Optimal. Leaf size=41 \[ x (1-x)^p \left (x^2+x+1\right )^p \left (1-x^3\right )^{-p} \, _2F_1\left (\frac{1}{3},-p;\frac{4}{3};x^3\right ) \]
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Rubi [A] time = 0.0326162, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ x (1-x)^p \left (x^2+x+1\right )^p \left (1-x^3\right )^{-p} \, _2F_1\left (\frac{1}{3},-p;\frac{4}{3};x^3\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 - x)^p*(1 + x + x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 5.72257, size = 31, normalized size = 0.76 \[ x \left (- x + 1\right )^{p} \left (- x^{3} + 1\right )^{- p} \left (x^{2} + x + 1\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{x^{3}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x)**p*(x**2+x+1)**p,x)
[Out]
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Mathematica [C] time = 0.213049, size = 133, normalized size = 3.24 \[ \frac{(x-1) (1-x)^p \left (\frac{-2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}\right )^{-p} \left (\frac{2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}\right )^{-p} \left (x^2+x+1\right )^p F_1\left (p+1;-p,-p;p+2;\frac{2 i (x-1)}{-3 i+\sqrt{3}},-\frac{2 i (x-1)}{3 i+\sqrt{3}}\right )}{p+1} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 - x)^p*(1 + x + x^2)^p,x]
[Out]
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Maple [F] time = 0.162, size = 0, normalized size = 0. \[ \int \left ( 1-x \right ) ^{p} \left ({x}^{2}+x+1 \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x)^p*(x^2+x+1)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{2} + x + 1\right )}^{p}{\left (-x + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + x + 1)^p*(-x + 1)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (x^{2} + x + 1\right )}^{p}{\left (-x + 1\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + x + 1)^p*(-x + 1)^p,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (- x + 1\right )^{p} \left (x^{2} + x + 1\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x)**p*(x**2+x+1)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{2} + x + 1\right )}^{p}{\left (-x + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + x + 1)^p*(-x + 1)^p,x, algorithm="giac")
[Out]