3.151 \(\int \frac{x^8}{(216+108 x^2+324 x^3+18 x^4+x^6)^2} \, dx\)

Optimal. Leaf size=1064 \[ \text{result too large to display} \]

[Out]

-((-1/3)^(1/3)*(9*(6 + (-3)^(1/3)*2^(2/3)) + (2 - 2^(2/3)*(6*(-6)^(2/3) + 27*(-3)^(1/3)))*x))/(162*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)) - ((-1/3)^(1/3)*(9*(6 - (-2)^(2
/3)*3^(1/3)) + (2 + 27*(-2)^(2/3)*3^(1/3) + 12*(-2)^(1/3)*3^(2/3))*x))/(729*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)) + (9*(6 - 2^(2/3)*3^(1/3)) + (2 + 2^(2/3)*(27*3^(1/3
) - 6*6^(2/3)))*x)/(1458*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)) - ((I/162)*(
(-2)^(2/3) + 6*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(2^(5/6)*3^(1
/3)*(1 + (-1)^(1/3))^5*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) - ((-1)^(1/3)*(2 + 27*(-2)^(2/3)*3^(1/3) + 12*(-2)^(1/3
)*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(162*2^(1/6)*3^(5/6)*(1 -
(-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(4 + 3*(-2)^(1/3)*3^(2/3))^(3/2)) - ((-1)^(1/3)*(6*(-6)^(2/3) + 27*(-3)^(1/3)
- 2^(1/3))*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(81*Sqrt[2]*3^(5/
6)*(1 + (-1)^(1/3))^4*(4 - 3*(-3)^(2/3)*2^(1/3))^(3/2)) + ((I*2^(2/3) - 9*3^(1/6) - (3*I)*3^(2/3))*ArcTan[(2^(
1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(162*2^(5/6)*3^(1/3)*(1 + (-1)^(1/3))^5*
Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((1 + 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))]])/(1458*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + ((2^(1/3) + 27*3^(1/3) - 6*6^(
2/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]])/(81*Sqrt[2]*3^(5/6)*(1 - (-
1)^(1/3))^2*(1 + (-1)^(1/3))^4*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) - Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2]/(972*2^
(1/3)*3^(2/3)*(1 + (-1)^(1/3))^4) + ((I/972)*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]/(8748*2^(1/3)*3^(2/3))

________________________________________________________________________________________

Rubi [A]  time = 2.50381, antiderivative size = 1064, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {2097, 638, 618, 204, 634, 628, 206} \[ -\frac{\sqrt [3]{-\frac{1}{3}} \left (\left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) x+9 \left (6-(-2)^{2/3} \sqrt [3]{3}\right )\right )}{729\ 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{\sqrt [3]{-1} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{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 )}{162 \sqrt [6]{2} 3^{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{i \left ((-2)^{2/3}+6\ 3^{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 )}{162\ 2^{5/6} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac{\left (i 2^{2/3}-9 \sqrt [6]{3}-3 i 3^{2/3}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{162\ 2^{5/6} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\sqrt [3]{-1} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{81 \sqrt{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}+\frac{\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{81 \sqrt{2} 3^{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 (1+3 \sqrt [3]{2} 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{1458 \sqrt [6]{2} 3^{5/6} \sqrt{-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac{\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{972 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac{i \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{972 \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 )}{8748 \sqrt [3]{2} 3^{2/3}}-\frac{\sqrt [3]{-\frac{1}{3}} \left (\left (2-3\ 2^{2/3} \left (2 (-6)^{2/3}+9 \sqrt [3]{-3}\right )\right ) x+9 \left (6+\sqrt [3]{-3} 2^{2/3}\right )\right )}{162\ 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{\left (2+2^{2/3} \left (27 \sqrt [3]{3}-6\ 6^{2/3}\right )\right ) x+9 \left (6-2^{2/3} \sqrt [3]{3}\right )}{1458\ 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 )} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]

[Out]

-((-1/3)^(1/3)*(9*(6 + (-3)^(1/3)*2^(2/3)) + (2 - 3*2^(2/3)*(2*(-6)^(2/3) + 9*(-3)^(1/3)))*x))/(162*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)) - ((-1/3)^(1/3)*(9*(6 - (-2)^(
2/3)*3^(1/3)) + (2 + 27*(-2)^(2/3)*3^(1/3) + 12*(-2)^(1/3)*3^(2/3))*x))/(729*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)) + (9*(6 - 2^(2/3)*3^(1/3)) + (2 + 2^(2/3)*(27*3^(1/
3) - 6*6^(2/3)))*x)/(1458*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)) - ((I/162)*
((-2)^(2/3) + 6*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(2^(5/6)*3^(
1/3)*(1 + (-1)^(1/3))^5*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) - ((-1)^(1/3)*(2 + 27*(-2)^(2/3)*3^(1/3) + 12*(-2)^(1/
3)*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(162*2^(1/6)*3^(5/6)*(1 -
(-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(4 + 3*(-2)^(1/3)*3^(2/3))^(3/2)) - ((-1)^(1/3)*(6*(-6)^(2/3) + 27*(-3)^(1/3
) - 2^(1/3))*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(81*Sqrt[2]*3^(5
/6)*(1 + (-1)^(1/3))^4*(4 - 3*(-3)^(2/3)*2^(1/3))^(3/2)) + ((I*2^(2/3) - 9*3^(1/6) - (3*I)*3^(2/3))*ArcTan[(2^
(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(162*2^(5/6)*3^(1/3)*(1 + (-1)^(1/3))^5
*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((1 + 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))]])/(1458*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + ((2^(1/3) + 27*3^(1/3) - 6*6^
(2/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]])/(81*Sqrt[2]*3^(5/6)*(1 - (
-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) - Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2]/(972*2
^(1/3)*3^(2/3)*(1 + (-1)^(1/3))^4) + ((I/972)*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]/(8748*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 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 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^8}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx &=1586874322944 \int \left (\frac{\sqrt [3]{-\frac{1}{3}} \left (-1+3 (-3)^{2/3} \sqrt [3]{2}+\left (9+\sqrt [3]{-3} 2^{2/3}\right ) x\right )}{42845606719488\ 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{27 \left (2+(-1)^{2/3}\right )-\left (1+\sqrt [3]{-1}\right ) x}{771220920950784 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac{\sqrt [3]{-\frac{1}{3}} \left (-1-3 \sqrt [3]{-2} 3^{2/3}+\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) x\right )}{42845606719488\ 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{i (-27+x)}{771220920950784 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac{1-3 \sqrt [3]{2} 3^{2/3}-\left (9-2^{2/3} \sqrt [3]{3}\right ) x}{42845606719488\ 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{-27+x}{6940988288557056 \sqrt [3]{2} 3^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=\frac{\sqrt [3]{-\frac{1}{3}} \int \frac{-1-3 \sqrt [3]{-2} 3^{2/3}+\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) x}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{243\ 2^{2/3}}-\frac{\int \frac{-27+x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{4374 \sqrt [3]{2} 3^{2/3}}+\frac{\int \frac{1-3 \sqrt [3]{2} 3^{2/3}-\left (9-2^{2/3} \sqrt [3]{3}\right ) x}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{243\ 2^{2/3} \sqrt [3]{3}}+\frac{\int \frac{27 \left (2+(-1)^{2/3}\right )-\left (1+\sqrt [3]{-1}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{486 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac{i \int \frac{-27+x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{486 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac{\sqrt [3]{-\frac{1}{3}} \int \frac{-1+3 (-3)^{2/3} \sqrt [3]{2}+\left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{\left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )^2} \, dx}{27\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}\\ &=-\frac{\sqrt [3]{-\frac{1}{3}} \left (9 \left (6+\sqrt [3]{-3} 2^{2/3}\right )+\left (2-3\ 2^{2/3} \left (2 (-6)^{2/3}+9 \sqrt [3]{-3}\right )\right ) x\right )}{162\ 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{\sqrt [3]{-\frac{1}{3}} \left (9 \left (6-(-2)^{2/3} \sqrt [3]{3}\right )+\left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{1458\ 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{9 \left (6-2^{2/3} \sqrt [3]{3}\right )+\left (2+2^{2/3} \left (27 \sqrt [3]{3}-6\ 6^{2/3}\right )\right ) x}{1458\ 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{\int \frac{3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{8748 \sqrt [3]{2} 3^{2/3}}+\frac{i \int \frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{972 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac{\int \frac{-3 \sqrt [3]{-3} 2^{2/3}+2 x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{972 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac{\left (\sqrt [3]{-\frac{1}{3}} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right )\right ) \int \frac{1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{162 \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}-\frac{\left (i \left ((-2)^{2/3}+6\ 3^{2/3}\right )\right ) \int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{108 \sqrt [3]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac{\left (\sqrt [3]{-\frac{1}{3}} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right )\right ) \int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1458\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}+\frac{\left (1+3 \sqrt [3]{2} 3^{2/3}\right ) \int \frac{1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1458\ 2^{2/3} \sqrt [3]{3}}+\frac{\left (3 \sqrt [3]{-3} 2^{2/3} \left (-1-\sqrt [3]{-1}\right )+54 \left (2+(-1)^{2/3}\right )\right ) \int \frac{1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{972 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac{\left (81+3^{2/3} \left (\sqrt [3]{2}-6\ 6^{2/3}\right )\right ) \int \frac{1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{4374 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}\\ &=-\frac{\sqrt [3]{-\frac{1}{3}} \left (9 \left (6+\sqrt [3]{-3} 2^{2/3}\right )+\left (2-3\ 2^{2/3} \left (2 (-6)^{2/3}+9 \sqrt [3]{-3}\right )\right ) x\right )}{162\ 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{\sqrt [3]{-\frac{1}{3}} \left (9 \left (6-(-2)^{2/3} \sqrt [3]{3}\right )+\left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{1458\ 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{9 \left (6-2^{2/3} \sqrt [3]{3}\right )+\left (2+2^{2/3} \left (27 \sqrt [3]{3}-6\ 6^{2/3}\right )\right ) x}{1458\ 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{\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{972 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac{i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{972 \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 )}{8748 \sqrt [3]{2} 3^{2/3}}-\frac{\left (\sqrt [3]{-\frac{1}{3}} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right )\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 )}{81 \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}+\frac{\left (i \left ((-2)^{2/3}+6\ 3^{2/3}\right )\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 )}{54 \sqrt [3]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac{\left (\sqrt [3]{-\frac{1}{3}} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right )\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 )}{729\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}-\frac{\left (1+3 \sqrt [3]{2} 3^{2/3}\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 )}{729\ 2^{2/3} \sqrt [3]{3}}-\frac{\left (3 \sqrt [3]{-3} 2^{2/3} \left (-1-\sqrt [3]{-1}\right )+54 \left (2+(-1)^{2/3}\right )\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 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac{\left (81+3^{2/3} \left (\sqrt [3]{2}-6\ 6^{2/3}\right )\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 )}{2187 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}\\ &=-\frac{\sqrt [3]{-\frac{1}{3}} \left (9 \left (6+\sqrt [3]{-3} 2^{2/3}\right )+\left (2-3\ 2^{2/3} \left (2 (-6)^{2/3}+9 \sqrt [3]{-3}\right )\right ) x\right )}{162\ 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{\sqrt [3]{-\frac{1}{3}} \left (9 \left (6-(-2)^{2/3} \sqrt [3]{3}\right )+\left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{1458\ 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{9 \left (6-2^{2/3} \sqrt [3]{3}\right )+\left (2+2^{2/3} \left (27 \sqrt [3]{3}-6\ 6^{2/3}\right )\right ) x}{1458\ 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{i \left ((-2)^{2/3}+6\ 3^{2/3}\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 )}{162\ 2^{5/6} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac{\sqrt [3]{-1} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\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 )}{1458 \sqrt [6]{2} 3^{5/6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac{\sqrt [3]{-1} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{81 \sqrt{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}+\frac{\left (i 3^{5/6}-9 \sqrt [3]{2} \left (2+(-1)^{2/3}\right )\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{486 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (1+3 \sqrt [3]{2} 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 )}{1458 \sqrt [6]{2} 3^{5/6} \sqrt{-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac{\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{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 )}{729 \sqrt{2} 3^{5/6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac{\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{972 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac{i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{972 \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 )}{8748 \sqrt [3]{2} 3^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0414178, size = 167, normalized size = 0.16 \[ \frac{-9 x^5-203 x^4-11610 x^3-3990 x^2+324 x-7884}{34182 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )}-\frac{\text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\& ,\frac{9 \text{$\#$1}^4 \log (x-\text{$\#$1})+406 \text{$\#$1}^3 \log (x-\text{$\#$1})+324 \text{$\#$1}^2 \log (x-\text{$\#$1})-96 \text{$\#$1} \log (x-\text{$\#$1})+324 \log (x-\text{$\#$1})}{\text{$\#$1}^5+12 \text{$\#$1}^3+162 \text{$\#$1}^2+36 \text{$\#$1}}\& \right ]}{205092} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^8/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]

[Out]

(-7884 + 324*x - 3990*x^2 - 11610*x^3 - 203*x^4 - 9*x^5)/(34182*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)) - Ro
otSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (324*Log[x - #1] - 96*Log[x - #1]*#1 + 324*Log[x - #1]*#1
^2 + 406*Log[x - #1]*#1^3 + 9*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/205092

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Maple [C]  time = 0.01, size = 122, normalized size = 0.1 \begin{align*}{\frac{1}{{x}^{6}+18\,{x}^{4}+324\,{x}^{3}+108\,{x}^{2}+216} \left ( -{\frac{{x}^{5}}{3798}}-{\frac{203\,{x}^{4}}{34182}}-{\frac{215\,{x}^{3}}{633}}-{\frac{665\,{x}^{2}}{5697}}+{\frac{2\,x}{211}}-{\frac{146}{633}} \right ) }+{\frac{1}{205092}\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 ( -9\,{{\it \_R}}^{4}-406\,{{\it \_R}}^{3}-324\,{{\it \_R}}^{2}+96\,{\it \_R}-324 \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^8/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x)

[Out]

(-1/3798*x^5-203/34182*x^4-215/633*x^3-665/5697*x^2+2/211*x-146/633)/(x^6+18*x^4+324*x^3+108*x^2+216)+1/205092
*sum((-9*_R^4-406*_R^3-324*_R^2+96*_R-324)/(_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{9 \, x^{5} + 203 \, x^{4} + 11610 \, x^{3} + 3990 \, x^{2} - 324 \, x + 7884}{34182 \,{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} - \frac{1}{34182} \, \int \frac{9 \, x^{4} + 406 \, x^{3} + 324 \, x^{2} - 96 \, x + 324}{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^8/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="maxima")

[Out]

-1/34182*(9*x^5 + 203*x^4 + 11610*x^3 + 3990*x^2 - 324*x + 7884)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216) - 1/
34182*integrate((9*x^4 + 406*x^3 + 324*x^2 - 96*x + 324)/(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^8/(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 = 0.372375, size = 112, normalized size = 0.11 \begin{align*} \operatorname{RootSum}{\left (85256017052964187415123360664576 t^{6} + 50105191533385434568704 t^{4} + 48885748051277486016 t^{3} + 865447782603408 t^{2} + 3220532460 t + 4513, \left ( t \mapsto t \log{\left (\frac{35492036204084174404119193135483487466590764032 t^{5}}{356900697070792948475845} - \frac{19474160067218837086826809631017022308224 t^{4}}{71380139414158589695169} + \frac{20779963076545132233894582764903396544 t^{3}}{356900697070792948475845} + \frac{20265219154367004972162198012037344 t^{2}}{356900697070792948475845} + \frac{275192468949210532049075145372 t}{356900697070792948475845} + x + \frac{1290285191292177289622012}{1070702091212378845427535} \right )} \right )\right )} - \frac{9 x^{5} + 203 x^{4} + 11610 x^{3} + 3990 x^{2} - 324 x + 7884}{34182 x^{6} + 615276 x^{4} + 11074968 x^{3} + 3691656 x^{2} + 7383312} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(x**6+18*x**4+324*x**3+108*x**2+216)**2,x)

[Out]

RootSum(85256017052964187415123360664576*_t**6 + 50105191533385434568704*_t**4 + 48885748051277486016*_t**3 +
865447782603408*_t**2 + 3220532460*_t + 4513, Lambda(_t, _t*log(3549203620408417440411919313548348746659076403
2*_t**5/356900697070792948475845 - 19474160067218837086826809631017022308224*_t**4/71380139414158589695169 + 2
0779963076545132233894582764903396544*_t**3/356900697070792948475845 + 20265219154367004972162198012037344*_t*
*2/356900697070792948475845 + 275192468949210532049075145372*_t/356900697070792948475845 + x + 129028519129217
7289622012/1070702091212378845427535))) - (9*x**5 + 203*x**4 + 11610*x**3 + 3990*x**2 - 324*x + 7884)/(34182*x
**6 + 615276*x**4 + 11074968*x**3 + 3691656*x**2 + 7383312)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{{\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^8/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="giac")

[Out]

integrate(x^8/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)^2, x)