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

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

[Out]

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

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Rubi [A]  time = 0.18, 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 = {} \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx \]

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*}

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

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

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)

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maple [A]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left ({\mathrm e}^{x}+1\right ) x}{\sqrt {x +{\mathrm e}^{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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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