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

Optimal. Leaf size=38 \[ \text {Int}\left (\frac {1}{\sqrt [3]{x+e^x}},x\right )+\text {Int}\left (\left (x+e^x\right )^{2/3},x\right )-\frac {3}{2} \left (x+e^x\right )^{2/3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {x}{\sqrt [3]{e^x+x}} \, dx \]

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 = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt [3]{e^x+x}} \, dx \]

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]

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (x +{\mathrm e}^{x}\right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{{\left (x+{\mathrm {e}}^x\right )}^{1/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x + exp(x))^(1/3),x)

[Out]

int(x/(x + exp(x))^(1/3), x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt [3]{x + e^{x}}}\, dx \]

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)

________________________________________________________________________________________