3.8.45 \(\int (\frac {(1+e^x) x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}) \, dx\) [745]

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

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {6874, 2305, 2294} \begin {gather*} 2 x \sqrt {x+e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + 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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 12, normalized size = 1.00 \begin {gather*} 2 x \sqrt {e^x+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + 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 = 0.03, size = 10, normalized size = 0.83

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+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)

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Maxima [A]
time = 0.33, size = 16, normalized size = 1.33 \begin {gather*} \frac {2 \, {\left (x^{2} + x e^{x}\right )}}{\sqrt {x + e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x^2 + x*e^x)/sqrt(x + e^x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x e^{x} + 3 x + 2 e^{x}}{\sqrt {x + e^{x}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*(e^x + 1)/sqrt(x + e^x) + 2*sqrt(x + e^x), x)

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Mupad [B]
time = 3.63, size = 9, normalized size = 0.75 \begin {gather*} 2\,x\,\sqrt {x+{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*(x + exp(x))^(1/2)

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