3.749 \(\int \frac{e^x x}{\sqrt{e^x+x}} \, dx\)

Optimal. Leaf size=50 \[ -\text{CannotIntegrate}\left (\frac{1}{\sqrt{x+e^x}},x\right )-3 \text{CannotIntegrate}\left (\sqrt{x+e^x},x\right )+2 \sqrt{x+e^x} x+2 \sqrt{x+e^x} \]

[Out]

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

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Rubi [A]  time = 0.0810275, 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 = {} \[ \int \frac{e^x x}{\sqrt{e^x+x}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^x*x)/Sqrt[E^x + x],x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{e^x x}{\sqrt{e^x+x}} \, dx &=2 x \sqrt{e^x+x}-2 \int \sqrt{e^x+x} \, dx-\int \frac{x}{\sqrt{e^x+x}} \, dx\\ &=2 \sqrt{e^x+x}+2 x \sqrt{e^x+x}-2 \int \sqrt{e^x+x} \, dx-\int \frac{1}{\sqrt{e^x+x}} \, dx-\int \sqrt{e^x+x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0970847, size = 0, normalized size = 0. \[ \int \frac{e^x x}{\sqrt{e^x+x}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(E^x*x)/Sqrt[E^x + x],x]

[Out]

Integrate[(E^x*x)/Sqrt[E^x + x], x]

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Maple [A]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{x}}x{\frac{1}{\sqrt{{{\rm e}^{x}}+x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*e^x/sqrt(x + e^x), 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(exp(x)*x/(x+exp(x))^(1/2),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 e^{x}}{\sqrt{x + e^{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*exp(x)/sqrt(x + exp(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*e^x/sqrt(x + e^x), x)