\(\int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx\) [145]

Optimal result

Integrand size = 26, antiderivative size = 361 \[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=-\frac {\arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{6 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{9\ 2^{2/3} 3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{18 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{36 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}} \]

[Out]

-1/216*(-1)^(2/3)*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(2/3)*3^(1/3)/(1+(-1)^(1/3))^2+1/648*(-1)^(2/3)*ln(6+3*(-
2)^(2/3)*3^(1/3)*x+x^2)*2^(2/3)*3^(1/3)+1/648*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)*2^(2/3)*3^(1/3)-1/36*arctan((3*(-3
)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2))*2^(5/6)*3^(1/6)/(1+(-1)^(1/3))^2/(4-3*(-3)^(2/3)*2^(1/3
))^(1/2)+1/108*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*2^(5/6)*3^(1/6)/(-4+3*2^(1
/3)*3^(2/3))^(1/2)+1/54*(-1)^(1/3)*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*2^(1/3)
*3^(1/6)/(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(1/2)

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2122, 648, 632, 210, 642, 212} \[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=-\frac {\arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{6 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{9\ 2^{2/3} 3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{18 \sqrt [6]{2} 3^{5/6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}-\frac {(-1)^{2/3} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{36 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{108 \sqrt [3]{2} 3^{2/3}}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{108 \sqrt [3]{2} 3^{2/3}} \]

[In]

Int[x^3/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]

[Out]

-1/6*ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]]/(2^(1/6)*3^(5/6)*(1 + (-1)^(1/3))
^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ((-1)^(1/3)*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*
3^(2/3))]])/(9*2^(2/3)*3^(5/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3)]) + ArcTanh[(2^(1/6)*(3*3^(1
/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(18*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) - ((-1)^
(2/3)*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(36*2^(1/3)*3^(2/3)*(1 + (-1)^(1/3))^2) + ((-1)^(2/3)*Log[6 + 3*(
-2)^(2/3)*3^(1/3)*x + x^2])/(108*2^(1/3)*3^(2/3)) + Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2]/(108*2^(1/3)*3^(2/3))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

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 642

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 648

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 2122

Int[(Q6_)^(p_)*(u_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coe
ff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Dist[1/(3^(3*p)*a^(2*p)), Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c,
3]*x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3*(-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x
+ b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] &&
EqQ[Coeff[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]

Rubi steps \begin{align*} \text {integral}& = 1259712 \int \left (\frac {(-1)^{2/3} x}{22674816 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )}-\frac {(-1)^{2/3} x}{22674816 \sqrt [3]{2} 3^{2/3} \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {x}{68024448 \sqrt [3]{2} 3^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{54 \sqrt [3]{2} 3^{2/3}}+\frac {(-1)^{2/3} \int \frac {x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{54 \sqrt [3]{2} 3^{2/3}}+\frac {(-1)^{2/3} \int \frac {x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{18 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2} \\ & = \frac {\sqrt [3]{-\frac {1}{3}} \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{18\ 2^{2/3}}+\frac {\int \frac {3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{108 \sqrt [3]{2} 3^{2/3}}+\frac {(-1)^{2/3} \int \frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{108 \sqrt [3]{2} 3^{2/3}}-\frac {\int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{18\ 2^{2/3} \sqrt [3]{3}}-\frac {(-1)^{2/3} \int \frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{36 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2} \\ & = -\frac {(-1)^{2/3} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{36 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}}-\frac {\sqrt [3]{-\frac {1}{3}} \text {Subst}\left (\int \frac {1}{-6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )-x^2} \, dx,x,3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )}{9\ 2^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{-6 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )-x^2} \, dx,x,3\ 2^{2/3} \sqrt [3]{3}+2 x\right )}{9\ 2^{2/3} \sqrt [3]{3}}+\frac {\text {Subst}\left (\int \frac {1}{-6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )-x^2} \, dx,x,3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{3\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2} \\ & = -\frac {\tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{6 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\sqrt [3]{-1} \tan ^{-1}\left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{18 \sqrt [6]{2} 3^{5/6} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{18 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{36 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{108 \sqrt [3]{2} 3^{2/3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.17 \[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^2}{36+162 \text {$\#$1}+12 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \]

[In]

Integrate[x^3/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]

[Out]

RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (Log[x - #1]*#1^2)/(36 + 162*#1 + 12*#1^2 + #1^4) & ]/6

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.16

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\)	\(56\)
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\)	\(56\)

[In]

int(x^3/(x^6+18*x^4+324*x^3+108*x^2+216),x,method=_RETURNVERBOSE)

[Out]

1/6*sum(_R^3/(_R^5+12*_R^3+162*_R^2+36*_R)*ln(x-_R),_R=RootOf(_Z^6+18*_Z^4+324*_Z^3+108*_Z^2+216))

Fricas [F(-1)]

Timed out. \[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Timed out} \]

[In]

integrate(x^3/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="fricas")

[Out]

Timed out

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.17 \[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (3390158631679488 t^{6} - 74384733888 t^{4} - 1332145440 t^{3} - 1417176 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {8482372214243328 t^{5}}{415817} + \frac {2216055910930560 t^{4}}{415817} - \frac {2062546612992 t^{3}}{415817} - \frac {57027208896 t^{2}}{415817} - \frac {416583756 t}{415817} + x - \frac {89938}{415817} \right )} \right )\right )} \]

[In]

integrate(x**3/(x**6+18*x**4+324*x**3+108*x**2+216),x)

[Out]

RootSum(3390158631679488*_t**6 - 74384733888*_t**4 - 1332145440*_t**3 - 1417176*_t**2 - 1, Lambda(_t, _t*log(-
8482372214243328*_t**5/415817 + 2216055910930560*_t**4/415817 - 2062546612992*_t**3/415817 - 57027208896*_t**2
/415817 - 416583756*_t/415817 + x - 89938/415817)))

Maxima [F]

\[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{3}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]

[In]

integrate(x^3/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima")

[Out]

integrate(x^3/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)

Giac [F]

\[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x^{3}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \]

[In]

integrate(x^3/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="giac")

[Out]

integrate(x^3/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.76 \[ \int \frac {x^3}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\sum _{k=1}^6\ln \left (-\frac {23328\,\left (297538935552\,x-7992872640\,x\,\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )+52488\,x\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^3+2904\,x\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^4+x\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^5-153055008\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^2-2764368\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^3-1620\,{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^4-3673320192\,\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )[phone]\right )}{{\mathrm {root}\left (z^6+1417176\,z^4+1332145440\,z^3+74384733888\,z^2-3390158631679488,z,k\right )}^5}\right )\,\mathrm {root}\left (z^6-\frac {z^4}{45576}-\frac {235\,z^3}{598048272}-\frac {z^2}{2392193088}-\frac {1}{3390158631679488},z,k\right ) \]

[In]

int(x^3/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216),x)

[Out]

symsum(log(-(23328*(297538935552*x - 7992872640*x*root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 -
3390158631679488, z, k) + 52488*x*root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 3390158631679488
, z, k)^3 + 2904*x*root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 3390158631679488, z, k)^4 + x*r
oot(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 3390158631679488, z, k)^5 - 153055008*root(z^6 + 14
17176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 3390158631679488, z, k)^2 - 2764368*root(z^6 + 1417176*z^4 + 13
32145440*z^3 + 74384733888*z^2 - 3390158631679488, z, k)^3 - 1620*root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74
384733888*z^2 - 3390158631679488, z, k)^4 - 3673320192*root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z
^2 - 3390158631679488, z, k) + 7240114098432))/root(z^6 + 1417176*z^4 + 1332145440*z^3 + 74384733888*z^2 - 339
0158631679488, z, k)^5)*root(z^6 - z^4/45576 - (235*z^3)/598048272 - z^2/2392193088 - 1/3390158631679488, z, k
), k, 1, 6)