\[ \int \frac {\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \]
Optimal antiderivative \[ \frac {\left (3072 a \,c^{4} x +4620 b c \right ) \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}+\left (-2688 a \,c^{3} x -5775 b \right ) \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}+\left (-4096 c^{5}+2464 a \,c^{2} x \right ) \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {2}{3}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}+\sqrt {a^{2} x^{2}-b}\, \left (3072 c^{4} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}-2688 c^{3} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}+2464 c^{2} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {2}{3}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}\right )}{6160 a \,c^{2} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {2}{3}}}-\frac {15 b \arctan \left (\frac {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {1}{4}}}{c^{\frac {1}{4}}}\right )}{32 a \,c^{\frac {9}{4}}}+\frac {15 b \arctanh \left (\frac {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {1}{4}}}{c^{\frac {1}{4}}}\right )}{32 a \,c^{\frac {9}{4}}} \]
command
Integrate[(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
\[ \frac {\frac {2 \sqrt [4]{c} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4} \left (-1155 b \left (-4 c+5 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )+32 c^2 \left (96 c^2 \sqrt {-b+a^2 x^2}-84 c \sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-128 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+77 \sqrt {-b+a^2 x^2} \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+a x \left (96 c^2-84 c \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}+77 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}}-5775 b \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )+5775 b \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{12320 a c^{9/4}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \]________________________________________________________________________________________