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

Optimal. Leaf size=1064 \[ -\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 (2-2^{2/3} \left (6 (-6)^{2/3}+27 \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 )} \]

[Out]

-1/972*(-1)^(1/3)*3^(2/3)*(54+9*(-3)^(1/3)*2^(2/3)+(2-2^(2/3)*(6*(-6)^(2/3)+27*(-3)^(1/3)))*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)-1/4374*(-1)^(1/3)*3^(2/3)*(54-9*(-2)^(2/3)*3
^(1/3)+(2+27*(-2)^(2/3)*3^(1/3)+12*(-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)+1/8748*(54-9*2^(2/3)*3^(1/3)+(2+2^(2/3)*(27*3^(1/3)-6*6^(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)-1/972*(-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)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*2^(5/6)*3^(1/6)/(1-(-1)^(1/3))^2/(1
+(-1)^(1/3))^4/(4+3*(-2)^(1/3)*3^(2/3))^(3/2)-1/5832*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(2/3)*3^(1/3)/(1+(-1)^
(1/3))^4+1/5832*I*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)*2^(2/3)*3^(5/6)/(1+(-1)^(1/3))^5-1/52488*ln(6+3*2^(2/3)*3^(
1/3)*x+x^2)*2^(2/3)*3^(1/3)-1/486*(-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)/(12-9*(-3)^(2/3)*2^(1/3))^(1/2))*3^(1/6)/(1+(-1)^(1/3))^4/(4-3*(-3)^(2/3)*2^(1/3))^(3/2)*2^(1/2)+1
/486*(2^(1/3)+27*3^(1/3)-6*6^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*3^(1/
6)/(1-(-1)^(1/3))^2/(1+(-1)^(1/3))^4/(-4+3*2^(1/3)*3^(2/3))^(3/2)*2^(1/2)+1/972*(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)/(12-9*(-3)^(2/3)*2^(1/3))^(1/2))*2^(1/6)*3^(2/3)/(1+(-1)^(1/3))^5/
(4-3*(-3)^(2/3)*2^(1/3))^(1/2)-1/972*I*((-2)^(2/3)+6*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1
/3)*3^(2/3))^(1/2))*2^(1/6)*3^(2/3)/(1+(-1)^(1/3))^5/(4+3*(-2)^(1/3)*3^(2/3))^(1/2)-1/8748*(1+3*2^(1/3)*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))*2^(5/6)*3^(1/6)/(-4+3*2^(1/3)*3^(2/3))
^(1/2)

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Rubi [A]  time = 2.50, antiderivative size = 1064, normalized size of antiderivative = 1.00, 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 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 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 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^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*}

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Mathematica [C]  time = 0.04, 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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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^8/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{8}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^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)

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maple [C]  time = 0.02, size = 122, normalized size = 0.11 \[ \frac {\left (-9 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{4}-406 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}-324 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}+96 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )-324\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+x \right )}{205092 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5}+2461104 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}+33224904 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}+7383312 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}+\frac {-\frac {1}{3798} x^{5}-\frac {203}{34182} x^{4}-\frac {215}{633} x^{3}-\frac {665}{5697} x^{2}+\frac {2}{211} x -\frac {146}{633}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {9 \, x^{5} + 203 \, x^{4} + 11610 \, x^{3} + 3990 \, x^{2} - 324 \, x + 7884}{34182 \, {\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} - \frac {1}{34182} \, \int \frac {9 \, x^{4} + 406 \, x^{3} + 324 \, x^{2} - 96 \, x + 324}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

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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mupad [B]  time = 2.34, size = 388, normalized size = 0.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log((239491904*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14149992416343982992 + (5171*z^2)/5093
99726988383387712 + (505*z)/13368686435083133627113728 + 4513/85256017052964187415123360664576, z, k)*x)/87630
6843 - (275536*x)/638827688547 - (3848128*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/141499924163439829
92 + (5171*z^2)/509399726988383387712 + (505*z)/13368686435083133627113728 + 4513/8525601705296418741512336066
4576, z, k))/3606201 - (152363520*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14149992416343982992 + (51
71*z^2)/509399726988383387712 + (505*z)/13368686435083133627113728 + 4513/85256017052964187415123360664576, z,
 k)^2*x)/44521 - (698075283456*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14149992416343982992 + (5171*
z^2)/509399726988383387712 + (505*z)/13368686435083133627113728 + 4513/85256017052964187415123360664576, z, k)
^3*x)/44521 + (130789789876224*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14149992416343982992 + (5171*
z^2)/509399726988383387712 + (505*z)/13368686435083133627113728 + 4513/85256017052964187415123360664576, z, k)
^4*x)/211 - 6940988288557056*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14149992416343982992 + (5171*z^
2)/509399726988383387712 + (505*z)/13368686435083133627113728 + 4513/85256017052964187415123360664576, z, k)^5
*x - (4264220928*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14149992416343982992 + (5171*z^2)/509399726
988383387712 + (505*z)/13368686435083133627113728 + 4513/85256017052964187415123360664576, z, k)^2)/44521 - (5
086414725120*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14149992416343982992 + (5171*z^2)/5093997269883
83387712 + (505*z)/13368686435083133627113728 + 4513/85256017052964187415123360664576, z, k)^3)/44521 + (24358
5208571904*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14149992416343982992 + (5171*z^2)/509399726988383
387712 + (505*z)/13368686435083133627113728 + 4513/85256017052964187415123360664576, z, k)^4)/211 - 1688973816
88221696*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14149992416343982992 + (5171*z^2)/50939972698838338
7712 + (505*z)/13368686435083133627113728 + 4513/85256017052964187415123360664576, z, k)^5 - 48160/23660284761
)*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14149992416343982992 + (5171*z^2)/509399726988383387712 +
(505*z)/13368686435083133627113728 + 4513/85256017052964187415123360664576, z, k), k, 1, 6) - ((665*x^2)/5697
- (2*x)/211 + (215*x^3)/633 + (203*x^4)/34182 + x^5/3798 + 146/633)/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216)

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sympy [A]  time = 0.40, size = 112, normalized size = 0.11 \[ \operatorname {RootSum} {\left (85256017052964187415123360664576 t^{6} + 50105191533385434568704 t^{4} + 48885748051277486016 t^{3} + 865447782603408 t^{2} + 3220532460 t + 4513, \left (t \mapsto t \log {\left (\frac {35492036204084174404119193135483487466590764032 t^{5}}{356900697070792948475845} - \frac {19474160067218837086826809631017022308224 t^{4}}{71380139414158589695169} + \frac {20779963076545132233894582764903396544 t^{3}}{356900697070792948475845} + \frac {20265219154367004972162198012037344 t^{2}}{356900697070792948475845} + \frac {275192468949210532049075145372 t}{356900697070792948475845} + x + \frac {1290285191292177289622012}{1070702091212378845427535} \right )} \right )\right )} + \frac {- 9 x^{5} - 203 x^{4} - 11610 x^{3} - 3990 x^{2} + 324 x - 7884}{34182 x^{6} + 615276 x^{4} + 11074968 x^{3} + 3691656 x^{2} + 7383312} \]

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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