3.2394 \(\int \frac{5-x}{(3+2 x)^3 (2+5 x+3 x^2)^2} \, dx\)

Optimal. Leaf size=77 \[ -\frac{3 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )}-\frac{2618}{125 (2 x+3)}-\frac{428}{25 (2 x+3)^2}-\log (x+1)+\frac{8104}{625} \log (2 x+3)-\frac{7479}{625} \log (3 x+2) \]

[Out]

-428/(25*(3 + 2*x)^2) - 2618/(125*(3 + 2*x)) - (3*(37 + 47*x))/(5*(3 + 2*x)^2*(2 + 5*x + 3*x^2)) - Log[1 + x]
+ (8104*Log[3 + 2*x])/625 - (7479*Log[2 + 3*x])/625

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Rubi [A]  time = 0.0502562, antiderivative size = 77, 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} \[ -\frac{3 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )}-\frac{2618}{125 (2 x+3)}-\frac{428}{25 (2 x+3)^2}-\log (x+1)+\frac{8104}{625} \log (2 x+3)-\frac{7479}{625} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-428/(25*(3 + 2*x)^2) - 2618/(125*(3 + 2*x)) - (3*(37 + 47*x))/(5*(3 + 2*x)^2*(2 + 5*x + 3*x^2)) - Log[1 + x]
+ (8104*Log[3 + 2*x])/625 - (7479*Log[2 + 3*x])/625

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)^3 \left (2+5 x+3 x^2\right )^2} \, dx &=-\frac{3 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )}-\frac{1}{5} \int \frac{841+846 x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{3 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )}-\frac{1}{5} \int \left (\frac{5}{1+x}-\frac{1712}{5 (3+2 x)^3}-\frac{5236}{25 (3+2 x)^2}-\frac{16208}{125 (3+2 x)}+\frac{22437}{125 (2+3 x)}\right ) \, dx\\ &=-\frac{428}{25 (3+2 x)^2}-\frac{2618}{125 (3+2 x)}-\frac{3 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )}-\log (1+x)+\frac{8104}{625} \log (3+2 x)-\frac{7479}{625} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0317155, size = 66, normalized size = 0.86 \[ \frac{1}{625} \left (-\frac{15 (903 x+653)}{3 x^2+5 x+2}-\frac{4060}{2 x+3}-\frac{650}{(2 x+3)^2}-7479 \log (-6 x-4)-625 \log (-2 (x+1))+8104 \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-650/(3 + 2*x)^2 - 4060/(3 + 2*x) - (15*(653 + 903*x))/(2 + 5*x + 3*x^2) - 7479*Log[-4 - 6*x] - 625*Log[-2*(1
+ x)] + 8104*Log[3 + 2*x])/625

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Maple [A]  time = 0.01, size = 58, normalized size = 0.8 \begin{align*} -6\, \left ( 1+x \right ) ^{-1}-\ln \left ( 1+x \right ) -{\frac{26}{25\, \left ( 3+2\,x \right ) ^{2}}}-{\frac{812}{375+250\,x}}+{\frac{8104\,\ln \left ( 3+2\,x \right ) }{625}}-{\frac{459}{250+375\,x}}-{\frac{7479\,\ln \left ( 2+3\,x \right ) }{625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/(1+x)-ln(1+x)-26/25/(3+2*x)^2-812/125/(3+2*x)+8104/625*ln(3+2*x)-459/125/(2+3*x)-7479/625*ln(2+3*x)

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Maxima [A]  time = 1.01042, size = 84, normalized size = 1.09 \begin{align*} -\frac{15708 \, x^{3} + 56162 \, x^{2} + 63967 \, x + 22763}{125 \,{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )}} - \frac{7479}{625} \, \log \left (3 \, x + 2\right ) + \frac{8104}{625} \, \log \left (2 \, x + 3\right ) - \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/(3*x^2+5*x+2)^2,x, algorithm="maxima")

[Out]

-1/125*(15708*x^3 + 56162*x^2 + 63967*x + 22763)/(12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18) - 7479/625*log(3*x + 2
) + 8104/625*log(2*x + 3) - log(x + 1)

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Fricas [A]  time = 1.2591, size = 359, normalized size = 4.66 \begin{align*} -\frac{78540 \, x^{3} + 280810 \, x^{2} + 7479 \,{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (3 \, x + 2\right ) - 8104 \,{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (2 \, x + 3\right ) + 625 \,{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (x + 1\right ) + 319835 \, x + 113815}{625 \,{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/625*(78540*x^3 + 280810*x^2 + 7479*(12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18)*log(3*x + 2) - 8104*(12*x^4 + 56*
x^3 + 95*x^2 + 69*x + 18)*log(2*x + 3) + 625*(12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18)*log(x + 1) + 319835*x + 11
3815)/(12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18)

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Sympy [A]  time = 0.227906, size = 60, normalized size = 0.78 \begin{align*} - \frac{15708 x^{3} + 56162 x^{2} + 63967 x + 22763}{1500 x^{4} + 7000 x^{3} + 11875 x^{2} + 8625 x + 2250} - \frac{7479 \log{\left (x + \frac{2}{3} \right )}}{625} - \log{\left (x + 1 \right )} + \frac{8104 \log{\left (x + \frac{3}{2} \right )}}{625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(15708*x**3 + 56162*x**2 + 63967*x + 22763)/(1500*x**4 + 7000*x**3 + 11875*x**2 + 8625*x + 2250) - 7479*log(x
+ 2/3)/625 - log(x + 1) + 8104*log(x + 3/2)/625

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Giac [A]  time = 1.14908, size = 84, normalized size = 1.09 \begin{align*} -\frac{15708 \, x^{3} + 56162 \, x^{2} + 63967 \, x + 22763}{125 \,{\left (3 \, x + 2\right )}{\left (2 \, x + 3\right )}^{2}{\left (x + 1\right )}} - \frac{7479}{625} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{8104}{625} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \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/(3*x^2+5*x+2)^2,x, algorithm="giac")

[Out]

-1/125*(15708*x^3 + 56162*x^2 + 63967*x + 22763)/((3*x + 2)*(2*x + 3)^2*(x + 1)) - 7479/625*log(abs(3*x + 2))
+ 8104/625*log(abs(2*x + 3)) - log(abs(x + 1))