3.3.67 \(\int \frac {x}{(2+x)^2} \, dx\) [267]

Optimal. Leaf size=12 \[ \frac {2}{2+x}+\log (2+x) \]

[Out]

2/(2+x)+ln(2+x)

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {45} \begin {gather*} \frac {2}{x+2}+\log (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(2 + x)^2,x]

[Out]

2/(2 + x) + Log[2 + x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {2}{2+x}+\log (2+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(2 + x)^2,x]

[Out]

2/(2 + x) + Log[2 + x]

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Mathics [A]
time = 1.65, size = 16, normalized size = 1.33 \begin {gather*} \frac {2+\text {Log}\left [2+x\right ] \left (2+x\right )}{2+x} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x/(2 + x)^2,x]')

[Out]

(2 + Log[2 + x] (2 + x)) / (2 + x)

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Maple [A]
time = 0.04, size = 13, normalized size = 1.08

method	result	size
default	\(\frac {2}{2+x}+\ln \left (2+x \right )\)	\(13\)
norman	\(\frac {2}{2+x}+\ln \left (2+x \right )\)	\(13\)
risch	\(\frac {2}{2+x}+\ln \left (2+x \right )\)	\(13\)
meijerg	\(-\frac {x}{2 \left (1+\frac {x}{2}\right )}+\ln \left (1+\frac {x}{2}\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(2+x)+ln(2+x)

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Maxima [A]
time = 0.26, size = 12, normalized size = 1.00 \begin {gather*} \frac {2}{x + 2} + \log \left (x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(x + 2) + log(x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x + 2)*log(x + 2) + 2)/(x + 2)

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Sympy [A]
time = 0.04, size = 8, normalized size = 0.67 \begin {gather*} \log {\left (x + 2 \right )} + \frac {2}{x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2+x)**2,x)

[Out]

log(x + 2) + 2/(x + 2)

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Giac [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {2}{x+2}+\ln \left |x+2\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2+x)^2,x)

[Out]

2/(x + 2) + log(abs(x + 2))

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Mupad [B]
time = 0.03, size = 12, normalized size = 1.00 \begin {gather*} \ln \left (x+2\right )+\frac {2}{x+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x + 2)^2,x)

[Out]

log(x + 2) + 2/(x + 2)

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