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

Optimal. Leaf size=140 \[ \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )}{4\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{4\ 2^{2/3}}+\frac {\left (x^3-1\right )^{2/3} \left (11 x^3+4\right )}{40 x^5}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right )}{8\ 2^{2/3}} \]

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Rubi [A]  time = 0.14, antiderivative size = 145, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {\log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{x^3-1}}\right )}{4\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3}}+\frac {\left (x^3-1\right )^{2/3}}{10 x^5}+\frac {11 \left (x^3-1\right )^{2/3}}{40 x^2}-\frac {\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {x^2}{\left (x^3-1\right )^{2/3}}+2^{2/3}\right )}{8\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 + x^3)^(2/3)/(10*x^5) + (11*(-1 + x^3)^(2/3))/(40*x^2) - (Sqrt[3]*ArcTan[(1 + (2^(2/3)*x)/(-1 + x^3)^(1/3)
)/Sqrt[3]])/(4*2^(2/3)) + Log[2^(1/3) - x/(-1 + x^3)^(1/3)]/(4*2^(2/3)) - Log[2^(2/3) + x^2/(-1 + x^3)^(2/3) +
(2^(1/3)*x)/(-1 + x^3)^(1/3)]/(8*2^(2/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 580

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*g*(m + 1)), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

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]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (-2+x^3\right )} \, dx &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {1}{10} \int \frac {11-13 x^3}{x^3 \left (-2+x^3\right ) \sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {1}{40} \int \frac {30}{\left (-2+x^3\right ) \sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {3}{4} \int \frac {1}{\left (-2+x^3\right ) \sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{-2+x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{2}+x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {-2 \sqrt [3]{2}-x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {\log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{2}+2 x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{8 \sqrt [3]{2}}\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {\log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}-\frac {\log \left (2^{2/3}+\frac {x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}-\frac {\log \left (2^{2/3}+\frac {x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 135, normalized size = 0.96 \begin {gather*} \left (\frac {1}{10 x^5}+\frac {11}{40 x^2}\right ) \left (x^3-1\right )^{2/3}-\frac {-2 \log \left (2-\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+\frac {\sqrt [3]{2} x^2}{\left (1-x^3\right )^{2/3}}+2\right )}{8\ 2^{2/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(1/(10*x^5) + 11/(40*x^2))*(-1 + x^3)^(2/3) - (2*Sqrt[3]*ArcTan[(1 + (2^(2/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]] - 2
*Log[2 - (2^(2/3)*x)/(1 - x^3)^(1/3)] + Log[2 + (2^(1/3)*x^2)/(1 - x^3)^(2/3) + (2^(2/3)*x)/(1 - x^3)^(1/3)])/
(8*2^(2/3))

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IntegrateAlgebraic [A]  time = 0.31, size = 140, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (4+11 x^3\right )}{40 x^5}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{8\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x^3)^(2/3)*(4 + 11*x^3))/(40*x^5) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))])/(4*2
^(2/3)) + Log[-x + 2^(1/3)*(-1 + x^3)^(1/3)]/(4*2^(2/3)) - Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1
+ x^3)^(2/3)]/(8*2^(2/3))

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fricas [B]  time = 3.15, size = 266, normalized size = 1.90 \begin {gather*} \frac {20 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )} + 12 \, {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 5 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) + 12 \, {\left (11 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{480 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(20*4^(1/6)*sqrt(3)*x^5*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(2*x^7 - 5*x^4 + 2*x)*(x^3 - 1)^(2/3) + 4
^(1/3)*(91*x^9 - 168*x^6 + 84*x^3 - 8) + 12*(19*x^8 - 22*x^5 + 4*x^2)*(x^3 - 1)^(1/3))/(53*x^9 - 48*x^6 - 12*x
^3 + 8)) + 10*4^(2/3)*x^5*log((6*4^(1/3)*(x^3 - 1)^(1/3)*x^2 + 4^(2/3)*(x^3 - 2) - 12*(x^3 - 1)^(2/3)*x)/(x^3
- 2)) - 5*4^(2/3)*x^5*log((6*4^(2/3)*(2*x^4 - x)*(x^3 - 1)^(2/3) + 4^(1/3)*(19*x^6 - 22*x^3 + 4) + 6*(5*x^5 -
4*x^2)*(x^3 - 1)^(1/3))/(x^6 - 4*x^3 + 4)) + 12*(11*x^3 + 4)*(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 )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 18.11, size = 889, normalized size = 6.35

method	result	size
risch	\(\frac {11 x^{6}-7 x^{3}-4}{40 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {\RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+15 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +2 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-3 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}+x \left (x^{3}-1\right )^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{3}-2\right )+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-2}\right )}{8}-\frac {\ln \left (\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}-18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-30 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x -\RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+24 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}+6 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-8 x \left (x^{3}-1\right )^{\frac {2}{3}}-4 \RootOf \left (\textit {\_Z}^{3}-2\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )}{8}-\frac {3 \ln \left (\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}-18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-30 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x -\RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+24 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}+6 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-8 x \left (x^{3}-1\right )^{\frac {2}{3}}-4 \RootOf \left (\textit {\_Z}^{3}-2\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-2}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{4}\)	\(889\)
trager	\(\text {Expression too large to display}\)	\(1116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(11*x^6-7*x^3-4)/x^5/(x^3-1)^(1/3)+1/8*RootOf(_Z^3-2)*ln((3*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+3
6*_Z^2)*RootOf(_Z^3-2)^3*x^3+27*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*x^3+15
*(x^3-1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^2*x+2*RootOf(_Z^3-2)^2*(x^3
-1)^(1/3)*x^2-3*(x^3-1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)*x^2-2*RootOf
(_Z^3-2)*x^3-18*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^3+x*(x^3-1)^(2/3)+2*RootOf(_Z^3-2)+18*R
ootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^3-2))-1/8*ln((6*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z
^3-2)+36*_Z^2)*RootOf(_Z^3-2)^3*x^3-18*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2
*x^3-30*(x^3-1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^2*x-RootOf(_Z^3-2)^2
*(x^3-1)^(1/3)*x^2+24*(x^3-1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)*x^2+6*
RootOf(_Z^3-2)*x^3-18*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^3-8*x*(x^3-1)^(2/3)-4*RootOf(_Z^3
-2)+12*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^3-2))*RootOf(_Z^3-2)-3/4*ln((6*RootOf(RootOf(_
Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^3*x^3-18*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_
Z^2)^2*RootOf(_Z^3-2)^2*x^3-30*(x^3-1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-
2)^2*x-RootOf(_Z^3-2)^2*(x^3-1)^(1/3)*x^2+24*(x^3-1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2
)*RootOf(_Z^3-2)*x^2+6*RootOf(_Z^3-2)*x^3-18*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^3-8*x*(x^3
-1)^(2/3)-4*RootOf(_Z^3-2)+12*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^3-2))*RootOf(RootOf(_Z^
3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^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 )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^3 + 1)*(x^3 - 1)^(2/3)/((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+1\right )}{x^6\,\left (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 + 1))/(x^6*(x^3 - 2)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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