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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]