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

Optimal. Leaf size=88 \[ -\frac {3 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )}-\frac {16522}{625 (2 x+3)}-\frac {2212}{125 (2 x+3)^2}-\frac {1258}{75 (2 x+3)^3}-13 \log (x+1)+\frac {65816 \log (2 x+3)}{3125}-\frac {25191 \log (3 x+2)}{3125} \]

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Rubi [A]  time = 0.06, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, 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)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )}-\frac {16522}{625 (2 x+3)}-\frac {2212}{125 (2 x+3)^2}-\frac {1258}{75 (2 x+3)^3}-13 \log (x+1)+\frac {65816 \log (2 x+3)}{3125}-\frac {25191 \log (3 x+2)}{3125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1258/(75*(3 + 2*x)^3) - 2212/(125*(3 + 2*x)^2) - 16522/(625*(3 + 2*x)) - (3*(37 + 47*x))/(5*(3 + 2*x)^3*(2 +
5*x + 3*x^2)) - 13*Log[1 + x] + (65816*Log[3 + 2*x])/3125 - (25191*Log[2 + 3*x])/3125

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)^4 \left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {3 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )}-\frac {1}{5} \int \frac {1063+1128 x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {3 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )}-\frac {1}{5} \int \left (\frac {65}{1+x}-\frac {2516}{5 (3+2 x)^4}-\frac {8848}{25 (3+2 x)^3}-\frac {33044}{125 (3+2 x)^2}-\frac {131632}{625 (3+2 x)}+\frac {75573}{625 (2+3 x)}\right ) \, dx\\ &=-\frac {1258}{75 (3+2 x)^3}-\frac {2212}{125 (3+2 x)^2}-\frac {16522}{625 (3+2 x)}-\frac {3 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )}-13 \log (1+x)+\frac {65816 \log (3+2 x)}{3125}-\frac {25191 \log (2+3 x)}{3125}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 75, normalized size = 0.85 \begin {gather*} \frac {-\frac {45 (4209 x+2959)}{3 x^2+5 x+2}-\frac {121560}{2 x+3}-\frac {30450}{(2 x+3)^2}-\frac {6500}{(2 x+3)^3}-75573 \log (-6 x-4)-121875 \log (-2 (x+1))+197448 \log (2 x+3)}{9375} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6500/(3 + 2*x)^3 - 30450/(3 + 2*x)^2 - 121560/(3 + 2*x) - (45*(2959 + 4209*x))/(2 + 5*x + 3*x^2) - 75573*Log
[-4 - 6*x] - 121875*Log[-2*(1 + x)] + 197448*Log[3 + 2*x])/9375

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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 )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 146, normalized size = 1.66 \begin {gather*} -\frac {2973960 \, x^{4} + 14873880 \, x^{3} + 27167700 \, x^{2} + 75573 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (3 \, x + 2\right ) - 197448 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (2 \, x + 3\right ) + 121875 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (x + 1\right ) + 21302995 \, x + 5978965}{9375 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\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)^2,x, algorithm="fricas")

[Out]

-1/9375*(2973960*x^4 + 14873880*x^3 + 27167700*x^2 + 75573*(24*x^5 + 148*x^4 + 358*x^3 + 423*x^2 + 243*x + 54)
*log(3*x + 2) - 197448*(24*x^5 + 148*x^4 + 358*x^3 + 423*x^2 + 243*x + 54)*log(2*x + 3) + 121875*(24*x^5 + 148
*x^4 + 358*x^3 + 423*x^2 + 243*x + 54)*log(x + 1) + 21302995*x + 5978965)/(24*x^5 + 148*x^4 + 358*x^3 + 423*x^
2 + 243*x + 54)

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giac [A]  time = 0.17, size = 67, normalized size = 0.76 \begin {gather*} -\frac {594792 \, x^{4} + 2974776 \, x^{3} + 5433540 \, x^{2} + 4260599 \, x + 1195793}{1875 \, {\left (3 \, x + 2\right )} {\left (2 \, x + 3\right )}^{3} {\left (x + 1\right )}} - \frac {25191}{3125} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {65816}{3125} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - 13 \, \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)^2,x, algorithm="giac")

[Out]

-1/1875*(594792*x^4 + 2974776*x^3 + 5433540*x^2 + 4260599*x + 1195793)/((3*x + 2)*(2*x + 3)^3*(x + 1)) - 25191
/3125*log(abs(3*x + 2)) + 65816/3125*log(abs(2*x + 3)) - 13*log(abs(x + 1))

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maple [A]  time = 0.06, size = 67, normalized size = 0.76 \begin {gather*} -\frac {25191 \ln \left (3 x +2\right )}{3125}+\frac {65816 \ln \left (2 x +3\right )}{3125}-13 \ln \left (x +1\right )-\frac {1377}{625 \left (3 x +2\right )}-\frac {52}{75 \left (2 x +3\right )^{3}}-\frac {406}{125 \left (2 x +3\right )^{2}}-\frac {8104}{625 \left (2 x +3\right )}-\frac {6}{x +1} \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)^2,x)

[Out]

-1377/625/(3*x+2)-25191/3125*ln(3*x+2)-52/75/(2*x+3)^3-406/125/(2*x+3)^2-8104/625/(2*x+3)+65816/3125*ln(2*x+3)
-6/(x+1)-13*ln(x+1)

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maxima [A]  time = 0.51, size = 72, normalized size = 0.82 \begin {gather*} -\frac {594792 \, x^{4} + 2974776 \, x^{3} + 5433540 \, x^{2} + 4260599 \, x + 1195793}{1875 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )}} - \frac {25191}{3125} \, \log \left (3 \, x + 2\right ) + \frac {65816}{3125} \, \log \left (2 \, x + 3\right ) - 13 \, \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)^2,x, algorithm="maxima")

[Out]

-1/1875*(594792*x^4 + 2974776*x^3 + 5433540*x^2 + 4260599*x + 1195793)/(24*x^5 + 148*x^4 + 358*x^3 + 423*x^2 +
243*x + 54) - 25191/3125*log(3*x + 2) + 65816/3125*log(2*x + 3) - 13*log(x + 1)

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mupad [B]  time = 2.37, size = 66, normalized size = 0.75 \begin {gather*} \frac {65816\,\ln \left (x+\frac {3}{2}\right )}{3125}-\frac {25191\,\ln \left (x+\frac {2}{3}\right )}{3125}-13\,\ln \left (x+1\right )-\frac {\frac {8261\,x^4}{625}+\frac {123949\,x^3}{1875}+\frac {90559\,x^2}{750}+\frac {4260599\,x}{45000}+\frac {1195793}{45000}}{x^5+\frac {37\,x^4}{6}+\frac {179\,x^3}{12}+\frac {141\,x^2}{8}+\frac {81\,x}{8}+\frac {9}{4}} \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)^2),x)

[Out]

(65816*log(x + 3/2))/3125 - (25191*log(x + 2/3))/3125 - 13*log(x + 1) - ((4260599*x)/45000 + (90559*x^2)/750 +
(123949*x^3)/1875 + (8261*x^4)/625 + 1195793/45000)/((81*x)/8 + (141*x^2)/8 + (179*x^3)/12 + (37*x^4)/6 + x^5
+ 9/4)

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sympy [A]  time = 0.23, size = 71, normalized size = 0.81 \begin {gather*} - \frac {594792 x^{4} + 2974776 x^{3} + 5433540 x^{2} + 4260599 x + 1195793}{45000 x^{5} + 277500 x^{4} + 671250 x^{3} + 793125 x^{2} + 455625 x + 101250} - \frac {25191 \log {\left (x + \frac {2}{3} \right )}}{3125} - 13 \log {\left (x + 1 \right )} + \frac {65816 \log {\left (x + \frac {3}{2} \right )}}{3125} \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)**2,x)

[Out]

-(594792*x**4 + 2974776*x**3 + 5433540*x**2 + 4260599*x + 1195793)/(45000*x**5 + 277500*x**4 + 671250*x**3 + 7
93125*x**2 + 455625*x + 101250) - 25191*log(x + 2/3)/3125 - 13*log(x + 1) + 65816*log(x + 3/2)/3125

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