3.2395 \(\int \frac{5-x}{(3+2 x)^4 (2+5 x+3 x^2)^2} \, dx\)

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.056415, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {822, 800} \[ -\frac{3 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )}-\frac{16522}{625 (2 x+3)}-\frac{2212}{125 (2 x+3)^2}-\frac{1258}{75 (2 x+3)^3}-13 \log (x+1)+\frac{65816 \log (2 x+3)}{3125}-\frac{25191 \log (3 x+2)}{3125} \]

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 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])

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 )^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*}

Mathematica [A]  time = 0.0432576, size = 75, normalized size = 0.85 \[ \frac{-\frac{45 (4209 x+2959)}{3 x^2+5 x+2}-\frac{121560}{2 x+3}-\frac{30450}{(2 x+3)^2}-\frac{6500}{(2 x+3)^3}-75573 \log (-6 x-4)-121875 \log (-2 (x+1))+197448 \log (2 x+3)}{9375} \]

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

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 67, normalized size = 0.8 \begin{align*} -6\, \left ( 1+x \right ) ^{-1}-13\,\ln \left ( 1+x \right ) -{\frac{52}{75\, \left ( 3+2\,x \right ) ^{3}}}-{\frac{406}{125\, \left ( 3+2\,x \right ) ^{2}}}-{\frac{8104}{1875+1250\,x}}+{\frac{65816\,\ln \left ( 3+2\,x \right ) }{3125}}-{\frac{1377}{1250+1875\,x}}-{\frac{25191\,\ln \left ( 2+3\,x \right ) }{3125}} \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)^2,x)

[Out]

-6/(1+x)-13*ln(1+x)-52/75/(3+2*x)^3-406/125/(3+2*x)^2-8104/625/(3+2*x)+65816/3125*ln(3+2*x)-1377/625/(2+3*x)-2
5191/3125*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 1.01436, size = 97, normalized size = 1.1 \begin{align*} -\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{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)^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)

________________________________________________________________________________________

Fricas [A]  time = 1.24801, size = 468, normalized size = 5.32 \begin{align*} -\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{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)^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)

________________________________________________________________________________________

Sympy [A]  time = 0.244492, size = 71, normalized size = 0.81 \begin{align*} - \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{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)**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

________________________________________________________________________________________

Giac [A]  time = 1.13367, size = 90, normalized size = 1.02 \begin{align*} -\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{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)^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))