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

Optimal. Leaf size=177 \[ -\frac{d-e+f-g+h-i}{36 (x+1)}+\frac{d+e+f+g+h+i}{12 (1-x)}+\frac{d+2 e+4 f+8 g+16 h+32 i}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f+11 g+14 h+17 i)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f+136 g+176 h+160 i)+\frac{1}{108} \log (x+1) (2 d+e-4 f+7 g-10 h+13 i)+\frac{1}{144} \log (x+2) (d-2 e+4 f-8 g+16 h-32 i) \]

[Out]

(d + e + f + g + h + i)/(12*(1 - x)) + (d + 2*e + 4*f + 8*g + 16*h + 32*i)/(36*(
2 - x)) - (d - e + f - g + h - i)/(36*(1 + x)) + ((2*d + 5*e + 8*f + 11*g + 14*h
+ 17*i)*Log[1 - x])/36 - ((35*d + 58*e + 92*f + 136*g + 176*h + 160*i)*Log[2 -
x])/432 + ((2*d + e - 4*f + 7*g - 10*h + 13*i)*Log[1 + x])/108 + ((d - 2*e + 4*f
- 8*g + 16*h - 32*i)*Log[2 + x])/144

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Rubi [A]  time = 0.724607, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049 \[ -\frac{d-e+f-g+h-i}{36 (x+1)}+\frac{d+e+f+g+h+i}{12 (1-x)}+\frac{d+2 e+4 f+8 g+16 h+32 i}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f+11 g+14 h+17 i)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f+136 g+176 h+160 i)+\frac{1}{108} \log (x+1) (2 d+e-4 f+7 g-10 h+13 i)+\frac{1}{144} \log (x+2) (d-2 e+4 f-8 g+16 h-32 i) \]

Antiderivative was successfully verified.

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

[Out]

(d + e + f + g + h + i)/(12*(1 - x)) + (d + 2*e + 4*f + 8*g + 16*h + 32*i)/(36*(
2 - x)) - (d - e + f - g + h - i)/(36*(1 + x)) + ((2*d + 5*e + 8*f + 11*g + 14*h
+ 17*i)*Log[1 - x])/36 - ((35*d + 58*e + 92*f + 136*g + 176*h + 160*i)*Log[2 -
x])/432 + ((2*d + e - 4*f + 7*g - 10*h + 13*i)*Log[1 + x])/108 + ((d - 2*e + 4*f
- 8*g + 16*h - 32*i)*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)*(i*x**5+h*x**4+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 = 0.225792, size = 195, normalized size = 1.1 \[ \frac{-5 d x^2+6 d x+5 d-4 e x^2+10 e-8 f x^2+6 f x+8 f-10 g x^2+16 g-20 h x^2+6 h x+20 h-34 i x^2+40 i}{36 \left (x^3-2 x^2-x+2\right )}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f+11 g+14 h+17 i)+\frac{1}{432} \log (2-x) (-35 d-58 e-92 f-136 g-176 h-160 i)+\frac{1}{108} \log (x+1) (2 d+e-4 f+7 g-10 h+13 i)+\frac{1}{144} \log (x+2) (d-2 e+4 f-8 g+16 h-32 i) \]

Antiderivative was successfully verified.

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

[Out]

(5*d + 10*e + 8*f + 16*g + 20*h + 40*i + 6*d*x + 6*f*x + 6*h*x - 5*d*x^2 - 4*e*x
^2 - 8*f*x^2 - 10*g*x^2 - 20*h*x^2 - 34*i*x^2)/(36*(2 - x - 2*x^2 + x^3)) + ((2*
d + 5*e + 8*f + 11*g + 14*h + 17*i)*Log[1 - x])/36 + ((-35*d - 58*e - 92*f - 136
*g - 176*h - 160*i)*Log[2 - x])/432 + ((2*d + e - 4*f + 7*g - 10*h + 13*i)*Log[1
+ x])/108 + ((d - 2*e + 4*f - 8*g + 16*h - 32*i)*Log[2 + x])/144

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Maple [A]  time = 0.025, size = 314, normalized size = 1.8 \[ -{\frac{8\,i}{9\,x-18}}+{\frac{i}{36+36\,x}}-{\frac{i}{-12+12\,x}}-{\frac{4\,h}{9\,x-18}}-{\frac{h}{36+36\,x}}-{\frac{h}{-12+12\,x}}+{\frac{g}{36+36\,x}}-{\frac{g}{-12+12\,x}}-{\frac{2\,g}{9\,x-18}}-{\frac{f}{36+36\,x}}-{\frac{d}{36+36\,x}}+{\frac{e}{36+36\,x}}-{\frac{d}{36\,x-72}}-{\frac{e}{18\,x-36}}-{\frac{f}{9\,x-18}}-{\frac{f}{-12+12\,x}}-{\frac{d}{-12+12\,x}}-{\frac{e}{-12+12\,x}}+{\frac{\ln \left ( 1+x \right ) d}{54}}+{\frac{\ln \left ( 1+x \right ) e}{108}}+{\frac{\ln \left ( -1+x \right ) d}{18}}+{\frac{5\,\ln \left ( -1+x \right ) e}{36}}-{\frac{10\,\ln \left ( x-2 \right ) i}{27}}+{\frac{13\,\ln \left ( 1+x \right ) i}{108}}+{\frac{17\,\ln \left ( -1+x \right ) i}{36}}-{\frac{2\,\ln \left ( 2+x \right ) i}{9}}-{\frac{11\,\ln \left ( x-2 \right ) h}{27}}-{\frac{5\,\ln \left ( 1+x \right ) h}{54}}+{\frac{\ln \left ( 2+x \right ) h}{9}}+{\frac{7\,\ln \left ( -1+x \right ) h}{18}}+{\frac{7\,\ln \left ( 1+x \right ) g}{108}}-{\frac{17\,\ln \left ( x-2 \right ) g}{54}}+{\frac{11\,\ln \left ( -1+x \right ) g}{36}}-{\frac{\ln \left ( 2+x \right ) g}{18}}-{\frac{35\,\ln \left ( x-2 \right ) d}{432}}-{\frac{29\,\ln \left ( x-2 \right ) e}{216}}-{\frac{\ln \left ( 2+x \right ) e}{72}}-{\frac{23\,\ln \left ( x-2 \right ) f}{108}}+{\frac{\ln \left ( 2+x \right ) d}{144}}-{\frac{\ln \left ( 1+x \right ) f}{27}}+{\frac{2\,\ln \left ( -1+x \right ) f}{9}}+{\frac{\ln \left ( 2+x \right ) f}{36}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/9/(x-2)*i+1/36/(1+x)*i-1/12/(-1+x)*i-4/9/(x-2)*h-1/36/(1+x)*h-1/12/(-1+x)*h+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-1/3
6/(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+1/5
4*ln(1+x)*d+1/108*ln(1+x)*e+1/18*ln(-1+x)*d+5/36*ln(-1+x)*e-10/27*ln(x-2)*i+13/1
08*ln(1+x)*i+17/36*ln(-1+x)*i-2/9*ln(2+x)*i-11/27*ln(x-2)*h-5/54*ln(1+x)*h+1/9*l
n(2+x)*h+7/18*ln(-1+x)*h+7/108*ln(1+x)*g-17/54*ln(x-2)*g+11/36*ln(-1+x)*g-1/18*l
n(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+1/36*ln(2+x)*f

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Maxima [A]  time = 0.706743, size = 220, normalized size = 1.24 \[ \frac{1}{144} \,{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} \log \left (x + 2\right ) + \frac{1}{108} \,{\left (2 \, d + e - 4 \, f + 7 \, g - 10 \, h + 13 \, i\right )} \log \left (x + 1\right ) + \frac{1}{36} \,{\left (2 \, d + 5 \, e + 8 \, f + 11 \, g + 14 \, h + 17 \, i\right )} \log \left (x - 1\right ) - \frac{1}{432} \,{\left (35 \, d + 58 \, e + 92 \, f + 136 \, g + 176 \, h + 160 \, i\right )} \log \left (x - 2\right ) - \frac{{\left (5 \, d + 4 \, e + 8 \, f + 10 \, g + 20 \, h + 34 \, i\right )} x^{2} - 6 \,{\left (d + f + h\right )} x - 5 \, d - 10 \, e - 8 \, f - 16 \, g - 20 \, h - 40 \, i}{36 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((i*x^5 + h*x^4 + 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 + 16*h - 32*i)*log(x + 2) + 1/108*(2*d + e - 4*f + 7*
g - 10*h + 13*i)*log(x + 1) + 1/36*(2*d + 5*e + 8*f + 11*g + 14*h + 17*i)*log(x
- 1) - 1/432*(35*d + 58*e + 92*f + 136*g + 176*h + 160*i)*log(x - 2) - 1/36*((5*
d + 4*e + 8*f + 10*g + 20*h + 34*i)*x^2 - 6*(d + f + h)*x - 5*d - 10*e - 8*f - 1
6*g - 20*h - 40*i)/(x^3 - 2*x^2 - x + 2)

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Fricas [A]  time = 26.7788, size = 581, normalized size = 3.28 \[ -\frac{12 \,{\left (5 \, d + 4 \, e + 8 \, f + 10 \, g + 20 \, h + 34 \, i\right )} x^{2} - 72 \,{\left (d + f + h\right )} x - 3 \,{\left ({\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} x^{3} - 2 \,{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} x^{2} -{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} x + 2 \, d - 4 \, e + 8 \, f - 16 \, g + 32 \, h - 64 \, i\right )} \log \left (x + 2\right ) - 4 \,{\left ({\left (2 \, d + e - 4 \, f + 7 \, g - 10 \, h + 13 \, i\right )} x^{3} - 2 \,{\left (2 \, d + e - 4 \, f + 7 \, g - 10 \, h + 13 \, i\right )} x^{2} -{\left (2 \, d + e - 4 \, f + 7 \, g - 10 \, h + 13 \, i\right )} x + 4 \, d + 2 \, e - 8 \, f + 14 \, g - 20 \, h + 26 \, i\right )} \log \left (x + 1\right ) - 12 \,{\left ({\left (2 \, d + 5 \, e + 8 \, f + 11 \, g + 14 \, h + 17 \, i\right )} x^{3} - 2 \,{\left (2 \, d + 5 \, e + 8 \, f + 11 \, g + 14 \, h + 17 \, i\right )} x^{2} -{\left (2 \, d + 5 \, e + 8 \, f + 11 \, g + 14 \, h + 17 \, i\right )} x + 4 \, d + 10 \, e + 16 \, f + 22 \, g + 28 \, h + 34 \, i\right )} \log \left (x - 1\right ) +{\left ({\left (35 \, d + 58 \, e + 92 \, f + 136 \, g + 176 \, h + 160 \, i\right )} x^{3} - 2 \,{\left (35 \, d + 58 \, e + 92 \, f + 136 \, g + 176 \, h + 160 \, i\right )} x^{2} -{\left (35 \, d + 58 \, e + 92 \, f + 136 \, g + 176 \, h + 160 \, i\right )} x + 70 \, d + 116 \, e + 184 \, f + 272 \, g + 352 \, h + 320 \, i\right )} \log \left (x - 2\right ) - 60 \, d - 120 \, e - 96 \, f - 192 \, g - 240 \, h - 480 \, i}{432 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((i*x^5 + h*x^4 + 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 + 20*h + 34*i)*x^2 - 72*(d + f + h)*x - 3*((d
- 2*e + 4*f - 8*g + 16*h - 32*i)*x^3 - 2*(d - 2*e + 4*f - 8*g + 16*h - 32*i)*x^
2 - (d - 2*e + 4*f - 8*g + 16*h - 32*i)*x + 2*d - 4*e + 8*f - 16*g + 32*h - 64*i
)*log(x + 2) - 4*((2*d + e - 4*f + 7*g - 10*h + 13*i)*x^3 - 2*(2*d + e - 4*f + 7
*g - 10*h + 13*i)*x^2 - (2*d + e - 4*f + 7*g - 10*h + 13*i)*x + 4*d + 2*e - 8*f
+ 14*g - 20*h + 26*i)*log(x + 1) - 12*((2*d + 5*e + 8*f + 11*g + 14*h + 17*i)*x^
3 - 2*(2*d + 5*e + 8*f + 11*g + 14*h + 17*i)*x^2 - (2*d + 5*e + 8*f + 11*g + 14*
h + 17*i)*x + 4*d + 10*e + 16*f + 22*g + 28*h + 34*i)*log(x - 1) + ((35*d + 58*e
+ 92*f + 136*g + 176*h + 160*i)*x^3 - 2*(35*d + 58*e + 92*f + 136*g + 176*h + 1
60*i)*x^2 - (35*d + 58*e + 92*f + 136*g + 176*h + 160*i)*x + 70*d + 116*e + 184*
f + 272*g + 352*h + 320*i)*log(x - 2) - 60*d - 120*e - 96*f - 192*g - 240*h - 48
0*i)/(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)*(i*x**5+h*x**4+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 = 0.28586, size = 234, normalized size = 1.32 \[ \frac{1}{144} \,{\left (d + 4 \, f - 8 \, g + 16 \, h - 32 \, i - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{108} \,{\left (2 \, d - 4 \, f + 7 \, g - 10 \, h + 13 \, i + e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{36} \,{\left (2 \, d + 8 \, f + 11 \, g + 14 \, h + 17 \, i + 5 \, e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) - \frac{1}{432} \,{\left (35 \, d + 92 \, f + 136 \, g + 176 \, h + 160 \, i + 58 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{{\left (5 \, d + 8 \, f + 10 \, g + 20 \, h + 34 \, i + 4 \, e\right )} x^{2} - 6 \,{\left (d + f + h\right )} x - 5 \, d - 8 \, f - 16 \, g - 20 \, h - 40 \, i - 10 \, e}{36 \,{\left (x + 1\right )}{\left (x - 1\right )}{\left (x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((i*x^5 + h*x^4 + 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 + 16*h - 32*i - 2*e)*ln(abs(x + 2)) + 1/108*(2*d - 4*f + 7*
g - 10*h + 13*i + e)*ln(abs(x + 1)) + 1/36*(2*d + 8*f + 11*g + 14*h + 17*i + 5*e
)*ln(abs(x - 1)) - 1/432*(35*d + 92*f + 136*g + 176*h + 160*i + 58*e)*ln(abs(x -
2)) - 1/36*((5*d + 8*f + 10*g + 20*h + 34*i + 4*e)*x^2 - 6*(d + f + h)*x - 5*d
- 8*f - 16*g - 20*h - 40*i - 10*e)/((x + 1)*(x - 1)*(x - 2))