\[ \int \frac {1}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \]
Optimal antiderivative \[ \frac {\left (1944 b \,c^{4}+3640 a b x \right ) \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {2}{3}}+\left (6561 a \,c^{7} x -2106 b \,c^{3}\right ) \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {2}{3}}+\left (-4374 a \,c^{6} x +2340 b \,c^{2}\right ) \sqrt {a x +\sqrt {a^{2} x^{2}-b}}\, \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {2}{3}}+\left (3645 a \,c^{5} x -2730 b c \right ) \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {3}{4}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {2}{3}}+\sqrt {a^{2} x^{2}-b}\, \left (3640 b \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {2}{3}}+6561 c^{7} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {2}{3}}-4374 c^{6} \sqrt {a x +\sqrt {a^{2} x^{2}-b}}\, \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {2}{3}}+3645 c^{5} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {3}{4}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {2}{3}}\right )}{4860 a \,c^{5} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {5}{4}}}+\frac {182 b \arctan \left (\frac {\sqrt {3}}{3}+\frac {2 \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}} \sqrt {3}}{3 c^{\frac {1}{3}}}\right ) \sqrt {3}}{729 a \,c^{\frac {16}{3}}}+\frac {182 b \ln \left (-c^{\frac {1}{3}}+\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}\right )}{729 a \,c^{\frac {16}{3}}}-\frac {91 b \ln \left (c^{\frac {2}{3}}+c^{\frac {1}{3}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}+\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {2}{3}}\right )}{729 a \,c^{\frac {16}{3}}} \]
command
Integrate[1/((a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)),x]
Mathematica 13.1 output
\[ \frac {\frac {3 \sqrt [3]{c} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3} \left (729 c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4} \left (9 c^2-6 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+5 \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )+2 b \left (972 c^4-1053 c^3 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+1170 c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}-1365 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}+1820 \left (a x+\sqrt {-b+a^2 x^2}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+3640 \sqrt {3} b \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+3640 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-1820 b \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{14580 a c^{16/3}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________