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

Optimal. Leaf size=69 \[ -\frac {3 (47 x+37)}{10 \left (3 x^2+5 x+2\right )^2}+\frac {11442 x+9587}{50 \left (3 x^2+5 x+2\right )}-233 \log (x+1)+\frac {208}{125} \log (2 x+3)+\frac {28917}{125} \log (3 x+2) \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {822, 800} \begin {gather*} -\frac {3 (47 x+37)}{10 \left (3 x^2+5 x+2\right )^2}+\frac {11442 x+9587}{50 \left (3 x^2+5 x+2\right )}-233 \log (x+1)+\frac {208}{125} \log (2 x+3)+\frac {28917}{125} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(37 + 47*x))/(10*(2 + 5*x + 3*x^2)^2) + (9587 + 11442*x)/(50*(2 + 5*x + 3*x^2)) - 233*Log[1 + x] + (208*Lo
g[3 + 2*x])/125 + (28917*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]

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])

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x) \left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {3 (37+47 x)}{10 \left (2+5 x+3 x^2\right )^2}-\frac {1}{10} \int \frac {1217+846 x}{(3+2 x) \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {3 (37+47 x)}{10 \left (2+5 x+3 x^2\right )^2}+\frac {9587+11442 x}{50 \left (2+5 x+3 x^2\right )}+\frac {1}{50} \int \frac {34534+22884 x}{(3+2 x) \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {3 (37+47 x)}{10 \left (2+5 x+3 x^2\right )^2}+\frac {9587+11442 x}{50 \left (2+5 x+3 x^2\right )}+\frac {1}{50} \int \left (-\frac {11650}{1+x}+\frac {832}{5 (3+2 x)}+\frac {173502}{5 (2+3 x)}\right ) \, dx\\ &=-\frac {3 (37+47 x)}{10 \left (2+5 x+3 x^2\right )^2}+\frac {9587+11442 x}{50 \left (2+5 x+3 x^2\right )}-233 \log (1+x)+\frac {208}{125} \log (3+2 x)+\frac {28917}{125} \log (2+3 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 68, normalized size = 0.99 \begin {gather*} \frac {1}{125} \left (-\frac {75 (47 x+37)}{2 \left (3 x^2+5 x+2\right )^2}+\frac {57210 x+47935}{6 x^2+10 x+4}+28917 \log (-6 x-4)-29125 \log (-2 (x+1))+208 \log (2 x+3)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-75*(37 + 47*x))/(2*(2 + 5*x + 3*x^2)^2) + (47935 + 57210*x)/(4 + 10*x + 6*x^2) + 28917*Log[-4 - 6*x] - 2912
5*Log[-2*(1 + x)] + 208*Log[3 + 2*x])/125

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 121, normalized size = 1.75 \begin {gather*} \frac {171630 \, x^{3} + 429855 \, x^{2} + 57834 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 416 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (2 \, x + 3\right ) - 58250 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) + 350570 \, x + 93095}{250 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/250*(171630*x^3 + 429855*x^2 + 57834*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(3*x + 2) + 416*(9*x^4 + 30*x^3
+ 37*x^2 + 20*x + 4)*log(2*x + 3) - 58250*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(x + 1) + 350570*x + 93095)
/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 55, normalized size = 0.80 \begin {gather*} \frac {34326 \, x^{3} + 85971 \, x^{2} + 70114 \, x + 18619}{50 \, {\left (3 \, x + 2\right )}^{2} {\left (x + 1\right )}^{2}} + \frac {28917}{125} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {208}{125} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - 233 \, \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)/(3*x^2+5*x+2)^3,x, algorithm="giac")

[Out]

1/50*(34326*x^3 + 85971*x^2 + 70114*x + 18619)/((3*x + 2)^2*(x + 1)^2) + 28917/125*log(abs(3*x + 2)) + 208/125
*log(abs(2*x + 3)) - 233*log(abs(x + 1))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 56, normalized size = 0.81 \begin {gather*} \frac {28917 \ln \left (3 x +2\right )}{125}+\frac {208 \ln \left (2 x +3\right )}{125}-233 \ln \left (x +1\right )-\frac {153}{10 \left (3 x +2\right )^{2}}+\frac {2646}{25 \left (3 x +2\right )}+\frac {3}{\left (x +1\right )^{2}}+\frac {41}{x +1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-153/10/(3*x+2)^2+2646/25/(3*x+2)+28917/125*ln(3*x+2)+208/125*ln(2*x+3)+3/(x+1)^2+41/(x+1)-233*ln(x+1)

________________________________________________________________________________________

maxima [A]  time = 0.59, size = 62, normalized size = 0.90 \begin {gather*} \frac {34326 \, x^{3} + 85971 \, x^{2} + 70114 \, x + 18619}{50 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + \frac {28917}{125} \, \log \left (3 \, x + 2\right ) + \frac {208}{125} \, \log \left (2 \, x + 3\right ) - 233 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50*(34326*x^3 + 85971*x^2 + 70114*x + 18619)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4) + 28917/125*log(3*x + 2) +
208/125*log(2*x + 3) - 233*log(x + 1)

________________________________________________________________________________________

mupad [B]  time = 2.32, size = 55, normalized size = 0.80 \begin {gather*} \frac {28917\,\ln \left (x+\frac {2}{3}\right )}{125}-233\,\ln \left (x+1\right )+\frac {208\,\ln \left (x+\frac {3}{2}\right )}{125}+\frac {\frac {1907\,x^3}{25}+\frac {28657\,x^2}{150}+\frac {35057\,x}{225}+\frac {18619}{450}}{x^4+\frac {10\,x^3}{3}+\frac {37\,x^2}{9}+\frac {20\,x}{9}+\frac {4}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(28917*log(x + 2/3))/125 - 233*log(x + 1) + (208*log(x + 3/2))/125 + ((35057*x)/225 + (28657*x^2)/150 + (1907*
x^3)/25 + 18619/450)/((20*x)/9 + (37*x^2)/9 + (10*x^3)/3 + x^4 + 4/9)

________________________________________________________________________________________

sympy [A]  time = 0.21, size = 63, normalized size = 0.91 \begin {gather*} - \frac {- 34326 x^{3} - 85971 x^{2} - 70114 x - 18619}{450 x^{4} + 1500 x^{3} + 1850 x^{2} + 1000 x + 200} + \frac {28917 \log {\left (x + \frac {2}{3} \right )}}{125} - 233 \log {\left (x + 1 \right )} + \frac {208 \log {\left (x + \frac {3}{2} \right )}}{125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-34326*x**3 - 85971*x**2 - 70114*x - 18619)/(450*x**4 + 1500*x**3 + 1850*x**2 + 1000*x + 200) + 28917*log(x
+ 2/3)/125 - 233*log(x + 1) + 208*log(x + 3/2)/125

________________________________________________________________________________________