Optimal. Leaf size=59 \[ -\text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x^3+x+1}-\text {$\#$1} x\right )-\text {$\#$1}^2 \log (x)}{2 \text {$\#$1}^3-1}\& \right ] \]
________________________________________________________________________________________
Rubi [F] time = 0.33, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(3+2 x) \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {(3+2 x) \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx &=\int \left (\frac {3 \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6}+\frac {2 x \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6}\right ) \, dx\\ &=2 \int \frac {x \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx+3 \int \frac {\left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+2 x) \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.00, size = 59, normalized size = 1.00 \begin {gather*} -\text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{1+x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+2 \text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + x + 1\right )}^{\frac {2}{3}} {\left (2 \, x + 3\right )}}{x^{6} + x^{4} + x^{3} + x^{2} + 2 \, x + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 5.94, size = 1207, normalized size = 20.46
method | result | size |
trager | \(18 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} \ln \left (-\frac {-18 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} x^{3}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x -9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} x -9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5}+2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x^{3}+\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-\left (x^{3}+x +1\right )^{\frac {2}{3}} x +\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x +\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2}}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-x^{3}+x +1}\right )-9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} \ln \left (-\frac {9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} x^{3}-9 \left (x^{3}+x +1\right )^{\frac {1}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{2}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x -2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x^{3}+\left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x -\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-x^{3}+x +1}\right )+9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} \ln \left (\frac {-18 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+6 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x +3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x +3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3}-2 x^{3}-x -1}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-2 x^{3}-x -1}\right )-3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} \ln \left (-\frac {-18 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} x^{3}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x -9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} x -9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5}+2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x^{3}+\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-\left (x^{3}+x +1\right )^{\frac {2}{3}} x +\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x +\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2}}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-x^{3}+x +1}\right )-2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) \ln \left (\frac {-18 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+6 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x +3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x +3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3}-2 x^{3}-x -1}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-2 x^{3}-x -1}\right )+\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) \ln \left (\frac {-27 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{7} x^{3}+9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} x^{3}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x -3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} x -3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4}-\left (x^{3}+x +1\right )^{\frac {2}{3}} x}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-2 x^{3}-x -1}\right )\) | \(1207\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + x + 1\right )}^{\frac {2}{3}} {\left (2 \, x + 3\right )}}{x^{6} + x^{4} + x^{3} + x^{2} + 2 \, x + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (2\,x+3\right )\,{\left (x^3+x+1\right )}^{2/3}}{x^6+x^4+x^3+x^2+2\,x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x + 3\right ) \left (x^{3} + x + 1\right )^{\frac {2}{3}}}{x^{6} + x^{4} + x^{3} + x^{2} + 2 x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________