Optimal. Leaf size=90 \[ \frac{x^3}{3}+\frac{x^2}{2}+\frac{2}{3} \log \left (x^2+x+1\right )-\frac{1}{24} \log \left (2 x^2-x+2\right )-\frac{3 x}{2}+\frac{5}{12} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )+\frac{8 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.122615, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2075, 634, 618, 204, 628} \[ \frac{x^3}{3}+\frac{x^2}{2}+\frac{2}{3} \log \left (x^2+x+1\right )-\frac{1}{24} \log \left (2 x^2-x+2\right )-\frac{3 x}{2}+\frac{5}{12} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )+\frac{8 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2075
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3 \left (5+x+3 x^2+2 x^3\right )}{2+x+3 x^2+x^3+2 x^4} \, dx &=\int \left (-\frac{3}{2}+x+x^2+\frac{2 (3+2 x)}{3 \left (1+x+x^2\right )}+\frac{-6-x}{6 \left (2-x+2 x^2\right )}\right ) \, dx\\ &=-\frac{3 x}{2}+\frac{x^2}{2}+\frac{x^3}{3}+\frac{1}{6} \int \frac{-6-x}{2-x+2 x^2} \, dx+\frac{2}{3} \int \frac{3+2 x}{1+x+x^2} \, dx\\ &=-\frac{3 x}{2}+\frac{x^2}{2}+\frac{x^3}{3}-\frac{1}{24} \int \frac{-1+4 x}{2-x+2 x^2} \, dx+\frac{2}{3} \int \frac{1+2 x}{1+x+x^2} \, dx-\frac{25}{24} \int \frac{1}{2-x+2 x^2} \, dx+\frac{4}{3} \int \frac{1}{1+x+x^2} \, dx\\ &=-\frac{3 x}{2}+\frac{x^2}{2}+\frac{x^3}{3}+\frac{2}{3} \log \left (1+x+x^2\right )-\frac{1}{24} \log \left (2-x+2 x^2\right )+\frac{25}{12} \operatorname{Subst}\left (\int \frac{1}{-15-x^2} \, dx,x,-1+4 x\right )-\frac{8}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-\frac{3 x}{2}+\frac{x^2}{2}+\frac{x^3}{3}+\frac{5}{12} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )+\frac{8 \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{2}{3} \log \left (1+x+x^2\right )-\frac{1}{24} \log \left (2-x+2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0229295, size = 78, normalized size = 0.87 \[ \frac{1}{72} \left (24 x^3+36 x^2+48 \log \left (x^2+x+1\right )-3 \log \left (2 x^2-x+2\right )-108 x+64 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )-10 \sqrt{15} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{15}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 69, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{3}}+{\frac{{x}^{2}}{2}}-{\frac{3\,x}{2}}-{\frac{\ln \left ( 2\,{x}^{2}-x+2 \right ) }{24}}-{\frac{5\,\sqrt{15}}{36}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{15}}{15}} \right ) }+{\frac{2\,\ln \left ({x}^{2}+x+1 \right ) }{3}}+{\frac{8\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.55046, size = 92, normalized size = 1.02 \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{5}{36} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) + \frac{8}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{3}{2} \, x - \frac{1}{24} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{2}{3} \, \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.40235, size = 246, normalized size = 2.73 \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{5}{36} \, \sqrt{5} \sqrt{3} \arctan \left (\frac{1}{15} \, \sqrt{5} \sqrt{3}{\left (4 \, x - 1\right )}\right ) + \frac{8}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{3}{2} \, x - \frac{1}{24} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{2}{3} \, \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.209759, size = 92, normalized size = 1.02 \begin{align*} \frac{x^{3}}{3} + \frac{x^{2}}{2} - \frac{3 x}{2} - \frac{\log{\left (x^{2} - \frac{x}{2} + 1 \right )}}{24} + \frac{2 \log{\left (x^{2} + x + 1 \right )}}{3} - \frac{5 \sqrt{15} \operatorname{atan}{\left (\frac{4 \sqrt{15} x}{15} - \frac{\sqrt{15}}{15} \right )}}{36} + \frac{8 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21717, size = 92, normalized size = 1.02 \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{5}{36} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) + \frac{8}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{3}{2} \, x - \frac{1}{24} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{2}{3} \, \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]