\(\int \frac {x^3}{(216+108 x^2+324 x^3+18 x^4+x^6)^2} \, dx\) [156]

Optimal result

Integrand size = 26, antiderivative size = 873 \[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\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}{157464 \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+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 \left (9 \sqrt [3]{2}-4 \sqrt [3]{3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\arctan \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 (2 i 2^{2/3}+9 \sqrt [6]{3}+2\ 2^{2/3} \sqrt {3}\right )\right ) \arctan \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 {\arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\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 (9 i+\sqrt [3]{3} \left (4 i 2^{2/3}-9 \sqrt [6]{3}\right )\right ) \arctan \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 {2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}-\frac {\text {arctanh}\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 ) \text {arctanh}\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}} \]

[Out]

1/157464*((-6)^(1/3)*(2*(-3)^(1/3)+9*2^(1/3))-3*x)/(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)+1/157464*(-(-6)^(1/3)*(9*(-2)^(1/3)+2*3^(1/3))-3*x)/(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))/(6
+3*(-2)^(2/3)*3^(1/3)*x+x^2)+1/104976*(-2*2^(1/3)+3*6^(2/3)+3^(1/3)*x)/(9*2^(1/3)-4*3^(1/3))/(6+3*2^(2/3)*3^(1
/3)*x+x^2)-1/139968*I*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(1/3)*3^(1/6)/(1+(-1)^(1/3))^5+1/3779136*ln(6+3*2^(2/
3)*3^(1/3)*x+x^2)*2^(1/3)*3^(2/3)+1/78732*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2))/
(8-9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(3/2)*3^(1/2)-1/78732*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^
(1/3)*3^(2/3))^(1/2))/(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(3/2)*3^(1/2)+1/279936*ln(6+3*(-2)^(2/3)*3^(1/
3)*x+x^2)*(3^(1/2)+I)*2^(1/3)*3^(1/6)/(1+(-1)^(1/3))^5-1/314928*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2
^(1/3)*3^(2/3))^(1/2))/(-4+3*2^(1/3)*3^(2/3))^(3/2)*6^(1/2)-1/209952*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(24-18*
(-3)^(2/3)*2^(1/3))^(1/2))*(9*I-3^(1/3)*(2*I*2^(2/3)+9*3^(1/6)+2*2^(2/3)*3^(1/2)))/(1+(-1)^(1/3))^5/(8-6*(-3)^
(2/3)*2^(1/3))^(1/2)+1/209952*(9*I+3^(1/3)*(4*I*2^(2/3)-9*3^(1/6)))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(
-2)^(1/3)*3^(2/3))^(1/2))/(1+(-1)^(1/3))^5/(8+6*(-2)^(1/3)*3^(2/3))^(1/2)+1/2834352*(2*2^(2/3)-3*3^(2/3))*arct
anh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*3^(5/6)/(-8+6*2^(1/3)*3^(2/3))^(1/2)

Rubi [A] (warning: unable to verify)

Time = 1.85 (sec) , antiderivative size = 873, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2122, 652, 632, 210, 648, 642, 212} \[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\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 ) \arctan \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 {\arctan \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 ) \arctan \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 {\arctan \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 ) \text {arctanh}\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 {\text {arctanh}\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 )} \]

[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 210

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

Rule 212

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

Rule 632

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 642

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 648

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 652

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)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), 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 2122

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

Rubi steps \begin{align*} \text {integral}& = 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 ) \text {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 {\text {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 {\text {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 {\text {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 ) \text {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 ) \text {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] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.02 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.19 \[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {972-3942 x+648 x^2+96 x^3-27 x^4+4 x^5}{3691656 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}+\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {1971 \log (x-\text {$\#$1})-162 \log (x-\text {$\#$1}) \text {$\#$1}+72 \log (x-\text {$\#$1}) \text {$\#$1}^2-27 \log (x-\text {$\#$1}) \text {$\#$1}^3+2 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{11074968} \]

[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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.14

method	result	size
default	\(\frac {\frac {1}{922914} x^{5}-\frac {1}{136728} x^{4}+\frac {4}{153819} x^{3}+\frac {1}{5697} x^{2}-\frac {73}{68364} x +\frac {1}{3798}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}-27 \textit {\_R}^{3}+72 \textit {\_R}^{2}-162 \textit {\_R} +1971\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{11074968}\)	\(122\)
risch	\(\frac {\frac {1}{922914} x^{5}-\frac {1}{136728} x^{4}+\frac {4}{153819} x^{3}+\frac {1}{5697} x^{2}-\frac {73}{68364} x +\frac {1}{3798}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}-27 \textit {\_R}^{3}+72 \textit {\_R}^{2}-162 \textit {\_R} +1971\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{11074968}\)	\(122\)

[In]

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

[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))

Fricas [F(-1)]

Timed out. \[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.13 \[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\operatorname {RootSum} {\left (1282755170017893101915524820582750453426552832 t^{6} - 906388465775544244426251149770752 t^{4} - 4300873166389987741684137984 t^{3} - 717000908921644962816 t^{2} + 135354162312576 t - 7197829, \left ( t \mapsto t \log {\left (\frac {17257935592810449901409556597891882995604001083339368041361480613888 t^{5}}{154206009791052044490694380303237521} + \frac {2389607400620985524376358853572652207181956324560587684052992 t^{4}}{154206009791052044490694380303237521} - \frac {12286072160883283930711715948878260078996992193488388096 t^{3}}{154206009791052044490694380303237521} - \frac {59490553573959173161125496013527909754156558410752 t^{2}}{154206009791052044490694380303237521} - \frac {17520149679836691112367064197713753004827200 t}{154206009791052044490694380303237521} + x + \frac {766422988707229615055855287040887332}{154206009791052044490694380303237521} \right )} \right )\right )} + \frac {4 x^{5} - 27 x^{4} + 96 x^{3} + 648 x^{2} - 3942 x + 972}{3691656 x^{6} + 66449808 x^{4} + 1196096544 x^{3} + 398698848 x^{2} + 797397696} \]

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [B] (verification not implemented)

Time = 9.31 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.44 \[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

symsum(log((11*x)/603554178896188848 - (14059*root(z^6 - (292589*z^4)/414082997094657024 - (11805253*z^3)/3520
970912943705975865344 - (2479189*z^2)/4435409310686141742269284220928 + (1989787*z)/18857266875842830829226645
405233577984 - 7197829/1282755170017893101915524820582750453426552832, z, k))/30663729050256 - (5658601*root(z
^6 - (292589*z^4)/414082997094657024 - (11805253*z^3)/3520970912943705975865344 - (2479189*z^2)/44354093106861
41742269284220928 + (1989787*z)/18857266875842830829226645405233577984 - 7197829/12827551700178931019155248205
82750453426552832, z, k)*x)/6623365474855296 + (6603523*root(z^6 - (292589*z^4)/414082997094657024 - (11805253
*z^3)/3520970912943705975865344 - (2479189*z^2)/4435409310686141742269284220928 + (1989787*z)/1885726687584283
0829226645405233577984 - 7197829/1282755170017893101915524820582750453426552832, z, k)^2*x)/584204562 - (17623
21104*root(z^6 - (292589*z^4)/414082997094657024 - (11805253*z^3)/3520970912943705975865344 - (2479189*z^2)/44
35409310686141742269284220928 + (1989787*z)/18857266875842830829226645405233577984 - 7197829/12827551700178931
01915524820582750453426552832, z, k)^3*x)/44521 - (59633904436992*root(z^6 - (292589*z^4)/414082997094657024 -
(11805253*z^3)/3520970912943705975865344 - (2479189*z^2)/4435409310686141742269284220928 + (1989787*z)/188572
66875842830829226645405233577984 - 7197829/1282755170017893101915524820582750453426552832, z, k)^4*x)/211 - 69
40988288557056*root(z^6 - (292589*z^4)/414082997094657024 - (11805253*z^3)/3520970912943705975865344 - (247918
9*z^2)/4435409310686141742269284220928 + (1989787*z)/18857266875842830829226645405233577984 - 7197829/12827551
70017893101915524820582750453426552832, z, k)^5*x + (166697*root(z^6 - (292589*z^4)/414082997094657024 - (1180
5253*z^3)/3520970912943705975865344 - (2479189*z^2)/4435409310686141742269284220928 + (1989787*z)/188572668758
42830829226645405233577984 - 7197829/1282755170017893101915524820582750453426552832, z, k)^2)/43274412 + (6391
93032*root(z^6 - (292589*z^4)/414082997094657024 - (11805253*z^3)/3520970912943705975865344 - (2479189*z^2)/44
35409310686141742269284220928 + (1989787*z)/18857266875842830829226645405233577984 - 7197829/12827551700178931
01915524820582750453426552832, z, k)^3)/44521 - (9815247601920*root(z^6 - (292589*z^4)/414082997094657024 - (1
1805253*z^3)/3520970912943705975865344 - (2479189*z^2)/4435409310686141742269284220928 + (1989787*z)/188572668
75842830829226645405233577984 - 7197829/1282755170017893101915524820582750453426552832, z, k)^4)/211 - 1688973
81688221696*root(z^6 - (292589*z^4)/414082997094657024 - (11805253*z^3)/3520970912943705975865344 - (2479189*z
^2)/4435409310686141742269284220928 + (1989787*z)/18857266875842830829226645405233577984 - 7197829/12827551700
17893101915524820582750453426552832, z, k)^5 + 661/28970600587017064704)*root(z^6 - (292589*z^4)/4140829970946
57024 - (11805253*z^3)/3520970912943705975865344 - (2479189*z^2)/4435409310686141742269284220928 + (1989787*z)
/18857266875842830829226645405233577984 - 7197829/1282755170017893101915524820582750453426552832, z, k), k, 1,
6) + (x^2/5697 - (73*x)/68364 + (4*x^3)/153819 - x^4/136728 + x^5/922914 + 1/3798)/(108*x^2 + 324*x^3 + 18*x^
4 + x^6 + 216)