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

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.05, size = 12, normalized size = 1.00 \[ 2 x \sqrt {x+e^x} \]

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]

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x e^{x}}{\sqrt {x + e^{x}}} + 2 \, \sqrt {x + e^{x}} + \frac {x}{\sqrt {x + e^{x}}}\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 10, normalized size = 0.83 \[ 2 \sqrt {x +{\mathrm e}^{x}}\, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.16, size = 9, normalized size = 0.75 \[ 2 \, \sqrt {x + e^{x}} x \]

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

________________________________________________________________________________________

mupad [B]  time = 3.35, size = 9, normalized size = 0.75 \[ 2\,x\,\sqrt {x+{\mathrm {e}}^x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*(x + exp(x))^(1/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x e^{x} + 3 x + 2 e^{x}}{\sqrt {x + e^{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)

________________________________________________________________________________________