\[ \int \frac {x^4}{\sqrt [4]{b+a x^4} \left (b+2 a x^4+2 x^8\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (-1+\left (-1\right )^{\frac {1}{4}}\right ) \arctan \! \left (\frac {\left (-1\right )^{\frac {7}{8}} \sqrt {2+\sqrt {2}}\, \left (a^{2}-2 b \right )^{\frac {1}{8}} x \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{\left (-1\right )^{\frac {3}{4}} \left (a^{2}-2 b \right )^{\frac {1}{4}} x^{2}+\sqrt {a \,x^{4}+b}}\right )}{8 \left (a^{2}-2 b \right )^{\frac {5}{8}}}+\frac {\mathrm {I} \left (\mathrm {I} \sqrt {2}+2+\sqrt {2}\right ) \arctan \! \left (\frac {\left (-1\right )^{\frac {7}{8}} \left (-2+\sqrt {2}\right ) \left (a^{2}-2 b \right )^{\frac {1}{8}} x \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{\left (-1\right )^{\frac {3}{4}} \sqrt {2-\sqrt {2}}\, \left (a^{2}-2 b \right )^{\frac {1}{4}} x^{2}+\sqrt {2-\sqrt {2}}\, \sqrt {a \,x^{4}+b}}\right )}{16 \left (a^{2}-2 b \right )^{\frac {5}{8}}}+\frac {\left (\sqrt {2}-\mathrm {I} \left (2+\sqrt {2}\right )\right ) \arctanh \! \left (\frac {\left (-1\right )^{\frac {7}{8}} \left (a^{2}-2 b \right )^{\frac {1}{4}} x^{2}-\left (-1\right )^{\frac {1}{8}} \sqrt {a \,x^{4}+b}}{\sqrt {2-\sqrt {2}}\, \left (a^{2}-2 b \right )^{\frac {1}{8}} x \left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )}{16 \left (a^{2}-2 b \right )^{\frac {5}{8}}}+\frac {\left (-1+\left (-1\right )^{\frac {1}{4}}\right ) \arctanh \! \left (\frac {\left (-1\right )^{\frac {7}{8}} \left (a^{2}-2 b \right )^{\frac {1}{4}} x^{2}-\left (-1\right )^{\frac {1}{8}} \sqrt {a \,x^{4}+b}}{\sqrt {2+\sqrt {2}}\, \left (a^{2}-2 b \right )^{\frac {1}{8}} x \left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )}{8 \left (a^{2}-2 b \right )^{\frac {5}{8}}} \]
command
Integrate[x^4/((b + a*x^4)^(1/4)*(b + 2*a*x^4 + 2*x^8)),x]
Mathematica 13.1 output
\[ \int \frac {x^4}{\sqrt [4]{b+a x^4} \left (b+2 a x^4+2 x^8\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {-\frac {\sqrt [4]{a-\sqrt {a^2-2 b}} \tan ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-2 b}} \sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}+\frac {\sqrt [4]{\sqrt {a^2-2 b}+a} \tan ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-2 b}+a} \sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}-\frac {\sqrt [4]{a-\sqrt {a^2-2 b}} \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-2 b}} \sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}+\frac {\sqrt [4]{\sqrt {a^2-2 b}+a} \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-2 b}+a} \sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}}{4 \sqrt {a^2-2 b}} \]