\(\int (1-x)^p (1+x+x^2)^p \, dx\) [2587]

Optimal result

Integrand size = 16, antiderivative size = 41 \[ \int (1-x)^p \left (1+x+x^2\right )^p \, dx=(1-x)^p x \left (1+x+x^2\right )^p \left (1-x^3\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},-p,\frac {4}{3},x^3\right ) \]

[Out]

(1-x)^p*x*(x^2+x+1)^p*hypergeom([1/3, -p],[4/3],x^3)/((-x^3+1)^p)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {727, 251} \[ \int (1-x)^p \left (1+x+x^2\right )^p \, dx=x (1-x)^p \left (x^2+x+1\right )^p \left (1-x^3\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},-p,\frac {4}{3},x^3\right ) \]

[In]

Int[(1 - x)^p*(1 + x + x^2)^p,x]

[Out]

((1 - x)^p*x*(1 + x + x^2)^p*Hypergeometric2F1[1/3, -p, 4/3, x^3])/(1 - x^3)^p

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 727

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(d + e*x)^FracPart[p]
*((a + b*x + c*x^2)^FracPart[p]/(a*d + c*e*x^3)^FracPart[p]), Int[(d + e*x)^(m - p)*(a*d + c*e*x^3)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0] && IGtQ[m - p + 1, 0] &&  !Intege
rQ[p]

Rubi steps \begin{align*} \text {integral}& = \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 \\ & = (1-x)^p x \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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.24 \[ \int (1-x)^p \left (1+x+x^2\right )^p \, dx=\frac {(1-x)^p \left (\frac {-i+\sqrt {3}-2 i x}{-3 i+\sqrt {3}}\right )^{-p} \left (\frac {i+\sqrt {3}+2 i x}{3 i+\sqrt {3}}\right )^{-p} (-1+x) \left (1+x+x^2\right )^p \operatorname {AppellF1}\left (1+p,-p,-p,2+p,\frac {2 i (-1+x)}{-3 i+\sqrt {3}},-\frac {2 i (-1+x)}{3 i+\sqrt {3}}\right )}{1+p} \]

[In]

Integrate[(1 - x)^p*(1 + x + x^2)^p,x]

[Out]

((1 - x)^p*(-1 + x)*(1 + x + x^2)^p*AppellF1[1 + p, -p, -p, 2 + p, ((2*I)*(-1 + x))/(-3*I + Sqrt[3]), ((-2*I)*
(-1 + x))/(3*I + Sqrt[3])])/((1 + p)*((-I + Sqrt[3] - (2*I)*x)/(-3*I + Sqrt[3]))^p*((I + Sqrt[3] + (2*I)*x)/(3
*I + Sqrt[3]))^p)

Maple [F]

\[\int \left (1-x \right )^{p} \left (x^{2}+x +1\right )^{p}d x\]

[In]

int((1-x)^p*(x^2+x+1)^p,x)

[Out]

int((1-x)^p*(x^2+x+1)^p,x)

Fricas [F]

\[ \int (1-x)^p \left (1+x+x^2\right )^p \, dx=\int { {\left (x^{2} + x + 1\right )}^{p} {\left (-x + 1\right )}^{p} \,d x } \]

[In]

integrate((1-x)^p*(x^2+x+1)^p,x, algorithm="fricas")

[Out]

integral((x^2 + x + 1)^p*(-x + 1)^p, x)

Sympy [F]

\[ \int (1-x)^p \left (1+x+x^2\right )^p \, dx=\int \left (1 - x\right )^{p} \left (x^{2} + x + 1\right )^{p}\, dx \]

[In]

integrate((1-x)**p*(x**2+x+1)**p,x)

[Out]

Integral((1 - x)**p*(x**2 + x + 1)**p, x)

Maxima [F]

\[ \int (1-x)^p \left (1+x+x^2\right )^p \, dx=\int { {\left (x^{2} + x + 1\right )}^{p} {\left (-x + 1\right )}^{p} \,d x } \]

[In]

integrate((1-x)^p*(x^2+x+1)^p,x, algorithm="maxima")

[Out]

integrate((x^2 + x + 1)^p*(-x + 1)^p, x)

Giac [F]

\[ \int (1-x)^p \left (1+x+x^2\right )^p \, dx=\int { {\left (x^{2} + x + 1\right )}^{p} {\left (-x + 1\right )}^{p} \,d x } \]

[In]

integrate((1-x)^p*(x^2+x+1)^p,x, algorithm="giac")

[Out]

integrate((x^2 + x + 1)^p*(-x + 1)^p, x)

Mupad [F(-1)]

Timed out. \[ \int (1-x)^p \left (1+x+x^2\right )^p \, dx=\int {\left (1-x\right )}^p\,{\left (x^2+x+1\right )}^p \,d x \]

[In]

int((1 - x)^p*(x + x^2 + 1)^p,x)

[Out]

int((1 - x)^p*(x + x^2 + 1)^p, x)