\[ \int \frac {\sqrt {-b+a^2 x^2} \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx \]
Optimal antiderivative \[ \frac {\left (-1205308188 a^{2} c^{12} x^{2}-1400169888 a^{3} c^{8} x^{3}+602654094 b \,c^{12}+1050127416 a b \,c^{8} x -573080508 b^{2} c^{4}-1815934120 a \,b^{2} x \right ) \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}+\left (3486784401 a \,c^{15} x +1004423490 a^{2} c^{11} x^{2}+1283489064 a^{3} c^{7} x^{3}-502211745 b \,c^{11}-12014883738 a b \,c^{7} x +668593926 b^{2} 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 {1}{3}}+\left (-1162261467 a \,c^{14} x -870500358 a^{2} c^{10} x^{2}-1188415800 a^{3} c^{6} x^{3}+435250179 b \,c^{10}+4575400830 a b \,c^{6} x -817170354 b^{2} 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 {1}{3}}+\left (774840978 a \,c^{13} x +773778096 a^{2} c^{9} x^{2}+1109188080 a^{3} c^{5} x^{3}-386889048 b \,c^{9}-3287950380 a b \,c^{5} x +1089560472 b^{2} 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 {1}{3}}+\sqrt {a^{2} x^{2}-b}\, \left (\left (-1205308188 a \,c^{12} x -1400169888 a^{2} c^{8} x^{2}+350042472 b \,c^{8}-1815934120 b^{2}\right ) \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}+\left (3486784401 c^{15}+1004423490 a \,c^{11} x +1283489064 a^{2} c^{7} x^{2}-11373139206 b \,c^{7}\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 (-1162261467 c^{14}-870500358 a \,c^{10} x -1188415800 a^{2} c^{6} x^{2}+3981192930 b \,c^{6}\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 (774840978 c^{13}+773778096 a \,c^{9} x +1109188080 a^{2} c^{5} x^{2}-2733356340 b \,c^{5}\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 )}{2865402540 a \,c^{5} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {5}{4}}}+\frac {308 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}}{729 a \,c^{\frac {17}{3}}}-\frac {308 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 )}{729 a \,c^{\frac {17}{3}}}+\frac {154 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 )}{729 a \,c^{\frac {17}{3}}} \]
command
Integrate[(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3),x]
Mathematica 13.1 output
\[ \frac {\frac {3 c^{2/3} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \left (-1179178 b^2 \left (486 c^4-567 c^3 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+693 c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}-924 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}+1540 \left (a x+\sqrt {-b+a^2 x^2}\right )\right )-729 b c^5 \left (-826686 c^7+688905 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-597051 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}}+530712 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}-480168 c^3 \left (3 a x+\sqrt {-b+a^2 x^2}\right )-81510 c \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (77 a x+67 \sqrt {-b+a^2 x^2}\right )+54340 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \left (83 a x+69 \sqrt {-b+a^2 x^2}\right )+48906 c^2 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (337 a x+319 \sqrt {-b+a^2 x^2}\right )\right )+729 c^5 \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (531441 c^8 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (9 c^2-3 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+2 \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )+10206 a c^4 x \left (-162 c^3+135 c^2 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-117 c \sqrt {a x+\sqrt {-b+a^2 x^2}}+104 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )+1976 a^2 x^2 \left (-972 c^3+891 c^2 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-825 c \sqrt {a x+\sqrt {-b+a^2 x^2}}+770 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+3631868240 \sqrt {3} b^2 \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )-3631868240 b^2 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )+1815934120 b^2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{8596207620 a c^{17/3}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {-b+a^2 x^2} \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx \]________________________________________________________________________________________