\(\int \frac {x}{\sqrt [3]{e^x+x}} \, dx\) [754]

Optimal result

Integrand size = 11, antiderivative size = 11 \[ \int \frac {x}{\sqrt [3]{e^x+x}} \, dx=-\frac {3}{2} \left (e^x+x\right )^{2/3}+\text {Int}\left (\frac {1}{\sqrt [3]{e^x+x}},x\right )+\text {Int}\left (\left (e^x+x\right )^{2/3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x}{\sqrt [3]{e^x+x}} \, dx=\int \frac {x}{\sqrt [3]{e^x+x}} \, dx \]

[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*} \text {integral}& = -\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 [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {x}{\sqrt [3]{e^x+x}} \, dx=\int \frac {x}{\sqrt [3]{e^x+x}} \, dx \]

[In]

Integrate[x/(E^x + x)^(1/3),x]

[Out]

Integrate[x/(E^x + x)^(1/3), x]

Maple [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73

\[\int \frac {x}{\left ({\mathrm e}^{x}+x \right )^{\frac {1}{3}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {x}{\sqrt [3]{e^x+x}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\sqrt [3]{e^x+x}} \, dx=\int \frac {x}{\sqrt [3]{x + e^{x}}}\, dx \]

[In]

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

[Out]

Integral(x/(x + exp(x))**(1/3), x)

Maxima [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\sqrt [3]{e^x+x}} \, dx=\int { \frac {x}{{\left (x + e^{x}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integrate(x/(x + e^x)^(1/3), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\sqrt [3]{e^x+x}} \, dx=\int { \frac {x}{{\left (x + e^{x}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integrate(x/(x + e^x)^(1/3), x)

Mupad [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\sqrt [3]{e^x+x}} \, dx=\int \frac {x}{{\left (x+{\mathrm {e}}^x\right )}^{1/3}} \,d x \]

[In]

int(x/(x + exp(x))^(1/3),x)

[Out]

int(x/(x + exp(x))^(1/3), x)