3.8.49 \(\int \frac {e^x x}{\sqrt {e^x+x}} \, dx\) [749]

Optimal. Leaf size=52 \[ 2 \sqrt {e^x+x}+2 x \sqrt {e^x+x}-\text {Int}\left (\frac {1}{\sqrt {e^x+x}},x\right )-3 \text {Int}\left (\sqrt {e^x+x},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.05, 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 {e^x x}{\sqrt {e^x+x}} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^x x}{\sqrt {e^x+x}} \, dx \end {gather*}

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

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 = 0.00, 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(exp(x)*x/(exp(x)+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(x*e^x/sqrt(x + e^x), x)

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Fricas [F(-2)]
time = 0.00, 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(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 e^{x}}{\sqrt {x + e^{x}}}\, dx \end {gather*}

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 = 0.00, 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(exp(x)*x/(exp(x)+x)^(1/2),x, algorithm="giac")

[Out]

integrate(x*e^x/sqrt(x + e^x), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x\,{\mathrm {e}}^x}{\sqrt {x+{\mathrm {e}}^x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(x))/(x + exp(x))^(1/2),x)

[Out]

int((x*exp(x))/(x + exp(x))^(1/2), x)

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