3.746 \(\int (\frac{x}{\sqrt{e^x+x}}+\frac{e^x x}{\sqrt{e^x+x}}+2 \sqrt{e^x+x}) \, dx\)

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

[Out]

2*x*Sqrt[E^x + x]

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Rubi [A]  time = 0.130266, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {2273, 2262} \[ 2 x \sqrt{x+e^x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*x*Sqrt[E^x + x]

Rule 2273

Int[(x_)^(m_.)*(E^(x_) + (x_)^(m_.))^(n_), x_Symbol] :> -Simp[(E^x + x^m)^(n + 1)/(n + 1), x] + (Dist[m, Int[x
^(m - 1)*(E^x + x^m)^n, x], x] + Int[(E^x + x^m)^(n + 1), x]) /; RationalQ[m, n] && GtQ[m, 0] && LtQ[n, 0] &&
NeQ[n, -1]

Rule 2262

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

Rubi steps

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

Mathematica [A]  time = 0.0446522, size = 12, normalized size = 1. \[ 2 x \sqrt{x+e^x} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[E^x + x] + (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 = 0.039, size = 10, normalized size = 0.8 \begin{align*} 2\,x\sqrt{{{\rm e}^{x}}+x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(exp(x)+x)^(1/2)+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 = 1.08326, size = 12, normalized size = 1. \begin{align*} 2 \, \sqrt{x + e^{x}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + e^x)*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(x/(x+exp(x))^(1/2)+exp(x)*x/(x+exp(x))^(1/2)+2*(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{x e^{x} + 3 x + 2 e^{x}}{\sqrt{x + e^{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(exp(x)+x)**(1/2)+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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x e^{x}}{\sqrt{x + e^{x}}} + 2 \, \sqrt{x + e^{x}} + \frac{x}{\sqrt{x + e^{x}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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