3.74 \(\int \frac{(d+e x) \left (2-3 x+x^2\right )}{4-5 x^2+x^4} \, dx\)

Optimal. Leaf size=22 \[ (d-e) \log (x+1)-(d-2 e) \log (x+2) \]

[Out]

(d - e)*Log[1 + x] - (d - 2*e)*Log[2 + x]

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Rubi [A]  time = 0.0504856, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ (d-e) \log (x+1)-(d-2 e) \log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)*(2 - 3*x + x^2))/(4 - 5*x^2 + x^4),x]

[Out]

(d - e)*Log[1 + x] - (d - 2*e)*Log[2 + x]

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Rubi in Sympy [A]  time = 8.67422, size = 17, normalized size = 0.77 \[ - \left (d - 2 e\right ) \log{\left (x + 2 \right )} + \left (d - e\right ) \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(x**2-3*x+2)/(x**4-5*x**2+4),x)

[Out]

-(d - 2*e)*log(x + 2) + (d - e)*log(x + 1)

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Mathematica [A]  time = 0.0133759, size = 23, normalized size = 1.05 \[ (d-e) \log (x+1)+(2 e-d) \log (x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)*(2 - 3*x + x^2))/(4 - 5*x^2 + x^4),x]

[Out]

(d - e)*Log[1 + x] + (-d + 2*e)*Log[2 + x]

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Maple [A]  time = 0.008, size = 29, normalized size = 1.3 \[ -\ln \left ( 2+x \right ) d+2\,\ln \left ( 2+x \right ) e+\ln \left ( 1+x \right ) d-\ln \left ( 1+x \right ) e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(x^2-3*x+2)/(x^4-5*x^2+4),x)

[Out]

-ln(2+x)*d+2*ln(2+x)*e+ln(1+x)*d-ln(1+x)*e

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Maxima [A]  time = 0.703036, size = 30, normalized size = 1.36 \[ -{\left (d - 2 \, e\right )} \log \left (x + 2\right ) +{\left (d - e\right )} \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 - 3*x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="maxima")

[Out]

-(d - 2*e)*log(x + 2) + (d - e)*log(x + 1)

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Fricas [A]  time = 0.257359, size = 30, normalized size = 1.36 \[ -{\left (d - 2 \, e\right )} \log \left (x + 2\right ) +{\left (d - e\right )} \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 - 3*x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="fricas")

[Out]

-(d - 2*e)*log(x + 2) + (d - e)*log(x + 1)

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Sympy [A]  time = 0.72595, size = 29, normalized size = 1.32 \[ \left (- d + 2 e\right ) \log{\left (x + \frac{4 d - 6 e}{2 d - 3 e} \right )} + \left (d - e\right ) \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(x**2-3*x+2)/(x**4-5*x**2+4),x)

[Out]

(-d + 2*e)*log(x + (4*d - 6*e)/(2*d - 3*e)) + (d - e)*log(x + 1)

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GIAC/XCAS [A]  time = 0.283366, size = 35, normalized size = 1.59 \[ -{\left (d - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) +{\left (d - e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 - 3*x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="giac")

[Out]

-(d - 2*e)*ln(abs(x + 2)) + (d - e)*ln(abs(x + 1))