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

Optimal. Leaf size=175 \[ -\frac{x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{(313 d+820 f) \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{1}{648} (13 d+25 f) \tanh ^{-1}(x)-\frac{1}{81} e \log \left (1-x^2\right )+\frac{1}{81} e \log \left (4-x^2\right )-\frac{e \left (5-2 x^2\right )}{54 \left (x^4-5 x^2+4\right )}+\frac{e \left (5-2 x^2\right )}{36 \left (x^4-5 x^2+4\right )^2} \]

[Out]

(e*(5 - 2*x^2))/(36*(4 - 5*x^2 + x^4)^2) + (x*(17*d + 20*f - (5*d + 8*f)*x^2))/(
144*(4 - 5*x^2 + x^4)^2) - (e*(5 - 2*x^2))/(54*(4 - 5*x^2 + x^4)) - (x*(59*d + 3
80*f - 35*(d + 4*f)*x^2))/(3456*(4 - 5*x^2 + x^4)) - ((313*d + 820*f)*ArcTanh[x/
2])/20736 + ((13*d + 25*f)*ArcTanh[x])/648 - (e*Log[1 - x^2])/81 + (e*Log[4 - x^
2])/81

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Rubi [A]  time = 0.452485, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ -\frac{x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{(313 d+820 f) \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{1}{648} (13 d+25 f) \tanh ^{-1}(x)-\frac{1}{81} e \log \left (1-x^2\right )+\frac{1}{81} e \log \left (4-x^2\right )-\frac{e \left (5-2 x^2\right )}{54 \left (x^4-5 x^2+4\right )}+\frac{e \left (5-2 x^2\right )}{36 \left (x^4-5 x^2+4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(e*(5 - 2*x^2))/(36*(4 - 5*x^2 + x^4)^2) + (x*(17*d + 20*f - (5*d + 8*f)*x^2))/(
144*(4 - 5*x^2 + x^4)^2) - (e*(5 - 2*x^2))/(54*(4 - 5*x^2 + x^4)) - (x*(59*d + 3
80*f - 35*(d + 4*f)*x^2))/(3456*(4 - 5*x^2 + x^4)) - ((313*d + 820*f)*ArcTanh[x/
2])/20736 + ((13*d + 25*f)*ArcTanh[x])/648 - (e*Log[1 - x^2])/81 + (e*Log[4 - x^
2])/81

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Rubi in Sympy [A]  time = 62.9354, size = 134, normalized size = 0.77 \[ - \frac{e \log{\left (- x^{2} + 1 \right )}}{81} + \frac{e \log{\left (- x^{2} + 4 \right )}}{81} - \frac{x \left (177 d - 150 e x^{3} + 366 e x + 1140 f - x^{2} \left (105 d + 420 f\right )\right )}{10368 \left (x^{4} - 5 x^{2} + 4\right )} + \frac{x \left (17 d - 5 e x^{3} + 17 e x + 20 f - x^{2} \left (5 d + 8 f\right )\right )}{144 \left (x^{4} - 5 x^{2} + 4\right )^{2}} - \left (\frac{313 d}{20736} + \frac{205 f}{5184}\right ) \operatorname{atanh}{\left (\frac{x}{2} \right )} + \left (\frac{13 d}{648} + \frac{25 f}{648}\right ) \operatorname{atanh}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**2+e*x+d)/(x**4-5*x**2+4)**3,x)

[Out]

-e*log(-x**2 + 1)/81 + e*log(-x**2 + 4)/81 - x*(177*d - 150*e*x**3 + 366*e*x + 1
140*f - x**2*(105*d + 420*f))/(10368*(x**4 - 5*x**2 + 4)) + x*(17*d - 5*e*x**3 +
17*e*x + 20*f - x**2*(5*d + 8*f))/(144*(x**4 - 5*x**2 + 4)**2) - (313*d/20736 +
205*f/5184)*atanh(x/2) + (13*d/648 + 25*f/648)*atanh(x)

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Mathematica [A]  time = 0.264777, size = 161, normalized size = 0.92 \[ \frac{\frac{12 \left (d x \left (35 x^2-59\right )+64 e \left (2 x^2-5\right )+20 f x \left (7 x^2-19\right )\right )}{x^4-5 x^2+4}+\frac{288 \left (-5 d x^3+17 d x+e \left (20-8 x^2\right )-8 f x^3+20 f x\right )}{\left (x^4-5 x^2+4\right )^2}-32 \log (1-x) (13 d+16 e+25 f)+\log (2-x) (313 d+512 e+820 f)+32 \log (x+1) (13 d-16 e+25 f)+\log (x+2) (-313 d+512 e-820 f)}{41472} \]

Antiderivative was successfully verified.

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

[Out]

((288*(17*d*x + 20*f*x - 5*d*x^3 - 8*f*x^3 + e*(20 - 8*x^2)))/(4 - 5*x^2 + x^4)^
2 + (12*(64*e*(-5 + 2*x^2) + 20*f*x*(-19 + 7*x^2) + d*x*(-59 + 35*x^2)))/(4 - 5*
x^2 + x^4) - 32*(13*d + 16*e + 25*f)*Log[1 - x] + (313*d + 512*e + 820*f)*Log[2
- x] + 32*(13*d - 16*e + 25*f)*Log[1 + x] + (-313*d + 512*e - 820*f)*Log[2 + x])
/41472

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Maple [A]  time = 0.029, size = 278, normalized size = 1.6 \[{\frac{d}{432\, \left ( -1+x \right ) ^{2}}}+{\frac{e}{432\, \left ( -1+x \right ) ^{2}}}-{\frac{d}{432\, \left ( 1+x \right ) ^{2}}}+{\frac{e}{432\, \left ( 1+x \right ) ^{2}}}-{\frac{d}{3456\, \left ( x-2 \right ) ^{2}}}-{\frac{e}{1728\, \left ( x-2 \right ) ^{2}}}+{\frac{d}{3456\, \left ( 2+x \right ) ^{2}}}-{\frac{e}{1728\, \left ( 2+x \right ) ^{2}}}-{\frac{f}{432\, \left ( 1+x \right ) ^{2}}}-{\frac{f}{864\, \left ( x-2 \right ) ^{2}}}+{\frac{f}{864\, \left ( 2+x \right ) ^{2}}}+{\frac{f}{432\, \left ( -1+x \right ) ^{2}}}+{\frac{5\,f}{432+432\,x}}+{\frac{d}{432+432\,x}}-{\frac{e}{144+144\,x}}+{\frac{19\,d}{6912\,x-13824}}+{\frac{17\,e}{3456\,x-6912}}+{\frac{5\,f}{576\,x-1152}}+{\frac{5\,f}{-432+432\,x}}+{\frac{19\,d}{13824+6912\,x}}-{\frac{17\,e}{6912+3456\,x}}+{\frac{d}{-432+432\,x}}+{\frac{e}{-144+144\,x}}+{\frac{5\,f}{1152+576\,x}}+{\frac{13\,\ln \left ( 1+x \right ) d}{1296}}-{\frac{\ln \left ( 1+x \right ) e}{81}}-{\frac{13\,\ln \left ( -1+x \right ) d}{1296}}-{\frac{\ln \left ( -1+x \right ) e}{81}}+{\frac{313\,\ln \left ( x-2 \right ) d}{41472}}+{\frac{\ln \left ( x-2 \right ) e}{81}}+{\frac{\ln \left ( 2+x \right ) e}{81}}+{\frac{205\,\ln \left ( x-2 \right ) f}{10368}}-{\frac{313\,\ln \left ( 2+x \right ) d}{41472}}+{\frac{25\,\ln \left ( 1+x \right ) f}{1296}}-{\frac{25\,\ln \left ( -1+x \right ) f}{1296}}-{\frac{205\,\ln \left ( 2+x \right ) f}{10368}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/432/(-1+x)^2*d+1/432/(-1+x)^2*e-1/432/(1+x)^2*d+1/432/(1+x)^2*e-1/3456/(x-2)^2
*d-1/1728/(x-2)^2*e+1/3456/(2+x)^2*d-1/1728/(2+x)^2*e-1/432/(1+x)^2*f-1/864/(x-2
)^2*f+1/864/(2+x)^2*f+1/432/(-1+x)^2*f+5/432/(1+x)*f+1/432/(1+x)*d-1/144/(1+x)*e
+19/6912/(x-2)*d+17/3456/(x-2)*e+5/576/(x-2)*f+5/432/(-1+x)*f+19/6912/(2+x)*d-17
/3456/(2+x)*e+1/432/(-1+x)*d+1/144/(-1+x)*e+5/576/(2+x)*f+13/1296*ln(1+x)*d-1/81
*ln(1+x)*e-13/1296*ln(-1+x)*d-1/81*ln(-1+x)*e+313/41472*ln(x-2)*d+1/81*ln(x-2)*e
+1/81*ln(2+x)*e+205/10368*ln(x-2)*f-313/41472*ln(2+x)*d+25/1296*ln(1+x)*f-25/129
6*ln(-1+x)*f-205/10368*ln(2+x)*f

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Maxima [A]  time = 0.705283, size = 209, normalized size = 1.19 \[ -\frac{1}{41472} \,{\left (313 \, d - 512 \, e + 820 \, f\right )} \log \left (x + 2\right ) + \frac{1}{1296} \,{\left (13 \, d - 16 \, e + 25 \, f\right )} \log \left (x + 1\right ) - \frac{1}{1296} \,{\left (13 \, d + 16 \, e + 25 \, f\right )} \log \left (x - 1\right ) + \frac{1}{41472} \,{\left (313 \, d + 512 \, e + 820 \, f\right )} \log \left (x - 2\right ) + \frac{35 \,{\left (d + 4 \, f\right )} x^{7} + 128 \, e x^{6} - 18 \,{\left (13 \, d + 60 \, f\right )} x^{5} - 960 \, e x^{4} + 63 \,{\left (5 \, d + 36 \, f\right )} x^{3} + 1920 \, e x^{2} + 4 \,{\left (43 \, d - 260 \, f\right )} x - 800 \, e}{3456 \,{\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/41472*(313*d - 512*e + 820*f)*log(x + 2) + 1/1296*(13*d - 16*e + 25*f)*log(x
+ 1) - 1/1296*(13*d + 16*e + 25*f)*log(x - 1) + 1/41472*(313*d + 512*e + 820*f)*
log(x - 2) + 1/3456*(35*(d + 4*f)*x^7 + 128*e*x^6 - 18*(13*d + 60*f)*x^5 - 960*e
*x^4 + 63*(5*d + 36*f)*x^3 + 1920*e*x^2 + 4*(43*d - 260*f)*x - 800*e)/(x^8 - 10*
x^6 + 33*x^4 - 40*x^2 + 16)

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Fricas [A]  time = 0.349195, size = 525, normalized size = 3. \[ \frac{420 \,{\left (d + 4 \, f\right )} x^{7} + 1536 \, e x^{6} - 216 \,{\left (13 \, d + 60 \, f\right )} x^{5} - 11520 \, e x^{4} + 756 \,{\left (5 \, d + 36 \, f\right )} x^{3} + 23040 \, e x^{2} + 48 \,{\left (43 \, d - 260 \, f\right )} x -{\left ({\left (313 \, d - 512 \, e + 820 \, f\right )} x^{8} - 10 \,{\left (313 \, d - 512 \, e + 820 \, f\right )} x^{6} + 33 \,{\left (313 \, d - 512 \, e + 820 \, f\right )} x^{4} - 40 \,{\left (313 \, d - 512 \, e + 820 \, f\right )} x^{2} + 5008 \, d - 8192 \, e + 13120 \, f\right )} \log \left (x + 2\right ) + 32 \,{\left ({\left (13 \, d - 16 \, e + 25 \, f\right )} x^{8} - 10 \,{\left (13 \, d - 16 \, e + 25 \, f\right )} x^{6} + 33 \,{\left (13 \, d - 16 \, e + 25 \, f\right )} x^{4} - 40 \,{\left (13 \, d - 16 \, e + 25 \, f\right )} x^{2} + 208 \, d - 256 \, e + 400 \, f\right )} \log \left (x + 1\right ) - 32 \,{\left ({\left (13 \, d + 16 \, e + 25 \, f\right )} x^{8} - 10 \,{\left (13 \, d + 16 \, e + 25 \, f\right )} x^{6} + 33 \,{\left (13 \, d + 16 \, e + 25 \, f\right )} x^{4} - 40 \,{\left (13 \, d + 16 \, e + 25 \, f\right )} x^{2} + 208 \, d + 256 \, e + 400 \, f\right )} \log \left (x - 1\right ) +{\left ({\left (313 \, d + 512 \, e + 820 \, f\right )} x^{8} - 10 \,{\left (313 \, d + 512 \, e + 820 \, f\right )} x^{6} + 33 \,{\left (313 \, d + 512 \, e + 820 \, f\right )} x^{4} - 40 \,{\left (313 \, d + 512 \, e + 820 \, f\right )} x^{2} + 5008 \, d + 8192 \, e + 13120 \, f\right )} \log \left (x - 2\right ) - 9600 \, e}{41472 \,{\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/41472*(420*(d + 4*f)*x^7 + 1536*e*x^6 - 216*(13*d + 60*f)*x^5 - 11520*e*x^4 +
756*(5*d + 36*f)*x^3 + 23040*e*x^2 + 48*(43*d - 260*f)*x - ((313*d - 512*e + 820
*f)*x^8 - 10*(313*d - 512*e + 820*f)*x^6 + 33*(313*d - 512*e + 820*f)*x^4 - 40*(
313*d - 512*e + 820*f)*x^2 + 5008*d - 8192*e + 13120*f)*log(x + 2) + 32*((13*d -
16*e + 25*f)*x^8 - 10*(13*d - 16*e + 25*f)*x^6 + 33*(13*d - 16*e + 25*f)*x^4 -
40*(13*d - 16*e + 25*f)*x^2 + 208*d - 256*e + 400*f)*log(x + 1) - 32*((13*d + 16
*e + 25*f)*x^8 - 10*(13*d + 16*e + 25*f)*x^6 + 33*(13*d + 16*e + 25*f)*x^4 - 40*
(13*d + 16*e + 25*f)*x^2 + 208*d + 256*e + 400*f)*log(x - 1) + ((313*d + 512*e +
820*f)*x^8 - 10*(313*d + 512*e + 820*f)*x^6 + 33*(313*d + 512*e + 820*f)*x^4 -
40*(313*d + 512*e + 820*f)*x^2 + 5008*d + 8192*e + 13120*f)*log(x - 2) - 9600*e)
/(x^8 - 10*x^6 + 33*x^4 - 40*x^2 + 16)

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Sympy [A]  time = 118.807, size = 2822, normalized size = 16.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**2+e*x+d)/(x**4-5*x**2+4)**3,x)

[Out]

(13*d - 16*e + 25*f)*log(x + (-1106258459719280*d**5*e - 13113710954343*d**5*(13
*d - 16*e + 25*f) - 12929482401572800*d**4*e*f - 107063904267900*d**4*f*(13*d -
16*e + 25*f) - 817263343042560*d**3*e**3 + 153628968222720*d**3*e**2*(13*d - 16*
e + 25*f) - 59478343838144000*d**3*e*f**2 + 9530197557248*d**3*e*(13*d - 16*e +
25*f)**2 - 324891412840800*d**3*f**2*(13*d - 16*e + 25*f) + 88038005760*d**3*(13
*d - 16*e + 25*f)**3 - 2885705898393600*d**2*e**3*f + 1014848673546240*d**2*e**2
*f*(13*d - 16*e + 25*f) - 134905286808320000*d**2*e*f**3 + 63469758382080*d**2*e
*f*(13*d - 16*e + 25*f)**2 - 422972724528000*d**2*f**3*(13*d - 16*e + 25*f) + 36
4616847360*d**2*f*(13*d - 16*e + 25*f)**3 + 5035763255214080*d*e**5 + 1426616337
03936*d*e**4*(13*d - 16*e + 25*f) - 2138314899456000*d*e**3*f**2 - 1967095021568
0*d*e**3*(13*d - 16*e + 25*f)**2 + 2257033730457600*d*e**2*f**2*(13*d - 16*e + 2
5*f) - 557272006656*d*e**2*(13*d - 16*e + 25*f)**3 - 151082645593600000*d*e*f**4
+ 141056507904000*d*e*f**2*(13*d - 16*e + 25*f)**2 - 167683154400000*d*f**4*(13
*d - 16*e + 25*f) + 339373670400*d*f**2*(13*d - 16*e + 25*f)**3 + 10643272556871
680*e**5*f + 214404767416320*e**4*f*(13*d - 16*e + 25*f) + 529992253440000*e**3*
f**3 - 41575283425280*e**3*f*(13*d - 16*e + 25*f)**2 + 1671759396864000*e**2*f**
3*(13*d - 16*e + 25*f) - 837518622720*e**2*f*(13*d - 16*e + 25*f)**3 - 668954521
08800000*e*f**5 + 104485486592000*e*f**3*(13*d - 16*e + 25*f)**2 + 5104192320000
0*f**5*(13*d - 16*e + 25*f) - 80289792000*f**3*(13*d - 16*e + 25*f)**3)/(2294125
6248261*d**6 + 197271407316645*d**5*f - 2312740746035200*d**4*e**2 + 61286291092
8900*d**4*f**2 - 20566607354920960*d**3*e**2*f + 767363353812000*d**3*f**3 + 447
3912813420544*d**2*e**4 - 68552762169753600*d**2*e**2*f**2 + 197499222000000*d**
2*f**4 + 20324472439439360*d*e**4*f - 101559983669248000*d*e**2*f**3 - 182883938
400000*d*f**5 + 22539988369408000*e**4*f**2 - 56422196838400000*e**2*f**4 + 2152
0080000000*f**6))/1296 - (13*d + 16*e + 25*f)*log(x + (-1106258459719280*d**5*e
+ 13113710954343*d**5*(13*d + 16*e + 25*f) - 12929482401572800*d**4*e*f + 107063
904267900*d**4*f*(13*d + 16*e + 25*f) - 817263343042560*d**3*e**3 - 153628968222
720*d**3*e**2*(13*d + 16*e + 25*f) - 59478343838144000*d**3*e*f**2 + 95301975572
48*d**3*e*(13*d + 16*e + 25*f)**2 + 324891412840800*d**3*f**2*(13*d + 16*e + 25*
f) - 88038005760*d**3*(13*d + 16*e + 25*f)**3 - 2885705898393600*d**2*e**3*f - 1
014848673546240*d**2*e**2*f*(13*d + 16*e + 25*f) - 134905286808320000*d**2*e*f**
3 + 63469758382080*d**2*e*f*(13*d + 16*e + 25*f)**2 + 422972724528000*d**2*f**3*
(13*d + 16*e + 25*f) - 364616847360*d**2*f*(13*d + 16*e + 25*f)**3 + 50357632552
14080*d*e**5 - 142661633703936*d*e**4*(13*d + 16*e + 25*f) - 2138314899456000*d*
e**3*f**2 - 19670950215680*d*e**3*(13*d + 16*e + 25*f)**2 - 2257033730457600*d*e
**2*f**2*(13*d + 16*e + 25*f) + 557272006656*d*e**2*(13*d + 16*e + 25*f)**3 - 15
1082645593600000*d*e*f**4 + 141056507904000*d*e*f**2*(13*d + 16*e + 25*f)**2 + 1
67683154400000*d*f**4*(13*d + 16*e + 25*f) - 339373670400*d*f**2*(13*d + 16*e +
25*f)**3 + 10643272556871680*e**5*f - 214404767416320*e**4*f*(13*d + 16*e + 25*f
) + 529992253440000*e**3*f**3 - 41575283425280*e**3*f*(13*d + 16*e + 25*f)**2 -
1671759396864000*e**2*f**3*(13*d + 16*e + 25*f) + 837518622720*e**2*f*(13*d + 16
*e + 25*f)**3 - 66895452108800000*e*f**5 + 104485486592000*e*f**3*(13*d + 16*e +
25*f)**2 - 51041923200000*f**5*(13*d + 16*e + 25*f) + 80289792000*f**3*(13*d +
16*e + 25*f)**3)/(22941256248261*d**6 + 197271407316645*d**5*f - 231274074603520
0*d**4*e**2 + 612862910928900*d**4*f**2 - 20566607354920960*d**3*e**2*f + 767363
353812000*d**3*f**3 + 4473912813420544*d**2*e**4 - 68552762169753600*d**2*e**2*f
**2 + 197499222000000*d**2*f**4 + 20324472439439360*d*e**4*f - 10155998366924800
0*d*e**2*f**3 - 182883938400000*d*f**5 + 22539988369408000*e**4*f**2 - 564221968
38400000*e**2*f**4 + 21520080000000*f**6))/1296 - (313*d - 512*e + 820*f)*log(x
+ (-1106258459719280*d**5*e + 13113710954343*d**5*(313*d - 512*e + 820*f)/32 - 1
2929482401572800*d**4*e*f + 26765976066975*d**4*f*(313*d - 512*e + 820*f)/8 - 81
7263343042560*d**3*e**3 - 4800905256960*d**3*e**2*(313*d - 512*e + 820*f) - 5947
8343838144000*d**3*e*f**2 + 9306833552*d**3*e*(313*d - 512*e + 820*f)**2 + 10152
856651275*d**3*f**2*(313*d - 512*e + 820*f) - 85974615*d**3*(313*d - 512*e + 820
*f)**3/32 - 2885705898393600*d**2*e**3*f - 31714021048320*d**2*e**2*f*(313*d - 5
12*e + 820*f) - 134905286808320000*d**2*e*f**3 + 61982185920*d**2*e*f*(313*d - 5
12*e + 820*f)**2 + 13217897641500*d**2*f**3*(313*d - 512*e + 820*f) - 89017785*d
**2*f*(313*d - 512*e + 820*f)**3/8 + 5035763255214080*d*e**5 - 4458176053248*d*e
**4*(313*d - 512*e + 820*f) - 2138314899456000*d*e**3*f**2 - 19209912320*d*e**3*
(313*d - 512*e + 820*f)**2 - 70532304076800*d*e**2*f**2*(313*d - 512*e + 820*f)
+ 17006592*d*e**2*(313*d - 512*e + 820*f)**3 - 151082645593600000*d*e*f**4 + 137
750496000*d*e*f**2*(313*d - 512*e + 820*f)**2 + 5240098575000*d*f**4*(313*d - 51
2*e + 820*f) - 20713725*d*f**2*(313*d - 512*e + 820*f)**3/2 + 10643272556871680*
e**5*f - 6700148981760*e**4*f*(313*d - 512*e + 820*f) + 529992253440000*e**3*f**
3 - 40600862720*e**3*f*(313*d - 512*e + 820*f)**2 - 52242481152000*e**2*f**3*(31
3*d - 512*e + 820*f) + 25559040*e**2*f*(313*d - 512*e + 820*f)**3 - 668954521088
00000*e*f**5 + 102036608000*e*f**3*(313*d - 512*e + 820*f)**2 - 1595060100000*f*
*5*(313*d - 512*e + 820*f) + 2450250*f**3*(313*d - 512*e + 820*f)**3)/(229412562
48261*d**6 + 197271407316645*d**5*f - 2312740746035200*d**4*e**2 + 6128629109289
00*d**4*f**2 - 20566607354920960*d**3*e**2*f + 767363353812000*d**3*f**3 + 44739
12813420544*d**2*e**4 - 68552762169753600*d**2*e**2*f**2 + 197499222000000*d**2*
f**4 + 20324472439439360*d*e**4*f - 101559983669248000*d*e**2*f**3 - 18288393840
0000*d*f**5 + 22539988369408000*e**4*f**2 - 56422196838400000*e**2*f**4 + 215200
80000000*f**6))/41472 + (313*d + 512*e + 820*f)*log(x + (-1106258459719280*d**5*
e - 13113710954343*d**5*(313*d + 512*e + 820*f)/32 - 12929482401572800*d**4*e*f
- 26765976066975*d**4*f*(313*d + 512*e + 820*f)/8 - 817263343042560*d**3*e**3 +
4800905256960*d**3*e**2*(313*d + 512*e + 820*f) - 59478343838144000*d**3*e*f**2
+ 9306833552*d**3*e*(313*d + 512*e + 820*f)**2 - 10152856651275*d**3*f**2*(313*d
+ 512*e + 820*f) + 85974615*d**3*(313*d + 512*e + 820*f)**3/32 - 28857058983936
00*d**2*e**3*f + 31714021048320*d**2*e**2*f*(313*d + 512*e + 820*f) - 1349052868
08320000*d**2*e*f**3 + 61982185920*d**2*e*f*(313*d + 512*e + 820*f)**2 - 1321789
7641500*d**2*f**3*(313*d + 512*e + 820*f) + 89017785*d**2*f*(313*d + 512*e + 820
*f)**3/8 + 5035763255214080*d*e**5 + 4458176053248*d*e**4*(313*d + 512*e + 820*f
) - 2138314899456000*d*e**3*f**2 - 19209912320*d*e**3*(313*d + 512*e + 820*f)**2
+ 70532304076800*d*e**2*f**2*(313*d + 512*e + 820*f) - 17006592*d*e**2*(313*d +
512*e + 820*f)**3 - 151082645593600000*d*e*f**4 + 137750496000*d*e*f**2*(313*d
+ 512*e + 820*f)**2 - 5240098575000*d*f**4*(313*d + 512*e + 820*f) + 20713725*d*
f**2*(313*d + 512*e + 820*f)**3/2 + 10643272556871680*e**5*f + 6700148981760*e**
4*f*(313*d + 512*e + 820*f) + 529992253440000*e**3*f**3 - 40600862720*e**3*f*(31
3*d + 512*e + 820*f)**2 + 52242481152000*e**2*f**3*(313*d + 512*e + 820*f) - 255
59040*e**2*f*(313*d + 512*e + 820*f)**3 - 66895452108800000*e*f**5 + 10203660800
0*e*f**3*(313*d + 512*e + 820*f)**2 + 1595060100000*f**5*(313*d + 512*e + 820*f)
- 2450250*f**3*(313*d + 512*e + 820*f)**3)/(22941256248261*d**6 + 1972714073166
45*d**5*f - 2312740746035200*d**4*e**2 + 612862910928900*d**4*f**2 - 20566607354
920960*d**3*e**2*f + 767363353812000*d**3*f**3 + 4473912813420544*d**2*e**4 - 68
552762169753600*d**2*e**2*f**2 + 197499222000000*d**2*f**4 + 20324472439439360*d
*e**4*f - 101559983669248000*d*e**2*f**3 - 182883938400000*d*f**5 + 225399883694
08000*e**4*f**2 - 56422196838400000*e**2*f**4 + 21520080000000*f**6))/41472 + (1
28*e*x**6 - 960*e*x**4 + 1920*e*x**2 - 800*e + x**7*(35*d + 140*f) + x**5*(-234*
d - 1080*f) + x**3*(315*d + 2268*f) + x*(172*d - 1040*f))/(3456*x**8 - 34560*x**
6 + 114048*x**4 - 138240*x**2 + 55296)

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GIAC/XCAS [A]  time = 0.271668, size = 212, normalized size = 1.21 \[ -\frac{1}{41472} \,{\left (313 \, d + 820 \, f - 512 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{1296} \,{\left (13 \, d + 25 \, f - 16 \, e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{1296} \,{\left (13 \, d + 25 \, f + 16 \, e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{41472} \,{\left (313 \, d + 820 \, f + 512 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) + \frac{35 \, d x^{7} + 140 \, f x^{7} + 128 \, x^{6} e - 234 \, d x^{5} - 1080 \, f x^{5} - 960 \, x^{4} e + 315 \, d x^{3} + 2268 \, f x^{3} + 1920 \, x^{2} e + 172 \, d x - 1040 \, f x - 800 \, e}{3456 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/41472*(313*d + 820*f - 512*e)*ln(abs(x + 2)) + 1/1296*(13*d + 25*f - 16*e)*ln
(abs(x + 1)) - 1/1296*(13*d + 25*f + 16*e)*ln(abs(x - 1)) + 1/41472*(313*d + 820
*f + 512*e)*ln(abs(x - 2)) + 1/3456*(35*d*x^7 + 140*f*x^7 + 128*x^6*e - 234*d*x^
5 - 1080*f*x^5 - 960*x^4*e + 315*d*x^3 + 2268*f*x^3 + 1920*x^2*e + 172*d*x - 104
0*f*x - 800*e)/(x^4 - 5*x^2 + 4)^2