3.18.98 \(\int x^6 \sqrt [3]{-x+x^3} \, dx\)

Optimal. Leaf size=122 \[ \frac {5}{243} \log \left (\sqrt [3]{x^3-x}-x\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x}+x}\right )}{81 \sqrt {3}}-\frac {5}{486} \log \left (\sqrt [3]{x^3-x} x+\left (x^3-x\right )^{2/3}+x^2\right )+\frac {1}{648} \sqrt [3]{x^3-x} \left (81 x^7-9 x^5-12 x^3-20 x\right ) \]

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Rubi [A]  time = 0.26, antiderivative size = 240, normalized size of antiderivative = 1.97, number of steps used = 14, 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}{54} \sqrt [3]{x^3-x} x^3-\frac {5}{162} \sqrt [3]{x^3-x} x+\frac {1}{8} \sqrt [3]{x^3-x} x^7-\frac {1}{72} \sqrt [3]{x^3-x} x^5+\frac {5 \left (x^2-1\right )^{2/3} x^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )}{243 \left (x^3-x\right )^{2/3}}-\frac {5 \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 )}{486 \left (x^3-x\right )^{2/3}}+\frac {5 \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 )}{81 \sqrt {3} \left (x^3-x\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6*(-x + x^3)^(1/3),x]

[Out]

(-5*x*(-x + x^3)^(1/3))/162 - (x^3*(-x + x^3)^(1/3))/54 - (x^5*(-x + x^3)^(1/3))/72 + (x^7*(-x + x^3)^(1/3))/8
+ (5*x^(2/3)*(-1 + x^2)^(2/3)*ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(81*Sqrt[3]*(-x + x^3)^(2/3
)) + (5*x^(2/3)*(-1 + x^2)^(2/3)*Log[1 - x^(2/3)/(-1 + x^2)^(1/3)])/(243*(-x + x^3)^(2/3)) - (5*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)])/(486*(-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^6 \sqrt [3]{-x+x^3} \, dx &=\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {1}{12} \int \frac {x^7}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {2}{27} \int \frac {x^5}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {5}{81} \int \frac {x^3}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {10}{243} \int \frac {x}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {\left (10 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right )^{2/3}} \, dx}{243 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {\left (10 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 )}{81 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {\left (5 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 )}{81 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {\left (5 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 )}{81 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {\left (5 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 )}{243 \left (-x+x^3\right )^{2/3}}+\frac {\left (5 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 )}{243 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}+\frac {5 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{243 \left (-x+x^3\right )^{2/3}}-\frac {\left (5 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 )}{486 \left (-x+x^3\right )^{2/3}}+\frac {\left (5 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 )}{162 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}+\frac {5 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{243 \left (-x+x^3\right )^{2/3}}-\frac {5 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 )}{486 \left (-x+x^3\right )^{2/3}}-\frac {\left (5 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 )}{81 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}+\frac {5 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 )}{81 \sqrt {3} \left (-x+x^3\right )^{2/3}}+\frac {5 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{243 \left (-x+x^3\right )^{2/3}}-\frac {5 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 )}{486 \left (-x+x^3\right )^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 72, normalized size = 0.59 \begin {gather*} \frac {x \sqrt [3]{x \left (x^2-1\right )} \left (20 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};x^2\right )+\sqrt [3]{1-x^2} \left (27 x^6-3 x^4-4 x^2-20\right )\right )}{216 \sqrt [3]{1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*(-x + x^3)^(1/3),x]

[Out]

(x*(x*(-1 + x^2))^(1/3)*((1 - x^2)^(1/3)*(-20 - 4*x^2 - 3*x^4 + 27*x^6) + 20*Hypergeometric2F1[-1/3, 2/3, 5/3,
x^2]))/(216*(1 - x^2)^(1/3))

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IntegrateAlgebraic [A]  time = 0.42, size = 122, normalized size = 1.00 \begin {gather*} \frac {1}{648} \sqrt [3]{-x+x^3} \left (-20 x-12 x^3-9 x^5+81 x^7\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (-x+\sqrt [3]{-x+x^3}\right )-\frac {5}{486} \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^6*(-x + x^3)^(1/3),x]

[Out]

((-x + x^3)^(1/3)*(-20*x - 12*x^3 - 9*x^5 + 81*x^7))/648 + (5*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/(8
1*Sqrt[3]) + (5*Log[-x + (-x + x^3)^(1/3)])/243 - (5*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/486

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fricas [A]  time = 0.86, size = 117, normalized size = 0.96 \begin {gather*} \frac {5}{243} \, \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}{648} \, {\left (81 \, x^{7} - 9 \, x^{5} - 12 \, x^{3} - 20 \, x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}} + \frac {5}{486} \, \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^6*(x^3-x)^(1/3),x, algorithm="fricas")

[Out]

5/243*sqrt(3)*arctan(-(44032959556*sqrt(3)*(x^3 - x)^(1/3)*x + sqrt(3)*(16754327161*x^2 - 2707204793) - 105243
05234*sqrt(3)*(x^3 - x)^(2/3))/(81835897185*x^2 - 1102302937)) + 1/648*(81*x^7 - 9*x^5 - 12*x^3 - 20*x)*(x^3 -
x)^(1/3) + 5/486*log(-3*(x^3 - x)^(1/3)*x + 3*(x^3 - x)^(2/3) + 1)

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giac [A]  time = 0.19, size = 127, normalized size = 1.04 \begin {gather*} -\frac {1}{648} \, {\left (20 \, {\left (\frac {1}{x^{2}} - 1\right )}^{3} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 72 \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 93 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} - 40 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{8} - \frac {5}{243} \, \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}{486} \, \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}{243} \, \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^6*(x^3-x)^(1/3),x, algorithm="giac")

[Out]

-1/648*(20*(1/x^2 - 1)^3*(-1/x^2 + 1)^(1/3) + 72*(1/x^2 - 1)^2*(-1/x^2 + 1)^(1/3) - 93*(-1/x^2 + 1)^(4/3) - 40
*(-1/x^2 + 1)^(1/3))*x^8 - 5/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-1/x^2 + 1)^(1/3) + 1)) - 5/486*log((-1/x^2 +
1)^(2/3) + (-1/x^2 + 1)^(1/3) + 1) + 5/243*log(abs((-1/x^2 + 1)^(1/3) - 1))

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maple [C]  time = 3.10, size = 33, normalized size = 0.27

method	result	size
meijerg	\(\frac {3 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {22}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], x^{2}\right )}{22 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}}}\)	\(33\)
trager	\(\frac {x \left (81 x^{6}-9 x^{4}-12 x^{2}-20\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{648}+\frac {5 \ln \left (4959 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+6768 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-22833 \left (x^{3}-x \right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +17718 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-19836 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}-5355 x \left (x^{3}-x \right )^{\frac {1}{3}}-1705 x^{2}-9711 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1085\right )}{243}+\frac {5 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-6354 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-16065 \left (x^{3}-x \right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +20715 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+25416 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}-7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+1550 x^{2}-4494 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-465\right )}{81}\)	\(322\)
risch	\(\frac {x \left (81 x^{6}-9 x^{4}-12 x^{2}-20\right ) \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}}}{648}+\frac {\left (\frac {5 \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 )}{1458}-\frac {5 \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 )}{1458}-\frac {5 \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 )}{243}\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 )}\)	\(803\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x^3-x)^(1/3),x,method=_RETURNVERBOSE)

[Out]

3/22*signum(x^2-1)^(1/3)/(-signum(x^2-1))^(1/3)*x^(22/3)*hypergeom([-1/3,11/3],[14/3],x^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{3} - x\right )}^{\frac {1}{3}} x^{6}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^3-x)^(1/3),x, algorithm="maxima")

[Out]

integrate((x^3 - x)^(1/3)*x^6, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^6\,{\left (x^3-x\right )}^{1/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x^3 - x)^(1/3),x)

[Out]

int(x^6*(x^3 - x)^(1/3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{6} \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**6*(x**3-x)**(1/3),x)

[Out]

Integral(x**6*(x*(x - 1)*(x + 1))**(1/3), x)

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