\(\int (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}) \, dx\) [746]

Optimal result

Integrand size = 37, antiderivative size = 12 \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=2 x \sqrt {e^x+x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2305, 2294} \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=2 x \sqrt {x+e^x} \]

[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 2294

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 2305

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*} \text {integral}& = 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] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=2 x \sqrt {e^x+x} \]

[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]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83

method	result	size
risch	\(2 x \sqrt {{\mathrm e}^{x}+x}\)	\(10\)

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=\int \frac {x e^{x} + 3 x + 2 e^{x}}{\sqrt {x + e^{x}}}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=2 \, \sqrt {x + e^{x}} x \]

[In]

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

[Out]

2*sqrt(x + e^x)*x

Giac [F]

\[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=\int { \frac {x e^{x}}{\sqrt {x + e^{x}}} + 2 \, \sqrt {x + e^{x}} + \frac {x}{\sqrt {x + e^{x}}} \,d x } \]

[In]

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

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=2\,x\,\sqrt {x+{\mathrm {e}}^x} \]

[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)