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