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

Optimal. Leaf size=89 \[ -\frac{x (3 d-4 e)+5 d-6 e}{12 \left (x^2+3 x+2\right )}-\frac{1}{36} (d+e) \log (1-x)+\frac{1}{144} (d+2 e) \log (2-x)-\frac{1}{36} (7 d-13 e) \log (x+1)+\frac{1}{144} (31 d-50 e) \log (x+2) \]

[Out]

-(5*d - 6*e + (3*d - 4*e)*x)/(12*(2 + 3*x + x^2)) - ((d + e)*Log[1 - x])/36 + ((
d + 2*e)*Log[2 - x])/144 - ((7*d - 13*e)*Log[1 + x])/36 + ((31*d - 50*e)*Log[2 +
x])/144

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Rubi [A]  time = 0.530986, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{x (3 d-4 e)+5 d-6 e}{12 \left (x^2+3 x+2\right )}-\frac{1}{36} (d+e) \log (1-x)+\frac{1}{144} (d+2 e) \log (2-x)-\frac{1}{36} (7 d-13 e) \log (x+1)+\frac{1}{144} (31 d-50 e) \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

-(5*d - 6*e + (3*d - 4*e)*x)/(12*(2 + 3*x + x^2)) - ((d + e)*Log[1 - x])/36 + ((
d + 2*e)*Log[2 - x])/144 - ((7*d - 13*e)*Log[1 + x])/36 + ((31*d - 50*e)*Log[2 +
x])/144

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Rubi in Sympy [A]  time = 48.1095, size = 78, normalized size = 0.88 \[ \left (\frac{d}{144} + \frac{e}{72}\right ) \log{\left (- x + 2 \right )} - \left (\frac{d}{36} + \frac{e}{36}\right ) \log{\left (- x + 1 \right )} - \left (\frac{7 d}{36} - \frac{13 e}{36}\right ) \log{\left (x + 1 \right )} + \left (\frac{31 d}{144} - \frac{25 e}{72}\right ) \log{\left (x + 2 \right )} - \frac{30 d - 36 e + x \left (18 d - 24 e\right )}{72 \left (x^{2} + 3 x + 2\right )} \]

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)**2,x)

[Out]

(d/144 + e/72)*log(-x + 2) - (d/36 + e/36)*log(-x + 1) - (7*d/36 - 13*e/36)*log(
x + 1) + (31*d/144 - 25*e/72)*log(x + 2) - (30*d - 36*e + x*(18*d - 24*e))/(72*(
x**2 + 3*x + 2))

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Mathematica [A]  time = 0.0891434, size = 80, normalized size = 0.9 \[ \frac{1}{144} \left (\frac{12 (-3 d x-5 d+4 e x+6 e)}{x^2+3 x+2}-4 (d+e) \log (1-x)+(d+2 e) \log (2-x)+4 (13 e-7 d) \log (x+1)+(31 d-50 e) \log (x+2)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((12*(-5*d + 6*e - 3*d*x + 4*e*x))/(2 + 3*x + x^2) - 4*(d + e)*Log[1 - x] + (d +
2*e)*Log[2 - x] + 4*(-7*d + 13*e)*Log[1 + x] + (31*d - 50*e)*Log[2 + x])/144

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Maple [A]  time = 0.02, size = 90, normalized size = 1. \[ -{\frac{d}{24+12\,x}}+{\frac{e}{12+6\,x}}+{\frac{31\,\ln \left ( 2+x \right ) d}{144}}-{\frac{25\,\ln \left ( 2+x \right ) e}{72}}-{\frac{\ln \left ( -1+x \right ) d}{36}}-{\frac{\ln \left ( -1+x \right ) e}{36}}-{\frac{7\,\ln \left ( 1+x \right ) d}{36}}+{\frac{13\,\ln \left ( 1+x \right ) e}{36}}-{\frac{d}{6+6\,x}}+{\frac{e}{6+6\,x}}+{\frac{\ln \left ( x-2 \right ) d}{144}}+{\frac{\ln \left ( x-2 \right ) e}{72}} \]

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)^2,x)

[Out]

-1/12/(2+x)*d+1/6/(2+x)*e+31/144*ln(2+x)*d-25/72*ln(2+x)*e-1/36*ln(-1+x)*d-1/36*
ln(-1+x)*e-7/36*ln(1+x)*d+13/36*ln(1+x)*e-1/6/(1+x)*d+1/6/(1+x)*e+1/144*ln(x-2)*
d+1/72*ln(x-2)*e

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Maxima [A]  time = 0.698071, size = 101, normalized size = 1.13 \[ \frac{1}{144} \,{\left (31 \, d - 50 \, e\right )} \log \left (x + 2\right ) - \frac{1}{36} \,{\left (7 \, d - 13 \, e\right )} \log \left (x + 1\right ) - \frac{1}{36} \,{\left (d + e\right )} \log \left (x - 1\right ) + \frac{1}{144} \,{\left (d + 2 \, e\right )} \log \left (x - 2\right ) - \frac{{\left (3 \, d - 4 \, e\right )} x + 5 \, d - 6 \, e}{12 \,{\left (x^{2} + 3 \, x + 2\right )}} \]

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)^2,x, algorithm="maxima")

[Out]

1/144*(31*d - 50*e)*log(x + 2) - 1/36*(7*d - 13*e)*log(x + 1) - 1/36*(d + e)*log
(x - 1) + 1/144*(d + 2*e)*log(x - 2) - 1/12*((3*d - 4*e)*x + 5*d - 6*e)/(x^2 + 3
*x + 2)

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Fricas [A]  time = 0.294017, size = 207, normalized size = 2.33 \[ -\frac{12 \,{\left (3 \, d - 4 \, e\right )} x -{\left ({\left (31 \, d - 50 \, e\right )} x^{2} + 3 \,{\left (31 \, d - 50 \, e\right )} x + 62 \, d - 100 \, e\right )} \log \left (x + 2\right ) + 4 \,{\left ({\left (7 \, d - 13 \, e\right )} x^{2} + 3 \,{\left (7 \, d - 13 \, e\right )} x + 14 \, d - 26 \, e\right )} \log \left (x + 1\right ) + 4 \,{\left ({\left (d + e\right )} x^{2} + 3 \,{\left (d + e\right )} x + 2 \, d + 2 \, e\right )} \log \left (x - 1\right ) -{\left ({\left (d + 2 \, e\right )} x^{2} + 3 \,{\left (d + 2 \, e\right )} x + 2 \, d + 4 \, e\right )} \log \left (x - 2\right ) + 60 \, d - 72 \, e}{144 \,{\left (x^{2} + 3 \, x + 2\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

-1/144*(12*(3*d - 4*e)*x - ((31*d - 50*e)*x^2 + 3*(31*d - 50*e)*x + 62*d - 100*e
)*log(x + 2) + 4*((7*d - 13*e)*x^2 + 3*(7*d - 13*e)*x + 14*d - 26*e)*log(x + 1)
+ 4*((d + e)*x^2 + 3*(d + e)*x + 2*d + 2*e)*log(x - 1) - ((d + 2*e)*x^2 + 3*(d +
2*e)*x + 2*d + 4*e)*log(x - 2) + 60*d - 72*e)/(x^2 + 3*x + 2)

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Sympy [A]  time = 17.0033, size = 1255, normalized size = 14.1 \[ \text{result too large to display} \]

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)**2,x)

[Out]

-(d + e)*log(x + (-24383100*d**6 + 187408066*d**5*e + 10439775*d**5*(d + e) - 51
1591980*d**4*e**2 - 94132290*d**4*e*(d + e) + 667200*d**4*(d + e)**2 + 469491120
*d**3*e**3 + 333672552*d**3*e**2*(d + e) - 2703328*d**3*e*(d + e)**2 - 198000*d*
*3*(d + e)**3 + 322778400*d**2*e**4 - 582497712*d**2*e**3*(d + e) + 1752768*d**2
*e**2*(d + e)**2 + 1107552*d**2*e*(d + e)**3 - 863493856*d*e**5 + 500776560*d*e*
*4*(d + e) + 4226944*d*e**3*(d + e)**2 - 1880640*d*e**2*(d + e)**3 + 429000000*e
**6 - 169242912*e**5*(d + e) - 4538112*e**4*(d + e)**2 + 964224*e**3*(d + e)**3)
/(13474125*d**6 - 102860175*d**5*e + 274190390*d**4*e**2 - 224142072*d**3*e**3 -
245084096*d**2*e**4 + 535797456*d*e**5 - 256183200*e**6))/36 + (d + 2*e)*log(x
+ (-24383100*d**6 + 187408066*d**5*e - 10439775*d**5*(d + 2*e)/4 - 511591980*d**
4*e**2 + 47066145*d**4*e*(d + 2*e)/2 + 41700*d**4*(d + 2*e)**2 + 469491120*d**3*
e**3 - 83418138*d**3*e**2*(d + 2*e) - 168958*d**3*e*(d + 2*e)**2 + 12375*d**3*(d
+ 2*e)**3/4 + 322778400*d**2*e**4 + 145624428*d**2*e**3*(d + 2*e) + 109548*d**2
*e**2*(d + 2*e)**2 - 34611*d**2*e*(d + 2*e)**3/2 - 863493856*d*e**5 - 125194140*
d*e**4*(d + 2*e) + 264184*d*e**3*(d + 2*e)**2 + 29385*d*e**2*(d + 2*e)**3 + 4290
00000*e**6 + 42310728*e**5*(d + 2*e) - 283632*e**4*(d + 2*e)**2 - 15066*e**3*(d
+ 2*e)**3)/(13474125*d**6 - 102860175*d**5*e + 274190390*d**4*e**2 - 224142072*d
**3*e**3 - 245084096*d**2*e**4 + 535797456*d*e**5 - 256183200*e**6))/144 - (7*d
- 13*e)*log(x + (-24383100*d**6 + 187408066*d**5*e + 10439775*d**5*(7*d - 13*e)
- 511591980*d**4*e**2 - 94132290*d**4*e*(7*d - 13*e) + 667200*d**4*(7*d - 13*e)*
*2 + 469491120*d**3*e**3 + 333672552*d**3*e**2*(7*d - 13*e) - 2703328*d**3*e*(7*
d - 13*e)**2 - 198000*d**3*(7*d - 13*e)**3 + 322778400*d**2*e**4 - 582497712*d**
2*e**3*(7*d - 13*e) + 1752768*d**2*e**2*(7*d - 13*e)**2 + 1107552*d**2*e*(7*d -
13*e)**3 - 863493856*d*e**5 + 500776560*d*e**4*(7*d - 13*e) + 4226944*d*e**3*(7*
d - 13*e)**2 - 1880640*d*e**2*(7*d - 13*e)**3 + 429000000*e**6 - 169242912*e**5*
(7*d - 13*e) - 4538112*e**4*(7*d - 13*e)**2 + 964224*e**3*(7*d - 13*e)**3)/(1347
4125*d**6 - 102860175*d**5*e + 274190390*d**4*e**2 - 224142072*d**3*e**3 - 24508
4096*d**2*e**4 + 535797456*d*e**5 - 256183200*e**6))/36 + (31*d - 50*e)*log(x +
(-24383100*d**6 + 187408066*d**5*e - 10439775*d**5*(31*d - 50*e)/4 - 511591980*d
**4*e**2 + 47066145*d**4*e*(31*d - 50*e)/2 + 41700*d**4*(31*d - 50*e)**2 + 46949
1120*d**3*e**3 - 83418138*d**3*e**2*(31*d - 50*e) - 168958*d**3*e*(31*d - 50*e)*
*2 + 12375*d**3*(31*d - 50*e)**3/4 + 322778400*d**2*e**4 + 145624428*d**2*e**3*(
31*d - 50*e) + 109548*d**2*e**2*(31*d - 50*e)**2 - 34611*d**2*e*(31*d - 50*e)**3
/2 - 863493856*d*e**5 - 125194140*d*e**4*(31*d - 50*e) + 264184*d*e**3*(31*d - 5
0*e)**2 + 29385*d*e**2*(31*d - 50*e)**3 + 429000000*e**6 + 42310728*e**5*(31*d -
50*e) - 283632*e**4*(31*d - 50*e)**2 - 15066*e**3*(31*d - 50*e)**3)/(13474125*d
**6 - 102860175*d**5*e + 274190390*d**4*e**2 - 224142072*d**3*e**3 - 245084096*d
**2*e**4 + 535797456*d*e**5 - 256183200*e**6))/144 - (5*d - 6*e + x*(3*d - 4*e))
/(12*x**2 + 36*x + 24)

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GIAC/XCAS [A]  time = 0.286233, size = 115, normalized size = 1.29 \[ \frac{1}{144} \,{\left (31 \, d - 50 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) - \frac{1}{36} \,{\left (7 \, d - 13 \, e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{36} \,{\left (d + e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{144} \,{\left (d + 2 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{{\left (3 \, d - 4 \, e\right )} x + 5 \, d - 6 \, e}{12 \,{\left (x + 2\right )}{\left (x + 1\right )}} \]

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)^2,x, algorithm="giac")

[Out]

1/144*(31*d - 50*e)*ln(abs(x + 2)) - 1/36*(7*d - 13*e)*ln(abs(x + 1)) - 1/36*(d
+ e)*ln(abs(x - 1)) + 1/144*(d + 2*e)*ln(abs(x - 2)) - 1/12*((3*d - 4*e)*x + 5*d
- 6*e)/((x + 2)*(x + 1))