3.150 \(\int \frac{1}{x^2 (216+108 x^2+324 x^3+18 x^4+x^6)} \, dx\)

Optimal. Leaf size=448 \[ -\frac{(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\left (3 (-6)^{2/3}+2 \sqrt [3]{-2}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{7776 \sqrt [3]{3}}-\frac{\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{3888 \sqrt [3]{6}}-\frac{1}{216 x}-\frac{\left (27 \sqrt [3]{-6}-(-2)^{2/3}+12\ 3^{2/3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{5832 \sqrt [6]{3} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac{(-1)^{2/3} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{1944 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{5832 \sqrt [6]{6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]

[Out]

-1/(216*x) - ((27*(-6)^(1/3) - (-2)^(2/3) + 12*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)
^(1/3)*3^(2/3))]])/(5832*3^(1/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3)]) - ((-1)^(2/3)*(6*(-6)^(2
/3) + 27*(-3)^(1/3) - 2^(1/3))*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]]
)/(1944*6^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((2^(1/3) + 27*3^(1/3) - 6*6^(2/3))*ArcTa
nh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]])/(5832*6^(1/6)*Sqrt[-4 + 3*2^(1/3)*3^(2
/3)]) - ((-1)^(2/3)*(9 + (-3)^(1/3)*2^(2/3))*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(1296*2^(1/3)*3^(2/3)*(1 +
(-1)^(1/3))^2) + ((3*(-6)^(2/3) + 2*(-2)^(1/3))*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(7776*3^(1/3)) - ((2^(
2/3) - 3*3^(2/3))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/(3888*6^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 1.10187, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2097, 634, 618, 204, 628, 206} \[ -\frac{(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\left (3 (-6)^{2/3}+2 \sqrt [3]{-2}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{7776 \sqrt [3]{3}}-\frac{\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{3888 \sqrt [3]{6}}-\frac{1}{216 x}-\frac{\left (27 \sqrt [3]{-6}-(-2)^{2/3}+12\ 3^{2/3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{5832 \sqrt [6]{3} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac{(-1)^{2/3} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{1944 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{5832 \sqrt [6]{6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)),x]

[Out]

-1/(216*x) - ((27*(-6)^(1/3) - (-2)^(2/3) + 12*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)
^(1/3)*3^(2/3))]])/(5832*3^(1/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3)]) - ((-1)^(2/3)*(6*(-6)^(2
/3) + 27*(-3)^(1/3) - 2^(1/3))*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]]
)/(1944*6^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((2^(1/3) + 27*3^(1/3) - 6*6^(2/3))*ArcTa
nh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]])/(5832*6^(1/6)*Sqrt[-4 + 3*2^(1/3)*3^(2
/3)]) - ((-1)^(2/3)*(9 + (-3)^(1/3)*2^(2/3))*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(1296*2^(1/3)*3^(2/3)*(1 +
(-1)^(1/3))^2) + ((3*(-6)^(2/3) + 2*(-2)^(1/3))*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(7776*3^(1/3)) - ((2^(
2/3) - 3*3^(2/3))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/(3888*6^(1/3))

Rule 2097

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

Rule 634

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

Rule 618

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

Rule 204

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

Rule 628

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

Rule 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{1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx &=1259712 \int \left (\frac{1}{272097792 x^2}+\frac{(-1)^{2/3} \left (-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}-\left (9+\sqrt [3]{-3} 2^{2/3}\right ) x\right )}{816293376 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac{(-1)^{2/3} \left (1+27 (-2)^{2/3} \sqrt [3]{3}+9 \sqrt [3]{-2} 3^{2/3}+\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) x\right )}{816293376 \sqrt [3]{2} 3^{2/3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac{-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [3]{2} 3^{2/3}-6^{2/3} \left (2^{2/3}-3\ 3^{2/3}\right ) x}{14693280768 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=-\frac{1}{216 x}+\frac{\int \frac{-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [3]{2} 3^{2/3}-6^{2/3} \left (2^{2/3}-3\ 3^{2/3}\right ) x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{11664}+\frac{(-1)^{2/3} \int \frac{1+27 (-2)^{2/3} \sqrt [3]{3}+9 \sqrt [3]{-2} 3^{2/3}+\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1944 \sqrt [3]{2} 3^{2/3}}+\frac{(-1)^{2/3} \int \frac{-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}-\left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{648 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=-\frac{1}{216 x}-\frac{\left ((-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right )\right ) \int \frac{-3 \sqrt [3]{-3} 2^{2/3}+2 x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{\left ((-1)^{2/3} \left ((-2)^{2/3}-3\ 3^{2/3}\right )\right ) \int \frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{3888 \sqrt [3]{6}}-\frac{\left (2^{2/3}-3\ 3^{2/3}\right ) \int \frac{3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{3888 \sqrt [3]{6}}+\frac{\left ((-1)^{2/3} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right )\right ) \int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{3888 \sqrt [3]{2} 3^{2/3}}+\frac{\left ((-1)^{2/3} \left (3 \sqrt [3]{-3} 2^{2/3} \left (-9-\sqrt [3]{-3} 2^{2/3}\right )+2 \left (-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}\right )\right )\right ) \int \frac{1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\left (18 \sqrt [3]{2} \left (2^{2/3}-3\ 3^{2/3}\right )+2 \left (-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [3]{2} 3^{2/3}\right )\right ) \int \frac{1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{23328}\\ &=-\frac{1}{216 x}-\frac{(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{(-1)^{2/3} \left ((-2)^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}}-\frac{\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}}-\frac{\left ((-1)^{2/3} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\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 )}{1944 \sqrt [3]{2} 3^{2/3}}-\frac{\left ((-1)^{2/3} \left (3 \sqrt [3]{-3} 2^{2/3} \left (-9-\sqrt [3]{-3} 2^{2/3}\right )+2 \left (-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}\right )\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 )}{648 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{\left (18 \sqrt [3]{2} \left (2^{2/3}-3\ 3^{2/3}\right )+2 \left (-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [3]{2} 3^{2/3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-6 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )-x^2} \, dx,x,3\ 2^{2/3} \sqrt [3]{3}+2 x\right )}{11664}\\ &=-\frac{1}{216 x}+\frac{(-1)^{2/3} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) \tan ^{-1}\left (\frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{5832\ 2^{5/6} \sqrt [6]{3} \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac{(-1)^{2/3} \left (2-12 (-3)^{2/3} \sqrt [3]{2}-27 \sqrt [3]{-3} 2^{2/3}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{1944\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac{\left (18\ 2^{2/3}-27\ 3^{2/3}-\sqrt [3]{6}\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 )}{5832 \sqrt [6]{2} \sqrt{3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac{(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{(-1)^{2/3} \left ((-2)^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}}-\frac{\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}}\\ \end{align*}

Mathematica [C]  time = 0.0188605, size = 109, normalized size = 0.24 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+18 \text{$\#$1}^2 \log (x-\text{$\#$1})+324 \text{$\#$1} \log (x-\text{$\#$1})+108 \log (x-\text{$\#$1})}{\text{$\#$1}^5+12 \text{$\#$1}^3+162 \text{$\#$1}^2+36 \text{$\#$1}}\& \right ]}{1296}-\frac{1}{216 x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)),x]

[Out]

-1/(216*x) - RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (108*Log[x - #1] + 324*Log[x - #1]*#1 + 18
*Log[x - #1]*#1^2 + Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/1296

________________________________________________________________________________________

Maple [C]  time = 0.007, size = 74, normalized size = 0.2 \begin{align*}{\frac{1}{1296}\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 ( -{{\it \_R}}^{4}-18\,{{\it \_R}}^{2}-324\,{\it \_R}-108 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}}-{\frac{1}{216\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(x^6+18*x^4+324*x^3+108*x^2+216),x)

[Out]

1/1296*sum((-_R^4-18*_R^2-324*_R-108)/(_R^5+12*_R^3+162*_R^2+36*_R)*ln(x-_R),_R=RootOf(_Z^6+18*_Z^4+324*_Z^3+1
08*_Z^2+216))-1/216/x

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{216 \, x} - \frac{1}{216} \, \int \frac{x^{4} + 18 \, x^{2} + 324 \, x + 108}{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(1/x^2/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima")

[Out]

-1/216/x - 1/216*integrate((x^4 + 18*x^2 + 324*x + 108)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [A]  time = 0.269043, size = 70, normalized size = 0.16 \begin{align*} \operatorname{RootSum}{\left (1594001683946413330255577088 t^{6} + 3791612026460331638784 t^{4} - 8643672699589509120 t^{3} - 10942820851968 t^{2} - 839808 t - 1, \left ( t \mapsto t \log{\left (- \frac{49875532761902496003293561236914468028416 t^{5}}{12350449784703991795} + \frac{12625489872431620388005975200497664 t^{4}}{12350449784703991795} - \frac{118637692607573771238550798852644864 t^{3}}{12350449784703991795} + \frac{270486324927832147818193778754816 t^{2}}{12350449784703991795} + \frac{273914194897479402961199352 t}{12350449784703991795} + x - \frac{12798926329353908292}{12350449784703991795} \right )} \right )\right )} - \frac{1}{216 x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(x**6+18*x**4+324*x**3+108*x**2+216),x)

[Out]

RootSum(1594001683946413330255577088*_t**6 + 3791612026460331638784*_t**4 - 8643672699589509120*_t**3 - 109428
20851968*_t**2 - 839808*_t - 1, Lambda(_t, _t*log(-49875532761902496003293561236914468028416*_t**5/12350449784
703991795 + 12625489872431620388005975200497664*_t**4/12350449784703991795 - 118637692607573771238550798852644
864*_t**3/12350449784703991795 + 270486324927832147818193778754816*_t**2/12350449784703991795 + 27391419489747
9402961199352*_t/12350449784703991795 + x - 12798926329353908292/12350449784703991795))) - 1/(216*x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="giac")

[Out]

integrate(1/((x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)*x^2), x)