Integrand size = 20, antiderivative size = 72 \[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\frac {\sqrt [4]{x^3+x^4} \left (7315-4180 x+3040 x^2-2432 x^3+2048 x^4\right )}{10240}+\frac {4389 \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )}{4096}-\frac {4389 \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )}{4096} \]
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Leaf count is larger than twice the leaf count of optimal. \(168\) vs. \(2(72)=144\).
Time = 0.14 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.33, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2081, 52, 65, 338, 304, 209, 212} \[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\frac {4389 \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4096 \sqrt [4]{x+1} x^{3/4}}-\frac {4389 \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4096 \sqrt [4]{x+1} x^{3/4}}+\frac {1}{5} \sqrt [4]{x^4+x^3} x^4-\frac {19}{80} \sqrt [4]{x^4+x^3} x^3-\frac {209}{512} \sqrt [4]{x^4+x^3} x+\frac {1463 \sqrt [4]{x^4+x^3}}{2048}+\frac {19}{64} \sqrt [4]{x^4+x^3} x^2 \]
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Rule 52
Rule 65
Rule 209
Rule 212
Rule 304
Rule 338
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^3+x^4} \int \frac {x^{19/4}}{(1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (19 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{15/4}}{(1+x)^{3/4}} \, dx}{20 x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}+\frac {\left (57 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{11/4}}{(1+x)^{3/4}} \, dx}{64 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (209 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{7/4}}{(1+x)^{3/4}} \, dx}{256 x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}+\frac {\left (1463 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(1+x)^{3/4}} \, dx}{2048 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{8192 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \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 )}{2048 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \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 )}{2048 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \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 )}{4096 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4389 \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 )}{4096 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}+\frac {4389 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4096 x^{3/4} \sqrt [4]{1+x}}-\frac {4389 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4096 x^{3/4} \sqrt [4]{1+x}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.32 \[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\frac {x^{9/4} \left (2 x^{3/4} \left (7315+3135 x-1140 x^2+608 x^3-384 x^4+2048 x^5\right )-21945 (1+x)^{3/4} \arctan \left (\frac {1}{\sqrt [4]{\frac {x}{1+x}}}\right )-21945 (1+x)^{3/4} \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {x}{1+x}}}\right )\right )}{20480 \left (x^3 (1+x)\right )^{3/4}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.57 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.21
method | result | size |
meijerg | \(\frac {4 x^{\frac {23}{4}} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {23}{4}\right ], \left [\frac {27}{4}\right ], -x \right )}{23}\) | \(15\) |
trager | \(\left (\frac {1}{5} x^{4}-\frac {19}{80} x^{3}+\frac {19}{64} x^{2}-\frac {209}{512} x +\frac {1463}{2048}\right ) \left (x^{4}+x^{3}\right )^{\frac {1}{4}}-\frac {4389 \ln \left (\frac {2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}+x^{3}}\, x +2 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} x^{2}+2 x^{3}+x^{2}}{x^{2}}\right )}{8192}-\frac {4389 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}-2 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} x^{2}}{x^{2}}\right )}{8192}\) | \(164\) |
pseudoelliptic | \(\frac {x^{15} \left (8192 x^{4} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-9728 x^{3} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+12160 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} x^{2}-16720 x \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+21945 \ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )-43890 \arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )-21945 \ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+x}{x}\right )+29260 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}\right )}{40960 {\left (\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x \right )}^{5} \left (x^{2}+\sqrt {x^{3} \left (1+x \right )}\right )^{5} {\left (\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+x \right )}^{5}}\) | \(169\) |
risch | \(\frac {\left (2048 x^{4}-2432 x^{3}+3040 x^{2}-4180 x +7315\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{10240}+\frac {\left (\frac {4389 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}-2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}}{\left (1+x \right )^{2}}\right )}{8192}+\frac {4389 \ln \left (\frac {2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, x +2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}-2 x^{3}-2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}+4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x -5 x^{2}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}-4 x -1}{\left (1+x \right )^{2}}\right )}{8192}\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (\left (1+x \right )^{3} x \right )^{\frac {1}{4}}}{x \left (1+x \right )}\) | \(390\) |
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Time = 0.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.21 \[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\frac {1}{10240} \, {\left (2048 \, x^{4} - 2432 \, x^{3} + 3040 \, x^{2} - 4180 \, x + 7315\right )} {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} - \frac {4389}{4096} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {4389}{8192} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {4389}{8192} \, \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
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\[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\int \frac {x^{4} \sqrt [4]{x^{3} \left (x + 1\right )}}{x + 1}\, dx \]
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\[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} x^{4}}{x + 1} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.21 \[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\frac {1}{10240} \, {\left (7315 \, {\left (\frac {1}{x} + 1\right )}^{\frac {17}{4}} - 33440 \, {\left (\frac {1}{x} + 1\right )}^{\frac {13}{4}} + 59470 \, {\left (\frac {1}{x} + 1\right )}^{\frac {9}{4}} - 50312 \, {\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 19015 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{5} - \frac {4389}{4096} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {4389}{8192} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {4389}{8192} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\int \frac {x^4\,{\left (x^4+x^3\right )}^{1/4}}{x+1} \,d x \]
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