\(\int \frac {(-1+x^3)^{2/3} (4+x^6)}{x^6 (4+2 x^3+x^6)} \, dx\) [1671]

Optimal result

Integrand size = 30, antiderivative size = 112 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-4+9 x^3\right )}{20 x^5}-\frac {1}{12} \text {RootSum}\left [7-10 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-7 \log (x)+7 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.43 (sec) , antiderivative size = 679, normalized size of antiderivative = 6.06, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6860, 270, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\frac {1}{12} \left (\sqrt {3}+i\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\frac {1}{12} \left (-\sqrt {3}+i\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\left (\sqrt {3}+i\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}} \sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{12 \left (\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}\right )^{2/3}}+\frac {\left (-\sqrt {3}+i\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+2 i}} \sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{12 \left (\frac {\sqrt {3}+i}{\sqrt {3}+2 i}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (2 x^3+2 \left (1-i \sqrt {3}\right )\right )}{72 \left (\frac {\sqrt {3}+i}{\sqrt {3}+2 i}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (2 x^3+2 \left (1+i \sqrt {3}\right )\right )}{72 \left (\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (-\sqrt [3]{x^3-1}+\frac {x}{\sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}}}\right )}{24 \left (\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (-\sqrt [3]{x^3-1}+\frac {x}{\sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+2 i}}}\right )}{24 \left (\frac {\sqrt {3}+i}{\sqrt {3}+2 i}\right )^{2/3}}-\frac {1}{24} \left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {1}{24} \left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {1}{4} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{4 x^2} \]

[In]

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

[Out]

(-1 + x^3)^(2/3)/(4*x^2) + (-1 + x^3)^(5/3)/(5*x^5) - ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(2*Sqrt[3])
- ((I - Sqrt[3])*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/12 + ((I + Sqrt[3])*ArcTan[(1 + (2*x)/(-1 + x^
3)^(1/3))/Sqrt[3]])/12 - ((I + Sqrt[3])*ArcTan[(1 + (2*x)/(((I - Sqrt[3])/(2*I - Sqrt[3]))^(1/3)*(-1 + x^3)^(1
/3)))/Sqrt[3]])/(12*((I - Sqrt[3])/(2*I - Sqrt[3]))^(2/3)) + ((I - Sqrt[3])*ArcTan[(1 + (2*x)/(((I + Sqrt[3])/
(2*I + Sqrt[3]))^(1/3)*(-1 + x^3)^(1/3)))/Sqrt[3]])/(12*((I + Sqrt[3])/(2*I + Sqrt[3]))^(2/3)) - ((3 - I*Sqrt[
3])*Log[2*(1 - I*Sqrt[3]) + 2*x^3])/(72*((I + Sqrt[3])/(2*I + Sqrt[3]))^(2/3)) - ((3 + I*Sqrt[3])*Log[2*(1 + I
*Sqrt[3]) + 2*x^3])/(72*((I - Sqrt[3])/(2*I - Sqrt[3]))^(2/3)) + ((3 + I*Sqrt[3])*Log[x/((I - Sqrt[3])/(2*I -
Sqrt[3]))^(1/3) - (-1 + x^3)^(1/3)])/(24*((I - Sqrt[3])/(2*I - Sqrt[3]))^(2/3)) + ((3 - I*Sqrt[3])*Log[x/((I +
Sqrt[3])/(2*I + Sqrt[3]))^(1/3) - (-1 + x^3)^(1/3)])/(24*((I + Sqrt[3])/(2*I + Sqrt[3]))^(2/3)) + Log[-x + (-
1 + x^3)^(1/3)]/4 - ((3 - I*Sqrt[3])*Log[-x + (-1 + x^3)^(1/3)])/24 - ((3 + I*Sqrt[3])*Log[-x + (-1 + x^3)^(1/
3)])/24

Rule 245

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 270

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 283

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 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 399

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

Rule 6860

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*} \text {integral}& = \int \left (\frac {\left (-1+x^3\right )^{2/3}}{x^6}-\frac {\left (-1+x^3\right )^{2/3}}{2 x^3}+\frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{2 \left (4+2 x^3+x^6\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\right )+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{4+2 x^3+x^6} \, dx+\int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{2} \int \left (\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3}+\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3}\right ) \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3} \, dx+\frac {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{12} \left (3-i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{12} \left (3+i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{6} \left (-3+5 i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (2-2 i \sqrt {3}+2 x^3\right )} \, dx-\frac {1}{6} \left (3+5 i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (2+2 i \sqrt {3}+2 x^3\right )} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{12} \left (i-\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {1}{12} \left (i+\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-\frac {\left (5-i \sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {i-\sqrt {3}}{2 i-\sqrt {3}}} \sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{12 \left (-i+\sqrt {3}\right )^{2/3} \sqrt [3]{-2 i+\sqrt {3}}}-\frac {\left (5+i \sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {i+\sqrt {3}}{2 i+\sqrt {3}}} \sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{12 \left (i+\sqrt {3}\right )^{2/3} \sqrt [3]{2 i+\sqrt {3}}}-\frac {\left (3 i+5 \sqrt {3}\right ) \log \left (2 \left (1-i \sqrt {3}\right )+2 x^3\right )}{72 \left (i+\sqrt {3}\right )^{2/3} \sqrt [3]{2 i+\sqrt {3}}}+\frac {\left (3 i-5 \sqrt {3}\right ) \log \left (2 \left (1+i \sqrt {3}\right )+2 x^3\right )}{72 \left (-i+\sqrt {3}\right )^{2/3} \sqrt [3]{-2 i+\sqrt {3}}}-\frac {\left (3 i-5 \sqrt {3}\right ) \log \left (\frac {x}{\sqrt [3]{\frac {i-\sqrt {3}}{2 i-\sqrt {3}}}}-\sqrt [3]{-1+x^3}\right )}{24 \left (-i+\sqrt {3}\right )^{2/3} \sqrt [3]{-2 i+\sqrt {3}}}+\frac {\left (3 i+5 \sqrt {3}\right ) \log \left (\frac {x}{\sqrt [3]{\frac {i+\sqrt {3}}{2 i+\sqrt {3}}}}-\sqrt [3]{-1+x^3}\right )}{24 \left (i+\sqrt {3}\right )^{2/3} \sqrt [3]{2 i+\sqrt {3}}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{24} \left (3-i \sqrt {3}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{24} \left (3+i \sqrt {3}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-4+9 x^3\right )}{20 x^5}-\frac {1}{12} \text {RootSum}\left [7-10 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-7 \log (x)+7 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]

[In]

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

[Out]

((-1 + x^3)^(2/3)*(-4 + 9*x^3))/(20*x^5) - RootSum[7 - 10*#1^3 + 4*#1^6 & , (-7*Log[x] + 7*Log[(-1 + x^3)^(1/3
) - x*#1] + 6*Log[x]*#1^3 - 6*Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/12

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 1.

Time = 5.32 (sec) , antiderivative size = 1803, normalized size of antiderivative = 16.10

\[\text {Expression too large to display}\]

[In]

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

[Out]

1/20*(9*x^6-13*x^3+4)/x^5/(x^3-1)^(1/3)+1/72*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)*ln(-(13
226976*RootOf(314928*_Z^6-4860*_Z^3+49)^6*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)*x^3+28512*
(x^3-1)^(1/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^3*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)^2*x
^2-175608*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)*RootOf(314928*_Z^6-4860*_Z^3+49)^3*x^3-196
7328*(x^3-1)^(2/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^3*x-99*(x^3-1)^(1/3)*RootOf(_Z^3+46656*RootOf(314928*_Z^6-
4860*_Z^3+49)^3-720)^2*x^2-99792*RootOf(314928*_Z^6-4860*_Z^3+49)^3*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*
_Z^3+49)^3-720)-340*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)*x^3+13728*x*(x^3-1)^(2/3)+1496*R
ootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720))/(486*RootOf(314928*_Z^6-4860*_Z^3+49)^3*x^3-x^3+11))
+27/22*ln((8817984*RootOf(314928*_Z^6-4860*_Z^3+49)^6*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720
)*x^3+17820*(x^3-1)^(1/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^3*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49
)^3-720)^2*x^2-314280*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)*RootOf(314928*_Z^6-4860*_Z^3+4
9)^3*x^3-684288*(x^3-1)^(2/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^3*x-319*(x^3-1)^(1/3)*RootOf(_Z^3+46656*RootOf(
314928*_Z^6-4860*_Z^3+49)^3-720)^2*x^2+99792*RootOf(314928*_Z^6-4860*_Z^3+49)^3*RootOf(_Z^3+46656*RootOf(31492
8*_Z^6-4860*_Z^3+49)^3-720)+1416*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)*x^3+19800*x*(x^3-1)
^(2/3)-528*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720))/(486*RootOf(314928*_Z^6-4860*_Z^3+49)^3*
x^3-x^3+11))*RootOf(314928*_Z^6-4860*_Z^3+49)^3*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)-13/7
92*ln((8817984*RootOf(314928*_Z^6-4860*_Z^3+49)^6*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)*x^
3+17820*(x^3-1)^(1/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^3*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-
720)^2*x^2-314280*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)*RootOf(314928*_Z^6-4860*_Z^3+49)^3
*x^3-684288*(x^3-1)^(2/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^3*x-319*(x^3-1)^(1/3)*RootOf(_Z^3+46656*RootOf(3149
28*_Z^6-4860*_Z^3+49)^3-720)^2*x^2+99792*RootOf(314928*_Z^6-4860*_Z^3+49)^3*RootOf(_Z^3+46656*RootOf(314928*_Z
^6-4860*_Z^3+49)^3-720)+1416*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)*x^3+19800*x*(x^3-1)^(2/
3)-528*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720))/(486*RootOf(314928*_Z^6-4860*_Z^3+49)^3*x^3-
x^3+11))*RootOf(_Z^3+46656*RootOf(314928*_Z^6-4860*_Z^3+49)^3-720)+1/2*RootOf(314928*_Z^6-4860*_Z^3+49)*ln(-(6
613488*x^3*RootOf(314928*_Z^6-4860*_Z^3+49)^7-513216*(x^3-1)^(1/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^5*x^2-1163
16*RootOf(314928*_Z^6-4860*_Z^3+49)^4*x^3+27324*(x^3-1)^(2/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^3*x+6138*(x^3-1
)^(1/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^2*x^2+49896*RootOf(314928*_Z^6-4860*_Z^3+49)^4+50*RootOf(314928*_Z^6-
4860*_Z^3+49)*x^3-231*x*(x^3-1)^(2/3)-22*RootOf(314928*_Z^6-4860*_Z^3+49))/(972*RootOf(314928*_Z^6-4860*_Z^3+4
9)^3*x^3-13*x^3-22))-486/11*ln((13226976*x^3*RootOf(314928*_Z^6-4860*_Z^3+49)^7-962280*(x^3-1)^(1/3)*RootOf(31
4928*_Z^6-4860*_Z^3+49)^5*x^2+63180*RootOf(314928*_Z^6-4860*_Z^3+49)^4*x^3+28512*(x^3-1)^(2/3)*RootOf(314928*_
Z^6-4860*_Z^3+49)^3*x-2376*(x^3-1)^(1/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^2*x^2-149688*RootOf(314928*_Z^6-4860
*_Z^3+49)^4-2001*RootOf(314928*_Z^6-4860*_Z^3+49)*x^3+385*x*(x^3-1)^(2/3)+1518*RootOf(314928*_Z^6-4860*_Z^3+49
))/(972*RootOf(314928*_Z^6-4860*_Z^3+49)^3*x^3-13*x^3-22))*RootOf(314928*_Z^6-4860*_Z^3+49)^4+1/11*ln((1322697
6*x^3*RootOf(314928*_Z^6-4860*_Z^3+49)^7-962280*(x^3-1)^(1/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^5*x^2+63180*Roo
tOf(314928*_Z^6-4860*_Z^3+49)^4*x^3+28512*(x^3-1)^(2/3)*RootOf(314928*_Z^6-4860*_Z^3+49)^3*x-2376*(x^3-1)^(1/3
)*RootOf(314928*_Z^6-4860*_Z^3+49)^2*x^2-149688*RootOf(314928*_Z^6-4860*_Z^3+49)^4-2001*RootOf(314928*_Z^6-486
0*_Z^3+49)*x^3+385*x*(x^3-1)^(2/3)+1518*RootOf(314928*_Z^6-4860*_Z^3+49))/(972*RootOf(314928*_Z^6-4860*_Z^3+49
)^3*x^3-13*x^3-22))*RootOf(314928*_Z^6-4860*_Z^3+49)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+4\right )}{x^6\,\left (x^6+2\,x^3+4\right )} \,d x \]

[In]

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

[Out]

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