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) + 10*Log[2 + x] - 10*Log[3 + x]

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Rubi [A]  time = 0.0182847, 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, Rules used = {44} \[ \frac{4}{x+2}+\frac{6}{x+3}-\frac{1}{2 (x+2)^2}+\frac{3}{2 (x+3)^2}+\frac{1}{3 (x+3)^3}+10 \log (x+2)-10 \log (x+3) \]

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) + 10*Log[2 + x] - 10*Log[3 + x]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(2+x)^3 (3+x)^4} \, dx &=\int \left (\frac{1}{(2+x)^3}-\frac{4}{(2+x)^2}+\frac{10}{2+x}-\frac{1}{(3+x)^4}-\frac{3}{(3+x)^3}-\frac{6}{(3+x)^2}-\frac{10}{3+x}\right ) \, dx\\ &=-\frac{1}{2 (2+x)^2}+\frac{4}{2+x}+\frac{1}{3 (3+x)^3}+\frac{3}{2 (3+x)^2}+\frac{6}{3+x}+10 \log (2+x)-10 \log (3+x)\\ \end{align*}

Mathematica [A]  time = 0.014764, 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) + 10*Log[2 + x] - 10*Log[3 + x]

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Maple [A]  time = 0., size = 49, normalized size = 0.9 \begin{align*} -{\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 ) \end{align*}

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 = 0.926761, size = 81, normalized size = 1.5 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+x)^3/(3+x)^4,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 [B]  time = 1.7558, size = 306, normalized size = 5.67 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+x)^3/(3+x)^4,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 + 1
3*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 = 0.160245, size = 58, normalized size = 1.07 \begin{align*} \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 )} \end{align*}

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 [A]  time = 1.05064, size = 63, normalized size = 1.17 \begin{align*} \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 \, \log \left ({\left | x + 3 \right |}\right ) + 10 \, \log \left ({\left | x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+x)^3/(3+x)^4,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*log(abs(x + 3)) + 10*log(abs(x +
2))