3.17.95 \(\int x^{10} \sqrt [3]{-1+x^3} \, dx\) [1695]

Optimal. Leaf size=114 \[ \frac {1}{972} \sqrt [3]{-1+x^3} \left (-20 x^2-12 x^5-9 x^8+81 x^{11}\right )+\frac {10 \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{243 \sqrt {3}}+\frac {10}{729} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {5}{729} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]

[Out]

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))

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Rubi [A]
time = 0.03, antiderivative size = 113, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {285, 327, 337} \begin {gather*} \frac {10 \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {5}{243} \log \left (x-\sqrt [3]{x^3-1}\right )+\frac {1}{12} \sqrt [3]{x^3-1} x^{11}-\frac {1}{108} \sqrt [3]{x^3-1} x^8-\frac {1}{81} \sqrt [3]{x^3-1} x^5-\frac {5}{243} \sqrt [3]{x^3-1} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^10*(-1 + x^3)^(1/3),x]

[Out]

(-5*x^2*(-1 + x^3)^(1/3))/243 - (x^5*(-1 + x^3)^(1/3))/81 - (x^8*(-1 + x^3)^(1/3))/108 + (x^11*(-1 + x^3)^(1/3
))/12 + (10*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(243*Sqrt[3]) + (5*Log[x - (-1 + x^3)^(1/3)])/243

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*} \int x^{10} \sqrt [3]{-1+x^3} \, dx &=\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {1}{12} \int \frac {x^{10}}{\left (-1+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {2}{27} \int \frac {x^7}{\left (-1+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {5}{81} \int \frac {x^4}{\left (-1+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {10}{243} \int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {10}{243} \text {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {10}{729} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {10}{729} \text {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}+\frac {10}{729} \log \left (1-\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {5}{729} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {5}{243} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}+\frac {10}{729} \log \left (1-\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {5}{729} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {10}{243} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}+\frac {10 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {10}{729} \log \left (1-\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {5}{729} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 109, normalized size = 0.96 \begin {gather*} \frac {3 x^2 \sqrt [3]{-1+x^3} \left (-20-12 x^3-9 x^6+81 x^9\right )+40 \sqrt {3} \text {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} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^10*(-1 + x^3)^(1/3),x]

[Out]

(3*x^2*(-1 + x^3)^(1/3)*(-20 - 12*x^3 - 9*x^6 + 81*x^9) + 40*Sqrt[3]*ArcTan[(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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.55, size = 33, normalized size = 0.29

method	result	size
meijerg	\(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} x^{11} \hypergeom \left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], x^{3}\right )}{11 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\)	\(33\)
risch	\(\frac {x^{2} \left (81 x^{9}-9 x^{6}-12 x^{3}-20\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{972}-\frac {5 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{243 \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(63\)
trager	\(\frac {x^{2} \left (81 x^{9}-9 x^{6}-12 x^{3}-20\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{972}+\frac {10 \ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{729}+\frac {10 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}-2 x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{729}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10*(x^3-1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

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)

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Maxima [A]
time = 0.45, size = 170, normalized size = 1.49 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(x^3-1)^(1/3),x, algorithm="maxima")

[Out]

-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)

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Fricas [A]
time = 0.41, size = 106, normalized size = 0.93 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(x^3-1)^(1/3),x, algorithm="fricas")

[Out]

-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)

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Sympy [C] Result contains complex when optimal does not.
time = 19.97, size = 32, normalized size = 0.28 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10*(x**3-1)**(1/3),x)

[Out]

x**11*exp(I*pi/3)*gamma(11/3)*hyper((-1/3, 11/3), (14/3,), x**3)/(3*gamma(14/3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(x^3-1)^(1/3),x, algorithm="giac")

[Out]

integrate((x^3 - 1)^(1/3)*x^10, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^{10}\,{\left (x^3-1\right )}^{1/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10*(x^3 - 1)^(1/3),x)

[Out]

int(x^10*(x^3 - 1)^(1/3), x)

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