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

Optimal. Leaf size=361 \[ \frac {(-1)^{2/3} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {(-1)^{2/3} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac {\tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{36 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\sqrt [3]{-1} \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{54\ 2^{2/3} 3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{108 \sqrt [6]{2} 3^{5/6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}} \]

[Out]

1/1296*(-1)^(2/3)*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(2/3)*3^(1/3)/(1+(-1)^(1/3))^2-1/3888*(-1)^(2/3)*ln(6+3*(
-2)^(2/3)*3^(1/3)*x+x^2)*2^(2/3)*3^(1/3)-1/3888*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)*2^(2/3)*3^(1/3)-1/216*arctan((3*
(-3)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2))*2^(5/6)*3^(1/6)/(1+(-1)^(1/3))^2/(4-3*(-3)^(2/3)*2^(
1/3))^(1/2)+1/648*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*2^(5/6)*3^(1/6)/(-4+3*2
^(1/3)*3^(2/3))^(1/2)+1/324*(-1)^(1/3)*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*2^(
1/3)*3^(1/6)/(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(1/2)

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Rubi [A]  time = 0.55, antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2097, 634, 618, 204, 628, 206} \[ \frac {(-1)^{2/3} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {(-1)^{2/3} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac {\tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{36 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\sqrt [3]{-1} \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{54\ 2^{2/3} 3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{108 \sqrt [6]{2} 3^{5/6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]]/(36*2^(1/6)*3^(5/6)*(1 + (-1)^(1/3))^
2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ((-1)^(1/3)*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3
^(2/3))]])/(54*2^(2/3)*3^(5/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*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))]]/(108*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + ((-1)
^(2/3)*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(216*2^(1/3)*3^(2/3)*(1 + (-1)^(1/3))^2) - ((-1)^(2/3)*Log[6 + 3
*(-2)^(2/3)*3^(1/3)*x + x^2])/(648*2^(1/3)*3^(2/3)) - Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2]/(648*2^(1/3)*3^(2/3))

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])

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 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 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 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]

Rubi steps

\begin {align*} \int \frac {x}{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}-x\right )}{136048896 \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}+x\right )}{136048896 \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]{3}+\sqrt [3]{2} x}{408146688\ 6^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=-\frac {(-1)^{2/3} \int \frac {3 (-2)^{2/3} \sqrt [3]{3}+x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{324 \sqrt [3]{2} 3^{2/3}}-\frac {\int \frac {6 \sqrt [3]{3}+\sqrt [3]{2} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{324\ 6^{2/3}}+\frac {(-1)^{2/3} \int \frac {3 \sqrt [3]{-3} 2^{2/3}-x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{108 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=\frac {\sqrt [3]{-\frac {1}{3}} \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{108\ 2^{2/3}}-\frac {\int \frac {3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{648 \sqrt [3]{2} 3^{2/3}}-\frac {(-1)^{2/3} \int \frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{648 \sqrt [3]{2} 3^{2/3}}-\frac {\int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{108\ 2^{2/3} \sqrt [3]{3}}+\frac {(-1)^{2/3} \int \frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{36\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=\frac {(-1)^{2/3} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {(-1)^{2/3} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac {\sqrt [3]{-\frac {1}{3}} \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 )}{54\ 2^{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 )}{54\ 2^{2/3} \sqrt [3]{3}}+\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 )}{18\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^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 )}{36 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\sqrt [3]{-1} \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 )}{108 \sqrt [6]{2} 3^{5/6} \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 )}{108 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {(-1)^{2/3} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {(-1)^{2/3} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{2} 3^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 57, normalized size = 0.16 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6+18 \text {$\#$1}^4+324 \text {$\#$1}^3+108 \text {$\#$1}^2+216\& ,\frac {\log (x-\text {$\#$1})}{\text {$\#$1}^4+12 \text {$\#$1}^2+162 \text {$\#$1}+36}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[x/(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]/(36 + 162*#1 + 12*#1^2 + #1^4) & ]/6

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.01, size = 54, normalized size = 0.15 \[ \frac {\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+x \right )}{6 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5}+72 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}+972 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}+216 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.42, size = 176, normalized size = 0.49 \[ \sum _{k=1}^6\ln \left (x+\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right )\,\left (216\,x+\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right )\,\left (51018336\,x-\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right )\,\left (277947894528\,x-\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right )\,\left (33192121254912\,x-\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right )\,\left (6940988288557056\,x+168897381688221696\right )[phone]\right )\right )\right )\right )\right )\,\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(x + root(z^6 - z^4/1640736 + (235*z^3)/129178426752 - z^2/3100282242048 - 1/158171241119638192128,
z, k)*(216*x + root(z^6 - z^4/1640736 + (235*z^3)/129178426752 - z^2/3100282242048 - 1/158171241119638192128,
z, k)*(51018336*x - root(z^6 - z^4/1640736 + (235*z^3)/129178426752 - z^2/3100282242048 - 1/158171241119638192
128, z, k)*(277947894528*x - root(z^6 - z^4/1640736 + (235*z^3)/129178426752 - z^2/3100282242048 - 1/158171241
119638192128, z, k)*(33192121254912*x - root(z^6 - z^4/1640736 + (235*z^3)/129178426752 - z^2/3100282242048 -
1/158171241119638192128, z, k)*(6940988288557056*x + 168897381688221696) + 28563737812992)))))*root(z^6 - z^4/
1640736 + (235*z^3)/129178426752 - z^2/3100282242048 - 1/158171241119638192128, z, k), k, 1, 6)

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sympy [A]  time = 0.26, size = 61, normalized size = 0.17 \[ \operatorname {RootSum} {\left (158171241119638192128 t^{6} - 96402615118848 t^{4} + 287743415040 t^{3} - 51018336 t^{2} - 1, \left (t \mapsto t \log {\left (\frac {65418399445721140961280 t^{5}}{415817} + \frac {2480926457425102848 t^{4}}{415817} - \frac {39451802929737984 t^{3}}{415817} + \frac {118071997444800 t^{2}}{415817} - \frac {16745884920 t}{415817} + x - \frac {268790}{415817} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(158171241119638192128*_t**6 - 96402615118848*_t**4 + 287743415040*_t**3 - 51018336*_t**2 - 1, Lambda(_
t, _t*log(65418399445721140961280*_t**5/415817 + 2480926457425102848*_t**4/415817 - 39451802929737984*_t**3/41
5817 + 118071997444800*_t**2/415817 - 16745884920*_t/415817 + x - 268790/415817)))

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