\[ \int \frac {1}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \]
Optimal antiderivative \[ -\frac {8 c \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}}{7 a}+\frac {3 b \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}}{8 a c \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {4}{3}}}-\frac {13 b \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}}{32 a \,c^{2} \left (a x +\sqrt {a^{2} x^{2}-b}\right )}+\frac {117 b \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}}{256 a \,c^{3} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {2}{3}}}-\frac {585 b \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}}{1024 a \,c^{4} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}}+\frac {6 \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}}{7 a}-\frac {585 b \arctan \left (\frac {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {1}{4}}}{c^{\frac {1}{4}}}\right )}{2048 a \,c^{\frac {17}{4}}}+\frac {585 b \arctanh \left (\frac {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {1}{4}}}{c^{\frac {1}{4}}}\right )}{2048 a \,c^{\frac {17}{4}}} \]
command
Integrate[1/((a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)),x]
Mathematica 13.1 output
\[ \frac {\frac {2 \sqrt [4]{c} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4} \left (2048 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{4/3} \left (-4 c+3 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )-7 b \left (-384 c^3+416 c^2 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-468 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+585 \left (a x+\sqrt {-b+a^2 x^2}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{4/3}}-4095 b \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )+4095 b \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{14336 a c^{17/4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________