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

Optimal. Leaf size=122 \[ \frac{d-2 e+4 f-8 g+16 h-32 i}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f+g+h+i)+\frac{1}{48} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac{1}{6} \log (x+1) (d-e+f-g+h-i)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f-8 g-80 h+352 i)+i x \]

[Out]

i*x + (d - 2*e + 4*f - 8*g + 16*h - 32*i)/(12*(2 + x)) - ((d + e + f + g + h + i
)*Log[1 - x])/18 + ((d + 2*e + 4*f + 8*g + 16*h + 32*i)*Log[2 - x])/48 + ((d - e
+ f - g + h - i)*Log[1 + x])/6 - ((19*d - 26*e + 28*f - 8*g - 80*h + 352*i)*Log
[2 + x])/144

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Rubi [A]  time = 0.603296, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.039 \[ \frac{d-2 e+4 f-8 g+16 h-32 i}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f+g+h+i)+\frac{1}{48} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac{1}{6} \log (x+1) (d-e+f-g+h-i)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f-8 g-80 h+352 i)+i x \]

Antiderivative was successfully verified.

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

[Out]

i*x + (d - 2*e + 4*f - 8*g + 16*h - 32*i)/(12*(2 + x)) - ((d + e + f + g + h + i
)*Log[1 - x])/18 + ((d + 2*e + 4*f + 8*g + 16*h + 32*i)*Log[2 - x])/48 + ((d - e
+ f - g + h - i)*Log[1 + x])/6 - ((19*d - 26*e + 28*f - 8*g - 80*h + 352*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((x**3-2*x**2-x+2)*(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 = 0.138592, size = 118, normalized size = 0.97 \[ \frac{1}{144} \left (\frac{12 (d-2 (e-2 f+4 g-8 h+16 i))}{x+2}-8 \log (1-x) (d+e+f+g+h+i)+3 \log (2-x) (d+2 e+4 (f+2 g+4 h+8 i))+24 \log (x+1) (d-e+f-g+h-i)+\log (x+2) (-19 d+26 e-28 f+8 g+80 h-352 i)+144 i x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(144*i*x + (12*(d - 2*(e - 2*f + 4*g - 8*h + 16*i)))/(2 + x) - 8*(d + e + f + g
+ h + i)*Log[1 - x] + 3*(d + 2*e + 4*(f + 2*g + 4*h + 8*i))*Log[2 - x] + 24*(d -
e + f - g + h - i)*Log[1 + x] + (-19*d + 26*e - 28*f + 8*g + 80*h - 352*i)*Log[
2 + x])/144

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Maple [A]  time = 0.018, size = 221, normalized size = 1.8 \[ -{\frac{8\,i}{6+3\,x}}+{\frac{4\,h}{6+3\,x}}-{\frac{2\,g}{6+3\,x}}+{\frac{d}{24+12\,x}}-{\frac{e}{12+6\,x}}+{\frac{f}{6+3\,x}}+{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}-{\frac{\ln \left ( -1+x \right ) d}{18}}-{\frac{\ln \left ( -1+x \right ) e}{18}}+{\frac{2\,\ln \left ( x-2 \right ) i}{3}}-{\frac{\ln \left ( 1+x \right ) i}{6}}-{\frac{\ln \left ( -1+x \right ) i}{18}}-{\frac{22\,\ln \left ( 2+x \right ) i}{9}}+{\frac{\ln \left ( x-2 \right ) h}{3}}+{\frac{\ln \left ( 1+x \right ) h}{6}}+{\frac{5\,\ln \left ( 2+x \right ) h}{9}}-{\frac{\ln \left ( -1+x \right ) h}{18}}-{\frac{\ln \left ( 1+x \right ) g}{6}}+{\frac{\ln \left ( x-2 \right ) g}{6}}-{\frac{\ln \left ( -1+x \right ) g}{18}}+{\frac{\ln \left ( 2+x \right ) g}{18}}+{\frac{\ln \left ( x-2 \right ) d}{48}}+{\frac{\ln \left ( x-2 \right ) e}{24}}+{\frac{13\,\ln \left ( 2+x \right ) e}{72}}+{\frac{\ln \left ( x-2 \right ) f}{12}}-{\frac{19\,\ln \left ( 2+x \right ) d}{144}}+{\frac{\ln \left ( 1+x \right ) f}{6}}-{\frac{\ln \left ( -1+x \right ) f}{18}}-{\frac{7\,\ln \left ( 2+x \right ) f}{36}}+ix \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/3/(2+x)*i+4/3/(2+x)*h-2/3/(2+x)*g+1/12/(2+x)*d-1/6/(2+x)*e+1/3/(2+x)*f+1/6*ln
(1+x)*d-1/6*ln(1+x)*e-1/18*ln(-1+x)*d-1/18*ln(-1+x)*e+2/3*ln(x-2)*i-1/6*ln(1+x)*
i-1/18*ln(-1+x)*i-22/9*ln(2+x)*i+1/3*ln(x-2)*h+1/6*ln(1+x)*h+5/9*ln(2+x)*h-1/18*
ln(-1+x)*h-1/6*ln(1+x)*g+1/6*ln(x-2)*g-1/18*ln(-1+x)*g+1/18*ln(2+x)*g+1/48*ln(x-
2)*d+1/24*ln(x-2)*e+13/72*ln(2+x)*e+1/12*ln(x-2)*f-19/144*ln(2+x)*d+1/6*ln(1+x)*
f-1/18*ln(-1+x)*f-7/36*ln(2+x)*f+i*x

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Maxima [A]  time = 0.715923, size = 146, normalized size = 1.2 \[ i x - \frac{1}{144} \,{\left (19 \, d - 26 \, e + 28 \, f - 8 \, g - 80 \, h + 352 \, i\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac{1}{18} \,{\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac{1}{48} \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) + \frac{d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i}{12 \,{\left (x + 2\right )}} \]

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^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4)^2,x, algorithm="maxima")

[Out]

i*x - 1/144*(19*d - 26*e + 28*f - 8*g - 80*h + 352*i)*log(x + 2) + 1/6*(d - e +
f - g + h - i)*log(x + 1) - 1/18*(d + e + f + g + h + i)*log(x - 1) + 1/48*(d +
2*e + 4*f + 8*g + 16*h + 32*i)*log(x - 2) + 1/12*(d - 2*e + 4*f - 8*g + 16*h - 3
2*i)/(x + 2)

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Fricas [A]  time = 25.4194, size = 270, normalized size = 2.21 \[ \frac{144 \, i x^{2} + 288 \, i x -{\left ({\left (19 \, d - 26 \, e + 28 \, f - 8 \, g - 80 \, h + 352 \, i\right )} x + 38 \, d - 52 \, e + 56 \, f - 16 \, g - 160 \, h + 704 \, i\right )} \log \left (x + 2\right ) + 24 \,{\left ({\left (d - e + f - g + h - i\right )} x + 2 \, d - 2 \, e + 2 \, f - 2 \, g + 2 \, h - 2 \, i\right )} \log \left (x + 1\right ) - 8 \,{\left ({\left (d + e + f + g + h + i\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g + 2 \, h + 2 \, i\right )} \log \left (x - 1\right ) + 3 \,{\left ({\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g + 32 \, h + 64 \, i\right )} \log \left (x - 2\right ) + 12 \, d - 24 \, e + 48 \, f - 96 \, g + 192 \, h - 384 \, i}{144 \,{\left (x + 2\right )}} \]

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^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4)^2,x, algorithm="fricas")

[Out]

1/144*(144*i*x^2 + 288*i*x - ((19*d - 26*e + 28*f - 8*g - 80*h + 352*i)*x + 38*d
- 52*e + 56*f - 16*g - 160*h + 704*i)*log(x + 2) + 24*((d - e + f - g + h - i)*
x + 2*d - 2*e + 2*f - 2*g + 2*h - 2*i)*log(x + 1) - 8*((d + e + f + g + h + i)*x
+ 2*d + 2*e + 2*f + 2*g + 2*h + 2*i)*log(x - 1) + 3*((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) + 12*d - 24*e + 4
8*f - 96*g + 192*h - 384*i)/(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**3-2*x**2-x+2)*(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 = 0.289118, size = 158, normalized size = 1.3 \[ i x - \frac{1}{144} \,{\left (19 \, d + 28 \, f - 8 \, g - 80 \, h + 352 \, i - 26 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{6} \,{\left (d + f - g + h - i - e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{18} \,{\left (d + f + g + h + i + e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{48} \,{\left (d + 4 \, f + 8 \, g + 16 \, h + 32 \, i + 2 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) + \frac{d + 4 \, f - 8 \, g + 16 \, h - 32 \, i - 2 \, e}{12 \,{\left (x + 2\right )}} \]

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^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4)^2,x, algorithm="giac")

[Out]

i*x - 1/144*(19*d + 28*f - 8*g - 80*h + 352*i - 26*e)*ln(abs(x + 2)) + 1/6*(d +
f - g + h - i - e)*ln(abs(x + 1)) - 1/18*(d + f + g + h + i + e)*ln(abs(x - 1))
+ 1/48*(d + 4*f + 8*g + 16*h + 32*i + 2*e)*ln(abs(x - 2)) + 1/12*(d + 4*f - 8*g
+ 16*h - 32*i - 2*e)/(x + 2)