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]

(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))

________________________________________________________________________________________

Rubi [A]  time = 1.55326, antiderivative size = 677, normalized size of antiderivative = 1., 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 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]

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 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 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 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 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])

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 = 0.0453387, 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

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Maple [C]  time = 0.01, size = 122, normalized size = 0.2 \begin{align*}{\frac{1}{{x}^{6}+18\,{x}^{4}+324\,{x}^{3}+108\,{x}^{2}+216} \left ( -{\frac{{x}^{5}}{22788}}+{\frac{73\,{x}^{4}}{68364}}-{\frac{2\,{x}^{3}}{1899}}-{\frac{16\,{x}^{2}}{17091}}+{\frac{x}{633}}-{\frac{8}{5697}} \right ) }+{\frac{1}{410184}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{ \left ( -3\,{{\it \_R}}^{4}+146\,{{\it \_R}}^{3}-108\,{{\it \_R}}^{2}+32\,{\it \_R}-108 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \end{align*}

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(x-_R),_R=RootOf(_Z^6+18*_Z^4+324*_Z^
3+108*_Z^2+216))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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} \end{align*}

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)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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Sympy [A]  time = 0.357636, size = 112, normalized size = 0.17 \begin{align*} \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} \end{align*}

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}}\,{d x} \end{align*}

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)