3.26.83 \(\int \frac {1}{(1+x)^{2/3} (1-x+x^2)^{2/3}} \, dx\) [2583]

Optimal. Leaf size=45 \[ \frac {x \left (1+x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-x^3\right )}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {727, 251} \begin {gather*} \frac {x \left (x^3+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-x^3\right )}{(x+1)^{2/3} \left (x^2-x+1\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 + x)^(2/3)*(1 - x + x^2)^(2/3)),x]

[Out]

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

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*} \int \frac {1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx &=\frac {\left (1+x^3\right )^{2/3} \int \frac {1}{\left (1+x^3\right )^{2/3}} \, dx}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}}\\ &=\frac {x \left (1+x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-x^3\right )}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.06, size = 143, normalized size = 3.18 \begin {gather*} \frac {3 \left (-i+\sqrt {3}+2 i x\right ) \sqrt [3]{1+x} \left (-\frac {3 i+\sqrt {3}+\left (-3 i+\sqrt {3}\right ) x}{-3 i+\sqrt {3}+\left (3 i+\sqrt {3}\right ) x}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {4 i \sqrt {3} (1+x)}{\left (3 i+\sqrt {3}\right ) \left (-i+\sqrt {3}+2 i x\right )}\right )}{\left (-3 i+\sqrt {3}\right ) \left (1-x+x^2\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x)^(2/3)*(1 - x + x^2)^(2/3)),x]

[Out]

(3*(-I + Sqrt[3] + (2*I)*x)*(1 + x)^(1/3)*(-((3*I + Sqrt[3] + (-3*I + Sqrt[3])*x)/(-3*I + Sqrt[3] + (3*I + Sqr
t[3])*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, ((4*I)*Sqrt[3]*(1 + x))/((3*I + Sqrt[3])*(-I + Sqrt[3] + (2*
I)*x))])/((-3*I + Sqrt[3])*(1 - x + x^2)^(2/3))

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Maple [F]
time = 0.35, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x +1\right )^{\frac {2}{3}} \left (x^{2}-x +1\right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x+1)^(2/3)/(x^2-x+1)^(2/3),x)

[Out]

int(1/(x+1)^(2/3)/(x^2-x+1)^(2/3),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)^(2/3)/(x^2-x+1)^(2/3),x, algorithm="maxima")

[Out]

integrate(1/((x^2 - x + 1)^(2/3)*(x + 1)^(2/3)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)^(2/3)/(x^2-x+1)^(2/3),x, algorithm="fricas")

[Out]

integral((x^2 - x + 1)^(1/3)*(x + 1)^(1/3)/(x^3 + 1), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x + 1\right )^{\frac {2}{3}} \left (x^{2} - x + 1\right )^{\frac {2}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)**(2/3)/(x**2-x+1)**(2/3),x)

[Out]

Integral(1/((x + 1)**(2/3)*(x**2 - x + 1)**(2/3)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)^(2/3)/(x^2-x+1)^(2/3),x, algorithm="giac")

[Out]

integrate(1/((x^2 - x + 1)^(2/3)*(x + 1)^(2/3)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (x+1\right )}^{2/3}\,{\left (x^2-x+1\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x + 1)^(2/3)*(x^2 - x + 1)^(2/3)),x)

[Out]

int(1/((x + 1)^(2/3)*(x^2 - x + 1)^(2/3)), x)

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