3.153 \(\int \frac{5+x}{-2+x+x^2} \, dx\)

Optimal. Leaf size=15 \[ 2 \log (1-x)-\log (x+2) \]

[Out]

2*Log[1 - x] - Log[2 + x]

________________________________________________________________________________________

Rubi [A]  time = 0.0036326, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {632, 31} \[ 2 \log (1-x)-\log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[(5 + x)/(-2 + x + x^2),x]

[Out]

2*Log[1 - x] - Log[2 + x]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{5+x}{-2+x+x^2} \, dx &=2 \int \frac{1}{-1+x} \, dx-\int \frac{1}{2+x} \, dx\\ &=2 \log (1-x)-\log (2+x)\\ \end{align*}

Mathematica [A]  time = 0.0033946, size = 15, normalized size = 1. \[ 2 \log (1-x)-\log (x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + x)/(-2 + x + x^2),x]

[Out]

2*Log[1 - x] - Log[2 + x]

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 14, normalized size = 0.9 \begin{align*} -\ln \left ( 2+x \right ) +2\,\ln \left ( -1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5+x)/(x^2+x-2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.921607, size = 18, normalized size = 1.2 \begin{align*} -\log \left (x + 2\right ) + 2 \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+x)/(x^2+x-2),x, algorithm="maxima")

[Out]

-log(x + 2) + 2*log(x - 1)

________________________________________________________________________________________

Fricas [A]  time = 2.12989, size = 38, normalized size = 2.53 \begin{align*} -\log \left (x + 2\right ) + 2 \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+x)/(x^2+x-2),x, algorithm="fricas")

[Out]

-log(x + 2) + 2*log(x - 1)

________________________________________________________________________________________

Sympy [A]  time = 0.086555, size = 10, normalized size = 0.67 \begin{align*} 2 \log{\left (x - 1 \right )} - \log{\left (x + 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+x)/(x**2+x-2),x)

[Out]

2*log(x - 1) - log(x + 2)

________________________________________________________________________________________

Giac [A]  time = 1.05226, size = 20, normalized size = 1.33 \begin{align*} -\log \left ({\left | x + 2 \right |}\right ) + 2 \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+x)/(x^2+x-2),x, algorithm="giac")

[Out]

-log(abs(x + 2)) + 2*log(abs(x - 1))