Optimal. Leaf size=41 \[ x (x+1)^p \left (x^2-x+1\right )^p \left (x^3+1\right )^{-p} \, _2F_1\left (\frac {1}{3},-p;\frac {4}{3};-x^3\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {713, 245} \[ x (x+1)^p \left (x^2-x+1\right )^p \left (x^3+1\right )^{-p} \, _2F_1\left (\frac {1}{3},-p;\frac {4}{3};-x^3\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 245
Rule 713
Rubi steps
\begin {align*} \int (1+x)^p \left (1-x+x^2\right )^p \, dx &=\left ((1+x)^p \left (1-x+x^2\right )^p \left (1+x^3\right )^{-p}\right ) \int \left (1+x^3\right )^p \, dx\\ &=x (1+x)^p \left (1-x+x^2\right )^p \left (1+x^3\right )^{-p} \, _2F_1\left (\frac {1}{3},-p;\frac {4}{3};-x^3\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.11, size = 132, normalized size = 3.22 \[ \frac {\left (\frac {-2 i x+\sqrt {3}+i}{\sqrt {3}+3 i}\right )^{-p} \left (\frac {2 i x+\sqrt {3}-i}{\sqrt {3}-3 i}\right )^{-p} (x+1)^{p+1} \left (x^2-x+1\right )^p F_1\left (p+1;-p,-p;p+2;\frac {2 i (x+1)}{3 i+\sqrt {3}},-\frac {2 i (x+1)}{-3 i+\sqrt {3}}\right )}{p+1} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (x^{2} - x + 1\right )}^{p} {\left (x + 1\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x^{2} - x + 1\right )}^{p} {\left (x + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.97, size = 0, normalized size = 0.00 \[ \int \left (x +1\right )^{p} \left (x^{2}-x +1\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x^{2} - x + 1\right )}^{p} {\left (x + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (x+1\right )}^p\,{\left (x^2-x+1\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (x + 1\right )^{p} \left (x^{2} - x + 1\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________