3.103 \(\int \frac{(1-x^3)^{2/3}}{(1+x+x^2)^2} \, dx\)

Optimal. Leaf size=43 \[ x^2 \left (-\, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{5}{3};x^3\right )\right )+\frac{x}{\sqrt [3]{1-x^3}}+\frac{1}{\sqrt [3]{1-x^3}} \]

[Out]

(1 - x^3)^(-1/3) + x/(1 - x^3)^(1/3) - x^2*Hypergeometric2F1[2/3, 4/3, 5/3, x^3]

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Rubi [F]  time = 0.425069, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-4*Defer[Int][(1 - x^3)^(2/3)/(-1 + I*Sqrt[3] - 2*x)^2, x])/3 + (((4*I)/3)*Defer[Int][(1 - x^3)^(2/3)/(-1 + I
*Sqrt[3] - 2*x), x])/Sqrt[3] - (4*Defer[Int][(1 - x^3)^(2/3)/(1 + I*Sqrt[3] + 2*x)^2, x])/3 + (((4*I)/3)*Defer
[Int][(1 - x^3)^(2/3)/(1 + I*Sqrt[3] + 2*x), x])/Sqrt[3]

Rubi steps

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

Mathematica [A]  time = 0.150557, size = 43, normalized size = 1. \[ x^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right )+\frac{(2 x+1) \left (1-x^3\right )^{2/3}}{x^2+x+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 2*x)*(1 - x^3)^(2/3))/(1 + x + x^2) + x^2*Hypergeometric2F1[1/3, 2/3, 5/3, x^3]

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Maple [A]  time = 0.036, size = 34, normalized size = 0.8 \begin{align*} -{ \left ( -1+x \right ) \left ( 1+2\,x \right ){\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}+{x}^{2}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{2}{3}};\,{\frac{5}{3}};\,{x}^{3})} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1+x)*(1+2*x)/(-x^3+1)^(1/3)+x^2*hypergeom([1/3,2/3],[5/3],x^3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{{\left (x^{2} + x + 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-x^3 + 1)^(2/3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{{\left (x^{2} + x + 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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