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

Optimal. Leaf size=11 \[ 2 \sqrt{x+e^x} \]

[Out]

2*Sqrt[E^x + x]

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Rubi [A]  time = 0.0650065, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ 2 \sqrt{x+e^x} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[E^x + x] + E^x/Sqrt[E^x + x],x]

[Out]

2*Sqrt[E^x + x]

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Rubi in Sympy [A]  time = 11.8331, size = 8, normalized size = 0.73 \[ 2 \sqrt{x + e^{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + exp(x))

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Mathematica [A]  time = 0.00707802, size = 11, normalized size = 1. \[ 2 \sqrt{x+e^x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[E^x + x] + E^x/Sqrt[E^x + x],x]

[Out]

2*Sqrt[E^x + x]

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Maple [A]  time = 0.032, size = 9, normalized size = 0.8 \[ 2\,\sqrt{{{\rm e}^{x}}+x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.848795, size = 11, normalized size = 1. \[ 2 \, \sqrt{x + e^{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*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(e^x/sqrt(x + e^x) + 1/sqrt(x + e^x),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{x} + 1}{\sqrt{x + e^{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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