3.99 \(\int \frac{(2+x) (d+e x+f x^2)}{(4-5 x^2+x^4)^2} \, dx\)
Optimal. Leaf size=122 \[ -\frac{d-e+f}{36 (x+1)}+\frac{d+e+f}{12 (1-x)}+\frac{d+2 e+4 f}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f)+\frac{1}{108} \log (x+1) (2 d+e-4 f)+\frac{1}{144} \log (x+2) (d-2 e+4 f) \]
[Out]
(d + e + f)/(12*(1 - x)) + (d + 2*e + 4*f)/(36*(2 - x)) - (d - e + f)/(36*(1 + x)) + ((2*d + 5*e + 8*f)*Log[1
- x])/36 - ((35*d + 58*e + 92*f)*Log[2 - x])/432 + ((2*d + e - 4*f)*Log[1 + x])/108 + ((d - 2*e + 4*f)*Log[2 +
x])/144
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Rubi [A] time = 0.221716, antiderivative size = 122, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used =
{1586, 6742} \[ -\frac{d-e+f}{36 (x+1)}+\frac{d+e+f}{12 (1-x)}+\frac{d+2 e+4 f}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f)+\frac{1}{108} \log (x+1) (2 d+e-4 f)+\frac{1}{144} \log (x+2) (d-2 e+4 f) \]
Antiderivative was successfully verified.
[In]
Int[((2 + x)*(d + e*x + f*x^2))/(4 - 5*x^2 + x^4)^2,x]
[Out]
(d + e + f)/(12*(1 - x)) + (d + 2*e + 4*f)/(36*(2 - x)) - (d - e + f)/(36*(1 + x)) + ((2*d + 5*e + 8*f)*Log[1
- x])/36 - ((35*d + 58*e + 92*f)*Log[2 - x])/432 + ((2*d + e - 4*f)*Log[1 + x])/108 + ((d - 2*e + 4*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{(2+x) \left (d+e x+f x^2\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac{d+e x+f x^2}{(2+x) \left (2-x-2 x^2+x^3\right )^2} \, dx\\ &=\int \left (\frac{d+2 e+4 f}{36 (-2+x)^2}+\frac{-35 d-58 e-92 f}{432 (-2+x)}+\frac{d+e+f}{12 (-1+x)^2}+\frac{2 d+5 e+8 f}{36 (-1+x)}+\frac{d-e+f}{36 (1+x)^2}+\frac{2 d+e-4 f}{108 (1+x)}+\frac{d-2 e+4 f}{144 (2+x)}\right ) \, dx\\ &=\frac{d+e+f}{12 (1-x)}+\frac{d+2 e+4 f}{36 (2-x)}-\frac{d-e+f}{36 (1+x)}+\frac{1}{36} (2 d+5 e+8 f) \log (1-x)-\frac{1}{432} (35 d+58 e+92 f) \log (2-x)+\frac{1}{108} (2 d+e-4 f) \log (1+x)+\frac{1}{144} (d-2 e+4 f) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0538168, size = 121, normalized size = 0.99 \[ \frac{1}{432} \left (\frac{12 \left (d \left (-5 x^2+6 x+5\right )+e \left (10-4 x^2\right )+2 f \left (-4 x^2+3 x+4\right )\right )}{x^3-2 x^2-x+2}+12 \log (1-x) (2 d+5 e+8 f)-\log (2-x) (35 d+58 e+92 f)+4 \log (x+1) (2 d+e-4 f)+3 \log (x+2) (d-2 e+4 f)\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[((2 + x)*(d + e*x + f*x^2))/(4 - 5*x^2 + x^4)^2,x]
[Out]
((12*(d*(5 + 6*x - 5*x^2) + e*(10 - 4*x^2) + 2*f*(4 + 3*x - 4*x^2)))/(2 - x - 2*x^2 + x^3) + 12*(2*d + 5*e + 8
*f)*Log[1 - x] - (35*d + 58*e + 92*f)*Log[2 - x] + 4*(2*d + e - 4*f)*Log[1 + x] + 3*(d - 2*e + 4*f)*Log[2 + x]
)/432
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Maple [A] time = 0.015, size = 158, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( 2+x \right ) d}{144}}-{\frac{\ln \left ( 2+x \right ) e}{72}}+{\frac{\ln \left ( 2+x \right ) f}{36}}-{\frac{d}{36+36\,x}}+{\frac{e}{36+36\,x}}-{\frac{f}{36+36\,x}}+{\frac{\ln \left ( 1+x \right ) d}{54}}+{\frac{\ln \left ( 1+x \right ) e}{108}}-{\frac{\ln \left ( 1+x \right ) f}{27}}-{\frac{35\,\ln \left ( x-2 \right ) d}{432}}-{\frac{29\,\ln \left ( x-2 \right ) e}{216}}-{\frac{23\,\ln \left ( x-2 \right ) f}{108}}-{\frac{d}{36\,x-72}}-{\frac{e}{18\,x-36}}-{\frac{f}{9\,x-18}}-{\frac{d}{12\,x-12}}-{\frac{e}{12\,x-12}}-{\frac{f}{12\,x-12}}+{\frac{\ln \left ( x-1 \right ) d}{18}}+{\frac{5\,\ln \left ( x-1 \right ) e}{36}}+{\frac{2\,\ln \left ( x-1 \right ) f}{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2+x)*(f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x)
[Out]
1/144*ln(2+x)*d-1/72*ln(2+x)*e+1/36*ln(2+x)*f-1/36/(1+x)*d+1/36/(1+x)*e-1/36/(1+x)*f+1/54*ln(1+x)*d+1/108*ln(1
+x)*e-1/27*ln(1+x)*f-35/432*ln(x-2)*d-29/216*ln(x-2)*e-23/108*ln(x-2)*f-1/36/(x-2)*d-1/18/(x-2)*e-1/9/(x-2)*f-
1/12/(x-1)*d-1/12/(x-1)*e-1/12/(x-1)*f+1/18*ln(x-1)*d+5/36*ln(x-1)*e+2/9*ln(x-1)*f
________________________________________________________________________________________
Maxima [A] time = 0.965564, size = 146, normalized size = 1.2 \begin{align*} \frac{1}{144} \,{\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) + \frac{1}{108} \,{\left (2 \, d + e - 4 \, f\right )} \log \left (x + 1\right ) + \frac{1}{36} \,{\left (2 \, d + 5 \, e + 8 \, f\right )} \log \left (x - 1\right ) - \frac{1}{432} \,{\left (35 \, d + 58 \, e + 92 \, f\right )} \log \left (x - 2\right ) - \frac{{\left (5 \, d + 4 \, e + 8 \, f\right )} x^{2} - 6 \,{\left (d + f\right )} x - 5 \, d - 10 \, e - 8 \, f}{36 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+x)*(f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="maxima")
[Out]
1/144*(d - 2*e + 4*f)*log(x + 2) + 1/108*(2*d + e - 4*f)*log(x + 1) + 1/36*(2*d + 5*e + 8*f)*log(x - 1) - 1/43
2*(35*d + 58*e + 92*f)*log(x - 2) - 1/36*((5*d + 4*e + 8*f)*x^2 - 6*(d + f)*x - 5*d - 10*e - 8*f)/(x^3 - 2*x^2
- x + 2)
________________________________________________________________________________________
Fricas [B] time = 2.15583, size = 698, normalized size = 5.72 \begin{align*} -\frac{12 \,{\left (5 \, d + 4 \, e + 8 \, f\right )} x^{2} - 72 \,{\left (d + f\right )} x - 3 \,{\left ({\left (d - 2 \, e + 4 \, f\right )} x^{3} - 2 \,{\left (d - 2 \, e + 4 \, f\right )} x^{2} -{\left (d - 2 \, e + 4 \, f\right )} x + 2 \, d - 4 \, e + 8 \, f\right )} \log \left (x + 2\right ) - 4 \,{\left ({\left (2 \, d + e - 4 \, f\right )} x^{3} - 2 \,{\left (2 \, d + e - 4 \, f\right )} x^{2} -{\left (2 \, d + e - 4 \, f\right )} x + 4 \, d + 2 \, e - 8 \, f\right )} \log \left (x + 1\right ) - 12 \,{\left ({\left (2 \, d + 5 \, e + 8 \, f\right )} x^{3} - 2 \,{\left (2 \, d + 5 \, e + 8 \, f\right )} x^{2} -{\left (2 \, d + 5 \, e + 8 \, f\right )} x + 4 \, d + 10 \, e + 16 \, f\right )} \log \left (x - 1\right ) +{\left ({\left (35 \, d + 58 \, e + 92 \, f\right )} x^{3} - 2 \,{\left (35 \, d + 58 \, e + 92 \, f\right )} x^{2} -{\left (35 \, d + 58 \, e + 92 \, f\right )} x + 70 \, d + 116 \, e + 184 \, f\right )} \log \left (x - 2\right ) - 60 \, d - 120 \, e - 96 \, f}{432 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+x)*(f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="fricas")
[Out]
-1/432*(12*(5*d + 4*e + 8*f)*x^2 - 72*(d + f)*x - 3*((d - 2*e + 4*f)*x^3 - 2*(d - 2*e + 4*f)*x^2 - (d - 2*e +
4*f)*x + 2*d - 4*e + 8*f)*log(x + 2) - 4*((2*d + e - 4*f)*x^3 - 2*(2*d + e - 4*f)*x^2 - (2*d + e - 4*f)*x + 4*
d + 2*e - 8*f)*log(x + 1) - 12*((2*d + 5*e + 8*f)*x^3 - 2*(2*d + 5*e + 8*f)*x^2 - (2*d + 5*e + 8*f)*x + 4*d +
10*e + 16*f)*log(x - 1) + ((35*d + 58*e + 92*f)*x^3 - 2*(35*d + 58*e + 92*f)*x^2 - (35*d + 58*e + 92*f)*x + 70
*d + 116*e + 184*f)*log(x - 2) - 60*d - 120*e - 96*f)/(x^3 - 2*x^2 - x + 2)
________________________________________________________________________________________
Sympy [B] time = 104.122, size = 5192, normalized size = 42.56 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+x)*(f*x**2+e*x+d)/(x**4-5*x**2+4)**2,x)
[Out]
(d - 2*e + 4*f)*log(x + (8710660*d**6 + 109305824*d**5*e + 136707258*d**5*f - 7579779*d**5*(d - 2*e + 4*f)/4 +
548679440*d**4*e**2 + 1278644860*d**4*e*f - 43835889*d**4*e*(d - 2*e + 4*f)/2 + 627558840*d**4*f**2 - 4049681
7*d**4*f*(d - 2*e + 4*f) - 83772*d**4*(d - 2*e + 4*f)**2 + 1416518400*d**3*e**3 + 4598750960*d**3*e**2*f - 965
52978*d**3*e**2*(d - 2*e + 4*f) + 3756616640*d**3*e*f**2 - 337816392*d**3*e*f*(d - 2*e + 4*f) - 765360*d**3*e*
(d - 2*e + 4*f)**2 + 42632000*d**3*f**3 - 290155704*d**3*f**2*(d - 2*e + 4*f) - 114822*d**3*f*(d - 2*e + 4*f)*
*2 + 65907*d**3*(d - 2*e + 4*f)**3/4 + 1987752640*d**2*e**4 + 7924847520*d**2*e**3*f - 206542956*d**2*e**3*(d
- 2*e + 4*f) + 7373599680*d**2*e**2*f**2 - 1045445256*d**2*e**2*f*(d - 2*e + 4*f) - 2695680*d**2*e**2*(d - 2*e
+ 4*f)**2 - 4656496000*d**2*e*f**3 - 1745170416*d**2*e*f**2*(d - 2*e + 4*f) - 1702188*d**2*e*f*(d - 2*e + 4*f
)**2 + 277587*d**2*e*(d - 2*e + 4*f)**3/2 - 6902995200*d**2*f**4 - 963557664*d**2*f**3*(d - 2*e + 4*f) + 44753
04*d**2*f**2*(d - 2*e + 4*f)**2 + 298323*d**2*f*(d - 2*e + 4*f)**3 + 1437185536*d*e**5 + 6500930720*d*e**4*f -
217034796*d*e**4*(d - 2*e + 4*f) + 4803912960*d*e**3*f**2 - 1422980448*d*e**3*f*(d - 2*e + 4*f) - 4196160*d*e
**3*(d - 2*e + 4*f)**2 - 17839947520*d*e**2*f**3 - 3475531872*d*e**2*f**2*(d - 2*e + 4*f) - 5674536*d*e**2*f*(
d - 2*e + 4*f)**2 + 339957*d*e**2*(d - 2*e + 4*f)**3 - 34607006720*d*e*f**4 - 3750241920*d*e*f**3*(d - 2*e + 4
*f) + 15048288*d*e*f**2*(d - 2*e + 4*f)**2 + 1193940*d*e*f*(d - 2*e + 4*f)**3 - 17540771328*d*f**5 - 150864787
2*d*f**4*(d - 2*e + 4*f) + 20676960*d*f**3*(d - 2*e + 4*f)**2 + 1012500*d*f**2*(d - 2*e + 4*f)**3 + 416940800*
e**6 + 2005475776*e**5*f - 90272712*e**5*(d - 2*e + 4*f) + 122654080*e**4*f**2 - 717642192*e**4*f*(d - 2*e + 4
*f) - 2392128*e**4*(d - 2*e + 4*f)**2 - 16377853440*e**3*f**3 - 2270414784*e**3*f**2*(d - 2*e + 4*f) - 5251536
*e**3*f*(d - 2*e + 4*f)**2 + 256554*e**3*(d - 2*e + 4*f)**3 - 39387581440*e**2*f**4 - 3567808896*e**2*f**3*(d
- 2*e + 4*f) + 12299040*e**2*f**2*(d - 2*e + 4*f)**2 + 1124604*e**2*f*(d - 2*e + 4*f)**3 - 37694227456*e*f**5
- 2778944640*e*f**4*(d - 2*e + 4*f) + 39067200*e*f**3*(d - 2*e + 4*f)**2 + 1575288*e*f**2*(d - 2*e + 4*f)**3 -
13332408320*f**6 - 856423680*f**5*(d - 2*e + 4*f) + 25200000*f**4*(d - 2*e + 4*f)**2 + 704592*f**3*(d - 2*e +
4*f)**3)/(3374210*d**6 + 45393715*d**5*e + 44170854*d**5*f + 247848970*d**4*e**2 + 464201768*d**4*e*f + 91507
752*d**4*f**2 + 703178520*d**3*e**3 + 1914915472*d**3*e**2*f + 607100704*d**3*e*f**2 - 999338816*d**3*f**3 + 1
094421680*d**2*e**4 + 3892700544*d**2*e**3*f + 1545770112*d**2*e**2*f**2 - 6739384832*d**2*e*f**3 - 5963752704
*d**2*f**4 + 887062640*d*e**5 + 3907683424*d*e**4*f + 1889544576*d*e**3*f**2 - 14086786304*d*e**2*f**3 - 24613
469440*d*e*f**4 - 11900367360*d*f**5 + 292932640*e**6 + 1550127488*e**5*f + 952305536*e**4*f**2 - 9235000320*e
**3*f**3 - 24236925440*e**2*f**4 - 23421052928*e*f**5 - 8154904576*f**6))/144 + (2*d + e - 4*f)*log(x + (87106
60*d**6 + 109305824*d**5*e + 136707258*d**5*f - 2526593*d**5*(2*d + e - 4*f) + 548679440*d**4*e**2 + 127864486
0*d**4*e*f - 29223926*d**4*e*(2*d + e - 4*f) + 627558840*d**4*f**2 - 53995756*d**4*f*(2*d + e - 4*f) - 148928*
d**4*(2*d + e - 4*f)**2 + 1416518400*d**3*e**3 + 4598750960*d**3*e**2*f - 128737304*d**3*e**2*(2*d + e - 4*f)
+ 3756616640*d**3*e*f**2 - 450421856*d**3*e*f*(2*d + e - 4*f) - 1360640*d**3*e*(2*d + e - 4*f)**2 + 42632000*d
**3*f**3 - 386874272*d**3*f**2*(2*d + e - 4*f) - 204128*d**3*f*(2*d + e - 4*f)**2 + 39056*d**3*(2*d + e - 4*f)
**3 + 1987752640*d**2*e**4 + 7924847520*d**2*e**3*f - 275390608*d**2*e**3*(2*d + e - 4*f) + 7373599680*d**2*e*
*2*f**2 - 1393927008*d**2*e**2*f*(2*d + e - 4*f) - 4792320*d**2*e**2*(2*d + e - 4*f)**2 - 4656496000*d**2*e*f*
*3 - 2326893888*d**2*e*f**2*(2*d + e - 4*f) - 3026112*d**2*e*f*(2*d + e - 4*f)**2 + 328992*d**2*e*(2*d + e - 4
*f)**3 - 6902995200*d**2*f**4 - 1284743552*d**2*f**3*(2*d + e - 4*f) + 7956096*d**2*f**2*(2*d + e - 4*f)**2 +
707136*d**2*f*(2*d + e - 4*f)**3 + 1437185536*d*e**5 + 6500930720*d*e**4*f - 289379728*d*e**4*(2*d + e - 4*f)
+ 4803912960*d*e**3*f**2 - 1897307264*d*e**3*f*(2*d + e - 4*f) - 7459840*d*e**3*(2*d + e - 4*f)**2 - 178399475
20*d*e**2*f**3 - 4634042496*d*e**2*f**2*(2*d + e - 4*f) - 10088064*d*e**2*f*(2*d + e - 4*f)**2 + 805824*d*e**2
*(2*d + e - 4*f)**3 - 34607006720*d*e*f**4 - 5000322560*d*e*f**3*(2*d + e - 4*f) + 26752512*d*e*f**2*(2*d + e
- 4*f)**2 + 2830080*d*e*f*(2*d + e - 4*f)**3 - 17540771328*d*f**5 - 2011530496*d*f**4*(2*d + e - 4*f) + 367590
40*d*f**3*(2*d + e - 4*f)**2 + 2400000*d*f**2*(2*d + e - 4*f)**3 + 416940800*e**6 + 2005475776*e**5*f - 120363
616*e**5*(2*d + e - 4*f) + 122654080*e**4*f**2 - 956856256*e**4*f*(2*d + e - 4*f) - 4252672*e**4*(2*d + e - 4*
f)**2 - 16377853440*e**3*f**3 - 3027219712*e**3*f**2*(2*d + e - 4*f) - 9336064*e**3*f*(2*d + e - 4*f)**2 + 608
128*e**3*(2*d + e - 4*f)**3 - 39387581440*e**2*f**4 - 4757078528*e**2*f**3*(2*d + e - 4*f) + 21864960*e**2*f**
2*(2*d + e - 4*f)**2 + 2665728*e**2*f*(2*d + e - 4*f)**3 - 37694227456*e*f**5 - 3705259520*e*f**4*(2*d + e - 4
*f) + 69452800*e*f**3*(2*d + e - 4*f)**2 + 3734016*e*f**2*(2*d + e - 4*f)**3 - 13332408320*f**6 - 1141898240*f
**5*(2*d + e - 4*f) + 44800000*f**4*(2*d + e - 4*f)**2 + 1670144*f**3*(2*d + e - 4*f)**3)/(3374210*d**6 + 4539
3715*d**5*e + 44170854*d**5*f + 247848970*d**4*e**2 + 464201768*d**4*e*f + 91507752*d**4*f**2 + 703178520*d**3
*e**3 + 1914915472*d**3*e**2*f + 607100704*d**3*e*f**2 - 999338816*d**3*f**3 + 1094421680*d**2*e**4 + 38927005
44*d**2*e**3*f + 1545770112*d**2*e**2*f**2 - 6739384832*d**2*e*f**3 - 5963752704*d**2*f**4 + 887062640*d*e**5
+ 3907683424*d*e**4*f + 1889544576*d*e**3*f**2 - 14086786304*d*e**2*f**3 - 24613469440*d*e*f**4 - 11900367360*
d*f**5 + 292932640*e**6 + 1550127488*e**5*f + 952305536*e**4*f**2 - 9235000320*e**3*f**3 - 24236925440*e**2*f*
*4 - 23421052928*e*f**5 - 8154904576*f**6))/108 + (2*d + 5*e + 8*f)*log(x + (8710660*d**6 + 109305824*d**5*e +
136707258*d**5*f - 7579779*d**5*(2*d + 5*e + 8*f) + 548679440*d**4*e**2 + 1278644860*d**4*e*f - 87671778*d**4
*e*(2*d + 5*e + 8*f) + 627558840*d**4*f**2 - 161987268*d**4*f*(2*d + 5*e + 8*f) - 1340352*d**4*(2*d + 5*e + 8*
f)**2 + 1416518400*d**3*e**3 + 4598750960*d**3*e**2*f - 386211912*d**3*e**2*(2*d + 5*e + 8*f) + 3756616640*d**
3*e*f**2 - 1351265568*d**3*e*f*(2*d + 5*e + 8*f) - 12245760*d**3*e*(2*d + 5*e + 8*f)**2 + 42632000*d**3*f**3 -
1160622816*d**3*f**2*(2*d + 5*e + 8*f) - 1837152*d**3*f*(2*d + 5*e + 8*f)**2 + 1054512*d**3*(2*d + 5*e + 8*f)
**3 + 1987752640*d**2*e**4 + 7924847520*d**2*e**3*f - 826171824*d**2*e**3*(2*d + 5*e + 8*f) + 7373599680*d**2*
e**2*f**2 - 4181781024*d**2*e**2*f*(2*d + 5*e + 8*f) - 43130880*d**2*e**2*(2*d + 5*e + 8*f)**2 - 4656496000*d*
*2*e*f**3 - 6980681664*d**2*e*f**2*(2*d + 5*e + 8*f) - 27235008*d**2*e*f*(2*d + 5*e + 8*f)**2 + 8882784*d**2*e
*(2*d + 5*e + 8*f)**3 - 6902995200*d**2*f**4 - 3854230656*d**2*f**3*(2*d + 5*e + 8*f) + 71604864*d**2*f**2*(2*
d + 5*e + 8*f)**2 + 19092672*d**2*f*(2*d + 5*e + 8*f)**3 + 1437185536*d*e**5 + 6500930720*d*e**4*f - 868139184
*d*e**4*(2*d + 5*e + 8*f) + 4803912960*d*e**3*f**2 - 5691921792*d*e**3*f*(2*d + 5*e + 8*f) - 67138560*d*e**3*(
2*d + 5*e + 8*f)**2 - 17839947520*d*e**2*f**3 - 13902127488*d*e**2*f**2*(2*d + 5*e + 8*f) - 90792576*d*e**2*f*
(2*d + 5*e + 8*f)**2 + 21757248*d*e**2*(2*d + 5*e + 8*f)**3 - 34607006720*d*e*f**4 - 15000967680*d*e*f**3*(2*d
+ 5*e + 8*f) + 240772608*d*e*f**2*(2*d + 5*e + 8*f)**2 + 76412160*d*e*f*(2*d + 5*e + 8*f)**3 - 17540771328*d*
f**5 - 6034591488*d*f**4*(2*d + 5*e + 8*f) + 330831360*d*f**3*(2*d + 5*e + 8*f)**2 + 64800000*d*f**2*(2*d + 5*
e + 8*f)**3 + 416940800*e**6 + 2005475776*e**5*f - 361090848*e**5*(2*d + 5*e + 8*f) + 122654080*e**4*f**2 - 28
70568768*e**4*f*(2*d + 5*e + 8*f) - 38274048*e**4*(2*d + 5*e + 8*f)**2 - 16377853440*e**3*f**3 - 9081659136*e*
*3*f**2*(2*d + 5*e + 8*f) - 84024576*e**3*f*(2*d + 5*e + 8*f)**2 + 16419456*e**3*(2*d + 5*e + 8*f)**3 - 393875
81440*e**2*f**4 - 14271235584*e**2*f**3*(2*d + 5*e + 8*f) + 196784640*e**2*f**2*(2*d + 5*e + 8*f)**2 + 7197465
6*e**2*f*(2*d + 5*e + 8*f)**3 - 37694227456*e*f**5 - 11115778560*e*f**4*(2*d + 5*e + 8*f) + 625075200*e*f**3*(
2*d + 5*e + 8*f)**2 + 100818432*e*f**2*(2*d + 5*e + 8*f)**3 - 13332408320*f**6 - 3425694720*f**5*(2*d + 5*e +
8*f) + 403200000*f**4*(2*d + 5*e + 8*f)**2 + 45093888*f**3*(2*d + 5*e + 8*f)**3)/(3374210*d**6 + 45393715*d**5
*e + 44170854*d**5*f + 247848970*d**4*e**2 + 464201768*d**4*e*f + 91507752*d**4*f**2 + 703178520*d**3*e**3 + 1
914915472*d**3*e**2*f + 607100704*d**3*e*f**2 - 999338816*d**3*f**3 + 1094421680*d**2*e**4 + 3892700544*d**2*e
**3*f + 1545770112*d**2*e**2*f**2 - 6739384832*d**2*e*f**3 - 5963752704*d**2*f**4 + 887062640*d*e**5 + 3907683
424*d*e**4*f + 1889544576*d*e**3*f**2 - 14086786304*d*e**2*f**3 - 24613469440*d*e*f**4 - 11900367360*d*f**5 +
292932640*e**6 + 1550127488*e**5*f + 952305536*e**4*f**2 - 9235000320*e**3*f**3 - 24236925440*e**2*f**4 - 2342
1052928*e*f**5 - 8154904576*f**6))/36 - (35*d + 58*e + 92*f)*log(x + (8710660*d**6 + 109305824*d**5*e + 136707
258*d**5*f + 2526593*d**5*(35*d + 58*e + 92*f)/4 + 548679440*d**4*e**2 + 1278644860*d**4*e*f + 14611963*d**4*e
*(35*d + 58*e + 92*f)/2 + 627558840*d**4*f**2 + 13498939*d**4*f*(35*d + 58*e + 92*f) - 9308*d**4*(35*d + 58*e
+ 92*f)**2 + 1416518400*d**3*e**3 + 4598750960*d**3*e**2*f + 32184326*d**3*e**2*(35*d + 58*e + 92*f) + 3756616
640*d**3*e*f**2 + 112605464*d**3*e*f*(35*d + 58*e + 92*f) - 85040*d**3*e*(35*d + 58*e + 92*f)**2 + 42632000*d*
*3*f**3 + 96718568*d**3*f**2*(35*d + 58*e + 92*f) - 12758*d**3*f*(35*d + 58*e + 92*f)**2 - 2441*d**3*(35*d + 5
8*e + 92*f)**3/4 + 1987752640*d**2*e**4 + 7924847520*d**2*e**3*f + 68847652*d**2*e**3*(35*d + 58*e + 92*f) + 7
373599680*d**2*e**2*f**2 + 348481752*d**2*e**2*f*(35*d + 58*e + 92*f) - 299520*d**2*e**2*(35*d + 58*e + 92*f)*
*2 - 4656496000*d**2*e*f**3 + 581723472*d**2*e*f**2*(35*d + 58*e + 92*f) - 189132*d**2*e*f*(35*d + 58*e + 92*f
)**2 - 10281*d**2*e*(35*d + 58*e + 92*f)**3/2 - 6902995200*d**2*f**4 + 321185888*d**2*f**3*(35*d + 58*e + 92*f
) + 497256*d**2*f**2*(35*d + 58*e + 92*f)**2 - 11049*d**2*f*(35*d + 58*e + 92*f)**3 + 1437185536*d*e**5 + 6500
930720*d*e**4*f + 72344932*d*e**4*(35*d + 58*e + 92*f) + 4803912960*d*e**3*f**2 + 474326816*d*e**3*f*(35*d + 5
8*e + 92*f) - 466240*d*e**3*(35*d + 58*e + 92*f)**2 - 17839947520*d*e**2*f**3 + 1158510624*d*e**2*f**2*(35*d +
58*e + 92*f) - 630504*d*e**2*f*(35*d + 58*e + 92*f)**2 - 12591*d*e**2*(35*d + 58*e + 92*f)**3 - 34607006720*d
*e*f**4 + 1250080640*d*e*f**3*(35*d + 58*e + 92*f) + 1672032*d*e*f**2*(35*d + 58*e + 92*f)**2 - 44220*d*e*f*(3
5*d + 58*e + 92*f)**3 - 17540771328*d*f**5 + 502882624*d*f**4*(35*d + 58*e + 92*f) + 2297440*d*f**3*(35*d + 58
*e + 92*f)**2 - 37500*d*f**2*(35*d + 58*e + 92*f)**3 + 416940800*e**6 + 2005475776*e**5*f + 30090904*e**5*(35*
d + 58*e + 92*f) + 122654080*e**4*f**2 + 239214064*e**4*f*(35*d + 58*e + 92*f) - 265792*e**4*(35*d + 58*e + 92
*f)**2 - 16377853440*e**3*f**3 + 756804928*e**3*f**2*(35*d + 58*e + 92*f) - 583504*e**3*f*(35*d + 58*e + 92*f)
**2 - 9502*e**3*(35*d + 58*e + 92*f)**3 - 39387581440*e**2*f**4 + 1189269632*e**2*f**3*(35*d + 58*e + 92*f) +
1366560*e**2*f**2*(35*d + 58*e + 92*f)**2 - 41652*e**2*f*(35*d + 58*e + 92*f)**3 - 37694227456*e*f**5 + 926314
880*e*f**4*(35*d + 58*e + 92*f) + 4340800*e*f**3*(35*d + 58*e + 92*f)**2 - 58344*e*f**2*(35*d + 58*e + 92*f)**
3 - 13332408320*f**6 + 285474560*f**5*(35*d + 58*e + 92*f) + 2800000*f**4*(35*d + 58*e + 92*f)**2 - 26096*f**3
*(35*d + 58*e + 92*f)**3)/(3374210*d**6 + 45393715*d**5*e + 44170854*d**5*f + 247848970*d**4*e**2 + 464201768*
d**4*e*f + 91507752*d**4*f**2 + 703178520*d**3*e**3 + 1914915472*d**3*e**2*f + 607100704*d**3*e*f**2 - 9993388
16*d**3*f**3 + 1094421680*d**2*e**4 + 3892700544*d**2*e**3*f + 1545770112*d**2*e**2*f**2 - 6739384832*d**2*e*f
**3 - 5963752704*d**2*f**4 + 887062640*d*e**5 + 3907683424*d*e**4*f + 1889544576*d*e**3*f**2 - 14086786304*d*e
**2*f**3 - 24613469440*d*e*f**4 - 11900367360*d*f**5 + 292932640*e**6 + 1550127488*e**5*f + 952305536*e**4*f**
2 - 9235000320*e**3*f**3 - 24236925440*e**2*f**4 - 23421052928*e*f**5 - 8154904576*f**6))/432 - (-5*d - 10*e -
8*f + x**2*(5*d + 4*e + 8*f) + x*(-6*d - 6*f))/(36*x**3 - 72*x**2 - 36*x + 72)
________________________________________________________________________________________
Giac [A] time = 1.07637, size = 159, normalized size = 1.3 \begin{align*} \frac{1}{144} \,{\left (d + 4 \, f - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{108} \,{\left (2 \, d - 4 \, f + e\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{36} \,{\left (2 \, d + 8 \, f + 5 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) - \frac{1}{432} \,{\left (35 \, d + 92 \, f + 58 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac{{\left (5 \, d + 8 \, f + 4 \, e\right )} x^{2} - 6 \,{\left (d + f\right )} x - 5 \, d - 8 \, f - 10 \, e}{36 \,{\left (x + 1\right )}{\left (x - 1\right )}{\left (x - 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+x)*(f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="giac")
[Out]
1/144*(d + 4*f - 2*e)*log(abs(x + 2)) + 1/108*(2*d - 4*f + e)*log(abs(x + 1)) + 1/36*(2*d + 8*f + 5*e)*log(abs
(x - 1)) - 1/432*(35*d + 92*f + 58*e)*log(abs(x - 2)) - 1/36*((5*d + 8*f + 4*e)*x^2 - 6*(d + f)*x - 5*d - 8*f
- 10*e)/((x + 1)*(x - 1)*(x - 2))