3.93 \(\int \frac{1}{\left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=34 \[ -\frac{6 x+5}{3 x^2+5 x+2}+6 \log (x+1)-6 \log (3 x+2) \]

[Out]

-((5 + 6*x)/(2 + 5*x + 3*x^2)) + 6*Log[1 + x] - 6*Log[2 + 3*x]

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Rubi [A]  time = 0.0218507, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{6 x+5}{3 x^2+5 x+2}+6 \log (x+1)-6 \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[(2 + 5*x + 3*x^2)^(-2),x]

[Out]

-((5 + 6*x)/(2 + 5*x + 3*x^2)) + 6*Log[1 + x] - 6*Log[2 + 3*x]

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Rubi in Sympy [A]  time = 1.95132, size = 29, normalized size = 0.85 \[ - \frac{6 x + 5}{3 x^{2} + 5 x + 2} + 6 \log{\left (x + 1 \right )} - 6 \log{\left (3 x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**2+5*x+2)**2,x)

[Out]

-(6*x + 5)/(3*x**2 + 5*x + 2) + 6*log(x + 1) - 6*log(3*x + 2)

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Mathematica [A]  time = 0.0188732, size = 33, normalized size = 0.97 \[ \frac{-6 x-5}{3 x^2+5 x+2}+6 \log (x+1)-6 \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

(-5 - 6*x)/(2 + 5*x + 3*x^2) + 6*Log[1 + x] - 6*Log[2 + 3*x]

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Maple [A]  time = 0.014, size = 32, normalized size = 0.9 \[ -3\, \left ( 2+3\,x \right ) ^{-1}-6\,\ln \left ( 2+3\,x \right ) - \left ( 1+x \right ) ^{-1}+6\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^2+5*x+2)^2,x)

[Out]

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

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Maxima [A]  time = 0.722376, size = 46, normalized size = 1.35 \[ -\frac{6 \, x + 5}{3 \, x^{2} + 5 \, x + 2} - 6 \, \log \left (3 \, x + 2\right ) + 6 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(6*x + 5)/(3*x^2 + 5*x + 2) - 6*log(3*x + 2) + 6*log(x + 1)

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Fricas [A]  time = 0.217436, size = 72, normalized size = 2.12 \[ -\frac{6 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 6 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 6 \, x + 5}{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(6*(3*x^2 + 5*x + 2)*log(3*x + 2) - 6*(3*x^2 + 5*x + 2)*log(x + 1) + 6*x + 5)/(
3*x^2 + 5*x + 2)

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Sympy [A]  time = 0.312059, size = 29, normalized size = 0.85 \[ - \frac{6 x + 5}{3 x^{2} + 5 x + 2} - 6 \log{\left (x + \frac{2}{3} \right )} + 6 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*x**2+5*x+2)**2,x)

[Out]

-(6*x + 5)/(3*x**2 + 5*x + 2) - 6*log(x + 2/3) + 6*log(x + 1)

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GIAC/XCAS [A]  time = 0.209539, size = 49, normalized size = 1.44 \[ -\frac{6 \, x + 5}{3 \, x^{2} + 5 \, x + 2} - 6 \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + 6 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(6*x + 5)/(3*x^2 + 5*x + 2) - 6*ln(abs(3*x + 2)) + 6*ln(abs(x + 1))