Integrand size = 15, antiderivative size = 59 \[ \int x^2 \sqrt [4]{x^2+x^4} \, dx=\frac {1}{16} \left (x+4 x^3\right ) \sqrt [4]{x^2+x^4}+\frac {3}{32} \arctan \left (\frac {x}{\sqrt [4]{x^2+x^4}}\right )-\frac {3}{32} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^2+x^4}}\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(125\) vs. \(2(59)=118\).
Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.12, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2046, 2049, 2057, 335, 338, 304, 209, 212} \[ \int x^2 \sqrt [4]{x^2+x^4} \, dx=\frac {3 \left (x^2+1\right )^{3/4} x^{3/2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{32 \left (x^4+x^2\right )^{3/4}}-\frac {3 \left (x^2+1\right )^{3/4} x^{3/2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{32 \left (x^4+x^2\right )^{3/4}}+\frac {1}{16} \sqrt [4]{x^4+x^2} x+\frac {1}{4} \sqrt [4]{x^4+x^2} x^3 \]
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 338
Rule 2046
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{8} \int \frac {x^4}{\left (x^2+x^4\right )^{3/4}} \, dx \\ & = \frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}-\frac {3}{32} \int \frac {x^2}{\left (x^2+x^4\right )^{3/4}} \, dx \\ & = \frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}-\frac {\left (3 x^{3/2} \left (1+x^2\right )^{3/4}\right ) \int \frac {\sqrt {x}}{\left (1+x^2\right )^{3/4}} \, dx}{32 \left (x^2+x^4\right )^{3/4}} \\ & = \frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}-\frac {\left (3 x^{3/2} \left (1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{16 \left (x^2+x^4\right )^{3/4}} \\ & = \frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}-\frac {\left (3 x^{3/2} \left (1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{16 \left (x^2+x^4\right )^{3/4}} \\ & = \frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}-\frac {\left (3 x^{3/2} \left (1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \left (x^2+x^4\right )^{3/4}}+\frac {\left (3 x^{3/2} \left (1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \left (x^2+x^4\right )^{3/4}} \\ & = \frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}+\frac {3 x^{3/2} \left (1+x^2\right )^{3/4} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \left (x^2+x^4\right )^{3/4}}-\frac {3 x^{3/2} \left (1+x^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \left (x^2+x^4\right )^{3/4}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.51 \[ \int x^2 \sqrt [4]{x^2+x^4} \, dx=\frac {\sqrt [4]{x^2+x^4} \left (2 x^{3/2} \sqrt [4]{1+x^2} \left (1+4 x^2\right )+3 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )-3 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )\right )}{32 \sqrt {x} \sqrt [4]{1+x^2}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.72 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.29
method | result | size |
meijerg | \(\frac {2 x^{\frac {7}{2}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], -x^{2}\right )}{7}\) | \(17\) |
pseudoelliptic | \(\frac {x^{4} \left (16 x^{3} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+4 x \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+3 \ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x}{x}\right )-6 \arctan \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )-3 \ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+x}{x}\right )\right )}{64 {\left (\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x \right )}^{2} \left (x^{2}+\sqrt {x^{2} \left (x^{2}+1\right )}\right )^{2} {\left (\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+x \right )}^{2}}\) | \(146\) |
trager | \(\frac {x \left (4 x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{16}+\frac {3 \ln \left (\frac {2 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}+x^{2}}\, x +2 x^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}}-2 x^{3}-x}{x}\right )}{64}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x}{x}\right )}{64}\) | \(151\) |
risch | \(\frac {x \left (4 x^{2}+1\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{16}+\frac {\left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} x^{4}-2 x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {3}{4}}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} x^{2}+2 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}\, x^{2}-5 x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}}+2 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}-4 x^{2}-1}{\left (x^{2}+1\right )^{2}}\right )}{64}-\frac {3 \ln \left (\frac {2 x^{6}+2 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} x^{4}+2 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}\, x^{2}+5 x^{4}+2 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {3}{4}}+4 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}} x^{2}+2 \sqrt {x^{8}+3 x^{6}+3 x^{4}+x^{2}}+4 x^{2}+2 \left (x^{8}+3 x^{6}+3 x^{4}+x^{2}\right )^{\frac {1}{4}}+1}{\left (x^{2}+1\right )^{2}}\right )}{64}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{2} \left (x^{2}+1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{2}+1\right )}\) | \(415\) |
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (47) = 94\).
Time = 0.77 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.73 \[ \int x^2 \sqrt [4]{x^2+x^4} \, dx=\frac {1}{16} \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (4 \, x^{3} + x\right )} + \frac {3}{64} \, \arctan \left (\frac {2 \, {\left ({\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x}\right ) + \frac {3}{64} \, \log \left (-\frac {2 \, x^{3} - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} + x^{2}} x + x - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x}\right ) \]
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\[ \int x^2 \sqrt [4]{x^2+x^4} \, dx=\int x^{2} \sqrt [4]{x^{2} \left (x^{2} + 1\right )}\, dx \]
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\[ \int x^2 \sqrt [4]{x^2+x^4} \, dx=\int { {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int x^2 \sqrt [4]{x^2+x^4} \, dx=\frac {1}{16} \, {\left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right )} x^{4} - \frac {3}{32} \, \arctan \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {3}{64} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {3}{64} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Timed out. \[ \int x^2 \sqrt [4]{x^2+x^4} \, dx=\int x^2\,{\left (x^4+x^2\right )}^{1/4} \,d x \]
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