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

Optimal. Leaf size=986 \[ \text{result too large to display} \]

[Out]

-(27*((-2)^(2/3) + 2*(-1)^(1/3)*3^(2/3)) - 6^(1/3)*(9 + (-3)^(1/3)*2^(2/3))*x)/(104976*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)) - (27*2^(2/3)*(1 + (-2)^(1/3)*3^(2/3)) - (-
1)^(1/3)*3^(2/3)*(2 + 3*(-2)^(1/3)*3^(2/3))*x)/(472392*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 - 3*2^(1/3)*3^(2/3))*x)/(314928*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)) - ((1 + I*Sqrt[3] + 3*2^(1/3)*3^(2/3))*ArcTan[
(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(8748*2^(2/3)*3^(5/6)*(1 + (-1)^(1/3))^4*(8
- (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) + ((3*(-3)^(2/3) + (-1)^(1/3)*2^(2/3))*ArcTan[(3*(-2)^(2/3
)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(17496*6^(5/6)*(1 - (-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(4
+ 3*(-2)^(1/3)*3^(2/3))^(3/2)) + ((I + Sqrt[3])*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3
^(2/3))]])/(34992*2^(1/6)*3^(1/3)*(1 + (-1)^(1/3))^5*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) + ((I/17496)*ArcTan[(2^(1
/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(2^(1/6)*3^(1/3)*(1 + (-1)^(1/3))^5*Sqrt[
4 - 3*(-3)^(2/3)*2^(1/3)]) - ((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))]])/(17496*6^(5/6)*(1 - (-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) - ArcT
anh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(157464*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^
(1/3)*3^(2/3)]) + ((I + Sqrt[3])*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(419904*2^(1/3)*3^(1/6)*(1 + (-1)^(1/3
))^5) - ((I/209952)*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]/(1889568*2^(1/3)*3^(2/3))

________________________________________________________________________________________

Rubi [A]  time = 1.92698, antiderivative size = 986, 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{27 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right )-\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{104976\ 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 (1+i \sqrt{3}+3 \sqrt [3]{2} 3^{2/3}\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 )}{8748\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac{\left (i+\sqrt{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 )}{34992 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac{\left (3 (-3)^{2/3}+\sqrt [3]{-1} 2^{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 )}{17496\ 6^{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 \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 )}{17496 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\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 )}{157464 \sqrt [6]{2} 3^{5/6} \sqrt{-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac{\left (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 )}{17496\ 6^{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 (i+\sqrt{3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{419904 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac{i \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{209952 \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 )}{1889568 \sqrt [3]{2} 3^{2/3}}-\frac{27\ 2^{2/3} \left (1+\sqrt [3]{-2} 3^{2/3}\right )-\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{472392\ 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{9 \left (6-2^{2/3} \sqrt [3]{3}\right )-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{314928\ 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^2/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]

[Out]

-(27*((-2)^(2/3) + 2*(-1)^(1/3)*3^(2/3)) - 6^(1/3)*(9 + (-3)^(1/3)*2^(2/3))*x)/(104976*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)) - (27*2^(2/3)*(1 + (-2)^(1/3)*3^(2/3)) - (-
1)^(1/3)*3^(2/3)*(2 + 3*(-2)^(1/3)*3^(2/3))*x)/(472392*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 - 3*2^(1/3)*3^(2/3))*x)/(314928*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)) - ((1 + I*Sqrt[3] + 3*2^(1/3)*3^(2/3))*ArcTan[
(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(8748*2^(2/3)*3^(5/6)*(1 + (-1)^(1/3))^4*(8
- (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) + ((3*(-3)^(2/3) + (-1)^(1/3)*2^(2/3))*ArcTan[(3*(-2)^(2/3
)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(17496*6^(5/6)*(1 - (-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(4
+ 3*(-2)^(1/3)*3^(2/3))^(3/2)) + ((I + Sqrt[3])*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3
^(2/3))]])/(34992*2^(1/6)*3^(1/3)*(1 + (-1)^(1/3))^5*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) + ((I/17496)*ArcTan[(2^(1
/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(2^(1/6)*3^(1/3)*(1 + (-1)^(1/3))^5*Sqrt[
4 - 3*(-3)^(2/3)*2^(1/3)]) - ((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))]])/(17496*6^(5/6)*(1 - (-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) - ArcT
anh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(157464*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^
(1/3)*3^(2/3)]) + ((I + Sqrt[3])*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(419904*2^(1/3)*3^(1/6)*(1 + (-1)^(1/3
))^5) - ((I/209952)*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]/(1889568*2^(1/3)*3^(2/3))

Rule 2097

Int[(Q6_)^(p_)*(u_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coe
ff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Dist[1/(3^(3*p)*a^(2*p)), Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c,
3]*x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3*(-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x
+ b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] &&
EqQ[Coeff[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^2}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx &=1586874322944 \int \left (\frac{2 \sqrt [3]{-1} 3^{2/3}+18 \sqrt [3]{6}+3 (-2)^{2/3} x}{55527906308456448\ 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{i \left (18\ 3^{5/6}+\sqrt [3]{2} \left (3 i-\sqrt{3}\right ) x\right )}{333167437850738688\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac{2 \sqrt [3]{-1} 3^{2/3}+18 (-1)^{2/3} \sqrt [3]{6}+3\ 2^{2/3} x}{55527906308456448\ 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{9+9 \sqrt [3]{-1}-i \sqrt [3]{2} \sqrt [6]{3} x}{166583718925369344\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac{-2+6 \sqrt [3]{2} 3^{2/3}+2^{2/3} \sqrt [3]{3} x}{18509302102818816\ 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{9 \sqrt [3]{3}+\sqrt [3]{2} x}{1499253470328324096\ 6^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{2 \sqrt [3]{-1} 3^{2/3}+18 (-1)^{2/3} \sqrt [3]{6}+3\ 2^{2/3} x}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{314928\ 2^{2/3}}+\frac{\int \frac{-2+6 \sqrt [3]{2} 3^{2/3}+2^{2/3} \sqrt [3]{3} x}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{104976\ 2^{2/3} \sqrt [3]{3}}+\frac{\int \frac{9 \sqrt [3]{3}+\sqrt [3]{2} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{944784\ 6^{2/3}}+\frac{\int \frac{9+9 \sqrt [3]{-1}-i \sqrt [3]{2} \sqrt [6]{3} x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{104976\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac{i \int \frac{18\ 3^{5/6}+\sqrt [3]{2} \left (3 i-\sqrt{3}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{209952\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac{\int \frac{2 \sqrt [3]{-1} 3^{2/3}+18 \sqrt [3]{6}+3 (-2)^{2/3} x}{\left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2} \, dx}{34992\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}\\ &=-\frac{27 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right )-\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{104976\ 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{27\ 2^{2/3} \left (1+\sqrt [3]{-2} 3^{2/3}\right )-\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{944784\ 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 (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{314928\ 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}{1889568 \sqrt [3]{2} 3^{2/3}}+\frac{\int \frac{1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{157464\ 2^{2/3} \sqrt [3]{3}}-\frac{i \int \frac{1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{5832\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\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}{209952 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac{\left (i+\sqrt{3}\right ) \int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{11664\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac{\left (i+\sqrt{3}\right ) \int \frac{-3 \sqrt [3]{-3} 2^{2/3}+2 x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{419904 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac{\left (2\ 3^{2/3}-9 \sqrt [3]{6}\right ) \int \frac{1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1889568 \left (2\ 2^{2/3}-3\ 3^{2/3}\right )}--\frac{\left (-18 \sqrt [3]{-6} (-1)^{2/3}-2 \left (2 \sqrt [3]{-1} 3^{2/3}+18 \sqrt [3]{6}\right )\right ) \int \frac{1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{34992\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (24-18 (-3)^{2/3} \sqrt [3]{2}\right )}+\frac{\left (-18 (-1)^{2/3} \sqrt [3]{6}+2 \left (2 \sqrt [3]{-1} 3^{2/3}+18 (-1)^{2/3} \sqrt [3]{6}\right )\right ) \int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{314928\ 2^{2/3} \left (24+18 \sqrt [3]{-2} 3^{2/3}\right )}\\ &=-\frac{27 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right )-\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{104976\ 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{27\ 2^{2/3} \left (1+\sqrt [3]{-2} 3^{2/3}\right )-\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{944784\ 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 (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{314928\ 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 (i+\sqrt{3}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{419904 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac{i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{209952 \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 )}{1889568 \sqrt [3]{2} 3^{2/3}}-\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 )}{78732\ 2^{2/3} \sqrt [3]{3}}+\frac{i \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 )}{2916\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac{\left (i+\sqrt{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 )}{5832\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac{\left (2\ 3^{2/3}-9 \sqrt [3]{6}\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 \left (2\ 2^{2/3}-3\ 3^{2/3}\right )}+-\frac{\left (-18 \sqrt [3]{-6} (-1)^{2/3}-2 \left (2 \sqrt [3]{-1} 3^{2/3}+18 \sqrt [3]{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 )}{17496\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (24-18 (-3)^{2/3} \sqrt [3]{2}\right )}-\frac{\left (-18 (-1)^{2/3} \sqrt [3]{6}+2 \left (2 \sqrt [3]{-1} 3^{2/3}+18 (-1)^{2/3} \sqrt [3]{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 )}{157464\ 2^{2/3} \left (24+18 \sqrt [3]{-2} 3^{2/3}\right )}\\ &=-\frac{27 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right )-\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{104976\ 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{27\ 2^{2/3} \left (1+\sqrt [3]{-2} 3^{2/3}\right )-\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{944784\ 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 (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{314928\ 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 (2 \sqrt [3]{-1}+3 \sqrt [3]{2} 3^{2/3}\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 )}{34992 \sqrt [6]{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 (3 (-3)^{2/3} \sqrt [6]{2}+\sqrt [3]{-1} 2^{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 )}{314928\ 3^{5/6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}+\frac{\left (i+\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 )}{34992 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac{i \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 )}{17496 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (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 )}{157464\ 6^{5/6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\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 )}{157464 \sqrt [6]{2} 3^{5/6} \sqrt{-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac{\left (i+\sqrt{3}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{419904 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac{i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{209952 \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 )}{1889568 \sqrt [3]{2} 3^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0340482, size = 167, normalized size = 0.17 \[ \frac{-9 x^5+8 x^4-216 x^3-2724 x^2+324 x-7884}{7383312 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )}-\frac{\text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\& ,\frac{9 \text{$\#$1}^4 \log (x-\text{$\#$1})-16 \text{$\#$1}^3 \log (x-\text{$\#$1})+324 \text{$\#$1}^2 \log (x-\text{$\#$1})+2436 \text{$\#$1} \log (x-\text{$\#$1})+324 \log (x-\text{$\#$1})}{\text{$\#$1}^5+12 \text{$\#$1}^3+162 \text{$\#$1}^2+36 \text{$\#$1}}\& \right ]}{44299872} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-7884 + 324*x - 2724*x^2 - 216*x^3 + 8*x^4 - 9*x^5)/(7383312*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)) - Root
Sum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (324*Log[x - #1] + 2436*Log[x - #1]*#1 + 324*Log[x - #1]*#1
^2 - 16*Log[x - #1]*#1^3 + 9*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/44299872

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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}}{820368}}+{\frac{{x}^{4}}{922914}}-{\frac{{x}^{3}}{34182}}-{\frac{227\,{x}^{2}}{615276}}+{\frac{x}{22788}}-{\frac{73}{68364}} \right ) }+{\frac{1}{44299872}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{ \left ( -9\,{{\it \_R}}^{4}+16\,{{\it \_R}}^{3}-324\,{{\it \_R}}^{2}-2436\,{\it \_R}-324 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/820368*x^5+1/922914*x^4-1/34182*x^3-227/615276*x^2+1/22788*x-73/68364)/(x^6+18*x^4+324*x^3+108*x^2+216)+1/
44299872*sum((-9*_R^4+16*_R^3-324*_R^2-2436*_R-324)/(_R^5+12*_R^3+162*_R^2+36*_R)*ln(x-_R),_R=RootOf(_Z^6+18*_
Z^4+324*_Z^3+108*_Z^2+216))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{9 \, x^{5} - 8 \, x^{4} + 216 \, x^{3} + 2724 \, x^{2} - 324 \, x + 7884}{7383312 \,{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} - \frac{1}{7383312} \, \int \frac{9 \, x^{4} - 16 \, x^{3} + 324 \, x^{2} + 2436 \, x + 324}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7383312*(9*x^5 - 8*x^4 + 216*x^3 + 2724*x^2 - 324*x + 7884)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216) - 1/73
83312*integrate((9*x^4 - 16*x^3 + 324*x^2 + 2436*x + 324)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(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 = 0.362038, size = 112, normalized size = 0.11 \begin{align*} \operatorname{RootSum}{\left (8658597397620778437929792538933565560629231616 t^{6} + 109068095871770168248838645612544 t^{4} - 492655707593366915713499136 t^{3} + 40378331745144603648 t^{2} - 695635011360 t + 4513, \left ( t \mapsto t \log{\left (\frac{101442531561804181113161287039859349851881619653631712165888 t^{5}}{356900697070792948475845} - \frac{149796550082359335112709434971975088967050210050048 t^{4}}{356900697070792948475845} + \frac{1222409754458272818505898777768670783617236992 t^{3}}{356900697070792948475845} - \frac{5775055524251595723022901938558261453824 t^{2}}{356900697070792948475845} + \frac{96165242200260265765603930470432 t}{71380139414158589695169} + x - \frac{17059152341129698120545584}{1070702091212378845427535} \right )} \right )\right )} - \frac{9 x^{5} - 8 x^{4} + 216 x^{3} + 2724 x^{2} - 324 x + 7884}{7383312 x^{6} + 132899616 x^{4} + 2392193088 x^{3} + 797397696 x^{2} + 1594795392} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(8658597397620778437929792538933565560629231616*_t**6 + 109068095871770168248838645612544*_t**4 - 49265
5707593366915713499136*_t**3 + 40378331745144603648*_t**2 - 695635011360*_t + 4513, Lambda(_t, _t*log(10144253
1561804181113161287039859349851881619653631712165888*_t**5/356900697070792948475845 - 149796550082359335112709
434971975088967050210050048*_t**4/356900697070792948475845 + 1222409754458272818505898777768670783617236992*_t
**3/356900697070792948475845 - 5775055524251595723022901938558261453824*_t**2/356900697070792948475845 + 96165
242200260265765603930470432*_t/71380139414158589695169 + x - 17059152341129698120545584/1070702091212378845427
535))) - (9*x**5 - 8*x**4 + 216*x**3 + 2724*x**2 - 324*x + 7884)/(7383312*x**6 + 132899616*x**4 + 2392193088*x
**3 + 797397696*x**2 + 1594795392)

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

[Out]

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