3.8.44 \(\int (\frac {1}{\sqrt {e^x+x}}+\frac {e^x}{\sqrt {e^x+x}}) \, dx\) [744]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 = {2293} \begin {gather*} 2 \sqrt {x+e^x} \end {gather*}

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 2293

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*}

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} 2 \sqrt {e^x+x} \end {gather*}

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 = 0.02, size = 9, normalized size = 0.82

method	result	size
risch	\(2 \sqrt {{\mathrm e}^{x}+x}\)	\(9\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 8, normalized size = 0.73 \begin {gather*} 2 \, \sqrt {x + e^{x}} \end {gather*}

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, algorithm="maxima")

[Out]

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

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 = 0.00, 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(exp(x)/(exp(x)+x)^(1/2)+1/(exp(x)+x)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 3.45, size = 8, normalized size = 0.73 \begin {gather*} 2\,\sqrt {x+{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x + exp(x))^(1/2)

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