3.87 \(\int \frac{(d+e x+f x^2) (2-x-2 x^2+x^3)}{(4-5 x^2+x^4)^2} \, dx\)
Optimal. Leaf size=82 \[ \frac{d-2 e+4 f}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f)+\frac{1}{48} \log (2-x) (d+2 e+4 f)+\frac{1}{6} \log (x+1) (d-e+f)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f) \]
[Out]
(d - 2*e + 4*f)/(12*(2 + x)) - ((d + e + f)*Log[1 - x])/18 + ((d + 2*e + 4*f)*Log[2 - x])/48 + ((d - e + f)*Lo
g[1 + x])/6 - ((19*d - 26*e + 28*f)*Log[2 + x])/144
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Rubi [A] time = 0.198761, antiderivative size = 82, 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, 6742} \[ \frac{d-2 e+4 f}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f)+\frac{1}{48} \log (2-x) (d+2 e+4 f)+\frac{1}{6} \log (x+1) (d-e+f)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f) \]
Antiderivative was successfully verified.
[In]
Int[((d + e*x + f*x^2)*(2 - x - 2*x^2 + x^3))/(4 - 5*x^2 + x^4)^2,x]
[Out]
(d - 2*e + 4*f)/(12*(2 + x)) - ((d + e + f)*Log[1 - x])/18 + ((d + 2*e + 4*f)*Log[2 - x])/48 + ((d - e + f)*Lo
g[1 + x])/6 - ((19*d - 26*e + 28*f)*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 (d+e x+f x^2\right ) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac{d+e x+f x^2}{(2+x)^2 \left (2-x-2 x^2+x^3\right )} \, dx\\ &=\int \left (\frac{d+2 e+4 f}{48 (-2+x)}+\frac{-d-e-f}{18 (-1+x)}+\frac{d-e+f}{6 (1+x)}+\frac{-d+2 e-4 f}{12 (2+x)^2}+\frac{-19 d+26 e-28 f}{144 (2+x)}\right ) \, dx\\ &=\frac{d-2 e+4 f}{12 (2+x)}-\frac{1}{18} (d+e+f) \log (1-x)+\frac{1}{48} (d+2 e+4 f) \log (2-x)+\frac{1}{6} (d-e+f) \log (1+x)-\frac{1}{144} (19 d-26 e+28 f) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0521707, size = 77, normalized size = 0.94 \[ \frac{1}{144} \left (\frac{12 (d-2 e+4 f)}{x+2}+24 \log (-x-1) (d-e+f)-8 \log (1-x) (d+e+f)+3 \log (2-x) (d+2 e+4 f)+\log (x+2) (-19 d+26 e-28 f)\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x + f*x^2)*(2 - x - 2*x^2 + x^3))/(4 - 5*x^2 + x^4)^2,x]
[Out]
((12*(d - 2*e + 4*f))/(2 + x) + 24*(d - e + f)*Log[-1 - x] - 8*(d + e + f)*Log[1 - x] + 3*(d + 2*e + 4*f)*Log[
2 - x] + (-19*d + 26*e - 28*f)*Log[2 + x])/144
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Maple [A] time = 0.01, size = 110, normalized size = 1.3 \begin{align*}{\frac{13\,\ln \left ( 2+x \right ) e}{72}}-{\frac{7\,\ln \left ( 2+x \right ) f}{36}}-{\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{\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 ( 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-1 \right ) d}{18}}-{\frac{\ln \left ( x-1 \right ) e}{18}}-{\frac{\ln \left ( x-1 \right ) f}{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x^2+e*x+d)*(x^3-2*x^2-x+2)/(x^4-5*x^2+4)^2,x)
[Out]
13/72*ln(2+x)*e-7/36*ln(2+x)*f-19/144*ln(2+x)*d+1/12/(2+x)*d-1/6/(2+x)*e+1/3/(2+x)*f+1/6*ln(1+x)*d-1/6*ln(1+x)
*e+1/6*ln(1+x)*f+1/48*ln(x-2)*d+1/24*ln(x-2)*e+1/12*ln(x-2)*f-1/18*ln(x-1)*d-1/18*ln(x-1)*e-1/18*ln(x-1)*f
________________________________________________________________________________________
Maxima [A] time = 0.983578, size = 92, normalized size = 1.12 \begin{align*} -\frac{1}{144} \,{\left (19 \, d - 26 \, e + 28 \, f\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e + f\right )} \log \left (x + 1\right ) - \frac{1}{18} \,{\left (d + e + f\right )} \log \left (x - 1\right ) + \frac{1}{48} \,{\left (d + 2 \, e + 4 \, f\right )} \log \left (x - 2\right ) + \frac{d - 2 \, e + 4 \, f}{12 \,{\left (x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x^2+e*x+d)*(x^3-2*x^2-x+2)/(x^4-5*x^2+4)^2,x, algorithm="maxima")
[Out]
-1/144*(19*d - 26*e + 28*f)*log(x + 2) + 1/6*(d - e + f)*log(x + 1) - 1/18*(d + e + f)*log(x - 1) + 1/48*(d +
2*e + 4*f)*log(x - 2) + 1/12*(d - 2*e + 4*f)/(x + 2)
________________________________________________________________________________________
Fricas [A] time = 2.18009, size = 335, normalized size = 4.09 \begin{align*} -\frac{{\left ({\left (19 \, d - 26 \, e + 28 \, f\right )} x + 38 \, d - 52 \, e + 56 \, f\right )} \log \left (x + 2\right ) - 24 \,{\left ({\left (d - e + f\right )} x + 2 \, d - 2 \, e + 2 \, f\right )} \log \left (x + 1\right ) + 8 \,{\left ({\left (d + e + f\right )} x + 2 \, d + 2 \, e + 2 \, f\right )} \log \left (x - 1\right ) - 3 \,{\left ({\left (d + 2 \, e + 4 \, f\right )} x + 2 \, d + 4 \, e + 8 \, f\right )} \log \left (x - 2\right ) - 12 \, d + 24 \, e - 48 \, f}{144 \,{\left (x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x^2+e*x+d)*(x^3-2*x^2-x+2)/(x^4-5*x^2+4)^2,x, algorithm="fricas")
[Out]
-1/144*(((19*d - 26*e + 28*f)*x + 38*d - 52*e + 56*f)*log(x + 2) - 24*((d - e + f)*x + 2*d - 2*e + 2*f)*log(x
+ 1) + 8*((d + e + f)*x + 2*d + 2*e + 2*f)*log(x - 1) - 3*((d + 2*e + 4*f)*x + 2*d + 4*e + 8*f)*log(x - 2) - 1
2*d + 24*e - 48*f)/(x + 2)
________________________________________________________________________________________
Sympy [B] time = 105.348, size = 4767, normalized size = 58.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x**2+e*x+d)*(x**3-2*x**2-x+2)/(x**4-5*x**2+4)**2,x)
[Out]
(d - e + f)*log(x + (-1534775*d**6 + 8032360*d**5*e - 12464043*d**5*f - 984027*d**5*(d - e + f) - 12991180*d**
4*e**2 + 53064770*d**4*e*f + 11797266*d**4*e*(d - e + f) - 41194260*d**4*f**2 - 7912356*d**4*f*(d - e + f) + 3
567168*d**4*(d - e + f)**2 + 1075200*d**3*e**3 - 67598200*d**3*e**2*f - 32721528*d**3*e**2*(d - e + f) + 13873
8400*d**3*e*f**2 + 59411424*d**3*e*f*(d - e + f) - 8725248*d**3*e*(d - e + f)**2 - 70926880*d**3*f**3 - 238463
04*d**3*f**2*(d - e + f) + 19430208*d**3*f*(d - e + f)**2 - 247104*d**3*(d - e + f)**3 + 16959280*d**2*e**4 +
8077200*d**2*e**3*f + 38977296*d**2*e**3*(d - e + f) - 132649440*d**2*e**2*f**2 - 117189216*d**2*e**2*f*(d - e
+ f) - 2820096*d**2*e**2*(d - e + f)**2 + 178427200*d**2*e*f**3 + 114278976*d**2*e*f**2*(d - e + f) - 4233945
6*d**2*e*f*(d - e + f)**2 - 10357632*d**2*e*(d - e + f)**3 - 67113600*d**2*f**4 - 34460544*d**2*f**3*(d - e +
f) + 38893824*d**2*f**2*(d - e + f)**2 + 2135808*d**2*f*(d - e + f)**3 - 15836800*d*e**5 + 36275600*d*e**4*f -
21294960*d*e**4*(d - e + f) + 17097600*d*e**3*f**2 + 89387904*d*e**3*f*(d - e + f) + 15436800*d*e**3*(d - e +
f)**2 - 114789760*d*e**2*f**3 - 143630208*d*e**2*f**2*(d - e + f) + 6780672*d*e**2*f*(d - e + f)**2 + 1627776
0*d*e**2*(d - e + f)**3 + 112386560*d*e*f**4 + 98370048*d*e*f**3*(d - e + f) - 63728640*d*e*f**2*(d - e + f)**
2 - 22643712*d*e*f*(d - e + f)**3 - 33092352*d*f**5 - 24472320*d*f**4*(d - e + f) + 33905664*d*f**3*(d - e + f
)**2 + 8045568*d*f**2*(d - e + f)**3 + 4283840*e**6 - 17164384*e**5*f + 3876000*e**5*(d - e + f) + 18093760*e*
*4*f**2 - 22632000*e**4*f*(d - e + f) - 6865920*e**4*(d - e + f)**2 + 10387200*e**3*f**3 + 52267776*e**3*f**2*
(d - e + f) + 13957632*e**3*f*(d - e + f)**2 - 4078080*e**3*(d - e + f)**3 - 36528640*e**2*f**4 - 59284992*e**
2*f**3*(d - e + f) + 12026880*e**2*f**2*(d - e + f)**2 + 13851648*e**2*f*(d - e + f)**3 + 27625984*e*f**5 + 32
033280*e*f**4*(d - e + f) - 30394368*e*f**3*(d - e + f)**2 - 14432256*e*f**2*(d - e + f)**3 - 6640640*f**6 - 6
988800*f**5*(d - e + f) + 10874880*f**4*(d - e + f)**2 + 6082560*f**3*(d - e + f)**3)/(801262*d**6 - 4662251*d
**5*e + 6598614*d**5*f + 7296938*d**4*e**2 - 32240296*d**4*e*f + 22080168*d**4*f**2 + 1388616*d**3*e**3 + 4420
7696*d**3*e**2*f - 87152288*d**3*e*f**2 + 38269376*d**3*f**3 - 12447440*d**2*e**4 - 4393344*d**2*e**3*f + 9721
4592*d**2*e**2*f**2 - 114767360*d**2*e*f**3 + 36053760*d**2*f**4 + 9990800*d*e**5 - 25278880*d*e**4*f - 190590
72*d*e**3*f**2 + 91652864*d*e**2*f**3 - 73388800*d*e*f**4 + 17395200*d*f**5 - 2380000*e**6 + 10947200*e**5*f -
11338880*e**4*f**2 - 13562880*e**3*f**3 + 31239680*e**2*f**4 - 18176000*e*f**5 + 3328000*f**6))/6 - (d + e +
f)*log(x + (-1534775*d**6 + 8032360*d**5*e - 12464043*d**5*f + 328009*d**5*(d + e + f) - 12991180*d**4*e**2 +
53064770*d**4*e*f - 3932422*d**4*e*(d + e + f) - 41194260*d**4*f**2 + 2637452*d**4*f*(d + e + f) + 396352*d**4
*(d + e + f)**2 + 1075200*d**3*e**3 - 67598200*d**3*e**2*f + 10907176*d**3*e**2*(d + e + f) + 138738400*d**3*e
*f**2 - 19803808*d**3*e*f*(d + e + f) - 969472*d**3*e*(d + e + f)**2 - 70926880*d**3*f**3 + 7948768*d**3*f**2*
(d + e + f) + 2158912*d**3*f*(d + e + f)**2 + 9152*d**3*(d + e + f)**3 + 16959280*d**2*e**4 + 8077200*d**2*e**
3*f - 12992432*d**2*e**3*(d + e + f) - 132649440*d**2*e**2*f**2 + 39063072*d**2*e**2*f*(d + e + f) - 313344*d*
*2*e**2*(d + e + f)**2 + 178427200*d**2*e*f**3 - 38092992*d**2*e*f**2*(d + e + f) - 4704384*d**2*e*f*(d + e +
f)**2 + 383616*d**2*e*(d + e + f)**3 - 67113600*d**2*f**4 + 11486848*d**2*f**3*(d + e + f) + 4321536*d**2*f**2
*(d + e + f)**2 - 79104*d**2*f*(d + e + f)**3 - 15836800*d*e**5 + 36275600*d*e**4*f + 7098320*d*e**4*(d + e +
f) + 17097600*d*e**3*f**2 - 29795968*d*e**3*f*(d + e + f) + 1715200*d*e**3*(d + e + f)**2 - 114789760*d*e**2*f
**3 + 47876736*d*e**2*f**2*(d + e + f) + 753408*d*e**2*f*(d + e + f)**2 - 602880*d*e**2*(d + e + f)**3 + 11238
6560*d*e*f**4 - 32790016*d*e*f**3*(d + e + f) - 7080960*d*e*f**2*(d + e + f)**2 + 838656*d*e*f*(d + e + f)**3
- 33092352*d*f**5 + 8157440*d*f**4*(d + e + f) + 3767296*d*f**3*(d + e + f)**2 - 297984*d*f**2*(d + e + f)**3
+ 4283840*e**6 - 17164384*e**5*f - 1292000*e**5*(d + e + f) + 18093760*e**4*f**2 + 7544000*e**4*f*(d + e + f)
- 762880*e**4*(d + e + f)**2 + 10387200*e**3*f**3 - 17422592*e**3*f**2*(d + e + f) + 1550848*e**3*f*(d + e + f
)**2 + 151040*e**3*(d + e + f)**3 - 36528640*e**2*f**4 + 19761664*e**2*f**3*(d + e + f) + 1336320*e**2*f**2*(d
+ e + f)**2 - 513024*e**2*f*(d + e + f)**3 + 27625984*e*f**5 - 10677760*e*f**4*(d + e + f) - 3377152*e*f**3*(
d + e + f)**2 + 534528*e*f**2*(d + e + f)**3 - 6640640*f**6 + 2329600*f**5*(d + e + f) + 1208320*f**4*(d + e +
f)**2 - 225280*f**3*(d + e + f)**3)/(801262*d**6 - 4662251*d**5*e + 6598614*d**5*f + 7296938*d**4*e**2 - 3224
0296*d**4*e*f + 22080168*d**4*f**2 + 1388616*d**3*e**3 + 44207696*d**3*e**2*f - 87152288*d**3*e*f**2 + 3826937
6*d**3*f**3 - 12447440*d**2*e**4 - 4393344*d**2*e**3*f + 97214592*d**2*e**2*f**2 - 114767360*d**2*e*f**3 + 360
53760*d**2*f**4 + 9990800*d*e**5 - 25278880*d*e**4*f - 19059072*d*e**3*f**2 + 91652864*d*e**2*f**3 - 73388800*
d*e*f**4 + 17395200*d*f**5 - 2380000*e**6 + 10947200*e**5*f - 11338880*e**4*f**2 - 13562880*e**3*f**3 + 312396
80*e**2*f**4 - 18176000*e*f**5 + 3328000*f**6))/18 + (d + 2*e + 4*f)*log(x + (-1534775*d**6 + 8032360*d**5*e -
12464043*d**5*f - 984027*d**5*(d + 2*e + 4*f)/8 - 12991180*d**4*e**2 + 53064770*d**4*e*f + 5898633*d**4*e*(d
+ 2*e + 4*f)/4 - 41194260*d**4*f**2 - 1978089*d**4*f*(d + 2*e + 4*f)/2 + 55737*d**4*(d + 2*e + 4*f)**2 + 10752
00*d**3*e**3 - 67598200*d**3*e**2*f - 4090191*d**3*e**2*(d + 2*e + 4*f) + 138738400*d**3*e*f**2 + 7426428*d**3
*e*f*(d + 2*e + 4*f) - 136332*d**3*e*(d + 2*e + 4*f)**2 - 70926880*d**3*f**3 - 2980788*d**3*f**2*(d + 2*e + 4*
f) + 303597*d**3*f*(d + 2*e + 4*f)**2 - 3861*d**3*(d + 2*e + 4*f)**3/8 + 16959280*d**2*e**4 + 8077200*d**2*e**
3*f + 4872162*d**2*e**3*(d + 2*e + 4*f) - 132649440*d**2*e**2*f**2 - 14648652*d**2*e**2*f*(d + 2*e + 4*f) - 44
064*d**2*e**2*(d + 2*e + 4*f)**2 + 178427200*d**2*e*f**3 + 14284872*d**2*e*f**2*(d + 2*e + 4*f) - 661554*d**2*
e*f*(d + 2*e + 4*f)**2 - 80919*d**2*e*(d + 2*e + 4*f)**3/4 - 67113600*d**2*f**4 - 4307568*d**2*f**3*(d + 2*e +
4*f) + 607716*d**2*f**2*(d + 2*e + 4*f)**2 + 8343*d**2*f*(d + 2*e + 4*f)**3/2 - 15836800*d*e**5 + 36275600*d*
e**4*f - 2661870*d*e**4*(d + 2*e + 4*f) + 17097600*d*e**3*f**2 + 11173488*d*e**3*f*(d + 2*e + 4*f) + 241200*d*
e**3*(d + 2*e + 4*f)**2 - 114789760*d*e**2*f**3 - 17953776*d*e**2*f**2*(d + 2*e + 4*f) + 105948*d*e**2*f*(d +
2*e + 4*f)**2 + 63585*d*e**2*(d + 2*e + 4*f)**3/2 + 112386560*d*e*f**4 + 12296256*d*e*f**3*(d + 2*e + 4*f) - 9
95760*d*e*f**2*(d + 2*e + 4*f)**2 - 44226*d*e*f*(d + 2*e + 4*f)**3 - 33092352*d*f**5 - 3059040*d*f**4*(d + 2*e
+ 4*f) + 529776*d*f**3*(d + 2*e + 4*f)**2 + 15714*d*f**2*(d + 2*e + 4*f)**3 + 4283840*e**6 - 17164384*e**5*f
+ 484500*e**5*(d + 2*e + 4*f) + 18093760*e**4*f**2 - 2829000*e**4*f*(d + 2*e + 4*f) - 107280*e**4*(d + 2*e + 4
*f)**2 + 10387200*e**3*f**3 + 6533472*e**3*f**2*(d + 2*e + 4*f) + 218088*e**3*f*(d + 2*e + 4*f)**2 - 7965*e**3
*(d + 2*e + 4*f)**3 - 36528640*e**2*f**4 - 7410624*e**2*f**3*(d + 2*e + 4*f) + 187920*e**2*f**2*(d + 2*e + 4*f
)**2 + 27054*e**2*f*(d + 2*e + 4*f)**3 + 27625984*e*f**5 + 4004160*e*f**4*(d + 2*e + 4*f) - 474912*e*f**3*(d +
2*e + 4*f)**2 - 28188*e*f**2*(d + 2*e + 4*f)**3 - 6640640*f**6 - 873600*f**5*(d + 2*e + 4*f) + 169920*f**4*(d
+ 2*e + 4*f)**2 + 11880*f**3*(d + 2*e + 4*f)**3)/(801262*d**6 - 4662251*d**5*e + 6598614*d**5*f + 7296938*d**
4*e**2 - 32240296*d**4*e*f + 22080168*d**4*f**2 + 1388616*d**3*e**3 + 44207696*d**3*e**2*f - 87152288*d**3*e*f
**2 + 38269376*d**3*f**3 - 12447440*d**2*e**4 - 4393344*d**2*e**3*f + 97214592*d**2*e**2*f**2 - 114767360*d**2
*e*f**3 + 36053760*d**2*f**4 + 9990800*d*e**5 - 25278880*d*e**4*f - 19059072*d*e**3*f**2 + 91652864*d*e**2*f**
3 - 73388800*d*e*f**4 + 17395200*d*f**5 - 2380000*e**6 + 10947200*e**5*f - 11338880*e**4*f**2 - 13562880*e**3*
f**3 + 31239680*e**2*f**4 - 18176000*e*f**5 + 3328000*f**6))/48 - (19*d - 26*e + 28*f)*log(x + (-1534775*d**6
+ 8032360*d**5*e - 12464043*d**5*f + 328009*d**5*(19*d - 26*e + 28*f)/8 - 12991180*d**4*e**2 + 53064770*d**4*e
*f - 1966211*d**4*e*(19*d - 26*e + 28*f)/4 - 41194260*d**4*f**2 + 659363*d**4*f*(19*d - 26*e + 28*f)/2 + 6193*
d**4*(19*d - 26*e + 28*f)**2 + 1075200*d**3*e**3 - 67598200*d**3*e**2*f + 1363397*d**3*e**2*(19*d - 26*e + 28*
f) + 138738400*d**3*e*f**2 - 2475476*d**3*e*f*(19*d - 26*e + 28*f) - 15148*d**3*e*(19*d - 26*e + 28*f)**2 - 70
926880*d**3*f**3 + 993596*d**3*f**2*(19*d - 26*e + 28*f) + 33733*d**3*f*(19*d - 26*e + 28*f)**2 + 143*d**3*(19
*d - 26*e + 28*f)**3/8 + 16959280*d**2*e**4 + 8077200*d**2*e**3*f - 1624054*d**2*e**3*(19*d - 26*e + 28*f) - 1
32649440*d**2*e**2*f**2 + 4882884*d**2*e**2*f*(19*d - 26*e + 28*f) - 4896*d**2*e**2*(19*d - 26*e + 28*f)**2 +
178427200*d**2*e*f**3 - 4761624*d**2*e*f**2*(19*d - 26*e + 28*f) - 73506*d**2*e*f*(19*d - 26*e + 28*f)**2 + 29
97*d**2*e*(19*d - 26*e + 28*f)**3/4 - 67113600*d**2*f**4 + 1435856*d**2*f**3*(19*d - 26*e + 28*f) + 67524*d**2
*f**2*(19*d - 26*e + 28*f)**2 - 309*d**2*f*(19*d - 26*e + 28*f)**3/2 - 15836800*d*e**5 + 36275600*d*e**4*f + 8
87290*d*e**4*(19*d - 26*e + 28*f) + 17097600*d*e**3*f**2 - 3724496*d*e**3*f*(19*d - 26*e + 28*f) + 26800*d*e**
3*(19*d - 26*e + 28*f)**2 - 114789760*d*e**2*f**3 + 5984592*d*e**2*f**2*(19*d - 26*e + 28*f) + 11772*d*e**2*f*
(19*d - 26*e + 28*f)**2 - 2355*d*e**2*(19*d - 26*e + 28*f)**3/2 + 112386560*d*e*f**4 - 4098752*d*e*f**3*(19*d
- 26*e + 28*f) - 110640*d*e*f**2*(19*d - 26*e + 28*f)**2 + 1638*d*e*f*(19*d - 26*e + 28*f)**3 - 33092352*d*f**
5 + 1019680*d*f**4*(19*d - 26*e + 28*f) + 58864*d*f**3*(19*d - 26*e + 28*f)**2 - 582*d*f**2*(19*d - 26*e + 28*
f)**3 + 4283840*e**6 - 17164384*e**5*f - 161500*e**5*(19*d - 26*e + 28*f) + 18093760*e**4*f**2 + 943000*e**4*f
*(19*d - 26*e + 28*f) - 11920*e**4*(19*d - 26*e + 28*f)**2 + 10387200*e**3*f**3 - 2177824*e**3*f**2*(19*d - 26
*e + 28*f) + 24232*e**3*f*(19*d - 26*e + 28*f)**2 + 295*e**3*(19*d - 26*e + 28*f)**3 - 36528640*e**2*f**4 + 24
70208*e**2*f**3*(19*d - 26*e + 28*f) + 20880*e**2*f**2*(19*d - 26*e + 28*f)**2 - 1002*e**2*f*(19*d - 26*e + 28
*f)**3 + 27625984*e*f**5 - 1334720*e*f**4*(19*d - 26*e + 28*f) - 52768*e*f**3*(19*d - 26*e + 28*f)**2 + 1044*e
*f**2*(19*d - 26*e + 28*f)**3 - 6640640*f**6 + 291200*f**5*(19*d - 26*e + 28*f) + 18880*f**4*(19*d - 26*e + 28
*f)**2 - 440*f**3*(19*d - 26*e + 28*f)**3)/(801262*d**6 - 4662251*d**5*e + 6598614*d**5*f + 7296938*d**4*e**2
- 32240296*d**4*e*f + 22080168*d**4*f**2 + 1388616*d**3*e**3 + 44207696*d**3*e**2*f - 87152288*d**3*e*f**2 + 3
8269376*d**3*f**3 - 12447440*d**2*e**4 - 4393344*d**2*e**3*f + 97214592*d**2*e**2*f**2 - 114767360*d**2*e*f**3
+ 36053760*d**2*f**4 + 9990800*d*e**5 - 25278880*d*e**4*f - 19059072*d*e**3*f**2 + 91652864*d*e**2*f**3 - 733
88800*d*e*f**4 + 17395200*d*f**5 - 2380000*e**6 + 10947200*e**5*f - 11338880*e**4*f**2 - 13562880*e**3*f**3 +
31239680*e**2*f**4 - 18176000*e*f**5 + 3328000*f**6))/144 + (d - 2*e + 4*f)/(12*x + 24)
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Giac [A] time = 1.08921, size = 104, normalized size = 1.27 \begin{align*} -\frac{1}{144} \,{\left (19 \, d + 28 \, f - 26 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{6} \,{\left (d + f - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{18} \,{\left (d + f + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{48} \,{\left (d + 4 \, f + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac{d + 4 \, f - 2 \, e}{12 \,{\left (x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x^2+e*x+d)*(x^3-2*x^2-x+2)/(x^4-5*x^2+4)^2,x, algorithm="giac")
[Out]
-1/144*(19*d + 28*f - 26*e)*log(abs(x + 2)) + 1/6*(d + f - e)*log(abs(x + 1)) - 1/18*(d + f + e)*log(abs(x - 1
)) + 1/48*(d + 4*f + 2*e)*log(abs(x - 2)) + 1/12*(d + 4*f - 2*e)/(x + 2)