3.2404 \(\int \frac{5-x}{(3+2 x)^3 (2+5 x+3 x^2)^3} \, dx\)

Optimal. Leaf size=105 \[ -\frac{3 (47 x+37)}{10 (2 x+3)^2 \left (3 x^2+5 x+2\right )^2}+\frac{10254 x+8999}{50 (2 x+3)^2 \left (3 x^2+5 x+2\right )}+\frac{35886}{625 (2 x+3)}+\frac{11856}{125 (2 x+3)^2}-141 \log (x+1)+\frac{68592 \log (2 x+3)}{3125}+\frac{372033 \log (3 x+2)}{3125} \]

[Out]

11856/(125*(3 + 2*x)^2) + 35886/(625*(3 + 2*x)) - (3*(37 + 47*x))/(10*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^2) + (8999
+ 10254*x)/(50*(3 + 2*x)^2*(2 + 5*x + 3*x^2)) - 141*Log[1 + x] + (68592*Log[3 + 2*x])/3125 + (372033*Log[2 +
3*x])/3125

________________________________________________________________________________________

Rubi [A]  time = 0.0700943, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {822, 800} \[ -\frac{3 (47 x+37)}{10 (2 x+3)^2 \left (3 x^2+5 x+2\right )^2}+\frac{10254 x+8999}{50 (2 x+3)^2 \left (3 x^2+5 x+2\right )}+\frac{35886}{625 (2 x+3)}+\frac{11856}{125 (2 x+3)^2}-141 \log (x+1)+\frac{68592 \log (2 x+3)}{3125}+\frac{372033 \log (3 x+2)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

11856/(125*(3 + 2*x)^2) + 35886/(625*(3 + 2*x)) - (3*(37 + 47*x))/(10*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^2) + (8999
+ 10254*x)/(50*(3 + 2*x)^2*(2 + 5*x + 3*x^2)) - 141*Log[1 + x] + (68592*Log[3 + 2*x])/3125 + (372033*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)^3 \left (2+5 x+3 x^2\right )^3} \, dx &=-\frac{3 (37+47 x)}{10 (3+2 x)^2 \left (2+5 x+3 x^2\right )^2}-\frac{1}{10} \int \frac{1661+1410 x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac{3 (37+47 x)}{10 (3+2 x)^2 \left (2+5 x+3 x^2\right )^2}+\frac{8999+10254 x}{50 (3+2 x)^2 \left (2+5 x+3 x^2\right )}+\frac{1}{50} \int \frac{68574+61524 x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{3 (37+47 x)}{10 (3+2 x)^2 \left (2+5 x+3 x^2\right )^2}+\frac{8999+10254 x}{50 (3+2 x)^2 \left (2+5 x+3 x^2\right )}+\frac{1}{50} \int \left (-\frac{7050}{1+x}-\frac{94848}{5 (3+2 x)^3}-\frac{143544}{25 (3+2 x)^2}+\frac{274368}{125 (3+2 x)}+\frac{2232198}{125 (2+3 x)}\right ) \, dx\\ &=\frac{11856}{125 (3+2 x)^2}+\frac{35886}{625 (3+2 x)}-\frac{3 (37+47 x)}{10 (3+2 x)^2 \left (2+5 x+3 x^2\right )^2}+\frac{8999+10254 x}{50 (3+2 x)^2 \left (2+5 x+3 x^2\right )}-141 \log (1+x)+\frac{68592 \log (3+2 x)}{3125}+\frac{372033 \log (2+3 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0513018, size = 86, normalized size = 0.82 \[ \frac{-\frac{75 (903 x+653)}{2 \left (3 x^2+5 x+2\right )^2}+\frac{611970 x+550495}{6 x^2+10 x+4}-\frac{24560}{2 x+3}-\frac{2600}{(2 x+3)^2}+372033 \log (-6 x-4)-440625 \log (-2 (x+1))+68592 \log (2 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2600/(3 + 2*x)^2 - 24560/(3 + 2*x) - (75*(653 + 903*x))/(2*(2 + 5*x + 3*x^2)^2) + (550495 + 611970*x)/(4 + 1
0*x + 6*x^2) + 372033*Log[-4 - 6*x] - 440625*Log[-2*(1 + x)] + 68592*Log[3 + 2*x])/3125

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 74, normalized size = 0.7 \begin{align*} 3\, \left ( 1+x \right ) ^{-2}+17\, \left ( 1+x \right ) ^{-1}-141\,\ln \left ( 1+x \right ) -{\frac{104}{125\, \left ( 3+2\,x \right ) ^{2}}}-{\frac{4912}{1875+1250\,x}}+{\frac{68592\,\ln \left ( 3+2\,x \right ) }{3125}}-{\frac{1377}{250\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{29322}{1250+1875\,x}}+{\frac{372033\,\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)^3/(3*x^2+5*x+2)^3,x)

[Out]

3/(1+x)^2+17/(1+x)-141*ln(1+x)-104/125/(3+2*x)^2-4912/625/(3+2*x)+68592/3125*ln(3+2*x)-1377/250/(2+3*x)^2+2932
2/625/(2+3*x)+372033/3125*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 0.999186, size = 111, normalized size = 1.06 \begin{align*} \frac{1291896 \, x^{5} + 7311204 \, x^{4} + 16096458 \, x^{3} + 17180967 \, x^{2} + 8871646 \, x + 1771579}{1250 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )}} + \frac{372033}{3125} \, \log \left (3 \, x + 2\right ) + \frac{68592}{3125} \, \log \left (2 \, x + 3\right ) - 141 \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1250*(1291896*x^5 + 7311204*x^4 + 16096458*x^3 + 17180967*x^2 + 8871646*x + 1771579)/(36*x^6 + 228*x^5 + 589
*x^4 + 794*x^3 + 589*x^2 + 228*x + 36) + 372033/3125*log(3*x + 2) + 68592/3125*log(2*x + 3) - 141*log(x + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.24242, size = 543, normalized size = 5.17 \begin{align*} \frac{6459480 \, x^{5} + 36556020 \, x^{4} + 80482290 \, x^{3} + 85904835 \, x^{2} + 744066 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (3 \, x + 2\right ) + 137184 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (2 \, x + 3\right ) - 881250 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (x + 1\right ) + 44358230 \, x + 8857895}{6250 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6250*(6459480*x^5 + 36556020*x^4 + 80482290*x^3 + 85904835*x^2 + 744066*(36*x^6 + 228*x^5 + 589*x^4 + 794*x^
3 + 589*x^2 + 228*x + 36)*log(3*x + 2) + 137184*(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)*
log(2*x + 3) - 881250*(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)*log(x + 1) + 44358230*x +
8857895)/(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)

________________________________________________________________________________________

Sympy [A]  time = 0.249623, size = 82, normalized size = 0.78 \begin{align*} \frac{1291896 x^{5} + 7311204 x^{4} + 16096458 x^{3} + 17180967 x^{2} + 8871646 x + 1771579}{45000 x^{6} + 285000 x^{5} + 736250 x^{4} + 992500 x^{3} + 736250 x^{2} + 285000 x + 45000} + \frac{372033 \log{\left (x + \frac{2}{3} \right )}}{3125} - 141 \log{\left (x + 1 \right )} + \frac{68592 \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)**3/(3*x**2+5*x+2)**3,x)

[Out]

(1291896*x**5 + 7311204*x**4 + 16096458*x**3 + 17180967*x**2 + 8871646*x + 1771579)/(45000*x**6 + 285000*x**5
+ 736250*x**4 + 992500*x**3 + 736250*x**2 + 285000*x + 45000) + 372033*log(x + 2/3)/3125 - 141*log(x + 1) + 68
592*log(x + 3/2)/3125

________________________________________________________________________________________

Giac [A]  time = 1.16747, size = 95, normalized size = 0.9 \begin{align*} \frac{1291896 \, x^{5} + 7311204 \, x^{4} + 16096458 \, x^{3} + 17180967 \, x^{2} + 8871646 \, x + 1771579}{1250 \,{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )}^{2}} + \frac{372033}{3125} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{68592}{3125} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - 141 \, \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)^3/(3*x^2+5*x+2)^3,x, algorithm="giac")

[Out]

1/1250*(1291896*x^5 + 7311204*x^4 + 16096458*x^3 + 17180967*x^2 + 8871646*x + 1771579)/(6*x^3 + 19*x^2 + 19*x
+ 6)^2 + 372033/3125*log(abs(3*x + 2)) + 68592/3125*log(abs(2*x + 3)) - 141*log(abs(x + 1))