\(\int x^{13} \sqrt [3]{-1+x^3} \, dx\) [1762]

Optimal result

Integrand size = 13, antiderivative size = 119 \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=\frac {\sqrt [3]{-1+x^3} \left (-220 x^2-132 x^5-99 x^8-81 x^{11}+972 x^{14}\right )}{14580}+\frac {22 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{729 \sqrt {3}}+\frac {22 \log \left (-x+\sqrt [3]{-1+x^3}\right )}{2187}-\frac {11 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{2187} \]

[Out]

1/14580*(x^3-1)^(1/3)*(972*x^14-81*x^11-99*x^8-132*x^5-220*x^2)+22/2187*arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3)))*
3^(1/2)+22/2187*ln(-x+(x^3-1)^(1/3))-11/2187*ln(x^2+x*(x^3-1)^(1/3)+(x^3-1)^(2/3))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {285, 327, 337} \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=\frac {22 \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{729 \sqrt {3}}+\frac {11}{729} \log \left (x-\sqrt [3]{x^3-1}\right )+\frac {1}{15} \sqrt [3]{x^3-1} x^{14}-\frac {1}{180} \sqrt [3]{x^3-1} x^{11}-\frac {11 \sqrt [3]{x^3-1} x^8}{1620}-\frac {11 \sqrt [3]{x^3-1} x^5}{1215}-\frac {11}{729} \sqrt [3]{x^3-1} x^2 \]

[In]

Int[x^13*(-1 + x^3)^(1/3),x]

[Out]

(-11*x^2*(-1 + x^3)^(1/3))/729 - (11*x^5*(-1 + x^3)^(1/3))/1215 - (11*x^8*(-1 + x^3)^(1/3))/1620 - (x^11*(-1 +
x^3)^(1/3))/180 + (x^14*(-1 + x^3)^(1/3))/15 + (22*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(729*Sqrt[3]
) + (11*Log[x - (-1 + x^3)^(1/3)])/729

Rule 285

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] + Dist[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]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

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] - Dist[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]

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{15} x^{14} \sqrt [3]{-1+x^3}-\frac {1}{15} \int \frac {x^{13}}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {1}{180} x^{11} \sqrt [3]{-1+x^3}+\frac {1}{15} x^{14} \sqrt [3]{-1+x^3}-\frac {11}{180} \int \frac {x^{10}}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {11 x^8 \sqrt [3]{-1+x^3}}{1620}-\frac {1}{180} x^{11} \sqrt [3]{-1+x^3}+\frac {1}{15} x^{14} \sqrt [3]{-1+x^3}-\frac {22}{405} \int \frac {x^7}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {11 x^5 \sqrt [3]{-1+x^3}}{1215}-\frac {11 x^8 \sqrt [3]{-1+x^3}}{1620}-\frac {1}{180} x^{11} \sqrt [3]{-1+x^3}+\frac {1}{15} x^{14} \sqrt [3]{-1+x^3}-\frac {11}{243} \int \frac {x^4}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {11}{729} x^2 \sqrt [3]{-1+x^3}-\frac {11 x^5 \sqrt [3]{-1+x^3}}{1215}-\frac {11 x^8 \sqrt [3]{-1+x^3}}{1620}-\frac {1}{180} x^{11} \sqrt [3]{-1+x^3}+\frac {1}{15} x^{14} \sqrt [3]{-1+x^3}-\frac {22}{729} \int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {11}{729} x^2 \sqrt [3]{-1+x^3}-\frac {11 x^5 \sqrt [3]{-1+x^3}}{1215}-\frac {11 x^8 \sqrt [3]{-1+x^3}}{1620}-\frac {1}{180} x^{11} \sqrt [3]{-1+x^3}+\frac {1}{15} x^{14} \sqrt [3]{-1+x^3}+\frac {22 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{729 \sqrt {3}}+\frac {11}{729} \log \left (x-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.96 \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=\frac {3 x^2 \sqrt [3]{-1+x^3} \left (-220-132 x^3-99 x^6-81 x^9+972 x^{12}\right )+440 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )+440 \log \left (-x+\sqrt [3]{-1+x^3}\right )-220 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{43740} \]

[In]

Integrate[x^13*(-1 + x^3)^(1/3),x]

[Out]

(3*x^2*(-1 + x^3)^(1/3)*(-220 - 132*x^3 - 99*x^6 - 81*x^9 + 972*x^12) + 440*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*
(-1 + x^3)^(1/3))] + 440*Log[-x + (-1 + x^3)^(1/3)] - 220*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/43
740

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.99 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28

method	result	size
meijerg	\(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} x^{14} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {14}{3}\right ], \left [\frac {17}{3}\right ], x^{3}\right )}{14 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\)	\(33\)
risch	\(\frac {x^{2} \left (972 x^{12}-81 x^{9}-99 x^{6}-132 x^{3}-220\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{14580}-\frac {11 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{729 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(68\)
pseudoelliptic	\(\frac {-220 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-440 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )+440 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )+\left (2916 x^{14}-243 x^{11}-297 x^{8}-396 x^{5}-660 x^{2}\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{43740 \left (x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )^{5} {\left (x -\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{5}}\)	\(143\)
trager	\(\frac {x^{2} \left (972 x^{12}-81 x^{9}-99 x^{6}-132 x^{3}-220\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{14580}+\frac {22 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{2187}-\frac {22 \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+2 x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{2187}-\frac {22 \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+2 x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{2187}\)	\(256\)

[In]

int(x^13*(x^3-1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/14*signum(x^3-1)^(1/3)/(-signum(x^3-1))^(1/3)*x^14*hypergeom([-1/3,14/3],[17/3],x^3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=-\frac {22}{2187} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{14580} \, {\left (972 \, x^{14} - 81 \, x^{11} - 99 \, x^{8} - 132 \, x^{5} - 220 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \frac {22}{2187} \, \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {11}{2187} \, \log \left (\frac {x^{2} + {\left (x^{3} - 1\right )}^{\frac {1}{3}} x + {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]

[In]

integrate(x^13*(x^3-1)^(1/3),x, algorithm="fricas")

[Out]

-22/2187*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - 1)^(1/3))/x) + 1/14580*(972*x^14 - 81*x^11 - 99*x^8
- 132*x^5 - 220*x^2)*(x^3 - 1)^(1/3) + 22/2187*log(-(x - (x^3 - 1)^(1/3))/x) - 11/2187*log((x^2 + (x^3 - 1)^(1
/3)*x + (x^3 - 1)^(2/3))/x^2)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 120.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.30 \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=- \frac {x^{14} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {14}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {14}{3} \\ \frac {17}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {17}{3}\right )} \]

[In]

integrate(x**13*(x**3-1)**(1/3),x)

[Out]

-x**14*exp(-2*I*pi/3)*gamma(14/3)*hyper((-1/3, 14/3), (17/3,), x**3)/(3*gamma(17/3))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.62 \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=-\frac {22}{2187} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {\frac {440 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {1555 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}}}{x^{4}} - \frac {1815 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}}}{x^{7}} + \frac {1012 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}}}{x^{10}} - \frac {220 \, {\left (x^{3} - 1\right )}^{\frac {13}{3}}}{x^{13}}}{14580 \, {\left (\frac {5 \, {\left (x^{3} - 1\right )}}{x^{3}} - \frac {10 \, {\left (x^{3} - 1\right )}^{2}}{x^{6}} + \frac {10 \, {\left (x^{3} - 1\right )}^{3}}{x^{9}} - \frac {5 \, {\left (x^{3} - 1\right )}^{4}}{x^{12}} + \frac {{\left (x^{3} - 1\right )}^{5}}{x^{15}} - 1\right )}} - \frac {11}{2187} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {22}{2187} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]

[In]

integrate(x^13*(x^3-1)^(1/3),x, algorithm="maxima")

[Out]

-22/2187*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3)/x + 1)) - 1/14580*(440*(x^3 - 1)^(1/3)/x + 1555*(x^3 -
1)^(4/3)/x^4 - 1815*(x^3 - 1)^(7/3)/x^7 + 1012*(x^3 - 1)^(10/3)/x^10 - 220*(x^3 - 1)^(13/3)/x^13)/(5*(x^3 - 1)
/x^3 - 10*(x^3 - 1)^2/x^6 + 10*(x^3 - 1)^3/x^9 - 5*(x^3 - 1)^4/x^12 + (x^3 - 1)^5/x^15 - 1) - 11/2187*log((x^3
- 1)^(1/3)/x + (x^3 - 1)^(2/3)/x^2 + 1) + 22/2187*log((x^3 - 1)^(1/3)/x - 1)

Giac [F]

\[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=\int { {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{13} \,d x } \]

[In]

integrate(x^13*(x^3-1)^(1/3),x, algorithm="giac")

[Out]

integrate((x^3 - 1)^(1/3)*x^13, x)

Mupad [F(-1)]

Timed out. \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=\int x^{13}\,{\left (x^3-1\right )}^{1/3} \,d x \]

[In]

int(x^13*(x^3 - 1)^(1/3),x)

[Out]

int(x^13*(x^3 - 1)^(1/3), x)