3.8.32 \(\int \frac {1+e^x}{\sqrt {e^x+x}} \, dx\) [732]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6818} \begin {gather*} 2 \sqrt {x+e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + E^x)/Sqrt[E^x + x],x]

[Out]

2*Sqrt[E^x + x]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} 2 \sqrt {e^x+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 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
derivativedivides	\(2 \sqrt {{\mathrm e}^{x}+x}\)	\(9\)
default	\(2 \sqrt {{\mathrm e}^{x}+x}\)	\(9\)
risch	\(2 \sqrt {{\mathrm e}^{x}+x}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, 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((1+exp(x))/(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((1+exp(x))/(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 [A]
time = 0.06, 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((1+exp(x))/(exp(x)+x)**(1/2),x)

[Out]

2*sqrt(x + exp(x))

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Giac [A]
time = 6.13, 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((1+exp(x))/(exp(x)+x)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(x + e^x)

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Mupad [B]
time = 3.50, 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((exp(x) + 1)/(x + exp(x))^(1/2),x)

[Out]

2*(x + exp(x))^(1/2)

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