\[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx \]
Optimal antiderivative \[ \text {Expression too large to display} \]
command
Integrate[(Sqrt[-b + a^2*x^2]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(a*x + Sqrt[-b + a^2*x^2])^(1/4),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 (-3272028760 b^3+494 b^2 \left (-646468452 c^8-1515591 c^7 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+1659933 c^6 \sqrt {a x+\sqrt {-b+a^2 x^2}}-1844370 c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}+2090286 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )+13247080 a x \left (a x+\sqrt {-b+a^2 x^2}\right )-2438667 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}+2980593 c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}-3974124 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}\right )+3188646 a c^8 x \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (55328 a^2 x^2+729 c^5 \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 )+14 a c 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 )\right )+531441 b c^8 \left (885248 a^2 x^2+7 a x \left (1458 c^4+150176 \sqrt {-b+a^2 x^2}-1215 c^3 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+1053 c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}-936 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )-3 c \left (-1134 c^3 \sqrt {-b+a^2 x^2}+6561 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+945 c^2 \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}-2187 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}}-819 c \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}+1458 c^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}+728 \sqrt {-b+a^2 x^2} \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 )^{9/4}}-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 )+323439244128 \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 )-323439244128 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 )+161719622064 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 )}{485158866192 a c^{26/3}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________