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

Optimal. Leaf size=102 \[ \frac {5}{3} \log \left (\sqrt [3]{x^3-2}+x\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-2}-x}\right )}{\sqrt {3}}+\frac {\left (x^3-2\right )^{2/3} \left (21 x^3+8\right )}{10 x^5}-\frac {5}{6} \log \left (-\sqrt [3]{x^3-2} x+\left (x^3-2\right )^{2/3}+x^2\right ) \]

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Rubi [A]  time = 0.14, antiderivative size = 111, normalized size of antiderivative = 1.09, 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, 618, 204, 628} \begin {gather*} \frac {5}{3} \log \left (\frac {x}{\sqrt [3]{x^3-2}}+1\right )-\frac {5 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-2}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {4 \left (x^3-2\right )^{2/3}}{5 x^5}+\frac {21 \left (x^3-2\right )^{2/3}}{10 x^2}-\frac {5}{6} \log \left (-\frac {x}{\sqrt [3]{x^3-2}}+\frac {x^2}{\left (x^3-2\right )^{2/3}}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-2 + x^3)^(2/3))/(5*x^5) + (21*(-2 + x^3)^(2/3))/(10*x^2) - (5*ArcTan[(1 - (2*x)/(-2 + x^3)^(1/3))/Sqrt[3]
])/Sqrt[3] - (5*Log[1 + x^2/(-2 + x^3)^(2/3) - x/(-2 + x^3)^(1/3)])/6 + (5*Log[1 + x/(-2 + x^3)^(1/3)])/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 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]

Rubi steps

\begin {align*} \int \frac {\left (-2+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-1+x^3\right )} \, dx &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {42-17 x^3}{x^3 \sqrt [3]{-2+x^3} \left (-1+x^3\right )} \, dx\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}+\frac {1}{20} \int -\frac {100}{\sqrt [3]{-2+x^3} \left (-1+x^3\right )} \, dx\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}-5 \int \frac {1}{\sqrt [3]{-2+x^3} \left (-1+x^3\right )} \, dx\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}-5 \operatorname {Subst}\left (\int \frac {1}{-1-x^3} \, dx,x,\frac {x}{\sqrt [3]{-2+x^3}}\right )\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}-\frac {5}{3} \operatorname {Subst}\left (\int \frac {1}{-1-x} \, dx,x,\frac {x}{\sqrt [3]{-2+x^3}}\right )-\frac {5}{3} \operatorname {Subst}\left (\int \frac {-2+x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-2+x^3}}\right )\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}+\frac {5}{3} \log \left (1+\frac {x}{\sqrt [3]{-2+x^3}}\right )-\frac {5}{6} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-2+x^3}}\right )+\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-2+x^3}}\right )\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}-\frac {5}{6} \log \left (1+\frac {x^2}{\left (-2+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-2+x^3}}\right )+\frac {5}{3} \log \left (1+\frac {x}{\sqrt [3]{-2+x^3}}\right )-5 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x}{\sqrt [3]{-2+x^3}}\right )\\ &=\frac {4 \left (-2+x^3\right )^{2/3}}{5 x^5}+\frac {21 \left (-2+x^3\right )^{2/3}}{10 x^2}-\frac {5 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{-2+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {5}{6} \log \left (1+\frac {x^2}{\left (-2+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-2+x^3}}\right )+\frac {5}{3} \log \left (1+\frac {x}{\sqrt [3]{-2+x^3}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 111, normalized size = 1.09 \begin {gather*} \frac {\left (x^3-2\right )^{2/3} \left (21 x^3+8\right )}{10 x^5}+\frac {5}{6} \left (2 \log \left (\frac {x}{\sqrt [3]{1-2 x^3}}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{1-2 x^3}}-1}{\sqrt {3}}\right )-\log \left (-\frac {x}{\sqrt [3]{1-2 x^3}}+\frac {x^2}{\left (1-2 x^3\right )^{2/3}}+1\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2 + x^3)^(2/3)*(8 + 21*x^3))/(10*x^5) + (5*(2*Sqrt[3]*ArcTan[(-1 + (2*x)/(1 - 2*x^3)^(1/3))/Sqrt[3]] - Log[
1 + x^2/(1 - 2*x^3)^(2/3) - x/(1 - 2*x^3)^(1/3)] + 2*Log[1 + x/(1 - 2*x^3)^(1/3)]))/6

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IntegrateAlgebraic [A]  time = 0.19, size = 102, normalized size = 1.00 \begin {gather*} \frac {\left (-2+x^3\right )^{2/3} \left (8+21 x^3\right )}{10 x^5}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-2+x^3}}\right )}{\sqrt {3}}+\frac {5}{3} \log \left (x+\sqrt [3]{-2+x^3}\right )-\frac {5}{6} \log \left (x^2-x \sqrt [3]{-2+x^3}+\left (-2+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2 + x^3)^(2/3)*(8 + 21*x^3))/(10*x^5) + (5*ArcTan[(Sqrt[3]*x)/(-x + 2*(-2 + x^3)^(1/3))])/Sqrt[3] + (5*Log[
x + (-2 + x^3)^(1/3)])/3 - (5*Log[x^2 - x*(-2 + x^3)^(1/3) + (-2 + x^3)^(2/3)])/6

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fricas [A]  time = 1.13, size = 122, normalized size = 1.20 \begin {gather*} -\frac {50 \, \sqrt {3} x^{5} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} - 2\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} - 2\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} - 2\right )}}{7 \, x^{3} + 2}\right ) - 25 \, x^{5} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} - 2\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 2\right )}^{\frac {2}{3}} x - 2}{x^{3} - 1}\right ) - 3 \, {\left (21 \, x^{3} + 8\right )} {\left (x^{3} - 2\right )}^{\frac {2}{3}}}{30 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(50*sqrt(3)*x^5*arctan((4*sqrt(3)*(x^3 - 2)^(1/3)*x^2 + 2*sqrt(3)*(x^3 - 2)^(2/3)*x + sqrt(3)*(x^3 - 2))
/(7*x^3 + 2)) - 25*x^5*log((2*x^3 + 3*(x^3 - 2)^(1/3)*x^2 + 3*(x^3 - 2)^(2/3)*x - 2)/(x^3 - 1)) - 3*(21*x^3 +
8)*(x^3 - 2)^(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} + 4\right )} {\left (x^{3} - 2\right )}^{\frac {2}{3}}}{{\left (x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 4.88, size = 338, normalized size = 3.31

method	result	size
trager	\(\frac {\left (x^{3}-2\right )^{\frac {2}{3}} \left (21 x^{3}+8\right )}{10 x^{5}}+\frac {5 \ln \left (-\frac {86436864 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+1075968 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x +6592608 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}+3456576 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}-57465 \left (x^{3}-2\right )^{\frac {2}{3}} x +11208 \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-6040 x^{3}-691494912 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-4242624 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+9060}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}+160 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \ln \left (-\frac {13916160 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}-1075968 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x +5516640 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-1335264 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}-68673 \left (x^{3}-2\right )^{\frac {2}{3}} x -11208 \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}+28137 x^{3}-111329280 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}+6913152 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+18758}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )\)	\(338\)
risch	\(\frac {21 x^{6}-34 x^{3}-16}{10 x^{5} \left (x^{3}-2\right )^{\frac {1}{3}}}-\frac {5 \ln \left (-\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 \left (x^{3}-2\right )^{\frac {2}{3}} x -2 \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-x^{3}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}-5 \ln \left (-\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 \left (x^{3}-2\right )^{\frac {2}{3}} x -2 \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-x^{3}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+5 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-2\right )^{\frac {1}{3}} x^{2}-\left (x^{3}-2\right )^{\frac {2}{3}} x +\left (x^{3}-2\right )^{\frac {1}{3}} x^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )\)	\(438\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(x^3-2)^(2/3)*(21*x^3+8)/x^5+5/3*ln(-(86436864*RootOf(9216*_Z^2+96*_Z+1)^2*x^3+1075968*RootOf(9216*_Z^2+9
6*_Z+1)*(x^3-2)^(2/3)*x+6592608*RootOf(9216*_Z^2+96*_Z+1)*(x^3-2)^(1/3)*x^2+3456576*RootOf(9216*_Z^2+96*_Z+1)*
x^3-57465*(x^3-2)^(2/3)*x+11208*(x^3-2)^(1/3)*x^2-6040*x^3-691494912*RootOf(9216*_Z^2+96*_Z+1)^2-4242624*RootO
f(9216*_Z^2+96*_Z+1)+9060)/(-1+x)/(x^2+x+1))+160*RootOf(9216*_Z^2+96*_Z+1)*ln(-(13916160*RootOf(9216*_Z^2+96*_
Z+1)^2*x^3-1075968*RootOf(9216*_Z^2+96*_Z+1)*(x^3-2)^(2/3)*x+5516640*RootOf(9216*_Z^2+96*_Z+1)*(x^3-2)^(1/3)*x
^2-1335264*RootOf(9216*_Z^2+96*_Z+1)*x^3-68673*(x^3-2)^(2/3)*x-11208*(x^3-2)^(1/3)*x^2+28137*x^3-111329280*Roo
tOf(9216*_Z^2+96*_Z+1)^2+6913152*RootOf(9216*_Z^2+96*_Z+1)+18758)/(-1+x)/(x^2+x+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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