3.2380 \(\int \frac{(5-x) (3+2 x)^2}{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 x^2}{3}+\frac{44 x}{9}-6 \log (x+1)+\frac{425}{27} \log (3 x+2) \]

[Out]

(44*x)/9 - (2*x^2)/3 - 6*Log[1 + x] + (425*Log[2 + 3*x])/27

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Rubi [A]  time = 0.0225528, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {800, 632, 31} \[ -\frac{2 x^2}{3}+\frac{44 x}{9}-6 \log (x+1)+\frac{425}{27} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(44*x)/9 - (2*x^2)/3 - 6*Log[1 + x] + (425*Log[2 + 3*x])/27

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + 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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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}{2+5 x+3 x^2} \, dx &=\int \left (\frac{44}{9}-\frac{4 x}{3}+\frac{317+263 x}{9 \left (2+5 x+3 x^2\right )}\right ) \, dx\\ &=\frac{44 x}{9}-\frac{2 x^2}{3}+\frac{1}{9} \int \frac{317+263 x}{2+5 x+3 x^2} \, dx\\ &=\frac{44 x}{9}-\frac{2 x^2}{3}-18 \int \frac{1}{3+3 x} \, dx+\frac{425}{9} \int \frac{1}{2+3 x} \, dx\\ &=\frac{44 x}{9}-\frac{2 x^2}{3}-6 \log (1+x)+\frac{425}{27} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.014763, size = 34, normalized size = 1.17 \[ -\frac{2 x^2}{3}+\frac{44 x}{9}+\frac{425}{27} \log (-6 x-4)-6 \log (-2 (x+1))+\frac{53}{6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

53/6 + (44*x)/9 - (2*x^2)/3 + (425*Log[-4 - 6*x])/27 - 6*Log[-2*(1 + x)]

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Maple [A]  time = 0.006, size = 24, normalized size = 0.8 \begin{align*}{\frac{44\,x}{9}}-{\frac{2\,{x}^{2}}{3}}-6\,\ln \left ( 1+x \right ) +{\frac{425\,\ln \left ( 2+3\,x \right ) }{27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

44/9*x-2/3*x^2-6*ln(1+x)+425/27*ln(2+3*x)

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Maxima [A]  time = 1.08969, size = 31, normalized size = 1.07 \begin{align*} -\frac{2}{3} \, x^{2} + \frac{44}{9} \, x + \frac{425}{27} \, \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)^2/(3*x^2+5*x+2),x, algorithm="maxima")

[Out]

-2/3*x^2 + 44/9*x + 425/27*log(3*x + 2) - 6*log(x + 1)

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Fricas [A]  time = 1.55615, size = 76, normalized size = 2.62 \begin{align*} -\frac{2}{3} \, x^{2} + \frac{44}{9} \, x + \frac{425}{27} \, \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)^2/(3*x^2+5*x+2),x, algorithm="fricas")

[Out]

-2/3*x^2 + 44/9*x + 425/27*log(3*x + 2) - 6*log(x + 1)

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Sympy [A]  time = 0.127855, size = 27, normalized size = 0.93 \begin{align*} - \frac{2 x^{2}}{3} + \frac{44 x}{9} + \frac{425 \log{\left (x + \frac{2}{3} \right )}}{27} - 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)**2/(3*x**2+5*x+2),x)

[Out]

-2*x**2/3 + 44*x/9 + 425*log(x + 2/3)/27 - 6*log(x + 1)

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Giac [A]  time = 1.11607, size = 34, normalized size = 1.17 \begin{align*} -\frac{2}{3} \, x^{2} + \frac{44}{9} \, x + \frac{425}{27} \, \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)^2/(3*x^2+5*x+2),x, algorithm="giac")

[Out]

-2/3*x^2 + 44/9*x + 425/27*log(abs(3*x + 2)) - 6*log(abs(x + 1))