Optimal. Leaf size=124 \[ \frac {x^5}{5}-2 x^2-\frac {1}{12} \log \left (x^2-x+1\right )+\frac {3}{4} 3^{2/3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )+\frac {1}{6} \log (x+1)-\frac {3}{2} 3^{2/3} \log \left (x+\sqrt [3]{3}\right )+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {9}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 124, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 31
Rule 204
Rule 292
Rule 617
Rule 618
Rule 628
Rule 634
Rule 1367
Rule 1502
Rule 1510
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 = 0.05, 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.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.82, size = 102, normalized size = 0.82 \[ \frac {1}{5} \, x^{5} - 2 \, x^{2} + \frac {3}{2} \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (2 \, \left (-9\right )^{\frac {1}{3}} x + 3\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {3}{4} \, \left (-9\right )^{\frac {1}{3}} \log \left (3 \, x^{2} - \left (-9\right )^{\frac {2}{3}} x - 3 \, \left (-9\right )^{\frac {1}{3}}\right ) + \frac {3}{2} \, \left (-9\right )^{\frac {1}{3}} \log \left (3 \, x + \left (-9\right )^{\frac {2}{3}}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.37, size = 96, normalized size = 0.77 \[ \frac {1}{5} \, x^{5} - 2 \, x^{2} + \frac {3}{4} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {3}{2} \cdot 3^{\frac {2}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {9}{2} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 94, normalized size = 0.76 \[ \frac {x^{5}}{5}-2 x^{2}+\frac {9 \,3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\ln \left (x +1\right )}{6}-\frac {3 \,3^{\frac {2}{3}} \ln \left (x +3^{\frac {1}{3}}\right )}{2}+\frac {3 \,3^{\frac {2}{3}} \ln \left (x^{2}-3^{\frac {1}{3}} x +3^{\frac {2}{3}}\right )}{4}-\frac {\ln \left (x^{2}-x +1\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.23, size = 94, normalized size = 0.76 \[ \frac {1}{5} \, x^{5} - 2 \, x^{2} + \frac {3}{4} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {3}{2} \cdot 3^{\frac {2}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {9}{2} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.24, size = 124, normalized size = 1.00 \[ \frac {\ln \left (x+1\right )}{6}-\frac {3\,3^{2/3}\,\ln \left (x+3^{1/3}\right )}{2}+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-2\,x^2+\frac {x^5}{5}-\frac {3\,{\left (-1\right )}^{1/3}\,\ln \left (x-\frac {{\left (-1\right )}^{1/3}\,3^{1/3}}{2}-\frac {{\left (-1\right )}^{1/6}\,3^{5/6}}{2}+\frac {3^{1/3}}{2}\right )\,\left (3^{2/3}+3^{1/6}\,3{}\mathrm {i}\right )}{4}+\frac {3\,{\left (-1\right )}^{1/3}\,3^{2/3}\,\ln \left (x+{\left (-1\right )}^{2/3}\,3^{1/3}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.62, size = 144, normalized size = 1.16 \[ \frac {x^{5}}{5} - 2 x^{2} + \frac {\log {\left (x + 1 \right )}}{6} + \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {3872 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{5}}{3281} + \frac {3188648 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{2}}{88587} \right )} + \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {3188648 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{2}}{88587} + \frac {3872 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{5}}{3281} \right )} + \operatorname {RootSum} {\left (8 t^{3} + 243, \left (t \mapsto t \log {\left (\frac {3872 t^{5}}{3281} + \frac {3188648 t^{2}}{88587} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________