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]

-1/4*x^2*AppellF1(2/3,1,1/3,5/3,x^3,-1/2*x^3)*2^(2/3)+1/3*arctan(1/3*(3^(1/3)+2*(x^3+2)^(1/3))*3^(1/6))*3^(1/6
)+2/3*arctan(1/3*(1+2*3^(1/3)*x/(x^3+2)^(1/3))*3^(1/2))*3^(1/6)+1/18*ln(-x^3+1)*3^(2/3)+1/6*ln(3^(1/3)-(x^3+2)
^(1/3))*3^(2/3)-1/3*ln(3^(1/3)*x-(x^3+2)^(1/3))*3^(2/3)

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Rubi [F]  time = 0.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{2+x^3}} \, dx \]

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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fricas [F]  time = 18.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x^{3} + 2\right )}^{\frac {2}{3}} {\left (x + 2\right )}}{x^{5} + x^{4} + x^{3} + 2 \, x^{2} + 2 \, x + 2}, x\right ) \]

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

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)

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maple [F]  time = 4.24, size = 0, normalized size = 0.00 \[ \int \frac {x +2}{\left (x^{2}+x +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x+2}{{\left (x^3+2\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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