3.747 \(\int \frac{(1+e^x) x}{\sqrt{e^x+x}} \, dx\)

Optimal. Leaf size=26 \[ 2 x \sqrt{x+e^x}-2 \text{CannotIntegrate}\left (\sqrt{x+e^x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.18311, 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 = {} \[ \int \frac{\left (1+e^x\right ) x}{\sqrt{e^x+x}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((1 + E^x)*x)/Sqrt[E^x + x],x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.121931, size = 0, normalized size = 0. \[ \int \frac{\left (1+e^x\right ) x}{\sqrt{e^x+x}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((1 + E^x)*x)/Sqrt[E^x + x],x]

[Out]

Integrate[((1 + E^x)*x)/Sqrt[E^x + x], x]

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Maple [A]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 1+{{\rm e}^{x}} \right ) x{\frac{1}{\sqrt{{{\rm e}^{x}}+x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((1+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{\left (e^{x} + 1\right )}}{\sqrt{x + e^{x}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(exp(x) + 1)/sqrt(x + exp(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*(e^x + 1)/sqrt(x + e^x), x)