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

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

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Rubi [A]  time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 1.00 \[ \frac {99}{x+3}+\frac {181}{x+4}+264 \log (x+3)-263 \log (x+4) \]

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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IntegrateAlgebraic [A]  time = 0.03, size = 29, normalized size = 1.07 \[ \frac {280 x+939}{(x+3) (x+4)}+264 \log (x+3)-263 \log (x+4) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(939 + 280*x)/((3 + x)*(4 + x)) + 264*Log[3 + x] - 263*Log[4 + x]

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fricas [A]  time = 0.85, size = 45, normalized size = 1.67 \[ -\frac {263 \, {\left (x^{2} + 7 \, x + 12\right )} \log \left (x + 4\right ) - 264 \, {\left (x^{2} + 7 \, x + 12\right )} \log \left (x + 3\right ) - 280 \, x - 939}{x^{2} + 7 \, x + 12} \]

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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giac [A]  time = 0.86, size = 37, normalized size = 1.37 \[ \frac {181}{x + 4} - \frac {99}{\frac {1}{x + 4} - 1} + \log \left ({\left | x + 4 \right |}\right ) + 264 \, \log \left ({\left | -\frac {1}{x + 4} + 1 \right |}\right ) \]

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))

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maple [A]  time = 0.31, size = 28, normalized size = 1.04

method	result	size
default	\(\frac {99}{3+x}+\frac {181}{4+x}+264 \ln \left (3+x \right )-263 \ln \left (4+x \right )\)	\(28\)
norman	\(\frac {280 x +939}{\left (4+x \right ) \left (3+x \right )}+264 \ln \left (3+x \right )-263 \ln \left (4+x \right )\)	\(30\)
risch	\(\frac {280 x +939}{\left (4+x \right ) \left (3+x \right )}+264 \ln \left (3+x \right )-263 \ln \left (4+x \right )\)	\(30\)

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,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 29, normalized size = 1.07 \[ \frac {280 \, x + 939}{x^{2} + 7 \, x + 12} - 263 \, \log \left (x + 4\right ) + 264 \, \log \left (x + 3\right ) \]

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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mupad [B]  time = 0.19, size = 29, normalized size = 1.07 \[ 264\,\ln \left (x+3\right )-263\,\ln \left (x+4\right )+\frac {280\,x+939}{x^2+7\,x+12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.15, size = 26, normalized size = 0.96 \[ \frac {280 x + 939}{x^{2} + 7 x + 12} + 264 \log {\left (x + 3 \right )} - 263 \log {\left (x + 4 \right )} \]

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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