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

Optimal. Leaf size=68 \[ \log (x+2) (d-2 e+4 f-8 g+16 h)+x (e-2 f+4 g-8 h)+\frac{1}{2} x^2 (f-2 g+4 h)+\frac{1}{3} x^3 (g-2 h)+\frac{h x^4}{4} \]

[Out]

(e - 2*f + 4*g - 8*h)*x + ((f - 2*g + 4*h)*x^2)/2 + ((g - 2*h)*x^3)/3 + (h*x^4)/
4 + (d - 2*e + 4*f - 8*g + 16*h)*Log[2 + x]

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Rubi [A]  time = 0.200283, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \log (x+2) (d-2 e+4 f-8 g+16 h)+x (e-2 f+4 g-8 h)+\frac{1}{2} x^2 (f-2 g+4 h)+\frac{1}{3} x^3 (g-2 h)+\frac{h x^4}{4} \]

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))/(4 - 5*x^2 + x^4),x]

[Out]

(e - 2*f + 4*g - 8*h)*x + ((f - 2*g + 4*h)*x^2)/2 + ((g - 2*h)*x^3)/3 + (h*x^4)/
4 + (d - 2*e + 4*f - 8*g + 16*h)*Log[2 + x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 4 f x + 12 g x - 32 h x + \frac{h \left (x + 2\right )^{4}}{4} + \left (\frac{g}{3} - \frac{8 h}{3}\right ) \left (x + 2\right )^{3} + \left (x + 2\right )^{2} \left (\frac{f}{2} - 3 g + 12 h\right ) + \left (d - 2 e + 4 f - 8 g + 16 h\right ) \log{\left (x + 2 \right )} + \int e\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*f*x + 12*g*x - 32*h*x + h*(x + 2)**4/4 + (g/3 - 8*h/3)*(x + 2)**3 + (x + 2)**
2*(f/2 - 3*g + 12*h) + (d - 2*e + 4*f - 8*g + 16*h)*log(x + 2) + Integral(e, x)

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Mathematica [A]  time = 0.0407927, size = 68, normalized size = 1. \[ \log (x+2) (d-2 e+4 f-8 g+16 h)+x (e-2 f+4 g-8 h)+\frac{1}{2} x^2 (f-2 g+4 h)+\frac{1}{3} x^3 (g-2 h)+\frac{h x^4}{4} \]

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))/(4 - 5*x^2 + x^4),x]

[Out]

(e - 2*f + 4*g - 8*h)*x + ((f - 2*g + 4*h)*x^2)/2 + ((g - 2*h)*x^3)/3 + (h*x^4)/
4 + (d - 2*e + 4*f - 8*g + 16*h)*Log[2 + x]

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Maple [A]  time = 0.005, size = 87, normalized size = 1.3 \[{\frac{h{x}^{4}}{4}}+{\frac{g{x}^{3}}{3}}-{\frac{2\,h{x}^{3}}{3}}+{\frac{f{x}^{2}}{2}}-g{x}^{2}+2\,h{x}^{2}+ex-2\,fx+4\,gx-8\,hx+\ln \left ( 2+x \right ) d-2\,\ln \left ( 2+x \right ) e+4\,\ln \left ( 2+x \right ) f-8\,\ln \left ( 2+x \right ) g+16\,\ln \left ( 2+x \right ) h \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*h*x^4+1/3*g*x^3-2/3*h*x^3+1/2*f*x^2-g*x^2+2*h*x^2+e*x-2*f*x+4*g*x-8*h*x+ln(2
+x)*d-2*ln(2+x)*e+4*ln(2+x)*f-8*ln(2+x)*g+16*ln(2+x)*h

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Maxima [A]  time = 0.700153, size = 84, normalized size = 1.24 \[ \frac{1}{4} \, h x^{4} + \frac{1}{3} \,{\left (g - 2 \, h\right )} x^{3} + \frac{1}{2} \,{\left (f - 2 \, g + 4 \, h\right )} x^{2} +{\left (e - 2 \, f + 4 \, g - 8 \, h\right )} x +{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h\right )} \log \left (x + 2\right ) \]

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

[Out]

1/4*h*x^4 + 1/3*(g - 2*h)*x^3 + 1/2*(f - 2*g + 4*h)*x^2 + (e - 2*f + 4*g - 8*h)*
x + (d - 2*e + 4*f - 8*g + 16*h)*log(x + 2)

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Fricas [A]  time = 0.246832, size = 84, normalized size = 1.24 \[ \frac{1}{4} \, h x^{4} + \frac{1}{3} \,{\left (g - 2 \, h\right )} x^{3} + \frac{1}{2} \,{\left (f - 2 \, g + 4 \, h\right )} x^{2} +{\left (e - 2 \, f + 4 \, g - 8 \, h\right )} x +{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h\right )} \log \left (x + 2\right ) \]

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

[Out]

1/4*h*x^4 + 1/3*(g - 2*h)*x^3 + 1/2*(f - 2*g + 4*h)*x^2 + (e - 2*f + 4*g - 8*h)*
x + (d - 2*e + 4*f - 8*g + 16*h)*log(x + 2)

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Sympy [A]  time = 1.3224, size = 63, normalized size = 0.93 \[ \frac{h x^{4}}{4} + x^{3} \left (\frac{g}{3} - \frac{2 h}{3}\right ) + x^{2} \left (\frac{f}{2} - g + 2 h\right ) + x \left (e - 2 f + 4 g - 8 h\right ) + \left (d - 2 e + 4 f - 8 g + 16 h\right ) \log{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

h*x**4/4 + x**3*(g/3 - 2*h/3) + x**2*(f/2 - g + 2*h) + x*(e - 2*f + 4*g - 8*h) +
(d - 2*e + 4*f - 8*g + 16*h)*log(x + 2)

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GIAC/XCAS [A]  time = 0.281569, size = 100, normalized size = 1.47 \[ \frac{1}{4} \, h x^{4} + \frac{1}{3} \, g x^{3} - \frac{2}{3} \, h x^{3} + \frac{1}{2} \, f x^{2} - g x^{2} + 2 \, h x^{2} - 2 \, f x + 4 \, g x - 8 \, h x + x e +{\left (d + 4 \, f - 8 \, g + 16 \, h - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) \]

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

[Out]

1/4*h*x^4 + 1/3*g*x^3 - 2/3*h*x^3 + 1/2*f*x^2 - g*x^2 + 2*h*x^2 - 2*f*x + 4*g*x
- 8*h*x + x*e + (d + 4*f - 8*g + 16*h - 2*e)*ln(abs(x + 2))