Integrand size = 22, antiderivative size = 173 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=-2 \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )}{2^{3/4}}+2 \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )}{2^{3/4}}-\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \]
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Result contains complex when optimal does not.
Time = 1.25 (sec) , antiderivative size = 616, normalized size of antiderivative = 3.56, number of steps used = 67, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.955, Rules used = {2081, 1600, 6865, 6874, 338, 304, 209, 212, 1254, 419, 243, 342, 281, 237, 416, 418, 1227, 551, 508, 6857, 1543} \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=-\frac {2 \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {(1-i)^{5/4} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{x+1}}+\frac {(1+i)^{5/4} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2^{3/4} x^{3/4} \sqrt [4]{x+1}}+\frac {2 \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {(1-i)^{5/4} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{x+1}}-\frac {(1+i)^{5/4} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2^{3/4} x^{3/4} \sqrt [4]{x+1}} \]
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Rule 209
Rule 212
Rule 237
Rule 243
Rule 281
Rule 304
Rule 338
Rule 342
Rule 416
Rule 418
Rule 419
Rule 508
Rule 551
Rule 1227
Rule 1254
Rule 1543
Rule 1600
Rule 2081
Rule 6857
Rule 6865
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^3+x^4} \int \frac {x^{11/4} \sqrt [4]{1+x}}{-1+x^4} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \int \frac {x^{11/4}}{(1+x)^{3/4} \left (-1+x-x^2+x^3\right )} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^{14}}{\left (1+x^4\right )^{3/4} \left (-1+x^4-x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2}{\left (1+x^4\right )^{3/4}}+\frac {x^2 \left (1-x^4+x^8\right )}{\left (1+x^4\right )^{3/4} \left (-1+x^4-x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (1-x^4+x^8\right )}{\left (1+x^4\right )^{3/4} \left (-1+x^4-x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{4 \left (-1+x^2\right ) \left (1+x^4\right )^{3/4}}+\frac {1}{4 \left (1+x^2\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2 \left (-1+x^4\right )}{2 \left (1+x^4\right )^{3/4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-1+x^4\right )}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \left (\frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \left (\frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \left (-\frac {x^2}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )}+\frac {x^6}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{1-x^4} \, dx,x,\sqrt [4]{x}\right )}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{2 x^{3/4} \sqrt [4]{1+x}}+2 \frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \left (-\frac {i x^2}{2 \left (-i+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {i x^2}{2 \left (i+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2}{2 \left (-i+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{2 \left (i+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (i \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (i \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{\left (-i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{\left (i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \left (-\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2} x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2} x^{3/4} \sqrt [4]{1+x}}\right )+\frac {\left (\sqrt {\frac {1}{1+x}} \sqrt [4]{1+x} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-2 x^4\right ) \sqrt {1-x^4}} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4}}+\frac {\left (\sqrt {\frac {1}{1+x}} \sqrt [4]{1+x} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^4} \left (-1+2 x^4\right )} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \left (\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}\right )-\frac {\left (i \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{i+(1-i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (i \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-i+(1+i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{i+(1-i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {x^2}{-i+(1+i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \left (\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}\right )-\frac {\left (i (1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (i (1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\left ((1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (i (1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (i (1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\left ((1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {2 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{1+x}}+\frac {(1-i)^{5/4} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{1+x}}+\frac {(1+i)^{5/4} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {(1-i)^{5/4} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {(1+i)^{5/4} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+2 \left (\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}\right ) \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=-\frac {x^{9/4} (1+x)^{3/4} \left (8 \left (4 \arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )-\sqrt [4]{2} \arctan \left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )-4 \text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )+\sqrt [4]{2} \text {arctanh}\left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )\right )+\text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{16 \left (x^3 (1+x)\right )^{3/4}} \]
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Time = 37.91 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.99
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{4}-\frac {\arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-1\right )}\right )}{4}+\ln \left (\frac {x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )-\ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )+2 \arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )\) | \(172\) |
trager | \(\text {Expression too large to display}\) | \(3984\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.32 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.36 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=-\frac {1}{16} \cdot 8^{\frac {3}{4}} \log \left (\frac {8^{\frac {3}{4}} x + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{16} \cdot 8^{\frac {3}{4}} \log \left (-\frac {8^{\frac {3}{4}} x - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{16} i \cdot 8^{\frac {3}{4}} \log \left (\frac {i \cdot 8^{\frac {3}{4}} x + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{16} i \cdot 8^{\frac {3}{4}} \log \left (\frac {-i \cdot 8^{\frac {3}{4}} x + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {i + 1}} \log \left (\frac {x \sqrt {-\sqrt {i + 1}} + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {-\sqrt {i + 1}} \log \left (-\frac {x \sqrt {-\sqrt {i + 1}} - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {-i + 1}} \log \left (\frac {x \sqrt {-\sqrt {-i + 1}} + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {-\sqrt {-i + 1}} \log \left (-\frac {x \sqrt {-\sqrt {-i + 1}} - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \left (i + 1\right )^{\frac {1}{4}} \log \left (\frac {\left (i + 1\right )^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \left (i + 1\right )^{\frac {1}{4}} \log \left (-\frac {\left (i + 1\right )^{\frac {1}{4}} x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \left (-i + 1\right )^{\frac {1}{4}} \log \left (\frac {\left (-i + 1\right )^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \left (-i + 1\right )^{\frac {1}{4}} \log \left (-\frac {\left (-i + 1\right )^{\frac {1}{4}} x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2 \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
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Not integrable
Time = 0.48 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.15 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=\int \frac {x^{2} \sqrt [4]{x^{3} \left (x + 1\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} x^{2}}{x^{4} - 1} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.32 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.49 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=\text {Too large to display} \]
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Not integrable
Time = 5.98 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx=\int \frac {x^2\,{\left (x^4+x^3\right )}^{1/4}}{x^4-1} \,d x \]
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