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

Optimal. Leaf size=42 \[ -\frac{5-6 x}{49 \left (-3 x^2+5 x+2\right )}-\frac{6}{343} \log (2-x)+\frac{6}{343} \log (3 x+1) \]

[Out]

-(5 - 6*x)/(49*(2 + 5*x - 3*x^2)) - (6*Log[2 - x])/343 + (6*Log[1 + 3*x])/343

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Rubi [A]  time = 0.0260591, antiderivative size = 42, 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 \[ -\frac{5-6 x}{49 \left (-3 x^2+5 x+2\right )}-\frac{6}{343} \log (2-x)+\frac{6}{343} \log (3 x+1) \]

Antiderivative was successfully verified.

[In]  Int[(2 + 5*x - 3*x^2)^(-2),x]

[Out]

-(5 - 6*x)/(49*(2 + 5*x - 3*x^2)) - (6*Log[2 - x])/343 + (6*Log[1 + 3*x])/343

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Rubi in Sympy [A]  time = 2.1557, size = 32, normalized size = 0.76 \[ - \frac{- 6 x + 5}{49 \left (- 3 x^{2} + 5 x + 2\right )} - \frac{6 \log{\left (- x + 2 \right )}}{343} + \frac{6 \log{\left (3 x + 1 \right )}}{343} \]

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)/(49*(-3*x**2 + 5*x + 2)) - 6*log(-x + 2)/343 + 6*log(3*x + 1)/343

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Mathematica [A]  time = 0.0231421, size = 42, normalized size = 1. \[ \frac{5-6 x}{49 \left (3 x^2-5 x-2\right )}-\frac{6}{343} \log (2-x)+\frac{6}{343} \log (3 x+1) \]

Antiderivative was successfully verified.

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

[Out]

(5 - 6*x)/(49*(-2 - 5*x + 3*x^2)) - (6*Log[2 - x])/343 + (6*Log[1 + 3*x])/343

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Maple [A]  time = 0.013, size = 32, normalized size = 0.8 \[ -{\frac{3}{147\,x+49}}+{\frac{6\,\ln \left ( 3\,x+1 \right ) }{343}}-{\frac{1}{49\,x-98}}-{\frac{6\,\ln \left ( x-2 \right ) }{343}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^2+5*x+2)^2,x)

[Out]

-3/49/(3*x+1)+6/343*ln(3*x+1)-1/49/(x-2)-6/343*ln(x-2)

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Maxima [A]  time = 0.715969, size = 46, normalized size = 1.1 \[ -\frac{6 \, x - 5}{49 \,{\left (3 \, x^{2} - 5 \, x - 2\right )}} + \frac{6}{343} \, \log \left (3 \, x + 1\right ) - \frac{6}{343} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/49*(6*x - 5)/(3*x^2 - 5*x - 2) + 6/343*log(3*x + 1) - 6/343*log(x - 2)

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Fricas [A]  time = 0.21972, size = 72, normalized size = 1.71 \[ \frac{6 \,{\left (3 \, x^{2} - 5 \, x - 2\right )} \log \left (3 \, x + 1\right ) - 6 \,{\left (3 \, x^{2} - 5 \, x - 2\right )} \log \left (x - 2\right ) - 42 \, x + 35}{343 \,{\left (3 \, x^{2} - 5 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/343*(6*(3*x^2 - 5*x - 2)*log(3*x + 1) - 6*(3*x^2 - 5*x - 2)*log(x - 2) - 42*x
+ 35)/(3*x^2 - 5*x - 2)

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Sympy [A]  time = 0.324176, size = 32, normalized size = 0.76 \[ - \frac{6 x - 5}{147 x^{2} - 245 x - 98} - \frac{6 \log{\left (x - 2 \right )}}{343} + \frac{6 \log{\left (x + \frac{1}{3} \right )}}{343} \]

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)/(147*x**2 - 245*x - 98) - 6*log(x - 2)/343 + 6*log(x + 1/3)/343

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GIAC/XCAS [A]  time = 0.209558, size = 49, normalized size = 1.17 \[ -\frac{6 \, x - 5}{49 \,{\left (3 \, x^{2} - 5 \, x - 2\right )}} + \frac{6}{343} \,{\rm ln}\left ({\left | 3 \, x + 1 \right |}\right ) - \frac{6}{343} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/49*(6*x - 5)/(3*x^2 - 5*x - 2) + 6/343*ln(abs(3*x + 1)) - 6/343*ln(abs(x - 2)
)