3.2383 \(\int \frac{5-x}{(3+2 x) (2+5 x+3 x^2)} \, dx\)

Optimal. Leaf size=27 \[ -6 \log (x+1)+\frac{13}{5} \log (2 x+3)+\frac{17}{5} \log (3 x+2) \]

[Out]

-6*Log[1 + x] + (13*Log[3 + 2*x])/5 + (17*Log[2 + 3*x])/5

________________________________________________________________________________________

Rubi [A]  time = 0.0257944, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {800} \[ -6 \log (x+1)+\frac{13}{5} \log (2 x+3)+\frac{17}{5} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-6*Log[1 + x] + (13*Log[3 + 2*x])/5 + (17*Log[2 + 3*x])/5

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]

Rubi steps

\begin{align*} \int \frac{5-x}{(3+2 x) \left (2+5 x+3 x^2\right )} \, dx &=\int \left (-\frac{6}{1+x}+\frac{26}{5 (3+2 x)}+\frac{51}{5 (2+3 x)}\right ) \, dx\\ &=-6 \log (1+x)+\frac{13}{5} \log (3+2 x)+\frac{17}{5} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0089317, size = 27, normalized size = 1. \[ -6 \log (x+1)+\frac{13}{5} \log (2 x+3)+\frac{17}{5} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-6*Log[1 + x] + (13*Log[3 + 2*x])/5 + (17*Log[2 + 3*x])/5

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 24, normalized size = 0.9 \begin{align*} -6\,\ln \left ( 1+x \right ) +{\frac{13\,\ln \left ( 3+2\,x \right ) }{5}}+{\frac{17\,\ln \left ( 2+3\,x \right ) }{5}} \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]

-6*ln(1+x)+13/5*ln(3+2*x)+17/5*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 1.24737, size = 31, normalized size = 1.15 \begin{align*} \frac{17}{5} \, \log \left (3 \, x + 2\right ) + \frac{13}{5} \, \log \left (2 \, x + 3\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]

17/5*log(3*x + 2) + 13/5*log(2*x + 3) - 6*log(x + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.27033, size = 73, normalized size = 2.7 \begin{align*} \frac{17}{5} \, \log \left (3 \, x + 2\right ) + \frac{13}{5} \, \log \left (2 \, x + 3\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]

17/5*log(3*x + 2) + 13/5*log(2*x + 3) - 6*log(x + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.135564, size = 26, normalized size = 0.96 \begin{align*} \frac{17 \log{\left (x + \frac{2}{3} \right )}}{5} - 6 \log{\left (x + 1 \right )} + \frac{13 \log{\left (x + \frac{3}{2} \right )}}{5} \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]

17*log(x + 2/3)/5 - 6*log(x + 1) + 13*log(x + 3/2)/5

________________________________________________________________________________________

Giac [A]  time = 1.12223, size = 35, normalized size = 1.3 \begin{align*} \frac{17}{5} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{13}{5} \, \log \left ({\left | 2 \, x + 3 \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]

17/5*log(abs(3*x + 2)) + 13/5*log(abs(2*x + 3)) - 6*log(abs(x + 1))