Integrand size = 25, antiderivative size = 255 \[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\frac {1}{8} (1+4 x) \sqrt [4]{x^3+x^4}-\frac {29}{16} \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )+\sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )+\frac {29}{16} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )-\sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right ) \]
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Time = 0.48 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.63, number of steps used = 26, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {1607, 2081, 917, 52, 65, 338, 304, 209, 212, 919, 925, 95} \[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=-\frac {29 \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{16 \sqrt [4]{x+1} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {5} \sqrt [4]{x+1} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {5} \sqrt [4]{x+1} x^{3/4}}+\frac {29 \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{16 \sqrt [4]{x+1} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {5} \sqrt [4]{x+1} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {5} \sqrt [4]{x+1} x^{3/4}}+\frac {1}{2} \sqrt [4]{x^4+x^3} x+\frac {1}{8} \sqrt [4]{x^4+x^3} \]
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Rule 52
Rule 65
Rule 95
Rule 209
Rule 212
Rule 304
Rule 338
Rule 917
Rule 919
Rule 925
Rule 1607
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \int \frac {x (1+x) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx \\ & = \frac {\sqrt [4]{x^3+x^4} \int \frac {x^{7/4} (1+x)^{5/4}}{-1+x+x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \int x^{3/4} \sqrt [4]{1+x} \, dx}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4} \sqrt [4]{1+x}}{-1+x+x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1}{2} x \sqrt [4]{x^3+x^4}+\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4}}{(1+x)^{3/4}} \, dx}{8 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (-1+x+x^2\right )} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}-\frac {\left (3 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{32 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \left (-\frac {2}{\sqrt {5} \left (-1+\sqrt {5}-2 x\right ) \sqrt [4]{x} (1+x)^{3/4}}-\frac {2}{\sqrt {5} \sqrt [4]{x} (1+x)^{3/4} \left (1+\sqrt {5}+2 x\right )}\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^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}-\frac {\left (3 \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 )}{8 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 (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\left (-1+\sqrt {5}-2 x\right ) \sqrt [4]{x} (1+x)^{3/4}} \, dx}{\sqrt {5} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (1+\sqrt {5}+2 x\right )} \, dx}{\sqrt {5} x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}-\frac {\left (3 \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 )}{8 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 (8 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1+\sqrt {5}-\left (-1+\sqrt {5}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {5} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (8 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-1+\sqrt {5}-\left (1+\sqrt {5}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {5} x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^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}}-\frac {\left (3 \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 )}{16 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (3 \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 )}{16 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt {\frac {2}{5}} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\left (-1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \sqrt {\frac {2}{5}} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\left (-1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt {\frac {2}{5}} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \sqrt {\frac {2}{5}} \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}-\frac {29 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{16 x^{3/4} \sqrt [4]{1+x}}+\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {5} x^{3/4} \sqrt [4]{1+x}}+\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {5} x^{3/4} \sqrt [4]{1+x}}+\frac {29 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{16 x^{3/4} \sqrt [4]{1+x}}-\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {5} x^{3/4} \sqrt [4]{1+x}}-\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {5} x^{3/4} \sqrt [4]{1+x}} \\ \end{align*}
Time = 0.70 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00 \[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\frac {x^{9/4} (1+x)^{3/4} \left (10 x^{3/4} \sqrt [4]{1+x}+40 x^{7/4} \sqrt [4]{1+x}-145 \arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )+16 \sqrt {10 \left (1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )+16 \sqrt {10 \left (-1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )+145 \text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )-16 \sqrt {10 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )-16 \sqrt {10 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt [4]{\frac {x}{1+x}}\right )\right )}{80 \left (x^3 (1+x)\right )^{3/4}} \]
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Time = 15.51 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(-\frac {x^{6} \left (\frac {5 \left (-\frac {1}{4}-x \right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{2}+\left (\operatorname {arctanh}\left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {2+2 \sqrt {5}}}\right )+\arctan \left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {2+2 \sqrt {5}}}\right )\right ) \sqrt {5}\, \sqrt {-2+2 \sqrt {5}}+\sqrt {2+2 \sqrt {5}}\, \left (\operatorname {arctanh}\left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {5}}}\right )+\arctan \left (\frac {2 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {5}}}\right )\right ) \sqrt {5}+\frac {145 \ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )}{32}-\frac {145 \ln \left (\frac {x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{32}-\frac {145 \arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{16}\right )}{5 {\left (x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}\right )}^{2} {\left (-\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+x \right )}^{2} \left (x^{2}+\sqrt {x^{3} \left (1+x \right )}\right )^{2}}\) | \(244\) |
trager | \(\text {Expression too large to display}\) | \(2153\) |
risch | \(\text {Expression too large to display}\) | \(4381\) |
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Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (177) = 354\).
Time = 0.28 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.84 \[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=-\frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (\frac {{\left (\sqrt {5} x - x\right )} \sqrt {2 \, \sqrt {5} + 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {{\left (\sqrt {5} x - x\right )} \sqrt {2 \, \sqrt {5} + 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (\frac {{\left (\sqrt {5} x + x\right )} \sqrt {2 \, \sqrt {5} - 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {{\left (\sqrt {5} x + x\right )} \sqrt {2 \, \sqrt {5} - 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} + 2} \log \left (\frac {{\left (\sqrt {5} x + x\right )} \sqrt {-2 \, \sqrt {5} + 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} + 2} \log \left (-\frac {{\left (\sqrt {5} x + x\right )} \sqrt {-2 \, \sqrt {5} + 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} - 2} \log \left (\frac {{\left (\sqrt {5} x - x\right )} \sqrt {-2 \, \sqrt {5} - 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} - 2} \log \left (-\frac {{\left (\sqrt {5} x - x\right )} \sqrt {-2 \, \sqrt {5} - 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x + 1\right )} + \frac {29}{16} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {29}{32} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {29}{32} \, \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
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\[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\int \frac {x \sqrt [4]{x^{3} \left (x + 1\right )} \left (x + 1\right )}{x^{2} + x - 1}\, dx \]
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\[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + x\right )}}{x^{2} + x - 1} \,d x } \]
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Time = 0.43 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.93 \[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\frac {1}{8} \, {\left ({\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} - \frac {1}{5} \, \sqrt {10 \, \sqrt {5} - 10} \arctan \left (\frac {{\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{5} \, \sqrt {10 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{10} \, \sqrt {10 \, \sqrt {5} - 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{10} \, \sqrt {10 \, \sqrt {5} + 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{10} \, \sqrt {10 \, \sqrt {5} - 10} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {1}{10} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {29}{16} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {29}{32} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {29}{32} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx=\int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (x^2+x\right )}{x^2+x-1} \,d x \]
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