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

Optimal. Leaf size=1005 \[ \text{result too large to display} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 2.40257, antiderivative size = 1005, 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, 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 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 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 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 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 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^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*}

Mathematica [C]  time = 0.0298419, 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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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{73\,{x}^{5}}{68364}}-{\frac{{x}^{4}}{3798}}+{\frac{227\,{x}^{3}}{17091}}+{\frac{4\,{x}^{2}}{633}}-{\frac{8\,x}{5697}}+{\frac{2}{211}} \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 ( 73\,{{\it \_R}}^{4}-36\,{{\it \_R}}^{3}+96\,{{\it \_R}}^{2}-216\,{\it \_R}+96 \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^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(x-_R),_R=RootOf(_Z^6+18*_Z^4+324*_Z^3+10
8*_Z^2+216))

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

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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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^7/(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 = 0.381625, size = 112, normalized size = 0.11 \begin{align*} \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} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{7}}{{\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^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)