3.899 \(\int \frac{7+3 x}{8+6 x+x^2} \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{2} \log (x+2)+\frac{5}{2} \log (x+4) \]

[Out]

Log[2 + x]/2 + (5*Log[4 + x])/2

________________________________________________________________________________________

Rubi [A]  time = 0.004653, antiderivative size = 17, 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} \[ \frac{1}{2} \log (x+2)+\frac{5}{2} \log (x+4) \]

Antiderivative was successfully verified.

[In]

Int[(7 + 3*x)/(8 + 6*x + x^2),x]

[Out]

Log[2 + x]/2 + (5*Log[4 + x])/2

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{7+3 x}{8+6 x+x^2} \, dx &=\frac{1}{2} \int \frac{1}{2+x} \, dx+\frac{5}{2} \int \frac{1}{4+x} \, dx\\ &=\frac{1}{2} \log (2+x)+\frac{5}{2} \log (4+x)\\ \end{align*}

Mathematica [A]  time = 0.0046759, size = 17, normalized size = 1. \[ \frac{1}{2} \log (x+2)+\frac{5}{2} \log (x+4) \]

Antiderivative was successfully verified.

[In]

Integrate[(7 + 3*x)/(8 + 6*x + x^2),x]

[Out]

Log[2 + x]/2 + (5*Log[4 + x])/2

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 14, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 2+x \right ) }{2}}+{\frac{5\,\ln \left ( 4+x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7+3*x)/(x^2+6*x+8),x)

[Out]

1/2*ln(2+x)+5/2*ln(4+x)

________________________________________________________________________________________

Maxima [A]  time = 1.06368, size = 18, normalized size = 1.06 \begin{align*} \frac{5}{2} \, \log \left (x + 4\right ) + \frac{1}{2} \, \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7+3*x)/(x^2+6*x+8),x, algorithm="maxima")

[Out]

5/2*log(x + 4) + 1/2*log(x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.54145, size = 45, normalized size = 2.65 \begin{align*} \frac{5}{2} \, \log \left (x + 4\right ) + \frac{1}{2} \, \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7+3*x)/(x^2+6*x+8),x, algorithm="fricas")

[Out]

5/2*log(x + 4) + 1/2*log(x + 2)

________________________________________________________________________________________

Sympy [A]  time = 0.236576, size = 14, normalized size = 0.82 \begin{align*} \frac{\log{\left (x + 2 \right )}}{2} + \frac{5 \log{\left (x + 4 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7+3*x)/(x**2+6*x+8),x)

[Out]

log(x + 2)/2 + 5*log(x + 4)/2

________________________________________________________________________________________

Giac [A]  time = 1.28152, size = 20, normalized size = 1.18 \begin{align*} \frac{5}{2} \, \log \left ({\left | x + 4 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7+3*x)/(x^2+6*x+8),x, algorithm="giac")

[Out]

5/2*log(abs(x + 4)) + 1/2*log(abs(x + 2))