3.104 \(\int \frac{-9+3 x-6 x^2+x^3}{(3+x)^2 (4+x)^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{99}{x+3}+\frac{181}{x+4}+264 \log (x+3)-263 \log (x+4) \]

[Out]

99/(3 + x) + 181/(4 + x) + 264*Log[3 + x] - 263*Log[4 + x]

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Rubi [A]  time = 0.0359, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1620} \[ \frac{99}{x+3}+\frac{181}{x+4}+264 \log (x+3)-263 \log (x+4) \]

Antiderivative was successfully verified.

[In]

Int[(-9 + 3*x - 6*x^2 + x^3)/((3 + x)^2*(4 + x)^2),x]

[Out]

99/(3 + x) + 181/(4 + x) + 264*Log[3 + x] - 263*Log[4 + x]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \frac{-9+3 x-6 x^2+x^3}{(3+x)^2 (4+x)^2} \, dx &=\int \left (-\frac{99}{(3+x)^2}+\frac{264}{3+x}-\frac{181}{(4+x)^2}-\frac{263}{4+x}\right ) \, dx\\ &=\frac{99}{3+x}+\frac{181}{4+x}+264 \log (3+x)-263 \log (4+x)\\ \end{align*}

Mathematica [A]  time = 0.015453, size = 27, normalized size = 1. \[ \frac{99}{x+3}+\frac{181}{x+4}+264 \log (x+3)-263 \log (x+4) \]

Antiderivative was successfully verified.

[In]

Integrate[(-9 + 3*x - 6*x^2 + x^3)/((3 + x)^2*(4 + x)^2),x]

[Out]

99/(3 + x) + 181/(4 + x) + 264*Log[3 + x] - 263*Log[4 + x]

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Maple [A]  time = 0.009, size = 28, normalized size = 1. \begin{align*} 99\, \left ( 3+x \right ) ^{-1}+181\, \left ( 4+x \right ) ^{-1}+264\,\ln \left ( 3+x \right ) -263\,\ln \left ( 4+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-6*x^2+3*x-9)/(3+x)^2/(4+x)^2,x)

[Out]

99/(3+x)+181/(4+x)+264*ln(3+x)-263*ln(4+x)

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Maxima [A]  time = 0.9427, size = 39, normalized size = 1.44 \begin{align*} \frac{280 \, x + 939}{x^{2} + 7 \, x + 12} - 263 \, \log \left (x + 4\right ) + 264 \, \log \left (x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-6*x^2+3*x-9)/(3+x)^2/(4+x)^2,x, algorithm="maxima")

[Out]

(280*x + 939)/(x^2 + 7*x + 12) - 263*log(x + 4) + 264*log(x + 3)

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Fricas [A]  time = 2.05338, size = 136, normalized size = 5.04 \begin{align*} -\frac{263 \,{\left (x^{2} + 7 \, x + 12\right )} \log \left (x + 4\right ) - 264 \,{\left (x^{2} + 7 \, x + 12\right )} \log \left (x + 3\right ) - 280 \, x - 939}{x^{2} + 7 \, x + 12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-6*x^2+3*x-9)/(3+x)^2/(4+x)^2,x, algorithm="fricas")

[Out]

-(263*(x^2 + 7*x + 12)*log(x + 4) - 264*(x^2 + 7*x + 12)*log(x + 3) - 280*x - 939)/(x^2 + 7*x + 12)

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Sympy [A]  time = 0.125058, size = 26, normalized size = 0.96 \begin{align*} \frac{280 x + 939}{x^{2} + 7 x + 12} + 264 \log{\left (x + 3 \right )} - 263 \log{\left (x + 4 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-6*x**2+3*x-9)/(3+x)**2/(4+x)**2,x)

[Out]

(280*x + 939)/(x**2 + 7*x + 12) + 264*log(x + 3) - 263*log(x + 4)

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Giac [A]  time = 1.05885, size = 50, normalized size = 1.85 \begin{align*} \frac{181}{x + 4} - \frac{99}{\frac{1}{x + 4} - 1} + \log \left ({\left | x + 4 \right |}\right ) + 264 \, \log \left ({\left | -\frac{1}{x + 4} + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-6*x^2+3*x-9)/(3+x)^2/(4+x)^2,x, algorithm="giac")

[Out]

181/(x + 4) - 99/(1/(x + 4) - 1) + log(abs(x + 4)) + 264*log(abs(-1/(x + 4) + 1))