3.80 \(\int \frac{(2+x) (d+e x)}{4-5 x^2+x^4} \, dx\)

Optimal. Leaf size=42 \[ -\frac{1}{2} (d+e) \log (1-x)+\frac{1}{3} (d+2 e) \log (2-x)+\frac{1}{6} (d-e) \log (x+1) \]

[Out]

-((d + e)*Log[1 - x])/2 + ((d + 2*e)*Log[2 - x])/3 + ((d - e)*Log[1 + x])/6

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Rubi [A]  time = 0.0984844, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{1}{2} (d+e) \log (1-x)+\frac{1}{3} (d+2 e) \log (2-x)+\frac{1}{6} (d-e) \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

-((d + e)*Log[1 - x])/2 + ((d + 2*e)*Log[2 - x])/3 + ((d - e)*Log[1 + x])/6

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Rubi in Sympy [A]  time = 23.5605, size = 36, normalized size = 0.86 \[ \left (\frac{d}{6} - \frac{e}{6}\right ) \log{\left (x + 1 \right )} + \left (\frac{d}{3} + \frac{2 e}{3}\right ) \log{\left (- x + 2 \right )} - \left (\frac{d}{2} + \frac{e}{2}\right ) \log{\left (- x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0282561, size = 39, normalized size = 0.93 \[ \frac{1}{6} (-3 (d+e) \log (1-x)+2 (d+2 e) \log (2-x)+(d-e) \log (x+1)) \]

Antiderivative was successfully verified.

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

[Out]

(-3*(d + e)*Log[1 - x] + 2*(d + 2*e)*Log[2 - x] + (d - e)*Log[1 + x])/6

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Maple [A]  time = 0.009, size = 44, normalized size = 1.1 \[ -{\frac{\ln \left ( -1+x \right ) d}{2}}-{\frac{\ln \left ( -1+x \right ) e}{2}}+{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}+{\frac{\ln \left ( x-2 \right ) d}{3}}+{\frac{2\,\ln \left ( x-2 \right ) e}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.699116, size = 43, normalized size = 1.02 \[ \frac{1}{6} \,{\left (d - e\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e\right )} \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.297686, size = 43, normalized size = 1.02 \[ \frac{1}{6} \,{\left (d - e\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e\right )} \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.68231, size = 304, normalized size = 7.24 \[ \frac{\left (d - e\right ) \log{\left (x + \frac{26 d^{3} + 66 d^{2} e - 9 d^{2} \left (d - e\right ) + 78 d e^{2} - 12 d e \left (d - e\right ) - 7 d \left (d - e\right )^{2} + 46 e^{3} + 3 e^{2} \left (d - e\right ) - 8 e \left (d - e\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d e^{2} + 35 e^{3}} \right )}}{6} - \frac{\left (d + e\right ) \log{\left (x + \frac{26 d^{3} + 66 d^{2} e + 27 d^{2} \left (d + e\right ) + 78 d e^{2} + 36 d e \left (d + e\right ) - 63 d \left (d + e\right )^{2} + 46 e^{3} - 9 e^{2} \left (d + e\right ) - 72 e \left (d + e\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d e^{2} + 35 e^{3}} \right )}}{2} + \frac{\left (d + 2 e\right ) \log{\left (x + \frac{26 d^{3} + 66 d^{2} e - 18 d^{2} \left (d + 2 e\right ) + 78 d e^{2} - 24 d e \left (d + 2 e\right ) - 28 d \left (d + 2 e\right )^{2} + 46 e^{3} + 6 e^{2} \left (d + 2 e\right ) - 32 e \left (d + 2 e\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d e^{2} + 35 e^{3}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d - e)*log(x + (26*d**3 + 66*d**2*e - 9*d**2*(d - e) + 78*d*e**2 - 12*d*e*(d -
e) - 7*d*(d - e)**2 + 46*e**3 + 3*e**2*(d - e) - 8*e*(d - e)**2)/(10*d**3 + 69*d
**2*e + 102*d*e**2 + 35*e**3))/6 - (d + e)*log(x + (26*d**3 + 66*d**2*e + 27*d**
2*(d + e) + 78*d*e**2 + 36*d*e*(d + e) - 63*d*(d + e)**2 + 46*e**3 - 9*e**2*(d +
e) - 72*e*(d + e)**2)/(10*d**3 + 69*d**2*e + 102*d*e**2 + 35*e**3))/2 + (d + 2*
e)*log(x + (26*d**3 + 66*d**2*e - 18*d**2*(d + 2*e) + 78*d*e**2 - 24*d*e*(d + 2*
e) - 28*d*(d + 2*e)**2 + 46*e**3 + 6*e**2*(d + 2*e) - 32*e*(d + 2*e)**2)/(10*d**
3 + 69*d**2*e + 102*d*e**2 + 35*e**3))/3

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GIAC/XCAS [A]  time = 0.284256, size = 51, normalized size = 1.21 \[ \frac{1}{6} \,{\left (d - e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\left (d + e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{3} \,{\left (d + 2 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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