3.122 \(\int \frac{2+x}{x+x^2} \, dx\)

Optimal. Leaf size=11 \[ 2 \log (x)-\log (x+1) \]

[Out]

2*Log[x] - Log[1 + x]

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Rubi [A]  time = 0.0051985, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {631} \[ 2 \log (x)-\log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(2 + x)/(x + x^2),x]

[Out]

2*Log[x] - Log[1 + x]

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.001957, size = 11, normalized size = 1. \[ 2 \log (x)-\log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x)/(x + x^2),x]

[Out]

2*Log[x] - Log[1 + x]

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Maple [A]  time = 0.005, size = 12, normalized size = 1.1 \begin{align*} 2\,\ln \left ( x \right ) -\ln \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*ln(x)-ln(1+x)

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Maxima [A]  time = 0.932655, size = 15, normalized size = 1.36 \begin{align*} -\log \left (x + 1\right ) + 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 1) + 2*log(x)

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Fricas [A]  time = 0.717003, size = 32, normalized size = 2.91 \begin{align*} -\log \left (x + 1\right ) + 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 1) + 2*log(x)

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Sympy [A]  time = 0.086883, size = 8, normalized size = 0.73 \begin{align*} 2 \log{\left (x \right )} - \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x) - log(x + 1)

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Giac [A]  time = 1.09685, size = 18, normalized size = 1.64 \begin{align*} -\log \left ({\left | x + 1 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(x + 1)) + 2*log(abs(x))