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

Optimal. Leaf size=94 \[ -\frac {3 (47 x+37)}{10 (2 x+3) \left (3 x^2+5 x+2\right )^2}+\frac {10848 x+9293}{50 (2 x+3) \left (3 x^2+5 x+2\right )}+\frac {12946}{125 (2 x+3)}-175 \log (x+1)+\frac {4912}{625} \log (2 x+3)+\frac {104463}{625} \log (3 x+2) \]

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Rubi [A]  time = 0.06, antiderivative size = 94, 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 (2 x+3) \left (3 x^2+5 x+2\right )^2}+\frac {10848 x+9293}{50 (2 x+3) \left (3 x^2+5 x+2\right )}+\frac {12946}{125 (2 x+3)}-175 \log (x+1)+\frac {4912}{625} \log (2 x+3)+\frac {104463}{625} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

12946/(125*(3 + 2*x)) - (3*(37 + 47*x))/(10*(3 + 2*x)*(2 + 5*x + 3*x^2)^2) + (9293 + 10848*x)/(50*(3 + 2*x)*(2
+ 5*x + 3*x^2)) - 175*Log[1 + x] + (4912*Log[3 + 2*x])/625 + (104463*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]

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)^2 \left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {3 (37+47 x)}{10 (3+2 x) \left (2+5 x+3 x^2\right )^2}-\frac {1}{10} \int \frac {1439+1128 x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {3 (37+47 x)}{10 (3+2 x) \left (2+5 x+3 x^2\right )^2}+\frac {9293+10848 x}{50 (3+2 x) \left (2+5 x+3 x^2\right )}+\frac {1}{50} \int \frac {52142+43392 x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {3 (37+47 x)}{10 (3+2 x) \left (2+5 x+3 x^2\right )^2}+\frac {9293+10848 x}{50 (3+2 x) \left (2+5 x+3 x^2\right )}+\frac {1}{50} \int \left (-\frac {8750}{1+x}-\frac {51784}{5 (3+2 x)^2}+\frac {19648}{25 (3+2 x)}+\frac {626778}{25 (2+3 x)}\right ) \, dx\\ &=\frac {12946}{125 (3+2 x)}-\frac {3 (37+47 x)}{10 (3+2 x) \left (2+5 x+3 x^2\right )^2}+\frac {9293+10848 x}{50 (3+2 x) \left (2+5 x+3 x^2\right )}-175 \log (1+x)+\frac {4912}{625} \log (3+2 x)+\frac {104463}{625} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 78, normalized size = 0.83 \begin {gather*} \frac {1}{625} \left (-\frac {75 (201 x+151)}{2 \left (3 x^2+5 x+2\right )^2}+\frac {5 (39462 x+33697)}{6 x^2+10 x+4}-\frac {1040}{2 x+3}+104463 \log (-6 x-4)-109375 \log (-2 (x+1))+4912 \log (2 x+3)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1040/(3 + 2*x) - (75*(151 + 201*x))/(2*(2 + 5*x + 3*x^2)^2) + (5*(33697 + 39462*x))/(4 + 10*x + 6*x^2) + 104
463*Log[-4 - 6*x] - 109375*Log[-2*(1 + x)] + 4912*Log[3 + 2*x])/625

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

[Out]

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

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fricas [A]  time = 0.40, size = 146, normalized size = 1.55 \begin {gather*} \frac {1165140 \, x^{4} + 4697400 \, x^{3} + 6842995 \, x^{2} + 208926 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (3 \, x + 2\right ) + 9824 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (2 \, x + 3\right ) - 218750 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (x + 1\right ) + 4275600 \, x + 968615}{1250 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1250*(1165140*x^4 + 4697400*x^3 + 6842995*x^2 + 208926*(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*log
(3*x + 2) + 9824*(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*log(2*x + 3) - 218750*(18*x^5 + 87*x^4 + 16
4*x^3 + 151*x^2 + 68*x + 12)*log(x + 1) + 4275600*x + 968615)/(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12
)

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giac [A]  time = 0.16, size = 95, normalized size = 1.01 \begin {gather*} -\frac {208}{125 \, {\left (2 \, x + 3\right )}} - \frac {2 \, {\left (\frac {168231}{2 \, x + 3} - \frac {211036}{{\left (2 \, x + 3\right )}^{2}} + \frac {82447}{{\left (2 \, x + 3\right )}^{3}} - 42642\right )}}{125 \, {\left (\frac {5}{2 \, x + 3} - 3\right )}^{2} {\left (\frac {1}{2 \, x + 3} - 1\right )}^{2}} - 175 \, \log \left ({\left | -\frac {1}{2 \, x + 3} + 1 \right |}\right ) + \frac {104463}{625} \, \log \left ({\left | -\frac {5}{2 \, x + 3} + 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-208/125/(2*x + 3) - 2/125*(168231/(2*x + 3) - 211036/(2*x + 3)^2 + 82447/(2*x + 3)^3 - 42642)/((5/(2*x + 3) -
3)^2*(1/(2*x + 3) - 1)^2) - 175*log(abs(-1/(2*x + 3) + 1)) + 104463/625*log(abs(-5/(2*x + 3) + 3))

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maple [A]  time = 0.05, size = 65, normalized size = 0.69 \begin {gather*} \frac {104463 \ln \left (3 x +2\right )}{625}+\frac {4912 \ln \left (2 x +3\right )}{625}-175 \ln \left (x +1\right )-\frac {459}{50 \left (3 x +2\right )^{2}}+\frac {8856}{125 \left (3 x +2\right )}-\frac {208}{125 \left (2 x +3\right )}+\frac {3}{\left (x +1\right )^{2}}+\frac {29}{x +1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-459/50/(3*x+2)^2+8856/125/(3*x+2)+104463/625*ln(3*x+2)-208/125/(2*x+3)+4912/625*ln(2*x+3)+3/(x+1)^2+29/(x+1)-
175*ln(x+1)

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maxima [A]  time = 0.69, size = 72, normalized size = 0.77 \begin {gather*} \frac {233028 \, x^{4} + 939480 \, x^{3} + 1368599 \, x^{2} + 855120 \, x + 193723}{250 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )}} + \frac {104463}{625} \, \log \left (3 \, x + 2\right ) + \frac {4912}{625} \, \log \left (2 \, x + 3\right ) - 175 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/250*(233028*x^4 + 939480*x^3 + 1368599*x^2 + 855120*x + 193723)/(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x
+ 12) + 104463/625*log(3*x + 2) + 4912/625*log(2*x + 3) - 175*log(x + 1)

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mupad [B]  time = 0.04, size = 65, normalized size = 0.69 \begin {gather*} \frac {104463\,\ln \left (x+\frac {2}{3}\right )}{625}-175\,\ln \left (x+1\right )+\frac {4912\,\ln \left (x+\frac {3}{2}\right )}{625}+\frac {\frac {6473\,x^4}{125}+\frac {15658\,x^3}{75}+\frac {1368599\,x^2}{4500}+\frac {14252\,x}{75}+\frac {193723}{4500}}{x^5+\frac {29\,x^4}{6}+\frac {82\,x^3}{9}+\frac {151\,x^2}{18}+\frac {34\,x}{9}+\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(104463*log(x + 2/3))/625 - 175*log(x + 1) + (4912*log(x + 3/2))/625 + ((14252*x)/75 + (1368599*x^2)/4500 + (1
5658*x^3)/75 + (6473*x^4)/125 + 193723/4500)/((34*x)/9 + (151*x^2)/18 + (82*x^3)/9 + (29*x^4)/6 + x^5 + 2/3)

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sympy [A]  time = 0.23, size = 73, normalized size = 0.78 \begin {gather*} - \frac {- 233028 x^{4} - 939480 x^{3} - 1368599 x^{2} - 855120 x - 193723}{4500 x^{5} + 21750 x^{4} + 41000 x^{3} + 37750 x^{2} + 17000 x + 3000} + \frac {104463 \log {\left (x + \frac {2}{3} \right )}}{625} - 175 \log {\left (x + 1 \right )} + \frac {4912 \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)**2/(3*x**2+5*x+2)**3,x)

[Out]

-(-233028*x**4 - 939480*x**3 - 1368599*x**2 - 855120*x - 193723)/(4500*x**5 + 21750*x**4 + 41000*x**3 + 37750*
x**2 + 17000*x + 3000) + 104463*log(x + 2/3)/625 - 175*log(x + 1) + 4912*log(x + 3/2)/625

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