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

Optimal. Leaf size=131 \[ -\frac{d-e+f-g+h}{6 (x+1)}-\frac{d-2 e+4 f-8 g+16 h}{12 (x+2)}-\frac{1}{36} \log (1-x) (d+e+f+g+h)+\frac{1}{144} \log (2-x) (d+2 e+4 f+8 g+16 h)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f-25 g+31 h)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f-104 g+112 h) \]

[Out]

-(d - e + f - g + h)/(6*(1 + x)) - (d - 2*e + 4*f - 8*g + 16*h)/(12*(2 + x)) - (
(d + e + f + g + h)*Log[1 - x])/36 + ((d + 2*e + 4*f + 8*g + 16*h)*Log[2 - x])/1
44 - ((7*d - 13*e + 19*f - 25*g + 31*h)*Log[1 + x])/36 + ((31*d - 50*e + 76*f -
104*g + 112*h)*Log[2 + x])/144

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Rubi [A]  time = 0.559654, antiderivative size = 131, 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 \[ -\frac{d-e+f-g+h}{6 (x+1)}-\frac{d-2 e+4 f-8 g+16 h}{12 (x+2)}-\frac{1}{36} \log (1-x) (d+e+f+g+h)+\frac{1}{144} \log (2-x) (d+2 e+4 f+8 g+16 h)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f-25 g+31 h)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f-104 g+112 h) \]

Antiderivative was successfully verified.

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

[Out]

-(d - e + f - g + h)/(6*(1 + x)) - (d - 2*e + 4*f - 8*g + 16*h)/(12*(2 + x)) - (
(d + e + f + g + h)*Log[1 - x])/36 + ((d + 2*e + 4*f + 8*g + 16*h)*Log[2 - x])/1
44 - ((7*d - 13*e + 19*f - 25*g + 31*h)*Log[1 + x])/36 + ((31*d - 50*e + 76*f -
104*g + 112*h)*Log[2 + x])/144

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Rubi in Sympy [A]  time = 137.068, size = 148, normalized size = 1.13 \[ \left (\frac{d}{144} + \frac{e}{72} + \frac{f}{36} + \frac{g}{18} + \frac{h}{9}\right ) \log{\left (- x + 2 \right )} - \left (\frac{d}{36} + \frac{e}{36} + \frac{f}{36} + \frac{g}{36} + \frac{h}{36}\right ) \log{\left (- x + 1 \right )} - \left (\frac{7 d}{36} - \frac{13 e}{36} + \frac{19 f}{36} - \frac{25 g}{36} + \frac{31 h}{36}\right ) \log{\left (x + 1 \right )} + \left (\frac{31 d}{144} - \frac{25 e}{72} + \frac{19 f}{36} - \frac{13 g}{18} + \frac{7 h}{9}\right ) \log{\left (x + 2 \right )} - \frac{\frac{d}{12} - \frac{e}{6} + \frac{f}{3} - \frac{2 g}{3} + \frac{4 h}{3}}{x + 2} - \frac{\frac{d}{6} - \frac{e}{6} + \frac{f}{6} - \frac{g}{6} + \frac{h}{6}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d/144 + e/72 + f/36 + g/18 + h/9)*log(-x + 2) - (d/36 + e/36 + f/36 + g/36 + h/
36)*log(-x + 1) - (7*d/36 - 13*e/36 + 19*f/36 - 25*g/36 + 31*h/36)*log(x + 1) +
(31*d/144 - 25*e/72 + 19*f/36 - 13*g/18 + 7*h/9)*log(x + 2) - (d/12 - e/6 + f/3
- 2*g/3 + 4*h/3)/(x + 2) - (d/6 - e/6 + f/6 - g/6 + h/6)/(x + 1)

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Mathematica [A]  time = 0.12316, size = 136, normalized size = 1.04 \[ \frac{1}{144} \left (-\frac{12 (d (3 x+5)+2 (-e (2 x+3)+3 f x+4 f-5 g x-6 g+9 h x+10 h))}{x^2+3 x+2}-4 \log (1-x) (d+e+f+g+h)+\log (2-x) (d+2 (e+2 f+4 g+8 h))-4 \log (x+1) (7 d-13 e+19 f-25 g+31 h)+\log (x+2) (31 d-50 e+76 f-104 g+112 h)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-12*(d*(5 + 3*x) + 2*(4*f - 6*g + 10*h + 3*f*x - 5*g*x + 9*h*x - e*(3 + 2*x)))
)/(2 + 3*x + x^2) - 4*(d + e + f + g + h)*Log[1 - x] + (d + 2*(e + 2*f + 4*g + 8
*h))*Log[2 - x] - 4*(7*d - 13*e + 19*f - 25*g + 31*h)*Log[1 + x] + (31*d - 50*e
+ 76*f - 104*g + 112*h)*Log[2 + x])/144

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Maple [A]  time = 0.021, size = 222, normalized size = 1.7 \[ -{\frac{h}{6+6\,x}}-{\frac{4\,h}{6+3\,x}}+{\frac{g}{6+6\,x}}+{\frac{2\,g}{6+3\,x}}-{\frac{f}{6+6\,x}}-{\frac{d}{6+6\,x}}+{\frac{e}{6+6\,x}}-{\frac{d}{24+12\,x}}+{\frac{e}{12+6\,x}}-{\frac{f}{6+3\,x}}-{\frac{7\,\ln \left ( 1+x \right ) d}{36}}+{\frac{13\,\ln \left ( 1+x \right ) e}{36}}-{\frac{\ln \left ( -1+x \right ) d}{36}}-{\frac{\ln \left ( -1+x \right ) e}{36}}+{\frac{\ln \left ( x-2 \right ) h}{9}}-{\frac{31\,\ln \left ( 1+x \right ) h}{36}}+{\frac{7\,\ln \left ( 2+x \right ) h}{9}}-{\frac{\ln \left ( -1+x \right ) h}{36}}+{\frac{25\,\ln \left ( 1+x \right ) g}{36}}+{\frac{\ln \left ( x-2 \right ) g}{18}}-{\frac{\ln \left ( -1+x \right ) g}{36}}-{\frac{13\,\ln \left ( 2+x \right ) g}{18}}+{\frac{\ln \left ( x-2 \right ) d}{144}}+{\frac{\ln \left ( x-2 \right ) e}{72}}-{\frac{25\,\ln \left ( 2+x \right ) e}{72}}+{\frac{\ln \left ( x-2 \right ) f}{36}}+{\frac{31\,\ln \left ( 2+x \right ) d}{144}}-{\frac{19\,\ln \left ( 1+x \right ) f}{36}}-{\frac{\ln \left ( -1+x \right ) f}{36}}+{\frac{19\,\ln \left ( 2+x \right ) f}{36}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/(1+x)*h-4/3/(2+x)*h+1/6/(1+x)*g+2/3/(2+x)*g-1/6/(1+x)*f-1/6/(1+x)*d+1/6/(1+
x)*e-1/12/(2+x)*d+1/6/(2+x)*e-1/3/(2+x)*f-7/36*ln(1+x)*d+13/36*ln(1+x)*e-1/36*ln
(-1+x)*d-1/36*ln(-1+x)*e+1/9*ln(x-2)*h-31/36*ln(1+x)*h+7/9*ln(2+x)*h-1/36*ln(-1+
x)*h+25/36*ln(1+x)*g+1/18*ln(x-2)*g-1/36*ln(-1+x)*g-13/18*ln(2+x)*g+1/144*ln(x-2
)*d+1/72*ln(x-2)*e-25/72*ln(2+x)*e+1/36*ln(x-2)*f+31/144*ln(2+x)*d-19/36*ln(1+x)
*f-1/36*ln(-1+x)*f+19/36*ln(2+x)*f

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Maxima [A]  time = 0.698622, size = 166, normalized size = 1.27 \[ \frac{1}{144} \,{\left (31 \, d - 50 \, e + 76 \, f - 104 \, g + 112 \, h\right )} \log \left (x + 2\right ) - \frac{1}{36} \,{\left (7 \, d - 13 \, e + 19 \, f - 25 \, g + 31 \, h\right )} \log \left (x + 1\right ) - \frac{1}{36} \,{\left (d + e + f + g + h\right )} \log \left (x - 1\right ) + \frac{1}{144} \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} \log \left (x - 2\right ) - \frac{{\left (3 \, d - 4 \, e + 6 \, f - 10 \, g + 18 \, h\right )} x + 5 \, d - 6 \, e + 8 \, f - 12 \, g + 20 \, h}{12 \,{\left (x^{2} + 3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^4 + g*x^3 + f*x^2 + 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 + 76*f - 104*g + 112*h)*log(x + 2) - 1/36*(7*d - 13*e + 19*f
- 25*g + 31*h)*log(x + 1) - 1/36*(d + e + f + g + h)*log(x - 1) + 1/144*(d + 2*e
+ 4*f + 8*g + 16*h)*log(x - 2) - 1/12*((3*d - 4*e + 6*f - 10*g + 18*h)*x + 5*d
- 6*e + 8*f - 12*g + 20*h)/(x^2 + 3*x + 2)

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Fricas [A]  time = 4.62455, size = 360, normalized size = 2.75 \[ -\frac{12 \,{\left (3 \, d - 4 \, e + 6 \, f - 10 \, g + 18 \, h\right )} x -{\left ({\left (31 \, d - 50 \, e + 76 \, f - 104 \, g + 112 \, h\right )} x^{2} + 3 \,{\left (31 \, d - 50 \, e + 76 \, f - 104 \, g + 112 \, h\right )} x + 62 \, d - 100 \, e + 152 \, f - 208 \, g + 224 \, h\right )} \log \left (x + 2\right ) + 4 \,{\left ({\left (7 \, d - 13 \, e + 19 \, f - 25 \, g + 31 \, h\right )} x^{2} + 3 \,{\left (7 \, d - 13 \, e + 19 \, f - 25 \, g + 31 \, h\right )} x + 14 \, d - 26 \, e + 38 \, f - 50 \, g + 62 \, h\right )} \log \left (x + 1\right ) + 4 \,{\left ({\left (d + e + f + g + h\right )} x^{2} + 3 \,{\left (d + e + f + g + h\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g + 2 \, h\right )} \log \left (x - 1\right ) -{\left ({\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} x^{2} + 3 \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g + 32 \, h\right )} \log \left (x - 2\right ) + 60 \, d - 72 \, e + 96 \, f - 144 \, g + 240 \, h}{144 \,{\left (x^{2} + 3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^4 + g*x^3 + f*x^2 + 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 + 6*f - 10*g + 18*h)*x - ((31*d - 50*e + 76*f - 104*g + 11
2*h)*x^2 + 3*(31*d - 50*e + 76*f - 104*g + 112*h)*x + 62*d - 100*e + 152*f - 208
*g + 224*h)*log(x + 2) + 4*((7*d - 13*e + 19*f - 25*g + 31*h)*x^2 + 3*(7*d - 13*
e + 19*f - 25*g + 31*h)*x + 14*d - 26*e + 38*f - 50*g + 62*h)*log(x + 1) + 4*((d
+ e + f + g + h)*x^2 + 3*(d + e + f + g + h)*x + 2*d + 2*e + 2*f + 2*g + 2*h)*l
og(x - 1) - ((d + 2*e + 4*f + 8*g + 16*h)*x^2 + 3*(d + 2*e + 4*f + 8*g + 16*h)*x
+ 2*d + 4*e + 8*f + 16*g + 32*h)*log(x - 2) + 60*d - 72*e + 96*f - 144*g + 240*
h)/(x^2 + 3*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((x**2-3*x+2)*(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 = 0.283873, size = 180, normalized size = 1.37 \[ \frac{1}{144} \,{\left (31 \, d + 76 \, f - 104 \, g + 112 \, h - 50 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) - \frac{1}{36} \,{\left (7 \, d + 19 \, f - 25 \, g + 31 \, h - 13 \, e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{36} \,{\left (d + f + g + h + e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{144} \,{\left (d + 4 \, f + 8 \, g + 16 \, h + 2 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) - \frac{{\left (3 \, d + 6 \, f - 10 \, g + 18 \, h - 4 \, e\right )} x + 5 \, d + 8 \, f - 12 \, g + 20 \, h - 6 \, e}{12 \,{\left (x + 2\right )}{\left (x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/144*(31*d + 76*f - 104*g + 112*h - 50*e)*ln(abs(x + 2)) - 1/36*(7*d + 19*f - 2
5*g + 31*h - 13*e)*ln(abs(x + 1)) - 1/36*(d + f + g + h + e)*ln(abs(x - 1)) + 1/
144*(d + 4*f + 8*g + 16*h + 2*e)*ln(abs(x - 2)) - 1/12*((3*d + 6*f - 10*g + 18*h
- 4*e)*x + 5*d + 8*f - 12*g + 20*h - 6*e)/((x + 2)*(x + 1))