\[ \int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx \]
Optimal antiderivative \[ \frac {\left (-16271660538 a^{2} c^{15} x^{2}-4687309620 a^{3} c^{11} x^{3}+8135830269 b \,c^{15}+3515482215 a b \,c^{11} x -748701954 b^{2} c^{7}-1204701498 a \,b^{2} c^{3} x \right ) \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}+\left (5423886846 a^{2} c^{14} x^{2}+4062335004 a^{3} c^{10} x^{3}-2711943423 b \,c^{14}-3046751253 a b \,c^{10} x +820006902 b^{2} c^{6}+1472412942 a \,b^{2} c^{2} x \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 {1}{3}}+\left (-3615924564 a^{2} c^{13} x^{2}-3610964448 a^{3} c^{9} x^{3}+1807962282 b \,c^{13}+2708223336 a b \,c^{9} x -911118780 b^{2} c^{5}-1963217256 a \,b^{2} c x \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 {1}{3}}+\left (2812385772 a^{2} c^{12} x^{2}+3267063072 a^{3} c^{8} x^{3}-1406192886 b \,c^{12}-2450297304 a b \,c^{8} x +1032601284 b^{2} c^{4}+3272028760 a \,b^{2} x \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 {1}{3}}+\sqrt {a^{2} x^{2}-b}\, \left (\left (-16271660538 a \,c^{15} x -4687309620 a^{2} c^{11} x^{2}+1171827405 b \,c^{11}-1204701498 b^{2} c^{3}\right ) \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}+\left (5423886846 a \,c^{14} x +4062335004 a^{2} c^{10} x^{2}-1015583751 b \,c^{10}+1472412942 b^{2} c^{2}\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 {1}{3}}+\left (-3615924564 a \,c^{13} x -3610964448 a^{2} c^{9} x^{2}+902741112 b \,c^{9}-1963217256 b^{2} c \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 {1}{3}}+\left (2812385772 a \,c^{12} x +3267063072 a^{2} c^{8} x^{2}-816765768 b \,c^{8}+3272028760 b^{2}\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 {1}{3}}\right )}{11979231264 a^{2} c^{8} x \sqrt {a^{2} x^{2}-b}+5989615632 a \,c^{8} \left (2 a^{2} x^{2}-b \right )}-\frac {21505 b^{2} \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}}{59049 a \,c^{\frac {26}{3}}}+\frac {2 \sqrt {3}\, 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 )}{a \,c^{\frac {2}{3}}}+\frac {21505 b^{2} \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 )}{59049 a \,c^{\frac {26}{3}}}-\frac {2 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 )}{a \,c^{\frac {2}{3}}}-\frac {21505 b^{2} \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 )}{118098 a \,c^{\frac {26}{3}}}+\frac {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 )}{a \,c^{\frac {2}{3}}} \]
command
Integrate[Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3),x]
Mathematica 13.1 output
\[ \frac {\frac {3 c^{2/3} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \left (826686 a c^8 x \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (-19683 c^7-5670 a c^3 x+6561 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+4914 a c^2 x \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-4374 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}}-4368 a c x \sqrt {a x+\sqrt {-b+a^2 x^2}}+3402 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}+3952 a x \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )+5434 b^2 \left (-137781 c^7+150903 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-167670 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}}+190026 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}-221697 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )+270963 c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}-361284 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+602140 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}\right )-413343 b c^8 \left (-19683 c^7+6561 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-4374 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}}+3402 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}-2835 c^3 \left (3 a x+\sqrt {-b+a^2 x^2}\right )+2457 c^2 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (3 a x+\sqrt {-b+a^2 x^2}\right )-2184 c \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (3 a x+\sqrt {-b+a^2 x^2}\right )+1976 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \left (3 a x+\sqrt {-b+a^2 x^2}\right )\right )\right )}{-b+2 a x \left (a x+\sqrt {-b+a^2 x^2}\right )}-6544057520 \sqrt {3} b^2 \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 )+35937693792 \sqrt {3} b c^8 \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 )+6544057520 b^2 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-35937693792 b c^8 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-3272028760 b^2 \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 )+17968846896 b c^8 \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 )}{17968846896 a c^{26/3}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx \]________________________________________________________________________________________