3.2381 \(\int \frac{(5-x) (3+2 x)}{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=22 \[ -\frac{2 x}{3}-6 \log (x+1)+\frac{85}{9} \log (3 x+2) \]

[Out]

(-2*x)/3 - 6*Log[1 + x] + (85*Log[2 + 3*x])/9

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Rubi [A]  time = 0.0145041, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {773, 632, 31} \[ -\frac{2 x}{3}-6 \log (x+1)+\frac{85}{9} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/3 - 6*Log[1 + x] + (85*Log[2 + 3*x])/9

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

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) (3+2 x)}{2+5 x+3 x^2} \, dx &=-\frac{2 x}{3}+\frac{1}{3} \int \frac{49+31 x}{2+5 x+3 x^2} \, dx\\ &=-\frac{2 x}{3}-18 \int \frac{1}{3+3 x} \, dx+\frac{85}{3} \int \frac{1}{2+3 x} \, dx\\ &=-\frac{2 x}{3}-6 \log (1+x)+\frac{85}{9} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0071633, size = 22, normalized size = 1. \[ -\frac{2 x}{3}-6 \log (x+1)+\frac{85}{9} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/3 - 6*Log[1 + x] + (85*Log[2 + 3*x])/9

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Maple [A]  time = 0.006, size = 19, normalized size = 0.9 \begin{align*} -{\frac{2\,x}{3}}-6\,\ln \left ( 1+x \right ) +{\frac{85\,\ln \left ( 2+3\,x \right ) }{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*x-6*ln(1+x)+85/9*ln(2+3*x)

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Maxima [A]  time = 1.28362, size = 24, normalized size = 1.09 \begin{align*} -\frac{2}{3} \, x + \frac{85}{9} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*x + 85/9*log(3*x + 2) - 6*log(x + 1)

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Fricas [A]  time = 1.36702, size = 58, normalized size = 2.64 \begin{align*} -\frac{2}{3} \, x + \frac{85}{9} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*x + 85/9*log(3*x + 2) - 6*log(x + 1)

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Sympy [A]  time = 0.119572, size = 20, normalized size = 0.91 \begin{align*} - \frac{2 x}{3} + \frac{85 \log{\left (x + \frac{2}{3} \right )}}{9} - 6 \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x/3 + 85*log(x + 2/3)/9 - 6*log(x + 1)

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Giac [A]  time = 1.15013, size = 27, normalized size = 1.23 \begin{align*} -\frac{2}{3} \, x + \frac{85}{9} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 6 \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*x + 85/9*log(abs(3*x + 2)) - 6*log(abs(x + 1))