Optimal. Leaf size=124 \[ \frac{x^5}{5}-2 x^2-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{3}{4} 3^{2/3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )+\frac{1}{6} \log (x+1)-\frac{3}{2} 3^{2/3} \log \left (x+\sqrt [3]{3}\right )+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{9}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]
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Rubi [A] time = 0.114143, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {1367, 1502, 1510, 292, 31, 634, 618, 204, 628, 617} \[ \frac{x^5}{5}-2 x^2-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{3}{4} 3^{2/3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )+\frac{1}{6} \log (x+1)-\frac{3}{2} 3^{2/3} \log \left (x+\sqrt [3]{3}\right )+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{9}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]
Antiderivative was successfully verified.
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Rule 1367
Rule 1502
Rule 1510
Rule 292
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rule 617
Rubi steps
\begin{align*} \int \frac{x^{10}}{3+4 x^3+x^6} \, dx &=\frac{x^5}{5}-\frac{1}{5} \int \frac{x^4 \left (15+20 x^3\right )}{3+4 x^3+x^6} \, dx\\ &=-2 x^2+\frac{x^5}{5}+\frac{1}{10} \int \frac{x \left (120+130 x^3\right )}{3+4 x^3+x^6} \, dx\\ &=-2 x^2+\frac{x^5}{5}-\frac{1}{2} \int \frac{x}{1+x^3} \, dx+\frac{27}{2} \int \frac{x}{3+x^3} \, dx\\ &=-2 x^2+\frac{x^5}{5}+\frac{1}{6} \int \frac{1}{1+x} \, dx-\frac{1}{6} \int \frac{1+x}{1-x+x^2} \, dx-\frac{1}{2} \left (3\ 3^{2/3}\right ) \int \frac{1}{\sqrt [3]{3}+x} \, dx+\frac{1}{2} \left (3\ 3^{2/3}\right ) \int \frac{\sqrt [3]{3}+x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx\\ &=-2 x^2+\frac{x^5}{5}+\frac{1}{6} \log (1+x)-\frac{3}{2} 3^{2/3} \log \left (\sqrt [3]{3}+x\right )-\frac{1}{12} \int \frac{-1+2 x}{1-x+x^2} \, dx-\frac{1}{4} \int \frac{1}{1-x+x^2} \, dx+\frac{27}{4} \int \frac{1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx+\frac{1}{4} \left (3\ 3^{2/3}\right ) \int \frac{-\sqrt [3]{3}+2 x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx\\ &=-2 x^2+\frac{x^5}{5}+\frac{1}{6} \log (1+x)-\frac{3}{2} 3^{2/3} \log \left (\sqrt [3]{3}+x\right )-\frac{1}{12} \log \left (1-x+x^2\right )+\frac{3}{4} 3^{2/3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac{1}{2} \left (9\ 3^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 x}{\sqrt [3]{3}}\right )\\ &=-2 x^2+\frac{x^5}{5}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{9}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+\frac{1}{6} \log (1+x)-\frac{3}{2} 3^{2/3} \log \left (\sqrt [3]{3}+x\right )-\frac{1}{12} \log \left (1-x+x^2\right )+\frac{3}{4} 3^{2/3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0535701, size = 118, normalized size = 0.95 \[ \frac{1}{60} \left (12 x^5-120 x^2-5 \log \left (x^2-x+1\right )+45\ 3^{2/3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+10 \log (x+1)-90\ 3^{2/3} \log \left (3^{2/3} x+3\right )-270 \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )-10 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 94, normalized size = 0.8 \begin{align*}{\frac{{x}^{5}}{5}}-2\,{x}^{2}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{3\,{3}^{2/3}\ln \left ( \sqrt [3]{3}+x \right ) }{2}}+{\frac{3\,{3}^{2/3}\ln \left ({3}^{2/3}-\sqrt [3]{3}x+{x}^{2} \right ) }{4}}+{\frac{9\,\sqrt [6]{3}}{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.63335, size = 127, normalized size = 1.02 \begin{align*} \frac{1}{5} \, x^{5} - 2 \, x^{2} + \frac{3}{4} \cdot 3^{\frac{2}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) - \frac{3}{2} \cdot 3^{\frac{2}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{9}{2} \cdot 3^{\frac{1}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, x - 3^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52044, size = 358, normalized size = 2.89 \begin{align*} \frac{1}{5} \, x^{5} - 2 \, x^{2} + \frac{3}{2} \, \sqrt{3} \left (-9\right )^{\frac{1}{3}} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (2 \, \left (-9\right )^{\frac{1}{3}} x + 3\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{3}{4} \, \left (-9\right )^{\frac{1}{3}} \log \left (3 \, x^{2} - \left (-9\right )^{\frac{2}{3}} x - 3 \, \left (-9\right )^{\frac{1}{3}}\right ) + \frac{3}{2} \, \left (-9\right )^{\frac{1}{3}} \log \left (3 \, x + \left (-9\right )^{\frac{2}{3}}\right ) - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.636538, size = 144, normalized size = 1.16 \begin{align*} \frac{x^{5}}{5} - 2 x^{2} + \frac{\log{\left (x + 1 \right )}}{6} + \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{3872 \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{5}}{3281} + \frac{3188648 \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{2}}{88587} \right )} + \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{3188648 \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{2}}{88587} + \frac{3872 \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{5}}{3281} \right )} + \operatorname{RootSum}{\left (8 t^{3} + 243, \left ( t \mapsto t \log{\left (\frac{3872 t^{5}}{3281} + \frac{3188648 t^{2}}{88587} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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