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

Optimal. Leaf size=122 \[ -\frac{d-e+f}{36 (x+1)}+\frac{d+e+f}{12 (1-x)}+\frac{d+2 e+4 f}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f)+\frac{1}{108} \log (x+1) (2 d+e-4 f)+\frac{1}{144} \log (x+2) (d-2 e+4 f) \]

[Out]

(d + e + f)/(12*(1 - x)) + (d + 2*e + 4*f)/(36*(2 - x)) - (d - e + f)/(36*(1 + x
)) + ((2*d + 5*e + 8*f)*Log[1 - x])/36 - ((35*d + 58*e + 92*f)*Log[2 - x])/432 +
((2*d + e - 4*f)*Log[1 + x])/108 + ((d - 2*e + 4*f)*Log[2 + x])/144

_______________________________________________________________________________________

Rubi [A]  time = 0.439293, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{d-e+f}{36 (x+1)}+\frac{d+e+f}{12 (1-x)}+\frac{d+2 e+4 f}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f)+\frac{1}{108} \log (x+1) (2 d+e-4 f)+\frac{1}{144} \log (x+2) (d-2 e+4 f) \]

Antiderivative was successfully verified.

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

[Out]

(d + e + f)/(12*(1 - x)) + (d + 2*e + 4*f)/(36*(2 - x)) - (d - e + f)/(36*(1 + x
)) + ((2*d + 5*e + 8*f)*Log[1 - x])/36 - ((35*d + 58*e + 92*f)*Log[2 - x])/432 +
((2*d + e - 4*f)*Log[1 + x])/108 + ((d - 2*e + 4*f)*Log[2 + x])/144

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.0951911, size = 121, normalized size = 0.99 \[ \frac{1}{432} \left (\frac{12 \left (d \left (-5 x^2+6 x+5\right )+e \left (10-4 x^2\right )+2 f \left (-4 x^2+3 x+4\right )\right )}{x^3-2 x^2-x+2}+12 \log (1-x) (2 d+5 e+8 f)-\log (2-x) (35 d+58 e+92 f)+4 \log (x+1) (2 d+e-4 f)+3 \log (x+2) (d-2 e+4 f)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((12*(d*(5 + 6*x - 5*x^2) + e*(10 - 4*x^2) + 2*f*(4 + 3*x - 4*x^2)))/(2 - x - 2*
x^2 + x^3) + 12*(2*d + 5*e + 8*f)*Log[1 - x] - (35*d + 58*e + 92*f)*Log[2 - x] +
4*(2*d + e - 4*f)*Log[1 + x] + 3*(d - 2*e + 4*f)*Log[2 + x])/432

_______________________________________________________________________________________

Maple [A]  time = 0.022, size = 158, normalized size = 1.3 \[{\frac{\ln \left ( 2+x \right ) d}{144}}-{\frac{\ln \left ( 2+x \right ) e}{72}}+{\frac{\ln \left ( 2+x \right ) f}{36}}-{\frac{d}{-12+12\,x}}-{\frac{e}{-12+12\,x}}-{\frac{f}{-12+12\,x}}+{\frac{\ln \left ( -1+x \right ) d}{18}}+{\frac{5\,\ln \left ( -1+x \right ) e}{36}}+{\frac{2\,\ln \left ( -1+x \right ) f}{9}}-{\frac{d}{36+36\,x}}+{\frac{e}{36+36\,x}}-{\frac{f}{36+36\,x}}+{\frac{\ln \left ( 1+x \right ) d}{54}}+{\frac{\ln \left ( 1+x \right ) e}{108}}-{\frac{\ln \left ( 1+x \right ) f}{27}}-{\frac{35\,\ln \left ( x-2 \right ) d}{432}}-{\frac{29\,\ln \left ( x-2 \right ) e}{216}}-{\frac{23\,\ln \left ( x-2 \right ) f}{108}}-{\frac{d}{36\,x-72}}-{\frac{e}{18\,x-36}}-{\frac{f}{9\,x-18}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/144*ln(2+x)*d-1/72*ln(2+x)*e+1/36*ln(2+x)*f-1/12/(-1+x)*d-1/12/(-1+x)*e-1/12/(
-1+x)*f+1/18*ln(-1+x)*d+5/36*ln(-1+x)*e+2/9*ln(-1+x)*f-1/36/(1+x)*d+1/36/(1+x)*e
-1/36/(1+x)*f+1/54*ln(1+x)*d+1/108*ln(1+x)*e-1/27*ln(1+x)*f-35/432*ln(x-2)*d-29/
216*ln(x-2)*e-23/108*ln(x-2)*f-1/36/(x-2)*d-1/18/(x-2)*e-1/9/(x-2)*f

_______________________________________________________________________________________

Maxima [A]  time = 0.705089, size = 146, normalized size = 1.2 \[ \frac{1}{144} \,{\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) + \frac{1}{108} \,{\left (2 \, d + e - 4 \, f\right )} \log \left (x + 1\right ) + \frac{1}{36} \,{\left (2 \, d + 5 \, e + 8 \, f\right )} \log \left (x - 1\right ) - \frac{1}{432} \,{\left (35 \, d + 58 \, e + 92 \, f\right )} \log \left (x - 2\right ) - \frac{{\left (5 \, d + 4 \, e + 8 \, f\right )} x^{2} - 6 \,{\left (d + f\right )} x - 5 \, d - 10 \, e - 8 \, f}{36 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/144*(d - 2*e + 4*f)*log(x + 2) + 1/108*(2*d + e - 4*f)*log(x + 1) + 1/36*(2*d
+ 5*e + 8*f)*log(x - 1) - 1/432*(35*d + 58*e + 92*f)*log(x - 2) - 1/36*((5*d + 4
*e + 8*f)*x^2 - 6*(d + f)*x - 5*d - 10*e - 8*f)/(x^3 - 2*x^2 - x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.367125, size = 360, normalized size = 2.95 \[ -\frac{12 \,{\left (5 \, d + 4 \, e + 8 \, f\right )} x^{2} - 72 \,{\left (d + f\right )} x - 3 \,{\left ({\left (d - 2 \, e + 4 \, f\right )} x^{3} - 2 \,{\left (d - 2 \, e + 4 \, f\right )} x^{2} -{\left (d - 2 \, e + 4 \, f\right )} x + 2 \, d - 4 \, e + 8 \, f\right )} \log \left (x + 2\right ) - 4 \,{\left ({\left (2 \, d + e - 4 \, f\right )} x^{3} - 2 \,{\left (2 \, d + e - 4 \, f\right )} x^{2} -{\left (2 \, d + e - 4 \, f\right )} x + 4 \, d + 2 \, e - 8 \, f\right )} \log \left (x + 1\right ) - 12 \,{\left ({\left (2 \, d + 5 \, e + 8 \, f\right )} x^{3} - 2 \,{\left (2 \, d + 5 \, e + 8 \, f\right )} x^{2} -{\left (2 \, d + 5 \, e + 8 \, f\right )} x + 4 \, d + 10 \, e + 16 \, f\right )} \log \left (x - 1\right ) +{\left ({\left (35 \, d + 58 \, e + 92 \, f\right )} x^{3} - 2 \,{\left (35 \, d + 58 \, e + 92 \, f\right )} x^{2} -{\left (35 \, d + 58 \, e + 92 \, f\right )} x + 70 \, d + 116 \, e + 184 \, f\right )} \log \left (x - 2\right ) - 60 \, d - 120 \, e - 96 \, f}{432 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/432*(12*(5*d + 4*e + 8*f)*x^2 - 72*(d + f)*x - 3*((d - 2*e + 4*f)*x^3 - 2*(d
- 2*e + 4*f)*x^2 - (d - 2*e + 4*f)*x + 2*d - 4*e + 8*f)*log(x + 2) - 4*((2*d + e
- 4*f)*x^3 - 2*(2*d + e - 4*f)*x^2 - (2*d + e - 4*f)*x + 4*d + 2*e - 8*f)*log(x
+ 1) - 12*((2*d + 5*e + 8*f)*x^3 - 2*(2*d + 5*e + 8*f)*x^2 - (2*d + 5*e + 8*f)*
x + 4*d + 10*e + 16*f)*log(x - 1) + ((35*d + 58*e + 92*f)*x^3 - 2*(35*d + 58*e +
92*f)*x^2 - (35*d + 58*e + 92*f)*x + 70*d + 116*e + 184*f)*log(x - 2) - 60*d -
120*e - 96*f)/(x^3 - 2*x^2 - x + 2)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.286379, size = 159, normalized size = 1.3 \[ \frac{1}{144} \,{\left (d + 4 \, f - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{108} \,{\left (2 \, d - 4 \, f + e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{36} \,{\left (2 \, d + 8 \, f + 5 \, e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) - \frac{1}{432} \,{\left (35 \, d + 92 \, f + 58 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{{\left (5 \, d + 8 \, f + 4 \, e\right )} x^{2} - 6 \,{\left (d + f\right )} x - 5 \, d - 8 \, f - 10 \, e}{36 \,{\left (x + 1\right )}{\left (x - 1\right )}{\left (x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/144*(d + 4*f - 2*e)*ln(abs(x + 2)) + 1/108*(2*d - 4*f + e)*ln(abs(x + 1)) + 1/
36*(2*d + 8*f + 5*e)*ln(abs(x - 1)) - 1/432*(35*d + 92*f + 58*e)*ln(abs(x - 2))
- 1/36*((5*d + 8*f + 4*e)*x^2 - 6*(d + f)*x - 5*d - 8*f - 10*e)/((x + 1)*(x - 1)
*(x - 2))