Integrand size = 15, antiderivative size = 117 \[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\frac {1}{108} \sqrt [3]{-x+x^3} \left (-5 x-3 x^3+18 x^5\right )+\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{54 \sqrt {3}}+\frac {5}{162} \log \left (-x+\sqrt [3]{-x+x^3}\right )-\frac {5}{324} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]
[Out]
Time = 0.10 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.36, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2046, 2049, 2057, 335, 281, 337} \[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\frac {5 \left (x^2-1\right )^{2/3} x^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{54 \sqrt {3} \left (x^3-x\right )^{2/3}}-\frac {1}{36} \sqrt [3]{x^3-x} x^3-\frac {5}{108} \sqrt [3]{x^3-x} x+\frac {1}{6} \sqrt [3]{x^3-x} x^5+\frac {5 \left (x^2-1\right )^{2/3} x^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{108 \left (x^3-x\right )^{2/3}} \]
[In]
[Out]
Rule 281
Rule 335
Rule 337
Rule 2046
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^5 \sqrt [3]{-x+x^3}-\frac {1}{9} \int \frac {x^5}{\left (-x+x^3\right )^{2/3}} \, dx \\ & = -\frac {1}{36} x^3 \sqrt [3]{-x+x^3}+\frac {1}{6} x^5 \sqrt [3]{-x+x^3}-\frac {5}{54} \int \frac {x^3}{\left (-x+x^3\right )^{2/3}} \, dx \\ & = -\frac {5}{108} x \sqrt [3]{-x+x^3}-\frac {1}{36} x^3 \sqrt [3]{-x+x^3}+\frac {1}{6} x^5 \sqrt [3]{-x+x^3}-\frac {5}{81} \int \frac {x}{\left (-x+x^3\right )^{2/3}} \, dx \\ & = -\frac {5}{108} x \sqrt [3]{-x+x^3}-\frac {1}{36} x^3 \sqrt [3]{-x+x^3}+\frac {1}{6} x^5 \sqrt [3]{-x+x^3}-\frac {\left (5 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right )^{2/3}} \, dx}{81 \left (-x+x^3\right )^{2/3}} \\ & = -\frac {5}{108} x \sqrt [3]{-x+x^3}-\frac {1}{36} x^3 \sqrt [3]{-x+x^3}+\frac {1}{6} x^5 \sqrt [3]{-x+x^3}-\frac {\left (5 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{27 \left (-x+x^3\right )^{2/3}} \\ & = -\frac {5}{108} x \sqrt [3]{-x+x^3}-\frac {1}{36} x^3 \sqrt [3]{-x+x^3}+\frac {1}{6} x^5 \sqrt [3]{-x+x^3}-\frac {\left (5 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{54 \left (-x+x^3\right )^{2/3}} \\ & = -\frac {5}{108} x \sqrt [3]{-x+x^3}-\frac {1}{36} x^3 \sqrt [3]{-x+x^3}+\frac {1}{6} x^5 \sqrt [3]{-x+x^3}+\frac {5 x^{2/3} \left (-1+x^2\right )^{2/3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{54 \sqrt {3} \left (-x+x^3\right )^{2/3}}+\frac {5 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{108 \left (-x+x^3\right )^{2/3}} \\ \end{align*}
Time = 1.14 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.44 \[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\frac {15 x^2-6 x^4-63 x^6+54 x^8+10 \sqrt {3} x^{2/3} \left (-1+x^2\right )^{2/3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+10 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-5 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )}{324 \left (x \left (-1+x^2\right )\right )^{2/3}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.88 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28
method | result | size |
meijerg | \(\frac {3 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {16}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], x^{2}\right )}{16 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
pseudoelliptic | \(-\frac {x^{3} \left (\left (-54 x^{5}+9 x^{3}+15 x \right ) \left (x^{3}-x \right )^{\frac {1}{3}}+10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right )+5 \ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )-10 \ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )\right )}{324 {\left (\left (x^{3}-x \right )^{\frac {2}{3}}+x \left (x +\left (x^{3}-x \right )^{\frac {1}{3}}\right )\right )}^{3} {\left (x -\left (x^{3}-x \right )^{\frac {1}{3}}\right )}^{3}}\) | \(149\) |
trager | \(\frac {x \left (18 x^{4}-3 x^{2}-5\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{108}-\frac {5 \ln \left (1395 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-6768 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -6303 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-5580 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+5355 x \left (x^{3}-x \right )^{\frac {1}{3}}+5510 x^{2}+8007 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1653\right )}{162}-\frac {5 \ln \left (1395 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-6768 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -6303 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-5580 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+5355 x \left (x^{3}-x \right )^{\frac {1}{3}}+5510 x^{2}+8007 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1653\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{54}+\frac {5 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (1395 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+6768 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +7233 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-5580 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}+7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+7766 x^{2}-11727 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4942\right )}{54}\) | \(468\) |
risch | \(\text {Expression too large to display}\) | \(798\) |
[In]
[Out]
none
Time = 0.43 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.96 \[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\frac {5}{162} \, \sqrt {3} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + \frac {1}{108} \, {\left (18 \, x^{5} - 3 \, x^{3} - 5 \, x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}} + \frac {5}{324} \, \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) \]
[In]
[Out]
\[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\int x^{4} \sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}\, dx \]
[In]
[Out]
\[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\int { {\left (x^{3} - x\right )}^{\frac {1}{3}} x^{4} \,d x } \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.93 \[ \int x^4 \sqrt [3]{-x+x^3} \, dx=-\frac {1}{108} \, {\left (5 \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 13 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} - 10 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{6} - \frac {5}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {5}{324} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{162} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
[In]
[Out]
Timed out. \[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\int x^4\,{\left (x^3-x\right )}^{1/3} \,d x \]
[In]
[Out]