3.728 \(\int \left (\frac{\left (1+e^x\right ) x}{\sqrt{e^x+x}}+2 \sqrt{e^x+x}\right ) \, dx\)

Optimal. Leaf size=12 \[ 2 x \sqrt{x+e^x} \]

[Out]

2*x*Sqrt[E^x + x]

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Rubi [A]  time = 0.395136, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ 2 x \sqrt{x+e^x} \]

Antiderivative was successfully verified.

[In]  Int[((1 + E^x)*x)/Sqrt[E^x + x] + 2*Sqrt[E^x + x],x]

[Out]

2*x*Sqrt[E^x + x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (e^{x} + 1\right )}{\sqrt{x + e^{x}}}\, dx + 2 \int \sqrt{x + e^{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+exp(x))*x/(exp(x)+x)**(1/2)+2*(exp(x)+x)**(1/2),x)

[Out]

Integral(x*(exp(x) + 1)/sqrt(x + exp(x)), x) + 2*Integral(sqrt(x + exp(x)), x)

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Mathematica [A]  time = 0.0250822, size = 12, normalized size = 1. \[ 2 x \sqrt{x+e^x} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 + E^x)*x)/Sqrt[E^x + x] + 2*Sqrt[E^x + x],x]

[Out]

2*x*Sqrt[E^x + x]

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Maple [A]  time = 0.046, size = 10, normalized size = 0.8 \[ 2\,x\sqrt{{{\rm e}^{x}}+x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.827253, size = 22, normalized size = 1.83 \[ \frac{2 \,{\left (x^{2} + x e^{x}\right )}}{\sqrt{x + e^{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(e^x + 1)/sqrt(x + e^x) + 2*sqrt(x + e^x),x, algorithm="maxima")

[Out]

2*(x^2 + x*e^x)/sqrt(x + e^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) + 2*sqrt(x + e^x),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x e^{x} + 3 x + 2 e^{x}}{\sqrt{x + e^{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+exp(x))*x/(exp(x)+x)**(1/2)+2*(exp(x)+x)**(1/2),x)

[Out]

Integral((x*exp(x) + 3*x + 2*exp(x))/sqrt(x + exp(x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x{\left (e^{x} + 1\right )}}{\sqrt{x + e^{x}}} + 2 \, \sqrt{x + e^{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(e^x + 1)/sqrt(x + e^x) + 2*sqrt(x + e^x),x, algorithm="giac")

[Out]

integrate(x*(e^x + 1)/sqrt(x + e^x) + 2*sqrt(x + e^x), x)