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

Optimal. Leaf size=395 \[ \frac{1}{216} \left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt{3}\right )\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )+\frac{1}{108} \left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )+\frac{1}{108} \left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )-\frac{\sqrt [3]{-2} \left (1+\sqrt [3]{-2} 3^{2/3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{3^{5/6} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac{\sqrt [6]{\frac{3}{2}} \left (1-(-3)^{2/3} \sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{\left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (1-\sqrt [3]{2} 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{\sqrt [6]{2} 3^{5/6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]

[Out]

-(((-2)^(1/3)*(1 + (-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))]]
)/(3^(5/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3)])) + ((3/2)^(1/6)*(1 - (-3)^(2/3)*2^(1/3))*ArcTa
n[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/((1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3
)^(2/3)*2^(1/3)]) - ((1 - 2^(1/3)*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))]])/(2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + ((36 + 2^(2/3)*3^(1/3)*(1 + I*Sqrt[3]))*Log[6 - 3*(
-3)^(1/3)*2^(2/3)*x + x^2])/216 + ((18 - (-2)^(2/3)*3^(1/3))*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/108 + ((18
- 2^(2/3)*3^(1/3))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/108

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Rubi [A]  time = 1.43847, antiderivative size = 395, 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} \[ \frac{1}{216} \left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt{3}\right )\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )+\frac{1}{108} \left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )+\frac{1}{108} \left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )-\frac{\sqrt [3]{-2} \left (1+\sqrt [3]{-2} 3^{2/3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{3^{5/6} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac{\sqrt [6]{\frac{3}{2}} \left (1-(-3)^{2/3} \sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{\left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (1-\sqrt [3]{2} 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{\sqrt [6]{2} 3^{5/6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((-2)^(1/3)*(1 + (-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))]]
)/(3^(5/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3)])) + ((3/2)^(1/6)*(1 - (-3)^(2/3)*2^(1/3))*ArcTa
n[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/((1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3
)^(2/3)*2^(1/3)]) - ((1 - 2^(1/3)*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))]])/(2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + ((36 + 2^(2/3)*3^(1/3)*(1 + I*Sqrt[3]))*Log[6 - 3*(
-3)^(1/3)*2^(2/3)*x + x^2])/216 + ((18 - (-2)^(2/3)*3^(1/3))*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/108 + ((18
- 2^(2/3)*3^(1/3))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/108

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{x^5}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx &=1259712 \int \left (\frac{(-1)^{2/3} \left (3 \sqrt [3]{-3} 2^{2/3}+\left (1-3 (-3)^{2/3} \sqrt [3]{2}\right ) x\right )}{3779136 \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 (3 (-2)^{2/3} \sqrt [3]{3}-\left (1+3 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{3779136 \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{6 \sqrt [3]{2} 3^{2/3}+\left (18-2^{2/3} \sqrt [3]{3}\right ) x}{68024448 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=\frac{1}{54} \int \frac{6 \sqrt [3]{2} 3^{2/3}+\left (18-2^{2/3} \sqrt [3]{3}\right ) x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx+\frac{(-1)^{2/3} \int \frac{3 (-2)^{2/3} \sqrt [3]{3}-\left (1+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{9 \sqrt [3]{2} 3^{2/3}}+\frac{(-1)^{2/3} \int \frac{3 \sqrt [3]{-3} 2^{2/3}+\left (1-3 (-3)^{2/3} \sqrt [3]{2}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{3 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=\frac{\left ((-1)^{2/3} \left (1-3 (-3)^{2/3} \sqrt [3]{2}\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}{6 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\left ((-1)^{2/3} \sqrt [3]{\frac{3}{2}} \left (6+\sqrt [3]{-3} 2^{2/3}\right )\right ) \int \frac{1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{2 \left (1+\sqrt [3]{-1}\right )^2}-\frac{\left ((-1)^{2/3} \left (6-(-2)^{2/3} \sqrt [3]{3}\right )\right ) \int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{2 \sqrt [3]{2} 3^{2/3}}+\frac{1}{108} \left (18-(-2)^{2/3} \sqrt [3]{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+\frac{1}{108} \left (18-2^{2/3} \sqrt [3]{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+\frac{\left (1-\sqrt [3]{2} 3^{2/3}\right ) \int \frac{1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{2^{2/3} \sqrt [3]{3}}\\ &=\frac{1}{216} \left (36+2^{2/3} \sqrt [3]{3}+i 2^{2/3} 3^{5/6}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )+\frac{1}{108} \left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )+\frac{1}{108} \left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )-\frac{\left ((-1)^{2/3} \sqrt [3]{6} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\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 )}{\left (1+\sqrt [3]{-1}\right )^2}-\frac{\left ((-1)^{2/3} \left ((-2)^{2/3}-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 )}{\sqrt [3]{6}}-\left (\sqrt [3]{\frac{2}{3}} \left (1-\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 )\\ &=\frac{(-1)^{2/3} \left ((-2)^{2/3}-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 )}{6^{5/6} \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac{(-1)^{2/3} \left (\sqrt [3]{-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 )}{\sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (1-\sqrt [3]{2} 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 )}{\sqrt [6]{2} 3^{5/6} \sqrt{-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac{1}{216} \left (36+2^{2/3} \sqrt [3]{3}+i 2^{2/3} 3^{5/6}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )+\frac{1}{108} \left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )+\frac{1}{108} \left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.0155941, size = 61, normalized size = 0.15 \[ \frac{1}{6} \text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})}{\text{$\#$1}^4+12 \text{$\#$1}^2+162 \text{$\#$1}+36}\& \right ] \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.007, size = 56, normalized size = 0.1 \begin{align*}{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{{{\it \_R}}^{5}\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),x)

[Out]

1/6*sum(_R^5/(_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*} \int \frac{x^{5}}{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),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Sympy [A]  time = 0.218052, size = 70, normalized size = 0.18 \begin{align*} \operatorname{RootSum}{\left (72662865048 t^{6} - 72662865048 t^{5} + 24163559388 t^{4} - 2646786132 t^{3} - 6626610 t^{2} - 4374 t - 1, \left ( t \mapsto t \log{\left (- \frac{89236417131047376 t^{5}}{833243797} + \frac{89301949532998128 t^{4}}{833243797} - \frac{29740560281805852 t^{3}}{833243797} + \frac{192466080408420 t^{2}}{49014341} + \frac{5867255361684 t}{833243797} + x + \frac{5365044886}{2499731391} \right )} \right )\right )} \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),x)

[Out]

RootSum(72662865048*_t**6 - 72662865048*_t**5 + 24163559388*_t**4 - 2646786132*_t**3 - 6626610*_t**2 - 4374*_t
- 1, Lambda(_t, _t*log(-89236417131047376*_t**5/833243797 + 89301949532998128*_t**4/833243797 - 2974056028180
5852*_t**3/833243797 + 192466080408420*_t**2/49014341 + 5867255361684*_t/833243797 + x + 5365044886/2499731391
)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{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),x, algorithm="giac")

[Out]

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