3.8.47 \(\int \frac {(1+e^x) x}{\sqrt {e^x+x}} \, dx\) [747]

Optimal. Leaf size=27 \[ 2 x \sqrt {e^x+x}-2 \text {Int}\left (\sqrt {e^x+x},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.12, 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 = {} \begin {gather*} \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx \end {gather*}

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+exp(x))*x/(exp(x)+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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (e^{x} + 1\right )}{\sqrt {x + e^{x}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x\,\left ({\mathrm {e}}^x+1\right )}{\sqrt {x+{\mathrm {e}}^x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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