3.2385 \(\int \frac{5-x}{(3+2 x)^3 (2+5 x+3 x^2)} \, dx\)

Optimal. Leaf size=49 \[ -\frac{99}{25 (2 x+3)}-\frac{13}{10 (2 x+3)^2}-6 \log (x+1)+\frac{597}{125} \log (2 x+3)+\frac{153}{125} \log (3 x+2) \]

[Out]

-13/(10*(3 + 2*x)^2) - 99/(25*(3 + 2*x)) - 6*Log[1 + x] + (597*Log[3 + 2*x])/125 + (153*Log[2 + 3*x])/125

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Rubi [A]  time = 0.0348011, antiderivative size = 49, 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} \[ -\frac{99}{25 (2 x+3)}-\frac{13}{10 (2 x+3)^2}-6 \log (x+1)+\frac{597}{125} \log (2 x+3)+\frac{153}{125} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-13/(10*(3 + 2*x)^2) - 99/(25*(3 + 2*x)) - 6*Log[1 + x] + (597*Log[3 + 2*x])/125 + (153*Log[2 + 3*x])/125

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 )} \, dx &=\int \left (-\frac{6}{1+x}+\frac{26}{5 (3+2 x)^3}+\frac{198}{25 (3+2 x)^2}+\frac{1194}{125 (3+2 x)}+\frac{459}{125 (2+3 x)}\right ) \, dx\\ &=-\frac{13}{10 (3+2 x)^2}-\frac{99}{25 (3+2 x)}-6 \log (1+x)+\frac{597}{125} \log (3+2 x)+\frac{153}{125} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0281965, size = 47, normalized size = 0.96 \[ \frac{1}{250} \left (-\frac{990}{2 x+3}-\frac{325}{(2 x+3)^2}+306 \log (-6 x-4)-1500 \log (-2 (x+1))+1194 \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-325/(3 + 2*x)^2 - 990/(3 + 2*x) + 306*Log[-4 - 6*x] - 1500*Log[-2*(1 + x)] + 1194*Log[3 + 2*x])/250

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Maple [A]  time = 0.009, size = 42, normalized size = 0.9 \begin{align*} -{\frac{13}{10\, \left ( 3+2\,x \right ) ^{2}}}-{\frac{99}{75+50\,x}}-6\,\ln \left ( 1+x \right ) +{\frac{597\,\ln \left ( 3+2\,x \right ) }{125}}+{\frac{153\,\ln \left ( 2+3\,x \right ) }{125}} \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),x)

[Out]

-13/10/(3+2*x)^2-99/25/(3+2*x)-6*ln(1+x)+597/125*ln(3+2*x)+153/125*ln(2+3*x)

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Maxima [A]  time = 1.31777, size = 57, normalized size = 1.16 \begin{align*} -\frac{396 \, x + 659}{50 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{153}{125} \, \log \left (3 \, x + 2\right ) + \frac{597}{125} \, \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/(3*x^2+5*x+2),x, algorithm="maxima")

[Out]

-1/50*(396*x + 659)/(4*x^2 + 12*x + 9) + 153/125*log(3*x + 2) + 597/125*log(2*x + 3) - 6*log(x + 1)

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Fricas [A]  time = 1.32548, size = 211, normalized size = 4.31 \begin{align*} \frac{306 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (3 \, x + 2\right ) + 1194 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (2 \, x + 3\right ) - 1500 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (x + 1\right ) - 1980 \, x - 3295}{250 \,{\left (4 \, x^{2} + 12 \, x + 9\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),x, algorithm="fricas")

[Out]

1/250*(306*(4*x^2 + 12*x + 9)*log(3*x + 2) + 1194*(4*x^2 + 12*x + 9)*log(2*x + 3) - 1500*(4*x^2 + 12*x + 9)*lo
g(x + 1) - 1980*x - 3295)/(4*x^2 + 12*x + 9)

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Sympy [A]  time = 0.186917, size = 41, normalized size = 0.84 \begin{align*} - \frac{396 x + 659}{200 x^{2} + 600 x + 450} + \frac{153 \log{\left (x + \frac{2}{3} \right )}}{125} - 6 \log{\left (x + 1 \right )} + \frac{597 \log{\left (x + \frac{3}{2} \right )}}{125} \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),x)

[Out]

-(396*x + 659)/(200*x**2 + 600*x + 450) + 153*log(x + 2/3)/125 - 6*log(x + 1) + 597*log(x + 3/2)/125

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Giac [A]  time = 1.12257, size = 54, normalized size = 1.1 \begin{align*} -\frac{396 \, x + 659}{50 \,{\left (2 \, x + 3\right )}^{2}} + \frac{153}{125} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{597}{125} \, \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/(3*x^2+5*x+2),x, algorithm="giac")

[Out]

-1/50*(396*x + 659)/(2*x + 3)^2 + 153/125*log(abs(3*x + 2)) + 597/125*log(abs(2*x + 3)) - 6*log(abs(x + 1))