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

Optimal. Leaf size=682 \[ \frac{\sqrt [3]{-\frac{1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac{\sqrt [3]{-\frac{1}{3}} \left ((-2)^{2/3} \sqrt [3]{3} x+4\right )}{8748\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac{2^{2/3} \sqrt [3]{3} x+4}{17496\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374 \sqrt{3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{4374 \sqrt{3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac{i \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1458\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{39366\ 2^{5/6} \sqrt [6]{3} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{8748 \sqrt{6} \left (3 \sqrt [3]{2} 3^{2/3}-4\right )^{3/2}} \]

[Out]

((-1/3)^(1/3)*(4 - (-3)^(1/3)*2^(2/3)*x))/(1944*2^(2/3)*(1 + (-1)^(1/3))^4*(4 -
3*(-3)^(2/3)*2^(1/3))*(6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2)) + ((-1/3)^(1/3)*(4 + (
-2)^(2/3)*3^(1/3)*x))/(8748*2^(2/3)*(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)) - (4 + 2^(2/3)*3^(1/3)*x)/(17496*2^(2/3)
*3^(1/3)*(4 - 3*2^(1/3)*3^(2/3))*(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))]]/(4374*2^(5/6)*3^(1/6
)*(1 + (-1)^(1/3))^4*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ArcTan[(3*(-3)^(1/3)*2^(2
/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]]/(4374*Sqrt[3]*(8 - (9*I)*2^(1/3)*
3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) - ((I/1458)*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2
*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(2^(5/6)*3^(2/3)*(1 + (-1)^(1/3))^5*Sqr
t[4 + 3*(-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))]]/(4374*Sqrt[3]*(8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2
/3))^(3/2)) - ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3
^(2/3))]]/(8748*Sqrt[6]*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) - ArcTanh[(2^(1/6)*(3*3^
(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(39366*2^(5/6)*3^(1/6)*Sqr
t[-4 + 3*2^(1/3)*3^(2/3)])

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Rubi [A]  time = 4.23325, antiderivative size = 682, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt [3]{-\frac{1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac{\sqrt [3]{-\frac{1}{3}} \left ((-2)^{2/3} \sqrt [3]{3} x+4\right )}{8748\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac{2^{2/3} \sqrt [3]{3} x+4}{17496\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374 \sqrt{3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{4374 \sqrt{3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac{i \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1458\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{39366\ 2^{5/6} \sqrt [6]{3} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{8748 \sqrt{6} \left (3 \sqrt [3]{2} 3^{2/3}-4\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-1/3)^(1/3)*(4 - (-3)^(1/3)*2^(2/3)*x))/(1944*2^(2/3)*(1 + (-1)^(1/3))^4*(4 -
3*(-3)^(2/3)*2^(1/3))*(6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2)) + ((-1/3)^(1/3)*(4 + (
-2)^(2/3)*3^(1/3)*x))/(8748*2^(2/3)*(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)) - (4 + 2^(2/3)*3^(1/3)*x)/(17496*2^(2/3)
*3^(1/3)*(4 - 3*2^(1/3)*3^(2/3))*(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))]]/(4374*2^(5/6)*3^(1/6
)*(1 + (-1)^(1/3))^4*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ArcTan[(3*(-3)^(1/3)*2^(2
/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]]/(4374*Sqrt[3]*(8 - (9*I)*2^(1/3)*
3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) - ((I/1458)*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2
*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(2^(5/6)*3^(2/3)*(1 + (-1)^(1/3))^5*Sqr
t[4 + 3*(-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))]]/(4374*Sqrt[3]*(8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2
/3))^(3/2)) - ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3
^(2/3))]]/(8748*Sqrt[6]*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) - ArcTanh[(2^(1/6)*(3*3^
(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(39366*2^(5/6)*3^(1/6)*Sqr
t[-4 + 3*2^(1/3)*3^(2/3)])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.0427513, size = 167, normalized size = 0.24 \[ \frac{\text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\&,\frac{4 \text{$\#$1}^4 \log (x-\text{$\#$1})-54 \text{$\#$1}^3 \log (x-\text{$\#$1})+2043 \text{$\#$1}^2 \log (x-\text{$\#$1})-324 \text{$\#$1} \log (x-\text{$\#$1})+144 \log (x-\text{$\#$1})}{\text{$\#$1}^5+12 \text{$\#$1}^3+162 \text{$\#$1}^2+36 \text{$\#$1}}\&\right ]}{3691656}+\frac{4 x^5-27 x^4+729 x^3+648 x^2-144 x+972}{615276 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )} \]

Antiderivative was successfully verified.

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

[Out]

(972 - 144*x + 648*x^2 + 729*x^3 - 27*x^4 + 4*x^5)/(615276*(216 + 108*x^2 + 324*
x^3 + 18*x^4 + x^6)) + RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (1
44*Log[x - #1] - 324*Log[x - #1]*#1 + 2043*Log[x - #1]*#1^2 - 54*Log[x - #1]*#1^
3 + 4*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/3691656

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Maple [C]  time = 0.015, size = 122, normalized size = 0.2 \[{\frac{1}{{x}^{6}+18\,{x}^{4}+324\,{x}^{3}+108\,{x}^{2}+216} \left ({\frac{{x}^{5}}{153819}}-{\frac{{x}^{4}}{22788}}+{\frac{{x}^{3}}{844}}+{\frac{2\,{x}^{2}}{1899}}-{\frac{4\,x}{17091}}+{\frac{1}{633}} \right ) }+{\frac{1}{3691656}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{ \left ( 4\,{{\it \_R}}^{4}-54\,{{\it \_R}}^{3}+2043\,{{\it \_R}}^{2}-324\,{\it \_R}+144 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \]

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)^2,x)

[Out]

(1/153819*x^5-1/22788*x^4+1/844*x^3+2/1899*x^2-4/17091*x+1/633)/(x^6+18*x^4+324*
x^3+108*x^2+216)+1/3691656*sum((4*_R^4-54*_R^3+2043*_R^2-324*_R+144)/(_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. \[ \frac{4 \, x^{5} - 27 \, x^{4} + 729 \, x^{3} + 648 \, x^{2} - 144 \, x + 972}{615276 \,{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} + \frac{1}{615276} \, \int \frac{4 \, x^{4} - 54 \, x^{3} + 2043 \, x^{2} - 324 \, x + 144}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

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)^2,x, algorithm="maxima")

[Out]

1/615276*(4*x^5 - 27*x^4 + 729*x^3 + 648*x^2 - 144*x + 972)/(x^6 + 18*x^4 + 324*
x^3 + 108*x^2 + 216) + 1/615276*integrate((4*x^4 - 54*x^3 + 2043*x^2 - 324*x + 1
44)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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)^2,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.872339, size = 104, normalized size = 0.15 \[ \operatorname{RootSum}{\left (27493895104978847349012449000830556700672 t^{6} - 1318718189226950088862983192576 t^{4} + 12120917704776776448 t^{2} - 39753025, \left ( t \mapsto t \log{\left (\frac{947842259001288723909832054550209950242045952 t^{5}}{61864539719962655} - \frac{243458646817775607639654889480814592 t^{4}}{9811980923071} - \frac{41682556475067500431787310779667456 t^{3}}{61864539719962655} + \frac{12026877442664328616462272 t^{2}}{9811980923071} + \frac{216142618488859793668428 t}{61864539719962655} + x - \frac{308574300024117}{39247923692284} \right )} \right )\right )} + \frac{4 x^{5} - 27 x^{4} + 729 x^{3} + 648 x^{2} - 144 x + 972}{615276 x^{6} + 11074968 x^{4} + 199349424 x^{3} + 66449808 x^{2} + 132899616} \]

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)**2,x)

[Out]

RootSum(27493895104978847349012449000830556700672*_t**6 - 1318718189226950088862
983192576*_t**4 + 12120917704776776448*_t**2 - 39753025, Lambda(_t, _t*log(94784
2259001288723909832054550209950242045952*_t**5/61864539719962655 - 2434586468177
75607639654889480814592*_t**4/9811980923071 - 4168255647506750043178731077966745
6*_t**3/61864539719962655 + 12026877442664328616462272*_t**2/9811980923071 + 216
142618488859793668428*_t/61864539719962655 + x - 308574300024117/39247923692284)
)) + (4*x**5 - 27*x**4 + 729*x**3 + 648*x**2 - 144*x + 972)/(615276*x**6 + 11074
968*x**4 + 199349424*x**3 + 66449808*x**2 + 132899616)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}}\,{d x} \]

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)^2,x, algorithm="giac")

[Out]

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