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

Optimal. Leaf size=103 \[ \frac {1}{3} \text {RootSum}\left [2 \text {$\#$1}^6-5 \text {$\#$1}^3+1\& ,\frac {-2 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+2 \text {$\#$1}^3 \log (x)+\log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-\log (x)}{4 \text {$\#$1}^4-5 \text {$\#$1}}\& \right ]-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \]

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Rubi [B]  time = 1.02, antiderivative size = 431, normalized size of antiderivative = 4.18, number of steps used = 22, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {6728, 277, 239, 1518, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \log \left (\sqrt [3]{1+\sqrt {17}}-\frac {\sqrt [3]{\sqrt {17}-3} x}{\sqrt [3]{x^3+1}}\right )}{3 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\sqrt [3]{\sqrt {17}-1}-\frac {\sqrt [3]{3+\sqrt {17}} x}{\sqrt [3]{x^3+1}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2 \left (5-\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {51}}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2}+\frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \log \left (\frac {\sqrt [3]{2 \left (7-\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+\frac {\left (\sqrt {17}-3\right )^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (1+\sqrt {17}\right )^{2/3}\right )}{6 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\frac {\sqrt [3]{2 \left (7+\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+\frac {\left (3+\sqrt {17}\right )^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (\sqrt {17}-1\right )^{2/3}\right )}{6 \sqrt {17}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(1 + x^3)^(2/3)/x^2 + (((-19 + 5*Sqrt[17])/2)^(1/3)*ArcTan[(1 + ((2*(5 - Sqrt[17]))^(1/3)*x)/(1 + x^3)^(1
/3))/Sqrt[3]])/Sqrt[51] + (((19 + 5*Sqrt[17])/2)^(1/3)*ArcTan[(1 + ((2*(5 + Sqrt[17]))^(1/3)*x)/(1 + x^3)^(1/3
))/Sqrt[3]])/Sqrt[51] + (((-19 + 5*Sqrt[17])/2)^(1/3)*Log[(1 + Sqrt[17])^(2/3) + ((-3 + Sqrt[17])^(2/3)*x^2)/(
1 + x^3)^(2/3) + ((2*(7 - Sqrt[17]))^(1/3)*x)/(1 + x^3)^(1/3)])/(6*Sqrt[17]) - (((-19 + 5*Sqrt[17])/2)^(1/3)*L
og[(1 + Sqrt[17])^(1/3) - ((-3 + Sqrt[17])^(1/3)*x)/(1 + x^3)^(1/3)])/(3*Sqrt[17]) - (((19 + 5*Sqrt[17])/2)^(1
/3)*Log[(-1 + Sqrt[17])^(1/3) - ((3 + Sqrt[17])^(1/3)*x)/(1 + x^3)^(1/3)])/(3*Sqrt[17]) + (((19 + 5*Sqrt[17])/
2)^(1/3)*Log[(-1 + Sqrt[17])^(2/3) + ((3 + Sqrt[17])^(2/3)*x^2)/(1 + x^3)^(2/3) + ((2*(7 + Sqrt[17]))^(1/3)*x)
/(1 + x^3)^(1/3)])/(6*Sqrt[17])

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 239

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

Rule 277

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

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 1518

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol
] :> Dist[(e*f^n)/c, Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1), x], x] - Dist[f^n/c, Int[((f*x)^(m - n)*(d + e*x^n
)^(q - 1)*Simp[a*e - (c*d - b*e)*x^n, x])/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && E
qQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !IntegerQ[q] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2*n
- 1]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-2+x^3+2 x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{x^3}-\frac {2 x^3 \left (1+x^3\right )^{2/3}}{-2+x^3+2 x^6}\right ) \, dx\\ &=-\left (2 \int \frac {x^3 \left (1+x^3\right )^{2/3}}{-2+x^3+2 x^6} \, dx\right )+\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \frac {-2-x^3}{\sqrt [3]{1+x^3} \left (-2+x^3+2 x^6\right )} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \left (\frac {-1-\frac {7}{\sqrt {17}}}{\sqrt [3]{1+x^3} \left (1-\sqrt {17}+4 x^3\right )}+\frac {-1+\frac {7}{\sqrt {17}}}{\sqrt [3]{1+x^3} \left (1+\sqrt {17}+4 x^3\right )}\right ) \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{17} \left (-17+7 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (1+\sqrt {17}+4 x^3\right )} \, dx-\frac {1}{17} \left (17+7 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (1-\sqrt {17}+4 x^3\right )} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{17} \left (-17+7 \sqrt {17}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {17}-\left (-3+\sqrt {17}\right ) x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{17} \left (17+7 \sqrt {17}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {17}-\left (-3-\sqrt {17}\right ) x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {\sqrt [3]{71+17 \sqrt {17}} \operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{-1+\sqrt {17}}+\sqrt [3]{3+\sqrt {17}} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}-\frac {\sqrt [3]{71+17 \sqrt {17}} \operatorname {Subst}\left (\int \frac {-2 \sqrt [3]{-1+\sqrt {17}}-\sqrt [3]{3+\sqrt {17}} x}{\left (-1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )} x+\left (3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{-289+71 \sqrt {17}} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{-3+\sqrt {17}} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}+\frac {\sqrt [3]{-289+71 \sqrt {17}} \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{1+\sqrt {17}}+\sqrt [3]{-3+\sqrt {17}} x}{\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )} x+\left (-3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\sqrt [3]{1+\sqrt {17}}-\frac {\sqrt [3]{-3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\sqrt [3]{-1+\sqrt {17}}-\frac {\sqrt [3]{3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{459-109 \sqrt {17}} \operatorname {Subst}\left (\int \frac {1}{\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )} x+\left (-3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{34^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )}+2 \left (-3+\sqrt {17}\right )^{2/3} x}{\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )} x+\left (-3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{6\ 17^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}+2 \left (3+\sqrt {17}\right )^{2/3} x}{\left (-1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )} x+\left (3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{6 \sqrt {17}}+\frac {\sqrt [3]{109+27 \sqrt {17}} \operatorname {Subst}\left (\int \frac {1}{\left (-1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )} x+\left (3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{2^{2/3} \sqrt {17}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\left (1+\sqrt {17}\right )^{2/3}+\frac {\left (-3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7-\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\sqrt [3]{1+\sqrt {17}}-\frac {\sqrt [3]{-3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\sqrt [3]{-1+\sqrt {17}}-\frac {\sqrt [3]{3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\left (-1+\sqrt {17}\right )^{2/3}+\frac {\left (3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {\sqrt [3]{10-2 \sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt {17}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2 \left (5-\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3} 17^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\left (1+\sqrt {17}\right )^{2/3}+\frac {\left (-3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7-\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\sqrt [3]{1+\sqrt {17}}-\frac {\sqrt [3]{-3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\sqrt [3]{-1+\sqrt {17}}-\frac {\sqrt [3]{3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\left (-1+\sqrt {17}\right )^{2/3}+\frac {\left (3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6 \sqrt {17}}\\ \end {align*}

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Mathematica [B]  time = 0.60, size = 373, normalized size = 3.62 \begin {gather*} \frac {-2 \sqrt [3]{19+5 \sqrt {17}} \log \left (\sqrt [3]{5-\sqrt {17}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}\right )-2 \sqrt [3]{5 \sqrt {17}-19} \log \left (\sqrt [3]{5+\sqrt {17}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}\right )+2 \sqrt {3} \sqrt [3]{5 \sqrt {17}-19} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+2 \sqrt {3} \sqrt [3]{19+5 \sqrt {17}} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\sqrt [3]{19+5 \sqrt {17}} \log \left (\frac {\sqrt [3]{10-2 \sqrt {17}} x}{\sqrt [3]{x^3+1}}+\frac {2^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (5-\sqrt {17}\right )^{2/3}\right )+\sqrt [3]{5 \sqrt {17}-19} \log \left (\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+\frac {2^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (5+\sqrt {17}\right )^{2/3}\right )}{6 \sqrt [3]{2} \sqrt {17}}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(1 + x^3)^(2/3)/x^2 + (2*Sqrt[3]*(-19 + 5*Sqrt[17])^(1/3)*ArcTan[(1 + (2*(2/(5 + Sqrt[17]))^(1/3)*x)/(1 +
x^3)^(1/3))/Sqrt[3]] + 2*Sqrt[3]*(19 + 5*Sqrt[17])^(1/3)*ArcTan[(1 + ((2*(5 + Sqrt[17]))^(1/3)*x)/(1 + x^3)^(
1/3))/Sqrt[3]] - 2*(19 + 5*Sqrt[17])^(1/3)*Log[(5 - Sqrt[17])^(1/3) - (2^(1/3)*x)/(1 + x^3)^(1/3)] - 2*(-19 +
5*Sqrt[17])^(1/3)*Log[(5 + Sqrt[17])^(1/3) - (2^(1/3)*x)/(1 + x^3)^(1/3)] + (19 + 5*Sqrt[17])^(1/3)*Log[(5 - S
qrt[17])^(2/3) + (2^(2/3)*x^2)/(1 + x^3)^(2/3) + ((10 - 2*Sqrt[17])^(1/3)*x)/(1 + x^3)^(1/3)] + (-19 + 5*Sqrt[
17])^(1/3)*Log[(5 + Sqrt[17])^(2/3) + (2^(2/3)*x^2)/(1 + x^3)^(2/3) + ((2*(5 + Sqrt[17]))^(1/3)*x)/(1 + x^3)^(
1/3)])/(6*2^(1/3)*Sqrt[17])

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IntegrateAlgebraic [A]  time = 0.00, size = 103, normalized size = 1.00 \begin {gather*} -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{3} \text {RootSum}\left [1-5 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(1 + x^3)^(2/3)/x^2 + RootSum[1 - 5*#1^3 + 2*#1^6 & , (-Log[x] + Log[(1 + x^3)^(1/3) - x*#1] + 2*Log[x]*#
1^3 - 2*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/3

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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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 (2 \, x^{6} + x^{3} - 2\right )} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 193.92, size = 6665, normalized size = 64.71

method	result	size
risch	\(\text {Expression too large to display}\)	\(6665\)
trager	\(\text {Expression too large to display}\)	\(8758\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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 (2 \, x^{6} + x^{3} - 2\right )} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

Timed out

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