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]

1/108*(18-(-2)^(2/3)*3^(1/3))*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)+1/108*(18-2^(2/3)*3^(1/3))*ln(6+3*2^(2/3)*3^(1/
3)*x+x^2)+1/216*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*(36+2^(2/3)*3^(1/3)*(1+I*3^(1/2)))+1/2*3^(1/6)*2^(5/6)*(1-(-3
)^(2/3)*2^(1/3))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/(12-9*(-3)^(2/3)*2^(1/3))^(1/2))/(1+(-1)^(1/3))^2/(4-
3*(-3)^(2/3)*2^(1/3))^(1/2)-1/6*(1-2^(1/3)*3^(2/3))*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)-1/3*(-2)^(1/3)*(1+(-2)^(1/3)*3^(2/3))*arctan((3*(-2)^(
2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*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 = 1.44, antiderivative size = 395, normalized size of antiderivative = 1.00, 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 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^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*}

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Mathematica [C]  time = 0.02, 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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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^5/(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 \[ \int \frac {x^{5}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^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)

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maple [C]  time = 0.01, size = 56, normalized size = 0.14 \[ \frac {\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5} \ln \left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+x \right )}{6 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5}+72 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}+972 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}+216 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^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(-_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 \[ \int \frac {x^{5}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^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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mupad [B]  time = 0.65, size = 427, normalized size = 1.08 \[ \sum _{k=1}^6\ln \left (\frac {362797056\,\left (19236852\,x\,\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )-19131876\,x-6482268\,x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^2+742851\,x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^3-4130\,x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^4+x\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^5-154944576\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^2+17047422\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^3+27054\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^4+9\,{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^5+465542316\,\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )-465542316\right )}{{\mathrm {root}\left (z^6+4374\,z^5+6626610\,z^4+2646786132\,z^3-24163559388\,z^2+72662865048\,z-72662865048,z,k\right )}^5}\right )\,\mathrm {root}\left (z^6-z^5+\frac {421\,z^4}{1266}-\frac {100853\,z^3}{2768742}-\frac {505\,z^2}{5537484}-\frac {z}{16612452}-\frac {1}{72662865048},z,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log((362797056*(19236852*x*root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662
865048*z - 72662865048, z, k) - 19131876*x - 6482268*x*root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 24
163559388*z^2 + 72662865048*z - 72662865048, z, k)^2 + 742851*x*root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786132
*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k)^3 - 4130*x*root(z^6 + 4374*z^5 + 6626610*z^4 + 264
6786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k)^4 + x*root(z^6 + 4374*z^5 + 6626610*z^4 + 2
646786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k)^5 - 154944576*root(z^6 + 4374*z^5 + 66266
10*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k)^2 + 17047422*root(z^6 + 4374*z^
5 + 6626610*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k)^3 + 27054*root(z^6 + 4
374*z^5 + 6626610*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k)^4 + 9*root(z^6 +
4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k)^5 + 465542316*
root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048, z, k) - 46
5542316))/root(z^6 + 4374*z^5 + 6626610*z^4 + 2646786132*z^3 - 24163559388*z^2 + 72662865048*z - 72662865048,
z, k)^5)*root(z^6 - z^5 + (421*z^4)/1266 - (100853*z^3)/2768742 - (505*z^2)/5537484 - z/16612452 - 1/726628650
48, z, k), k, 1, 6)

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sympy [A]  time = 0.26, size = 70, normalized size = 0.18 \[ \operatorname {RootSum} {\left (72662865048 t^{6} - 72662865048 t^{5} + 24163559388 t^{4} - 2646786132 t^{3} - 6626610 t^{2} - 4374 t - 1, \left (t \mapsto t \log {\left (- \frac {89236417131047376 t^{5}}{833243797} + \frac {89301949532998128 t^{4}}{833243797} - \frac {29740560281805852 t^{3}}{833243797} + \frac {192466080408420 t^{2}}{49014341} + \frac {5867255361684 t}{833243797} + x + \frac {5365044886}{2499731391} \right )} \right )\right )} \]

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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