3.2386 \(\int \frac{5-x}{(3+2 x)^4 (2+5 x+3 x^2)} \, dx\)

Optimal. Leaf size=60 \[ -\frac{597}{125 (2 x+3)}-\frac{99}{50 (2 x+3)^2}-\frac{13}{15 (2 x+3)^3}-6 \log (x+1)+\frac{3291}{625} \log (2 x+3)+\frac{459}{625} \log (3 x+2) \]

[Out]

-13/(15*(3 + 2*x)^3) - 99/(50*(3 + 2*x)^2) - 597/(125*(3 + 2*x)) - 6*Log[1 + x] + (3291*Log[3 + 2*x])/625 + (4
59*Log[2 + 3*x])/625

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Rubi [A]  time = 0.0379009, antiderivative size = 60, 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} \[ -\frac{597}{125 (2 x+3)}-\frac{99}{50 (2 x+3)^2}-\frac{13}{15 (2 x+3)^3}-6 \log (x+1)+\frac{3291}{625} \log (2 x+3)+\frac{459}{625} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)/((3 + 2*x)^4*(2 + 5*x + 3*x^2)),x]

[Out]

-13/(15*(3 + 2*x)^3) - 99/(50*(3 + 2*x)^2) - 597/(125*(3 + 2*x)) - 6*Log[1 + x] + (3291*Log[3 + 2*x])/625 + (4
59*Log[2 + 3*x])/625

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)^4 \left (2+5 x+3 x^2\right )} \, dx &=\int \left (-\frac{6}{1+x}+\frac{26}{5 (3+2 x)^4}+\frac{198}{25 (3+2 x)^3}+\frac{1194}{125 (3+2 x)^2}+\frac{6582}{625 (3+2 x)}+\frac{1377}{625 (2+3 x)}\right ) \, dx\\ &=-\frac{13}{15 (3+2 x)^3}-\frac{99}{50 (3+2 x)^2}-\frac{597}{125 (3+2 x)}-6 \log (1+x)+\frac{3291}{625} \log (3+2 x)+\frac{459}{625} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0372284, size = 62, normalized size = 1.03 \[ -\frac{597}{125 (2 x+3)}-\frac{99}{50 (2 x+3)^2}-\frac{13}{15 (2 x+3)^3}+\frac{459}{625} \log (-6 x-4)-6 \log (-2 (x+1))+\frac{3291}{625} \log (2 x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/((3 + 2*x)^4*(2 + 5*x + 3*x^2)),x]

[Out]

-13/(15*(3 + 2*x)^3) - 99/(50*(3 + 2*x)^2) - 597/(125*(3 + 2*x)) + (459*Log[-4 - 6*x])/625 - 6*Log[-2*(1 + x)]
+ (3291*Log[3 + 2*x])/625

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Maple [A]  time = 0.009, size = 51, normalized size = 0.9 \begin{align*} -{\frac{13}{15\, \left ( 3+2\,x \right ) ^{3}}}-{\frac{99}{50\, \left ( 3+2\,x \right ) ^{2}}}-{\frac{597}{375+250\,x}}-6\,\ln \left ( 1+x \right ) +{\frac{3291\,\ln \left ( 3+2\,x \right ) }{625}}+{\frac{459\,\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)^4/(3*x^2+5*x+2),x)

[Out]

-13/15/(3+2*x)^3-99/50/(3+2*x)^2-597/125/(3+2*x)-6*ln(1+x)+3291/625*ln(3+2*x)+459/625*ln(2+3*x)

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Maxima [A]  time = 1.09702, size = 70, normalized size = 1.17 \begin{align*} -\frac{14328 \, x^{2} + 45954 \, x + 37343}{750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac{459}{625} \, \log \left (3 \, x + 2\right ) + \frac{3291}{625} \, \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)^4/(3*x^2+5*x+2),x, algorithm="maxima")

[Out]

-1/750*(14328*x^2 + 45954*x + 37343)/(8*x^3 + 36*x^2 + 54*x + 27) + 459/625*log(3*x + 2) + 3291/625*log(2*x +
3) - 6*log(x + 1)

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Fricas [A]  time = 1.34061, size = 293, normalized size = 4.88 \begin{align*} -\frac{71640 \, x^{2} - 2754 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (3 \, x + 2\right ) - 19746 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (2 \, x + 3\right ) + 22500 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (x + 1\right ) + 229770 \, x + 186715}{3750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^4/(3*x^2+5*x+2),x, algorithm="fricas")

[Out]

-1/3750*(71640*x^2 - 2754*(8*x^3 + 36*x^2 + 54*x + 27)*log(3*x + 2) - 19746*(8*x^3 + 36*x^2 + 54*x + 27)*log(2
*x + 3) + 22500*(8*x^3 + 36*x^2 + 54*x + 27)*log(x + 1) + 229770*x + 186715)/(8*x^3 + 36*x^2 + 54*x + 27)

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Sympy [A]  time = 0.211237, size = 51, normalized size = 0.85 \begin{align*} - \frac{14328 x^{2} + 45954 x + 37343}{6000 x^{3} + 27000 x^{2} + 40500 x + 20250} + \frac{459 \log{\left (x + \frac{2}{3} \right )}}{625} - 6 \log{\left (x + 1 \right )} + \frac{3291 \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)**4/(3*x**2+5*x+2),x)

[Out]

-(14328*x**2 + 45954*x + 37343)/(6000*x**3 + 27000*x**2 + 40500*x + 20250) + 459*log(x + 2/3)/625 - 6*log(x +
1) + 3291*log(x + 3/2)/625

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Giac [A]  time = 1.10703, size = 61, normalized size = 1.02 \begin{align*} -\frac{14328 \, x^{2} + 45954 \, x + 37343}{750 \,{\left (2 \, x + 3\right )}^{3}} + \frac{459}{625} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{3291}{625} \, \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)^4/(3*x^2+5*x+2),x, algorithm="giac")

[Out]

-1/750*(14328*x^2 + 45954*x + 37343)/(2*x + 3)^3 + 459/625*log(abs(3*x + 2)) + 3291/625*log(abs(2*x + 3)) - 6*
log(abs(x + 1))