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

Optimal. Leaf size=14 \[ \frac{1}{x}+2 \log (x)+3 \log (x+2) \]

[Out]

x^(-1) + 2*Log[x] + 3*Log[2 + x]

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Rubi [A]  time = 0.0420218, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{x}+2 \log (x)+3 \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

x^(-1) + 2*Log[x] + 3*Log[2 + x]

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Rubi in Sympy [A]  time = 3.16199, size = 14, normalized size = 1. \[ 2 \log{\left (x \right )} + 3 \log{\left (x + 2 \right )} + \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(x) + 3*log(x + 2) + 1/x

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Mathematica [A]  time = 0.00573666, size = 14, normalized size = 1. \[ \frac{1}{x}+2 \log (x)+3 \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

x^(-1) + 2*Log[x] + 3*Log[2 + x]

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Maple [A]  time = 0.01, size = 15, normalized size = 1.1 \[{x}^{-1}+2\,\ln \left ( x \right ) +3\,\ln \left ( 2+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37504, size = 19, normalized size = 1.36 \[ \frac{1}{x} + 3 \, \log \left (x + 2\right ) + 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/x + 3*log(x + 2) + 2*log(x)

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Fricas [A]  time = 0.205905, size = 24, normalized size = 1.71 \[ \frac{3 \, x \log \left (x + 2\right ) + 2 \, x \log \left (x\right ) + 1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*x*log(x + 2) + 2*x*log(x) + 1)/x

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Sympy [A]  time = 0.115328, size = 14, normalized size = 1. \[ 2 \log{\left (x \right )} + 3 \log{\left (x + 2 \right )} + \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(x) + 3*log(x + 2) + 1/x

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GIAC/XCAS [A]  time = 0.211322, size = 22, normalized size = 1.57 \[ \frac{1}{x} + 3 \,{\rm ln}\left ({\left | x + 2 \right |}\right ) + 2 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/x + 3*ln(abs(x + 2)) + 2*ln(abs(x))