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

Optimal. Leaf size=14 \[ (d-2 e) \log (x+2)+e x \]

[Out]

e*x + (d - 2*e)*Log[2 + x]

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Rubi [A]  time = 0.0397892, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ (d-2 e) \log (x+2)+e x \]

Antiderivative was successfully verified.

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

[Out]

e*x + (d - 2*e)*Log[2 + x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \left (d - 2 e\right ) \log{\left (x + 2 \right )} + \int e\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d - 2*e)*log(x + 2) + Integral(e, x)

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Mathematica [A]  time = 0.00731641, size = 16, normalized size = 1.14 \[ (d-2 e) \log (x+2)+e (x+2) \]

Antiderivative was successfully verified.

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

[Out]

e*(2 + x) + (d - 2*e)*Log[2 + x]

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Maple [A]  time = 0.003, size = 18, normalized size = 1.3 \[ ex+\ln \left ( 2+x \right ) d-2\,\ln \left ( 2+x \right ) e \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.708374, size = 19, normalized size = 1.36 \[ e x +{\left (d - 2 \, e\right )} \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*x + (d - 2*e)*log(x + 2)

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Fricas [A]  time = 0.264118, size = 19, normalized size = 1.36 \[ e x +{\left (d - 2 \, e\right )} \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*x + (d - 2*e)*log(x + 2)

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Sympy [A]  time = 1.026, size = 12, normalized size = 0.86 \[ e x + \left (d - 2 e\right ) \log{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*x + (d - 2*e)*log(x + 2)

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GIAC/XCAS [A]  time = 0.280264, size = 23, normalized size = 1.64 \[ x e +{\left (d - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*e + (d - 2*e)*ln(abs(x + 2))