3.154 \(\int \frac{x^5}{(216+108 x^2+324 x^3+18 x^4+x^6)^2} \, dx\)

Optimal. Leaf size=682 \[ \frac{\sqrt [3]{-\frac{1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac{\sqrt [3]{-\frac{1}{3}} \left ((-2)^{2/3} \sqrt [3]{3} x+4\right )}{8748\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac{2^{2/3} \sqrt [3]{3} x+4}{17496\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374 \sqrt{3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{4374 \sqrt{3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac{i \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1458\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{39366\ 2^{5/6} \sqrt [6]{3} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{8748 \sqrt{6} \left (3 \sqrt [3]{2} 3^{2/3}-4\right )^{3/2}} \]

[Out]

((-1/3)^(1/3)*(4 - (-3)^(1/3)*2^(2/3)*x))/(1944*2^(2/3)*(1 + (-1)^(1/3))^4*(4 - 3*(-3)^(2/3)*2^(1/3))*(6 - 3*(
-3)^(1/3)*2^(2/3)*x + x^2)) + ((-1/3)^(1/3)*(4 + (-2)^(2/3)*3^(1/3)*x))/(8748*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)) - (4 + 2^(2/3)*3^(1/3)*x)/(17496*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)) - ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-
3)^(2/3)*2^(1/3))]]/(4374*2^(5/6)*3^(1/6)*(1 + (-1)^(1/3))^4*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ArcTan[(3*(-3)^
(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]]/(4374*Sqrt[3]*(8 - (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*
3^(2/3))^(3/2)) - ((I/1458)*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(2^(5/6)*
3^(2/3)*(1 + (-1)^(1/3))^5*Sqrt[4 + 3*(-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))]]/(4374*Sqrt[3]*(8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) - ArcTanh[(2^(1/6)
*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(8748*Sqrt[6]*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) - Ar
cTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(39366*2^(5/6)*3^(1/6)*Sqrt[-4 + 3*2
^(1/3)*3^(2/3)])

________________________________________________________________________________________

Rubi [A]  time = 1.23518, antiderivative size = 682, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2097, 638, 618, 204, 206} \[ \frac{\sqrt [3]{-\frac{1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac{\sqrt [3]{-\frac{1}{3}} \left ((-2)^{2/3} \sqrt [3]{3} x+4\right )}{8748\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac{2^{2/3} \sqrt [3]{3} x+4}{17496\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374 \sqrt{3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{4374 \sqrt{3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac{i \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1458\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{39366\ 2^{5/6} \sqrt [6]{3} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{8748 \sqrt{6} \left (3 \sqrt [3]{2} 3^{2/3}-4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]

[Out]

((-1/3)^(1/3)*(4 - (-3)^(1/3)*2^(2/3)*x))/(1944*2^(2/3)*(1 + (-1)^(1/3))^4*(4 - 3*(-3)^(2/3)*2^(1/3))*(6 - 3*(
-3)^(1/3)*2^(2/3)*x + x^2)) + ((-1/3)^(1/3)*(4 + (-2)^(2/3)*3^(1/3)*x))/(8748*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)) - (4 + 2^(2/3)*3^(1/3)*x)/(17496*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)) - ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-
3)^(2/3)*2^(1/3))]]/(4374*2^(5/6)*3^(1/6)*(1 + (-1)^(1/3))^4*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ArcTan[(3*(-3)^
(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]]/(4374*Sqrt[3]*(8 - (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*
3^(2/3))^(3/2)) - ((I/1458)*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(2^(5/6)*
3^(2/3)*(1 + (-1)^(1/3))^5*Sqrt[4 + 3*(-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))]]/(4374*Sqrt[3]*(8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) - ArcTanh[(2^(1/6)
*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(8748*Sqrt[6]*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) - Ar
cTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(39366*2^(5/6)*3^(1/6)*Sqrt[-4 + 3*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 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^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx &=1586874322944 \int \left (-\frac{\sqrt [3]{-\frac{1}{3}} x}{1542441841901568\ 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{1}{4627325525704704 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )}-\frac{\sqrt [3]{-\frac{1}{3}} x}{1542441841901568\ 2^{2/3} \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2}-\frac{i}{4627325525704704 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac{x}{1542441841901568\ 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{1}{41645929731342336 \sqrt [3]{2} 3^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=-\frac{\sqrt [3]{-\frac{1}{3}} \int \frac{x}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{8748\ 2^{2/3}}+\frac{\int \frac{1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{26244 \sqrt [3]{2} 3^{2/3}}+\frac{\int \frac{x}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{8748\ 2^{2/3} \sqrt [3]{3}}-\frac{i \int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{2916 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac{\sqrt [3]{-\frac{1}{3}} \int \frac{x}{\left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2} \, dx}{972\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}-\frac{\int \frac{1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{2916 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}\\ &=\frac{\sqrt [3]{-\frac{1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac{\sqrt [3]{-\frac{1}{3}} \left (4+(-2)^{2/3} \sqrt [3]{3} x\right )}{17496\ 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{4+2^{2/3} \sqrt [3]{3} x}{17496\ 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{\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 )}{13122 \sqrt [3]{2} 3^{2/3}}+\frac{i \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 )}{1458 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\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 )}{1458 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}-\frac{\int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{17496 \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}{17496 \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}{8748 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}\\ &=\frac{\sqrt [3]{-\frac{1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac{\sqrt [3]{-\frac{1}{3}} \left (4+(-2)^{2/3} \sqrt [3]{3} x\right )}{17496\ 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{4+2^{2/3} \sqrt [3]{3} x}{17496\ 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{\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 )}{4374\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{i \tan ^{-1}\left (\frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1458\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}-\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 )}{39366\ 2^{5/6} \sqrt [6]{3} \sqrt{-4+3 \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 )}{8748 \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 )}{8748 \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 )}{4374 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}\\ &=\frac{\sqrt [3]{-\frac{1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac{\sqrt [3]{-\frac{1}{3}} \left (4+(-2)^{2/3} \sqrt [3]{3} x\right )}{17496\ 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{4+2^{2/3} \sqrt [3]{3} x}{17496\ 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{\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 )}{4374\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}+\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 )}{4374 \sqrt{3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\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 )}{8748 \sqrt{6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac{i \tan ^{-1}\left (\frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1458\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}-\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 )}{8748 \sqrt{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 )}{39366\ 2^{5/6} \sqrt [6]{3} \sqrt{-4+3 \sqrt [3]{2} 3^{2/3}}}\\ \end{align*}

Mathematica [C]  time = 0.0247175, size = 167, normalized size = 0.24 \[ \frac{\text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\& ,\frac{4 \text{$\#$1}^4 \log (x-\text{$\#$1})-54 \text{$\#$1}^3 \log (x-\text{$\#$1})+2043 \text{$\#$1}^2 \log (x-\text{$\#$1})-324 \text{$\#$1} \log (x-\text{$\#$1})+144 \log (x-\text{$\#$1})}{\text{$\#$1}^5+12 \text{$\#$1}^3+162 \text{$\#$1}^2+36 \text{$\#$1}}\& \right ]}{3691656}+\frac{4 x^5-27 x^4+729 x^3+648 x^2-144 x+972}{615276 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^5/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]

[Out]

(972 - 144*x + 648*x^2 + 729*x^3 - 27*x^4 + 4*x^5)/(615276*(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 & , (144*Log[x - #1] - 324*Log[x - #1]*#1 + 2043*Log[x - #1]*#1^2
- 54*Log[x - #1]*#1^3 + 4*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/3691656

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Maple [C]  time = 0.008, size = 122, normalized size = 0.2 \begin{align*}{\frac{1}{{x}^{6}+18\,{x}^{4}+324\,{x}^{3}+108\,{x}^{2}+216} \left ({\frac{{x}^{5}}{153819}}-{\frac{{x}^{4}}{22788}}+{\frac{{x}^{3}}{844}}+{\frac{2\,{x}^{2}}{1899}}-{\frac{4\,x}{17091}}+{\frac{1}{633}} \right ) }+{\frac{1}{3691656}\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 ( 4\,{{\it \_R}}^{4}-54\,{{\it \_R}}^{3}+2043\,{{\it \_R}}^{2}-324\,{\it \_R}+144 \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^5/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x)

[Out]

(1/153819*x^5-1/22788*x^4+1/844*x^3+2/1899*x^2-4/17091*x+1/633)/(x^6+18*x^4+324*x^3+108*x^2+216)+1/3691656*sum
((4*_R^4-54*_R^3+2043*_R^2-324*_R+144)/(_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} + 729 \, x^{3} + 648 \, x^{2} - 144 \, x + 972}{615276 \,{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} + \frac{1}{615276} \, \int \frac{4 \, x^{4} - 54 \, x^{3} + 2043 \, x^{2} - 324 \, x + 144}{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^5/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="maxima")

[Out]

1/615276*(4*x^5 - 27*x^4 + 729*x^3 + 648*x^2 - 144*x + 972)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216) + 1/61527
6*integrate((4*x^4 - 54*x^3 + 2043*x^2 - 324*x + 144)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)

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Fricas [B]  time = 9.74688, size = 8841, normalized size = 12.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="fricas")

[Out]

1/28041818976*(182304*x^5 - 1230552*x^4 + 33224904*x^3 + 422*sqrt(1/633)*(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 2
16)*sqrt(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)*log(2/1982119441*sqrt(1/633)*(7238020557*(5034474*18^
(2/3) + 9367856*18^(1/3) + 44687457)^2 - 4479023748400406176979673*18^(2/3) - 8334306522507661258645112*18^(1/
3) - 26862559811422885347120477)*sqrt(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457) - 7383041510/9393931*(50
34474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 247458158879850620*x + 5132255454960803463351330/9393931*18^
(2/3) + 9549802036377046040753520/9393931*18^(1/3) + 27278928233033940032425830/9393931) - 422*sqrt(1/633)*(x^
6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)*sqrt(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)*log(-2/1982119441*s
qrt(1/633)*(7238020557*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 - 4479023748400406176979673*18^(2/3)
- 8334306522507661258645112*18^(1/3) - 26862559811422885347120477)*sqrt(5034474*18^(2/3) + 9367856*18^(1/3) +
44687457) - 7383041510/9393931*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 247458158879850620*x + 51
32255454960803463351330/9393931*18^(2/3) + 9549802036377046040753520/9393931*18^(1/3) + 2727892823303394003242
5830/9393931) - 9*sqrt(422)*(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)*sqrt(-20718*18^(2/3) + sqrt(-1/19683*(503
4474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*18^(1/3) + 2739749626
99) - 9367856/243*18^(1/3) + 367798)*log(14766083020/211*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 +
3064230/211*sqrt(-1/19683*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 34454787
01088/81*18^(1/3) + 273974962699)*(5895278433468*18^(2/3) + 10969590754592*18^(1/3) + 57028339027521) + 9/9393
931*(14476041114*sqrt(422)*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 243*sqrt(-1/19683*(5034474*18^
(2/3) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*18^(1/3) + 273974962699)*(144
76041114*sqrt(422)*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457) - 161351097450615865*sqrt(422)) - 17793412
96985705429*sqrt(422)*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457) + 26505855880569051992480475*sqrt(422))
*sqrt(-20718*18^(2/3) + sqrt(-1/19683*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3
) + 3445478701088/81*18^(1/3) + 273974962699) - 9367856/243*18^(1/3) + 367798) + 44068338765959317812080*x - 1
0264510909921606926702660/211*18^(2/3) - 19099604072754092081507040/211*18^(1/3) - 54557856466067880064851660/
211) + 9*sqrt(422)*(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)*sqrt(-20718*18^(2/3) + sqrt(-1/19683*(5034474*18^(
2/3) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*18^(1/3) + 273974962699) - 936
7856/243*18^(1/3) + 367798)*log(14766083020/211*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 3064230/2
11*sqrt(-1/19683*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*
18^(1/3) + 273974962699)*(5895278433468*18^(2/3) + 10969590754592*18^(1/3) + 57028339027521) - 9/9393931*(1447
6041114*sqrt(422)*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 243*sqrt(-1/19683*(5034474*18^(2/3) + 9
367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*18^(1/3) + 273974962699)*(14476041114*
sqrt(422)*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457) - 161351097450615865*sqrt(422)) - 17793412969857054
29*sqrt(422)*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457) + 26505855880569051992480475*sqrt(422))*sqrt(-20
718*18^(2/3) + sqrt(-1/19683*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 34454
78701088/81*18^(1/3) + 273974962699) - 9367856/243*18^(1/3) + 367798) + 44068338765959317812080*x - 1026451090
9921606926702660/211*18^(2/3) - 19099604072754092081507040/211*18^(1/3) - 54557856466067880064851660/211) - 9*
sqrt(422)*(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)*sqrt(-20718*18^(2/3) - sqrt(-1/19683*(5034474*18^(2/3) + 93
67856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*18^(1/3) + 273974962699) - 9367856/243*
18^(1/3) + 367798)*log(14766083020/211*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 - 3064230/211*sqrt(-
1/19683*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*18^(1/3)
+ 273974962699)*(5895278433468*18^(2/3) + 10969590754592*18^(1/3) + 57028339027521) + 9/9393931*(14476041114*s
qrt(422)*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 - 243*sqrt(-1/19683*(5034474*18^(2/3) + 9367856*18
^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*18^(1/3) + 273974962699)*(14476041114*sqrt(422)
*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457) - 161351097450615865*sqrt(422)) - 1779341296985705429*sqrt(4
22)*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457) + 26505855880569051992480475*sqrt(422))*sqrt(-20718*18^(2
/3) - sqrt(-1/19683*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/
81*18^(1/3) + 273974962699) - 9367856/243*18^(1/3) + 367798) + 44068338765959317812080*x - 1026451090992160692
6702660/211*18^(2/3) - 19099604072754092081507040/211*18^(1/3) - 54557856466067880064851660/211) + 9*sqrt(422)
*(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)*sqrt(-20718*18^(2/3) - sqrt(-1/19683*(5034474*18^(2/3) + 9367856*18^
(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*18^(1/3) + 273974962699) - 9367856/243*18^(1/3)
+ 367798)*log(14766083020/211*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 - 3064230/211*sqrt(-1/19683*(
5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*18^(1/3) + 2739749
62699)*(5895278433468*18^(2/3) + 10969590754592*18^(1/3) + 57028339027521) - 9/9393931*(14476041114*sqrt(422)*
(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 - 243*sqrt(-1/19683*(5034474*18^(2/3) + 9367856*18^(1/3) +
44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*18^(1/3) + 273974962699)*(14476041114*sqrt(422)*(5034474
*18^(2/3) + 9367856*18^(1/3) + 44687457) - 161351097450615865*sqrt(422)) - 1779341296985705429*sqrt(422)*(5034
474*18^(2/3) + 9367856*18^(1/3) + 44687457) + 26505855880569051992480475*sqrt(422))*sqrt(-20718*18^(2/3) - sqr
t(-1/19683*(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/81*18^(1/
3) + 273974962699) - 9367856/243*18^(1/3) + 367798) + 44068338765959317812080*x - 10264510909921606926702660/2
11*18^(2/3) - 19099604072754092081507040/211*18^(1/3) - 54557856466067880064851660/211) + 29533248*x^2 - 65629
44*x + 44299872)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)

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Sympy [A]  time = 0.281599, size = 104, normalized size = 0.15 \begin{align*} \operatorname{RootSum}{\left (27493895104978847349012449000830556700672 t^{6} - 1318718189226950088862983192576 t^{4} + 12120917704776776448 t^{2} - 39753025, \left ( t \mapsto t \log{\left (\frac{947842259001288723909832054550209950242045952 t^{5}}{61864539719962655} - \frac{243458646817775607639654889480814592 t^{4}}{9811980923071} - \frac{41682556475067500431787310779667456 t^{3}}{61864539719962655} + \frac{12026877442664328616462272 t^{2}}{9811980923071} + \frac{216142618488859793668428 t}{61864539719962655} + x - \frac{308574300024117}{39247923692284} \right )} \right )\right )} + \frac{4 x^{5} - 27 x^{4} + 729 x^{3} + 648 x^{2} - 144 x + 972}{615276 x^{6} + 11074968 x^{4} + 199349424 x^{3} + 66449808 x^{2} + 132899616} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(x**6+18*x**4+324*x**3+108*x**2+216)**2,x)

[Out]

RootSum(27493895104978847349012449000830556700672*_t**6 - 1318718189226950088862983192576*_t**4 + 121209177047
76776448*_t**2 - 39753025, Lambda(_t, _t*log(947842259001288723909832054550209950242045952*_t**5/6186453971996
2655 - 243458646817775607639654889480814592*_t**4/9811980923071 - 41682556475067500431787310779667456*_t**3/61
864539719962655 + 12026877442664328616462272*_t**2/9811980923071 + 216142618488859793668428*_t/618645397199626
55 + x - 308574300024117/39247923692284))) + (4*x**5 - 27*x**4 + 729*x**3 + 648*x**2 - 144*x + 972)/(615276*x*
*6 + 11074968*x**4 + 199349424*x**3 + 66449808*x**2 + 132899616)

________________________________________________________________________________________

Giac [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^5/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="giac")

[Out]

Timed out