Optimal. Leaf size=873 \[ \text{result too large to display} \]
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Rubi [A] time = 6.34642, antiderivative size = 873, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{\sqrt [3]{-6} \left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right )-3 x}{157464 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac{\left (9 \sqrt [6]{3}+i \left (2\ 2^{2/3}-2 i 2^{2/3} \sqrt{3}-3\ 3^{2/3}\right )\right ) \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 )}{69984\ 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\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 )}{26244 \sqrt{3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac{\left (9 \sqrt [6]{3}-i \left (4\ 2^{2/3}+3\ 3^{2/3}\right )\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 )}{69984\ 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}-\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 )}{26244 \sqrt{3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac{\left (2\ 2^{2/3}-3\ 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{944784 \sqrt [6]{3} \sqrt{2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{52488 \sqrt{6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac{i \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{23328\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac{\left (i+\sqrt{3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{46656\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{629856\ 2^{2/3} \sqrt [3]{3}}-\frac{3 x+\sqrt [3]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )}{157464 \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{-\sqrt [3]{3} x-3\ 6^{2/3}+2 \sqrt [3]{2}}{104976 \left (9 \sqrt [3]{2}-4 \sqrt [3]{3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )} \]
Warning: Unable to verify antiderivative.
[In] Int[x^3/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]
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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**3/(x**6+18*x**4+324*x**3+108*x**2+216)**2,x)
[Out]
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Mathematica [C] time = 0.0422749, size = 167, normalized size = 0.19 \[ \frac{\text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\&,\frac{2 \text{$\#$1}^4 \log (x-\text{$\#$1})-27 \text{$\#$1}^3 \log (x-\text{$\#$1})+72 \text{$\#$1}^2 \log (x-\text{$\#$1})-162 \text{$\#$1} \log (x-\text{$\#$1})+1971 \log (x-\text{$\#$1})}{\text{$\#$1}^5+12 \text{$\#$1}^3+162 \text{$\#$1}^2+36 \text{$\#$1}}\&\right ]}{11074968}+\frac{4 x^5-27 x^4+96 x^3+648 x^2-3942 x+972}{3691656 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]
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Maple [C] time = 0.014, size = 122, normalized size = 0.1 \[{\frac{1}{{x}^{6}+18\,{x}^{4}+324\,{x}^{3}+108\,{x}^{2}+216} \left ({\frac{{x}^{5}}{922914}}-{\frac{{x}^{4}}{136728}}+{\frac{4\,{x}^{3}}{153819}}+{\frac{{x}^{2}}{5697}}-{\frac{73\,x}{68364}}+{\frac{1}{3798}} \right ) }+{\frac{1}{11074968}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{ \left ( 2\,{{\it \_R}}^{4}-27\,{{\it \_R}}^{3}+72\,{{\it \_R}}^{2}-162\,{\it \_R}+1971 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{4 \, x^{5} - 27 \, x^{4} + 96 \, x^{3} + 648 \, x^{2} - 3942 \, x + 972}{3691656 \,{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} + \frac{1}{1845828} \, \int \frac{2 \, x^{4} - 27 \, x^{3} + 72 \, x^{2} - 162 \, x + 1971}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)^2,x, algorithm="maxima")
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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^3/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)^2,x, algorithm="fricas")
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Sympy [A] time = 1.20717, size = 112, normalized size = 0.13 \[ \operatorname{RootSum}{\left (1282755170017893101915524820582750453426552832 t^{6} - 906388465775544244426251149770752 t^{4} - 4300873166389987741684137984 t^{3} - 717000908921644962816 t^{2} + 135354162312576 t - 7197829, \left ( t \mapsto t \log{\left (\frac{17257935592810449901409556597891882995604001083339368041361480613888 t^{5}}{154206009791052044490694380303237521} + \frac{2389607400620985524376358853572652207181956324560587684052992 t^{4}}{154206009791052044490694380303237521} - \frac{12286072160883283930711715948878260078996992193488388096 t^{3}}{154206009791052044490694380303237521} - \frac{59490553573959173161125496013527909754156558410752 t^{2}}{154206009791052044490694380303237521} - \frac{17520149679836691112367064197713753004827200 t}{154206009791052044490694380303237521} + x + \frac{766422988707229615055855287040887332}{154206009791052044490694380303237521} \right )} \right )\right )} + \frac{4 x^{5} - 27 x^{4} + 96 x^{3} + 648 x^{2} - 3942 x + 972}{3691656 x^{6} + 66449808 x^{4} + 1196096544 x^{3} + 398698848 x^{2} + 797397696} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(x**6+18*x**4+324*x**3+108*x**2+216)**2,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)^2,x, algorithm="giac")
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