Integrand size = 13, antiderivative size = 114 \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\frac {1}{972} \sqrt [3]{-1+x^3} \left (-20 x^2-12 x^5-9 x^8+81 x^{11}\right )+\frac {10 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{243 \sqrt {3}}+\frac {10}{729} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {5}{729} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Output:
1/972*(x^3-1)^(1/3)*(81*x^11-9*x^8-12*x^5-20*x^2)+10/729*arctan(3^(1/2)*x/ (x+2*(x^3-1)^(1/3)))*3^(1/2)+10/729*ln(-x+(x^3-1)^(1/3))-5/729*ln(x^2+x*(x ^3-1)^(1/3)+(x^3-1)^(2/3))
Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\frac {3 x^2 \sqrt [3]{-1+x^3} \left (-20-12 x^3-9 x^6+81 x^9\right )+40 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )+40 \log \left (-x+\sqrt [3]{-1+x^3}\right )-20 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{2916} \] Input:
Integrate[x^10*(-1 + x^3)^(1/3),x]
Output:
(3*x^2*(-1 + x^3)^(1/3)*(-20 - 12*x^3 - 9*x^6 + 81*x^9) + 40*Sqrt[3]*ArcTa n[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))] + 40*Log[-x + (-1 + x^3)^(1/3)] - 20*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/2916
Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {811, 843, 843, 843, 853}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^{10} \sqrt [3]{x^3-1} \, dx\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {1}{12} x^{11} \sqrt [3]{x^3-1}-\frac {1}{12} \int \frac {x^{10}}{\left (x^3-1\right )^{2/3}}dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {1}{12} \left (-\frac {8}{9} \int \frac {x^7}{\left (x^3-1\right )^{2/3}}dx-\frac {1}{9} \sqrt [3]{x^3-1} x^8\right )+\frac {1}{12} \sqrt [3]{x^3-1} x^{11}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {1}{12} \left (-\frac {8}{9} \left (\frac {5}{6} \int \frac {x^4}{\left (x^3-1\right )^{2/3}}dx+\frac {1}{6} \sqrt [3]{x^3-1} x^5\right )-\frac {1}{9} \sqrt [3]{x^3-1} x^8\right )+\frac {1}{12} \sqrt [3]{x^3-1} x^{11}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {1}{12} \left (-\frac {8}{9} \left (\frac {5}{6} \left (\frac {2}{3} \int \frac {x}{\left (x^3-1\right )^{2/3}}dx+\frac {1}{3} \sqrt [3]{x^3-1} x^2\right )+\frac {1}{6} \sqrt [3]{x^3-1} x^5\right )-\frac {1}{9} \sqrt [3]{x^3-1} x^8\right )+\frac {1}{12} \sqrt [3]{x^3-1} x^{11}\) |
| \(\Big \downarrow \) 853 |
| \(\displaystyle \frac {1}{12} \left (-\frac {8}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3-1}\right )\right )+\frac {1}{3} \sqrt [3]{x^3-1} x^2\right )+\frac {1}{6} \sqrt [3]{x^3-1} x^5\right )-\frac {1}{9} \sqrt [3]{x^3-1} x^8\right )+\frac {1}{12} \sqrt [3]{x^3-1} x^{11}\) |
Input:
Int[x^10*(-1 + x^3)^(1/3),x]
Output:
(x^11*(-1 + x^3)^(1/3))/12 + (-1/9*(x^8*(-1 + x^3)^(1/3)) - (8*((x^5*(-1 + x^3)^(1/3))/6 + (5*((x^2*(-1 + x^3)^(1/3))/3 + (2*(-(ArcTan[(1 + (2*x)/(- 1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3]) - Log[x - (-1 + x^3)^(1/3)]/2))/3))/6))/ 9)/12
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp [Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.29
| method | result | size |
| meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} x^{11} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], x^{3}\right )}{11 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
| risch | \(\frac {x^{2} \left (81 x^{9}-9 x^{6}-12 x^{3}-20\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{972}-\frac {5 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{243 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(63\) |
| pseudoelliptic | \(\frac {-20 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-40 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )+40 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )+\left (243 x^{11}-27 x^{8}-36 x^{5}-60 x^{2}\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{2916 \left (x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )^{4} {\left (-\left (x^{3}-1\right )^{\frac {1}{3}}+x \right )}^{4}}\) | \(138\) |
| trager | \(\frac {x^{2} \left (81 x^{9}-9 x^{6}-12 x^{3}-20\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{972}+\frac {10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{729}-\frac {10 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+4 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{729}-\frac {10 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+4 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right )}{729}\) | \(325\) |
Input:
int(x^10*(x^3-1)^(1/3),x,method=_RETURNVERBOSE)
Output:
1/11*signum(x^3-1)^(1/3)/(-signum(x^3-1))^(1/3)*x^11*hypergeom([-1/3,11/3] ,[14/3],x^3)
Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.93 \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=-\frac {10}{729} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{972} \, {\left (81 \, x^{11} - 9 \, x^{8} - 12 \, x^{5} - 20 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \frac {10}{729} \, \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {5}{729} \, \log \left (\frac {x^{2} + {\left (x^{3} - 1\right )}^{\frac {1}{3}} x + {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \] Input:
integrate(x^10*(x^3-1)^(1/3),x, algorithm="fricas")
Output:
-10/729*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - 1)^(1/3))/x) + 1/ 972*(81*x^11 - 9*x^8 - 12*x^5 - 20*x^2)*(x^3 - 1)^(1/3) + 10/729*log(-(x - (x^3 - 1)^(1/3))/x) - 5/729*log((x^2 + (x^3 - 1)^(1/3)*x + (x^3 - 1)^(2/3 ))/x^2)
Result contains complex when optimal does not.
Time = 19.41 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.28 \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\frac {x^{11} e^{\frac {i \pi }{3}} \Gamma \left (\frac {11}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {11}{3} \\ \frac {14}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {14}{3}\right )} \] Input:
integrate(x**10*(x**3-1)**(1/3),x)
Output:
x**11*exp(I*pi/3)*gamma(11/3)*hyper((-1/3, 11/3), (14/3,), x**3)/(3*gamma( 14/3))
Time = 0.12 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.49 \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=-\frac {10}{729} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {\frac {40 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {93 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}}}{x^{4}} - \frac {72 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}}}{x^{7}} + \frac {20 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}}}{x^{10}}}{972 \, {\left (\frac {4 \, {\left (x^{3} - 1\right )}}{x^{3}} - \frac {6 \, {\left (x^{3} - 1\right )}^{2}}{x^{6}} + \frac {4 \, {\left (x^{3} - 1\right )}^{3}}{x^{9}} - \frac {{\left (x^{3} - 1\right )}^{4}}{x^{12}} - 1\right )}} - \frac {5}{729} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {10}{729} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \] Input:
integrate(x^10*(x^3-1)^(1/3),x, algorithm="maxima")
Output:
-10/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3)/x + 1)) - 1/972*(40* (x^3 - 1)^(1/3)/x + 93*(x^3 - 1)^(4/3)/x^4 - 72*(x^3 - 1)^(7/3)/x^7 + 20*( x^3 - 1)^(10/3)/x^10)/(4*(x^3 - 1)/x^3 - 6*(x^3 - 1)^2/x^6 + 4*(x^3 - 1)^3 /x^9 - (x^3 - 1)^4/x^12 - 1) - 5/729*log((x^3 - 1)^(1/3)/x + (x^3 - 1)^(2/ 3)/x^2 + 1) + 10/729*log((x^3 - 1)^(1/3)/x - 1)
\[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\int { {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{10} \,d x } \] Input:
integrate(x^10*(x^3-1)^(1/3),x, algorithm="giac")
Output:
integrate((x^3 - 1)^(1/3)*x^10, x)
Timed out. \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\int x^{10}\,{\left (x^3-1\right )}^{1/3} \,d x \] Input:
int(x^10*(x^3 - 1)^(1/3),x)
Output:
int(x^10*(x^3 - 1)^(1/3), x)
\[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\frac {\left (x^{3}-1\right )^{\frac {1}{3}} x^{11}}{12}-\frac {\left (x^{3}-1\right )^{\frac {1}{3}} x^{8}}{108}-\frac {\left (x^{3}-1\right )^{\frac {1}{3}} x^{5}}{81}-\frac {5 \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}}{243}-\frac {10 \left (\int \frac {x}{\left (x^{3}-1\right )^{\frac {2}{3}}}d x \right )}{243} \] Input:
int(x^10*(x^3-1)^(1/3),x)
Output:
(81*(x**3 - 1)**(1/3)*x**11 - 9*(x**3 - 1)**(1/3)*x**8 - 12*(x**3 - 1)**(1 /3)*x**5 - 20*(x**3 - 1)**(1/3)*x**2 - 40*int(((x**3 - 1)**(1/3)*x)/(x**3 - 1),x))/972