3.187 \(\int \frac{1}{(2+x)^3 (3+x)^4} \, dx\)

Optimal. Leaf size=54 \[ \frac{4}{x+2}+\frac{6}{x+3}-\frac{1}{2 (x+2)^2}+\frac{3}{2 (x+3)^2}+\frac{1}{3 (x+3)^3}+10 \log (x+2)-10 \log (x+3) \]

[Out]

-1/(2*(2 + x)^2) + 4/(2 + x) + 1/(3*(3 + x)^3) + 3/(2*(3 + x)^2) + 6/(3 + x) + 1
0*Log[2 + x] - 10*Log[3 + x]

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Rubi [A]  time = 0.0536938, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{4}{x+2}+\frac{6}{x+3}-\frac{1}{2 (x+2)^2}+\frac{3}{2 (x+3)^2}+\frac{1}{3 (x+3)^3}+10 \log (x+2)-10 \log (x+3) \]

Antiderivative was successfully verified.

[In]  Int[1/((2 + x)^3*(3 + x)^4),x]

[Out]

-1/(2*(2 + x)^2) + 4/(2 + x) + 1/(3*(3 + x)^3) + 3/(2*(3 + x)^2) + 6/(3 + x) + 1
0*Log[2 + x] - 10*Log[3 + x]

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Rubi in Sympy [A]  time = 3.36141, size = 48, normalized size = 0.89 \[ 10 \log{\left (x + 2 \right )} - 10 \log{\left (x + 3 \right )} + \frac{6}{x + 3} + \frac{3}{2 \left (x + 3\right )^{2}} + \frac{1}{3 \left (x + 3\right )^{3}} + \frac{4}{x + 2} - \frac{1}{2 \left (x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2+x)**3/(3+x)**4,x)

[Out]

10*log(x + 2) - 10*log(x + 3) + 6/(x + 3) + 3/(2*(x + 3)**2) + 1/(3*(x + 3)**3)
+ 4/(x + 2) - 1/(2*(x + 2)**2)

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Mathematica [A]  time = 0.022466, size = 54, normalized size = 1. \[ \frac{4}{x+2}+\frac{6}{x+3}-\frac{1}{2 (x+2)^2}+\frac{3}{2 (x+3)^2}+\frac{1}{3 (x+3)^3}+10 \log (x+2)-10 \log (x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((2 + x)^3*(3 + x)^4),x]

[Out]

-1/(2*(2 + x)^2) + 4/(2 + x) + 1/(3*(3 + x)^3) + 3/(2*(3 + x)^2) + 6/(3 + x) + 1
0*Log[2 + x] - 10*Log[3 + x]

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Maple [A]  time = 0., size = 49, normalized size = 0.9 \[ -{\frac{1}{2\, \left ( 2+x \right ) ^{2}}}+4\, \left ( 2+x \right ) ^{-1}+{\frac{1}{3\, \left ( 3+x \right ) ^{3}}}+{\frac{3}{2\, \left ( 3+x \right ) ^{2}}}+6\, \left ( 3+x \right ) ^{-1}+10\,\ln \left ( 2+x \right ) -10\,\ln \left ( 3+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2+x)^3/(3+x)^4,x)

[Out]

-1/2/(2+x)^2+4/(2+x)+1/3/(3+x)^3+3/2/(3+x)^2+6/(3+x)+10*ln(2+x)-10*ln(3+x)

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Maxima [A]  time = 1.41008, size = 81, normalized size = 1.5 \[ \frac{60 \, x^{4} + 630 \, x^{3} + 2450 \, x^{2} + 4175 \, x + 2627}{6 \,{\left (x^{5} + 13 \, x^{4} + 67 \, x^{3} + 171 \, x^{2} + 216 \, x + 108\right )}} - 10 \, \log \left (x + 3\right ) + 10 \, \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 3)^4*(x + 2)^3),x, algorithm="maxima")

[Out]

1/6*(60*x^4 + 630*x^3 + 2450*x^2 + 4175*x + 2627)/(x^5 + 13*x^4 + 67*x^3 + 171*x
^2 + 216*x + 108) - 10*log(x + 3) + 10*log(x + 2)

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Fricas [A]  time = 0.197954, size = 142, normalized size = 2.63 \[ \frac{60 \, x^{4} + 630 \, x^{3} + 2450 \, x^{2} - 60 \,{\left (x^{5} + 13 \, x^{4} + 67 \, x^{3} + 171 \, x^{2} + 216 \, x + 108\right )} \log \left (x + 3\right ) + 60 \,{\left (x^{5} + 13 \, x^{4} + 67 \, x^{3} + 171 \, x^{2} + 216 \, x + 108\right )} \log \left (x + 2\right ) + 4175 \, x + 2627}{6 \,{\left (x^{5} + 13 \, x^{4} + 67 \, x^{3} + 171 \, x^{2} + 216 \, x + 108\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 3)^4*(x + 2)^3),x, algorithm="fricas")

[Out]

1/6*(60*x^4 + 630*x^3 + 2450*x^2 - 60*(x^5 + 13*x^4 + 67*x^3 + 171*x^2 + 216*x +
108)*log(x + 3) + 60*(x^5 + 13*x^4 + 67*x^3 + 171*x^2 + 216*x + 108)*log(x + 2)
+ 4175*x + 2627)/(x^5 + 13*x^4 + 67*x^3 + 171*x^2 + 216*x + 108)

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Sympy [A]  time = 0.211755, size = 58, normalized size = 1.07 \[ \frac{60 x^{4} + 630 x^{3} + 2450 x^{2} + 4175 x + 2627}{6 x^{5} + 78 x^{4} + 402 x^{3} + 1026 x^{2} + 1296 x + 648} + 10 \log{\left (x + 2 \right )} - 10 \log{\left (x + 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2+x)**3/(3+x)**4,x)

[Out]

(60*x**4 + 630*x**3 + 2450*x**2 + 4175*x + 2627)/(6*x**5 + 78*x**4 + 402*x**3 +
1026*x**2 + 1296*x + 648) + 10*log(x + 2) - 10*log(x + 3)

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GIAC/XCAS [A]  time = 0.212501, size = 63, normalized size = 1.17 \[ \frac{60 \, x^{4} + 630 \, x^{3} + 2450 \, x^{2} + 4175 \, x + 2627}{6 \,{\left (x + 3\right )}^{3}{\left (x + 2\right )}^{2}} - 10 \,{\rm ln}\left ({\left | x + 3 \right |}\right ) + 10 \,{\rm ln}\left ({\left | x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 3)^4*(x + 2)^3),x, algorithm="giac")

[Out]

1/6*(60*x^4 + 630*x^3 + 2450*x^2 + 4175*x + 2627)/((x + 3)^3*(x + 2)^2) - 10*ln(
abs(x + 3)) + 10*ln(abs(x + 2))