3.2392 \(\int \frac{5-x}{(3+2 x) (2+5 x+3 x^2)^2} \, dx\)

Optimal. Leaf size=48 \[ -\frac{3 (47 x+37)}{5 \left (3 x^2+5 x+2\right )}+23 \log (x+1)+\frac{52}{25} \log (2 x+3)-\frac{627}{25} \log (3 x+2) \]

[Out]

(-3*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)) + 23*Log[1 + x] + (52*Log[3 + 2*x])/25 - (627*Log[2 + 3*x])/25

________________________________________________________________________________________

Rubi [A]  time = 0.0380793, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {822, 800} \[ -\frac{3 (47 x+37)}{5 \left (3 x^2+5 x+2\right )}+23 \log (x+1)+\frac{52}{25} \log (2 x+3)-\frac{627}{25} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)) + 23*Log[1 + x] + (52*Log[3 + 2*x])/25 - (627*Log[2 + 3*x])/25

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

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 )^2} \, dx &=-\frac{3 (37+47 x)}{5 \left (2+5 x+3 x^2\right )}-\frac{1}{5} \int \frac{397+282 x}{(3+2 x) \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{3 (37+47 x)}{5 \left (2+5 x+3 x^2\right )}-\frac{1}{5} \int \left (-\frac{115}{1+x}-\frac{104}{5 (3+2 x)}+\frac{1881}{5 (2+3 x)}\right ) \, dx\\ &=-\frac{3 (37+47 x)}{5 \left (2+5 x+3 x^2\right )}+23 \log (1+x)+\frac{52}{25} \log (3+2 x)-\frac{627}{25} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0321799, size = 48, normalized size = 1. \[ \frac{1}{25} \left (-\frac{15 (47 x+37)}{3 x^2+5 x+2}-627 \log (-6 x-4)+575 \log (-2 (x+1))+52 \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-15*(37 + 47*x))/(2 + 5*x + 3*x^2) - 627*Log[-4 - 6*x] + 575*Log[-2*(1 + x)] + 52*Log[3 + 2*x])/25

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 40, normalized size = 0.8 \begin{align*} -6\, \left ( 1+x \right ) ^{-1}+23\,\ln \left ( 1+x \right ) +{\frac{52\,\ln \left ( 3+2\,x \right ) }{25}}-{\frac{51}{10+15\,x}}-{\frac{627\,\ln \left ( 2+3\,x \right ) }{25}} \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)^2,x)

[Out]

-6/(1+x)+23*ln(1+x)+52/25*ln(3+2*x)-51/5/(2+3*x)-627/25*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 1.16206, size = 57, normalized size = 1.19 \begin{align*} -\frac{3 \,{\left (47 \, x + 37\right )}}{5 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} - \frac{627}{25} \, \log \left (3 \, x + 2\right ) + \frac{52}{25} \, \log \left (2 \, x + 3\right ) + 23 \, \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)^2,x, algorithm="maxima")

[Out]

-3/5*(47*x + 37)/(3*x^2 + 5*x + 2) - 627/25*log(3*x + 2) + 52/25*log(2*x + 3) + 23*log(x + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.28606, size = 198, normalized size = 4.12 \begin{align*} -\frac{627 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 52 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (2 \, x + 3\right ) - 575 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 705 \, x + 555}{25 \,{\left (3 \, x^{2} + 5 \, x + 2\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)^2,x, algorithm="fricas")

[Out]

-1/25*(627*(3*x^2 + 5*x + 2)*log(3*x + 2) - 52*(3*x^2 + 5*x + 2)*log(2*x + 3) - 575*(3*x^2 + 5*x + 2)*log(x +
1) + 705*x + 555)/(3*x^2 + 5*x + 2)

________________________________________________________________________________________

Sympy [A]  time = 0.192668, size = 41, normalized size = 0.85 \begin{align*} - \frac{141 x + 111}{15 x^{2} + 25 x + 10} - \frac{627 \log{\left (x + \frac{2}{3} \right )}}{25} + 23 \log{\left (x + 1 \right )} + \frac{52 \log{\left (x + \frac{3}{2} \right )}}{25} \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)**2,x)

[Out]

-(141*x + 111)/(15*x**2 + 25*x + 10) - 627*log(x + 2/3)/25 + 23*log(x + 1) + 52*log(x + 3/2)/25

________________________________________________________________________________________

Giac [A]  time = 1.12355, size = 61, normalized size = 1.27 \begin{align*} -\frac{3 \,{\left (47 \, x + 37\right )}}{5 \,{\left (3 \, x + 2\right )}{\left (x + 1\right )}} - \frac{627}{25} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{52}{25} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) + 23 \, \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)^2,x, algorithm="giac")

[Out]

-3/5*(47*x + 37)/((3*x + 2)*(x + 1)) - 627/25*log(abs(3*x + 2)) + 52/25*log(abs(2*x + 3)) + 23*log(abs(x + 1))