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

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {800} \begin {gather*} -\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) \end {gather*}

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 = 0.03, size = 62, normalized size = 1.03 \begin {gather*} -\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) \end {gather*}

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 96, normalized size = 1.60 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 45, normalized size = 0.75 \begin {gather*} -\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 {gather*}

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))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 51, normalized size = 0.85 \begin {gather*} \frac {459 \ln \left (3 x +2\right )}{625}+\frac {3291 \ln \left (2 x +3\right )}{625}-6 \ln \left (x +1\right )-\frac {13}{15 \left (2 x +3\right )^{3}}-\frac {99}{50 \left (2 x +3\right )^{2}}-\frac {597}{125 \left (2 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(2*x+3)^4/(3*x^2+5*x+2),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.55, size = 52, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 46, normalized size = 0.77 \begin {gather*} \frac {459\,\ln \left (x+\frac {2}{3}\right )}{625}-6\,\ln \left (x+1\right )+\frac {3291\,\ln \left (x+\frac {3}{2}\right )}{625}-\frac {\frac {597\,x^2}{250}+\frac {7659\,x}{1000}+\frac {37343}{6000}}{x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5)/((2*x + 3)^4*(5*x + 3*x^2 + 2)),x)

[Out]

(459*log(x + 2/3))/625 - 6*log(x + 1) + (3291*log(x + 3/2))/625 - ((7659*x)/1000 + (597*x^2)/250 + 37343/6000)
/((27*x)/4 + (9*x^2)/2 + x^3 + 27/8)

________________________________________________________________________________________

sympy [A]  time = 0.20, size = 51, normalized size = 0.85 \begin {gather*} - \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 {gather*}

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

________________________________________________________________________________________