3.22.92 \(\int \frac {(-4+x^3) (-2+x^3) (-1+x^3)^{2/3}}{x^6 (-2+x^3+x^6)} \, dx\)

Optimal. Leaf size=162 \[ -\sqrt [3]{\frac {2}{3}} \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3-1}-3 x\right )+\sqrt [3]{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+\sqrt [3]{3} x}\right )+\frac {\left (x^3-1\right )^{2/3} \left (8-13 x^3\right )}{10 x^5}+\frac {\log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3-1} x+2^{2/3} \sqrt [3]{3} \left (x^3-1\right )^{2/3}+3 x^2\right )}{2^{2/3} \sqrt [3]{3}} \]

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Rubi [A]  time = 0.60, antiderivative size = 157, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 11, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1586, 6725, 271, 264, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -\sqrt [3]{\frac {2}{3}} \log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3-1}}\right )+\sqrt [3]{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {2^{2/3} x}{\sqrt [6]{3} \sqrt [3]{x^3-1}}+\frac {1}{\sqrt {3}}\right )+\frac {4 \left (x^3-1\right )^{2/3}}{5 x^5}-\frac {13 \left (x^3-1\right )^{2/3}}{10 x^2}+\frac {\log \left (\frac {\sqrt [3]{6} x}{\sqrt [3]{x^3-1}}+\frac {3^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+2^{2/3}\right )}{2^{2/3} \sqrt [3]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-4 + x^3)*(-2 + x^3)*(-1 + x^3)^(2/3))/(x^6*(-2 + x^3 + x^6)),x]

[Out]

(4*(-1 + x^3)^(2/3))/(5*x^5) - (13*(-1 + x^3)^(2/3))/(10*x^2) + 2^(1/3)*3^(1/6)*ArcTan[1/Sqrt[3] + (2^(2/3)*x)
/(3^(1/6)*(-1 + x^3)^(1/3))] - (2/3)^(1/3)*Log[2^(1/3) - (3^(1/3)*x)/(-1 + x^3)^(1/3)] + Log[2^(2/3) + (3^(2/3
)*x^2)/(-1 + x^3)^(2/3) + (6^(1/3)*x)/(-1 + x^3)^(1/3)]/(2^(2/3)*3^(1/3))

Rule 31

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

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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 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 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 271

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

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^6 \left (-2+x^3+x^6\right )} \, dx &=\int \frac {\left (-4+x^3\right ) \left (-2+x^3\right )}{x^6 \sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx\\ &=\int \left (\frac {4}{x^6 \sqrt [3]{-1+x^3}}-\frac {5}{x^3 \sqrt [3]{-1+x^3}}+\frac {6}{\sqrt [3]{-1+x^3} \left (2+x^3\right )}\right ) \, dx\\ &=4 \int \frac {1}{x^6 \sqrt [3]{-1+x^3}} \, dx-5 \int \frac {1}{x^3 \sqrt [3]{-1+x^3}} \, dx+6 \int \frac {1}{\sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx\\ &=\frac {4 \left (-1+x^3\right )^{2/3}}{5 x^5}-\frac {5 \left (-1+x^3\right )^{2/3}}{2 x^2}+\frac {12}{5} \int \frac {1}{x^3 \sqrt [3]{-1+x^3}} \, dx+6 \operatorname {Subst}\left (\int \frac {1}{2-3 x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=\frac {4 \left (-1+x^3\right )^{2/3}}{5 x^5}-\frac {13 \left (-1+x^3\right )^{2/3}}{10 x^2}+\sqrt [3]{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2}-\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\sqrt [3]{2} \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{2}+\sqrt [3]{3} x}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=\frac {4 \left (-1+x^3\right )^{2/3}}{5 x^5}-\frac {13 \left (-1+x^3\right )^{2/3}}{10 x^2}-\sqrt [3]{\frac {2}{3}} \log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )+\frac {3 \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{\sqrt [3]{2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{6}+2\ 3^{2/3} x}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{2^{2/3} \sqrt [3]{3}}\\ &=\frac {4 \left (-1+x^3\right )^{2/3}}{5 x^5}-\frac {13 \left (-1+x^3\right )^{2/3}}{10 x^2}-\sqrt [3]{\frac {2}{3}} \log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )+\frac {\log \left (2^{2/3}+\frac {3^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{6} x}{\sqrt [3]{-1+x^3}}\right )}{2^{2/3} \sqrt [3]{3}}-\left (\sqrt [3]{2} 3^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )\\ &=\frac {4 \left (-1+x^3\right )^{2/3}}{5 x^5}-\frac {13 \left (-1+x^3\right )^{2/3}}{10 x^2}+\sqrt [3]{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-\sqrt [3]{\frac {2}{3}} \log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )+\frac {\log \left (2^{2/3}+\frac {3^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{6} x}{\sqrt [3]{-1+x^3}}\right )}{2^{2/3} \sqrt [3]{3}}\\ \end {align*}

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Mathematica [F]  time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^6 \left (-2+x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-4 + x^3)*(-2 + x^3)*(-1 + x^3)^(2/3))/(x^6*(-2 + x^3 + x^6)),x]

[Out]

Integrate[((-4 + x^3)*(-2 + x^3)*(-1 + x^3)^(2/3))/(x^6*(-2 + x^3 + x^6)), x]

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IntegrateAlgebraic [A]  time = 0.45, size = 162, normalized size = 1.00 \begin {gather*} \frac {\left (8-13 x^3\right ) \left (-1+x^3\right )^{2/3}}{10 x^5}+\sqrt [3]{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )-\sqrt [3]{\frac {2}{3}} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )}{2^{2/3} \sqrt [3]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-4 + x^3)*(-2 + x^3)*(-1 + x^3)^(2/3))/(x^6*(-2 + x^3 + x^6)),x]

[Out]

((8 - 13*x^3)*(-1 + x^3)^(2/3))/(10*x^5) + 2^(1/3)*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(-1 + x^3
)^(1/3))] - (2/3)^(1/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + x^3)^(1/3)] + Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(-1 + x^3
)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + x^3)^(2/3)]/(2^(2/3)*3^(1/3))

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fricas [B]  time = 2.89, size = 291, normalized size = 1.80 \begin {gather*} \frac {10 \cdot 3^{\frac {2}{3}} \left (-2\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {9 \cdot 3^{\frac {1}{3}} \left (-2\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3^{\frac {2}{3}} \left (-2\right )^{\frac {1}{3}} {\left (x^{3} + 2\right )} - 18 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 2}\right ) - 5 \cdot 3^{\frac {2}{3}} \left (-2\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {12 \cdot 3^{\frac {2}{3}} \left (-2\right )^{\frac {1}{3}} {\left (4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 3^{\frac {1}{3}} \left (-2\right )^{\frac {2}{3}} {\left (55 \, x^{6} - 50 \, x^{3} + 4\right )} - 18 \, {\left (7 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{3} + 4}\right ) - 30 \cdot 3^{\frac {1}{6}} \left (-2\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3^{\frac {1}{6}} {\left (12 \cdot 3^{\frac {2}{3}} \left (-2\right )^{\frac {2}{3}} {\left (4 \, x^{7} + 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 18 \, \left (-2\right )^{\frac {1}{3}} {\left (55 \, x^{8} - 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 3^{\frac {1}{3}} {\left (377 \, x^{9} - 600 \, x^{6} + 204 \, x^{3} - 8\right )}\right )}}{3 \, {\left (487 \, x^{9} - 480 \, x^{6} + 12 \, x^{3} + 8\right )}}\right ) - 9 \, {\left (13 \, x^{3} - 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4)*(x^3-2)*(x^3-1)^(2/3)/x^6/(x^6+x^3-2),x, algorithm="fricas")

[Out]

1/90*(10*3^(2/3)*(-2)^(1/3)*x^5*log(-(9*3^(1/3)*(-2)^(2/3)*(x^3 - 1)^(1/3)*x^2 + 3^(2/3)*(-2)^(1/3)*(x^3 + 2)
- 18*(x^3 - 1)^(2/3)*x)/(x^3 + 2)) - 5*3^(2/3)*(-2)^(1/3)*x^5*log(-(12*3^(2/3)*(-2)^(1/3)*(4*x^4 - x)*(x^3 - 1
)^(2/3) - 3^(1/3)*(-2)^(2/3)*(55*x^6 - 50*x^3 + 4) - 18*(7*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 + 4*x^3 + 4)) -
30*3^(1/6)*(-2)^(1/3)*x^5*arctan(1/3*3^(1/6)*(12*3^(2/3)*(-2)^(2/3)*(4*x^7 + 7*x^4 - 2*x)*(x^3 - 1)^(2/3) + 18
*(-2)^(1/3)*(55*x^8 - 50*x^5 + 4*x^2)*(x^3 - 1)^(1/3) - 3^(1/3)*(377*x^9 - 600*x^6 + 204*x^3 - 8))/(487*x^9 -
480*x^6 + 12*x^3 + 8)) - 9*(13*x^3 - 8)*(x^3 - 1)^(2/3))/x^5

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )} {\left (x^{3} - 4\right )}}{{\left (x^{6} + x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4)*(x^3-2)*(x^3-1)^(2/3)/x^6/(x^6+x^3-2),x, algorithm="giac")

[Out]

integrate((x^3 - 1)^(2/3)*(x^3 - 2)*(x^3 - 4)/((x^6 + x^3 - 2)*x^6), x)

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maple [C]  time = 17.70, size = 890, normalized size = 5.49

method	result	size
risch	\(-\frac {13 x^{6}-21 x^{3}+8}{10 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}-\frac {\ln \left (-\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}-162 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+42 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x +\left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}+144 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{2}-10 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}+270 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{3}-48 x \left (x^{3}-1\right )^{\frac {2}{3}}+4 \RootOf \left (\textit {\_Z}^{3}+18\right )-108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )}{x^{3}+2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )}{3}-6 \ln \left (-\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}-162 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+42 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x +\left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}+144 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{2}-10 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}+270 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{3}-48 x \left (x^{3}-1\right )^{\frac {2}{3}}+4 \RootOf \left (\textit {\_Z}^{3}+18\right )-108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )}{x^{3}+2}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )+\frac {\RootOf \left (\textit {\_Z}^{3}+18\right ) \ln \left (\frac {-3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}-135 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+21 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x -4 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}-9 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}-90 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{3}+18\right )+90 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+324 \textit {\_Z}^{2}\right )}{x^{3}+2}\right )}{3}\)	\(890\)
trager	\(\text {Expression too large to display}\)	\(1096\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-4)*(x^3-2)*(x^3-1)^(2/3)/x^6/(x^6+x^3-2),x,method=_RETURNVERBOSE)

[Out]

-1/10*(13*x^6-21*x^3+8)/x^5/(x^3-1)^(1/3)-1/3*ln(-(6*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*
RootOf(_Z^3+18)^3*x^3-162*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)^2*RootOf(_Z^3+18)^2*x^3+42*
(x^3-1)^(2/3)*RootOf(_Z^3+18)^2*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x+(x^3-1)^(1/3)*RootO
f(_Z^3+18)^2*x^2+144*(x^3-1)^(1/3)*RootOf(_Z^3+18)*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^
2-10*RootOf(_Z^3+18)*x^3+270*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^3-48*x*(x^3-1)^(2/3)+4
*RootOf(_Z^3+18)-108*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2))/(x^3+2))*RootOf(_Z^3+18)-6*ln(-
(6*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootOf(_Z^3+18)^3*x^3-162*RootOf(RootOf(_Z^3+18)^2
+18*_Z*RootOf(_Z^3+18)+324*_Z^2)^2*RootOf(_Z^3+18)^2*x^3+42*(x^3-1)^(2/3)*RootOf(_Z^3+18)^2*RootOf(RootOf(_Z^3
+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x+(x^3-1)^(1/3)*RootOf(_Z^3+18)^2*x^2+144*(x^3-1)^(1/3)*RootOf(_Z^3+18)
*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^2-10*RootOf(_Z^3+18)*x^3+270*RootOf(RootOf(_Z^3+18
)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^3-48*x*(x^3-1)^(2/3)+4*RootOf(_Z^3+18)-108*RootOf(RootOf(_Z^3+18)^2+18*_
Z*RootOf(_Z^3+18)+324*_Z^2))/(x^3+2))*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)+1/3*RootOf(_Z^3
+18)*ln((-3*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootOf(_Z^3+18)^3*x^3-135*RootOf(RootOf(_
Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)^2*RootOf(_Z^3+18)^2*x^3+21*(x^3-1)^(2/3)*RootOf(_Z^3+18)^2*RootOf(Ro
otOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x-4*(x^3-1)^(1/3)*RootOf(_Z^3+18)^2*x^2-9*(x^3-1)^(1/3)*RootOf
(_Z^3+18)*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^2-2*RootOf(_Z^3+18)*x^3-90*RootOf(RootOf(
_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^3+3*x*(x^3-1)^(2/3)+2*RootOf(_Z^3+18)+90*RootOf(RootOf(_Z^3+18)^2
+18*_Z*RootOf(_Z^3+18)+324*_Z^2))/(x^3+2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )} {\left (x^{3} - 4\right )}}{{\left (x^{6} + x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4)*(x^3-2)*(x^3-1)^(2/3)/x^6/(x^6+x^3-2),x, algorithm="maxima")

[Out]

integrate((x^3 - 1)^(2/3)*(x^3 - 2)*(x^3 - 4)/((x^6 + x^3 - 2)*x^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3-2\right )\,\left (x^3-4\right )}{x^6\,\left (x^6+x^3-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(2/3)*(x^3 - 2)*(x^3 - 4))/(x^6*(x^3 + x^6 - 2)),x)

[Out]

int(((x^3 - 1)^(2/3)*(x^3 - 2)*(x^3 - 4))/(x^6*(x^3 + x^6 - 2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-4)*(x**3-2)*(x**3-1)**(2/3)/x**6/(x**6+x**3-2),x)

[Out]

Timed out

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