Integrand size = 17, antiderivative size = 73 \[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\frac {\left (-77-44 x-32 x^2+384 x^3\right ) \sqrt [4]{-x^3+x^4}}{1536}+\frac {77 \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{1024}-\frac {77 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{1024} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(73)=146\).
Time = 0.19 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.22, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2046, 2049, 2057, 65, 246, 218, 212, 209} \[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=-\frac {77 (x-1)^{3/4} x^{9/4} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{1024 \left (x^4-x^3\right )^{3/4}}-\frac {77 (x-1)^{3/4} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{1024 \left (x^4-x^3\right )^{3/4}}+\frac {1}{4} \sqrt [4]{x^4-x^3} x^3-\frac {11}{384} \sqrt [4]{x^4-x^3} x-\frac {77 \sqrt [4]{x^4-x^3}}{1536}-\frac {1}{48} \sqrt [4]{x^4-x^3} x^2 \]
[In]
[Out]
Rule 65
Rule 209
Rule 212
Rule 218
Rule 246
Rule 2046
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {1}{16} \int \frac {x^5}{\left (-x^3+x^4\right )^{3/4}} \, dx \\ & = -\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {11}{192} \int \frac {x^4}{\left (-x^3+x^4\right )^{3/4}} \, dx \\ & = -\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {77 \int \frac {x^3}{\left (-x^3+x^4\right )^{3/4}} \, dx}{1536} \\ & = -\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {77 \int \frac {x^2}{\left (-x^3+x^4\right )^{3/4}} \, dx}{2048} \\ & = -\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {\left (77 (-1+x)^{3/4} x^{9/4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{2048 \left (-x^3+x^4\right )^{3/4}} \\ & = -\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {\left (77 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{512 \left (-x^3+x^4\right )^{3/4}} \\ & = -\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {\left (77 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{512 \left (-x^3+x^4\right )^{3/4}} \\ & = -\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {\left (77 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{1024 \left (-x^3+x^4\right )^{3/4}}-\frac {\left (77 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{1024 \left (-x^3+x^4\right )^{3/4}} \\ & = -\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {77 (-1+x)^{3/4} x^{9/4} \arctan \left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{1024 \left (-x^3+x^4\right )^{3/4}}-\frac {77 (-1+x)^{3/4} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{1024 \left (-x^3+x^4\right )^{3/4}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.16 \[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\frac {(-1+x)^{3/4} x^{9/4} \left (2 \sqrt [4]{-1+x} x^{3/4} \left (-77-44 x-32 x^2+384 x^3\right )+231 \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-231 \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{3072 \left ((-1+x) x^3\right )^{3/4}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.37
method | result | size |
meijerg | \(\frac {4 \operatorname {signum}\left (x -1\right )^{\frac {1}{4}} x^{\frac {15}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {15}{4}\right ], \left [\frac {19}{4}\right ], x\right )}{15 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{4}}}\) | \(27\) |
pseudoelliptic | \(\frac {x^{12} \left (1536 \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}} x^{3}-128 x^{2} \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}-176 x \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}+231 \ln \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}-x}{x}\right )-462 \arctan \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{x}\right )-231 \ln \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}+x}{x}\right )-308 \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}\right )}{6144 {\left (\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}-x \right )}^{4} \left (x^{2}+\sqrt {x^{3} \left (x -1\right )}\right )^{4} {\left (\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}+x \right )}^{4}}\) | \(155\) |
trager | \(\left (\frac {1}{4} x^{3}-\frac {1}{48} x^{2}-\frac {11}{384} x -\frac {77}{1536}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+\frac {77 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-2 x^{3}+x^{2}}{x^{2}}\right )}{2048}+\frac {77 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{2048}\) | \(173\) |
risch | \(\frac {\left (384 x^{3}-32 x^{2}-44 x -77\right ) \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{1536}+\frac {\left (-\frac {77 \ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+2 x^{3}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+4 x -1}{\left (x -1\right )^{2}}\right )}{2048}+\frac {77 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x -1\right )^{2}}\right )}{2048}\right ) \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}} \left (x \left (x -1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x -1\right )}\) | \(407\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.23 \[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\frac {1}{1536} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (384 \, x^{3} - 32 \, x^{2} - 44 \, x - 77\right )} - \frac {77}{1024} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {77}{2048} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {77}{2048} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
[In]
[Out]
\[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\int x^{2} \sqrt [4]{x^{3} \left (x - 1\right )}\, dx \]
[In]
[Out]
\[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\int { {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x^{2} \,d x } \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.45 \[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=-\frac {1}{1536} \, {\left (77 \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 275 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 351 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - 231 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{4} - \frac {77}{1024} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {77}{2048} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {77}{2048} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
[In]
[Out]
Timed out. \[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\int x^2\,{\left (x^4-x^3\right )}^{1/4} \,d x \]
[In]
[Out]