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

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

[Out]

2*Sqrt[E^x + x]

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Rubi [A]  time = 0.0252986, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {6686} \[ 2 \sqrt{x+e^x} \]

Antiderivative was successfully verified.

[In]

Int[(1 + E^x)/Sqrt[E^x + x],x]

[Out]

2*Sqrt[E^x + x]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1+e^x}{\sqrt{e^x+x}} \, dx &=2 \sqrt{e^x+x}\\ \end{align*}

Mathematica [A]  time = 0.0075249, size = 11, normalized size = 1. \[ 2 \sqrt{x+e^x} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^x)/Sqrt[E^x + x],x]

[Out]

2*Sqrt[E^x + x]

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Maple [A]  time = 0.066, size = 9, normalized size = 0.8 \begin{align*} 2\,\sqrt{{{\rm e}^{x}}+x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.96614, size = 11, normalized size = 1. \begin{align*} 2 \, \sqrt{x + e^{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + e^x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+exp(x))/(x+exp(x))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0.147997, size = 8, normalized size = 0.73 \begin{align*} 2 \sqrt{x + e^{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+exp(x))/(exp(x)+x)**(1/2),x)

[Out]

2*sqrt(x + exp(x))

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Giac [A]  time = 1.28738, size = 11, normalized size = 1. \begin{align*} 2 \, \sqrt{x + e^{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + e^x)