3.27 \(\int \frac{d+e x+f x^2}{(4-5 x^2+x^4)^2} \, dx\)
Optimal. Leaf size=115 \[ \frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{72 \left (x^4-5 x^2+4\right )}+\frac{1}{432} (19 d+52 f) \tanh ^{-1}\left (\frac{x}{2}\right )-\frac{1}{54} (d+7 f) \tanh ^{-1}(x)+\frac{e \left (5-2 x^2\right )}{18 \left (x^4-5 x^2+4\right )}+\frac{1}{27} e \log \left (1-x^2\right )-\frac{1}{27} e \log \left (4-x^2\right ) \]
[Out]
(e*(5 - 2*x^2))/(18*(4 - 5*x^2 + x^4)) + (x*(17*d + 20*f - (5*d + 8*f)*x^2))/(72*(4 - 5*x^2 + x^4)) + ((19*d +
52*f)*ArcTanh[x/2])/432 - ((d + 7*f)*ArcTanh[x])/54 + (e*Log[1 - x^2])/27 - (e*Log[4 - x^2])/27
________________________________________________________________________________________
Rubi [A] time = 0.140115, antiderivative size = 115, normalized size of antiderivative = 1.,
number of steps used = 11, 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} \[ \frac{x \left (x^2 (-(5 d+8 f))+17 d+20 f\right )}{72 \left (x^4-5 x^2+4\right )}+\frac{1}{432} (19 d+52 f) \tanh ^{-1}\left (\frac{x}{2}\right )-\frac{1}{54} (d+7 f) \tanh ^{-1}(x)+\frac{e \left (5-2 x^2\right )}{18 \left (x^4-5 x^2+4\right )}+\frac{1}{27} e \log \left (1-x^2\right )-\frac{1}{27} e \log \left (4-x^2\right ) \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x + f*x^2)/(4 - 5*x^2 + x^4)^2,x]
[Out]
(e*(5 - 2*x^2))/(18*(4 - 5*x^2 + x^4)) + (x*(17*d + 20*f - (5*d + 8*f)*x^2))/(72*(4 - 5*x^2 + x^4)) + ((19*d +
52*f)*ArcTanh[x/2])/432 - ((d + 7*f)*ArcTanh[x])/54 + (e*Log[1 - x^2])/27 - (e*Log[4 - x^2])/27
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 )^2} \, dx &=\int \frac{e x}{\left (4-5 x^2+x^4\right )^2} \, dx+\int \frac{d+f x^2}{\left (4-5 x^2+x^4\right )^2} \, dx\\ &=\frac{x \left (17 d+20 f-(5 d+8 f) x^2\right )}{72 \left (4-5 x^2+x^4\right )}-\frac{1}{72} \int \frac{-d+20 f+(5 d+8 f) x^2}{4-5 x^2+x^4} \, dx+e \int \frac{x}{\left (4-5 x^2+x^4\right )^2} \, dx\\ &=\frac{x \left (17 d+20 f-(5 d+8 f) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{\left (4-5 x+x^2\right )^2} \, dx,x,x^2\right )-\frac{1}{54} (-d-7 f) \int \frac{1}{-1+x^2} \, dx-\frac{1}{216} (19 d+52 f) \int \frac{1}{-4+x^2} \, dx\\ &=\frac{e \left (5-2 x^2\right )}{18 \left (4-5 x^2+x^4\right )}+\frac{x \left (17 d+20 f-(5 d+8 f) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac{1}{432} (19 d+52 f) \tanh ^{-1}\left (\frac{x}{2}\right )-\frac{1}{54} (d+7 f) \tanh ^{-1}(x)-\frac{1}{9} 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 )}{18 \left (4-5 x^2+x^4\right )}+\frac{x \left (17 d+20 f-(5 d+8 f) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac{1}{432} (19 d+52 f) \tanh ^{-1}\left (\frac{x}{2}\right )-\frac{1}{54} (d+7 f) \tanh ^{-1}(x)-\frac{1}{27} e \operatorname{Subst}\left (\int \frac{1}{-4+x} \, dx,x,x^2\right )+\frac{1}{27} e \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,x^2\right )\\ &=\frac{e \left (5-2 x^2\right )}{18 \left (4-5 x^2+x^4\right )}+\frac{x \left (17 d+20 f-(5 d+8 f) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac{1}{432} (19 d+52 f) \tanh ^{-1}\left (\frac{x}{2}\right )-\frac{1}{54} (d+7 f) \tanh ^{-1}(x)+\frac{1}{27} e \log \left (1-x^2\right )-\frac{1}{27} e \log \left (4-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0808132, size = 112, normalized size = 0.97 \[ \frac{1}{864} \left (\frac{12 \left (-5 d x^3+17 d x+e \left (20-8 x^2\right )-8 f x^3+20 f x\right )}{x^4-5 x^2+4}+8 \log (1-x) (d+4 e+7 f)-\log (2-x) (19 d+32 e+52 f)-8 \log (x+1) (d-4 e+7 f)+\log (x+2) (19 d-32 e+52 f)\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x + f*x^2)/(4 - 5*x^2 + x^4)^2,x]
[Out]
((12*(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) + 8*(d + 4*e + 7*f)*Log[1 - x]
- (19*d + 32*e + 52*f)*Log[2 - x] - 8*(d - 4*e + 7*f)*Log[1 + x] + (19*d - 32*e + 52*f)*Log[2 + x])/864
________________________________________________________________________________________
Maple [A] time = 0.018, size = 182, normalized size = 1.6 \begin{align*} -{\frac{d}{288+144\,x}}+{\frac{e}{144+72\,x}}-{\frac{f}{72+36\,x}}+{\frac{19\,\ln \left ( 2+x \right ) d}{864}}-{\frac{\ln \left ( 2+x \right ) e}{27}}+{\frac{13\,\ln \left ( 2+x \right ) f}{216}}-{\frac{\ln \left ( 1+x \right ) d}{108}}+{\frac{\ln \left ( 1+x \right ) e}{27}}-{\frac{7\,\ln \left ( 1+x \right ) f}{108}}-{\frac{d}{36+36\,x}}+{\frac{e}{36+36\,x}}-{\frac{f}{36+36\,x}}-{\frac{19\,\ln \left ( x-2 \right ) d}{864}}-{\frac{\ln \left ( x-2 \right ) e}{27}}-{\frac{13\,\ln \left ( x-2 \right ) f}{216}}-{\frac{d}{144\,x-288}}-{\frac{e}{72\,x-144}}-{\frac{f}{36\,x-72}}-{\frac{d}{36\,x-36}}-{\frac{e}{36\,x-36}}-{\frac{f}{36\,x-36}}+{\frac{\ln \left ( x-1 \right ) d}{108}}+{\frac{\ln \left ( x-1 \right ) e}{27}}+{\frac{7\,\ln \left ( x-1 \right ) f}{108}} \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)^2,x)
[Out]
-1/144/(2+x)*d+1/72/(2+x)*e-1/36/(2+x)*f+19/864*ln(2+x)*d-1/27*ln(2+x)*e+13/216*ln(2+x)*f-1/108*ln(1+x)*d+1/27
*ln(1+x)*e-7/108*ln(1+x)*f-1/36/(1+x)*d+1/36/(1+x)*e-1/36/(1+x)*f-19/864*ln(x-2)*d-1/27*ln(x-2)*e-13/216*ln(x-
2)*f-1/144/(x-2)*d-1/72/(x-2)*e-1/36/(x-2)*f-1/36/(x-1)*d-1/36/(x-1)*e-1/36/(x-1)*f+1/108*ln(x-1)*d+1/27*ln(x-
1)*e+7/108*ln(x-1)*f
________________________________________________________________________________________
Maxima [A] time = 0.942326, size = 143, normalized size = 1.24 \begin{align*} \frac{1}{864} \,{\left (19 \, d - 32 \, e + 52 \, f\right )} \log \left (x + 2\right ) - \frac{1}{108} \,{\left (d - 4 \, e + 7 \, f\right )} \log \left (x + 1\right ) + \frac{1}{108} \,{\left (d + 4 \, e + 7 \, f\right )} \log \left (x - 1\right ) - \frac{1}{864} \,{\left (19 \, d + 32 \, e + 52 \, f\right )} \log \left (x - 2\right ) - \frac{{\left (5 \, d + 8 \, f\right )} x^{3} + 8 \, e x^{2} -{\left (17 \, d + 20 \, f\right )} x - 20 \, e}{72 \,{\left (x^{4} - 5 \, x^{2} + 4\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)^2,x, algorithm="maxima")
[Out]
1/864*(19*d - 32*e + 52*f)*log(x + 2) - 1/108*(d - 4*e + 7*f)*log(x + 1) + 1/108*(d + 4*e + 7*f)*log(x - 1) -
1/864*(19*d + 32*e + 52*f)*log(x - 2) - 1/72*((5*d + 8*f)*x^3 + 8*e*x^2 - (17*d + 20*f)*x - 20*e)/(x^4 - 5*x^2
+ 4)
________________________________________________________________________________________
Fricas [B] time = 3.0018, size = 585, normalized size = 5.09 \begin{align*} -\frac{12 \,{\left (5 \, d + 8 \, f\right )} x^{3} + 96 \, e x^{2} - 12 \,{\left (17 \, d + 20 \, f\right )} x -{\left ({\left (19 \, d - 32 \, e + 52 \, f\right )} x^{4} - 5 \,{\left (19 \, d - 32 \, e + 52 \, f\right )} x^{2} + 76 \, d - 128 \, e + 208 \, f\right )} \log \left (x + 2\right ) + 8 \,{\left ({\left (d - 4 \, e + 7 \, f\right )} x^{4} - 5 \,{\left (d - 4 \, e + 7 \, f\right )} x^{2} + 4 \, d - 16 \, e + 28 \, f\right )} \log \left (x + 1\right ) - 8 \,{\left ({\left (d + 4 \, e + 7 \, f\right )} x^{4} - 5 \,{\left (d + 4 \, e + 7 \, f\right )} x^{2} + 4 \, d + 16 \, e + 28 \, f\right )} \log \left (x - 1\right ) +{\left ({\left (19 \, d + 32 \, e + 52 \, f\right )} x^{4} - 5 \,{\left (19 \, d + 32 \, e + 52 \, f\right )} x^{2} + 76 \, d + 128 \, e + 208 \, f\right )} \log \left (x - 2\right ) - 240 \, e}{864 \,{\left (x^{4} - 5 \, x^{2} + 4\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)^2,x, algorithm="fricas")
[Out]
-1/864*(12*(5*d + 8*f)*x^3 + 96*e*x^2 - 12*(17*d + 20*f)*x - ((19*d - 32*e + 52*f)*x^4 - 5*(19*d - 32*e + 52*f
)*x^2 + 76*d - 128*e + 208*f)*log(x + 2) + 8*((d - 4*e + 7*f)*x^4 - 5*(d - 4*e + 7*f)*x^2 + 4*d - 16*e + 28*f)
*log(x + 1) - 8*((d + 4*e + 7*f)*x^4 - 5*(d + 4*e + 7*f)*x^2 + 4*d + 16*e + 28*f)*log(x - 1) + ((19*d + 32*e +
52*f)*x^4 - 5*(19*d + 32*e + 52*f)*x^2 + 76*d + 128*e + 208*f)*log(x - 2) - 240*e)/(x^4 - 5*x^2 + 4)
________________________________________________________________________________________
Sympy [B] time = 38.346, size = 2689, normalized size = 23.38 \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)**2,x)
[Out]
-(d - 4*e + 7*f)*log(x + (-6006260*d**5*e + 2341251*d**5*(d - 4*e + 7*f) - 246016240*d**4*e*f + 31626180*d**4*
f*(d - 4*e + 7*f) - 18247680*d**3*e**3 + 24099840*d**3*e**2*(d - 4*e + 7*f) - 2758371200*d**3*e*f**2 + 7387904
*d**3*e*(d - 4*e + 7*f)**2 + 171122976*d**3*f**2*(d - 4*e + 7*f) - 665280*d**3*(d - 4*e + 7*f)**3 + 298598400*
d**2*e**3*f + 369487872*d**2*e**2*f*(d - 4*e + 7*f) - 13192256000*d**2*e*f**3 + 90885120*d**2*e*f*(d - 4*e + 7
*f)**2 + 441486720*d**2*f**3*(d - 4*e + 7*f) - 5536512*d**2*f*(d - 4*e + 7*f)**3 + 587202560*d*e**5 - 12582912
*d*e**4*(d - 4*e + 7*f) + 1353646080*d*e**3*f**2 - 36700160*d*e**3*(d - 4*e + 7*f)**2 + 1448755200*d*e**2*f**2
*(d - 4*e + 7*f) + 786432*d*e**2*(d - 4*e + 7*f)**3 - 28282393600*d*e*f**4 + 362729472*d*e*f**2*(d - 4*e + 7*f
)**2 + 399575808*d*f**4*(d - 4*e + 7*f) - 10368000*d*f**2*(d - 4*e + 7*f)**3 + 2751463424*e**5*f + 251658240*e
**4*f*(d - 4*e + 7*f) - 530841600*e**3*f**3 - 171966464*e**3*f*(d - 4*e + 7*f)**2 + 1935212544*e**2*f**3*(d -
4*e + 7*f) - 15728640*e**2*f*(d - 4*e + 7*f)**3 - 21886889984*e*f**5 + 483737600*e*f**3*(d - 4*e + 7*f)**2 - 2
12474880*f**5*(d - 4*e + 7*f) + 4534272*f**3*(d - 4*e + 7*f)**3)/(1675971*d**6 + 28507545*d**5*f - 66150400*d*
*4*e**2 + 168075324*d**4*f**2 - 1091117056*d**3*e**2*f + 384095520*d**3*f**3 + 318767104*d**2*e**4 - 652886016
0*d**2*e**2*f**2 + 162082944*d**2*f**4 + 3103784960*d*e**4*f - 17414619136*d*e**2*f**3 - 305130240*d*f**5 + 61
06906624*e**4*f**2 - 17414225920*e**2*f**4 + 67931136*f**6))/108 + (d + 4*e + 7*f)*log(x + (-6006260*d**5*e -
2341251*d**5*(d + 4*e + 7*f) - 246016240*d**4*e*f - 31626180*d**4*f*(d + 4*e + 7*f) - 18247680*d**3*e**3 - 240
99840*d**3*e**2*(d + 4*e + 7*f) - 2758371200*d**3*e*f**2 + 7387904*d**3*e*(d + 4*e + 7*f)**2 - 171122976*d**3*
f**2*(d + 4*e + 7*f) + 665280*d**3*(d + 4*e + 7*f)**3 + 298598400*d**2*e**3*f - 369487872*d**2*e**2*f*(d + 4*e
+ 7*f) - 13192256000*d**2*e*f**3 + 90885120*d**2*e*f*(d + 4*e + 7*f)**2 - 441486720*d**2*f**3*(d + 4*e + 7*f)
+ 5536512*d**2*f*(d + 4*e + 7*f)**3 + 587202560*d*e**5 + 12582912*d*e**4*(d + 4*e + 7*f) + 1353646080*d*e**3*
f**2 - 36700160*d*e**3*(d + 4*e + 7*f)**2 - 1448755200*d*e**2*f**2*(d + 4*e + 7*f) - 786432*d*e**2*(d + 4*e +
7*f)**3 - 28282393600*d*e*f**4 + 362729472*d*e*f**2*(d + 4*e + 7*f)**2 - 399575808*d*f**4*(d + 4*e + 7*f) + 10
368000*d*f**2*(d + 4*e + 7*f)**3 + 2751463424*e**5*f - 251658240*e**4*f*(d + 4*e + 7*f) - 530841600*e**3*f**3
- 171966464*e**3*f*(d + 4*e + 7*f)**2 - 1935212544*e**2*f**3*(d + 4*e + 7*f) + 15728640*e**2*f*(d + 4*e + 7*f)
**3 - 21886889984*e*f**5 + 483737600*e*f**3*(d + 4*e + 7*f)**2 + 212474880*f**5*(d + 4*e + 7*f) - 4534272*f**3
*(d + 4*e + 7*f)**3)/(1675971*d**6 + 28507545*d**5*f - 66150400*d**4*e**2 + 168075324*d**4*f**2 - 1091117056*d
**3*e**2*f + 384095520*d**3*f**3 + 318767104*d**2*e**4 - 6528860160*d**2*e**2*f**2 + 162082944*d**2*f**4 + 310
3784960*d*e**4*f - 17414619136*d*e**2*f**3 - 305130240*d*f**5 + 6106906624*e**4*f**2 - 17414225920*e**2*f**4 +
67931136*f**6))/108 + (19*d - 32*e + 52*f)*log(x + (-6006260*d**5*e - 2341251*d**5*(19*d - 32*e + 52*f)/8 - 2
46016240*d**4*e*f - 7906545*d**4*f*(19*d - 32*e + 52*f)/2 - 18247680*d**3*e**3 - 3012480*d**3*e**2*(19*d - 32*
e + 52*f) - 2758371200*d**3*e*f**2 + 115436*d**3*e*(19*d - 32*e + 52*f)**2 - 21390372*d**3*f**2*(19*d - 32*e +
52*f) + 10395*d**3*(19*d - 32*e + 52*f)**3/8 + 298598400*d**2*e**3*f - 46185984*d**2*e**2*f*(19*d - 32*e + 52
*f) - 13192256000*d**2*e*f**3 + 1420080*d**2*e*f*(19*d - 32*e + 52*f)**2 - 55185840*d**2*f**3*(19*d - 32*e + 5
2*f) + 21627*d**2*f*(19*d - 32*e + 52*f)**3/2 + 587202560*d*e**5 + 1572864*d*e**4*(19*d - 32*e + 52*f) + 13536
46080*d*e**3*f**2 - 573440*d*e**3*(19*d - 32*e + 52*f)**2 - 181094400*d*e**2*f**2*(19*d - 32*e + 52*f) - 1536*
d*e**2*(19*d - 32*e + 52*f)**3 - 28282393600*d*e*f**4 + 5667648*d*e*f**2*(19*d - 32*e + 52*f)**2 - 49946976*d*
f**4*(19*d - 32*e + 52*f) + 20250*d*f**2*(19*d - 32*e + 52*f)**3 + 2751463424*e**5*f - 31457280*e**4*f*(19*d -
32*e + 52*f) - 530841600*e**3*f**3 - 2686976*e**3*f*(19*d - 32*e + 52*f)**2 - 241901568*e**2*f**3*(19*d - 32*
e + 52*f) + 30720*e**2*f*(19*d - 32*e + 52*f)**3 - 21886889984*e*f**5 + 7558400*e*f**3*(19*d - 32*e + 52*f)**2
+ 26559360*f**5*(19*d - 32*e + 52*f) - 8856*f**3*(19*d - 32*e + 52*f)**3)/(1675971*d**6 + 28507545*d**5*f - 6
6150400*d**4*e**2 + 168075324*d**4*f**2 - 1091117056*d**3*e**2*f + 384095520*d**3*f**3 + 318767104*d**2*e**4 -
6528860160*d**2*e**2*f**2 + 162082944*d**2*f**4 + 3103784960*d*e**4*f - 17414619136*d*e**2*f**3 - 305130240*d
*f**5 + 6106906624*e**4*f**2 - 17414225920*e**2*f**4 + 67931136*f**6))/864 - (19*d + 32*e + 52*f)*log(x + (-60
06260*d**5*e + 2341251*d**5*(19*d + 32*e + 52*f)/8 - 246016240*d**4*e*f + 7906545*d**4*f*(19*d + 32*e + 52*f)/
2 - 18247680*d**3*e**3 + 3012480*d**3*e**2*(19*d + 32*e + 52*f) - 2758371200*d**3*e*f**2 + 115436*d**3*e*(19*d
+ 32*e + 52*f)**2 + 21390372*d**3*f**2*(19*d + 32*e + 52*f) - 10395*d**3*(19*d + 32*e + 52*f)**3/8 + 29859840
0*d**2*e**3*f + 46185984*d**2*e**2*f*(19*d + 32*e + 52*f) - 13192256000*d**2*e*f**3 + 1420080*d**2*e*f*(19*d +
32*e + 52*f)**2 + 55185840*d**2*f**3*(19*d + 32*e + 52*f) - 21627*d**2*f*(19*d + 32*e + 52*f)**3/2 + 58720256
0*d*e**5 - 1572864*d*e**4*(19*d + 32*e + 52*f) + 1353646080*d*e**3*f**2 - 573440*d*e**3*(19*d + 32*e + 52*f)**
2 + 181094400*d*e**2*f**2*(19*d + 32*e + 52*f) + 1536*d*e**2*(19*d + 32*e + 52*f)**3 - 28282393600*d*e*f**4 +
5667648*d*e*f**2*(19*d + 32*e + 52*f)**2 + 49946976*d*f**4*(19*d + 32*e + 52*f) - 20250*d*f**2*(19*d + 32*e +
52*f)**3 + 2751463424*e**5*f + 31457280*e**4*f*(19*d + 32*e + 52*f) - 530841600*e**3*f**3 - 2686976*e**3*f*(19
*d + 32*e + 52*f)**2 + 241901568*e**2*f**3*(19*d + 32*e + 52*f) - 30720*e**2*f*(19*d + 32*e + 52*f)**3 - 21886
889984*e*f**5 + 7558400*e*f**3*(19*d + 32*e + 52*f)**2 - 26559360*f**5*(19*d + 32*e + 52*f) + 8856*f**3*(19*d
+ 32*e + 52*f)**3)/(1675971*d**6 + 28507545*d**5*f - 66150400*d**4*e**2 + 168075324*d**4*f**2 - 1091117056*d**
3*e**2*f + 384095520*d**3*f**3 + 318767104*d**2*e**4 - 6528860160*d**2*e**2*f**2 + 162082944*d**2*f**4 + 31037
84960*d*e**4*f - 17414619136*d*e**2*f**3 - 305130240*d*f**5 + 6106906624*e**4*f**2 - 17414225920*e**2*f**4 + 6
7931136*f**6))/864 - (8*e*x**2 - 20*e + x**3*(5*d + 8*f) + x*(-17*d - 20*f))/(72*x**4 - 360*x**2 + 288)
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Giac [A] time = 1.08665, size = 155, normalized size = 1.35 \begin{align*} \frac{1}{864} \,{\left (19 \, d + 52 \, f - 32 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac{1}{108} \,{\left (d + 7 \, f - 4 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{108} \,{\left (d + 7 \, f + 4 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) - \frac{1}{864} \,{\left (19 \, d + 52 \, f + 32 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac{5 \, d x^{3} + 8 \, f x^{3} + 8 \, x^{2} e - 17 \, d x - 20 \, f x - 20 \, e}{72 \,{\left (x^{4} - 5 \, x^{2} + 4\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)^2,x, algorithm="giac")
[Out]
1/864*(19*d + 52*f - 32*e)*log(abs(x + 2)) - 1/108*(d + 7*f - 4*e)*log(abs(x + 1)) + 1/108*(d + 7*f + 4*e)*log
(abs(x - 1)) - 1/864*(19*d + 52*f + 32*e)*log(abs(x - 2)) - 1/72*(5*d*x^3 + 8*f*x^3 + 8*x^2*e - 17*d*x - 20*f*
x - 20*e)/(x^4 - 5*x^2 + 4)