3.2403 \(\int \frac{5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^3} \, dx\)

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)) - 1
41*Log[1 + x] + (68592*Log[3 + 2*x])/3125 + (372033*Log[2 + 3*x])/3125

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Rubi [A]  time = 0.166153, 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 \[ -\frac{3 (47 x+37)}{10 (2 x+3)^2 \left (3 x^2+5 x+2\right )^2}+\frac{10254 x+8999}{50 (2 x+3)^2 \left (3 x^2+5 x+2\right )}+\frac{35886}{625 (2 x+3)}+\frac{11856}{125 (2 x+3)^2}-141 \log (x+1)+\frac{68592 \log (2 x+3)}{3125}+\frac{372033 \log (3 x+2)}{3125} \]

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)) - 1
41*Log[1 + x] + (68592*Log[3 + 2*x])/3125 + (372033*Log[2 + 3*x])/3125

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Rubi in Sympy [A]  time = 30.7173, size = 92, normalized size = 0.88 \[ - 141 \log{\left (x + 1 \right )} + \frac{68592 \log{\left (2 x + 3 \right )}}{3125} + \frac{372033 \log{\left (3 x + 2 \right )}}{3125} + \frac{35886}{625 \left (2 x + 3\right )} - \frac{141 x + 111}{10 \left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{2}} + \frac{10254 x + 8999}{50 \left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )} + \frac{11856}{125 \left (2 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)/(3+2*x)**3/(3*x**2+5*x+2)**3,x)

[Out]

-141*log(x + 1) + 68592*log(2*x + 3)/3125 + 372033*log(3*x + 2)/3125 + 35886/(62
5*(2*x + 3)) - (141*x + 111)/(10*(2*x + 3)**2*(3*x**2 + 5*x + 2)**2) + (10254*x
+ 8999)/(50*(2*x + 3)**2*(3*x**2 + 5*x + 2)) + 11856/(125*(2*x + 3)**2)

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Mathematica [A]  time = 0.10061, 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 + 10*x + 6*x^2) + 372033*Log[-4 - 6*x] - 440625*Log[-
2*(1 + x)] + 68592*Log[3 + 2*x])/3125

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Maple [A]  time = 0.022, size = 74, normalized size = 0.7 \[ -{\frac{1377}{250\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{29322}{1250+1875\,x}}+{\frac{372033\,\ln \left ( 2+3\,x \right ) }{3125}}-{\frac{104}{125\, \left ( 3+2\,x \right ) ^{2}}}-{\frac{4912}{1875+1250\,x}}+{\frac{68592\,\ln \left ( 3+2\,x \right ) }{3125}}+3\, \left ( 1+x \right ) ^{-2}+17\, \left ( 1+x \right ) ^{-1}-141\,\ln \left ( 1+x \right ) \]

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]

-1377/250/(2+3*x)^2+29322/625/(2+3*x)+372033/3125*ln(2+3*x)-104/125/(3+2*x)^2-49
12/625/(3+2*x)+68592/3125*ln(3+2*x)+3/(1+x)^2+17/(1+x)-141*ln(1+x)

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Maxima [A]  time = 0.693647, size = 111, normalized size = 1.06 \[ \frac{1291896 \, x^{5} + 7311204 \, x^{4} + 16096458 \, x^{3} + 17180967 \, x^{2} + 8871646 \, x + 1771579}{1250 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )}} + \frac{372033}{3125} \, \log \left (3 \, x + 2\right ) + \frac{68592}{3125} \, \log \left (2 \, x + 3\right ) - 141 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1250*(1291896*x^5 + 7311204*x^4 + 16096458*x^3 + 17180967*x^2 + 8871646*x + 17
71579)/(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36) + 372033/31
25*log(3*x + 2) + 68592/3125*log(2*x + 3) - 141*log(x + 1)

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Fricas [A]  time = 0.26766, size = 231, normalized size = 2.2 \[ \frac{6459480 \, x^{5} + 36556020 \, x^{4} + 80482290 \, x^{3} + 85904835 \, x^{2} + 744066 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (3 \, x + 2\right ) + 137184 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (2 \, x + 3\right ) - 881250 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (x + 1\right ) + 44358230 \, x + 8857895}{6250 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/((3*x^2 + 5*x + 2)^3*(2*x + 3)^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*(3
6*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)*log(2*x + 3) - 88125
0*(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)*log(x + 1) + 443
58230*x + 8857895)/(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)

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Sympy [A]  time = 0.711832, size = 82, normalized size = 0.78 \[ \frac{1291896 x^{5} + 7311204 x^{4} + 16096458 x^{3} + 17180967 x^{2} + 8871646 x + 1771579}{45000 x^{6} + 285000 x^{5} + 736250 x^{4} + 992500 x^{3} + 736250 x^{2} + 285000 x + 45000} + \frac{372033 \log{\left (x + \frac{2}{3} \right )}}{3125} - 141 \log{\left (x + 1 \right )} + \frac{68592 \log{\left (x + \frac{3}{2} \right )}}{3125} \]

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 + 17715
79)/(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) + 68592*log(x + 3/2)/312
5

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GIAC/XCAS [A]  time = 0.264904, size = 95, normalized size = 0.9 \[ \frac{1291896 \, x^{5} + 7311204 \, x^{4} + 16096458 \, x^{3} + 17180967 \, x^{2} + 8871646 \, x + 1771579}{1250 \,{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )}^{2}} + \frac{372033}{3125} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{68592}{3125} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - 141 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1250*(1291896*x^5 + 7311204*x^4 + 16096458*x^3 + 17180967*x^2 + 8871646*x + 17
71579)/(6*x^3 + 19*x^2 + 19*x + 6)^2 + 372033/3125*ln(abs(3*x + 2)) + 68592/3125
*ln(abs(2*x + 3)) - 141*ln(abs(x + 1))