a^{3/4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a^5 \log (x)-a^3 b \log (x)-8 a^5 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^3 b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-a^4 \log (x) \text {$\#$1}^4+a^2 b \log (x) \text {$\#$1}^4-b^2 \log (x) \text {$\#$1}^4+4 a^4 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4-4 a^2 b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4+4 b^2 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ]\right )}{16 a^{3/4} \left (x^3 (b+a x)\right )^{3/4}} \]
Mathematica 12.3 output
\[ \int \frac {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx \]________________________________________________________________________________________