3.61 \(\int \frac{2+x}{(1+x+x^2) \sqrt [3]{2+x^3}} \, dx\)

Optimal. Leaf size=168 \[ -\frac{x^2 F_1\left (\frac{2}{3};1,\frac{1}{3};\frac{5}{3};x^3,-\frac{x^3}{2}\right )}{2 \sqrt [3]{2}}+\frac{\log \left (1-x^3\right )}{6 \sqrt [3]{3}}+\frac{\log \left (\sqrt [3]{3}-\sqrt [3]{x^3+2}\right )}{2 \sqrt [3]{3}}-\frac{\log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+2}\right )}{\sqrt [3]{3}}+\frac{2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{3^{5/6}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{x^3+2}+\sqrt [3]{3}}{3^{5/6}}\right )}{3^{5/6}} \]

[Out]

-(x^2*AppellF1[2/3, 1, 1/3, 5/3, x^3, -x^3/2])/(2*2^(1/3)) + (2*ArcTan[(1 + (2*3^(1/3)*x)/(2 + x^3)^(1/3))/Sqr
t[3]])/3^(5/6) + ArcTan[(3^(1/3) + 2*(2 + x^3)^(1/3))/3^(5/6)]/3^(5/6) + Log[1 - x^3]/(6*3^(1/3)) + Log[3^(1/3
) - (2 + x^3)^(1/3)]/(2*3^(1/3)) - Log[3^(1/3)*x - (2 + x^3)^(1/3)]/3^(1/3)

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Rubi [F]  time = 0.204262, 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 = {} \[ \int \frac{2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0.137935, size = 0, normalized size = 0. \[ \int \frac{2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{\frac{2+x}{{x}^{2}+x+1}{\frac{1}{\sqrt [3]{{x}^{3}+2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (x^{3} + 2\right )}^{\frac{2}{3}}{\left (x + 2\right )}}{x^{5} + x^{4} + x^{3} + 2 \, x^{2} + 2 \, x + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((x^3 + 2)^(2/3)*(x + 2)/(x^5 + x^4 + x^3 + 2*x^2 + 2*x + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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