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

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

[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 + 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 - (e*Log[1 - x^2])/81 + (e*Log[4 - x^2])/81

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Rubi [A]  time = 0.223885, 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, Rules used = {1673, 1178, 1166, 207, 12, 1107, 614, 616, 31} \[ -\frac{x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{(313 d+820 f) \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{1}{648} (13 d+25 f) \tanh ^{-1}(x)-\frac{e \left (5-2 x^2\right )}{54 \left (x^4-5 x^2+4\right )}+\frac{e \left (5-2 x^2\right )}{36 \left (x^4-5 x^2+4\right )^2}-\frac{1}{81} e \log \left (1-x^2\right )+\frac{1}{81} e \log \left (4-x^2\right ) \]

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 + 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 - (e*Log[1 - x^2])/81 + (e*Log[4 - x^2])/81

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1107

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

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}{\left (4-5 x^2+x^4\right )^3} \, dx &=\int \frac{e x}{\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\\ &=\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+e \int \frac{x}{\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{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}{2} e \operatorname{Subst}\left (\int \frac{1}{\left (4-5 x+x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac{e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}+\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{x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac{1}{6} e \operatorname{Subst}\left (\int \frac{1}{\left (4-5 x+x^2\right )^2} \, dx,x,x^2\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{e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}+\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{e \left (5-2 x^2\right )}{54 \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}{27} e \operatorname{Subst}\left (\int \frac{1}{4-5 x+x^2} \, dx,x,x^2\right )\\ &=\frac{e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}+\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{e \left (5-2 x^2\right )}{54 \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}{81} e \operatorname{Subst}\left (\int \frac{1}{-4+x} \, dx,x,x^2\right )-\frac{1}{81} e \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,x^2\right )\\ &=\frac{e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}+\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{e \left (5-2 x^2\right )}{54 \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}{81} e \log \left (1-x^2\right )+\frac{1}{81} e \log \left (4-x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.1285, size = 161, normalized size = 0.92 \[ \frac{\frac{288 \left (-5 d x^3+17 d x+e \left (20-8 x^2\right )-8 f x^3+20 f x\right )}{\left (x^4-5 x^2+4\right )^2}+\frac{12 \left (d x \left (35 x^2-59\right )+64 e \left (2 x^2-5\right )+20 f x \left (7 x^2-19\right )\right )}{x^4-5 x^2+4}-32 \log (1-x) (13 d+16 e+25 f)+\log (2-x) (313 d+512 e+820 f)+32 \log (x+1) (13 d-16 e+25 f)+\log (x+2) (-313 d+512 e-820 f)}{41472} \]

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) + 2
0*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 + 5
12*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 = 0.018, size = 278, normalized size = 1.6 \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{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{d}{432+432\,x}}-{\frac{e}{144+144\,x}}+{\frac{19\,d}{6912\,x-13824}}+{\frac{17\,e}{3456\,x-6912}}+{\frac{d}{432\,x-432}}+{\frac{e}{144\,x-144}}+{\frac{19\,d}{13824+6912\,x}}-{\frac{17\,e}{6912+3456\,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{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((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/(1+x)^2*f+1/864/(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+1/432/(1+x)*d-1/144/(1+x)*e+19/6912/(x-2)*d+17/3456/(x-2)*e+1/432/(x-1)*d+1/144/(x-1)*e+19/6912/(2+x)*d-17/
3456/(2+x)*e+5/432/(1+x)*f+5/576/(x-2)*f+5/432/(x-1)*f+5/576/(2+x)*f+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 = 0.953977, size = 209, normalized size = 1.19 \begin{align*} -\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 )}} \end{align*}

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 + 2
5*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 [B]  time = 2.3227, size = 1134, normalized size = 6.48 \begin{align*} \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 )}} \end{align*}

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 - 25
6*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*(31
3*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 [B]  time = 43.3403, size = 2822, normalized size = 16.13 \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**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) - 1292948240
1572800*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 + 63469758382
080*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*(1
3*d - 16*e + 25*f)**3 + 5035763255214080*d*e**5 + 142661633703936*d*e**4*(13*d - 16*e + 25*f) - 21383148994560
00*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 + 2
5*f) - 557272006656*d*e**2*(13*d - 16*e + 25*f)**3 - 151082645593600000*d*e*f**4 + 141056507904000*d*e*f**2*(1
3*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 + 10643272556871680*e**5*f + 214404767416320*e**4*f*(13*d - 16*e + 25*f) + 529992253440000*e**3*f**3 - 415
75283425280*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 + 510
41923200000*f**5*(13*d - 16*e + 25*f) - 80289792000*f**3*(13*d - 16*e + 25*f)**3)/(22941256248261*d**6 + 19727
1407316645*d**5*f - 2312740746035200*d**4*e**2 + 612862910928900*d**4*f**2 - 20566607354920960*d**3*e**2*f + 7
67363353812000*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 - 101559983669248000*d*e**2*f**3 - 182883938400000*d*f**5 + 22539988369408
000*e**4*f**2 - 56422196838400000*e**2*f**4 + 21520080000000*f**6))/1296 - (13*d + 16*e + 25*f)*log(x + (-1106
258459719280*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) - 594
78343838144000*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) - 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 - 151082645593600000*d*e*f**4 + 141056507904000*d*e*f**2*(13*d + 16*e + 25*f)**2 + 16768315440
0000*d*f**4*(13*d + 16*e + 25*f) - 339373670400*d*f**2*(13*d + 16*e + 25*f)**3 + 10643272556871680*e**5*f - 21
4404767416320*e**4*f*(13*d + 16*e + 25*f) + 529992253440000*e**3*f**3 - 41575283425280*e**3*f*(13*d + 16*e + 2
5*f)**2 - 1671759396864000*e**2*f**3*(13*d + 16*e + 25*f) + 837518622720*e**2*f*(13*d + 16*e + 25*f)**3 - 6689
5452108800000*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 + 767363353812000*d**3*f**3 + 44739128
13420544*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 - 182883938400000*d*f**5 + 22539988369408000*e**4*f**2 - 56422196838400000*e
**2*f**4 + 21520080000000*f**6))/1296 - (313*d - 512*e + 820*f)*log(x + (-1106258459719280*d**5*e + 1311371095
4343*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) - 8597
4615*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 - 512*e + 820*f)**2 + 13217897641
500*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 - 5
12*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 - 67001489817
60*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*(313*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) + 2
450250*f**3*(313*d - 512*e + 820*f)**3)/(22941256248261*d**6 + 197271407316645*d**5*f - 2312740746035200*d**4*
e**2 + 612862910928900*d**4*f**2 - 20566607354920960*d**3*e**2*f + 767363353812000*d**3*f**3 + 447391281342054
4*d**2*e**4 - 68552762169753600*d**2*e**2*f**2 + 197499222000000*d**2*f**4 + 20324472439439360*d*e**4*f - 1015
59983669248000*d*e**2*f**3 - 182883938400000*d*f**5 + 22539988369408000*e**4*f**2 - 56422196838400000*e**2*f**
4 + 21520080000000*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 - 2885705898393600*d**2*e**3*f + 31714021048320*d**2*e**2*f*(313*d + 512*e +
820*f) - 134905286808320000*d**2*e*f**3 + 61982185920*d**2*e*f*(313*d + 512*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 + 445
8176053248*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*(313*d + 512*e + 820*f)**2 + 5224
2481152000*e**2*f**3*(313*d + 512*e + 820*f) - 25559040*e**2*f*(313*d + 512*e + 820*f)**3 - 66895452108800000*
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)/(22941256248261*d**6 + 197271407316645*d**5*f - 2312740746035200*d**4*e**2 +
612862910928900*d**4*f**2 - 20566607354920960*d**3*e**2*f + 767363353812000*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 - 1015599836
69248000*d*e**2*f**3 - 182883938400000*d*f**5 + 22539988369408000*e**4*f**2 - 56422196838400000*e**2*f**4 + 21
520080000000*f**6))/41472 + (128*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 [A]  time = 1.0935, size = 212, normalized size = 1.21 \begin{align*} -\frac{1}{41472} \,{\left (313 \, d + 820 \, f - 512 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{1296} \,{\left (13 \, d + 25 \, f - 16 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{1296} \,{\left (13 \, d + 25 \, f + 16 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{41472} \,{\left (313 \, d + 820 \, f + 512 \, e\right )} \log \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}} \end{align*}

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)*log(abs(x + 2)) + 1/1296*(13*d + 25*f - 16*e)*log(abs(x + 1)) - 1/1296*(13*d
+ 25*f + 16*e)*log(abs(x - 1)) + 1/41472*(313*d + 820*f + 512*e)*log(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 - 1040*f*x
- 800*e)/(x^4 - 5*x^2 + 4)^2