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

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

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Rubi [A]  time = 0.25, antiderivative size = 117, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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) \end {gather*}

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*}

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Mathematica [A]  time = 0.06, size = 114, normalized size = 0.97 \begin {gather*} \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 ) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 3.20, size = 229, normalized size = 1.96 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.38, size = 117, normalized size = 1.00 \begin {gather*} \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 {gather*}

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))

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maple [A]  time = 0.02, size = 178, normalized size = 1.52 \begin {gather*} \frac {31 d \ln \left (x +2\right )}{144}+\frac {d \ln \left (x -2\right )}{144}-\frac {d \ln \left (x -1\right )}{36}-\frac {7 d \ln \left (x +1\right )}{36}-\frac {25 e \ln \left (x +2\right )}{72}+\frac {e \ln \left (x -2\right )}{72}-\frac {e \ln \left (x -1\right )}{36}+\frac {13 e \ln \left (x +1\right )}{36}+\frac {19 f \ln \left (x +2\right )}{36}+\frac {f \ln \left (x -2\right )}{36}-\frac {f \ln \left (x -1\right )}{36}-\frac {19 f \ln \left (x +1\right )}{36}-\frac {13 g \ln \left (x +2\right )}{18}+\frac {g \ln \left (x -2\right )}{18}-\frac {g \ln \left (x -1\right )}{36}+\frac {25 g \ln \left (x +1\right )}{36}-\frac {d}{6 \left (x +1\right )}-\frac {d}{12 \left (x +2\right )}+\frac {e}{6 x +6}+\frac {e}{6 x +12}-\frac {f}{6 \left (x +1\right )}-\frac {f}{3 \left (x +2\right )}+\frac {g}{6 x +6}+\frac {2 g}{3 \left (x +2\right )} \end {gather*}

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]

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

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maxima [A]  time = 0.44, size = 107, normalized size = 0.91 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.91, size = 115, normalized size = 0.98 \begin {gather*} \ln \left (x-2\right )\,\left (\frac {d}{144}+\frac {e}{72}+\frac {f}{36}+\frac {g}{18}\right )-\ln \left (x+1\right )\,\left (\frac {7\,d}{36}-\frac {13\,e}{36}+\frac {19\,f}{36}-\frac {25\,g}{36}\right )-\ln \left (x-1\right )\,\left (\frac {d}{36}+\frac {e}{36}+\frac {f}{36}+\frac {g}{36}\right )+\ln \left (x+2\right )\,\left (\frac {31\,d}{144}-\frac {25\,e}{72}+\frac {19\,f}{36}-\frac {13\,g}{18}\right )-\frac {\frac {5\,d}{12}-\frac {e}{2}+\frac {2\,f}{3}-g+x\,\left (\frac {d}{4}-\frac {e}{3}+\frac {f}{2}-\frac {5\,g}{6}\right )}{x^2+3\,x+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - 2)*(d/144 + e/72 + f/36 + g/18) - log(x + 1)*((7*d)/36 - (13*e)/36 + (19*f)/36 - (25*g)/36) - log(x -
1)*(d/36 + e/36 + f/36 + g/36) + log(x + 2)*((31*d)/144 - (25*e)/72 + (19*f)/36 - (13*g)/18) - ((5*d)/12 - e/2
+ (2*f)/3 - g + x*(d/4 - e/3 + f/2 - (5*g)/6))/(3*x + x^2 + 2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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