3.107 \(\int \frac{5+2 x}{-3+2 x+x^2} \, dx\)

Optimal. Leaf size=19 \[ \frac{7}{4} \log (1-x)+\frac{1}{4} \log (x+3) \]

[Out]

(7*Log[1 - x])/4 + Log[3 + x]/4

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Rubi [A]  time = 0.0044599, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {632, 31} \[ \frac{7}{4} \log (1-x)+\frac{1}{4} \log (x+3) \]

Antiderivative was successfully verified.

[In]

Int[(5 + 2*x)/(-3 + 2*x + x^2),x]

[Out]

(7*Log[1 - x])/4 + Log[3 + x]/4

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+2 x}{-3+2 x+x^2} \, dx &=\frac{1}{4} \int \frac{1}{3+x} \, dx+\frac{7}{4} \int \frac{1}{-1+x} \, dx\\ &=\frac{7}{4} \log (1-x)+\frac{1}{4} \log (3+x)\\ \end{align*}

Mathematica [A]  time = 0.0032404, size = 19, normalized size = 1. \[ \frac{7}{4} \log (1-x)+\frac{1}{4} \log (x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + 2*x)/(-3 + 2*x + x^2),x]

[Out]

(7*Log[1 - x])/4 + Log[3 + x]/4

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Maple [A]  time = 0.004, size = 14, normalized size = 0.7 \begin{align*}{\frac{7\,\ln \left ( -1+x \right ) }{4}}+{\frac{\ln \left ( 3+x \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5+2*x)/(x^2+2*x-3),x)

[Out]

7/4*ln(-1+x)+1/4*ln(3+x)

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Maxima [A]  time = 0.931335, size = 18, normalized size = 0.95 \begin{align*} \frac{1}{4} \, \log \left (x + 3\right ) + \frac{7}{4} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)/(x^2+2*x-3),x, algorithm="maxima")

[Out]

1/4*log(x + 3) + 7/4*log(x - 1)

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Fricas [A]  time = 0.635538, size = 45, normalized size = 2.37 \begin{align*} \frac{1}{4} \, \log \left (x + 3\right ) + \frac{7}{4} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)/(x^2+2*x-3),x, algorithm="fricas")

[Out]

1/4*log(x + 3) + 7/4*log(x - 1)

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Sympy [A]  time = 0.096212, size = 14, normalized size = 0.74 \begin{align*} \frac{7 \log{\left (x - 1 \right )}}{4} + \frac{\log{\left (x + 3 \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)/(x**2+2*x-3),x)

[Out]

7*log(x - 1)/4 + log(x + 3)/4

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Giac [A]  time = 1.0666, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{4} \, \log \left ({\left | x + 3 \right |}\right ) + \frac{7}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)/(x^2+2*x-3),x, algorithm="giac")

[Out]

1/4*log(abs(x + 3)) + 7/4*log(abs(x - 1))