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

Optimal. Leaf size=47 \[ \log (x+1) (d-e+f-g)-\log (x+2) (d-2 e+4 f-8 g)+x (f-3 g)+\frac{g x^2}{2} \]

[Out]

(f - 3*g)*x + (g*x^2)/2 + (d - e + f - g)*Log[1 + x] - (d - 2*e + 4*f - 8*g)*Log
[2 + x]

_______________________________________________________________________________________

Rubi [A]  time = 0.122068, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \log (x+1) (d-e+f-g)-\log (x+2) (d-2 e+4 f-8 g)+x (f-3 g)+\frac{g x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

(f - 3*g)*x + (g*x^2)/2 + (d - e + f - g)*Log[1 + x] - (d - 2*e + 4*f - 8*g)*Log
[2 + x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 3 g x + g \int x\, dx - \left (d - 2 e + 4 f - 8 g\right ) \log{\left (x + 2 \right )} + \left (d - e + f - g\right ) \log{\left (x + 1 \right )} + \int f\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*g*x + g*Integral(x, x) - (d - 2*e + 4*f - 8*g)*log(x + 2) + (d - e + f - g)*l
og(x + 1) + Integral(f, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0413789, size = 44, normalized size = 0.94 \[ \log (x+1) (d-e+f-g)-\log (x+2) (d-2 e+4 f-8 g)+f x+\frac{1}{2} g (x-6) x \]

Antiderivative was successfully verified.

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

[Out]

f*x + (g*(-6 + x)*x)/2 + (d - e + f - g)*Log[1 + x] - (d - 2*e + 4*f - 8*g)*Log[
2 + x]

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 69, normalized size = 1.5 \[{\frac{g{x}^{2}}{2}}+fx-3\,gx-\ln \left ( 2+x \right ) d+2\,\ln \left ( 2+x \right ) e-4\,\ln \left ( 2+x \right ) f+8\,\ln \left ( 2+x \right ) g+\ln \left ( 1+x \right ) d-\ln \left ( 1+x \right ) e+\ln \left ( 1+x \right ) f-\ln \left ( 1+x \right ) g \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.704417, size = 61, normalized size = 1.3 \[ \frac{1}{2} \, g x^{2} +{\left (f - 3 \, g\right )} x -{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) +{\left (d - e + f - g\right )} \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.255763, size = 61, normalized size = 1.3 \[ \frac{1}{2} \, g x^{2} +{\left (f - 3 \, g\right )} x -{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) +{\left (d - e + f - g\right )} \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.35304, size = 66, normalized size = 1.4 \[ \frac{g x^{2}}{2} + x \left (f - 3 g\right ) + \left (- d + 2 e - 4 f + 8 g\right ) \log{\left (x + \frac{4 d - 6 e + 10 f - 18 g}{2 d - 3 e + 5 f - 9 g} \right )} + \left (d - e + f - g\right ) \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

g*x**2/2 + x*(f - 3*g) + (-d + 2*e - 4*f + 8*g)*log(x + (4*d - 6*e + 10*f - 18*g
)/(2*d - 3*e + 5*f - 9*g)) + (d - e + f - g)*log(x + 1)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.286063, size = 66, normalized size = 1.4 \[ \frac{1}{2} \, g x^{2} + f x - 3 \, g x -{\left (d + 4 \, f - 8 \, g - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) +{\left (d + f - g - e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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