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

Optimal. Leaf size=141 \[ -\frac{d-e+f-g}{36 (x+1)}+\frac{d+e+f+g}{12 (1-x)}+\frac{d+2 e+4 f+8 g}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f+11 g)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f+136 g)+\frac{1}{108} \log (x+1) (2 d+e-4 f+7 g)+\frac{1}{144} \log (x+2) (d-2 e+4 f-8 g) \]

[Out]

(d + e + f + g)/(12*(1 - x)) + (d + 2*e + 4*f + 8*g)/(36*(2 - x)) - (d - e + f -
g)/(36*(1 + x)) + ((2*d + 5*e + 8*f + 11*g)*Log[1 - x])/36 - ((35*d + 58*e + 92
*f + 136*g)*Log[2 - x])/432 + ((2*d + e - 4*f + 7*g)*Log[1 + x])/108 + ((d - 2*e
+ 4*f - 8*g)*Log[2 + x])/144

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Rubi [A]  time = 0.508023, antiderivative size = 141, 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 \[ -\frac{d-e+f-g}{36 (x+1)}+\frac{d+e+f+g}{12 (1-x)}+\frac{d+2 e+4 f+8 g}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f+11 g)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f+136 g)+\frac{1}{108} \log (x+1) (2 d+e-4 f+7 g)+\frac{1}{144} \log (x+2) (d-2 e+4 f-8 g) \]

Antiderivative was successfully verified.

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

[Out]

(d + e + f + g)/(12*(1 - x)) + (d + 2*e + 4*f + 8*g)/(36*(2 - x)) - (d - e + f -
g)/(36*(1 + x)) + ((2*d + 5*e + 8*f + 11*g)*Log[1 - x])/36 - ((35*d + 58*e + 92
*f + 136*g)*Log[2 - x])/432 + ((2*d + e - 4*f + 7*g)*Log[1 + x])/108 + ((d - 2*e
+ 4*f - 8*g)*Log[2 + x])/144

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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)*(g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.150077, size = 144, normalized size = 1.02 \[ \frac{1}{432} \left (\frac{12 \left (d \left (-5 x^2+6 x+5\right )+2 \left (e \left (5-2 x^2\right )+f \left (-4 x^2+3 x+4\right )+g \left (8-5 x^2\right )\right )\right )}{x^3-2 x^2-x+2}+12 \log (1-x) (2 d+5 e+8 f+11 g)-\log (2-x) (35 d+58 e+92 f+136 g)+4 \log (x+1) (2 d+e-4 f+7 g)+3 \log (x+2) (d-2 e+4 f-8 g)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((12*(d*(5 + 6*x - 5*x^2) + 2*(g*(8 - 5*x^2) + f*(4 + 3*x - 4*x^2) + e*(5 - 2*x^
2))))/(2 - x - 2*x^2 + x^3) + 12*(2*d + 5*e + 8*f + 11*g)*Log[1 - x] - (35*d + 5
8*e + 92*f + 136*g)*Log[2 - x] + 4*(2*d + e - 4*f + 7*g)*Log[1 + x] + 3*(d - 2*e
+ 4*f - 8*g)*Log[2 + x])/432

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Maple [A]  time = 0.024, size = 210, normalized size = 1.5 \[{\frac{g}{36+36\,x}}-{\frac{g}{-12+12\,x}}-{\frac{2\,g}{9\,x-18}}-{\frac{f}{36+36\,x}}-{\frac{d}{36+36\,x}}+{\frac{e}{36+36\,x}}-{\frac{d}{36\,x-72}}-{\frac{e}{18\,x-36}}-{\frac{f}{9\,x-18}}-{\frac{f}{-12+12\,x}}-{\frac{d}{-12+12\,x}}-{\frac{e}{-12+12\,x}}+{\frac{\ln \left ( 1+x \right ) d}{54}}+{\frac{\ln \left ( 1+x \right ) e}{108}}+{\frac{\ln \left ( -1+x \right ) d}{18}}+{\frac{5\,\ln \left ( -1+x \right ) e}{36}}+{\frac{7\,\ln \left ( 1+x \right ) g}{108}}-{\frac{17\,\ln \left ( x-2 \right ) g}{54}}+{\frac{11\,\ln \left ( -1+x \right ) g}{36}}-{\frac{\ln \left ( 2+x \right ) g}{18}}-{\frac{35\,\ln \left ( x-2 \right ) d}{432}}-{\frac{29\,\ln \left ( x-2 \right ) e}{216}}-{\frac{\ln \left ( 2+x \right ) e}{72}}-{\frac{23\,\ln \left ( x-2 \right ) f}{108}}+{\frac{\ln \left ( 2+x \right ) d}{144}}-{\frac{\ln \left ( 1+x \right ) f}{27}}+{\frac{2\,\ln \left ( -1+x \right ) f}{9}}+{\frac{\ln \left ( 2+x \right ) f}{36}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/36/(1+x)*g-1/12/(-1+x)*g-2/9/(x-2)*g-1/36/(1+x)*f-1/36/(1+x)*d+1/36/(1+x)*e-1/
36/(x-2)*d-1/18/(x-2)*e-1/9/(x-2)*f-1/12/(-1+x)*f-1/12/(-1+x)*d-1/12/(-1+x)*e+1/
54*ln(1+x)*d+1/108*ln(1+x)*e+1/18*ln(-1+x)*d+5/36*ln(-1+x)*e+7/108*ln(1+x)*g-17/
54*ln(x-2)*g+11/36*ln(-1+x)*g-1/18*ln(2+x)*g-35/432*ln(x-2)*d-29/216*ln(x-2)*e-1
/72*ln(2+x)*e-23/108*ln(x-2)*f+1/144*ln(2+x)*d-1/27*ln(1+x)*f+2/9*ln(-1+x)*f+1/3
6*ln(2+x)*f

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Maxima [A]  time = 0.715162, size = 170, normalized size = 1.21 \[ \frac{1}{144} \,{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) + \frac{1}{108} \,{\left (2 \, d + e - 4 \, f + 7 \, g\right )} \log \left (x + 1\right ) + \frac{1}{36} \,{\left (2 \, d + 5 \, e + 8 \, f + 11 \, g\right )} \log \left (x - 1\right ) - \frac{1}{432} \,{\left (35 \, d + 58 \, e + 92 \, f + 136 \, g\right )} \log \left (x - 2\right ) - \frac{{\left (5 \, d + 4 \, e + 8 \, f + 10 \, g\right )} x^{2} - 6 \,{\left (d + f\right )} x - 5 \, d - 10 \, e - 8 \, f - 16 \, g}{36 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^3 + 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 - 8*g)*log(x + 2) + 1/108*(2*d + e - 4*f + 7*g)*log(x + 1)
+ 1/36*(2*d + 5*e + 8*f + 11*g)*log(x - 1) - 1/432*(35*d + 58*e + 92*f + 136*g)*
log(x - 2) - 1/36*((5*d + 4*e + 8*f + 10*g)*x^2 - 6*(d + f)*x - 5*d - 10*e - 8*f
- 16*g)/(x^3 - 2*x^2 - x + 2)

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Fricas [A]  time = 0.913155, size = 433, normalized size = 3.07 \[ -\frac{12 \,{\left (5 \, d + 4 \, e + 8 \, f + 10 \, g\right )} x^{2} - 72 \,{\left (d + f\right )} x - 3 \,{\left ({\left (d - 2 \, e + 4 \, f - 8 \, g\right )} x^{3} - 2 \,{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} x^{2} -{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} x + 2 \, d - 4 \, e + 8 \, f - 16 \, g\right )} \log \left (x + 2\right ) - 4 \,{\left ({\left (2 \, d + e - 4 \, f + 7 \, g\right )} x^{3} - 2 \,{\left (2 \, d + e - 4 \, f + 7 \, g\right )} x^{2} -{\left (2 \, d + e - 4 \, f + 7 \, g\right )} x + 4 \, d + 2 \, e - 8 \, f + 14 \, g\right )} \log \left (x + 1\right ) - 12 \,{\left ({\left (2 \, d + 5 \, e + 8 \, f + 11 \, g\right )} x^{3} - 2 \,{\left (2 \, d + 5 \, e + 8 \, f + 11 \, g\right )} x^{2} -{\left (2 \, d + 5 \, e + 8 \, f + 11 \, g\right )} x + 4 \, d + 10 \, e + 16 \, f + 22 \, g\right )} \log \left (x - 1\right ) +{\left ({\left (35 \, d + 58 \, e + 92 \, f + 136 \, g\right )} x^{3} - 2 \,{\left (35 \, d + 58 \, e + 92 \, f + 136 \, g\right )} x^{2} -{\left (35 \, d + 58 \, e + 92 \, f + 136 \, g\right )} x + 70 \, d + 116 \, e + 184 \, f + 272 \, g\right )} \log \left (x - 2\right ) - 60 \, d - 120 \, e - 96 \, f - 192 \, g}{432 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^3 + 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 + 10*g)*x^2 - 72*(d + f)*x - 3*((d - 2*e + 4*f - 8*g
)*x^3 - 2*(d - 2*e + 4*f - 8*g)*x^2 - (d - 2*e + 4*f - 8*g)*x + 2*d - 4*e + 8*f
- 16*g)*log(x + 2) - 4*((2*d + e - 4*f + 7*g)*x^3 - 2*(2*d + e - 4*f + 7*g)*x^2
- (2*d + e - 4*f + 7*g)*x + 4*d + 2*e - 8*f + 14*g)*log(x + 1) - 12*((2*d + 5*e
+ 8*f + 11*g)*x^3 - 2*(2*d + 5*e + 8*f + 11*g)*x^2 - (2*d + 5*e + 8*f + 11*g)*x
+ 4*d + 10*e + 16*f + 22*g)*log(x - 1) + ((35*d + 58*e + 92*f + 136*g)*x^3 - 2*(
35*d + 58*e + 92*f + 136*g)*x^2 - (35*d + 58*e + 92*f + 136*g)*x + 70*d + 116*e
+ 184*f + 272*g)*log(x - 2) - 60*d - 120*e - 96*f - 192*g)/(x^3 - 2*x^2 - x + 2)

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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)*(g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.28452, size = 184, normalized size = 1.3 \[ \frac{1}{144} \,{\left (d + 4 \, f - 8 \, g - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{108} \,{\left (2 \, d - 4 \, f + 7 \, g + e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{36} \,{\left (2 \, d + 8 \, f + 11 \, g + 5 \, e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) - \frac{1}{432} \,{\left (35 \, d + 92 \, f + 136 \, g + 58 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{{\left (5 \, d + 8 \, f + 10 \, g + 4 \, e\right )} x^{2} - 6 \,{\left (d + f\right )} x - 5 \, d - 8 \, f - 16 \, g - 10 \, e}{36 \,{\left (x + 1\right )}{\left (x - 1\right )}{\left (x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/144*(d + 4*f - 8*g - 2*e)*ln(abs(x + 2)) + 1/108*(2*d - 4*f + 7*g + e)*ln(abs(
x + 1)) + 1/36*(2*d + 8*f + 11*g + 5*e)*ln(abs(x - 1)) - 1/432*(35*d + 92*f + 13
6*g + 58*e)*ln(abs(x - 2)) - 1/36*((5*d + 8*f + 10*g + 4*e)*x^2 - 6*(d + f)*x -
5*d - 8*f - 16*g - 10*e)/((x + 1)*(x - 1)*(x - 2))