Optimal. Leaf size=361 \[ \frac{(-1)^{2/3} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{(-1)^{2/3} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{36 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac{\sqrt [3]{-1} \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{54\ 2^{2/3} 3^{5/6} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{108 \sqrt [6]{2} 3^{5/6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.547554, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2097, 634, 618, 204, 628, 206} \[ \frac{(-1)^{2/3} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{(-1)^{2/3} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{36 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac{\sqrt [3]{-1} \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{54\ 2^{2/3} 3^{5/6} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{108 \sqrt [6]{2} 3^{5/6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2097
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx &=1259712 \int \left (\frac{(-1)^{2/3} \left (3 \sqrt [3]{-3} 2^{2/3}-x\right )}{136048896 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )}+\frac{(-1)^{2/3} \left (3 (-2)^{2/3} \sqrt [3]{3}+x\right )}{136048896 \sqrt [3]{2} 3^{2/3} \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac{6 \sqrt [3]{3}+\sqrt [3]{2} x}{408146688\ 6^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=-\frac{(-1)^{2/3} \int \frac{3 (-2)^{2/3} \sqrt [3]{3}+x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{324 \sqrt [3]{2} 3^{2/3}}-\frac{\int \frac{6 \sqrt [3]{3}+\sqrt [3]{2} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{324\ 6^{2/3}}+\frac{(-1)^{2/3} \int \frac{3 \sqrt [3]{-3} 2^{2/3}-x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{108 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=\frac{\sqrt [3]{-\frac{1}{3}} \int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{108\ 2^{2/3}}-\frac{\int \frac{3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{648 \sqrt [3]{2} 3^{2/3}}-\frac{(-1)^{2/3} \int \frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\int \frac{1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{108\ 2^{2/3} \sqrt [3]{3}}+\frac{(-1)^{2/3} \int \frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{\int \frac{1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{36\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=\frac{(-1)^{2/3} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{(-1)^{2/3} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\sqrt [3]{-\frac{1}{3}} \operatorname{Subst}\left (\int \frac{1}{-6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )-x^2} \, dx,x,3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )}{54\ 2^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-6 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )-x^2} \, dx,x,3\ 2^{2/3} \sqrt [3]{3}+2 x\right )}{54\ 2^{2/3} \sqrt [3]{3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )-x^2} \, dx,x,3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{18\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^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 )}{36 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac{\sqrt [3]{-1} \tan ^{-1}\left (\frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{108 \sqrt [6]{2} 3^{5/6} \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt{3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{108 \sqrt [6]{2} 3^{5/6} \sqrt{-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac{(-1)^{2/3} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{(-1)^{2/3} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{2} 3^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0116398, size = 57, normalized size = 0.16 \[ \frac{1}{6} \text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\& ,\frac{\log (x-\text{$\#$1})}{\text{$\#$1}^4+12 \text{$\#$1}^2+162 \text{$\#$1}+36}\& \right ] \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.005, size = 54, normalized size = 0.2 \begin{align*}{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{{\it \_R}\,\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.215503, size = 61, normalized size = 0.17 \begin{align*} \operatorname{RootSum}{\left (158171241119638192128 t^{6} - 96402615118848 t^{4} + 287743415040 t^{3} - 51018336 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{65418399445721140961280 t^{5}}{415817} + \frac{2480926457425102848 t^{4}}{415817} - \frac{39451802929737984 t^{3}}{415817} + \frac{118071997444800 t^{2}}{415817} - \frac{16745884920 t}{415817} + x - \frac{268790}{415817} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]