3.152 \(\int \frac {x^7}{(216+108 x^2+324 x^3+18 x^4+x^6)^2} \, dx\)

Optimal. Leaf size=1005 \[ \frac {2 \left (2-3 \sqrt [3]{2} 3^{2/3}\right )-3 \left (6-2^{2/3} \sqrt [3]{3}\right ) x}{2916\ 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 {\left (9 i+\sqrt [3]{3} \left (2 i 2^{2/3}-9 \sqrt [6]{3}+2\ 2^{2/3} \sqrt {3}\right )\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 )}{5832 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac {\left (9 \sqrt [6]{3}+i \left (4\ 2^{2/3}-3\ 3^{2/3}\right )\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 )}{1944\ 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}+\frac {\left (1+\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 )}{54 \sqrt {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 {\sqrt [3]{-1} \left (\sqrt [3]{-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 )}{54 \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 (2\ 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 (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{26244 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}+\frac {\left (1-\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 )}{54 \sqrt {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 \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{648\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (i+\sqrt {3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{1296\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{17496\ 2^{2/3} \sqrt [3]{3}}-\frac {2 \left (2 \sqrt [3]{-1} 3^{2/3}+9 \sqrt [3]{6}\right )-9 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right ) x}{972\ 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]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )-9 \left (1+\sqrt [3]{-2} 3^{2/3}\right ) x}{4374 \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 )} \]

[Out]

1/1944*(-4*(-1)^(1/3)*3^(2/3)-18*6^(1/3)+9*((-2)^(2/3)+2*(-1)^(1/3)*3^(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)+1/4374*(-(-6)^(1/3)*(9*(-2)^(1/3)+2*3^(1/3))+9*(1+(-2)^(1/3
)*3^(2/3))*x)/(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)+1/17496*(4-6*2^(1/3)*3^
(2/3)-3*(6-2^(2/3)*3^(1/3))*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)+1/3888*I*ln(6
-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(1/3)*3^(1/6)/(1+(-1)^(1/3))^5-1/104976*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)*2^(1/3)*3
^(2/3)-1/324*(-1)^(1/3)*((-3)^(1/3)+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)-1/7776*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x
^2)*(3^(1/2)+I)*2^(1/3)*3^(1/6)/(1+(-1)^(1/3))^5+1/324*(1+(-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))/(1-(-1)^(1/3))^2/(1+(-1)^(1/3))^4/(4+3*(-2)^(1/3)*3^(2/3))^(3/2)*6^(1/2)+1
/324*(1-2^(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))/(1-(-1)^(1/3))^2
/(1+(-1)^(1/3))^4/(-4+3*2^(1/3)*3^(2/3))^(3/2)*6^(1/2)+1/5832*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2
/3)*2^(1/3))^(1/2))*(9*I+3^(1/3)*(2*I*2^(2/3)-9*3^(1/6)+2*2^(2/3)*3^(1/2)))/(1+(-1)^(1/3))^5/(8-6*(-3)^(2/3)*2
^(1/3))^(1/2)+1/5832*(9*3^(1/6)+I*(4*2^(2/3)-3*3^(2/3)))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3
^(2/3))^(1/2))*3^(1/3)/(1+(-1)^(1/3))^5/(8+6*(-2)^(1/3)*3^(2/3))^(1/2)+1/78732*(2*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))*3^(5/6)/(-8+6*2^(1/3)*3^(2/3))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.40, antiderivative size = 1005, 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, 634, 618, 204, 628, 638, 206} \[ \frac {2 \left (2-3 \sqrt [3]{2} 3^{2/3}\right )-3 \left (6-2^{2/3} \sqrt [3]{3}\right ) x}{2916\ 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 {\left (9 i+\sqrt [3]{3} \left (2 i 2^{2/3}-9 \sqrt [6]{3}+2\ 2^{2/3} \sqrt {3}\right )\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 )}{5832 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac {\left (9 \sqrt [6]{3}+i \left (4\ 2^{2/3}-3\ 3^{2/3}\right )\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 )}{1944\ 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}+\frac {\left (1+\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 )}{54 \sqrt {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 {\sqrt [3]{-1} \left (\sqrt [3]{-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 )}{54 \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 (2\ 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 (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{26244 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}+\frac {\left (1-\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 )}{54 \sqrt {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 \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{648\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (i+\sqrt {3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{1296\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{17496\ 2^{2/3} \sqrt [3]{3}}-\frac {2 \left (2 \sqrt [3]{-1} 3^{2/3}+9 \sqrt [3]{6}\right )-9 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right ) x}{972\ 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]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )-9 \left (1+\sqrt [3]{-2} 3^{2/3}\right ) x}{4374 \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 )} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]

[Out]

-(2*(2*(-1)^(1/3)*3^(2/3) + 9*6^(1/3)) - 9*((-2)^(2/3) + 2*(-1)^(1/3)*3^(2/3))*x)/(972*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)) - ((-6)^(1/3)*(9*(-2)^(1/3) + 2*3^(1/3)) -
9*(1 + (-2)^(1/3)*3^(2/3))*x)/(4374*(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)) + (2*(2 - 3*2^(1/3)*3^(2/3)) - 3*(6 - 2^(2/3)*3^(1/3))*x)/(2916*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)) + ((9*I + 3^(1/3)*((2*I)*2^(2/3) - 9*3^(1/6) + 2*2^(2/3)*Sqrt[3]))*ArcTan
[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(5832*(1 + (-1)^(1/3))^5*Sqrt[2*(4 - 3*(-3)
^(2/3)*2^(1/3))]) + ((1 + (-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))]])/(54*Sqrt[6]*(1 - (-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(4 + 3*(-2)^(1/3)*3^(2/3))^(3/2)) + ((9*3^(1/6) + I
*(4*2^(2/3) - 3*3^(2/3)))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(1944*3^(2/
3)*(1 + (-1)^(1/3))^5*Sqrt[2*(4 + 3*(-2)^(1/3)*3^(2/3))]) - ((-1)^(1/3)*((-3)^(1/3) + 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))]])/(54*Sqrt[2]*3^(5/6)*(1 + (-1)^(1/3))^4*(4
- 3*(-3)^(2/3)*2^(1/3))^(3/2)) + ((1 - 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))]])/(54*Sqrt[6]*(1 - (-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) + ((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))]])/(26244*3^(1
/6)*Sqrt[2*(-4 + 3*2^(1/3)*3^(2/3))]) + ((I/648)*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(2^(2/3)*3^(5/6)*(1 +
(-1)^(1/3))^5) - ((I + Sqrt[3])*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(1296*2^(2/3)*3^(5/6)*(1 + (-1)^(1/3))^
5) - Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2]/(17496*2^(2/3)*3^(1/3))

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^7}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx &=1586874322944 \int \left (\frac {-27+3\ 2^{2/3} \sqrt [3]{3}+9 i \sqrt {3}+i 2^{2/3} 3^{5/6}-3 i \sqrt [3]{2} \sqrt [6]{3} x}{9254651051409408 \left (1+\sqrt [3]{-1}\right )^5 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )}+\frac {9 (-2)^{2/3}+\sqrt [3]{3} \left (\sqrt [3]{-3}+9 \sqrt [3]{2}\right ) x}{771220920950784\ 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 {9\ 2^{2/3}+\sqrt [3]{-1} 3^{2/3} \left (1+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{771220920950784\ 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 {2 \left (27-9 i \sqrt {3}+2 i 2^{2/3} 3^{5/6}\right )-3 \sqrt [3]{2} \sqrt [6]{3} \left (i+\sqrt {3}\right ) x}{18509302102818816 \left (1+\sqrt [3]{-1}\right )^5 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (1-3 \sqrt [3]{2} 3^{2/3}\right ) x}{257073640316928\ 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 {-18-2\ 2^{2/3} \sqrt [3]{3}-\sqrt [3]{2} 3^{2/3} x}{83291859462684672 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {-18-2\ 2^{2/3} \sqrt [3]{3}-\sqrt [3]{2} 3^{2/3} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{52488}+\frac {\int \frac {9\ 2^{2/3}+\sqrt [3]{-1} 3^{2/3} \left (1+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{4374\ 2^{2/3}}+\frac {\int \frac {3\ 2^{2/3} \sqrt [3]{3}-\left (1-3 \sqrt [3]{2} 3^{2/3}\right ) x}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{1458\ 2^{2/3} \sqrt [3]{3}}+\frac {\int \frac {2 \left (27-9 i \sqrt {3}+2 i 2^{2/3} 3^{5/6}\right )-3 \sqrt [3]{2} \sqrt [6]{3} \left (i+\sqrt {3}\right ) x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{11664 \left (1+\sqrt [3]{-1}\right )^5}+\frac {\int \frac {-27+3\ 2^{2/3} \sqrt [3]{3}+9 i \sqrt {3}+i 2^{2/3} 3^{5/6}-3 i \sqrt [3]{2} \sqrt [6]{3} x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{5832 \left (1+\sqrt [3]{-1}\right )^5}+\frac {\int \frac {9 (-2)^{2/3}+\sqrt [3]{3} \left (\sqrt [3]{-3}+9 \sqrt [3]{2}\right ) x}{\left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )^2} \, dx}{486\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}\\ &=-\frac {2 \left (2 \sqrt [3]{-1} 3^{2/3}+9 \sqrt [3]{6}\right )-9 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right ) x}{972\ 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]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )-9 \left (1+\sqrt [3]{-2} 3^{2/3}\right ) x}{8748 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {2 \left (2-3 \sqrt [3]{2} 3^{2/3}\right )-3 \left (6-2^{2/3} \sqrt [3]{3}\right ) x}{2916\ 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}{17496\ 2^{2/3} \sqrt [3]{3}}+\frac {i \int \frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{648\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (9+2\ 2^{2/3} \sqrt [3]{3}\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{52488}-\frac {\left (i+\sqrt {3}\right ) \int \frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1296\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (1+\sqrt [3]{-2} 3^{2/3}\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{972 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}+\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{972 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}+\frac {\left (18 (-2)^{2/3}+3 \sqrt [3]{-1} 6^{2/3} \left (\sqrt [3]{-3}+9 \sqrt [3]{2}\right )\right ) \int \frac {1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{486\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (24-18 (-3)^{2/3} \sqrt [3]{2}\right )}+\frac {\left (-18 (-1)^{5/6} \sqrt {3}+2 \left (-27+3\ 2^{2/3} \sqrt [3]{3}+9 i \sqrt {3}+i 2^{2/3} 3^{5/6}\right )\right ) \int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{11664 \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (18 (-1)^{2/3} \sqrt {3} \left (i+\sqrt {3}\right )+4 \left (27-9 i \sqrt {3}+2 i 2^{2/3} 3^{5/6}\right )\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{23328 \left (1+\sqrt [3]{-1}\right )^5}\\ &=-\frac {2 \left (2 \sqrt [3]{-1} 3^{2/3}+9 \sqrt [3]{6}\right )-9 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right ) x}{972\ 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]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )-9 \left (1+\sqrt [3]{-2} 3^{2/3}\right ) x}{8748 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {2 \left (2-3 \sqrt [3]{2} 3^{2/3}\right )-3 \left (6-2^{2/3} \sqrt [3]{3}\right ) x}{2916\ 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 \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{648\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (i+\sqrt {3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{1296\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{17496\ 2^{2/3} \sqrt [3]{3}}+\frac {\left (9+2\ 2^{2/3} \sqrt [3]{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 )}{26244}-\frac {\left (1+\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 )}{486 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}-\frac {\left (1-\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 )}{486 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}-\frac {\left (18 (-2)^{2/3}+3 \sqrt [3]{-1} 6^{2/3} \left (\sqrt [3]{-3}+9 \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 )}{243\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (24-18 (-3)^{2/3} \sqrt [3]{2}\right )}-\frac {\left (-18 (-1)^{5/6} \sqrt {3}+2 \left (-27+3\ 2^{2/3} \sqrt [3]{3}+9 i \sqrt {3}+i 2^{2/3} 3^{5/6}\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 )}{5832 \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (18 (-1)^{2/3} \sqrt {3} \left (i+\sqrt {3}\right )+4 \left (27-9 i \sqrt {3}+2 i 2^{2/3} 3^{5/6}\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 )}{11664 \left (1+\sqrt [3]{-1}\right )^5}\\ &=-\frac {2 \left (2 \sqrt [3]{-1} 3^{2/3}+9 \sqrt [3]{6}\right )-9 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right ) x}{972\ 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]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )-9 \left (1+\sqrt [3]{-2} 3^{2/3}\right ) x}{8748 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {2 \left (2-3 \sqrt [3]{2} 3^{2/3}\right )-3 \left (6-2^{2/3} \sqrt [3]{3}\right ) x}{2916\ 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 {\left (27-6\ 2^{2/3} \sqrt [3]{3}-9 i \sqrt {3}-2 i 2^{2/3} 3^{5/6}\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 )}{5832 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac {\left (1+\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 )}{486 \sqrt {6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (9 i-4 i 2^{2/3} \sqrt [3]{3}-9 \sqrt {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 )}{5832 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}-\frac {\sqrt [3]{-1} \left (\sqrt [3]{-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 )}{54 \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 (1-\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 )}{486 \sqrt {6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (2\ 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 )}{26244 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}+\frac {i \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{648\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (i+\sqrt {3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{1296\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{17496\ 2^{2/3} \sqrt [3]{3}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 167, normalized size = 0.17 \[ \frac {\text {RootSum}\left [\text {$\#$1}^6+18 \text {$\#$1}^4+324 \text {$\#$1}^3+108 \text {$\#$1}^2+216\& ,\frac {73 \text {$\#$1}^4 \log (x-\text {$\#$1})-36 \text {$\#$1}^3 \log (x-\text {$\#$1})+96 \text {$\#$1}^2 \log (x-\text {$\#$1})-216 \text {$\#$1} \log (x-\text {$\#$1})+96 \log (x-\text {$\#$1})}{\text {$\#$1}^5+12 \text {$\#$1}^3+162 \text {$\#$1}^2+36 \text {$\#$1}}\& \right ]}{410184}+\frac {73 x^5-18 x^4+908 x^3+432 x^2-96 x+648}{68364 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^7/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]

[Out]

(648 - 96*x + 432*x^2 + 908*x^3 - 18*x^4 + 73*x^5)/(68364*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)) + RootSum[
216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (96*Log[x - #1] - 216*Log[x - #1]*#1 + 96*Log[x - #1]*#1^2 - 36
*Log[x - #1]*#1^3 + 73*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/410184

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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^7/(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 \[ \int \frac {x^{7}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="giac")

[Out]

integrate(x^7/(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 = 0.12 \[ \frac {\left (73 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{4}-36 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}+96 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}-216 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+96\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+x \right )}{410184 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5}+4922208 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}[phone] \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}[phone] \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}+\frac {\frac {73}{68364} x^{5}-\frac {1}{3798} x^{4}+\frac {227}{17091} x^{3}+\frac {4}{633} x^{2}-\frac {8}{5697} x +\frac {2}{211}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x)

[Out]

(73/68364*x^5-1/3798*x^4+227/17091*x^3+4/633*x^2-8/5697*x+2/211)/(x^6+18*x^4+324*x^3+108*x^2+216)+1/410184*sum
((73*_R^4-36*_R^3+96*_R^2-216*_R+96)/(_R^5+12*_R^3+162*_R^2+36*_R)*ln(-_R+x),_R=RootOf(_Z^6+18*_Z^4+324*_Z^3+1
08*_Z^2+216))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {73 \, x^{5} - 18 \, x^{4} + 908 \, x^{3} + 432 \, x^{2} - 96 \, x + 648}{68364 \, {\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} + \frac {1}{68364} \, \int \frac {73 \, x^{4} - 36 \, x^{3} + 96 \, x^{2} - 216 \, x + 96}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="maxima")

[Out]

1/68364*(73*x^5 - 18*x^4 + 908*x^3 + 432*x^2 - 96*x + 648)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216) + 1/68364*
integrate((73*x^4 - 36*x^3 + 96*x^2 - 216*x + 96)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)

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mupad [B]  time = 2.30, size = 387, normalized size = 0.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216)^2,x)

[Out]

symsum(log((8336932*root(z^6 - (292589*z^4)/319508485412544 + (11805253*z^3)/75466626220501242624 - (2479189*z
^2)/2640728184707779481899008 - (1989787*z)/311864717157619341253309046784 - 7197829/5892895898700884634133326
68913549312, z, k))/97367427 - (480227*x)/851770251396 - (759164282*root(z^6 - (292589*z^4)/319508485412544 +
(11805253*z^3)/75466626220501242624 - (2479189*z^2)/2640728184707779481899008 - (1989787*z)/311864717157619341
253309046784 - 7197829/589289589870088463413332668913549312, z, k)*x)/7886761587 - (207565888*root(z^6 - (2925
89*z^4)/319508485412544 + (11805253*z^3)/75466626220501242624 - (2479189*z^2)/2640728184707779481899008 - (198
9787*z)/311864717157619341253309046784 - 7197829/589289589870088463413332668913549312, z, k)^2*x)/400689 - (10
8430970112*root(z^6 - (292589*z^4)/319508485412544 + (11805253*z^3)/75466626220501242624 - (2479189*z^2)/26407
28184707779481899008 - (1989787*z)/311864717157619341253309046784 - 7197829/5892895898700884634133326689135493
12, z, k)^3*x)/44521 - (147138513610752*root(z^6 - (292589*z^4)/319508485412544 + (11805253*z^3)/7546662622050
1242624 - (2479189*z^2)/2640728184707779481899008 - (1989787*z)/311864717157619341253309046784 - 7197829/58928
9589870088463413332668913549312, z, k)^4*x)/211 - 6940988288557056*root(z^6 - (292589*z^4)/319508485412544 + (
11805253*z^3)/75466626220501242624 - (2479189*z^2)/2640728184707779481899008 - (1989787*z)/3118647171576193412
53309046784 - 7197829/589289589870088463413332668913549312, z, k)^5*x - (1156135728*root(z^6 - (292589*z^4)/31
9508485412544 + (11805253*z^3)/75466626220501242624 - (2479189*z^2)/2640728184707779481899008 - (1989787*z)/31
1864717157619341253309046784 - 7197829/589289589870088463413332668913549312, z, k)^2)/44521 + (6458021903232*r
oot(z^6 - (292589*z^4)/319508485412544 + (11805253*z^3)/75466626220501242624 - (2479189*z^2)/26407281847077794
81899008 - (1989787*z)/311864717157619341253309046784 - 7197829/589289589870088463413332668913549312, z, k)^3)
/44521 - (102226052063232*root(z^6 - (292589*z^4)/319508485412544 + (11805253*z^3)/75466626220501242624 - (247
9189*z^2)/2640728184707779481899008 - (1989787*z)/311864717157619341253309046784 - 7197829/5892895898700884634
13332668913549312, z, k)^4)/211 - 168897381688221696*root(z^6 - (292589*z^4)/319508485412544 + (11805253*z^3)/
75466626220501242624 - (2479189*z^2)/2640728184707779481899008 - (1989787*z)/311864717157619341253309046784 -
7197829/589289589870088463413332668913549312, z, k)^5 + 2207561/7665932262564)*root(z^6 - (292589*z^4)/3195084
85412544 + (11805253*z^3)/75466626220501242624 - (2479189*z^2)/2640728184707779481899008 - (1989787*z)/3118647
17157619341253309046784 - 7197829/589289589870088463413332668913549312, z, k), k, 1, 6) + ((4*x^2)/633 - (8*x)
/5697 + (227*x^3)/17091 - x^4/3798 + (73*x^5)/68364 + 2/211)/(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 \[ \operatorname {RootSum} {\left (589289589870088463413332668913549312 t^{6} - 539640290266075248405737472 t^{4} + 92182638168509682392064 t^{3} - 553241442069170496 t^{2} - 3759837842016 t - 7197829, \left (t \mapsto t \log {\left (\frac {42996027639727447714003743305160746111018438501025999323136 t^{5}}{154206009791052044490694380303237521} - \frac {42584766259508194684689715474422251405157209835847680 t^{4}}{154206009791052044490694380303237521} - \frac {37512446128849588150108369449323754078317341082112 t^{3}}{154206009791052044490694380303237521} + \frac {7152037594021675267638890715531672481920222144 t^{2}}{154206009791052044490694380303237521} - \frac {44227546998835297723830291794974310524032 t}{154206009791052044490694380303237521} + x - \frac {174573349036676047734132569583024855}{154206009791052044490694380303237521} \right )} \right )\right )} + \frac {73 x^{5} - 18 x^{4} + 908 x^{3} + 432 x^{2} - 96 x + 648}{68364 x^{6} + 1230552 x^{4} + 22149936 x^{3} + 7383312 x^{2} + 14766624} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(x**6+18*x**4+324*x**3+108*x**2+216)**2,x)

[Out]

RootSum(589289589870088463413332668913549312*_t**6 - 539640290266075248405737472*_t**4 + 921826381685096823920
64*_t**3 - 553241442069170496*_t**2 - 3759837842016*_t - 7197829, Lambda(_t, _t*log(42996027639727447714003743
305160746111018438501025999323136*_t**5/154206009791052044490694380303237521 - 4258476625950819468468971547442
2251405157209835847680*_t**4/154206009791052044490694380303237521 - 375124461288495881501083694493237540783173
41082112*_t**3/154206009791052044490694380303237521 + 7152037594021675267638890715531672481920222144*_t**2/154
206009791052044490694380303237521 - 44227546998835297723830291794974310524032*_t/15420600979105204449069438030
3237521 + x - 174573349036676047734132569583024855/154206009791052044490694380303237521))) + (73*x**5 - 18*x**
4 + 908*x**3 + 432*x**2 - 96*x + 648)/(68364*x**6 + 1230552*x**4 + 22149936*x**3 + 7383312*x**2 + 14766624)

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