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

Optimal. Leaf size=873 \[ \text{result too large to display} \]

[Out]

((-6)^(1/3)*(2*(-3)^(1/3) + 9*2^(1/3)) - 3*x)/(157464*(8 - (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))*(6 - 3*(
-3)^(1/3)*2^(2/3)*x + x^2)) - ((-6)^(1/3)*(9*(-2)^(1/3) + 2*3^(1/3)) + 3*x)/(157464*(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^(1/3) - 3*6^(2/3) - 3^(1/3)*x)/(104976*(9*2^(
1/3) - 4*3^(1/3))*(6 + 3*2^(2/3)*3^(1/3)*x + x^2)) + ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2
/3)*2^(1/3))]]/(26244*Sqrt[3]*(8 - (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) - ((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)
)]])/(209952*(1 + (-1)^(1/3))^5*Sqrt[2*(4 - 3*(-3)^(2/3)*2^(1/3))]) - ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt
[6*(4 + 3*(-2)^(1/3)*3^(2/3))]]/(26244*Sqrt[3]*(8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) + ((9*I
+ 3^(1/3)*((4*I)*2^(2/3) - 9*3^(1/6)))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]]
)/(209952*(1 + (-1)^(1/3))^5*Sqrt[2*(4 + 3*(-2)^(1/3)*3^(2/3))]) - ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/S
qrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(52488*Sqrt[6]*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) + ((2*2^(2/3) - 3*3^(2/3))*Arc
Tanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]])/(944784*3^(1/6)*Sqrt[2*(-4 + 3*2^(1/
3)*3^(2/3))]) - ((I/23328)*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])/(46656*2^(2/3)*3^(5/6)*(1 + (-1)^(1/3))^5) + Log[6 + 3*2^(2/3
)*3^(1/3)*x + x^2]/(629856*2^(2/3)*3^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 1.91551, antiderivative size = 873, 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]{-6} \left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right )-3 x}{157464 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} 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 )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt{2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac{\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 )}{26244 \sqrt{3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac{\left (9 i+\sqrt [3]{3} \left (4 i 2^{2/3}-9 \sqrt [6]{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 )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt{2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}-\frac{\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 )}{26244 \sqrt{3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+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 (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{944784 \sqrt [6]{3} \sqrt{2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac{\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 )}{52488 \sqrt{6} \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 )}{23328\ 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 )}{46656\ 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 )}{629856\ 2^{2/3} \sqrt [3]{3}}-\frac{3 x+\sqrt [3]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )}{157464 \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]{3} x-3\ 6^{2/3}+2 \sqrt [3]{2}}{104976 \left (9 \sqrt [3]{2}-4 \sqrt [3]{3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-6)^(1/3)*(2*(-3)^(1/3) + 9*2^(1/3)) - 3*x)/(157464*(8 - (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))*(6 - 3*(
-3)^(1/3)*2^(2/3)*x + x^2)) - ((-6)^(1/3)*(9*(-2)^(1/3) + 2*3^(1/3)) + 3*x)/(157464*(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^(1/3) - 3*6^(2/3) - 3^(1/3)*x)/(104976*(9*2^(
1/3) - 4*3^(1/3))*(6 + 3*2^(2/3)*3^(1/3)*x + x^2)) + ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2
/3)*2^(1/3))]]/(26244*Sqrt[3]*(8 - (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) - ((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)
)]])/(209952*(1 + (-1)^(1/3))^5*Sqrt[2*(4 - 3*(-3)^(2/3)*2^(1/3))]) - ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt
[6*(4 + 3*(-2)^(1/3)*3^(2/3))]]/(26244*Sqrt[3]*(8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) + ((9*I
+ 3^(1/3)*((4*I)*2^(2/3) - 9*3^(1/6)))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]]
)/(209952*(1 + (-1)^(1/3))^5*Sqrt[2*(4 + 3*(-2)^(1/3)*3^(2/3))]) - ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/S
qrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(52488*Sqrt[6]*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) + ((2*2^(2/3) - 3*3^(2/3))*Arc
Tanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]])/(944784*3^(1/6)*Sqrt[2*(-4 + 3*2^(1/
3)*3^(2/3))]) - ((I/23328)*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])/(46656*2^(2/3)*3^(5/6)*(1 + (-1)^(1/3))^5) + Log[6 + 3*2^(2/3
)*3^(1/3)*x + x^2]/(629856*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 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^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx &=1586874322944 \int \left (-\frac{9 (-2)^{2/3}-\sqrt [3]{-1} 3^{2/3} x}{27763953154228224\ 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+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}{333167437850738688 \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]{-1} 3^{2/3} x}{27763953154228224\ 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-i \left (9 \sqrt{3}+2\ 2^{2/3} 3^{5/6}\right )\right )+3 \sqrt [3]{2} \sqrt [6]{3} \left (i+\sqrt{3}\right ) x}{666334875701477376 \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}+x}{9254651051409408\ 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}{2998506940656648192 \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}{1889568}-\frac{\int \frac{9\ 2^{2/3}-\sqrt [3]{-1} 3^{2/3} x}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{157464\ 2^{2/3}}-\frac{\int \frac{3\ 2^{2/3} \sqrt [3]{3}+x}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{52488\ 2^{2/3} \sqrt [3]{3}}+\frac{\int \frac{-2 \left (27-i \left (9 \sqrt{3}+2\ 2^{2/3} 3^{5/6}\right )\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}{419904 \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}{209952 \left (1+\sqrt [3]{-1}\right )^5}-\frac{\int \frac{9 (-2)^{2/3}-\sqrt [3]{-1} 3^{2/3} x}{\left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2} \, dx}{17496\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}\\ &=\frac{\sqrt [3]{-6} \left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right )-3 x}{157464 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\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 )+3 x}{314928 \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 \sqrt [3]{2}-3\ 6^{2/3}-\sqrt [3]{3} x}{104976 \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}{629856\ 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}{23328\ 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}{1889568}+\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}{46656\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac{\int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{104976 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}-\frac{\int \frac{1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{104976 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}+\frac{\int \frac{1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{52488 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\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}{419904 \left (1+\sqrt [3]{-1}\right )^5}-\frac{\left (18 (-1)^{2/3} \sqrt{3} \left (i+\sqrt{3}\right )+4 \left (27-i \left (9 \sqrt{3}+2\ 2^{2/3} 3^{5/6}\right )\right )\right ) \int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{839808 \left (1+\sqrt [3]{-1}\right )^5}\\ &=\frac{\sqrt [3]{-6} \left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right )-3 x}{157464 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\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 )+3 x}{314928 \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 \sqrt [3]{2}-3\ 6^{2/3}-\sqrt [3]{3} x}{104976 \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 )}{23328\ 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 )}{46656\ 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 )}{629856\ 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 )}{944784}+\frac{\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 )}{52488 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}+\frac{\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 )}{52488 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}-\frac{\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 )}{26244 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\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 )}{209952 \left (1+\sqrt [3]{-1}\right )^5}+\frac{\left (18 (-1)^{2/3} \sqrt{3} \left (i+\sqrt{3}\right )+4 \left (27-i \left (9 \sqrt{3}+2\ 2^{2/3} 3^{5/6}\right )\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 )}{419904 \left (1+\sqrt [3]{-1}\right )^5}\\ &=\frac{\sqrt [3]{-6} \left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right )-3 x}{157464 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\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 )+3 x}{314928 \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 \sqrt [3]{2}-3\ 6^{2/3}-\sqrt [3]{3} x}{104976 \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{\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 )}{26244 \sqrt{3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\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 )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac{\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 )}{52488 \sqrt{6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac{\left (27-9 i \sqrt{3}-4 i 2^{2/3} 3^{5/6}\right ) \tan ^{-1}\left (\frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}-\frac{\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 )}{52488 \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 )}{944784 \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 )}{23328\ 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 )}{46656\ 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 )}{629856\ 2^{2/3} \sqrt [3]{3}}\\ \end{align*}

Mathematica [C]  time = 0.0273065, size = 167, normalized size = 0.19 \[ \frac{\text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\& ,\frac{2 \text{$\#$1}^4 \log (x-\text{$\#$1})-27 \text{$\#$1}^3 \log (x-\text{$\#$1})+72 \text{$\#$1}^2 \log (x-\text{$\#$1})-162 \text{$\#$1} \log (x-\text{$\#$1})+1971 \log (x-\text{$\#$1})}{\text{$\#$1}^5+12 \text{$\#$1}^3+162 \text{$\#$1}^2+36 \text{$\#$1}}\& \right ]}{11074968}+\frac{4 x^5-27 x^4+96 x^3+648 x^2-3942 x+972}{3691656 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(972 - 3942*x + 648*x^2 + 96*x^3 - 27*x^4 + 4*x^5)/(3691656*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)) + RootSu
m[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (1971*Log[x - #1] - 162*Log[x - #1]*#1 + 72*Log[x - #1]*#1^2
- 27*Log[x - #1]*#1^3 + 2*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/11074968

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Maple [C]  time = 0.009, 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}}{922914}}-{\frac{{x}^{4}}{136728}}+{\frac{4\,{x}^{3}}{153819}}+{\frac{{x}^{2}}{5697}}-{\frac{73\,x}{68364}}+{\frac{1}{3798}} \right ) }+{\frac{1}{11074968}\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 ( 2\,{{\it \_R}}^{4}-27\,{{\it \_R}}^{3}+72\,{{\it \_R}}^{2}-162\,{\it \_R}+1971 \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^3/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x)

[Out]

(1/922914*x^5-1/136728*x^4+4/153819*x^3+1/5697*x^2-73/68364*x+1/3798)/(x^6+18*x^4+324*x^3+108*x^2+216)+1/11074
968*sum((2*_R^4-27*_R^3+72*_R^2-162*_R+1971)/(_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{4 \, x^{5} - 27 \, x^{4} + 96 \, x^{3} + 648 \, x^{2} - 3942 \, x + 972}{3691656 \,{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} + \frac{1}{1845828} \, \int \frac{2 \, x^{4} - 27 \, x^{3} + 72 \, x^{2} - 162 \, x + 1971}{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^3/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="maxima")

[Out]

1/3691656*(4*x^5 - 27*x^4 + 96*x^3 + 648*x^2 - 3942*x + 972)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216) + 1/1845
828*integrate((2*x^4 - 27*x^3 + 72*x^2 - 162*x + 1971)/(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^3/(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 = 0.371063, size = 112, normalized size = 0.13 \begin{align*} \operatorname{RootSum}{\left (1282755170017893101915524820582750453426552832 t^{6} - 906388465775544244426251149770752 t^{4} - 4300873166389987741684137984 t^{3} - 717000908921644962816 t^{2} + 135354162312576 t - 7197829, \left ( t \mapsto t \log{\left (\frac{17257935592810449901409556597891882995604001083339368041361480613888 t^{5}}{154206009791052044490694380303237521} + \frac{2389607400620985524376358853572652207181956324560587684052992 t^{4}}{154206009791052044490694380303237521} - \frac{12286072160883283930711715948878260078996992193488388096 t^{3}}{154206009791052044490694380303237521} - \frac{59490553573959173161125496013527909754156558410752 t^{2}}{154206009791052044490694380303237521} - \frac{17520149679836691112367064197713753004827200 t}{154206009791052044490694380303237521} + x + \frac{766422988707229615055855287040887332}{154206009791052044490694380303237521} \right )} \right )\right )} + \frac{4 x^{5} - 27 x^{4} + 96 x^{3} + 648 x^{2} - 3942 x + 972}{3691656 x^{6} + 66449808 x^{4} + 1196096544 x^{3} + 398698848 x^{2} + 797397696} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(1282755170017893101915524820582750453426552832*_t**6 - 906388465775544244426251149770752*_t**4 - 43008
73166389987741684137984*_t**3 - 717000908921644962816*_t**2 + 135354162312576*_t - 7197829, Lambda(_t, _t*log(
17257935592810449901409556597891882995604001083339368041361480613888*_t**5/15420600979105204449069438030323752
1 + 2389607400620985524376358853572652207181956324560587684052992*_t**4/154206009791052044490694380303237521 -
12286072160883283930711715948878260078996992193488388096*_t**3/154206009791052044490694380303237521 - 5949055
3573959173161125496013527909754156558410752*_t**2/154206009791052044490694380303237521 - 175201496798366911123
67064197713753004827200*_t/154206009791052044490694380303237521 + x + 766422988707229615055855287040887332/154
206009791052044490694380303237521))) + (4*x**5 - 27*x**4 + 96*x**3 + 648*x**2 - 3942*x + 972)/(3691656*x**6 +
66449808*x**4 + 1196096544*x**3 + 398698848*x**2 + 797397696)

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

[Out]

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