3.17.51 \(\int x^2 \sqrt [3]{-x+x^3} \, dx\)

Optimal. Leaf size=112 \[ \frac {1}{12} \sqrt [3]{x^3-x} \left (3 x^3-x\right )+\frac {1}{18} \log \left (\sqrt [3]{x^3-x}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x}+x}\right )}{6 \sqrt {3}}-\frac {1}{36} \log \left (\sqrt [3]{x^3-x} x+\left (x^3-x\right )^{2/3}+x^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 204, normalized size of antiderivative = 1.82, number of steps used = 12, number of rules used = 12, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {2021, 2024, 2032, 329, 275, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {1}{4} \sqrt [3]{x^3-x} x^3-\frac {1}{12} \sqrt [3]{x^3-x} x+\frac {\left (x^2-1\right )^{2/3} x^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )}{18 \left (x^3-x\right )^{2/3}}-\frac {\left (x^2-1\right )^{2/3} x^{2/3} \log \left (\frac {x^{4/3}}{\left (x^2-1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2-1}}+1\right )}{36 \left (x^3-x\right )^{2/3}}+\frac {\left (x^2-1\right )^{2/3} x^{2/3} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \left (x^3-x\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(-x + x^3)^(1/3),x]

[Out]

-1/12*(x*(-x + x^3)^(1/3)) + (x^3*(-x + x^3)^(1/3))/4 + (x^(2/3)*(-1 + x^2)^(2/3)*ArcTan[(1 + (2*x^(2/3))/(-1
+ x^2)^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(-x + x^3)^(2/3)) + (x^(2/3)*(-1 + x^2)^(2/3)*Log[1 - x^(2/3)/(-1 + x^2)^(1
/3)])/(18*(-x + x^3)^(2/3)) - (x^(2/3)*(-1 + x^2)^(2/3)*Log[1 + x^(4/3)/(-1 + x^2)^(2/3) + x^(2/3)/(-1 + x^2)^
(1/3)])/(36*(-x + x^3)^(2/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 2021

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b
*x^n)^p)/(c*(m + n*p + 1)), x] + Dist[(a*(n - j)*p)/(c^j*(m + n*p + 1)), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2024

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n +
1)*(a*x^j + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^(n - j)*(m + j*p - n + j + 1))/(b*(m + n*p + 1)
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rubi steps

\begin {align*} \int x^2 \sqrt [3]{-x+x^3} \, dx &=\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {1}{6} \int \frac {x^3}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {1}{9} \int \frac {x}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right )^{2/3}} \, dx}{9 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{3 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{6 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{18 \left (-x+x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{18 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{18 \left (-x+x^3\right )^{2/3}}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{36 \left (-x+x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{18 \left (-x+x^3\right )^{2/3}}-\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{36 \left (-x+x^3\right )^{2/3}}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{6 \sqrt {3} \left (-x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{18 \left (-x+x^3\right )^{2/3}}-\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{36 \left (-x+x^3\right )^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.02, size = 54, normalized size = 0.48 \begin {gather*} \frac {x \sqrt [3]{x \left (x^2-1\right )} \left (\, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};x^2\right )-\left (1-x^2\right )^{4/3}\right )}{4 \sqrt [3]{1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(-x + x^3)^(1/3),x]

[Out]

(x*(x*(-1 + x^2))^(1/3)*(-(1 - x^2)^(4/3) + Hypergeometric2F1[-1/3, 2/3, 5/3, x^2]))/(4*(1 - x^2)^(1/3))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.24, size = 112, normalized size = 1.00 \begin {gather*} \frac {1}{12} \sqrt [3]{-x+x^3} \left (-x+3 x^3\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{6 \sqrt {3}}+\frac {1}{18} \log \left (-x+\sqrt [3]{-x+x^3}\right )-\frac {1}{36} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2*(-x + x^3)^(1/3),x]

[Out]

((-x + x^3)^(1/3)*(-x + 3*x^3))/12 + ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))]/(6*Sqrt[3]) + Log[-x + (-x +
x^3)^(1/3)]/18 - Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)]/36

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 107, normalized size = 0.96 \begin {gather*} \frac {1}{18} \, \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}{12} \, {\left (3 \, x^{3} - x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}} + \frac {1}{36} \, \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3-x)^(1/3),x, algorithm="fricas")

[Out]

1/18*sqrt(3)*arctan(-(44032959556*sqrt(3)*(x^3 - x)^(1/3)*x + sqrt(3)*(16754327161*x^2 - 2707204793) - 1052430
5234*sqrt(3)*(x^3 - x)^(2/3))/(81835897185*x^2 - 1102302937)) + 1/12*(3*x^3 - x)*(x^3 - x)^(1/3) + 1/36*log(-3
*(x^3 - x)^(1/3)*x + 3*(x^3 - x)^(2/3) + 1)

________________________________________________________________________________________

giac [A]  time = 2.01, size = 89, normalized size = 0.79 \begin {gather*} \frac {1}{12} \, {\left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{4} - \frac {1}{18} \, \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 {1}{36} \, \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 {1}{18} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3-x)^(1/3),x, algorithm="giac")

[Out]

1/12*((-1/x^2 + 1)^(4/3) + 2*(-1/x^2 + 1)^(1/3))*x^4 - 1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-1/x^2 + 1)^(1/3) +
1)) - 1/36*log((-1/x^2 + 1)^(2/3) + (-1/x^2 + 1)^(1/3) + 1) + 1/18*log(abs((-1/x^2 + 1)^(1/3) - 1))

________________________________________________________________________________________

maple [C]  time = 3.14, size = 33, normalized size = 0.29

method	result	size
meijerg	\(\frac {3 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {10}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], x^{2}\right )}{10 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}}}\)	\(33\)
trager	\(\frac {x \left (3 x^{2}-1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{12}+\frac {\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (1395 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+6768 \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}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +7233 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}+7611 x \left (x^{3}-x \right )^{\frac {1}{3}}-5580 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+7766 x^{2}-11727 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4942\right )}{6}-\frac {\ln \left (1395 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \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}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -6303 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+5355 x \left (x^{3}-x \right )^{\frac {1}{3}}-5580 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+5510 x^{2}+8007 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1653\right )}{18}-\frac {\ln \left (1395 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \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}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -6303 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+5355 x \left (x^{3}-x \right )^{\frac {1}{3}}-5580 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+5510 x^{2}+8007 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1653\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{6}\)	\(463\)
risch	\(\frac {x \left (3 x^{2}-1\right ) \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}}}{12}+\frac {\left (\frac {\RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \ln \left (-\frac {47 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}+3207 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-235 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+6930 x^{4}+2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-5601 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+5238 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+188 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+5238 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-11340 x^{2}+2394 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )-5238 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+4410}{\left (-1+x \right ) \left (1+x \right )}\right )}{108}-\frac {\ln \left (-\frac {47 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-2643 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-235 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-10620 x^{4}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+2781 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}-12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+188 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}+2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+13806 x^{2}-138 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-3186}{\left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )}{108}-\frac {\ln \left (-\frac {47 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-2643 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-235 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-10620 x^{4}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+2781 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}-12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+188 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}+2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+13806 x^{2}-138 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-3186}{\left (-1+x \right ) \left (1+x \right )}\right )}{18}\right ) \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}} \left (x^{2} \left (x^{2}-1\right )^{2}\right )^{\frac {1}{3}}}{x \left (x^{2}-1\right )}\)	\(793\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^3-x)^(1/3),x,method=_RETURNVERBOSE)

[Out]

3/10*signum(x^2-1)^(1/3)/(-signum(x^2-1))^(1/3)*x^(10/3)*hypergeom([-1/3,5/3],[8/3],x^2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{3} - x\right )}^{\frac {1}{3}} x^{2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3-x)^(1/3),x, algorithm="maxima")

[Out]

integrate((x^3 - x)^(1/3)*x^2, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\left (x^3-x\right )}^{1/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^3 - x)^(1/3),x)

[Out]

int(x^2*(x^3 - x)^(1/3), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(x**3-x)**(1/3),x)

[Out]

Integral(x**2*(x*(x - 1)*(x + 1))**(1/3), x)

________________________________________________________________________________________