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

Optimal. Leaf size=239 \[ -\frac{x \left (-5 x^2 (7 d+28 f+64 h)+59 d+380 f+848 h\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{x \left (x^2 (-(5 d+8 f+20 h))+17 d+20 f+32 h\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{\tanh ^{-1}\left (\frac{x}{2}\right ) (313 d+820 f+1936 h)}{20736}+\frac{1}{648} \tanh ^{-1}(x) (13 d+25 f+61 h)-\frac{1}{162} \log \left (1-x^2\right ) (2 e+5 g+11 i)+\frac{1}{162} \log \left (4-x^2\right ) (2 e+5 g+11 i)-\frac{\left (5-2 x^2\right ) (2 e+5 g+11 i)}{108 \left (x^4-5 x^2+4\right )}+\frac{x^2 (-(2 e+5 g+17 i))+5 e+8 g+20 i}{36 \left (x^4-5 x^2+4\right )^2} \]

[Out]

(x*(17*d + 20*f + 32*h - (5*d + 8*f + 20*h)*x^2))/(144*(4 - 5*x^2 + x^4)^2) + (5
*e + 8*g + 20*i - (2*e + 5*g + 17*i)*x^2)/(36*(4 - 5*x^2 + x^4)^2) - ((2*e + 5*g
+ 11*i)*(5 - 2*x^2))/(108*(4 - 5*x^2 + x^4)) - (x*(59*d + 380*f + 848*h - 5*(7*
d + 28*f + 64*h)*x^2))/(3456*(4 - 5*x^2 + x^4)) - ((313*d + 820*f + 1936*h)*ArcT
anh[x/2])/20736 + ((13*d + 25*f + 61*h)*ArcTanh[x])/648 - ((2*e + 5*g + 11*i)*Lo
g[1 - x^2])/162 + ((2*e + 5*g + 11*i)*Log[4 - x^2])/162

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Rubi [A]  time = 0.660687, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29 \[ -\frac{x \left (-5 x^2 (7 d+28 f+64 h)+59 d+380 f+848 h\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{x \left (x^2 (-(5 d+8 f+20 h))+17 d+20 f+32 h\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{\tanh ^{-1}\left (\frac{x}{2}\right ) (313 d+820 f+1936 h)}{20736}+\frac{1}{648} \tanh ^{-1}(x) (13 d+25 f+61 h)-\frac{1}{162} \log \left (1-x^2\right ) (2 e+5 g+11 i)+\frac{1}{162} \log \left (4-x^2\right ) (2 e+5 g+11 i)-\frac{\left (5-2 x^2\right ) (2 e+5 g+11 i)}{108 \left (x^4-5 x^2+4\right )}+\frac{x^2 (-(2 e+5 g+17 i))+5 e+8 g+20 i}{36 \left (x^4-5 x^2+4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(17*d + 20*f + 32*h - (5*d + 8*f + 20*h)*x^2))/(144*(4 - 5*x^2 + x^4)^2) + (5
*e + 8*g + 20*i - (2*e + 5*g + 17*i)*x^2)/(36*(4 - 5*x^2 + x^4)^2) - ((2*e + 5*g
+ 11*i)*(5 - 2*x^2))/(108*(4 - 5*x^2 + x^4)) - (x*(59*d + 380*f + 848*h - 5*(7*
d + 28*f + 64*h)*x^2))/(3456*(4 - 5*x^2 + x^4)) - ((313*d + 820*f + 1936*h)*ArcT
anh[x/2])/20736 + ((13*d + 25*f + 61*h)*ArcTanh[x])/648 - ((2*e + 5*g + 11*i)*Lo
g[1 - x^2])/162 + ((2*e + 5*g + 11*i)*Log[4 - x^2])/162

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Rubi in Sympy [A]  time = 104.153, size = 196, normalized size = 0.82 \[ - \frac{x \left (22125 d + 142500 f + 318000 h - x^{3} \left (18750 e + 57000 g + 120000\right ) - x^{2} \left (13125 d + 52500 f + 120000 h\right ) + x \left (45750 e + 165000 g + 336000\right )\right )}{1296000 \left (x^{4} - 5 x^{2} + 4\right )} + \frac{x \left (2125 d + 2500 f + 4000 h - x^{3} \left (625 e + 1000 g + 2500\right ) - x^{2} \left (625 d + 1000 f + 2500 h\right ) + x \left (2125 e + 2500 g + 4000\right )\right )}{18000 \left (x^{4} - 5 x^{2} + 4\right )^{2}} - \left (\frac{313 d}{20736} + \frac{205 f}{5184} + \frac{121 h}{1296}\right ) \operatorname{atanh}{\left (\frac{x}{2} \right )} + \left (\frac{13 d}{648} + \frac{25 f}{648} + \frac{61 h}{648}\right ) \operatorname{atanh}{\left (x \right )} - \left (\frac{e}{81} + \frac{5 g}{162} + \frac{11}{162}\right ) \log{\left (- x^{2} + 1 \right )} + \left (\frac{e}{81} + \frac{5 g}{162} + \frac{11}{162}\right ) \log{\left (- x^{2} + 4 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(22125*d + 142500*f + 318000*h - x**3*(18750*e + 57000*g + 120000) - x**2*(13
125*d + 52500*f + 120000*h) + x*(45750*e + 165000*g + 336000))/(1296000*(x**4 -
5*x**2 + 4)) + x*(2125*d + 2500*f + 4000*h - x**3*(625*e + 1000*g + 2500) - x**2
*(625*d + 1000*f + 2500*h) + x*(2125*e + 2500*g + 4000))/(18000*(x**4 - 5*x**2 +
4)**2) - (313*d/20736 + 205*f/5184 + 121*h/1296)*atanh(x/2) + (13*d/648 + 25*f/
648 + 61*h/648)*atanh(x) - (e/81 + 5*g/162 + 11/162)*log(-x**2 + 1) + (e/81 + 5*
g/162 + 11/162)*log(-x**2 + 4)

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Mathematica [A]  time = 0.304763, size = 261, normalized size = 1.09 \[ \frac{-5 d x^3+17 d x-8 e x^2+20 e-8 f x^3+20 f x-20 g x^2+32 g-20 h x^3+32 h x-68 i x^2+80 i}{144 \left (x^4-5 x^2+4\right )^2}+\frac{35 d x^3-59 d x+128 e x^2-320 e+140 f x^3-380 f x+320 g x^2-800 g+320 h x^3-848 h x+704 i x^2-1760 i}{3456 \left (x^4-5 x^2+4\right )}+\frac{\log (1-x) (-13 d-16 e-25 f-40 g-61 h-88 i)}{1296}+\frac{\log (2-x) (313 d+512 e+820 f+1280 g+1936 h+2816 i)}{41472}+\frac{\log (x+1) (13 d-16 e+25 f-40 g+61 h-88 i)}{1296}+\frac{\log (x+2) (-313 d+512 e-820 f+1280 g-1936 h+2816 i)}{41472} \]

Antiderivative was successfully verified.

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

[Out]

(20*e + 32*g + 80*i + 17*d*x + 20*f*x + 32*h*x - 8*e*x^2 - 20*g*x^2 - 68*i*x^2 -
5*d*x^3 - 8*f*x^3 - 20*h*x^3)/(144*(4 - 5*x^2 + x^4)^2) + (-320*e - 800*g - 176
0*i - 59*d*x - 380*f*x - 848*h*x + 128*e*x^2 + 320*g*x^2 + 704*i*x^2 + 35*d*x^3
+ 140*f*x^3 + 320*h*x^3)/(3456*(4 - 5*x^2 + x^4)) + ((-13*d - 16*e - 25*f - 40*g
- 61*h - 88*i)*Log[1 - x])/1296 + ((313*d + 512*e + 820*f + 1280*g + 1936*h + 2
816*i)*Log[2 - x])/41472 + ((13*d - 16*e + 25*f - 40*g + 61*h - 88*i)*Log[1 + x]
)/1296 + ((-313*d + 512*e - 820*f + 1280*g - 1936*h + 2816*i)*Log[2 + x])/41472

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Maple [B]  time = 0.034, size = 554, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/108/(2+x)^2*i-1/108/(x-2)^2*i+1/432/(1+x)^2*i+1/432/(-1+x)^2*i-1/216/(x-2)^2*
h+1/432/(-1+x)^2*h-1/432/(1+x)^2*h+1/216/(2+x)^2*h-1/432/(x-2)^2*g+1/432/(1+x)^2
*g+1/432/(-1+x)^2*g-1/432/(2+x)^2*g+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+1/24/
(x-2)*i-11/432/(1+x)*i+11/432/(-1+x)*i-1/24/(2+x)*i+11/432/(x-2)*h+1/48/(1+x)*h+
1/48/(-1+x)*h+11/432/(2+x)*h-7/432/(1+x)*g+7/432/(-1+x)*g-13/864/(2+x)*g+13/864/
(x-2)*g+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+11/162*ln(x-2)*i-11/162*ln(1+x)*i-11/162*ln(-1+x)*i+11/162*ln(2+x
)*i+121/2592*ln(x-2)*h+61/1296*ln(1+x)*h-121/2592*ln(2+x)*h-61/1296*ln(-1+x)*h-5
/162*ln(1+x)*g+5/162*ln(x-2)*g-5/162*ln(-1+x)*g+5/162*ln(2+x)*g+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+25/129
6*ln(1+x)*f-25/1296*ln(-1+x)*f-205/10368*ln(2+x)*f

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Maxima [A]  time = 0.709746, size = 321, normalized size = 1.34 \[ -\frac{1}{41472} \,{\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g + 1936 \, h - 2816 \, i\right )} \log \left (x + 2\right ) + \frac{1}{1296} \,{\left (13 \, d - 16 \, e + 25 \, f - 40 \, g + 61 \, h - 88 \, i\right )} \log \left (x + 1\right ) - \frac{1}{1296} \,{\left (13 \, d + 16 \, e + 25 \, f + 40 \, g + 61 \, h + 88 \, i\right )} \log \left (x - 1\right ) + \frac{1}{41472} \,{\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g + 1936 \, h + 2816 \, i\right )} \log \left (x - 2\right ) + \frac{5 \,{\left (7 \, d + 28 \, f + 64 \, h\right )} x^{7} + 64 \,{\left (2 \, e + 5 \, g + 11 \, i\right )} x^{6} - 18 \,{\left (13 \, d + 60 \, f + 136 \, h\right )} x^{5} - 480 \,{\left (2 \, e + 5 \, g + 11 \, i\right )} x^{4} + 63 \,{\left (5 \, d + 36 \, f + 80 \, h\right )} x^{3} + 192 \,{\left (10 \, e + 25 \, g + 52 \, i\right )} x^{2} + 4 \,{\left (43 \, d - 260 \, f - 656 \, h\right )} x - 800 \, e - 2432 \, g - 5120 \, i}{3456 \,{\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((i*x^5 + h*x^4 + g*x^3 + 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 - 1280*g + 1936*h - 2816*i)*log(x + 2) + 1/1296*
(13*d - 16*e + 25*f - 40*g + 61*h - 88*i)*log(x + 1) - 1/1296*(13*d + 16*e + 25*
f + 40*g + 61*h + 88*i)*log(x - 1) + 1/41472*(313*d + 512*e + 820*f + 1280*g + 1
936*h + 2816*i)*log(x - 2) + 1/3456*(5*(7*d + 28*f + 64*h)*x^7 + 64*(2*e + 5*g +
11*i)*x^6 - 18*(13*d + 60*f + 136*h)*x^5 - 480*(2*e + 5*g + 11*i)*x^4 + 63*(5*d
+ 36*f + 80*h)*x^3 + 192*(10*e + 25*g + 52*i)*x^2 + 4*(43*d - 260*f - 656*h)*x
- 800*e - 2432*g - 5120*i)/(x^8 - 10*x^6 + 33*x^4 - 40*x^2 + 16)

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Fricas [A]  time = 8.88531, size = 832, normalized size = 3.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/41472*(60*(7*d + 28*f + 64*h)*x^7 + 768*(2*e + 5*g + 11*i)*x^6 - 216*(13*d + 6
0*f + 136*h)*x^5 - 5760*(2*e + 5*g + 11*i)*x^4 + 756*(5*d + 36*f + 80*h)*x^3 + 2
304*(10*e + 25*g + 52*i)*x^2 + 48*(43*d - 260*f - 656*h)*x - ((313*d - 512*e + 8
20*f - 1280*g + 1936*h - 2816*i)*x^8 - 10*(313*d - 512*e + 820*f - 1280*g + 1936
*h - 2816*i)*x^6 + 33*(313*d - 512*e + 820*f - 1280*g + 1936*h - 2816*i)*x^4 - 4
0*(313*d - 512*e + 820*f - 1280*g + 1936*h - 2816*i)*x^2 + 5008*d - 8192*e + 131
20*f - 20480*g + 30976*h - 45056*i)*log(x + 2) + 32*((13*d - 16*e + 25*f - 40*g
+ 61*h - 88*i)*x^8 - 10*(13*d - 16*e + 25*f - 40*g + 61*h - 88*i)*x^6 + 33*(13*d
- 16*e + 25*f - 40*g + 61*h - 88*i)*x^4 - 40*(13*d - 16*e + 25*f - 40*g + 61*h
- 88*i)*x^2 + 208*d - 256*e + 400*f - 640*g + 976*h - 1408*i)*log(x + 1) - 32*((
13*d + 16*e + 25*f + 40*g + 61*h + 88*i)*x^8 - 10*(13*d + 16*e + 25*f + 40*g + 6
1*h + 88*i)*x^6 + 33*(13*d + 16*e + 25*f + 40*g + 61*h + 88*i)*x^4 - 40*(13*d +
16*e + 25*f + 40*g + 61*h + 88*i)*x^2 + 208*d + 256*e + 400*f + 640*g + 976*h +
1408*i)*log(x - 1) + ((313*d + 512*e + 820*f + 1280*g + 1936*h + 2816*i)*x^8 - 1
0*(313*d + 512*e + 820*f + 1280*g + 1936*h + 2816*i)*x^6 + 33*(313*d + 512*e + 8
20*f + 1280*g + 1936*h + 2816*i)*x^4 - 40*(313*d + 512*e + 820*f + 1280*g + 1936
*h + 2816*i)*x^2 + 5008*d + 8192*e + 13120*f + 20480*g + 30976*h + 45056*i)*log(
x - 2) - 9600*e - 29184*g - 61440*i)/(x^8 - 10*x^6 + 33*x^4 - 40*x^2 + 16)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.273092, size = 347, normalized size = 1.45 \[ -\frac{1}{41472} \,{\left (313 \, d + 820 \, f - 1280 \, g + 1936 \, h - 2816 \, i - 512 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{1296} \,{\left (13 \, d + 25 \, f - 40 \, g + 61 \, h - 88 \, i - 16 \, e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{1296} \,{\left (13 \, d + 25 \, f + 40 \, g + 61 \, h + 88 \, i + 16 \, e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{41472} \,{\left (313 \, d + 820 \, f + 1280 \, g + 1936 \, h + 2816 \, i + 512 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) + \frac{35 \, d x^{7} + 140 \, f x^{7} + 320 \, h x^{7} + 320 \, g x^{6} + 704 \, i x^{6} + 128 \, x^{6} e - 234 \, d x^{5} - 1080 \, f x^{5} - 2448 \, h x^{5} - 2400 \, g x^{4} - 5280 \, i x^{4} - 960 \, x^{4} e + 315 \, d x^{3} + 2268 \, f x^{3} + 5040 \, h x^{3} + 4800 \, g x^{2} + 9984 \, i x^{2} + 1920 \, x^{2} e + 172 \, d x - 1040 \, f x - 2624 \, h x - 2432 \, g - 5120 \, i - 800 \, e}{3456 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/41472*(313*d + 820*f - 1280*g + 1936*h - 2816*i - 512*e)*ln(abs(x + 2)) + 1/1
296*(13*d + 25*f - 40*g + 61*h - 88*i - 16*e)*ln(abs(x + 1)) - 1/1296*(13*d + 25
*f + 40*g + 61*h + 88*i + 16*e)*ln(abs(x - 1)) + 1/41472*(313*d + 820*f + 1280*g
+ 1936*h + 2816*i + 512*e)*ln(abs(x - 2)) + 1/3456*(35*d*x^7 + 140*f*x^7 + 320*
h*x^7 + 320*g*x^6 + 704*i*x^6 + 128*x^6*e - 234*d*x^5 - 1080*f*x^5 - 2448*h*x^5
- 2400*g*x^4 - 5280*i*x^4 - 960*x^4*e + 315*d*x^3 + 2268*f*x^3 + 5040*h*x^3 + 48
00*g*x^2 + 9984*i*x^2 + 1920*x^2*e + 172*d*x - 1040*f*x - 2624*h*x - 2432*g - 51
20*i - 800*e)/(x^4 - 5*x^2 + 4)^2