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

Optimal. Leaf size=204 \[ -\frac{x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{(313 d+820 f) \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{1}{648} (13 d+25 f) \tanh ^{-1}(x)-\frac{\left (5-2 x^2\right ) (2 e+5 g)}{108 \left (x^4-5 x^2+4\right )}+\frac{x^2 (-(2 e+5 g))+5 e+8 g}{36 \left (x^4-5 x^2+4\right )^2}-\frac{1}{162} (2 e+5 g) \log \left (1-x^2\right )+\frac{1}{162} (2 e+5 g) \log \left (4-x^2\right ) \]

[Out]

(x*(17*d + 20*f - (5*d + 8*f)*x^2))/(144*(4 - 5*x^2 + x^4)^2) + (5*e + 8*g - (2*e + 5*g)*x^2)/(36*(4 - 5*x^2 +
x^4)^2) - ((2*e + 5*g)*(5 - 2*x^2))/(108*(4 - 5*x^2 + x^4)) - (x*(59*d + 380*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 - ((2*e + 5*g)*Log[1 -
x^2])/162 + ((2*e + 5*g)*Log[4 - x^2])/162

________________________________________________________________________________________

Rubi [A]  time = 0.251946, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {1673, 1178, 1166, 207, 1247, 638, 614, 616, 31} \[ -\frac{x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{(313 d+820 f) \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{1}{648} (13 d+25 f) \tanh ^{-1}(x)-\frac{\left (5-2 x^2\right ) (2 e+5 g)}{108 \left (x^4-5 x^2+4\right )}+\frac{x^2 (-(2 e+5 g))+5 e+8 g}{36 \left (x^4-5 x^2+4\right )^2}-\frac{1}{162} (2 e+5 g) \log \left (1-x^2\right )+\frac{1}{162} (2 e+5 g) \log \left (4-x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(17*d + 20*f - (5*d + 8*f)*x^2))/(144*(4 - 5*x^2 + x^4)^2) + (5*e + 8*g - (2*e + 5*g)*x^2)/(36*(4 - 5*x^2 +
x^4)^2) - ((2*e + 5*g)*(5 - 2*x^2))/(108*(4 - 5*x^2 + x^4)) - (x*(59*d + 380*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 - ((2*e + 5*g)*Log[1 -
x^2])/162 + ((2*e + 5*g)*Log[4 - x^2])/162

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{d+e x+f x^2+g x^3}{\left (4-5 x^2+x^4\right )^3} \, dx &=\int \frac{d+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx+\int \frac{x \left (e+g x^2\right )}{\left (4-5 x^2+x^4\right )^3} \, dx\\ &=\frac{x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac{1}{144} \int \frac{-19 d+20 f+5 (5 d+8 f) x^2}{\left (4-5 x^2+x^4\right )^2} \, dx+\frac{1}{2} \operatorname{Subst}\left (\int \frac{e+g x}{\left (4-5 x+x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac{x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac{5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}-\frac{x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}+\frac{\int \frac{3 (173 d+260 f)+105 (d+4 f) x^2}{4-5 x^2+x^4} \, dx}{10368}+\frac{1}{12} (-2 e-5 g) \operatorname{Subst}\left (\int \frac{1}{\left (4-5 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac{5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}-\frac{(2 e+5 g) \left (5-2 x^2\right )}{108 \left (4-5 x^2+x^4\right )}-\frac{x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}+\frac{1}{648} (-13 d-25 f) \int \frac{1}{-1+x^2} \, dx+\frac{(313 d+820 f) \int \frac{1}{-4+x^2} \, dx}{10368}+\frac{1}{54} (2 e+5 g) \operatorname{Subst}\left (\int \frac{1}{4-5 x+x^2} \, dx,x,x^2\right )\\ &=\frac{x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac{5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}-\frac{(2 e+5 g) \left (5-2 x^2\right )}{108 \left (4-5 x^2+x^4\right )}-\frac{x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\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}{162} (-2 e-5 g) \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,x^2\right )+\frac{1}{162} (2 e+5 g) \operatorname{Subst}\left (\int \frac{1}{-4+x} \, dx,x,x^2\right )\\ &=\frac{x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac{5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}-\frac{(2 e+5 g) \left (5-2 x^2\right )}{108 \left (4-5 x^2+x^4\right )}-\frac{x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\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}{162} (2 e+5 g) \log \left (1-x^2\right )+\frac{1}{162} (2 e+5 g) \log \left (4-x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0914973, size = 193, normalized size = 0.95 \[ \frac{\frac{288 \left (-5 d x^3+17 d x+e \left (20-8 x^2\right )-8 f x^3+20 f x-4 g \left (5 x^2-8\right )\right )}{\left (x^4-5 x^2+4\right )^2}+\frac{12 \left (d x \left (35 x^2-59\right )+64 e \left (2 x^2-5\right )+20 f x \left (7 x^2-19\right )+160 g \left (2 x^2-5\right )\right )}{x^4-5 x^2+4}-32 \log (1-x) (13 d+16 e+25 f+40 g)+\log (2-x) (313 d+512 e+820 f+1280 g)+32 \log (x+1) (13 d-16 e+25 f-40 g)+\log (x+2) (-313 d+512 e-820 f+1280 g)}{41472} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*x^2 + g*x^3)/(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*g*(-8 + 5*x^2)))/(4 - 5*x^2 + x^4)^2 + (12*(64
*e*(-5 + 2*x^2) + 160*g*(-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 + 40*g)*Log[1 - x] + (313*d + 512*e + 820*f + 1280*g)*Log[2 - x] + 32*(13*d - 16*e + 25*f - 40
*g)*Log[1 + x] + (-313*d + 512*e - 820*f + 1280*g)*Log[2 + x])/41472

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Maple [A]  time = 0.022, size = 370, normalized size = 1.8 \begin{align*} -{\frac{313\,\ln \left ( 2+x \right ) d}{41472}}+{\frac{\ln \left ( 2+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{13\,\ln \left ( x-1 \right ) d}{1296}}-{\frac{\ln \left ( x-1 \right ) e}{81}}-{\frac{g}{432\, \left ( 2+x \right ) ^{2}}}+{\frac{g}{432\, \left ( 1+x \right ) ^{2}}}-{\frac{g}{432\, \left ( x-2 \right ) ^{2}}}+{\frac{g}{432\, \left ( x-1 \right ) ^{2}}}-{\frac{f}{432\, \left ( 1+x \right ) ^{2}}}+{\frac{f}{864\, \left ( 2+x \right ) ^{2}}}+{\frac{d}{432\, \left ( x-1 \right ) ^{2}}}+{\frac{e}{432\, \left ( x-1 \right ) ^{2}}}+{\frac{d}{3456\, \left ( 2+x \right ) ^{2}}}-{\frac{e}{1728\, \left ( 2+x \right ) ^{2}}}-{\frac{f}{864\, \left ( x-2 \right ) ^{2}}}-{\frac{d}{432\, \left ( 1+x \right ) ^{2}}}+{\frac{e}{432\, \left ( 1+x \right ) ^{2}}}+{\frac{f}{432\, \left ( x-1 \right ) ^{2}}}-{\frac{d}{3456\, \left ( x-2 \right ) ^{2}}}-{\frac{e}{1728\, \left ( x-2 \right ) ^{2}}}-{\frac{13\,g}{1728+864\,x}}+{\frac{d}{432+432\,x}}-{\frac{e}{144+144\,x}}+{\frac{13\,g}{864\,x-1728}}+{\frac{19\,d}{6912\,x-13824}}+{\frac{17\,e}{3456\,x-6912}}+{\frac{7\,g}{432\,x-432}}+{\frac{d}{432\,x-432}}+{\frac{e}{144\,x-144}}+{\frac{19\,d}{13824+6912\,x}}-{\frac{17\,e}{6912+3456\,x}}-{\frac{7\,g}{432+432\,x}}+{\frac{5\,f}{432+432\,x}}+{\frac{5\,f}{576\,x-1152}}+{\frac{5\,f}{432\,x-432}}+{\frac{5\,f}{1152+576\,x}}+{\frac{5\,\ln \left ( 2+x \right ) g}{162}}-{\frac{5\,\ln \left ( 1+x \right ) g}{162}}+{\frac{5\,\ln \left ( x-2 \right ) g}{162}}-{\frac{5\,\ln \left ( x-1 \right ) g}{162}}+{\frac{205\,\ln \left ( x-2 \right ) f}{10368}}-{\frac{25\,\ln \left ( x-1 \right ) f}{1296}}-{\frac{205\,\ln \left ( 2+x \right ) f}{10368}}+{\frac{25\,\ln \left ( 1+x \right ) f}{1296}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-313/41472*ln(2+x)*d+1/81*ln(2+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-13/129
6*ln(x-1)*d-1/81*ln(x-1)*e-1/432/(2+x)^2*g+1/432/(1+x)^2*g-1/432/(x-2)^2*g+1/432/(x-1)^2*g-1/432/(1+x)^2*f+1/8
64/(2+x)^2*f+1/432/(x-1)^2*d+1/432/(x-1)^2*e+1/3456/(2+x)^2*d-1/1728/(2+x)^2*e-1/864/(x-2)^2*f-1/432/(1+x)^2*d
+1/432/(1+x)^2*e+1/432/(x-1)^2*f-1/3456/(x-2)^2*d-1/1728/(x-2)^2*e-13/864/(2+x)*g+1/432/(1+x)*d-1/144/(1+x)*e+
13/864/(x-2)*g+19/6912/(x-2)*d+17/3456/(x-2)*e+7/432/(x-1)*g+1/432/(x-1)*d+1/144/(x-1)*e+19/6912/(2+x)*d-17/34
56/(2+x)*e-7/432/(1+x)*g+5/432/(1+x)*f+5/576/(x-2)*f+5/432/(x-1)*f+5/576/(2+x)*f+5/162*ln(2+x)*g-5/162*ln(1+x)
*g+5/162*ln(x-2)*g-5/162*ln(x-1)*g+205/10368*ln(x-2)*f-25/1296*ln(x-1)*f-205/10368*ln(2+x)*f+25/1296*ln(1+x)*f

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Maxima [A]  time = 0.946445, size = 254, normalized size = 1.25 \begin{align*} -\frac{1}{41472} \,{\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g\right )} \log \left (x + 2\right ) + \frac{1}{1296} \,{\left (13 \, d - 16 \, e + 25 \, f - 40 \, g\right )} \log \left (x + 1\right ) - \frac{1}{1296} \,{\left (13 \, d + 16 \, e + 25 \, f + 40 \, g\right )} \log \left (x - 1\right ) + \frac{1}{41472} \,{\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g\right )} \log \left (x - 2\right ) + \frac{35 \,{\left (d + 4 \, f\right )} x^{7} + 64 \,{\left (2 \, e + 5 \, g\right )} x^{6} - 18 \,{\left (13 \, d + 60 \, f\right )} x^{5} - 480 \,{\left (2 \, e + 5 \, g\right )} x^{4} + 63 \,{\left (5 \, d + 36 \, f\right )} x^{3} + 960 \,{\left (2 \, e + 5 \, g\right )} x^{2} + 4 \,{\left (43 \, d - 260 \, f\right )} x - 800 \, e - 2432 \, g}{3456 \,{\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)*log(x + 2) + 1/1296*(13*d - 16*e + 25*f - 40*g)*log(x + 1) - 1/1296*
(13*d + 16*e + 25*f + 40*g)*log(x - 1) + 1/41472*(313*d + 512*e + 820*f + 1280*g)*log(x - 2) + 1/3456*(35*(d +
4*f)*x^7 + 64*(2*e + 5*g)*x^6 - 18*(13*d + 60*f)*x^5 - 480*(2*e + 5*g)*x^4 + 63*(5*d + 36*f)*x^3 + 960*(2*e +
5*g)*x^2 + 4*(43*d - 260*f)*x - 800*e - 2432*g)/(x^8 - 10*x^6 + 33*x^4 - 40*x^2 + 16)

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Fricas [B]  time = 4.31871, size = 1407, normalized size = 6.9 \begin{align*} \frac{420 \,{\left (d + 4 \, f\right )} x^{7} + 768 \,{\left (2 \, e + 5 \, g\right )} x^{6} - 216 \,{\left (13 \, d + 60 \, f\right )} x^{5} - 5760 \,{\left (2 \, e + 5 \, g\right )} x^{4} + 756 \,{\left (5 \, d + 36 \, f\right )} x^{3} + 11520 \,{\left (2 \, e + 5 \, g\right )} x^{2} + 48 \,{\left (43 \, d - 260 \, f\right )} x -{\left ({\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g\right )} x^{8} - 10 \,{\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g\right )} x^{6} + 33 \,{\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g\right )} x^{4} - 40 \,{\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g\right )} x^{2} + 5008 \, d - 8192 \, e + 13120 \, f - 20480 \, g\right )} \log \left (x + 2\right ) + 32 \,{\left ({\left (13 \, d - 16 \, e + 25 \, f - 40 \, g\right )} x^{8} - 10 \,{\left (13 \, d - 16 \, e + 25 \, f - 40 \, g\right )} x^{6} + 33 \,{\left (13 \, d - 16 \, e + 25 \, f - 40 \, g\right )} x^{4} - 40 \,{\left (13 \, d - 16 \, e + 25 \, f - 40 \, g\right )} x^{2} + 208 \, d - 256 \, e + 400 \, f - 640 \, g\right )} \log \left (x + 1\right ) - 32 \,{\left ({\left (13 \, d + 16 \, e + 25 \, f + 40 \, g\right )} x^{8} - 10 \,{\left (13 \, d + 16 \, e + 25 \, f + 40 \, g\right )} x^{6} + 33 \,{\left (13 \, d + 16 \, e + 25 \, f + 40 \, g\right )} x^{4} - 40 \,{\left (13 \, d + 16 \, e + 25 \, f + 40 \, g\right )} x^{2} + 208 \, d + 256 \, e + 400 \, f + 640 \, g\right )} \log \left (x - 1\right ) +{\left ({\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g\right )} x^{8} - 10 \,{\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g\right )} x^{6} + 33 \,{\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g\right )} x^{4} - 40 \,{\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g\right )} x^{2} + 5008 \, d + 8192 \, e + 13120 \, f + 20480 \, g\right )} \log \left (x - 2\right ) - 9600 \, e - 29184 \, g}{41472 \,{\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+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 + 768*(2*e + 5*g)*x^6 - 216*(13*d + 60*f)*x^5 - 5760*(2*e + 5*g)*x^4 + 756*(5*d + 3
6*f)*x^3 + 11520*(2*e + 5*g)*x^2 + 48*(43*d - 260*f)*x - ((313*d - 512*e + 820*f - 1280*g)*x^8 - 10*(313*d - 5
12*e + 820*f - 1280*g)*x^6 + 33*(313*d - 512*e + 820*f - 1280*g)*x^4 - 40*(313*d - 512*e + 820*f - 1280*g)*x^2
+ 5008*d - 8192*e + 13120*f - 20480*g)*log(x + 2) + 32*((13*d - 16*e + 25*f - 40*g)*x^8 - 10*(13*d - 16*e + 2
5*f - 40*g)*x^6 + 33*(13*d - 16*e + 25*f - 40*g)*x^4 - 40*(13*d - 16*e + 25*f - 40*g)*x^2 + 208*d - 256*e + 40
0*f - 640*g)*log(x + 1) - 32*((13*d + 16*e + 25*f + 40*g)*x^8 - 10*(13*d + 16*e + 25*f + 40*g)*x^6 + 33*(13*d
+ 16*e + 25*f + 40*g)*x^4 - 40*(13*d + 16*e + 25*f + 40*g)*x^2 + 208*d + 256*e + 400*f + 640*g)*log(x - 1) + (
(313*d + 512*e + 820*f + 1280*g)*x^8 - 10*(313*d + 512*e + 820*f + 1280*g)*x^6 + 33*(313*d + 512*e + 820*f + 1
280*g)*x^4 - 40*(313*d + 512*e + 820*f + 1280*g)*x^2 + 5008*d + 8192*e + 13120*f + 20480*g)*log(x - 2) - 9600*
e - 29184*g)/(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. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 [A]  time = 1.12401, size = 257, normalized size = 1.26 \begin{align*} -\frac{1}{41472} \,{\left (313 \, d + 820 \, f - 1280 \, g - 512 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{1296} \,{\left (13 \, d + 25 \, f - 40 \, g - 16 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{1296} \,{\left (13 \, d + 25 \, f + 40 \, g + 16 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{41472} \,{\left (313 \, d + 820 \, f + 1280 \, g + 512 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac{35 \, d x^{7} + 140 \, f x^{7} + 320 \, g x^{6} + 128 \, x^{6} e - 234 \, d x^{5} - 1080 \, f x^{5} - 2400 \, g x^{4} - 960 \, x^{4} e + 315 \, d x^{3} + 2268 \, f x^{3} + 4800 \, g x^{2} + 1920 \, x^{2} e + 172 \, d x - 1040 \, f x - 2432 \, g - 800 \, e}{3456 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 - 512*e)*log(abs(x + 2)) + 1/1296*(13*d + 25*f - 40*g - 16*e)*log(abs(x + 1))
- 1/1296*(13*d + 25*f + 40*g + 16*e)*log(abs(x - 1)) + 1/41472*(313*d + 820*f + 1280*g + 512*e)*log(abs(x - 2
)) + 1/3456*(35*d*x^7 + 140*f*x^7 + 320*g*x^6 + 128*x^6*e - 234*d*x^5 - 1080*f*x^5 - 2400*g*x^4 - 960*x^4*e +
315*d*x^3 + 2268*f*x^3 + 4800*g*x^2 + 1920*x^2*e + 172*d*x - 1040*f*x - 2432*g - 800*e)/(x^4 - 5*x^2 + 4)^2