\[ \int x^2 \sqrt [4]{x^2+x^4} \, dx \]
Optimal antiderivative \[ \frac {\left (4 x^{3}+x \right ) \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{16}+\frac {3 \arctan \! \left (\frac {x}{\left (x^{4}+x^{2}\right )^{\frac {1}{4}}}\right )}{32}-\frac {3 \arctanh \! \left (\frac {x}{\left (x^{4}+x^{2}\right )^{\frac {1}{4}}}\right )}{32} \]
command
int(x^2*(x^4+x^2)^(1/4),x)
Maple 2022.1 output
\[\int x^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}}\, dx\]
Maple 2021.1 output
| method | result | size |
| meijerg | \(\frac {2 x^{\frac {7}{2}} \hypergeom \left (\left [-\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], -x^{2}\right )}{7}\) | \(17\) |
| trager | \(\frac {x \left (4 x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{16}-\frac {3 \ln \left (\frac {2 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}+x^{2}}\, x +2 x^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}}+2 x^{3}+x}{x}\right )}{64}-\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+x^{2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x}{x}\right )}{64}\) | \(149\) |
| risch | \(\frac {x \left (4 x^{2}+1\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{16}+\frac {\left (-\frac {3 \ln \left (\frac {2 x^{6}+2 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} x^{4}+2 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}\, x^{2}+5 x^{4}+2 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {3}{4}}+4 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} x^{2}+2 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}+4 x^{2}+2 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}}+1}{\left (x^{2}+1\right )^{2}}\right )}{64}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} x^{4}+2 x^{6}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {3}{4}}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} x^{2}-2 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}\, x^{2}+5 x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}}-2 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}+4 x^{2}+1}{\left (x^{2}+1\right )^{2}}\right )}{64}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{2} \left (x^{2}+1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{2}+1\right )}\) | \(416\) |