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

Optimal. Leaf size=677 \[ \frac {\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x+9 (-2)^{2/3}}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x+9\ 2^{2/3}}{13122\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{8748\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{5832 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{5832 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{52488 \sqrt [3]{2} 3^{2/3}}+\frac {\sqrt [3]{-1} \left (3 (-3)^{2/3}-2^{2/3}\right ) \tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{486\ 6^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}+\frac {\left (3 (-3)^{2/3}+\sqrt [3]{-1} 2^{2/3}\right ) \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{486\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{486\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )^{3/2}} \]

[Out]

1/5832*(9*(-2)^(2/3)+6^(1/3)*(9+(-3)^(1/3)*2^(2/3))*x)*2^(1/3)/(1+(-1)^(1/3))^4/(4-3*(-3)^(2/3)*2^(1/3))/(6-3*
(-3)^(1/3)*2^(2/3)*x+x^2)+1/26244*(9*2^(2/3)+(-1)^(1/3)*3^(2/3)*(2+3*(-2)^(1/3)*3^(2/3))*x)*2^(1/3)/(8+9*I*2^(
1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))/(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)+1/52488*(3*2^(2/3)*3^(1/3)-(2-3*2^(1/3)*3^(2/3)
)*x)*2^(1/3)*3^(2/3)/(4-3*2^(1/3)*3^(2/3))/(6+3*2^(2/3)*3^(1/3)*x+x^2)+1/2916*(-1)^(1/3)*(3*(-3)^(2/3)-2^(2/3)
)*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2))*6^(1/6)/(1+(-1)^(1/3))^4/(4-3*(-3)^(2/3)
*2^(1/3))^(3/2)+1/2916*(3*(-3)^(2/3)+(-1)^(1/3)*2^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3
^(2/3))^(1/2))*6^(1/6)/(1-(-1)^(1/3))^2/(1+(-1)^(1/3))^4/(4+3*(-2)^(1/3)*3^(2/3))^(3/2)-1/2916*(2^(2/3)-3*3^(2
/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*6^(1/6)/(1-(-1)^(1/3))^2/(1+(-1)^(1/
3))^4/(-4+3*2^(1/3)*3^(2/3))^(3/2)+1/34992*(-1)^(1/6)*3^(5/6)*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(2/3)/(1+(-1)
^(1/3))^5-1/34992*I*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)*2^(2/3)*3^(5/6)/(1+(-1)^(1/3))^5+1/314928*ln(6+3*2^(2/3)*
3^(1/3)*x+x^2)*2^(2/3)*3^(1/3)

________________________________________________________________________________________

Rubi [A]  time = 1.55, antiderivative size = 677, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2097, 638, 618, 204, 628, 206} \[ \frac {\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x+9 (-2)^{2/3}}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x+9\ 2^{2/3}}{13122\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{8748\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{5832 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{5832 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{52488 \sqrt [3]{2} 3^{2/3}}+\frac {\sqrt [3]{-1} \left (3 (-3)^{2/3}-2^{2/3}\right ) \tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{486\ 6^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}+\frac {\left (3 (-3)^{2/3}+\sqrt [3]{-1} 2^{2/3}\right ) \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{486\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{486\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*(-2)^(2/3) + 6^(1/3)*(9 + (-3)^(1/3)*2^(2/3))*x)/(2916*2^(2/3)*(1 + (-1)^(1/3))^4*(4 - 3*(-3)^(2/3)*2^(1/3)
)*(6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2)) + (9*2^(2/3) + (-1)^(1/3)*3^(2/3)*(2 + 3*(-2)^(1/3)*3^(2/3))*x)/(13122*2
^(2/3)*(8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))*(6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2)) + (3*2^(2/3)*3^(1/3
) - (2 - 3*2^(1/3)*3^(2/3))*x)/(8748*2^(2/3)*3^(1/3)*(4 - 3*2^(1/3)*3^(2/3))*(6 + 3*2^(2/3)*3^(1/3)*x + x^2))
+ ((-1)^(1/3)*(3*(-3)^(2/3) - 2^(2/3))*ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]]
)/(486*6^(5/6)*(1 + (-1)^(1/3))^4*(4 - 3*(-3)^(2/3)*2^(1/3))^(3/2)) + ((3*(-3)^(2/3) + (-1)^(1/3)*2^(2/3))*Arc
Tan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(486*6^(5/6)*(1 - (-1)^(1/3))^2*(1 + (-1
)^(1/3))^4*(4 + 3*(-2)^(1/3)*3^(2/3))^(3/2)) - ((2^(2/3) - 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))]])/(486*6^(5/6)*(1 - (-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(-4 + 3*2^(1/3)*3^(2/3
))^(3/2)) + ((-1/3)^(1/6)*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(5832*2^(1/3)*(1 + (-1)^(1/3))^5) - ((I/5832)
*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(2^(1/3)*3^(1/6)*(1 + (-1)^(1/3))^5) + Log[6 + 3*2^(2/3)*3^(1/3)*x + x
^2]/(52488*2^(1/3)*3^(2/3))

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 206

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

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 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 2097

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*} \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx &=1586874322944 \int \left (-\frac {-2 \sqrt [3]{-1} 3^{2/3}+3 (-2)^{2/3} x}{1542441841901568\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2}+\frac {-3 i 3^{5/6}+\left (\sqrt [3]{-2}+\sqrt [3]{2}\right ) x}{4627325525704704\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}-\frac {-2 \sqrt [3]{-1} 3^{2/3}+3\ 2^{2/3} x}{1542441841901568\ 2^{2/3} \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2}+\frac {3+3 \sqrt [3]{-1}-i \sqrt [3]{2} \sqrt [6]{3} x}{4627325525704704\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {2+2^{2/3} \sqrt [3]{3} x}{514147280633856\ 2^{2/3} \sqrt [3]{3} \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2}+\frac {3 \sqrt [3]{3}+\sqrt [3]{2} x}{41645929731342336\ 6^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=-\frac {\int \frac {-2 \sqrt [3]{-1} 3^{2/3}+3\ 2^{2/3} x}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{8748\ 2^{2/3}}-\frac {\int \frac {2+2^{2/3} \sqrt [3]{3} x}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{2916\ 2^{2/3} \sqrt [3]{3}}+\frac {\int \frac {3 \sqrt [3]{3}+\sqrt [3]{2} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{26244\ 6^{2/3}}+\frac {\int \frac {3+3 \sqrt [3]{-1}-i \sqrt [3]{2} \sqrt [6]{3} x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{2916\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\int \frac {-3 i 3^{5/6}+\left (\sqrt [3]{-2}+\sqrt [3]{2}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{2916\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\int \frac {-2 \sqrt [3]{-1} 3^{2/3}+3 (-2)^{2/3} x}{\left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2} \, dx}{972\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}\\ &=\frac {9 (-2)^{2/3}+\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {9\ 2^{2/3}+\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{26244\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{8748\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{5832 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{5832 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{52488 \sqrt [3]{2} 3^{2/3}}-\frac {\left (-18 \sqrt [3]{-6} (-1)^{2/3}+4 \sqrt [3]{-1} 3^{2/3}\right ) \int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{972\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (24-18 (-3)^{2/3} \sqrt [3]{2}\right )}-\frac {\left (2\ 3^{2/3}-9 \sqrt [3]{6}\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{52488 \left (2\ 2^{2/3}-3\ 3^{2/3}\right )}+-\frac {\left (-4 \sqrt [3]{-1} 3^{2/3}-18 (-1)^{2/3} \sqrt [3]{6}\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{8748\ 2^{2/3} \left (24+18 \sqrt [3]{-2} 3^{2/3}\right )}\\ &=\frac {9 (-2)^{2/3}+\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {9\ 2^{2/3}+\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{26244\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{8748\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{5832 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{5832 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{52488 \sqrt [3]{2} 3^{2/3}}+\frac {\left (-18 \sqrt [3]{-6} (-1)^{2/3}+4 \sqrt [3]{-1} 3^{2/3}\right ) \operatorname {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 )}{486\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (24-18 (-3)^{2/3} \sqrt [3]{2}\right )}+\frac {\left (2\ 3^{2/3}-9 \sqrt [3]{6}\right ) \operatorname {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 )}{26244 \left (2\ 2^{2/3}-3\ 3^{2/3}\right )}--\frac {\left (-4 \sqrt [3]{-1} 3^{2/3}-18 (-1)^{2/3} \sqrt [3]{6}\right ) \operatorname {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 )}{4374\ 2^{2/3} \left (24+18 \sqrt [3]{-2} 3^{2/3}\right )}\\ &=\frac {9 (-2)^{2/3}+\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {9\ 2^{2/3}+\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{26244\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{8748\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\sqrt [3]{-1} \left (3 (-3)^{2/3}-2^{2/3}\right ) \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 )}{486\ 6^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}+\frac {\left (3 (-3)^{2/3} \sqrt [6]{2}+\sqrt [3]{-1} 2^{5/6}\right ) \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 )}{8748\ 3^{5/6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \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 )}{4374\ 6^{5/6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{5832 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{5832 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{52488 \sqrt [3]{2} 3^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.05, size = 167, normalized size = 0.25 \[ \frac {-3 x^5+73 x^4-72 x^3-64 x^2+108 x-96}{68364 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )}-\frac {\text {RootSum}\left [\text {$\#$1}^6+18 \text {$\#$1}^4+324 \text {$\#$1}^3+108 \text {$\#$1}^2+216\& ,\frac {3 \text {$\#$1}^4 \log (x-\text {$\#$1})-146 \text {$\#$1}^3 \log (x-\text {$\#$1})+108 \text {$\#$1}^2 \log (x-\text {$\#$1})-32 \text {$\#$1} \log (x-\text {$\#$1})+108 \log (x-\text {$\#$1})}{\text {$\#$1}^5+12 \text {$\#$1}^3+162 \text {$\#$1}^2+36 \text {$\#$1}}\& \right ]}{410184} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-96 + 108*x - 64*x^2 - 72*x^3 + 73*x^4 - 3*x^5)/(68364*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)) - RootSum[21
6 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (108*Log[x - #1] - 32*Log[x - #1]*#1 + 108*Log[x - #1]*#1^2 - 146
*Log[x - #1]*#1^3 + 3*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/410184

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.02, size = 122, normalized size = 0.18 \[ \frac {\left (-3 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{4}+146 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}-108 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}+32 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )-108\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+x \right )}{410184 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5}+4922208 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}[phone] \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}[phone] \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}+\frac {-\frac {1}{22788} x^{5}+\frac {73}{68364} x^{4}-\frac {2}{1899} x^{3}-\frac {16}{17091} x^{2}+\frac {1}{633} x -\frac {8}{5697}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/22788*x^5+73/68364*x^4-2/1899*x^3-16/17091*x^2+1/633*x-8/5697)/(x^6+18*x^4+324*x^3+108*x^2+216)+1/410184*s
um((-3*_R^4+146*_R^3-108*_R^2+32*_R-108)/(_R^5+12*_R^3+162*_R^2+36*_R)*ln(-_R+x),_R=RootOf(_Z^6+18*_Z^4+324*_Z
^3+108*_Z^2+216))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {3 \, x^{5} - 73 \, x^{4} + 72 \, x^{3} + 64 \, x^{2} - 108 \, x + 96}{68364 \, {\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} - \frac {1}{68364} \, \int \frac {3 \, x^{4} - 146 \, x^{3} + 108 \, x^{2} - 32 \, x + 108}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/68364*(3*x^5 - 73*x^4 + 72*x^3 + 64*x^2 - 108*x + 96)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216) - 1/68364*in
tegrate((3*x^4 - 146*x^3 + 108*x^2 - 32*x + 108)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)

________________________________________________________________________________________

mupad [B]  time = 2.33, size = 388, normalized size = 0.57 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log((7028852*root(z^6 - (60865*z^4)/239631364059408 - (15496909*z^3)/3056398361930300326272 - (168169*z
^2)/5941638415592503834272768 - (3971*z)/311864717157619341253309046784 - 880007/39777047316230971280399955151
66457856, z, k))/2628920529 - (1980083*x)/310470256633842 - (235710556*root(z^6 - (60865*z^4)/239631364059408
- (15496909*z^3)/3056398361930300326272 - (168169*z^2)/5941638415592503834272768 - (3971*z)/311864717157619341
253309046784 - 880007/3977704731623097128039995515166457856, z, k)*x)/70980854283 - (6628544*root(z^6 - (60865
*z^4)/239631364059408 - (15496909*z^3)/3056398361930300326272 - (168169*z^2)/5941638415592503834272768 - (3971
*z)/311864717157619341253309046784 - 880007/3977704731623097128039995515166457856, z, k)^2*x)/44521 - (1417767
59808*root(z^6 - (60865*z^4)/239631364059408 - (15496909*z^3)/3056398361930300326272 - (168169*z^2)/5941638415
592503834272768 - (3971*z)/311864717157619341253309046784 - 880007/3977704731623097128039995515166457856, z, k
)^3*x)/44521 + (183701926508544*root(z^6 - (60865*z^4)/239631364059408 - (15496909*z^3)/3056398361930300326272
- (168169*z^2)/5941638415592503834272768 - (3971*z)/311864717157619341253309046784 - 880007/39777047316230971
28039995515166457856, z, k)^4*x)/211 - 6940988288557056*root(z^6 - (60865*z^4)/239631364059408 - (15496909*z^3
)/3056398361930300326272 - (168169*z^2)/5941638415592503834272768 - (3971*z)/311864717157619341253309046784 -
880007/3977704731623097128039995515166457856, z, k)^5*x + (100886752*root(z^6 - (60865*z^4)/239631364059408 -
(15496909*z^3)/3056398361930300326272 - (168169*z^2)/5941638415592503834272768 - (3971*z)/31186471715761934125
3309046784 - 880007/3977704731623097128039995515166457856, z, k)^2)/133563 + (1715052538368*root(z^6 - (60865*
z^4)/239631364059408 - (15496909*z^3)/3056398361930300326272 - (168169*z^2)/5941638415592503834272768 - (3971*
z)/311864717157619341253309046784 - 880007/3977704731623097128039995515166457856, z, k)^3)/44521 + (1150043085
71136*root(z^6 - (60865*z^4)/239631364059408 - (15496909*z^3)/3056398361930300326272 - (168169*z^2)/5941638415
592503834272768 - (3971*z)/311864717157619341253309046784 - 880007/3977704731623097128039995515166457856, z, k
)^4)/211 - 168897381688221696*root(z^6 - (60865*z^4)/239631364059408 - (15496909*z^3)/3056398361930300326272 -
(168169*z^2)/5941638415592503834272768 - (3971*z)/311864717157619341253309046784 - 880007/3977704731623097128
039995515166457856, z, k)^5 - 265/5749449196923)*root(z^6 - (60865*z^4)/239631364059408 - (15496909*z^3)/30563
98361930300326272 - (168169*z^2)/5941638415592503834272768 - (3971*z)/311864717157619341253309046784 - 880007/
3977704731623097128039995515166457856, z, k), k, 1, 6) - ((16*x^2)/17091 - x/633 + (2*x^3)/1899 - (73*x^4)/683
64 + x^5/22788 + 8/5697)/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216)

________________________________________________________________________________________

sympy [A]  time = 0.38, size = 112, normalized size = 0.17 \[ \operatorname {RootSum} {\left (3977704731623097128039995515166457856 t^{6} - 1010314319415295961050951680 t^{4} - 20168224477093957151232 t^{3} - 112582856818899648 t^{2} - 50648453064 t - 880007, \left (t \mapsto t \log {\left (- \frac {273655567090018991570649941414395560986199688040644608 t^{5}}{49797855396139900267573395695} + \frac {11837008470196046085308646230764354292805044570112 t^{4}}{49797855396139900267573395695} - \frac {10570581900446717266374077482873315047787008 t^{3}}{49797855396139900267573395695} - \frac {1552547411569469872387563218792789323392 t^{2}}{49797855396139900267573395695} - \frac {12542923791159140826909003250295928 t}{49797855396139900267573395695} + x - \frac {23066533870320322410834348296}{49797855396139900267573395695} \right )} \right )\right )} + \frac {- 3 x^{5} + 73 x^{4} - 72 x^{3} - 64 x^{2} + 108 x - 96}{68364 x^{6} + 1230552 x^{4} + 22149936 x^{3} + 7383312 x^{2} + 14766624} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(3977704731623097128039995515166457856*_t**6 - 1010314319415295961050951680*_t**4 - 2016822447709395715
1232*_t**3 - 112582856818899648*_t**2 - 50648453064*_t - 880007, Lambda(_t, _t*log(-27365556709001899157064994
1414395560986199688040644608*_t**5/49797855396139900267573395695 + 1183700847019604608530864623076435429280504
4570112*_t**4/49797855396139900267573395695 - 10570581900446717266374077482873315047787008*_t**3/4979785539613
9900267573395695 - 1552547411569469872387563218792789323392*_t**2/49797855396139900267573395695 - 125429237911
59140826909003250295928*_t/49797855396139900267573395695 + x - 23066533870320322410834348296/49797855396139900
267573395695))) + (-3*x**5 + 73*x**4 - 72*x**3 - 64*x**2 + 108*x - 96)/(68364*x**6 + 1230552*x**4 + 22149936*x
**3 + 7383312*x**2 + 14766624)

________________________________________________________________________________________