3.2381 \(\int \frac{5-x}{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=17 \[ \frac{17}{3} \log (3 x+2)-6 \log (x+1) \]

[Out]

-6*Log[1 + x] + (17*Log[2 + 3*x])/3

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Rubi [A]  time = 0.0166199, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{17}{3} \log (3 x+2)-6 \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

-6*Log[1 + x] + (17*Log[2 + 3*x])/3

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Rubi in Sympy [A]  time = 5.32295, size = 15, normalized size = 0.88 \[ - 6 \log{\left (x + 1 \right )} + \frac{17 \log{\left (3 x + 2 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*log(x + 1) + 17*log(3*x + 2)/3

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Mathematica [A]  time = 0.00696123, size = 17, normalized size = 1. \[ \frac{17}{3} \log (3 x+2)-6 \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

-6*Log[1 + x] + (17*Log[2 + 3*x])/3

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Maple [A]  time = 0.007, size = 16, normalized size = 0.9 \[ -6\,\ln \left ( 1+x \right ) +{\frac{17\,\ln \left ( 2+3\,x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(3*x^2+5*x+2),x)

[Out]

-6*ln(1+x)+17/3*ln(2+3*x)

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Maxima [A]  time = 0.686616, size = 20, normalized size = 1.18 \[ \frac{17}{3} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17/3*log(3*x + 2) - 6*log(x + 1)

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Fricas [A]  time = 0.271096, size = 20, normalized size = 1.18 \[ \frac{17}{3} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17/3*log(3*x + 2) - 6*log(x + 1)

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Sympy [A]  time = 0.211859, size = 15, normalized size = 0.88 \[ \frac{17 \log{\left (x + \frac{2}{3} \right )}}{3} - 6 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17*log(x + 2/3)/3 - 6*log(x + 1)

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GIAC/XCAS [A]  time = 0.299223, size = 23, normalized size = 1.35 \[ \frac{17}{3} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - 6 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17/3*ln(abs(3*x + 2)) - 6*ln(abs(x + 1))