3.730 \(\int \frac{\left (1+e^x\right ) x}{\sqrt{e^x+x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.282509, 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. \[ \text{Int}\left (\frac{\left (1+e^x\right ) x}{\sqrt{e^x+x}},x\right ) \]

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]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [A]  time = 0.187918, 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 = 0.01, size = 0, normalized size = 0. \[ \int{ \left ( 1+{{\rm e}^{x}} \right ) x{\frac{1}{\sqrt{{{\rm e}^{x}}+x}}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(e^x + 1)/sqrt(x + e^x),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. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(e^x + 1)/sqrt(x + e^x),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(e^x + 1)/sqrt(x + e^x),x, algorithm="giac")

[Out]

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