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

Optimal. Leaf size=105 \[ -\frac{d-e}{36 (x+1)}+\frac{d+e}{12 (1-x)}+\frac{d+2 e}{36 (2-x)}+\frac{1}{36} (2 d+5 e) \log (1-x)-\frac{1}{432} (35 d+58 e) \log (2-x)+\frac{1}{108} (2 d+e) \log (x+1)+\frac{1}{144} (d-2 e) \log (x+2) \]

[Out]

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

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Rubi [A]  time = 0.386524, antiderivative size = 105, 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 \[ -\frac{d-e}{36 (x+1)}+\frac{d+e}{12 (1-x)}+\frac{d+2 e}{36 (2-x)}+\frac{1}{36} (2 d+5 e) \log (1-x)-\frac{1}{432} (35 d+58 e) \log (2-x)+\frac{1}{108} (2 d+e) \log (x+1)+\frac{1}{144} (d-2 e) \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 0.165186, size = 97, normalized size = 0.92 \[ \frac{1}{432} \left (\frac{12 \left (d \left (-5 x^2+6 x+5\right )+2 e \left (5-2 x^2\right )\right )}{x^3-2 x^2-x+2}+12 (2 d+5 e) \log (1-x)-(35 d+58 e) \log (2-x)+4 (2 d+e) \log (x+1)+3 (d-2 e) \log (x+2)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 106, normalized size = 1. \[{\frac{\ln \left ( 2+x \right ) d}{144}}-{\frac{\ln \left ( 2+x \right ) e}{72}}-{\frac{d}{-12+12\,x}}-{\frac{e}{-12+12\,x}}+{\frac{\ln \left ( -1+x \right ) d}{18}}+{\frac{5\,\ln \left ( -1+x \right ) e}{36}}-{\frac{d}{36+36\,x}}+{\frac{e}{36+36\,x}}+{\frac{\ln \left ( 1+x \right ) d}{54}}+{\frac{\ln \left ( 1+x \right ) e}{108}}-{\frac{35\,\ln \left ( x-2 \right ) d}{432}}-{\frac{29\,\ln \left ( x-2 \right ) e}{216}}-{\frac{d}{36\,x-72}}-{\frac{e}{18\,x-36}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.700365, size = 119, normalized size = 1.13 \[ \frac{1}{144} \,{\left (d - 2 \, e\right )} \log \left (x + 2\right ) + \frac{1}{108} \,{\left (2 \, d + e\right )} \log \left (x + 1\right ) + \frac{1}{36} \,{\left (2 \, d + 5 \, e\right )} \log \left (x - 1\right ) - \frac{1}{432} \,{\left (35 \, d + 58 \, e\right )} \log \left (x - 2\right ) - \frac{{\left (5 \, d + 4 \, e\right )} x^{2} - 6 \, d x - 5 \, d - 10 \, e}{36 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.277462, size = 285, normalized size = 2.71 \[ -\frac{12 \,{\left (5 \, d + 4 \, e\right )} x^{2} - 72 \, d x - 3 \,{\left ({\left (d - 2 \, e\right )} x^{3} - 2 \,{\left (d - 2 \, e\right )} x^{2} -{\left (d - 2 \, e\right )} x + 2 \, d - 4 \, e\right )} \log \left (x + 2\right ) - 4 \,{\left ({\left (2 \, d + e\right )} x^{3} - 2 \,{\left (2 \, d + e\right )} x^{2} -{\left (2 \, d + e\right )} x + 4 \, d + 2 \, e\right )} \log \left (x + 1\right ) - 12 \,{\left ({\left (2 \, d + 5 \, e\right )} x^{3} - 2 \,{\left (2 \, d + 5 \, e\right )} x^{2} -{\left (2 \, d + 5 \, e\right )} x + 4 \, d + 10 \, e\right )} \log \left (x - 1\right ) +{\left ({\left (35 \, d + 58 \, e\right )} x^{3} - 2 \,{\left (35 \, d + 58 \, e\right )} x^{2} -{\left (35 \, d + 58 \, e\right )} x + 70 \, d + 116 \, e\right )} \log \left (x - 2\right ) - 60 \, d - 120 \, e}{432 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 16.4521, size = 1032, normalized size = 9.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d - 2*e)*log(x + (8710660*d**5 + 91884504*d**4*e - 7579779*d**4*(d - 2*e)/4 + 3
64910432*d**3*e**2 - 18128055*d**3*e*(d - 2*e) - 83772*d**3*(d - 2*e)**2 + 68669
7536*d**2*e**3 - 60296868*d**2*e**2*(d - 2*e) - 597816*d**2*e*(d - 2*e)**2 + 659
07*d**2*(d - 2*e)**3/4 + 614357568*d*e**4 - 85949220*d*e**3*(d - 2*e) - 1500048*
d*e**2*(d - 2*e)**2 + 105840*d*e*(d - 2*e)**3 + 208470400*e**5 - 45136356*e**4*(
d - 2*e) - 1196064*e**3*(d - 2*e)**2 + 128277*e**2*(d - 2*e)**3)/(3374210*d**5 +
38645295*d**4*e + 170558380*d**3*e**2 + 362061760*d**2*e**3 + 370298160*d*e**4
+ 146466320*e**5))/144 + (2*d + e)*log(x + (8710660*d**5 + 91884504*d**4*e - 252
6593*d**4*(2*d + e) + 364910432*d**3*e**2 - 24170740*d**3*e*(2*d + e) - 148928*d
**3*(2*d + e)**2 + 686697536*d**2*e**3 - 80395824*d**2*e**2*(2*d + e) - 1062784*
d**2*e*(2*d + e)**2 + 39056*d**2*(2*d + e)**3 + 614357568*d*e**4 - 114598960*d*e
**3*(2*d + e) - 2666752*d*e**2*(2*d + e)**2 + 250880*d*e*(2*d + e)**3 + 20847040
0*e**5 - 60181808*e**4*(2*d + e) - 2126336*e**3*(2*d + e)**2 + 304064*e**2*(2*d
+ e)**3)/(3374210*d**5 + 38645295*d**4*e + 170558380*d**3*e**2 + 362061760*d**2*
e**3 + 370298160*d*e**4 + 146466320*e**5))/108 + (2*d + 5*e)*log(x + (8710660*d*
*5 + 91884504*d**4*e - 7579779*d**4*(2*d + 5*e) + 364910432*d**3*e**2 - 72512220
*d**3*e*(2*d + 5*e) - 1340352*d**3*(2*d + 5*e)**2 + 686697536*d**2*e**3 - 241187
472*d**2*e**2*(2*d + 5*e) - 9565056*d**2*e*(2*d + 5*e)**2 + 1054512*d**2*(2*d +
5*e)**3 + 614357568*d*e**4 - 343796880*d*e**3*(2*d + 5*e) - 24000768*d*e**2*(2*d
+ 5*e)**2 + 6773760*d*e*(2*d + 5*e)**3 + 208470400*e**5 - 180545424*e**4*(2*d +
5*e) - 19137024*e**3*(2*d + 5*e)**2 + 8209728*e**2*(2*d + 5*e)**3)/(3374210*d**
5 + 38645295*d**4*e + 170558380*d**3*e**2 + 362061760*d**2*e**3 + 370298160*d*e*
*4 + 146466320*e**5))/36 - (35*d + 58*e)*log(x + (8710660*d**5 + 91884504*d**4*e
+ 2526593*d**4*(35*d + 58*e)/4 + 364910432*d**3*e**2 + 6042685*d**3*e*(35*d + 5
8*e) - 9308*d**3*(35*d + 58*e)**2 + 686697536*d**2*e**3 + 20098956*d**2*e**2*(35
*d + 58*e) - 66424*d**2*e*(35*d + 58*e)**2 - 2441*d**2*(35*d + 58*e)**3/4 + 6143
57568*d*e**4 + 28649740*d*e**3*(35*d + 58*e) - 166672*d*e**2*(35*d + 58*e)**2 -
3920*d*e*(35*d + 58*e)**3 + 208470400*e**5 + 15045452*e**4*(35*d + 58*e) - 13289
6*e**3*(35*d + 58*e)**2 - 4751*e**2*(35*d + 58*e)**3)/(3374210*d**5 + 38645295*d
**4*e + 170558380*d**3*e**2 + 362061760*d**2*e**3 + 370298160*d*e**4 + 146466320
*e**5))/432 - (-6*d*x - 5*d - 10*e + x**2*(5*d + 4*e))/(36*x**3 - 72*x**2 - 36*x
+ 72)

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GIAC/XCAS [A]  time = 0.286671, size = 132, normalized size = 1.26 \[ \frac{1}{144} \,{\left (d - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{108} \,{\left (2 \, d + e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{36} \,{\left (2 \, d + 5 \, e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) - \frac{1}{432} \,{\left (35 \, d + 58 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{{\left (5 \, d + 4 \, e\right )} x^{2} - 6 \, d x - 5 \, d - 10 \, e}{36 \,{\left (x + 1\right )}{\left (x - 1\right )}{\left (x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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