\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \]
Optimal antiderivative \[ \frac {\left (1409286144 a^{3} c^{9} x^{3}-1056964608 a b \,c^{9} x +5752750080 a^{2} b \,c^{6} x^{2}-3081830400 b^{2} c^{6}+238761600 a \,b^{2} c^{3} x -727476750 a^{2} b^{2} x^{2}+363738375 b^{3}\right ) \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}+\left (1073741824 a^{2} c^{11} x^{2}-1291845632 a^{3} c^{8} x^{3}-536870912 b \,c^{11}+968884224 a b \,c^{8} x +214016000 b^{2} c^{5}-258658400 a \,b^{2} c^{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 (-805306368 a^{2} c^{10} x^{2}+1211105280 a^{3} c^{7} x^{3}+402653184 b \,c^{10}-908328960 a b \,c^{7} x -224716800 b^{2} c^{4}+290990700 a \,b^{2} c 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 (\left (1409286144 a^{2} c^{9} x^{2}-352321536 b \,c^{9}+5752750080 a b \,c^{6} x +238761600 b^{2} c^{3}-727476750 a \,b^{2} x \right ) \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {3}{4}}+\left (1073741824 a \,c^{11} x -1291845632 a^{2} c^{8} x^{2}+322961408 b \,c^{8}-258658400 b^{2} c^{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 (-805306368 a \,c^{10} x +1211105280 a^{2} c^{7} x^{2}-302776320 b \,c^{7}+290990700 b^{2} c \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 )}{1917583360 a \,c^{7} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {7}{3}}}-\frac {49725 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 )}{524288 a \,c^{\frac {29}{4}}}+\frac {3 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 )}{4 a \,c^{\frac {5}{4}}}+\frac {49725 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 )}{524288 a \,c^{\frac {29}{4}}}-\frac {3 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 )}{4 a \,c^{\frac {5}{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
\[ \frac {\left (363738375 b^3-3081830400 b^2 c^6+238761600 a b^2 c^3 x-1056964608 a b c^9 x-727476750 a^2 b^2 x^2+5752750080 a^2 b c^6 x^2+1409286144 a^3 c^9 x^3\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (214016000 b^2 c^5-536870912 b c^{11}-258658400 a b^2 c^2 x+968884224 a b c^8 x+1073741824 a^2 c^{11} x^2-1291845632 a^3 c^8 x^3\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-224716800 b^2 c^4+402653184 b c^{10}+290990700 a b^2 c x-908328960 a b c^7 x-805306368 a^2 c^{10} x^2+1211105280 a^3 c^7 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\sqrt {-b+a^2 x^2} \left (\left (238761600 b^2 c^3-352321536 b c^9-727476750 a b^2 x+5752750080 a b c^6 x+1409286144 a^2 c^9 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-258658400 b^2 c^2+322961408 b c^8+1073741824 a c^{11} x-1291845632 a^2 c^8 x^2\right ) \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (290990700 b^2 c-302776320 b c^7-805306368 a c^{10} x+1211105280 a^2 c^7 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}\right )}{1917583360 a c^7 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/3}}-\frac {49725 b^2 \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}+\frac {3 b \text {ArcTan}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}}+\frac {49725 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{524288 a c^{29/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{4 a c^{5/4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________