3.754 \(\int \frac{x}{\sqrt [3]{e^x+x}} \, dx\)

Optimal. Leaf size=36 \[ \text{CannotIntegrate}\left (\frac{1}{\sqrt [3]{x+e^x}},x\right )+\text{CannotIntegrate}\left (\left (x+e^x\right )^{2/3},x\right )-\frac{3}{2} \left (x+e^x\right )^{2/3} \]

[Out]

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

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Rubi [A]  time = 0.0396365, 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 = {} \[ \int \frac{x}{\sqrt [3]{e^x+x}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x/(E^x + x)^(1/3),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0477577, size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt [3]{e^x+x}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x/(E^x + x)^(1/3),x]

[Out]

Integrate[x/(E^x + x)^(1/3), x]

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Maple [A]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{x{\frac{1}{\sqrt [3]{{{\rm e}^{x}}+x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(x + e^x)^(1/3), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt [3]{x + e^{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(x + exp(x))**(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(x + e^x)^(1/3), x)