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

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

[Out]

2*Sqrt[E^x + x]

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Rubi [A]  time = 0.0408029, 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, Rules used = {2261} \[ 2 \sqrt{x+e^x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[E^x + x]

Rule 2261

Int[(F_)^((e_.)*((c_.) + (d_.)*(x_)))*((b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))) + (a_.)*(x_)^(n_.))^(p_.), x_Sy
mbol] :> Simp[(a*x^n + b*F^(e*(c + d*x)))^(p + 1)/(b*d*e*(p + 1)*Log[F]), x] - Dist[(a*n)/(b*d*e*Log[F]), Int[
x^(n - 1)*(a*x^n + b*F^(e*(c + d*x)))^p, x], x] /; FreeQ[{F, a, b, c, d, e, n, p}, x] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0047456, 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 = 0.042, 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(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 = 1.09435, 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(exp(x)/(x+exp(x))^(1/2)+1/(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(exp(x)/(x+exp(x))^(1/2)+1/(x+exp(x))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{x}}{\sqrt{x + e^{x}}} + \frac{1}{\sqrt{x + e^{x}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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