3.62 \(\int \frac{1}{(8+24 x+8 x^2-15 x^3+8 x^4)^2} \, dx\)

Optimal. Leaf size=366 \[ \frac{73}{208} \sqrt{\frac{3}{13}} \tan ^{-1}\left (\frac{-5 x^2+12 x+8}{\sqrt{39} x^2}\right )-\frac{3 \left (3359-107 \left (\frac{4}{x}+3\right )^2\right )}{208 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right )}+\frac{\left (3327931-129631 \left (\frac{4}{x}+3\right )^2\right ) \left (\frac{4}{x}+3\right )}{322608 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right )}-\frac{\sqrt{\frac{2623170438295 \sqrt{517}-59644114671451}{40326}} \log \left (\left (\frac{4}{x}+3\right )^2-\sqrt{2 \left (19+\sqrt{517}\right )} \left (\frac{4}{x}+3\right )+\sqrt{517}\right )}{645216}+\frac{\sqrt{\frac{2623170438295 \sqrt{517}-59644114671451}{40326}} \log \left (\left (\frac{4}{x}+3\right )^2+\sqrt{2 \left (19+\sqrt{517}\right )} \left (\frac{4}{x}+3\right )+\sqrt{517}\right )}{645216}-\frac{\sqrt{\frac{19+\sqrt{517}}{40326}} \left (1678181+74897 \sqrt{517}\right ) \tan ^{-1}\left (\frac{\frac{8}{x}-\sqrt{2 \left (19+\sqrt{517}\right )}+6}{\sqrt{2 \left (\sqrt{517}-19\right )}}\right )}{645216}-\frac{\sqrt{\frac{19+\sqrt{517}}{40326}} \left (1678181+74897 \sqrt{517}\right ) \tan ^{-1}\left (\frac{\frac{8}{x}+\sqrt{2 \left (19+\sqrt{517}\right )}+6}{\sqrt{2 \left (\sqrt{517}-19\right )}}\right )}{645216} \]

[Out]

(-3*(3359 - 107*(3 + 4/x)^2))/(208*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)) + ((3327931 - 129631*(3 + 4/x)^2)*(3
+ 4/x))/(322608*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)) - (Sqrt[(19 + Sqrt[517])/40326]*(1678181 + 74897*Sqrt[51
7])*ArcTan[(6 - Sqrt[2*(19 + Sqrt[517])] + 8/x)/Sqrt[2*(-19 + Sqrt[517])]])/645216 - (Sqrt[(19 + Sqrt[517])/40
326]*(1678181 + 74897*Sqrt[517])*ArcTan[(6 + Sqrt[2*(19 + Sqrt[517])] + 8/x)/Sqrt[2*(-19 + Sqrt[517])]])/64521
6 + (73*Sqrt[3/13]*ArcTan[(8 + 12*x - 5*x^2)/(Sqrt[39]*x^2)])/208 - (Sqrt[(-59644114671451 + 2623170438295*Sqr
t[517])/40326]*Log[Sqrt[517] - Sqrt[2*(19 + Sqrt[517])]*(3 + 4/x) + (3 + 4/x)^2])/645216 + (Sqrt[(-59644114671
451 + 2623170438295*Sqrt[517])/40326]*Log[Sqrt[517] + Sqrt[2*(19 + Sqrt[517])]*(3 + 4/x) + (3 + 4/x)^2])/64521
6

________________________________________________________________________________________

Rubi [A]  time = 0.507629, antiderivative size = 366, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2069, 12, 1673, 1678, 1169, 634, 618, 204, 628, 1663, 1660} \[ \frac{73}{208} \sqrt{\frac{3}{13}} \tan ^{-1}\left (\frac{-5 x^2+12 x+8}{\sqrt{39} x^2}\right )-\frac{3 \left (3359-107 \left (\frac{4}{x}+3\right )^2\right )}{208 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right )}+\frac{\left (3327931-129631 \left (\frac{4}{x}+3\right )^2\right ) \left (\frac{4}{x}+3\right )}{322608 \left (\left (\frac{4}{x}+3\right )^4-38 \left (\frac{4}{x}+3\right )^2+517\right )}-\frac{\sqrt{\frac{2623170438295 \sqrt{517}-59644114671451}{40326}} \log \left (\left (\frac{4}{x}+3\right )^2-\sqrt{2 \left (19+\sqrt{517}\right )} \left (\frac{4}{x}+3\right )+\sqrt{517}\right )}{645216}+\frac{\sqrt{\frac{2623170438295 \sqrt{517}-59644114671451}{40326}} \log \left (\left (\frac{4}{x}+3\right )^2+\sqrt{2 \left (19+\sqrt{517}\right )} \left (\frac{4}{x}+3\right )+\sqrt{517}\right )}{645216}-\frac{\sqrt{\frac{19+\sqrt{517}}{40326}} \left (1678181+74897 \sqrt{517}\right ) \tan ^{-1}\left (\frac{\frac{8}{x}-\sqrt{2 \left (19+\sqrt{517}\right )}+6}{\sqrt{2 \left (\sqrt{517}-19\right )}}\right )}{645216}-\frac{\sqrt{\frac{19+\sqrt{517}}{40326}} \left (1678181+74897 \sqrt{517}\right ) \tan ^{-1}\left (\frac{\frac{8}{x}+\sqrt{2 \left (19+\sqrt{517}\right )}+6}{\sqrt{2 \left (\sqrt{517}-19\right )}}\right )}{645216} \]

Antiderivative was successfully verified.

[In]

Int[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-2),x]

[Out]

(-3*(3359 - 107*(3 + 4/x)^2))/(208*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)) + ((3327931 - 129631*(3 + 4/x)^2)*(3
+ 4/x))/(322608*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)) - (Sqrt[(19 + Sqrt[517])/40326]*(1678181 + 74897*Sqrt[51
7])*ArcTan[(6 - Sqrt[2*(19 + Sqrt[517])] + 8/x)/Sqrt[2*(-19 + Sqrt[517])]])/645216 - (Sqrt[(19 + Sqrt[517])/40
326]*(1678181 + 74897*Sqrt[517])*ArcTan[(6 + Sqrt[2*(19 + Sqrt[517])] + 8/x)/Sqrt[2*(-19 + Sqrt[517])]])/64521
6 + (73*Sqrt[3/13]*ArcTan[(8 + 12*x - 5*x^2)/(Sqrt[39]*x^2)])/208 - (Sqrt[(-59644114671451 + 2623170438295*Sqr
t[517])/40326]*Log[Sqrt[517] - Sqrt[2*(19 + Sqrt[517])]*(3 + 4/x) + (3 + 4/x)^2])/645216 + (Sqrt[(-59644114671
451 + 2623170438295*Sqrt[517])/40326]*Log[Sqrt[517] + Sqrt[2*(19 + Sqrt[517])]*(3 + 4/x) + (3 + 4/x)^2])/64521
6

Rule 2069

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Dist[-16*a^2, Subst[Int[(1*((a*(-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 256*a^3*e -
32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4))/(b - 4*a*x)^4)^p)/(b - 4*a*x)^2, x], x, b/(4*a) + 1/x], x] /; NeQ[a
, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] &&  !
IGtQ[p, 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 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 1678

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2
+ c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*
a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1169

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

Rule 634

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

Rule 618

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

Rule 204

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1663

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)
*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Inte
gerQ[(m - 1)/2]

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rubi steps

\begin{align*} \int \frac{1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx &=-\left (1024 \operatorname{Subst}\left (\int \frac{(24-32 x)^6}{64 \left (2117632-2490368 x^2+1048576 x^4\right )^2} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )\right )\\ &=-\left (16 \operatorname{Subst}\left (\int \frac{(24-32 x)^6}{\left (2117632-2490368 x^2+1048576 x^4\right )^2} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )\right )\\ &=-\left (16 \operatorname{Subst}\left (\int \frac{x \left (-1528823808-9059696640 x^2-4831838208 x^4\right )}{\left (2117632-2490368 x^2+1048576 x^4\right )^2} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )\right )-16 \operatorname{Subst}\left (\int \frac{191102976+5096079360 x^2+9059696640 x^4+1073741824 x^6}{\left (2117632-2490368 x^2+1048576 x^4\right )^2} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )\\ &=\frac{\left (3327931-129631 \left (3+\frac{4}{x}\right )^2\right ) \left (3+\frac{4}{x}\right )}{322608 \left (517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{120925685220163941564416+86350361930539017961472 x^2}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{709422494427119616}-8 \operatorname{Subst}\left (\int \frac{-1528823808-9059696640 x-4831838208 x^2}{\left (2117632-2490368 x+1048576 x^2\right )^2} \, dx,x,\left (\frac{3}{4}+\frac{1}{x}\right )^2\right )\\ &=-\frac{3 \left (3359-107 \left (3+\frac{4}{x}\right )^2\right )}{208 \left (517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4\right )}+\frac{\left (3327931-129631 \left (3+\frac{4}{x}\right )^2\right ) \left (3+\frac{4}{x}\right )}{322608 \left (517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4\right )}-\frac{\operatorname{Subst}\left (\int -\frac{46232264924725248}{2117632-2490368 x+1048576 x^2} \, dx,x,\left (\frac{3}{4}+\frac{1}{x}\right )^2\right )}{335007449088}-\frac{\operatorname{Subst}\left (\int \frac{30231421305040985391104 \sqrt{2 \left (19+\sqrt{517}\right )}-\left (120925685220163941564416-5396897620658688622592 \sqrt{517}\right ) x}{\frac{\sqrt{517}}{16}-\frac{1}{2} \sqrt{\frac{1}{2} \left (19+\sqrt{517}\right )} x+x^2} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{23246356297387855577088 \sqrt{1034 \left (19+\sqrt{517}\right )}}-\frac{\operatorname{Subst}\left (\int \frac{30231421305040985391104 \sqrt{2 \left (19+\sqrt{517}\right )}+\left (120925685220163941564416-5396897620658688622592 \sqrt{517}\right ) x}{\frac{\sqrt{517}}{16}+\frac{1}{2} \sqrt{\frac{1}{2} \left (19+\sqrt{517}\right )} x+x^2} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{23246356297387855577088 \sqrt{1034 \left (19+\sqrt{517}\right )}}\\ &=-\frac{3 \left (3359-107 \left (3+\frac{4}{x}\right )^2\right )}{208 \left (517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4\right )}+\frac{\left (3327931-129631 \left (3+\frac{4}{x}\right )^2\right ) \left (3+\frac{4}{x}\right )}{322608 \left (517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4\right )}+\frac{1794048}{13} \operatorname{Subst}\left (\int \frac{1}{2117632-2490368 x+1048576 x^2} \, dx,x,\left (\frac{3}{4}+\frac{1}{x}\right )^2\right )+\frac{\left (1678181-74897 \sqrt{517}\right ) \operatorname{Subst}\left (\int \frac{-\frac{1}{2} \sqrt{\frac{1}{2} \left (19+\sqrt{517}\right )}+2 x}{\frac{\sqrt{517}}{16}-\frac{1}{2} \sqrt{\frac{1}{2} \left (19+\sqrt{517}\right )} x+x^2} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{645216 \sqrt{1034 \left (19+\sqrt{517}\right )}}-\frac{\left (1678181-74897 \sqrt{517}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \sqrt{\frac{1}{2} \left (19+\sqrt{517}\right )}+2 x}{\frac{\sqrt{517}}{16}+\frac{1}{2} \sqrt{\frac{1}{2} \left (19+\sqrt{517}\right )} x+x^2} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{645216 \sqrt{1034 \left (19+\sqrt{517}\right )}}-\frac{\left (38721749+1678181 \sqrt{517}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{517}}{16}-\frac{1}{2} \sqrt{\frac{1}{2} \left (19+\sqrt{517}\right )} x+x^2} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{1334306688}-\frac{\left (38721749+1678181 \sqrt{517}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{517}}{16}+\frac{1}{2} \sqrt{\frac{1}{2} \left (19+\sqrt{517}\right )} x+x^2} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )}{1334306688}\\ &=-\frac{3 \left (3359-107 \left (3+\frac{4}{x}\right )^2\right )}{208 \left (517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4\right )}+\frac{\left (3327931-129631 \left (3+\frac{4}{x}\right )^2\right ) \left (3+\frac{4}{x}\right )}{322608 \left (517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4\right )}-\frac{\sqrt{-\frac{59644114671451}{40326}+\frac{5073830635 \sqrt{517}}{78}} \log \left (\sqrt{517}-\sqrt{2 \left (19+\sqrt{517}\right )} \left (3+\frac{4}{x}\right )+\left (3+\frac{4}{x}\right )^2\right )}{645216}+\frac{\sqrt{-\frac{59644114671451}{40326}+\frac{5073830635 \sqrt{517}}{78}} \log \left (\sqrt{517}+\sqrt{2 \left (19+\sqrt{517}\right )} \left (3+\frac{4}{x}\right )+\left (3+\frac{4}{x}\right )^2\right )}{645216}-\frac{3588096}{13} \operatorname{Subst}\left (\int \frac{1}{-2680059592704-x^2} \, dx,x,-2490368+2097152 \left (\frac{3}{4}+\frac{1}{x}\right )^2\right )+\frac{\left (38721749+1678181 \sqrt{517}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{8} \left (19-\sqrt{517}\right )-x^2} \, dx,x,-\frac{1}{2} \sqrt{\frac{1}{2} \left (19+\sqrt{517}\right )}+2 \left (\frac{3}{4}+\frac{1}{x}\right )\right )}{667153344}+\frac{\left (38721749+1678181 \sqrt{517}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{8} \left (19-\sqrt{517}\right )-x^2} \, dx,x,\frac{1}{4} \left (6+\sqrt{2 \left (19+\sqrt{517}\right )}+\frac{8}{x}\right )\right )}{667153344}\\ &=-\frac{3 \left (3359-107 \left (3+\frac{4}{x}\right )^2\right )}{208 \left (517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4\right )}+\frac{\left (3327931-129631 \left (3+\frac{4}{x}\right )^2\right ) \left (3+\frac{4}{x}\right )}{322608 \left (517-38 \left (3+\frac{4}{x}\right )^2+\left (3+\frac{4}{x}\right )^4\right )}-\frac{73}{208} \sqrt{\frac{3}{13}} \tan ^{-1}\left (\frac{19-\left (3+\frac{4}{x}\right )^2}{2 \sqrt{39}}\right )-\frac{\left (1678181+74897 \sqrt{517}\right ) \tan ^{-1}\left (\frac{6+\sqrt{2 \left (19+\sqrt{517}\right )}+\frac{8}{x}}{\sqrt{2 \left (-19+\sqrt{517}\right )}}\right )}{322608 \sqrt{1034 \left (-19+\sqrt{517}\right )}}-\frac{\left (1678181+74897 \sqrt{517}\right ) \tan ^{-1}\left (\frac{8+\left (6-\sqrt{2 \left (19+\sqrt{517}\right )}\right ) x}{\sqrt{2 \left (-19+\sqrt{517}\right )} x}\right )}{322608 \sqrt{1034 \left (-19+\sqrt{517}\right )}}-\frac{\sqrt{-\frac{59644114671451}{40326}+\frac{5073830635 \sqrt{517}}{78}} \log \left (\sqrt{517}-\sqrt{2 \left (19+\sqrt{517}\right )} \left (3+\frac{4}{x}\right )+\left (3+\frac{4}{x}\right )^2\right )}{645216}+\frac{\sqrt{-\frac{59644114671451}{40326}+\frac{5073830635 \sqrt{517}}{78}} \log \left (\sqrt{517}+\sqrt{2 \left (19+\sqrt{517}\right )} \left (3+\frac{4}{x}\right )+\left (3+\frac{4}{x}\right )^2\right )}{645216}\\ \end{align*}

Mathematica [C]  time = 0.0185367, size = 128, normalized size = 0.35 \[ \frac{\text{RootSum}\left [8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\& ,\frac{19640 \text{$\#$1}^2 \log (x-\text{$\#$1})-57489 \text{$\#$1} \log (x-\text{$\#$1})+74897 \log (x-\text{$\#$1})}{32 \text{$\#$1}^3-45 \text{$\#$1}^2+16 \text{$\#$1}+24}\& \right ]}{80652}+\frac{39280 x^3-94314 x^2+89033 x+72888}{161304 \left (8 x^4-15 x^3+8 x^2+24 x+8\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-2),x]

[Out]

(72888 + 89033*x - 94314*x^2 + 39280*x^3)/(161304*(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)) + RootSum[8 + 24*#1 + 8
*#1^2 - 15*#1^3 + 8*#1^4 & , (74897*Log[x - #1] - 57489*Log[x - #1]*#1 + 19640*Log[x - #1]*#1^2)/(24 + 16*#1 -
45*#1^2 + 32*#1^3) & ]/80652

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Maple [C]  time = 0.007, size = 96, normalized size = 0.3 \begin{align*}{ \left ({\frac{2455\,{x}^{3}}{80652}}-{\frac{1429\,{x}^{2}}{19552}}+{\frac{89033\,x}{1290432}}+{\frac{3037}{53768}} \right ) \left ({x}^{4}-{\frac{15\,{x}^{3}}{8}}+{x}^{2}+3\,x+1 \right ) ^{-1}}+{\frac{1}{80652}\sum _{{\it \_R}={\it RootOf} \left ( 8\,{{\it \_Z}}^{4}-15\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}+24\,{\it \_Z}+8 \right ) }{\frac{ \left ( 19640\,{{\it \_R}}^{2}-57489\,{\it \_R}+74897 \right ) \ln \left ( x-{\it \_R} \right ) }{32\,{{\it \_R}}^{3}-45\,{{\it \_R}}^{2}+16\,{\it \_R}+24}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(8*x^4-15*x^3+8*x^2+24*x+8)^2,x)

[Out]

(2455/80652*x^3-1429/19552*x^2+89033/1290432*x+3037/53768)/(x^4-15/8*x^3+x^2+3*x+1)+1/80652*sum((19640*_R^2-57
489*_R+74897)/(32*_R^3-45*_R^2+16*_R+24)*ln(x-_R),_R=RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{39280 \, x^{3} - 94314 \, x^{2} + 89033 \, x + 72888}{161304 \,{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}} + \frac{1}{80652} \, \int \frac{19640 \, x^{2} - 57489 \, x + 74897}{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8)^2,x, algorithm="maxima")

[Out]

1/161304*(39280*x^3 - 94314*x^2 + 89033*x + 72888)/(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8) + 1/80652*integrate((19
640*x^2 - 57489*x + 74897)/(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8), x)

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Fricas [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(1/(8*x^4-15*x^3+8*x^2+24*x+8)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 0.956513, size = 76, normalized size = 0.21 \begin{align*} \frac{39280 x^{3} - 94314 x^{2} + 89033 x + 72888}{1290432 x^{4} - 2419560 x^{3} + 1290432 x^{2} + 3871296 x + 1290432} + \operatorname{RootSum}{\left (1991678427489244336128 t^{4} + 56610734087162189376 t^{2} + 20948104645409331 t + 1938464112640, \left ( t \mapsto t \log{\left (- \frac{705077742393966388453254545830232274432 t^{3}}{50310177134331359960511301071755} + \frac{126981475823989945260152267904580608 t^{2}}{50310177134331359960511301071755} - \frac{20040865325746858989799932658629535256 t}{50310177134331359960511301071755} + x - \frac{18148095975820500157416495488749859}{241488850244790527810454245144424} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*x**4-15*x**3+8*x**2+24*x+8)**2,x)

[Out]

(39280*x**3 - 94314*x**2 + 89033*x + 72888)/(1290432*x**4 - 2419560*x**3 + 1290432*x**2 + 3871296*x + 1290432)
+ RootSum(1991678427489244336128*_t**4 + 56610734087162189376*_t**2 + 20948104645409331*_t + 1938464112640, L
ambda(_t, _t*log(-705077742393966388453254545830232274432*_t**3/50310177134331359960511301071755 + 12698147582
3989945260152267904580608*_t**2/50310177134331359960511301071755 - 20040865325746858989799932658629535256*_t/5
0310177134331359960511301071755 + x - 18148095975820500157416495488749859/241488850244790527810454245144424)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8)^2,x, algorithm="giac")

[Out]

undef