\[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \]
Optimal antiderivative \[ \frac {\left (-4026531840 a^{2} c^{10} x^{2}+6055526400 a^{3} c^{7} x^{3}+2013265920 b \,c^{10}-4541644800 a b \,c^{7} x -1860655104 b^{2} c^{4}+2409402996 a \,b^{2} c x \right ) \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}+\left (3523215360 a^{2} c^{9} x^{2}-5752750080 a^{3} c^{6} x^{3}-1761607680 b \,c^{9}+4314562560 a b \,c^{6} x +1976946048 b^{2} c^{3}-3011753745 a \,b^{2} x \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 (2684354560 a \,c^{11} x -3229614080 a^{2} c^{8} x^{2}+5513052160 a^{3} c^{5} x^{3}+1614807040 b \,c^{8}-28943523840 a b \,c^{5} x -2141691552 b^{2} c^{2}\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 (\left (-4026531840 a \,c^{10} x +6055526400 a^{2} c^{7} x^{2}-1513881600 b \,c^{7}+2409402996 b^{2} c \right ) \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}+\left (3523215360 a \,c^{9} x -5752750080 a^{2} c^{6} x^{2}+1438187520 b \,c^{6}-3011753745 b^{2}\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 (2684354560 c^{11}-3229614080 a \,c^{8} x +5513052160 a^{2} c^{5} x^{2}-26186997760 b \,c^{5}\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}}\right )}{12404367360 a \,c^{5} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {5}{3}}}-\frac {1989 b^{2} \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 )}{16384 a \,c^{\frac {21}{4}}}+\frac {1989 b^{2} \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 )}{16384 a \,c^{\frac {21}{4}}} \]
command
Integrate[(Sqrt[-b + a^2*x^2]*(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
\[ \frac {\frac {2 \sqrt [4]{c} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4} \left (-302841 b^2 \left (6144 c^4-6528 c^3 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}+7072 c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}-7956 c \left (a x+\sqrt {-b+a^2 x^2}\right )+9945 \left (a x+\sqrt {-b+a^2 x^2}\right )^{4/3}\right )-40960 b c^5 \left (-49152 c^5+43008 c^4 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}-39424 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}+36960 c^2 \left (3 a x+\sqrt {-b+a^2 x^2}\right )-35112 c \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (3 a x+\sqrt {-b+a^2 x^2}\right )+33649 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (21 a x+19 \sqrt {-b+a^2 x^2}\right )\right )+163840 c^5 \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (16384 c^6 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}-256 a c^3 x \left (96 c^2-84 c \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}+77 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}\right )+77 a^2 x^2 \left (480 c^2-456 c \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}+437 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{5/3}}-3011753745 b^2 \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )+3011753745 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{24808734720 a c^{21/4}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \]________________________________________________________________________________________