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

Optimal. Leaf size=95 \[ \frac{d-2 e+4 f-8 g}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f+g)+\frac{1}{48} \log (2-x) (d+2 e+4 f+8 g)+\frac{1}{6} \log (x+1) (d-e+f-g)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f-8 g) \]

[Out]

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

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Rubi [A]  time = 0.221323, antiderivative size = 95, 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, Rules used = {1586, 6742} \[ \frac{d-2 e+4 f-8 g}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f+g)+\frac{1}{48} \log (2-x) (d+2 e+4 f+8 g)+\frac{1}{6} \log (x+1) (d-e+f-g)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f-8 g) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin{align*} \int \frac{\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac{d+e x+f x^2+g x^3}{(2+x)^2 \left (2-x-2 x^2+x^3\right )} \, dx\\ &=\int \left (\frac{d+2 e+4 f+8 g}{48 (-2+x)}+\frac{-d-e-f-g}{18 (-1+x)}+\frac{d-e+f-g}{6 (1+x)}+\frac{-d+2 e-4 f+8 g}{12 (2+x)^2}+\frac{-19 d+26 e-28 f+8 g}{144 (2+x)}\right ) \, dx\\ &=\frac{d-2 e+4 f-8 g}{12 (2+x)}-\frac{1}{18} (d+e+f+g) \log (1-x)+\frac{1}{48} (d+2 e+4 f+8 g) \log (2-x)+\frac{1}{6} (d-e+f-g) \log (1+x)-\frac{1}{144} (19 d-26 e+28 f-8 g) \log (2+x)\\ \end{align*}

Mathematica [A]  time = 0.0466772, size = 90, normalized size = 0.95 \[ \frac{1}{144} \left (\frac{12 (d-2 e+4 f-8 g)}{x+2}+24 \log (-x-1) (d-e+f-g)-8 \log (1-x) (d+e+f+g)+3 \log (2-x) (d+2 e+4 f+8 g)+\log (x+2) (-19 d+26 e-28 f+8 g)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 146, normalized size = 1.5 \begin{align*}{\frac{13\,\ln \left ( 2+x \right ) e}{72}}-{\frac{7\,\ln \left ( 2+x \right ) f}{36}}+{\frac{\ln \left ( 2+x \right ) g}{18}}-{\frac{19\,\ln \left ( 2+x \right ) d}{144}}+{\frac{d}{24+12\,x}}-{\frac{e}{12+6\,x}}+{\frac{f}{6+3\,x}}-{\frac{2\,g}{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 ) f}{6}}-{\frac{\ln \left ( 1+x \right ) g}{6}}+{\frac{\ln \left ( x-2 \right ) d}{48}}+{\frac{\ln \left ( x-2 \right ) e}{24}}+{\frac{\ln \left ( x-2 \right ) f}{12}}+{\frac{\ln \left ( x-2 \right ) g}{6}}-{\frac{\ln \left ( x-1 \right ) d}{18}}-{\frac{\ln \left ( x-1 \right ) e}{18}}-{\frac{\ln \left ( x-1 \right ) f}{18}}-{\frac{\ln \left ( x-1 \right ) g}{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13/72*ln(2+x)*e-7/36*ln(2+x)*f+1/18*ln(2+x)*g-19/144*ln(2+x)*d+1/12/(2+x)*d-1/6/(2+x)*e+1/3/(2+x)*f-2/3/(2+x)*
g+1/6*ln(1+x)*d-1/6*ln(1+x)*e+1/6*ln(1+x)*f-1/6*ln(1+x)*g+1/48*ln(x-2)*d+1/24*ln(x-2)*e+1/12*ln(x-2)*f+1/6*ln(
x-2)*g-1/18*ln(x-1)*d-1/18*ln(x-1)*e-1/18*ln(x-1)*f-1/18*ln(x-1)*g

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Maxima [A]  time = 0.988366, size = 109, normalized size = 1.15 \begin{align*} -\frac{1}{144} \,{\left (19 \, d - 26 \, e + 28 \, f - 8 \, g\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e + f - g\right )} \log \left (x + 1\right ) - \frac{1}{18} \,{\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac{1}{48} \,{\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) + \frac{d - 2 \, e + 4 \, f - 8 \, g}{12 \,{\left (x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(19*d - 26*e + 28*f - 8*g)*log(x + 2) + 1/6*(d - e + f - g)*log(x + 1) - 1/18*(d + e + f + g)*log(x - 1
) + 1/48*(d + 2*e + 4*f + 8*g)*log(x - 2) + 1/12*(d - 2*e + 4*f - 8*g)/(x + 2)

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Fricas [A]  time = 6.17876, size = 406, normalized size = 4.27 \begin{align*} -\frac{{\left ({\left (19 \, d - 26 \, e + 28 \, f - 8 \, g\right )} x + 38 \, d - 52 \, e + 56 \, f - 16 \, g\right )} \log \left (x + 2\right ) - 24 \,{\left ({\left (d - e + f - g\right )} x + 2 \, d - 2 \, e + 2 \, f - 2 \, g\right )} \log \left (x + 1\right ) + 8 \,{\left ({\left (d + e + f + g\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g\right )} \log \left (x - 1\right ) - 3 \,{\left ({\left (d + 2 \, e + 4 \, f + 8 \, g\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g\right )} \log \left (x - 2\right ) - 12 \, d + 24 \, e - 48 \, f + 96 \, g}{144 \,{\left (x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(((19*d - 26*e + 28*f - 8*g)*x + 38*d - 52*e + 56*f - 16*g)*log(x + 2) - 24*((d - e + f - g)*x + 2*d -
2*e + 2*f - 2*g)*log(x + 1) + 8*((d + e + f + g)*x + 2*d + 2*e + 2*f + 2*g)*log(x - 1) - 3*((d + 2*e + 4*f + 8
*g)*x + 2*d + 4*e + 8*f + 16*g)*log(x - 2) - 12*d + 24*e - 48*f + 96*g)/(x + 2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-2*x**2-x+2)*(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 [A]  time = 1.08241, size = 122, normalized size = 1.28 \begin{align*} -\frac{1}{144} \,{\left (19 \, d + 28 \, f - 8 \, g - 26 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{6} \,{\left (d + f - g - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{18} \,{\left (d + f + g + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{48} \,{\left (d + 4 \, f + 8 \, g + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac{d + 4 \, f - 8 \, g - 2 \, e}{12 \,{\left (x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(19*d + 28*f - 8*g - 26*e)*log(abs(x + 2)) + 1/6*(d + f - g - e)*log(abs(x + 1)) - 1/18*(d + f + g + e)
*log(abs(x - 1)) + 1/48*(d + 4*f + 8*g + 2*e)*log(abs(x - 2)) + 1/12*(d + 4*f - 8*g - 2*e)/(x + 2)