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

Optimal. Leaf size=117 \[ -\frac{d-2 e+4 f-8 g}{12 (x+2)}-\frac{d-e+f-g}{6 (x+1)}-\frac{1}{36} \log (1-x) (d+e+f+g)+\frac{1}{144} \log (2-x) (d+2 e+4 f+8 g)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f-25 g)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f-104 g) \]

[Out]

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

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Rubi [A]  time = 0.245746, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {1586, 6728} \[ -\frac{d-2 e+4 f-8 g}{12 (x+2)}-\frac{d-e+f-g}{6 (x+1)}-\frac{1}{36} \log (1-x) (d+e+f+g)+\frac{1}{144} \log (2-x) (d+2 e+4 f+8 g)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f-25 g)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f-104 g) \]

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)^2,x]

[Out]

-(d - e + f - g)/(6*(1 + x)) - (d - 2*e + 4*f - 8*g)/(12*(2 + x)) - ((d + e + f + g)*Log[1 - x])/36 + ((d + 2*
e + 4*f + 8*g)*Log[2 - x])/144 - ((7*d - 13*e + 19*f - 25*g)*Log[1 + x])/36 + ((31*d - 50*e + 76*f - 104*g)*Lo
g[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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{\left (2-3 x+x^2\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}{\left (2-3 x+x^2\right ) \left (2+3 x+x^2\right )^2} \, dx\\ &=\int \left (\frac{d+2 e+4 f+8 g}{144 (-2+x)}+\frac{-d-e-f-g}{36 (-1+x)}+\frac{d-e+f-g}{6 (1+x)^2}+\frac{-7 d+13 e-19 f+25 g}{36 (1+x)}+\frac{d-2 e+4 f-8 g}{12 (2+x)^2}+\frac{31 d-50 e+76 f-104 g}{144 (2+x)}\right ) \, dx\\ &=-\frac{d-e+f-g}{6 (1+x)}-\frac{d-2 e+4 f-8 g}{12 (2+x)}-\frac{1}{36} (d+e+f+g) \log (1-x)+\frac{1}{144} (d+2 e+4 f+8 g) \log (2-x)-\frac{1}{36} (7 d-13 e+19 f-25 g) \log (1+x)+\frac{1}{144} (31 d-50 e+76 f-104 g) \log (2+x)\\ \end{align*}

Mathematica [A]  time = 0.0599166, size = 114, normalized size = 0.97 \[ \frac{1}{144} \left (\frac{12 (-3 d x-5 d+4 e x+6 e-6 f x-8 f+10 g x+12 g)}{x^2+3 x+2}-4 \log (1-x) (d+e+f+g)+\log (2-x) (d+2 e+4 f+8 g)+4 \log (x+1) (-7 d+13 e-19 f+25 g)+\log (x+2) (31 d-50 e+76 f-104 g)\right ) \]

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)^2,x]

[Out]

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

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Maple [A]  time = 0.013, size = 178, normalized size = 1.5 \begin{align*} -{\frac{d}{24+12\,x}}+{\frac{e}{12+6\,x}}-{\frac{f}{6+3\,x}}+{\frac{2\,g}{6+3\,x}}+{\frac{31\,\ln \left ( 2+x \right ) d}{144}}-{\frac{25\,\ln \left ( 2+x \right ) e}{72}}+{\frac{19\,\ln \left ( 2+x \right ) f}{36}}-{\frac{13\,\ln \left ( 2+x \right ) g}{18}}-{\frac{7\,\ln \left ( 1+x \right ) d}{36}}+{\frac{13\,\ln \left ( 1+x \right ) e}{36}}-{\frac{19\,\ln \left ( 1+x \right ) f}{36}}+{\frac{25\,\ln \left ( 1+x \right ) g}{36}}-{\frac{d}{6+6\,x}}+{\frac{e}{6+6\,x}}-{\frac{f}{6+6\,x}}+{\frac{g}{6+6\,x}}+{\frac{\ln \left ( x-2 \right ) d}{144}}+{\frac{\ln \left ( x-2 \right ) e}{72}}+{\frac{\ln \left ( x-2 \right ) f}{36}}+{\frac{\ln \left ( x-2 \right ) g}{18}}-{\frac{\ln \left ( x-1 \right ) d}{36}}-{\frac{\ln \left ( x-1 \right ) e}{36}}-{\frac{\ln \left ( x-1 \right ) f}{36}}-{\frac{\ln \left ( x-1 \right ) g}{36}} \end{align*}

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)^2,x)

[Out]

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

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Maxima [A]  time = 0.94504, size = 144, normalized size = 1.23 \begin{align*} \frac{1}{144} \,{\left (31 \, d - 50 \, e + 76 \, f - 104 \, g\right )} \log \left (x + 2\right ) - \frac{1}{36} \,{\left (7 \, d - 13 \, e + 19 \, f - 25 \, g\right )} \log \left (x + 1\right ) - \frac{1}{36} \,{\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac{1}{144} \,{\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) - \frac{{\left (3 \, d - 4 \, e + 6 \, f - 10 \, g\right )} x + 5 \, d - 6 \, e + 8 \, f - 12 \, g}{12 \,{\left (x^{2} + 3 \, x + 2\right )}} \end{align*}

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)^2,x, algorithm="maxima")

[Out]

1/144*(31*d - 50*e + 76*f - 104*g)*log(x + 2) - 1/36*(7*d - 13*e + 19*f - 25*g)*log(x + 1) - 1/36*(d + e + f +
g)*log(x - 1) + 1/144*(d + 2*e + 4*f + 8*g)*log(x - 2) - 1/12*((3*d - 4*e + 6*f - 10*g)*x + 5*d - 6*e + 8*f -
12*g)/(x^2 + 3*x + 2)

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Fricas [B]  time = 7.25608, size = 655, normalized size = 5.6 \begin{align*} -\frac{12 \,{\left (3 \, d - 4 \, e + 6 \, f - 10 \, g\right )} x -{\left ({\left (31 \, d - 50 \, e + 76 \, f - 104 \, g\right )} x^{2} + 3 \,{\left (31 \, d - 50 \, e + 76 \, f - 104 \, g\right )} x + 62 \, d - 100 \, e + 152 \, f - 208 \, g\right )} \log \left (x + 2\right ) + 4 \,{\left ({\left (7 \, d - 13 \, e + 19 \, f - 25 \, g\right )} x^{2} + 3 \,{\left (7 \, d - 13 \, e + 19 \, f - 25 \, g\right )} x + 14 \, d - 26 \, e + 38 \, f - 50 \, g\right )} \log \left (x + 1\right ) + 4 \,{\left ({\left (d + e + f + g\right )} x^{2} + 3 \,{\left (d + e + f + g\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g\right )} \log \left (x - 1\right ) -{\left ({\left (d + 2 \, e + 4 \, f + 8 \, g\right )} x^{2} + 3 \,{\left (d + 2 \, e + 4 \, f + 8 \, g\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g\right )} \log \left (x - 2\right ) + 60 \, d - 72 \, e + 96 \, f - 144 \, g}{144 \,{\left (x^{2} + 3 \, x + 2\right )}} \end{align*}

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)^2,x, algorithm="fricas")

[Out]

-1/144*(12*(3*d - 4*e + 6*f - 10*g)*x - ((31*d - 50*e + 76*f - 104*g)*x^2 + 3*(31*d - 50*e + 76*f - 104*g)*x +
62*d - 100*e + 152*f - 208*g)*log(x + 2) + 4*((7*d - 13*e + 19*f - 25*g)*x^2 + 3*(7*d - 13*e + 19*f - 25*g)*x
+ 14*d - 26*e + 38*f - 50*g)*log(x + 1) + 4*((d + e + f + g)*x^2 + 3*(d + e + f + g)*x + 2*d + 2*e + 2*f + 2*
g)*log(x - 1) - ((d + 2*e + 4*f + 8*g)*x^2 + 3*(d + 2*e + 4*f + 8*g)*x + 2*d + 4*e + 8*f + 16*g)*log(x - 2) +
60*d - 72*e + 96*f - 144*g)/(x^2 + 3*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**2-3*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 = 1.10134, size = 158, normalized size = 1.35 \begin{align*} \frac{1}{144} \,{\left (31 \, d + 76 \, f - 104 \, g - 50 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac{1}{36} \,{\left (7 \, d + 19 \, f - 25 \, g - 13 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{36} \,{\left (d + f + g + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{144} \,{\left (d + 4 \, f + 8 \, g + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac{{\left (3 \, d + 6 \, f - 10 \, g - 4 \, e\right )} x + 5 \, d + 8 \, f - 12 \, g - 6 \, e}{12 \,{\left (x + 2\right )}{\left (x + 1\right )}} \end{align*}

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)^2,x, algorithm="giac")

[Out]

1/144*(31*d + 76*f - 104*g - 50*e)*log(abs(x + 2)) - 1/36*(7*d + 19*f - 25*g - 13*e)*log(abs(x + 1)) - 1/36*(d
+ f + g + e)*log(abs(x - 1)) + 1/144*(d + 4*f + 8*g + 2*e)*log(abs(x - 2)) - 1/12*((3*d + 6*f - 10*g - 4*e)*x
+ 5*d + 8*f - 12*g - 6*e)/((x + 2)*(x + 1))