3.486 \(\int \frac {1}{x^{5/2} (a+b x^2) (c+d x^2)^3} \, dx\)

Optimal. Leaf size=681 \[ \frac {b^{15/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{15/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}-\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}-\frac {77 a^2 d^2-133 a b c d+32 b^2 c^2}{48 a c^3 x^{3/2} (b c-a d)^2}-\frac {d (19 b c-11 a d)}{16 c^2 x^{3/2} \left (c+d x^2\right ) (b c-a d)^2}-\frac {d}{4 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

1/48*(-77*a^2*d^2+133*a*b*c*d-32*b^2*c^2)/a/c^3/(-a*d+b*c)^2/x^(3/2)-1/4*d/c/(-a*d+b*c)/x^(3/2)/(d*x^2+c)^2-1/
16*d*(-11*a*d+19*b*c)/c^2/(-a*d+b*c)^2/x^(3/2)/(d*x^2+c)+1/2*b^(15/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4)
)/a^(7/4)/(-a*d+b*c)^3*2^(1/2)-1/2*b^(15/4)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/(-a*d+b*c)^3*2^(
1/2)-1/64*d^(7/4)*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(15/4)/(-a*
d+b*c)^3*2^(1/2)+1/64*d^(7/4)*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c
^(15/4)/(-a*d+b*c)^3*2^(1/2)+1/4*b^(15/4)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(7/4)/(-a*d+
b*c)^3*2^(1/2)-1/4*b^(15/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(7/4)/(-a*d+b*c)^3*2^(1/2)
-1/128*d^(7/4)*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(1
5/4)/(-a*d+b*c)^3*2^(1/2)+1/128*d^(7/4)*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1
/4)*2^(1/2)*x^(1/2))/c^(15/4)/(-a*d+b*c)^3*2^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.93, antiderivative size = 681, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {466, 472, 579, 583, 522, 211, 1165, 628, 1162, 617, 204} \[ -\frac {77 a^2 d^2-133 a b c d+32 b^2 c^2}{48 a c^3 x^{3/2} (b c-a d)^2}-\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}-\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {b^{15/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{15/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d (19 b c-11 a d)}{16 c^2 x^{3/2} \left (c+d x^2\right ) (b c-a d)^2}-\frac {d}{4 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

-(32*b^2*c^2 - 133*a*b*c*d + 77*a^2*d^2)/(48*a*c^3*(b*c - a*d)^2*x^(3/2)) - d/(4*c*(b*c - a*d)*x^(3/2)*(c + d*
x^2)^2) - (d*(19*b*c - 11*a*d))/(16*c^2*(b*c - a*d)^2*x^(3/2)*(c + d*x^2)) + (b^(15/4)*ArcTan[1 - (Sqrt[2]*b^(
1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(15/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/
4)])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d
^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(15/4)*(b*c - a*d)^3) + (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d
^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(15/4)*(b*c - a*d)^3) + (b^(15/4)*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(15/4)*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210*a
*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(b*c - a
*d)^3) + (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqr
t[d]*x])/(64*Sqrt[2]*c^(15/4)*(b*c - a*d)^3)

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 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 472

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

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 579

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 1162

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {8 b c-11 a d-11 b d x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 c (b c-a d)}\\ &=-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {32 b^2 c^2-133 a b c d+77 a^2 d^2-7 b d (19 b c-11 a d) x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {3 \left (32 b^3 c^3+32 a b^2 c^2 d-133 a^2 b c d^2+77 a^3 d^3\right )+3 b d \left (32 b^2 c^2-133 a b c d+77 a^2 d^2\right ) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{48 a c^3 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac {\left (2 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^3}+\frac {\left (d^2 \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^3 (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac {b^4 \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} (b c-a d)^3}-\frac {b^4 \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} (b c-a d)^3}+\frac {\left (d^2 \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{7/2} (b c-a d)^3}+\frac {\left (d^2 \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{7/2} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac {b^{7/2} \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^{3/2} (b c-a d)^3}-\frac {b^{7/2} \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^{3/2} (b c-a d)^3}+\frac {b^{15/4} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{15/4} \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {\left (d^{3/2} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{7/2} (b c-a d)^3}+\frac {\left (d^{3/2} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{7/2} (b c-a d)^3}-\frac {\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}-\frac {\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac {b^{15/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}-\frac {b^{15/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{15/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}+\frac {\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}-\frac {\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac {b^{15/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {b^{15/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A]  time = 6.18, size = 701, normalized size = 1.03 \[ \frac {b^{15/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \tan ^{-1}\left (\frac {2 \sqrt [4]{b} \sqrt {x}-\sqrt {2} \sqrt [4]{a}}{\sqrt {2} \sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a}+2 \sqrt [4]{b} \sqrt {x}}{\sqrt {2} \sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}+\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (a d-b c)^3}-\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (a d-b c)^3}-\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (\frac {2 \sqrt [4]{d} \sqrt {x}-\sqrt {2} \sqrt [4]{c}}{\sqrt {2} \sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (a d-b c)^3}-\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c}+2 \sqrt [4]{d} \sqrt {x}}{\sqrt {2} \sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (a d-b c)^3}+\frac {d^2 \sqrt {x} (23 b c-15 a d)}{16 c^3 \left (c+d x^2\right ) (b c-a d)^2}+\frac {d^2 \sqrt {x}}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)}-\frac {2}{3 a c^3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

-2/(3*a*c^3*x^(3/2)) + (d^2*Sqrt[x])/(4*c^2*(b*c - a*d)*(c + d*x^2)^2) + (d^2*(23*b*c - 15*a*d)*Sqrt[x])/(16*c
^3*(b*c - a*d)^2*(c + d*x^2)) - (b^(15/4)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/
(Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(15/4)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(
Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*ArcTan[(-(Sqrt[2]*c^(1/4))
+ 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(15/4)*(-(b*c) + a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210*
a*b*c*d + 77*a^2*d^2)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(32*Sqrt[2]*c^(15/4)*(-
(b*c) + a*d)^3) + (b^(15/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*
c - a*d)^3) - (b^(15/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c -
a*d)^3) + (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sq
rt[d]*x])/(64*Sqrt[2]*c^(15/4)*(-(b*c) + a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[
c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(-(b*c) + a*d)^3)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 1.71, size = 995, normalized size = 1.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

-(a*b^3)^(1/4)*b^3*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^3*c^3 - 3*
sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2 - sqrt(2)*a^5*d^3) - (a*b^3)^(1/4)*b^3*arctan(-1/2*sqrt(2)*(sqrt
(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d
^2 - sqrt(2)*a^5*d^3) - 1/2*(a*b^3)^(1/4)*b^3*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^
3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2 - sqrt(2)*a^5*d^3) + 1/2*(a*b^3)^(1/4)*b^3*log(-sqrt(2
)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2
- sqrt(2)*a^5*d^3) + 1/32*(165*(c*d^3)^(1/4)*b^2*c^2*d - 210*(c*d^3)^(1/4)*a*b*c*d^2 + 77*(c*d^3)^(1/4)*a^2*d^
3)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d
+ 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) + 1/32*(165*(c*d^3)^(1/4)*b^2*c^2*d - 210*(c*d^3)^(1/4)*a*b*c
*d^2 + 77*(c*d^3)^(1/4)*a^2*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b
^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) + 1/64*(165*(c*d^3)^(1/4)*b^2*
c^2*d - 210*(c*d^3)^(1/4)*a*b*c*d^2 + 77*(c*d^3)^(1/4)*a^2*d^3)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d
))/(sqrt(2)*b^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) - 1/64*(165*(c*d^
3)^(1/4)*b^2*c^2*d - 210*(c*d^3)^(1/4)*a*b*c*d^2 + 77*(c*d^3)^(1/4)*a^2*d^3)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4)
+ x + sqrt(c/d))/(sqrt(2)*b^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) + 1
/16*(23*b*c*d^3*x^(5/2) - 15*a*d^4*x^(5/2) + 27*b*c^2*d^2*sqrt(x) - 19*a*c*d^3*sqrt(x))/((b^2*c^5 - 2*a*b*c^4*
d + a^2*c^3*d^2)*(d*x^2 + c)^2) - 2/3/(a*c^3*x^(3/2))

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maple [A]  time = 0.03, size = 906, normalized size = 1.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x)

[Out]

1/4/a^2*b^4/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1
/2)*x^(1/2)+(a/b)^(1/2)))+1/2/a^2*b^4/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+1/
2/a^2*b^4/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-15/16*d^5/c^3/(a*d-b*c)^3/(d*x
^2+c)^2*x^(5/2)*a^2+19/8*d^4/c^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(5/2)*a*b-23/16*d^3/c/(a*d-b*c)^3/(d*x^2+c)^2*x^(5/
2)*b^2-19/16*d^4/c^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(1/2)*a^2+23/8*d^3/c/(a*d-b*c)^3/(d*x^2+c)^2*x^(1/2)*a*b-27/16*
d^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(1/2)*b^2-77/64*d^4/c^4/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/
4)*x^(1/2)+1)*a^2+105/32*d^3/c^3/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b-165
/64*d^2/c^2/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2-77/64*d^4/c^4/(a*d-b*c)^
3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2+105/32*d^3/c^3/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-165/64*d^2/c^2/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)
^(1/4)*x^(1/2)-1)*b^2-77/128*d^4/c^4/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(
1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a^2+105/64*d^3/c^3/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*ln((x+(c
/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a*b-165/128*d^2/c^2/(a*d-b
*c)^3*(c/d)^(1/4)*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^
(1/2)))*b^2-2/3/a/c^3/x^(3/2)

________________________________________________________________________________________

maxima [A]  time = 2.62, size = 755, normalized size = 1.11 \[ -\frac {32 \, b^{2} c^{4} - 64 \, a b c^{3} d + 32 \, a^{2} c^{2} d^{2} + {\left (32 \, b^{2} c^{2} d^{2} - 133 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} x^{4} + {\left (64 \, b^{2} c^{3} d - 209 \, a b c^{2} d^{2} + 121 \, a^{2} c d^{3}\right )} x^{2}}{48 \, {\left ({\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{\frac {11}{2}} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{\frac {7}{2}} + {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} x^{\frac {3}{2}}\right )}} - \frac {\frac {2 \, \sqrt {2} b^{4} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b^{4} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {15}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {15}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}}{4 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/48*(32*b^2*c^4 - 64*a*b*c^3*d + 32*a^2*c^2*d^2 + (32*b^2*c^2*d^2 - 133*a*b*c*d^3 + 77*a^2*d^4)*x^4 + (64*b^
2*c^3*d - 209*a*b*c^2*d^2 + 121*a^2*c*d^3)*x^2)/((a*b^2*c^5*d^2 - 2*a^2*b*c^4*d^3 + a^3*c^3*d^4)*x^(11/2) + 2*
(a*b^2*c^6*d - 2*a^2*b*c^5*d^2 + a^3*c^4*d^3)*x^(7/2) + (a*b^2*c^7 - 2*a^2*b*c^6*d + a^3*c^5*d^2)*x^(3/2)) - 1
/4*(2*sqrt(2)*b^4*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqr
t(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*b^4*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/
sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(15/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x)
+ sqrt(b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^(15/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/
a^(3/4))/(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3) + 1/128*(2*sqrt(2)*(165*b^2*c^2*d^2 - 210*a*b
*c*d^3 + 77*a^2*d^4)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(
sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(165*b^2*c^2*d^2 - 210*a*b*c*d^3 + 77*a^2*d^4)*arctan(-1/2*sqrt(2)*
(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)
*(165*b^2*c^2*d^2 - 210*a*b*c*d^3 + 77*a^2*d^4)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^
(3/4)*d^(1/4)) - sqrt(2)*(165*b^2*c^2*d^2 - 210*a*b*c*d^3 + 77*a^2*d^4)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) +
sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)

________________________________________________________________________________________

mupad [B]  time = 10.81, size = 44524, normalized size = 65.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^3),x)

[Out]

atan(((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 1
1394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c
^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 +
1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^
7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 36909
87520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15
+ 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^1
0 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*
c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*
d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a
^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1
107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 -
3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4
*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^2
7 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c
^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555
072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^
10 - 201326592*a*b^11*c^26*d))^(3/4)*(x^(1/2)*(18446744073709551616*a^11*b^39*c^68*d^4 - 479615345916448342016
*a^12*b^38*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*d^6 - 47961534591644834201600*a^14*b^36*c^65*d^7 +
275778823901957796659200*a^15*b^35*c^64*d^8 - 1212936383169193658286080*a^16*b^34*c^63*d^9 + 4232993998288506
144686080*a^17*b^33*c^62*d^10 - 11941164077799654041845760*a^18*b^32*c^61*d^11 + 27104869321333056471040000*a^
19*b^31*c^60*d^12 - 46637619173392487079215104*a^20*b^30*c^59*d^13 + 43611606538557895133364224*a^21*b^29*c^58
*d^14 + 72781112360087761599856640*a^22*b^28*c^57*d^15 - 523234066593179210717593600*a^23*b^27*c^56*d^16 + 172
3753001020797184743833600*a^24*b^26*c^55*d^17 - 4269437167365872814842183680*a^25*b^25*c^54*d^18 + 87273227578
49829186700574720*a^26*b^24*c^53*d^19 - 15215326043975142249374679040*a^27*b^23*c^52*d^20 + 229626584632465196
25580544000*a^28*b^22*c^51*d^21 - 30231538828274701475145318400*a^29*b^21*c^50*d^22 + 348701630317663899528829
33760*a^30*b^20*c^49*d^23 - 35316718238336158489724846080*a^31*b^19*c^48*d^24 + 31433146498544749041648926720*
a^32*b^18*c^47*d^25 - 24575140799491012895231180800*a^33*b^17*c^46*d^26 + 16850754961433442876234137600*a^34*b
^16*c^45*d^27 - 10105200492115418262179676160*a^35*b^15*c^44*d^28 + 5278011312905736232783314944*a^36*b^14*c^4
3*d^29 - 2387248399405916166169821184*a^37*b^13*c^42*d^30 + 927828632312674738870681600*a^38*b^12*c^41*d^31 -
306693733103726739901644800*a^39*b^11*c^40*d^32 + 85038075959446046066606080*a^40*b^10*c^39*d^33 - 19409595119
210898894356480*a^41*b^9*c^38*d^34 + 3551400405635812871372800*a^42*b^8*c^37*d^35 - 500844593983932480880640*a
^43*b^7*c^36*d^36 + 51111802530990496153600*a^44*b^6*c^35*d^37 - 3359577235627333124096*a^45*b^5*c^34*d^38 + 1
06807368762718683136*a^46*b^4*c^33*d^39) + (-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7
*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^
5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^1
5*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*
a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7
+ 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11
*c^26*d))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 1069911156275153993728*a^14*b^37*c^70*d^5 + 1497875
6187852155912192*a^15*b^36*c^69*d^6 - 134999037738929532960768*a^16*b^35*c^68*d^7 + 882016079904862321508352*a
^17*b^34*c^67*d^8 - 4465630463459278708539392*a^18*b^33*c^66*d^9 + 18321125205332103390035968*a^19*b^32*c^65*d
^10 - 63021545228377166868119552*a^20*b^31*c^64*d^11 + 187018029382071665408606208*a^21*b^30*c^63*d^12 - 49071
3180393588600090918912*a^22*b^29*c^62*d^13 + 1161438545048511890042388480*a^23*b^28*c^61*d^14 - 25129740563090
66269898833920*a^24*b^27*c^60*d^15 + 4997541469898172697285754880*a^25*b^26*c^59*d^16 - 9119889428539397211967
979520*a^26*b^25*c^58*d^17 + 15181544306461039744285409280*a^27*b^24*c^57*d^18 - 22888317982577902576352624640
*a^28*b^23*c^56*d^19 + 31049708276802113763866050560*a^29*b^22*c^55*d^20 - 37706614244767692268335267840*a^30*
b^21*c^54*d^21 + 40833216619792283792163471360*a^31*b^20*c^53*d^22 - 39312168062751093709382615040*a^32*b^19*c
^52*d^23 + 33557805042801128843488788480*a^33*b^18*c^51*d^24 - 25329188887155786370693201920*a^34*b^17*c^50*d^
25 + 16851463310911481777624186880*a^35*b^16*c^49*d^26 - 9843609097631363291959787520*a^36*b^15*c^48*d^27 + 50
23816147465636127472353280*a^37*b^14*c^47*d^28 - 2226054577272365612261179392*a^38*b^13*c^46*d^29 + 8494197527
18963326077370368*a^39*b^12*c^45*d^30 - 276172923601113041340465152*a^40*b^11*c^44*d^31 + 75441341408208223215
812608*a^41*b^10*c^43*d^32 - 16988052798101408932954112*a^42*b^9*c^42*d^33 + 3070410975444256772063232*a^43*b^
8*c^41*d^34 - 428198505575496787427328*a^44*b^7*c^40*d^35 + 43254156088335077998592*a^45*b^6*c^39*d^36 - 28165
87235754527162368*a^46*b^5*c^38*d^37 + 88774955854727217152*a^47*b^4*c^37*d^38)) - 11889503016258109440*a^9*b^
38*c^56*d^7 + 217253646024352727040*a^10*b^37*c^55*d^8 - 1879766455667426066432*a^11*b^36*c^54*d^9 + 102371503
27374383939584*a^12*b^35*c^53*d^10 - 37711511320670913953792*a^13*b^34*c^52*d^11 + 77353208427556875796480*a^1
4*b^33*c^51*d^12 + 127627238172719495249920*a^15*b^32*c^50*d^13 - 2130084466030987427446784*a^16*b^31*c^49*d^1
4 + 11885048527140028256616448*a^17*b^30*c^48*d^15 - 45690531361686842972831744*a^18*b^29*c^47*d^16 + 13585192
9384595950057553920*a^19*b^28*c^46*d^17 - 326376775711477371051704320*a^20*b^27*c^45*d^18 + 648353352496064059
760705536*a^21*b^26*c^44*d^19 - 1080394184249474617790431232*a^22*b^25*c^43*d^20 + 152472533992863002915346841
6*a^23*b^24*c^42*d^21 - 1834102420924176937716285440*a^24*b^23*c^41*d^22 + 1888062742223171008426147840*a^25*b
^22*c^40*d^23 - 1666588213584359199850102784*a^26*b^21*c^39*d^24 + 1261562453800014779376467968*a^27*b^20*c^38
*d^25 - 817528072151542384572760064*a^28*b^19*c^37*d^26 + 451847698934426396681830400*a^29*b^18*c^36*d^27 - 21
1721890947778234390937600*a^30*b^17*c^35*d^28 + 83366248780838000977248256*a^31*b^16*c^34*d^29 - 2724126662404
4306322685952*a^32*b^15*c^33*d^30 + 7257515800860571589410816*a^33*b^14*c^32*d^31 - 1536699518639901947985920*
a^34*b^13*c^31*d^32 + 248859486128715197317120*a^35*b^12*c^30*d^33 - 28961642042172523937792*a^36*b^11*c^29*d^
34 + 2157438443758953693184*a^37*b^10*c^28*d^35 - 77302354662372933632*a^38*b^9*c^27*d^36) + x^(1/2)*(30652624
963790438400*a^9*b^37*c^51*d^9 - 507161613037259980800*a^10*b^36*c^50*d^10 + 4774956969550613053440*a^11*b^35*
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592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 858730950
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1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 2013265
92*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4
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d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5
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8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*
c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c
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1*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b
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720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*
b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 155
02147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^
18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(3/4)*(x^(1/2)*(18446744073709551616*a^11*b
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591644834201600*a^14*b^36*c^65*d^7 + 275778823901957796659200*a^15*b^35*c^64*d^8 - 1212936383169193658286080*a
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d^11 + 27104869321333056471040000*a^19*b^31*c^60*d^12 - 46637619173392487079215104*a^20*b^30*c^59*d^13 + 43611
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0717593600*a^23*b^27*c^56*d^16 + 1723753001020797184743833600*a^24*b^26*c^55*d^17 - 42694371673658728148421836
80*a^25*b^25*c^54*d^18 + 8727322757849829186700574720*a^26*b^24*c^53*d^19 - 15215326043975142249374679040*a^27
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c^50*d^22 + 34870163031766389952882933760*a^30*b^20*c^49*d^23 - 35316718238336158489724846080*a^31*b^19*c^48*d
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16850754961433442876234137600*a^34*b^16*c^45*d^27 - 10105200492115418262179676160*a^35*b^15*c^44*d^28 + 527801
1312905736232783314944*a^36*b^14*c^43*d^29 - 2387248399405916166169821184*a^37*b^13*c^42*d^30 + 92782863231267
4738870681600*a^38*b^12*c^41*d^31 - 306693733103726739901644800*a^39*b^11*c^40*d^32 + 850380759594460460666060
80*a^40*b^10*c^39*d^33 - 19409595119210898894356480*a^41*b^9*c^38*d^34 + 3551400405635812871372800*a^42*b^8*c^
37*d^35 - 500844593983932480880640*a^43*b^7*c^36*d^36 + 51111802530990496153600*a^44*b^6*c^35*d^37 - 335957723
5627333124096*a^45*b^5*c^34*d^38 + 106807368762718683136*a^46*b^4*c^33*d^39) + (-(35153041*a^8*d^15 + 74120062
5*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798
150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16
777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 36909
87520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21
*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a
^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 106991115627515
3993728*a^14*b^37*c^70*d^5 + 14978756187852155912192*a^15*b^36*c^69*d^6 - 134999037738929532960768*a^16*b^35*c
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5205332103390035968*a^19*b^32*c^65*d^10 - 63021545228377166868119552*a^20*b^31*c^64*d^11 + 1870180293820716654
08606208*a^21*b^30*c^63*d^12 - 490713180393588600090918912*a^22*b^29*c^62*d^13 + 1161438545048511890042388480*
a^23*b^28*c^61*d^14 - 2512974056309066269898833920*a^24*b^27*c^60*d^15 + 4997541469898172697285754880*a^25*b^2
6*c^59*d^16 - 9119889428539397211967979520*a^26*b^25*c^58*d^17 + 15181544306461039744285409280*a^27*b^24*c^57*
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168062751093709382615040*a^32*b^19*c^52*d^23 + 33557805042801128843488788480*a^33*b^18*c^51*d^24 - 25329188887
155786370693201920*a^34*b^17*c^50*d^25 + 16851463310911481777624186880*a^35*b^16*c^49*d^26 - 98436090976313632
91959787520*a^36*b^15*c^48*d^27 + 5023816147465636127472353280*a^37*b^14*c^47*d^28 - 2226054577272365612261179
392*a^38*b^13*c^46*d^29 + 849419752718963326077370368*a^39*b^12*c^45*d^30 - 276172923601113041340465152*a^40*b
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d^38)) - 11889503016258109440*a^9*b^38*c^56*d^7 + 217253646024352727040*a^10*b^37*c^55*d^8 - 18797664556674260
66432*a^11*b^36*c^54*d^9 + 10237150327374383939584*a^12*b^35*c^53*d^10 - 37711511320670913953792*a^13*b^34*c^5
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66030987427446784*a^16*b^31*c^49*d^14 + 11885048527140028256616448*a^17*b^30*c^48*d^15 - 456905313616868429728
31744*a^18*b^29*c^47*d^16 + 135851929384595950057553920*a^19*b^28*c^46*d^17 - 326376775711477371051704320*a^20
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3*d^20 + 1524725339928630029153468416*a^23*b^24*c^42*d^21 - 1834102420924176937716285440*a^24*b^23*c^41*d^22 +
1888062742223171008426147840*a^25*b^22*c^40*d^23 - 1666588213584359199850102784*a^26*b^21*c^39*d^24 + 1261562
453800014779376467968*a^27*b^20*c^38*d^25 - 817528072151542384572760064*a^28*b^19*c^37*d^26 + 4518476989344263
96681830400*a^29*b^18*c^36*d^27 - 211721890947778234390937600*a^30*b^17*c^35*d^28 + 83366248780838000977248256
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32*d^31 - 1536699518639901947985920*a^34*b^13*c^31*d^32 + 248859486128715197317120*a^35*b^12*c^30*d^33 - 28961
642042172523937792*a^36*b^11*c^29*d^34 + 2157438443758953693184*a^37*b^10*c^28*d^35 - 77302354662372933632*a^3
8*b^9*c^27*d^36) + x^(1/2)*(30652624963790438400*a^9*b^37*c^51*d^9 - 507161613037259980800*a^10*b^36*c^50*d^10
+ 4774956969550613053440*a^11*b^35*c^49*d^11 - 34948190471081762488320*a^12*b^34*c^48*d^12 + 2084099627864836
28670976*a^13*b^33*c^47*d^13 - 990271368055602664177664*a^14*b^32*c^46*d^14 + 3711631588079120800546816*a^15*b
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7 - 51805174836472540920020992*a^18*b^28*c^42*d^18 + 83617663209148864427720704*a^19*b^27*c^41*d^19 - 11235043
0315654120415952896*a^20*b^26*c^40*d^20 + 126417217514830317658046464*a^21*b^25*c^39*d^21 - 119537906081128203
174281216*a^22*b^24*c^38*d^22 + 95089864774620999552335872*a^23*b^23*c^37*d^23 - 63545506634457987380412416*a^
24*b^22*c^36*d^24 + 35529578846146774008070144*a^25*b^21*c^35*d^25 - 16501565732136655819636736*a^26*b^20*c^34
*d^26 + 6295808856090071441342464*a^27*b^19*c^33*d^27 - 1940847984249953081884672*a^28*b^18*c^32*d^28 + 471738
031694568778366976*a^29*b^17*c^31*d^29 - 87073083559063809163264*a^30*b^16*c^30*d^30 + 11476570419434950230016
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999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d
^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 110
7296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^
22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 369098752
0*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 7
41200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 +
9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^
14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2
- 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b
^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 11072
96256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 377
3385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^1
1 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 +
16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*
d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*
a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 -
201326592*a*b^11*c^26*d))^(3/4)*(x^(1/2)*(18446744073709551616*a^11*b^39*c^68*d^4 - 479615345916448342016*a^1
2*b^38*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*d^6 - 47961534591644834201600*a^14*b^36*c^65*d^7 + 275
778823901957796659200*a^15*b^35*c^64*d^8 - 1212936383169193658286080*a^16*b^34*c^63*d^9 + 42329939982885061446
86080*a^17*b^33*c^62*d^10 - 11941164077799654041845760*a^18*b^32*c^61*d^11 + 27104869321333056471040000*a^19*b
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4 + 72781112360087761599856640*a^22*b^28*c^57*d^15 - 523234066593179210717593600*a^23*b^27*c^56*d^16 + 1723753
001020797184743833600*a^24*b^26*c^55*d^17 - 4269437167365872814842183680*a^25*b^25*c^54*d^18 + 872732275784982
9186700574720*a^26*b^24*c^53*d^19 - 15215326043975142249374679040*a^27*b^23*c^52*d^20 + 2296265846324651962558
0544000*a^28*b^22*c^51*d^21 - 30231538828274701475145318400*a^29*b^21*c^50*d^22 + 3487016303176638995288293376
0*a^30*b^20*c^49*d^23 - 35316718238336158489724846080*a^31*b^19*c^48*d^24 + 31433146498544749041648926720*a^32
*b^18*c^47*d^25 - 24575140799491012895231180800*a^33*b^17*c^46*d^26 + 16850754961433442876234137600*a^34*b^16*
c^45*d^27 - 10105200492115418262179676160*a^35*b^15*c^44*d^28 + 5278011312905736232783314944*a^36*b^14*c^43*d^
29 - 2387248399405916166169821184*a^37*b^13*c^42*d^30 + 927828632312674738870681600*a^38*b^12*c^41*d^31 - 3066
93733103726739901644800*a^39*b^11*c^40*d^32 + 85038075959446046066606080*a^40*b^10*c^39*d^33 - 194095951192108
98894356480*a^41*b^9*c^38*d^34 + 3551400405635812871372800*a^42*b^8*c^37*d^35 - 500844593983932480880640*a^43*
b^7*c^36*d^36 + 51111802530990496153600*a^44*b^6*c^35*d^37 - 3359577235627333124096*a^45*b^5*c^34*d^38 + 10680
7368762718683136*a^46*b^4*c^33*d^39) - (-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8
+ 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^
3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^
12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*
b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 83
04721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^2
6*d))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 1069911156275153993728*a^14*b^37*c^70*d^5 + 14978756187
852155912192*a^15*b^36*c^69*d^6 - 134999037738929532960768*a^16*b^35*c^68*d^7 + 882016079904862321508352*a^17*
b^34*c^67*d^8 - 4465630463459278708539392*a^18*b^33*c^66*d^9 + 18321125205332103390035968*a^19*b^32*c^65*d^10
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393588600090918912*a^22*b^29*c^62*d^13 + 1161438545048511890042388480*a^23*b^28*c^61*d^14 - 251297405630906626
9898833920*a^24*b^27*c^60*d^15 + 4997541469898172697285754880*a^25*b^26*c^59*d^16 - 91198894285393972119679795
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5754527162368*a^46*b^5*c^38*d^37 + 88774955854727217152*a^47*b^4*c^37*d^38)) + 11889503016258109440*a^9*b^38*c
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13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2
*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*2i - 2*atan(((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 37733
85000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11
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777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^
3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^
7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 2
01326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587
309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d
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1326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^2
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1/4)*((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 1
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7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 36909
87520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(3/4)*(x^(1/2)*(18446744073
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5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a
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584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^
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302354662372933632*a^38*b^9*c^27*d^36)*1i - x^(1/2)*(30652624963790438400*a^9*b^37*c^51*d^9 - 5071616130372599
80800*a^10*b^36*c^50*d^10 + 4774956969550613053440*a^11*b^35*c^49*d^11 - 34948190471081762488320*a^12*b^34*c^4
8*d^12 + 208409962786483628670976*a^13*b^33*c^47*d^13 - 990271368055602664177664*a^14*b^32*c^46*d^14 + 3711631
588079120800546816*a^15*b^31*c^45*d^15 - 11050795179720929846493184*a^16*b^30*c^44*d^16 + 26487755718620581216
649216*a^17*b^29*c^43*d^17 - 51805174836472540920020992*a^18*b^28*c^42*d^18 + 83617663209148864427720704*a^19*
b^27*c^41*d^19 - 112350430315654120415952896*a^20*b^26*c^40*d^20 + 126417217514830317658046464*a^21*b^25*c^39*
d^21 - 119537906081128203174281216*a^22*b^24*c^38*d^22 + 95089864774620999552335872*a^23*b^23*c^37*d^23 - 6354
5506634457987380412416*a^24*b^22*c^36*d^24 + 35529578846146774008070144*a^25*b^21*c^35*d^25 - 1650156573213665
5819636736*a^26*b^20*c^34*d^26 + 6295808856090071441342464*a^27*b^19*c^33*d^27 - 1940847984249953081884672*a^2
8*b^18*c^32*d^28 + 471738031694568778366976*a^29*b^17*c^31*d^29 - 87073083559063809163264*a^30*b^16*c^30*d^30
+ 11476570419434950230016*a^31*b^15*c^29*d^31 - 962765689885917446144*a^32*b^14*c^28*d^32 + 386511773311864668
16*a^33*b^13*c^27*d^33)) + (-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 858730950
0*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 +
1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 2013265
92*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4
- 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8
*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*
((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 113949
99000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^
13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107
296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^2
2*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520
*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 74
1200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9
636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^1
4)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 -
3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^
6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 110729
6256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(3/4)*(x^(1/2)*(18446744073709551616*a^11*b^39*c^68*d^4 -
479615345916448342016*a^12*b^38*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*d^6 - 47961534591644834201600
*a^14*b^36*c^65*d^7 + 275778823901957796659200*a^15*b^35*c^64*d^8 - 1212936383169193658286080*a^16*b^34*c^63*d
^9 + 4232993998288506144686080*a^17*b^33*c^62*d^10 - 11941164077799654041845760*a^18*b^32*c^61*d^11 + 27104869
321333056471040000*a^19*b^31*c^60*d^12 - 46637619173392487079215104*a^20*b^30*c^59*d^13 + 43611606538557895133
364224*a^21*b^29*c^58*d^14 + 72781112360087761599856640*a^22*b^28*c^57*d^15 - 523234066593179210717593600*a^23
*b^27*c^56*d^16 + 1723753001020797184743833600*a^24*b^26*c^55*d^17 - 4269437167365872814842183680*a^25*b^25*c^
54*d^18 + 8727322757849829186700574720*a^26*b^24*c^53*d^19 - 15215326043975142249374679040*a^27*b^23*c^52*d^20
+ 22962658463246519625580544000*a^28*b^22*c^51*d^21 - 30231538828274701475145318400*a^29*b^21*c^50*d^22 + 348
70163031766389952882933760*a^30*b^20*c^49*d^23 - 35316718238336158489724846080*a^31*b^19*c^48*d^24 + 314331464
98544749041648926720*a^32*b^18*c^47*d^25 - 24575140799491012895231180800*a^33*b^17*c^46*d^26 + 168507549614334
42876234137600*a^34*b^16*c^45*d^27 - 10105200492115418262179676160*a^35*b^15*c^44*d^28 + 527801131290573623278
3314944*a^36*b^14*c^43*d^29 - 2387248399405916166169821184*a^37*b^13*c^42*d^30 + 927828632312674738870681600*a
^38*b^12*c^41*d^31 - 306693733103726739901644800*a^39*b^11*c^40*d^32 + 85038075959446046066606080*a^40*b^10*c^
39*d^33 - 19409595119210898894356480*a^41*b^9*c^38*d^34 + 3551400405635812871372800*a^42*b^8*c^37*d^35 - 50084
4593983932480880640*a^43*b^7*c^36*d^36 + 51111802530990496153600*a^44*b^6*c^35*d^37 - 3359577235627333124096*a
^45*b^5*c^34*d^38 + 106807368762718683136*a^46*b^4*c^33*d^39) + (-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 -
3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4
*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^2
7 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c
^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555
072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^
10 - 201326592*a*b^11*c^26*d))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 1069911156275153993728*a^14*b^
37*c^70*d^5 + 14978756187852155912192*a^15*b^36*c^69*d^6 - 134999037738929532960768*a^16*b^35*c^68*d^7 + 88201
6079904862321508352*a^17*b^34*c^67*d^8 - 4465630463459278708539392*a^18*b^33*c^66*d^9 + 1832112520533210339003
5968*a^19*b^32*c^65*d^10 - 63021545228377166868119552*a^20*b^31*c^64*d^11 + 187018029382071665408606208*a^21*b
^30*c^63*d^12 - 490713180393588600090918912*a^22*b^29*c^62*d^13 + 1161438545048511890042388480*a^23*b^28*c^61*
d^14 - 2512974056309066269898833920*a^24*b^27*c^60*d^15 + 4997541469898172697285754880*a^25*b^26*c^59*d^16 - 9
119889428539397211967979520*a^26*b^25*c^58*d^17 + 15181544306461039744285409280*a^27*b^24*c^57*d^18 - 22888317
982577902576352624640*a^28*b^23*c^56*d^19 + 31049708276802113763866050560*a^29*b^22*c^55*d^20 - 37706614244767
692268335267840*a^30*b^21*c^54*d^21 + 40833216619792283792163471360*a^31*b^20*c^53*d^22 - 39312168062751093709
382615040*a^32*b^19*c^52*d^23 + 33557805042801128843488788480*a^33*b^18*c^51*d^24 - 25329188887155786370693201
920*a^34*b^17*c^50*d^25 + 16851463310911481777624186880*a^35*b^16*c^49*d^26 - 9843609097631363291959787520*a^3
6*b^15*c^48*d^27 + 5023816147465636127472353280*a^37*b^14*c^47*d^28 - 2226054577272365612261179392*a^38*b^13*c
^46*d^29 + 849419752718963326077370368*a^39*b^12*c^45*d^30 - 276172923601113041340465152*a^40*b^11*c^44*d^31 +
75441341408208223215812608*a^41*b^10*c^43*d^32 - 16988052798101408932954112*a^42*b^9*c^42*d^33 + 307041097544
4256772063232*a^43*b^8*c^41*d^34 - 428198505575496787427328*a^44*b^7*c^40*d^35 + 43254156088335077998592*a^45*
b^6*c^39*d^36 - 2816587235754527162368*a^46*b^5*c^38*d^37 + 88774955854727217152*a^47*b^4*c^37*d^38)*1i)*1i +
11889503016258109440*a^9*b^38*c^56*d^7 - 217253646024352727040*a^10*b^37*c^55*d^8 + 1879766455667426066432*a^1
1*b^36*c^54*d^9 - 10237150327374383939584*a^12*b^35*c^53*d^10 + 37711511320670913953792*a^13*b^34*c^52*d^11 -
77353208427556875796480*a^14*b^33*c^51*d^12 - 127627238172719495249920*a^15*b^32*c^50*d^13 + 21300844660309874
27446784*a^16*b^31*c^49*d^14 - 11885048527140028256616448*a^17*b^30*c^48*d^15 + 45690531361686842972831744*a^1
8*b^29*c^47*d^16 - 135851929384595950057553920*a^19*b^28*c^46*d^17 + 326376775711477371051704320*a^20*b^27*c^4
5*d^18 - 648353352496064059760705536*a^21*b^26*c^44*d^19 + 1080394184249474617790431232*a^22*b^25*c^43*d^20 -
1524725339928630029153468416*a^23*b^24*c^42*d^21 + 1834102420924176937716285440*a^24*b^23*c^41*d^22 - 18880627
42223171008426147840*a^25*b^22*c^40*d^23 + 1666588213584359199850102784*a^26*b^21*c^39*d^24 - 1261562453800014
779376467968*a^27*b^20*c^38*d^25 + 817528072151542384572760064*a^28*b^19*c^37*d^26 - 4518476989344263966818304
00*a^29*b^18*c^36*d^27 + 211721890947778234390937600*a^30*b^17*c^35*d^28 - 83366248780838000977248256*a^31*b^1
6*c^34*d^29 + 27241266624044306322685952*a^32*b^15*c^33*d^30 - 7257515800860571589410816*a^33*b^14*c^32*d^31 +
1536699518639901947985920*a^34*b^13*c^31*d^32 - 248859486128715197317120*a^35*b^12*c^30*d^33 + 28961642042172
523937792*a^36*b^11*c^29*d^34 - 2157438443758953693184*a^37*b^10*c^28*d^35 + 77302354662372933632*a^38*b^9*c^2
7*d^36)*1i - x^(1/2)*(30652624963790438400*a^9*b^37*c^51*d^9 - 507161613037259980800*a^10*b^36*c^50*d^10 + 477
4956969550613053440*a^11*b^35*c^49*d^11 - 34948190471081762488320*a^12*b^34*c^48*d^12 + 2084099627864836286709
76*a^13*b^33*c^47*d^13 - 990271368055602664177664*a^14*b^32*c^46*d^14 + 3711631588079120800546816*a^15*b^31*c^
45*d^15 - 11050795179720929846493184*a^16*b^30*c^44*d^16 + 26487755718620581216649216*a^17*b^29*c^43*d^17 - 51
805174836472540920020992*a^18*b^28*c^42*d^18 + 83617663209148864427720704*a^19*b^27*c^41*d^19 - 11235043031565
4120415952896*a^20*b^26*c^40*d^20 + 126417217514830317658046464*a^21*b^25*c^39*d^21 - 119537906081128203174281
216*a^22*b^24*c^38*d^22 + 95089864774620999552335872*a^23*b^23*c^37*d^23 - 63545506634457987380412416*a^24*b^2
2*c^36*d^24 + 35529578846146774008070144*a^25*b^21*c^35*d^25 - 16501565732136655819636736*a^26*b^20*c^34*d^26
+ 6295808856090071441342464*a^27*b^19*c^33*d^27 - 1940847984249953081884672*a^28*b^18*c^32*d^28 + 471738031694
568778366976*a^29*b^17*c^31*d^29 - 87073083559063809163264*a^30*b^16*c^30*d^30 + 11476570419434950230016*a^31*
b^15*c^29*d^31 - 962765689885917446144*a^32*b^14*c^28*d^32 + 38651177331186466816*a^33*b^13*c^27*d^33)))/((-(3
5153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000
*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 -
383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 110729625
6*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5
+ 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*
b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 7412006
25*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 963679
8150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(1
6777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690
987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^2
1*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*
a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 377338500
0*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 53
17666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 167772
16*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 +
8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^
5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 20132
6592*a*b^11*c^26*d))^(3/4)*(x^(1/2)*(18446744073709551616*a^11*b^39*c^68*d^4 - 479615345916448342016*a^12*b^38
*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*d^6 - 47961534591644834201600*a^14*b^36*c^65*d^7 + 275778823
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7309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*
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20*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^
(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 1069911156275153993728*a^14*b^37*c^70*d^5 + 14978756187852155
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33557805042801128843488788480*a^33*b^18*c^51*d^24 - 25329188887155786370693201920*a^34*b^17*c^50*d^25 + 16851
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5636127472353280*a^37*b^14*c^47*d^28 - 2226054577272365612261179392*a^38*b^13*c^46*d^29 + 84941975271896332607
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7162368*a^46*b^5*c^38*d^37 + 88774955854727217152*a^47*b^4*c^37*d^38)*1i)*1i - 11889503016258109440*a^9*b^38*c
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4383939584*a^12*b^35*c^53*d^10 - 37711511320670913953792*a^13*b^34*c^52*d^11 + 77353208427556875796480*a^14*b^
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11885048527140028256616448*a^17*b^30*c^48*d^15 - 45690531361686842972831744*a^18*b^29*c^47*d^16 + 135851929384
595950057553920*a^19*b^28*c^46*d^17 - 326376775711477371051704320*a^20*b^27*c^45*d^18 + 6483533524960640597607
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890947778234390937600*a^30*b^17*c^35*d^28 + 83366248780838000977248256*a^31*b^16*c^34*d^29 - 27241266624044306
322685952*a^32*b^15*c^33*d^30 + 7257515800860571589410816*a^33*b^14*c^32*d^31 - 1536699518639901947985920*a^34
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2157438443758953693184*a^37*b^10*c^28*d^35 - 77302354662372933632*a^38*b^9*c^27*d^36)*1i - x^(1/2)*(306526249
63790438400*a^9*b^37*c^51*d^9 - 507161613037259980800*a^10*b^36*c^50*d^10 + 4774956969550613053440*a^11*b^35*c
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368055602664177664*a^14*b^32*c^46*d^14 + 3711631588079120800546816*a^15*b^31*c^45*d^15 - 110507951797209298464
93184*a^16*b^30*c^44*d^16 + 26487755718620581216649216*a^17*b^29*c^43*d^17 - 51805174836472540920020992*a^18*b
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20 + 126417217514830317658046464*a^21*b^25*c^39*d^21 - 119537906081128203174281216*a^22*b^24*c^38*d^22 + 95089
864774620999552335872*a^23*b^23*c^37*d^23 - 63545506634457987380412416*a^24*b^22*c^36*d^24 + 35529578846146774
008070144*a^25*b^21*c^35*d^25 - 16501565732136655819636736*a^26*b^20*c^34*d^26 + 6295808856090071441342464*a^2
7*b^19*c^33*d^27 - 1940847984249953081884672*a^28*b^18*c^32*d^28 + 471738031694568778366976*a^29*b^17*c^31*d^2
9 - 87073083559063809163264*a^30*b^16*c^30*d^30 + 11476570419434950230016*a^31*b^15*c^29*d^31 - 96276568988591
7446144*a^32*b^14*c^28*d^32 + 38651177331186466816*a^33*b^13*c^27*d^33))*1i - (-(35153041*a^8*d^15 + 741200625
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50*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(167
77216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 369098
7520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*
d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^
10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*
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666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216
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04721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*
c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 2013265
92*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500
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1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 20132659
2*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4
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b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(3/4)*(
x^(1/2)*(18446744073709551616*a^11*b^39*c^68*d^4 - 479615345916448342016*a^12*b^38*c^67*d^5 + 5995191823955604
275200*a^13*b^37*c^66*d^6 - 47961534591644834201600*a^14*b^36*c^65*d^7 + 275778823901957796659200*a^15*b^35*c^
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64077799654041845760*a^18*b^32*c^61*d^11 + 27104869321333056471040000*a^19*b^31*c^60*d^12 - 466376191733924870
79215104*a^20*b^30*c^59*d^13 + 43611606538557895133364224*a^21*b^29*c^58*d^14 + 72781112360087761599856640*a^2
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9676160*a^35*b^15*c^44*d^28 + 5278011312905736232783314944*a^36*b^14*c^43*d^29 - 2387248399405916166169821184*
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c^40*d^32 + 85038075959446046066606080*a^40*b^10*c^39*d^33 - 19409595119210898894356480*a^41*b^9*c^38*d^34 + 3
551400405635812871372800*a^42*b^8*c^37*d^35 - 500844593983932480880640*a^43*b^7*c^36*d^36 + 511118025309904961
53600*a^44*b^6*c^35*d^37 - 3359577235627333124096*a^45*b^5*c^34*d^38 + 106807368762718683136*a^46*b^4*c^33*d^3
9) + (-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11
394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^
2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 +
1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7
*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 369098
7520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*(36893488147419103232*
a^13*b^38*c^71*d^4 - 1069911156275153993728*a^14*b^37*c^70*d^5 + 14978756187852155912192*a^15*b^36*c^69*d^6 -
134999037738929532960768*a^16*b^35*c^68*d^7 + 882016079904862321508352*a^17*b^34*c^67*d^8 - 446563046345927870
8539392*a^18*b^33*c^66*d^9 + 18321125205332103390035968*a^19*b^32*c^65*d^10 - 63021545228377166868119552*a^20*
b^31*c^64*d^11 + 187018029382071665408606208*a^21*b^30*c^63*d^12 - 490713180393588600090918912*a^22*b^29*c^62*
d^13 + 1161438545048511890042388480*a^23*b^28*c^61*d^14 - 2512974056309066269898833920*a^24*b^27*c^60*d^15 + 4
997541469898172697285754880*a^25*b^26*c^59*d^16 - 9119889428539397211967979520*a^26*b^25*c^58*d^17 + 151815443
06461039744285409280*a^27*b^24*c^57*d^18 - 22888317982577902576352624640*a^28*b^23*c^56*d^19 + 310497082768021
13763866050560*a^29*b^22*c^55*d^20 - 37706614244767692268335267840*a^30*b^21*c^54*d^21 + 408332166197922837921
63471360*a^31*b^20*c^53*d^22 - 39312168062751093709382615040*a^32*b^19*c^52*d^23 + 335578050428011288434887884
80*a^33*b^18*c^51*d^24 - 25329188887155786370693201920*a^34*b^17*c^50*d^25 + 16851463310911481777624186880*a^3
5*b^16*c^49*d^26 - 9843609097631363291959787520*a^36*b^15*c^48*d^27 + 5023816147465636127472353280*a^37*b^14*c
^47*d^28 - 2226054577272365612261179392*a^38*b^13*c^46*d^29 + 849419752718963326077370368*a^39*b^12*c^45*d^30
- 276172923601113041340465152*a^40*b^11*c^44*d^31 + 75441341408208223215812608*a^41*b^10*c^43*d^32 - 169880527
98101408932954112*a^42*b^9*c^42*d^33 + 3070410975444256772063232*a^43*b^8*c^41*d^34 - 428198505575496787427328
*a^44*b^7*c^40*d^35 + 43254156088335077998592*a^45*b^6*c^39*d^36 - 2816587235754527162368*a^46*b^5*c^38*d^37 +
88774955854727217152*a^47*b^4*c^37*d^38)*1i)*1i + 11889503016258109440*a^9*b^38*c^56*d^7 - 217253646024352727
040*a^10*b^37*c^55*d^8 + 1879766455667426066432*a^11*b^36*c^54*d^9 - 10237150327374383939584*a^12*b^35*c^53*d^
10 + 37711511320670913953792*a^13*b^34*c^52*d^11 - 77353208427556875796480*a^14*b^33*c^51*d^12 - 1276272381727
19495249920*a^15*b^32*c^50*d^13 + 2130084466030987427446784*a^16*b^31*c^49*d^14 - 11885048527140028256616448*a
^17*b^30*c^48*d^15 + 45690531361686842972831744*a^18*b^29*c^47*d^16 - 135851929384595950057553920*a^19*b^28*c^
46*d^17 + 326376775711477371051704320*a^20*b^27*c^45*d^18 - 648353352496064059760705536*a^21*b^26*c^44*d^19 +
1080394184249474617790431232*a^22*b^25*c^43*d^20 - 1524725339928630029153468416*a^23*b^24*c^42*d^21 + 18341024
20924176937716285440*a^24*b^23*c^41*d^22 - 1888062742223171008426147840*a^25*b^22*c^40*d^23 + 1666588213584359
199850102784*a^26*b^21*c^39*d^24 - 1261562453800014779376467968*a^27*b^20*c^38*d^25 + 817528072151542384572760
064*a^28*b^19*c^37*d^26 - 451847698934426396681830400*a^29*b^18*c^36*d^27 + 211721890947778234390937600*a^30*b
^17*c^35*d^28 - 83366248780838000977248256*a^31*b^16*c^34*d^29 + 27241266624044306322685952*a^32*b^15*c^33*d^3
0 - 7257515800860571589410816*a^33*b^14*c^32*d^31 + 1536699518639901947985920*a^34*b^13*c^31*d^32 - 2488594861
28715197317120*a^35*b^12*c^30*d^33 + 28961642042172523937792*a^36*b^11*c^29*d^34 - 2157438443758953693184*a^37
*b^10*c^28*d^35 + 77302354662372933632*a^38*b^9*c^27*d^36)*1i - x^(1/2)*(30652624963790438400*a^9*b^37*c^51*d^
9 - 507161613037259980800*a^10*b^36*c^50*d^10 + 4774956969550613053440*a^11*b^35*c^49*d^11 - 34948190471081762
488320*a^12*b^34*c^48*d^12 + 208409962786483628670976*a^13*b^33*c^47*d^13 - 990271368055602664177664*a^14*b^32
*c^46*d^14 + 3711631588079120800546816*a^15*b^31*c^45*d^15 - 11050795179720929846493184*a^16*b^30*c^44*d^16 +
26487755718620581216649216*a^17*b^29*c^43*d^17 - 51805174836472540920020992*a^18*b^28*c^42*d^18 + 836176632091
48864427720704*a^19*b^27*c^41*d^19 - 112350430315654120415952896*a^20*b^26*c^40*d^20 + 12641721751483031765804
6464*a^21*b^25*c^39*d^21 - 119537906081128203174281216*a^22*b^24*c^38*d^22 + 95089864774620999552335872*a^23*b
^23*c^37*d^23 - 63545506634457987380412416*a^24*b^22*c^36*d^24 + 35529578846146774008070144*a^25*b^21*c^35*d^2
5 - 16501565732136655819636736*a^26*b^20*c^34*d^26 + 6295808856090071441342464*a^27*b^19*c^33*d^27 - 194084798
4249953081884672*a^28*b^18*c^32*d^28 + 471738031694568778366976*a^29*b^17*c^31*d^29 - 87073083559063809163264*
a^30*b^16*c^30*d^30 + 11476570419434950230016*a^31*b^15*c^29*d^31 - 962765689885917446144*a^32*b^14*c^28*d^32
+ 38651177331186466816*a^33*b^13*c^27*d^33))*1i))*(-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*
b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 531766
6200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a
^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304
721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^
20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592
*a*b^11*c^26*d))^(1/4) + atan(((-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c
^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12
672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d
^11))^(1/4)*(x^(1/2)*(30652624963790438400*a^9*b^37*c^51*d^9 - 507161613037259980800*a^10*b^36*c^50*d^10 + 477
4956969550613053440*a^11*b^35*c^49*d^11 - 34948190471081762488320*a^12*b^34*c^48*d^12 + 2084099627864836286709
76*a^13*b^33*c^47*d^13 - 990271368055602664177664*a^14*b^32*c^46*d^14 + 3711631588079120800546816*a^15*b^31*c^
45*d^15 - 11050795179720929846493184*a^16*b^30*c^44*d^16 + 26487755718620581216649216*a^17*b^29*c^43*d^17 - 51
805174836472540920020992*a^18*b^28*c^42*d^18 + 83617663209148864427720704*a^19*b^27*c^41*d^19 - 11235043031565
4120415952896*a^20*b^26*c^40*d^20 + 126417217514830317658046464*a^21*b^25*c^39*d^21 - 119537906081128203174281
216*a^22*b^24*c^38*d^22 + 95089864774620999552335872*a^23*b^23*c^37*d^23 - 63545506634457987380412416*a^24*b^2
2*c^36*d^24 + 35529578846146774008070144*a^25*b^21*c^35*d^25 - 16501565732136655819636736*a^26*b^20*c^34*d^26
+ 6295808856090071441342464*a^27*b^19*c^33*d^27 - 1940847984249953081884672*a^28*b^18*c^32*d^28 + 471738031694
568778366976*a^29*b^17*c^31*d^29 - 87073083559063809163264*a^30*b^16*c^30*d^30 + 11476570419434950230016*a^31*
b^15*c^29*d^31 - 962765689885917446144*a^32*b^14*c^28*d^32 + 38651177331186466816*a^33*b^13*c^27*d^33) + (-b^1
5/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 79
20*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4
*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*((x^(1/2)*(1844674407370
9551616*a^11*b^39*c^68*d^4 - 479615345916448342016*a^12*b^38*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*
d^6 - 47961534591644834201600*a^14*b^36*c^65*d^7 + 275778823901957796659200*a^15*b^35*c^64*d^8 - 1212936383169
193658286080*a^16*b^34*c^63*d^9 + 4232993998288506144686080*a^17*b^33*c^62*d^10 - 11941164077799654041845760*a
^18*b^32*c^61*d^11 + 27104869321333056471040000*a^19*b^31*c^60*d^12 - 46637619173392487079215104*a^20*b^30*c^5
9*d^13 + 43611606538557895133364224*a^21*b^29*c^58*d^14 + 72781112360087761599856640*a^22*b^28*c^57*d^15 - 523
234066593179210717593600*a^23*b^27*c^56*d^16 + 1723753001020797184743833600*a^24*b^26*c^55*d^17 - 426943716736
5872814842183680*a^25*b^25*c^54*d^18 + 8727322757849829186700574720*a^26*b^24*c^53*d^19 - 15215326043975142249
374679040*a^27*b^23*c^52*d^20 + 22962658463246519625580544000*a^28*b^22*c^51*d^21 - 30231538828274701475145318
400*a^29*b^21*c^50*d^22 + 34870163031766389952882933760*a^30*b^20*c^49*d^23 - 35316718238336158489724846080*a^
31*b^19*c^48*d^24 + 31433146498544749041648926720*a^32*b^18*c^47*d^25 - 24575140799491012895231180800*a^33*b^1
7*c^46*d^26 + 16850754961433442876234137600*a^34*b^16*c^45*d^27 - 10105200492115418262179676160*a^35*b^15*c^44
*d^28 + 5278011312905736232783314944*a^36*b^14*c^43*d^29 - 2387248399405916166169821184*a^37*b^13*c^42*d^30 +
927828632312674738870681600*a^38*b^12*c^41*d^31 - 306693733103726739901644800*a^39*b^11*c^40*d^32 + 8503807595
9446046066606080*a^40*b^10*c^39*d^33 - 19409595119210898894356480*a^41*b^9*c^38*d^34 + 35514004056358128713728
00*a^42*b^8*c^37*d^35 - 500844593983932480880640*a^43*b^7*c^36*d^36 + 51111802530990496153600*a^44*b^6*c^35*d^
37 - 3359577235627333124096*a^45*b^5*c^34*d^38 + 106807368762718683136*a^46*b^4*c^33*d^39) + (-b^15/(16*a^19*d
^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*
c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3
520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(36893488147419103232*a^13*b^38*c^71
*d^4 - 1069911156275153993728*a^14*b^37*c^70*d^5 + 14978756187852155912192*a^15*b^36*c^69*d^6 - 13499903773892
9532960768*a^16*b^35*c^68*d^7 + 882016079904862321508352*a^17*b^34*c^67*d^8 - 4465630463459278708539392*a^18*b
^33*c^66*d^9 + 18321125205332103390035968*a^19*b^32*c^65*d^10 - 63021545228377166868119552*a^20*b^31*c^64*d^11
+ 187018029382071665408606208*a^21*b^30*c^63*d^12 - 490713180393588600090918912*a^22*b^29*c^62*d^13 + 1161438
545048511890042388480*a^23*b^28*c^61*d^14 - 2512974056309066269898833920*a^24*b^27*c^60*d^15 + 499754146989817
2697285754880*a^25*b^26*c^59*d^16 - 9119889428539397211967979520*a^26*b^25*c^58*d^17 + 15181544306461039744285
409280*a^27*b^24*c^57*d^18 - 22888317982577902576352624640*a^28*b^23*c^56*d^19 + 31049708276802113763866050560
*a^29*b^22*c^55*d^20 - 37706614244767692268335267840*a^30*b^21*c^54*d^21 + 40833216619792283792163471360*a^31*
b^20*c^53*d^22 - 39312168062751093709382615040*a^32*b^19*c^52*d^23 + 33557805042801128843488788480*a^33*b^18*c
^51*d^24 - 25329188887155786370693201920*a^34*b^17*c^50*d^25 + 16851463310911481777624186880*a^35*b^16*c^49*d^
26 - 9843609097631363291959787520*a^36*b^15*c^48*d^27 + 5023816147465636127472353280*a^37*b^14*c^47*d^28 - 222
6054577272365612261179392*a^38*b^13*c^46*d^29 + 849419752718963326077370368*a^39*b^12*c^45*d^30 - 276172923601
113041340465152*a^40*b^11*c^44*d^31 + 75441341408208223215812608*a^41*b^10*c^43*d^32 - 16988052798101408932954
112*a^42*b^9*c^42*d^33 + 3070410975444256772063232*a^43*b^8*c^41*d^34 - 428198505575496787427328*a^44*b^7*c^40
*d^35 + 43254156088335077998592*a^45*b^6*c^39*d^36 - 2816587235754527162368*a^46*b^5*c^38*d^37 + 8877495585472
7217152*a^47*b^4*c^37*d^38))*(-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^1
0*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 1267
2*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^1
1))^(3/4) - 11889503016258109440*a^9*b^38*c^56*d^7 + 217253646024352727040*a^10*b^37*c^55*d^8 - 18797664556674
26066432*a^11*b^36*c^54*d^9 + 10237150327374383939584*a^12*b^35*c^53*d^10 - 37711511320670913953792*a^13*b^34*
c^52*d^11 + 77353208427556875796480*a^14*b^33*c^51*d^12 + 127627238172719495249920*a^15*b^32*c^50*d^13 - 21300
84466030987427446784*a^16*b^31*c^49*d^14 + 11885048527140028256616448*a^17*b^30*c^48*d^15 - 456905313616868429
72831744*a^18*b^29*c^47*d^16 + 135851929384595950057553920*a^19*b^28*c^46*d^17 - 326376775711477371051704320*a
^20*b^27*c^45*d^18 + 648353352496064059760705536*a^21*b^26*c^44*d^19 - 1080394184249474617790431232*a^22*b^25*
c^43*d^20 + 1524725339928630029153468416*a^23*b^24*c^42*d^21 - 1834102420924176937716285440*a^24*b^23*c^41*d^2
2 + 1888062742223171008426147840*a^25*b^22*c^40*d^23 - 1666588213584359199850102784*a^26*b^21*c^39*d^24 + 1261
562453800014779376467968*a^27*b^20*c^38*d^25 - 817528072151542384572760064*a^28*b^19*c^37*d^26 + 4518476989344
26396681830400*a^29*b^18*c^36*d^27 - 211721890947778234390937600*a^30*b^17*c^35*d^28 + 83366248780838000977248
256*a^31*b^16*c^34*d^29 - 27241266624044306322685952*a^32*b^15*c^33*d^30 + 7257515800860571589410816*a^33*b^14
*c^32*d^31 - 1536699518639901947985920*a^34*b^13*c^31*d^32 + 248859486128715197317120*a^35*b^12*c^30*d^33 - 28
961642042172523937792*a^36*b^11*c^29*d^34 + 2157438443758953693184*a^37*b^10*c^28*d^35 - 77302354662372933632*
a^38*b^9*c^27*d^36))*1i + (-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d
^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a
^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))
^(1/4)*(x^(1/2)*(30652624963790438400*a^9*b^37*c^51*d^9 - 507161613037259980800*a^10*b^36*c^50*d^10 + 47749569
69550613053440*a^11*b^35*c^49*d^11 - 34948190471081762488320*a^12*b^34*c^48*d^12 + 208409962786483628670976*a^
13*b^33*c^47*d^13 - 990271368055602664177664*a^14*b^32*c^46*d^14 + 3711631588079120800546816*a^15*b^31*c^45*d^
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4836472540920020992*a^18*b^28*c^42*d^18 + 83617663209148864427720704*a^19*b^27*c^41*d^19 - 1123504303156541204
15952896*a^20*b^26*c^40*d^20 + 126417217514830317658046464*a^21*b^25*c^39*d^21 - 119537906081128203174281216*a
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5808856090071441342464*a^27*b^19*c^33*d^27 - 1940847984249953081884672*a^28*b^18*c^32*d^28 + 47173803169456877
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c^29*d^31 - 962765689885917446144*a^32*b^14*c^28*d^32 + 38651177331186466816*a^33*b^13*c^27*d^33) + (-b^15/(16
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11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*
d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*((x^(1/2)*(184467440737095516
16*a^11*b^39*c^68*d^4 - 479615345916448342016*a^12*b^38*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*d^6 -
47961534591644834201600*a^14*b^36*c^65*d^7 + 275778823901957796659200*a^15*b^35*c^64*d^8 - 121293638316919365
8286080*a^16*b^34*c^63*d^9 + 4232993998288506144686080*a^17*b^33*c^62*d^10 - 11941164077799654041845760*a^18*b
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3 + 43611606538557895133364224*a^21*b^29*c^58*d^14 + 72781112360087761599856640*a^22*b^28*c^57*d^15 - 52323406
6593179210717593600*a^23*b^27*c^56*d^16 + 1723753001020797184743833600*a^24*b^26*c^55*d^17 - 42694371673658728
14842183680*a^25*b^25*c^54*d^18 + 8727322757849829186700574720*a^26*b^24*c^53*d^19 - 1521532604397514224937467
9040*a^27*b^23*c^52*d^20 + 22962658463246519625580544000*a^28*b^22*c^51*d^21 - 30231538828274701475145318400*a
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19*c^48*d^24 + 31433146498544749041648926720*a^32*b^18*c^47*d^25 - 24575140799491012895231180800*a^33*b^17*c^4
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8632312674738870681600*a^38*b^12*c^41*d^31 - 306693733103726739901644800*a^39*b^11*c^40*d^32 + 850380759594460
46066606080*a^40*b^10*c^39*d^33 - 19409595119210898894356480*a^41*b^9*c^38*d^34 + 3551400405635812871372800*a^
42*b^8*c^37*d^35 - 500844593983932480880640*a^43*b^7*c^36*d^36 + 51111802530990496153600*a^44*b^6*c^35*d^37 -
3359577235627333124096*a^45*b^5*c^34*d^38 + 106807368762718683136*a^46*b^4*c^33*d^39) - (-b^15/(16*a^19*d^12 +
16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d
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^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4
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7018029382071665408606208*a^21*b^30*c^63*d^12 - 490713180393588600090918912*a^22*b^29*c^62*d^13 + 116143854504
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0*a^27*b^24*c^57*d^18 - 22888317982577902576352624640*a^28*b^23*c^56*d^19 + 31049708276802113763866050560*a^29
*b^22*c^55*d^20 - 37706614244767692268335267840*a^30*b^21*c^54*d^21 + 40833216619792283792163471360*a^31*b^20*
c^53*d^22 - 39312168062751093709382615040*a^32*b^19*c^52*d^23 + 33557805042801128843488788480*a^33*b^18*c^51*d
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9843609097631363291959787520*a^36*b^15*c^48*d^27 + 5023816147465636127472353280*a^37*b^14*c^47*d^28 - 22260545
77272365612261179392*a^38*b^13*c^46*d^29 + 849419752718963326077370368*a^39*b^12*c^45*d^30 - 27617292360111304
1340465152*a^40*b^11*c^44*d^31 + 75441341408208223215812608*a^41*b^10*c^43*d^32 - 16988052798101408932954112*a
^42*b^9*c^42*d^33 + 3070410975444256772063232*a^43*b^8*c^41*d^34 - 428198505575496787427328*a^44*b^7*c^40*d^35
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52*a^47*b^4*c^37*d^38))*(-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2
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4*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(
3/4) + 11889503016258109440*a^9*b^38*c^56*d^7 - 217253646024352727040*a^10*b^37*c^55*d^8 + 1879766455667426066
432*a^11*b^36*c^54*d^9 - 10237150327374383939584*a^12*b^35*c^53*d^10 + 37711511320670913953792*a^13*b^34*c^52*
d^11 - 77353208427556875796480*a^14*b^33*c^51*d^12 - 127627238172719495249920*a^15*b^32*c^50*d^13 + 2130084466
030987427446784*a^16*b^31*c^49*d^14 - 11885048527140028256616448*a^17*b^30*c^48*d^15 + 45690531361686842972831
744*a^18*b^29*c^47*d^16 - 135851929384595950057553920*a^19*b^28*c^46*d^17 + 326376775711477371051704320*a^20*b
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888062742223171008426147840*a^25*b^22*c^40*d^23 + 1666588213584359199850102784*a^26*b^21*c^39*d^24 - 126156245
3800014779376467968*a^27*b^20*c^38*d^25 + 817528072151542384572760064*a^28*b^19*c^37*d^26 - 451847698934426396
681830400*a^29*b^18*c^36*d^27 + 211721890947778234390937600*a^30*b^17*c^35*d^28 - 83366248780838000977248256*a
^31*b^16*c^34*d^29 + 27241266624044306322685952*a^32*b^15*c^33*d^30 - 7257515800860571589410816*a^33*b^14*c^32
*d^31 + 1536699518639901947985920*a^34*b^13*c^31*d^32 - 248859486128715197317120*a^35*b^12*c^30*d^33 + 2896164
2042172523937792*a^36*b^11*c^29*d^34 - 2157438443758953693184*a^37*b^10*c^28*d^35 + 77302354662372933632*a^38*
b^9*c^27*d^36))*1i)/((-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 -
3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b
^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4
)*(x^(1/2)*(30652624963790438400*a^9*b^37*c^51*d^9 - 507161613037259980800*a^10*b^36*c^50*d^10 + 4774956969550
613053440*a^11*b^35*c^49*d^11 - 34948190471081762488320*a^12*b^34*c^48*d^12 + 208409962786483628670976*a^13*b^
33*c^47*d^13 - 990271368055602664177664*a^14*b^32*c^46*d^14 + 3711631588079120800546816*a^15*b^31*c^45*d^15 -
11050795179720929846493184*a^16*b^30*c^44*d^16 + 26487755718620581216649216*a^17*b^29*c^43*d^17 - 518051748364
72540920020992*a^18*b^28*c^42*d^18 + 83617663209148864427720704*a^19*b^27*c^41*d^19 - 112350430315654120415952
896*a^20*b^26*c^40*d^20 + 126417217514830317658046464*a^21*b^25*c^39*d^21 - 119537906081128203174281216*a^22*b
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4 + 35529578846146774008070144*a^25*b^21*c^35*d^25 - 16501565732136655819636736*a^26*b^20*c^34*d^26 + 62958088
56090071441342464*a^27*b^19*c^33*d^27 - 1940847984249953081884672*a^28*b^18*c^32*d^28 + 4717380316945687783669
76*a^29*b^17*c^31*d^29 - 87073083559063809163264*a^30*b^16*c^30*d^30 + 11476570419434950230016*a^31*b^15*c^29*
d^31 - 962765689885917446144*a^32*b^14*c^28*d^32 + 38651177331186466816*a^33*b^13*c^27*d^33) + (-b^15/(16*a^19
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8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 -
3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*((x^(1/2)*(18446744073709551616*a^
11*b^39*c^68*d^4 - 479615345916448342016*a^12*b^38*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*d^6 - 4796
1534591644834201600*a^14*b^36*c^65*d^7 + 275778823901957796659200*a^15*b^35*c^64*d^8 - 12129363831691936582860
80*a^16*b^34*c^63*d^9 + 4232993998288506144686080*a^17*b^33*c^62*d^10 - 11941164077799654041845760*a^18*b^32*c
^61*d^11 + 27104869321333056471040000*a^19*b^31*c^60*d^12 - 46637619173392487079215104*a^20*b^30*c^59*d^13 + 4
3611606538557895133364224*a^21*b^29*c^58*d^14 + 72781112360087761599856640*a^22*b^28*c^57*d^15 - 5232340665931
79210717593600*a^23*b^27*c^56*d^16 + 1723753001020797184743833600*a^24*b^26*c^55*d^17 - 4269437167365872814842
183680*a^25*b^25*c^54*d^18 + 8727322757849829186700574720*a^26*b^24*c^53*d^19 - 15215326043975142249374679040*
a^27*b^23*c^52*d^20 + 22962658463246519625580544000*a^28*b^22*c^51*d^21 - 30231538828274701475145318400*a^29*b
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48*d^24 + 31433146498544749041648926720*a^32*b^18*c^47*d^25 - 24575140799491012895231180800*a^33*b^17*c^46*d^2
6 + 16850754961433442876234137600*a^34*b^16*c^45*d^27 - 10105200492115418262179676160*a^35*b^15*c^44*d^28 + 52
78011312905736232783314944*a^36*b^14*c^43*d^29 - 2387248399405916166169821184*a^37*b^13*c^42*d^30 + 9278286323
12674738870681600*a^38*b^12*c^41*d^31 - 306693733103726739901644800*a^39*b^11*c^40*d^32 + 85038075959446046066
606080*a^40*b^10*c^39*d^33 - 19409595119210898894356480*a^41*b^9*c^38*d^34 + 3551400405635812871372800*a^42*b^
8*c^37*d^35 - 500844593983932480880640*a^43*b^7*c^36*d^36 + 51111802530990496153600*a^44*b^6*c^35*d^37 - 33595
77235627333124096*a^45*b^5*c^34*d^38 + 106807368762718683136*a^46*b^4*c^33*d^39) + (-b^15/(16*a^19*d^12 + 16*a
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12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b
^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 106
9911156275153993728*a^14*b^37*c^70*d^5 + 14978756187852155912192*a^15*b^36*c^69*d^6 - 134999037738929532960768
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^9 + 18321125205332103390035968*a^19*b^32*c^65*d^10 - 63021545228377166868119552*a^20*b^31*c^64*d^11 + 1870180
29382071665408606208*a^21*b^30*c^63*d^12 - 490713180393588600090918912*a^22*b^29*c^62*d^13 + 11614385450485118
90042388480*a^23*b^28*c^61*d^14 - 2512974056309066269898833920*a^24*b^27*c^60*d^15 + 4997541469898172697285754
880*a^25*b^26*c^59*d^16 - 9119889428539397211967979520*a^26*b^25*c^58*d^17 + 15181544306461039744285409280*a^2
7*b^24*c^57*d^18 - 22888317982577902576352624640*a^28*b^23*c^56*d^19 + 31049708276802113763866050560*a^29*b^22
*c^55*d^20 - 37706614244767692268335267840*a^30*b^21*c^54*d^21 + 40833216619792283792163471360*a^31*b^20*c^53*
d^22 - 39312168062751093709382615040*a^32*b^19*c^52*d^23 + 33557805042801128843488788480*a^33*b^18*c^51*d^24 -
25329188887155786370693201920*a^34*b^17*c^50*d^25 + 16851463310911481777624186880*a^35*b^16*c^49*d^26 - 98436
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254156088335077998592*a^45*b^6*c^39*d^36 - 2816587235754527162368*a^46*b^5*c^38*d^37 + 88774955854727217152*a^
47*b^4*c^37*d^38))*(-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 35
20*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5
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7427446784*a^16*b^31*c^49*d^14 + 11885048527140028256616448*a^17*b^30*c^48*d^15 - 45690531361686842972831744*a
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14779376467968*a^27*b^20*c^38*d^25 - 817528072151542384572760064*a^28*b^19*c^37*d^26 + 45184769893442639668183
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72523937792*a^36*b^11*c^29*d^34 + 2157438443758953693184*a^37*b^10*c^28*d^35 - 77302354662372933632*a^38*b^9*c
^27*d^36)) - (-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^1
0*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d
^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(x^(1/
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0*a^11*b^35*c^49*d^11 - 34948190471081762488320*a^12*b^34*c^48*d^12 + 208409962786483628670976*a^13*b^33*c^47*
d^13 - 990271368055602664177664*a^14*b^32*c^46*d^14 + 3711631588079120800546816*a^15*b^31*c^45*d^15 - 11050795
179720929846493184*a^16*b^30*c^44*d^16 + 26487755718620581216649216*a^17*b^29*c^43*d^17 - 51805174836472540920
020992*a^18*b^28*c^42*d^18 + 83617663209148864427720704*a^19*b^27*c^41*d^19 - 112350430315654120415952896*a^20
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9578846146774008070144*a^25*b^21*c^35*d^25 - 16501565732136655819636736*a^26*b^20*c^34*d^26 + 6295808856090071
441342464*a^27*b^19*c^33*d^27 - 1940847984249953081884672*a^28*b^18*c^32*d^28 + 471738031694568778366976*a^29*
b^17*c^31*d^29 - 87073083559063809163264*a^30*b^16*c^30*d^30 + 11476570419434950230016*a^31*b^15*c^29*d^31 - 9
62765689885917446144*a^32*b^14*c^28*d^32 + 38651177331186466816*a^33*b^13*c^27*d^33) + (-b^15/(16*a^19*d^12 +
16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^
4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^
16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*((x^(1/2)*(18446744073709551616*a^11*b^39*
c^68*d^4 - 479615345916448342016*a^12*b^38*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*d^6 - 479615345916
44834201600*a^14*b^36*c^65*d^7 + 275778823901957796659200*a^15*b^35*c^64*d^8 - 1212936383169193658286080*a^16*
b^34*c^63*d^9 + 4232993998288506144686080*a^17*b^33*c^62*d^10 - 11941164077799654041845760*a^18*b^32*c^61*d^11
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38557895133364224*a^21*b^29*c^58*d^14 + 72781112360087761599856640*a^22*b^28*c^57*d^15 - 523234066593179210717
593600*a^23*b^27*c^56*d^16 + 1723753001020797184743833600*a^24*b^26*c^55*d^17 - 4269437167365872814842183680*a
^25*b^25*c^54*d^18 + 8727322757849829186700574720*a^26*b^24*c^53*d^19 - 15215326043975142249374679040*a^27*b^2
3*c^52*d^20 + 22962658463246519625580544000*a^28*b^22*c^51*d^21 - 30231538828274701475145318400*a^29*b^21*c^50
*d^22 + 34870163031766389952882933760*a^30*b^20*c^49*d^23 - 35316718238336158489724846080*a^31*b^19*c^48*d^24
+ 31433146498544749041648926720*a^32*b^18*c^47*d^25 - 24575140799491012895231180800*a^33*b^17*c^46*d^26 + 1685
0754961433442876234137600*a^34*b^16*c^45*d^27 - 10105200492115418262179676160*a^35*b^15*c^44*d^28 + 5278011312
905736232783314944*a^36*b^14*c^43*d^29 - 2387248399405916166169821184*a^37*b^13*c^42*d^30 + 927828632312674738
870681600*a^38*b^12*c^41*d^31 - 306693733103726739901644800*a^39*b^11*c^40*d^32 + 85038075959446046066606080*a
^40*b^10*c^39*d^33 - 19409595119210898894356480*a^41*b^9*c^38*d^34 + 3551400405635812871372800*a^42*b^8*c^37*d
^35 - 500844593983932480880640*a^43*b^7*c^36*d^36 + 51111802530990496153600*a^44*b^6*c^35*d^37 - 3359577235627
333124096*a^45*b^5*c^34*d^38 + 106807368762718683136*a^46*b^4*c^33*d^39) - (-b^15/(16*a^19*d^12 + 16*a^7*b^12*
c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^
12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d
^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 10699111562
75153993728*a^14*b^37*c^70*d^5 + 14978756187852155912192*a^15*b^36*c^69*d^6 - 134999037738929532960768*a^16*b^
35*c^68*d^7 + 882016079904862321508352*a^17*b^34*c^67*d^8 - 4465630463459278708539392*a^18*b^33*c^66*d^9 + 183
21125205332103390035968*a^19*b^32*c^65*d^10 - 63021545228377166868119552*a^20*b^31*c^64*d^11 + 187018029382071
665408606208*a^21*b^30*c^63*d^12 - 490713180393588600090918912*a^22*b^29*c^62*d^13 + 1161438545048511890042388
480*a^23*b^28*c^61*d^14 - 2512974056309066269898833920*a^24*b^27*c^60*d^15 + 4997541469898172697285754880*a^25
*b^26*c^59*d^16 - 9119889428539397211967979520*a^26*b^25*c^58*d^17 + 15181544306461039744285409280*a^27*b^24*c
^57*d^18 - 22888317982577902576352624640*a^28*b^23*c^56*d^19 + 31049708276802113763866050560*a^29*b^22*c^55*d^
20 - 37706614244767692268335267840*a^30*b^21*c^54*d^21 + 40833216619792283792163471360*a^31*b^20*c^53*d^22 - 3
9312168062751093709382615040*a^32*b^19*c^52*d^23 + 33557805042801128843488788480*a^33*b^18*c^51*d^24 - 2532918
8887155786370693201920*a^34*b^17*c^50*d^25 + 16851463310911481777624186880*a^35*b^16*c^49*d^26 - 9843609097631
363291959787520*a^36*b^15*c^48*d^27 + 5023816147465636127472353280*a^37*b^14*c^47*d^28 - 222605457727236561226
1179392*a^38*b^13*c^46*d^29 + 849419752718963326077370368*a^39*b^12*c^45*d^30 - 276172923601113041340465152*a^
40*b^11*c^44*d^31 + 75441341408208223215812608*a^41*b^10*c^43*d^32 - 16988052798101408932954112*a^42*b^9*c^42*
d^33 + 3070410975444256772063232*a^43*b^8*c^41*d^34 - 428198505575496787427328*a^44*b^7*c^40*d^35 + 4325415608
8335077998592*a^45*b^6*c^39*d^36 - 2816587235754527162368*a^46*b^5*c^38*d^37 + 88774955854727217152*a^47*b^4*c
^37*d^38))*(-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*
b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7
+ 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(3/4) + 118895
03016258109440*a^9*b^38*c^56*d^7 - 217253646024352727040*a^10*b^37*c^55*d^8 + 1879766455667426066432*a^11*b^36
*c^54*d^9 - 10237150327374383939584*a^12*b^35*c^53*d^10 + 37711511320670913953792*a^13*b^34*c^52*d^11 - 773532
08427556875796480*a^14*b^33*c^51*d^12 - 127627238172719495249920*a^15*b^32*c^50*d^13 + 21300844660309874274467
84*a^16*b^31*c^49*d^14 - 11885048527140028256616448*a^17*b^30*c^48*d^15 + 45690531361686842972831744*a^18*b^29
*c^47*d^16 - 135851929384595950057553920*a^19*b^28*c^46*d^17 + 326376775711477371051704320*a^20*b^27*c^45*d^18
- 648353352496064059760705536*a^21*b^26*c^44*d^19 + 1080394184249474617790431232*a^22*b^25*c^43*d^20 - 152472
5339928630029153468416*a^23*b^24*c^42*d^21 + 1834102420924176937716285440*a^24*b^23*c^41*d^22 - 18880627422231
71008426147840*a^25*b^22*c^40*d^23 + 1666588213584359199850102784*a^26*b^21*c^39*d^24 - 1261562453800014779376
467968*a^27*b^20*c^38*d^25 + 817528072151542384572760064*a^28*b^19*c^37*d^26 - 451847698934426396681830400*a^2
9*b^18*c^36*d^27 + 211721890947778234390937600*a^30*b^17*c^35*d^28 - 83366248780838000977248256*a^31*b^16*c^34
*d^29 + 27241266624044306322685952*a^32*b^15*c^33*d^30 - 7257515800860571589410816*a^33*b^14*c^32*d^31 + 15366
99518639901947985920*a^34*b^13*c^31*d^32 - 248859486128715197317120*a^35*b^12*c^30*d^33 + 28961642042172523937
792*a^36*b^11*c^29*d^34 - 2157438443758953693184*a^37*b^10*c^28*d^35 + 77302354662372933632*a^38*b^9*c^27*d^36
))))*(-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^
9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 792
0*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*2i - 2*atan(((
-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3
+ 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15
*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(x^(1/2)*(3065262496
3790438400*a^9*b^37*c^51*d^9 - 507161613037259980800*a^10*b^36*c^50*d^10 + 4774956969550613053440*a^11*b^35*c^
49*d^11 - 34948190471081762488320*a^12*b^34*c^48*d^12 + 208409962786483628670976*a^13*b^33*c^47*d^13 - 9902713
68055602664177664*a^14*b^32*c^46*d^14 + 3711631588079120800546816*a^15*b^31*c^45*d^15 - 1105079517972092984649
3184*a^16*b^30*c^44*d^16 + 26487755718620581216649216*a^17*b^29*c^43*d^17 - 51805174836472540920020992*a^18*b^
28*c^42*d^18 + 83617663209148864427720704*a^19*b^27*c^41*d^19 - 112350430315654120415952896*a^20*b^26*c^40*d^2
0 + 126417217514830317658046464*a^21*b^25*c^39*d^21 - 119537906081128203174281216*a^22*b^24*c^38*d^22 + 950898
64774620999552335872*a^23*b^23*c^37*d^23 - 63545506634457987380412416*a^24*b^22*c^36*d^24 + 355295788461467740
08070144*a^25*b^21*c^35*d^25 - 16501565732136655819636736*a^26*b^20*c^34*d^26 + 6295808856090071441342464*a^27
*b^19*c^33*d^27 - 1940847984249953081884672*a^28*b^18*c^32*d^28 + 471738031694568778366976*a^29*b^17*c^31*d^29
- 87073083559063809163264*a^30*b^16*c^30*d^30 + 11476570419434950230016*a^31*b^15*c^29*d^31 - 962765689885917
446144*a^32*b^14*c^28*d^32 + 38651177331186466816*a^33*b^13*c^27*d^33) - (-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^
12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12
*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9
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615345916448342016*a^12*b^38*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*d^6 - 47961534591644834201600*a^
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*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 1069911156275153993728*a^
14*b^37*c^70*d^5 + 14978756187852155912192*a^15*b^36*c^69*d^6 - 134999037738929532960768*a^16*b^35*c^68*d^7 +
882016079904862321508352*a^17*b^34*c^67*d^8 - 4465630463459278708539392*a^18*b^33*c^66*d^9 + 18321125205332103
390035968*a^19*b^32*c^65*d^10 - 63021545228377166868119552*a^20*b^31*c^64*d^11 + 187018029382071665408606208*a
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c^61*d^14 - 2512974056309066269898833920*a^24*b^27*c^60*d^15 + 4997541469898172697285754880*a^25*b^26*c^59*d^1
6 - 9119889428539397211967979520*a^26*b^25*c^58*d^17 + 15181544306461039744285409280*a^27*b^24*c^57*d^18 - 228
88317982577902576352624640*a^28*b^23*c^56*d^19 + 31049708276802113763866050560*a^29*b^22*c^55*d^20 - 377066142
44767692268335267840*a^30*b^21*c^54*d^21 + 40833216619792283792163471360*a^31*b^20*c^53*d^22 - 393121680627510
93709382615040*a^32*b^19*c^52*d^23 + 33557805042801128843488788480*a^33*b^18*c^51*d^24 - 253291888871557863706
93201920*a^34*b^17*c^50*d^25 + 16851463310911481777624186880*a^35*b^16*c^49*d^26 - 984360909763136329195978752
0*a^36*b^15*c^48*d^27 + 5023816147465636127472353280*a^37*b^14*c^47*d^28 - 2226054577272365612261179392*a^38*b
^13*c^46*d^29 + 849419752718963326077370368*a^39*b^12*c^45*d^30 - 276172923601113041340465152*a^40*b^11*c^44*d
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975444256772063232*a^43*b^8*c^41*d^34 - 428198505575496787427328*a^44*b^7*c^40*d^35 + 43254156088335077998592*
a^45*b^6*c^39*d^36 - 2816587235754527162368*a^46*b^5*c^38*d^37 + 88774955854727217152*a^47*b^4*c^37*d^38)*1i)*
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5*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(3/4)*1i - 11889503016258
109440*a^9*b^38*c^56*d^7 + 217253646024352727040*a^10*b^37*c^55*d^8 - 1879766455667426066432*a^11*b^36*c^54*d^
9 + 10237150327374383939584*a^12*b^35*c^53*d^10 - 37711511320670913953792*a^13*b^34*c^52*d^11 + 77353208427556
875796480*a^14*b^33*c^51*d^12 + 127627238172719495249920*a^15*b^32*c^50*d^13 - 2130084466030987427446784*a^16*
b^31*c^49*d^14 + 11885048527140028256616448*a^17*b^30*c^48*d^15 - 45690531361686842972831744*a^18*b^29*c^47*d^
16 + 135851929384595950057553920*a^19*b^28*c^46*d^17 - 326376775711477371051704320*a^20*b^27*c^45*d^18 + 64835
3352496064059760705536*a^21*b^26*c^44*d^19 - 1080394184249474617790431232*a^22*b^25*c^43*d^20 + 15247253399286
30029153468416*a^23*b^24*c^42*d^21 - 1834102420924176937716285440*a^24*b^23*c^41*d^22 + 1888062742223171008426
147840*a^25*b^22*c^40*d^23 - 1666588213584359199850102784*a^26*b^21*c^39*d^24 + 1261562453800014779376467968*a
^27*b^20*c^38*d^25 - 817528072151542384572760064*a^28*b^19*c^37*d^26 + 451847698934426396681830400*a^29*b^18*c
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27241266624044306322685952*a^32*b^15*c^33*d^30 + 7257515800860571589410816*a^33*b^14*c^32*d^31 - 1536699518639
901947985920*a^34*b^13*c^31*d^32 + 248859486128715197317120*a^35*b^12*c^30*d^33 - 28961642042172523937792*a^36
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(-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3
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5*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(x^(1/2)*(306526249
63790438400*a^9*b^37*c^51*d^9 - 507161613037259980800*a^10*b^36*c^50*d^10 + 4774956969550613053440*a^11*b^35*c
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368055602664177664*a^14*b^32*c^46*d^14 + 3711631588079120800546816*a^15*b^31*c^45*d^15 - 110507951797209298464
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20 + 126417217514830317658046464*a^21*b^25*c^39*d^21 - 119537906081128203174281216*a^22*b^24*c^38*d^22 + 95089
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7446144*a^32*b^14*c^28*d^32 + 38651177331186466816*a^33*b^13*c^27*d^33) - (-b^15/(16*a^19*d^12 + 16*a^7*b^12*c
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2*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^
9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*((x^(1/2)*(18446744073709551616*a^11*b^39*c^68*d^4 - 47
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5*b^5*c^34*d^38 + 106807368762718683136*a^46*b^4*c^33*d^39) + (-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^
8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^
5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^1
7*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 1069911156275153993728*a
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882016079904862321508352*a^17*b^34*c^67*d^8 - 4465630463459278708539392*a^18*b^33*c^66*d^9 + 1832112520533210
3390035968*a^19*b^32*c^65*d^10 - 63021545228377166868119552*a^20*b^31*c^64*d^11 + 187018029382071665408606208*
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244767692268335267840*a^30*b^21*c^54*d^21 + 40833216619792283792163471360*a^31*b^20*c^53*d^22 - 39312168062751
093709382615040*a^32*b^19*c^52*d^23 + 33557805042801128843488788480*a^33*b^18*c^51*d^24 - 25329188887155786370
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20*a^36*b^15*c^48*d^27 + 5023816147465636127472353280*a^37*b^14*c^47*d^28 - 2226054577272365612261179392*a^38*
b^13*c^46*d^29 + 849419752718963326077370368*a^39*b^12*c^45*d^30 - 276172923601113041340465152*a^40*b^11*c^44*
d^31 + 75441341408208223215812608*a^41*b^10*c^43*d^32 - 16988052798101408932954112*a^42*b^9*c^42*d^33 + 307041
0975444256772063232*a^43*b^8*c^41*d^34 - 428198505575496787427328*a^44*b^7*c^40*d^35 + 43254156088335077998592
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*(-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^
3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^
15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(3/4)*1i + 1188950301625
8109440*a^9*b^38*c^56*d^7 - 217253646024352727040*a^10*b^37*c^55*d^8 + 1879766455667426066432*a^11*b^36*c^54*d
^9 - 10237150327374383939584*a^12*b^35*c^53*d^10 + 37711511320670913953792*a^13*b^34*c^52*d^11 - 7735320842755
6875796480*a^14*b^33*c^51*d^12 - 127627238172719495249920*a^15*b^32*c^50*d^13 + 2130084466030987427446784*a^16
*b^31*c^49*d^14 - 11885048527140028256616448*a^17*b^30*c^48*d^15 + 45690531361686842972831744*a^18*b^29*c^47*d
^16 - 135851929384595950057553920*a^19*b^28*c^46*d^17 + 326376775711477371051704320*a^20*b^27*c^45*d^18 - 6483
53352496064059760705536*a^21*b^26*c^44*d^19 + 1080394184249474617790431232*a^22*b^25*c^43*d^20 - 1524725339928
630029153468416*a^23*b^24*c^42*d^21 + 1834102420924176937716285440*a^24*b^23*c^41*d^22 - 188806274222317100842
6147840*a^25*b^22*c^40*d^23 + 1666588213584359199850102784*a^26*b^21*c^39*d^24 - 1261562453800014779376467968*
a^27*b^20*c^38*d^25 + 817528072151542384572760064*a^28*b^19*c^37*d^26 - 451847698934426396681830400*a^29*b^18*
c^36*d^27 + 211721890947778234390937600*a^30*b^17*c^35*d^28 - 83366248780838000977248256*a^31*b^16*c^34*d^29 +
27241266624044306322685952*a^32*b^15*c^33*d^30 - 7257515800860571589410816*a^33*b^14*c^32*d^31 + 153669951863
9901947985920*a^34*b^13*c^31*d^32 - 248859486128715197317120*a^35*b^12*c^30*d^33 + 28961642042172523937792*a^3
6*b^11*c^29*d^34 - 2157438443758953693184*a^37*b^10*c^28*d^35 + 77302354662372933632*a^38*b^9*c^27*d^36)*1i))/
((-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^
3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^
15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(x^(1/2)*(30652624
963790438400*a^9*b^37*c^51*d^9 - 507161613037259980800*a^10*b^36*c^50*d^10 + 4774956969550613053440*a^11*b^35*
c^49*d^11 - 34948190471081762488320*a^12*b^34*c^48*d^12 + 208409962786483628670976*a^13*b^33*c^47*d^13 - 99027
1368055602664177664*a^14*b^32*c^46*d^14 + 3711631588079120800546816*a^15*b^31*c^45*d^15 - 11050795179720929846
493184*a^16*b^30*c^44*d^16 + 26487755718620581216649216*a^17*b^29*c^43*d^17 - 51805174836472540920020992*a^18*
b^28*c^42*d^18 + 83617663209148864427720704*a^19*b^27*c^41*d^19 - 112350430315654120415952896*a^20*b^26*c^40*d
^20 + 126417217514830317658046464*a^21*b^25*c^39*d^21 - 119537906081128203174281216*a^22*b^24*c^38*d^22 + 9508
9864774620999552335872*a^23*b^23*c^37*d^23 - 63545506634457987380412416*a^24*b^22*c^36*d^24 + 3552957884614677
4008070144*a^25*b^21*c^35*d^25 - 16501565732136655819636736*a^26*b^20*c^34*d^26 + 6295808856090071441342464*a^
27*b^19*c^33*d^27 - 1940847984249953081884672*a^28*b^18*c^32*d^28 + 471738031694568778366976*a^29*b^17*c^31*d^
29 - 87073083559063809163264*a^30*b^16*c^30*d^30 + 11476570419434950230016*a^31*b^15*c^29*d^31 - 9627656898859
17446144*a^32*b^14*c^28*d^32 + 38651177331186466816*a^33*b^13*c^27*d^33) - (-b^15/(16*a^19*d^12 + 16*a^7*b^12*
c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^
12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d
^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*((x^(1/2)*(18446744073709551616*a^11*b^39*c^68*d^4 - 4
79615345916448342016*a^12*b^38*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*d^6 - 47961534591644834201600*
a^14*b^36*c^65*d^7 + 275778823901957796659200*a^15*b^35*c^64*d^8 - 1212936383169193658286080*a^16*b^34*c^63*d^
9 + 4232993998288506144686080*a^17*b^33*c^62*d^10 - 11941164077799654041845760*a^18*b^32*c^61*d^11 + 271048693
21333056471040000*a^19*b^31*c^60*d^12 - 46637619173392487079215104*a^20*b^30*c^59*d^13 + 436116065385578951333
64224*a^21*b^29*c^58*d^14 + 72781112360087761599856640*a^22*b^28*c^57*d^15 - 523234066593179210717593600*a^23*
b^27*c^56*d^16 + 1723753001020797184743833600*a^24*b^26*c^55*d^17 - 4269437167365872814842183680*a^25*b^25*c^5
4*d^18 + 8727322757849829186700574720*a^26*b^24*c^53*d^19 - 15215326043975142249374679040*a^27*b^23*c^52*d^20
+ 22962658463246519625580544000*a^28*b^22*c^51*d^21 - 30231538828274701475145318400*a^29*b^21*c^50*d^22 + 3487
0163031766389952882933760*a^30*b^20*c^49*d^23 - 35316718238336158489724846080*a^31*b^19*c^48*d^24 + 3143314649
8544749041648926720*a^32*b^18*c^47*d^25 - 24575140799491012895231180800*a^33*b^17*c^46*d^26 + 1685075496143344
2876234137600*a^34*b^16*c^45*d^27 - 10105200492115418262179676160*a^35*b^15*c^44*d^28 + 5278011312905736232783
314944*a^36*b^14*c^43*d^29 - 2387248399405916166169821184*a^37*b^13*c^42*d^30 + 927828632312674738870681600*a^
38*b^12*c^41*d^31 - 306693733103726739901644800*a^39*b^11*c^40*d^32 + 85038075959446046066606080*a^40*b^10*c^3
9*d^33 - 19409595119210898894356480*a^41*b^9*c^38*d^34 + 3551400405635812871372800*a^42*b^8*c^37*d^35 - 500844
593983932480880640*a^43*b^7*c^36*d^36 + 51111802530990496153600*a^44*b^6*c^35*d^37 - 3359577235627333124096*a^
45*b^5*c^34*d^38 + 106807368762718683136*a^46*b^4*c^33*d^39) - (-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a
^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d
^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^
17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 1069911156275153993728*
a^14*b^37*c^70*d^5 + 14978756187852155912192*a^15*b^36*c^69*d^6 - 134999037738929532960768*a^16*b^35*c^68*d^7
+ 882016079904862321508352*a^17*b^34*c^67*d^8 - 4465630463459278708539392*a^18*b^33*c^66*d^9 + 183211252053321
03390035968*a^19*b^32*c^65*d^10 - 63021545228377166868119552*a^20*b^31*c^64*d^11 + 187018029382071665408606208
*a^21*b^30*c^63*d^12 - 490713180393588600090918912*a^22*b^29*c^62*d^13 + 1161438545048511890042388480*a^23*b^2
8*c^61*d^14 - 2512974056309066269898833920*a^24*b^27*c^60*d^15 + 4997541469898172697285754880*a^25*b^26*c^59*d
^16 - 9119889428539397211967979520*a^26*b^25*c^58*d^17 + 15181544306461039744285409280*a^27*b^24*c^57*d^18 - 2
2888317982577902576352624640*a^28*b^23*c^56*d^19 + 31049708276802113763866050560*a^29*b^22*c^55*d^20 - 3770661
4244767692268335267840*a^30*b^21*c^54*d^21 + 40833216619792283792163471360*a^31*b^20*c^53*d^22 - 3931216806275
1093709382615040*a^32*b^19*c^52*d^23 + 33557805042801128843488788480*a^33*b^18*c^51*d^24 - 2532918888715578637
0693201920*a^34*b^17*c^50*d^25 + 16851463310911481777624186880*a^35*b^16*c^49*d^26 - 9843609097631363291959787
520*a^36*b^15*c^48*d^27 + 5023816147465636127472353280*a^37*b^14*c^47*d^28 - 2226054577272365612261179392*a^38
*b^13*c^46*d^29 + 849419752718963326077370368*a^39*b^12*c^45*d^30 - 276172923601113041340465152*a^40*b^11*c^44
*d^31 + 75441341408208223215812608*a^41*b^10*c^43*d^32 - 16988052798101408932954112*a^42*b^9*c^42*d^33 + 30704
10975444256772063232*a^43*b^8*c^41*d^34 - 428198505575496787427328*a^44*b^7*c^40*d^35 + 4325415608833507799859
2*a^45*b^6*c^39*d^36 - 2816587235754527162368*a^46*b^5*c^38*d^37 + 88774955854727217152*a^47*b^4*c^37*d^38)*1i
)*(-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d
^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a
^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(3/4)*1i - 118895030162
58109440*a^9*b^38*c^56*d^7 + 217253646024352727040*a^10*b^37*c^55*d^8 - 1879766455667426066432*a^11*b^36*c^54*
d^9 + 10237150327374383939584*a^12*b^35*c^53*d^10 - 37711511320670913953792*a^13*b^34*c^52*d^11 + 773532084275
56875796480*a^14*b^33*c^51*d^12 + 127627238172719495249920*a^15*b^32*c^50*d^13 - 2130084466030987427446784*a^1
6*b^31*c^49*d^14 + 11885048527140028256616448*a^17*b^30*c^48*d^15 - 45690531361686842972831744*a^18*b^29*c^47*
d^16 + 135851929384595950057553920*a^19*b^28*c^46*d^17 - 326376775711477371051704320*a^20*b^27*c^45*d^18 + 648
353352496064059760705536*a^21*b^26*c^44*d^19 - 1080394184249474617790431232*a^22*b^25*c^43*d^20 + 152472533992
8630029153468416*a^23*b^24*c^42*d^21 - 1834102420924176937716285440*a^24*b^23*c^41*d^22 + 18880627422231710084
26147840*a^25*b^22*c^40*d^23 - 1666588213584359199850102784*a^26*b^21*c^39*d^24 + 1261562453800014779376467968
*a^27*b^20*c^38*d^25 - 817528072151542384572760064*a^28*b^19*c^37*d^26 + 451847698934426396681830400*a^29*b^18
*c^36*d^27 - 211721890947778234390937600*a^30*b^17*c^35*d^28 + 83366248780838000977248256*a^31*b^16*c^34*d^29
- 27241266624044306322685952*a^32*b^15*c^33*d^30 + 7257515800860571589410816*a^33*b^14*c^32*d^31 - 15366995186
39901947985920*a^34*b^13*c^31*d^32 + 248859486128715197317120*a^35*b^12*c^30*d^33 - 28961642042172523937792*a^
36*b^11*c^29*d^34 + 2157438443758953693184*a^37*b^10*c^28*d^35 - 77302354662372933632*a^38*b^9*c^27*d^36)*1i)*
1i - (-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^
9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 792
0*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(x^(1/2)*(3065
2624963790438400*a^9*b^37*c^51*d^9 - 507161613037259980800*a^10*b^36*c^50*d^10 + 4774956969550613053440*a^11*b
^35*c^49*d^11 - 34948190471081762488320*a^12*b^34*c^48*d^12 + 208409962786483628670976*a^13*b^33*c^47*d^13 - 9
90271368055602664177664*a^14*b^32*c^46*d^14 + 3711631588079120800546816*a^15*b^31*c^45*d^15 - 1105079517972092
9846493184*a^16*b^30*c^44*d^16 + 26487755718620581216649216*a^17*b^29*c^43*d^17 - 51805174836472540920020992*a
^18*b^28*c^42*d^18 + 83617663209148864427720704*a^19*b^27*c^41*d^19 - 112350430315654120415952896*a^20*b^26*c^
40*d^20 + 126417217514830317658046464*a^21*b^25*c^39*d^21 - 119537906081128203174281216*a^22*b^24*c^38*d^22 +
95089864774620999552335872*a^23*b^23*c^37*d^23 - 63545506634457987380412416*a^24*b^22*c^36*d^24 + 355295788461
46774008070144*a^25*b^21*c^35*d^25 - 16501565732136655819636736*a^26*b^20*c^34*d^26 + 629580885609007144134246
4*a^27*b^19*c^33*d^27 - 1940847984249953081884672*a^28*b^18*c^32*d^28 + 471738031694568778366976*a^29*b^17*c^3
1*d^29 - 87073083559063809163264*a^30*b^16*c^30*d^30 + 11476570419434950230016*a^31*b^15*c^29*d^31 - 962765689
885917446144*a^32*b^14*c^28*d^32 + 38651177331186466816*a^33*b^13*c^27*d^33) - (-b^15/(16*a^19*d^12 + 16*a^7*b
^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 1267
2*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c
^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*((x^(1/2)*(18446744073709551616*a^11*b^39*c^68*d^4
- 479615345916448342016*a^12*b^38*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*d^6 - 47961534591644834201
600*a^14*b^36*c^65*d^7 + 275778823901957796659200*a^15*b^35*c^64*d^8 - 1212936383169193658286080*a^16*b^34*c^6
3*d^9 + 4232993998288506144686080*a^17*b^33*c^62*d^10 - 11941164077799654041845760*a^18*b^32*c^61*d^11 + 27104
869321333056471040000*a^19*b^31*c^60*d^12 - 46637619173392487079215104*a^20*b^30*c^59*d^13 + 43611606538557895
133364224*a^21*b^29*c^58*d^14 + 72781112360087761599856640*a^22*b^28*c^57*d^15 - 523234066593179210717593600*a
^23*b^27*c^56*d^16 + 1723753001020797184743833600*a^24*b^26*c^55*d^17 - 4269437167365872814842183680*a^25*b^25
*c^54*d^18 + 8727322757849829186700574720*a^26*b^24*c^53*d^19 - 15215326043975142249374679040*a^27*b^23*c^52*d
^20 + 22962658463246519625580544000*a^28*b^22*c^51*d^21 - 30231538828274701475145318400*a^29*b^21*c^50*d^22 +
34870163031766389952882933760*a^30*b^20*c^49*d^23 - 35316718238336158489724846080*a^31*b^19*c^48*d^24 + 314331
46498544749041648926720*a^32*b^18*c^47*d^25 - 24575140799491012895231180800*a^33*b^17*c^46*d^26 + 168507549614
33442876234137600*a^34*b^16*c^45*d^27 - 10105200492115418262179676160*a^35*b^15*c^44*d^28 + 527801131290573623
2783314944*a^36*b^14*c^43*d^29 - 2387248399405916166169821184*a^37*b^13*c^42*d^30 + 92782863231267473887068160
0*a^38*b^12*c^41*d^31 - 306693733103726739901644800*a^39*b^11*c^40*d^32 + 85038075959446046066606080*a^40*b^10
*c^39*d^33 - 19409595119210898894356480*a^41*b^9*c^38*d^34 + 3551400405635812871372800*a^42*b^8*c^37*d^35 - 50
0844593983932480880640*a^43*b^7*c^36*d^36 + 51111802530990496153600*a^44*b^6*c^35*d^37 - 335957723562733312409
6*a^45*b^5*c^34*d^38 + 106807368762718683136*a^46*b^4*c^33*d^39) + (-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 1
92*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c
^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 105
6*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 1069911156275153993
728*a^14*b^37*c^70*d^5 + 14978756187852155912192*a^15*b^36*c^69*d^6 - 134999037738929532960768*a^16*b^35*c^68*
d^7 + 882016079904862321508352*a^17*b^34*c^67*d^8 - 4465630463459278708539392*a^18*b^33*c^66*d^9 + 18321125205
332103390035968*a^19*b^32*c^65*d^10 - 63021545228377166868119552*a^20*b^31*c^64*d^11 + 18701802938207166540860
6208*a^21*b^30*c^63*d^12 - 490713180393588600090918912*a^22*b^29*c^62*d^13 + 1161438545048511890042388480*a^23
*b^28*c^61*d^14 - 2512974056309066269898833920*a^24*b^27*c^60*d^15 + 4997541469898172697285754880*a^25*b^26*c^
59*d^16 - 9119889428539397211967979520*a^26*b^25*c^58*d^17 + 15181544306461039744285409280*a^27*b^24*c^57*d^18
- 22888317982577902576352624640*a^28*b^23*c^56*d^19 + 31049708276802113763866050560*a^29*b^22*c^55*d^20 - 377
06614244767692268335267840*a^30*b^21*c^54*d^21 + 40833216619792283792163471360*a^31*b^20*c^53*d^22 - 393121680
62751093709382615040*a^32*b^19*c^52*d^23 + 33557805042801128843488788480*a^33*b^18*c^51*d^24 - 253291888871557
86370693201920*a^34*b^17*c^50*d^25 + 16851463310911481777624186880*a^35*b^16*c^49*d^26 - 984360909763136329195
9787520*a^36*b^15*c^48*d^27 + 5023816147465636127472353280*a^37*b^14*c^47*d^28 - 2226054577272365612261179392*
a^38*b^13*c^46*d^29 + 849419752718963326077370368*a^39*b^12*c^45*d^30 - 276172923601113041340465152*a^40*b^11*
c^44*d^31 + 75441341408208223215812608*a^41*b^10*c^43*d^32 - 16988052798101408932954112*a^42*b^9*c^42*d^33 + 3
070410975444256772063232*a^43*b^8*c^41*d^34 - 428198505575496787427328*a^44*b^7*c^40*d^35 + 432541560883350779
98592*a^45*b^6*c^39*d^36 - 2816587235754527162368*a^46*b^5*c^38*d^37 + 88774955854727217152*a^47*b^4*c^37*d^38
)*1i)*(-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^9*c
^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 + 79
20*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(3/4)*1i + 11889503
016258109440*a^9*b^38*c^56*d^7 - 217253646024352727040*a^10*b^37*c^55*d^8 + 1879766455667426066432*a^11*b^36*c
^54*d^9 - 10237150327374383939584*a^12*b^35*c^53*d^10 + 37711511320670913953792*a^13*b^34*c^52*d^11 - 77353208
427556875796480*a^14*b^33*c^51*d^12 - 127627238172719495249920*a^15*b^32*c^50*d^13 + 2130084466030987427446784
*a^16*b^31*c^49*d^14 - 11885048527140028256616448*a^17*b^30*c^48*d^15 + 45690531361686842972831744*a^18*b^29*c
^47*d^16 - 135851929384595950057553920*a^19*b^28*c^46*d^17 + 326376775711477371051704320*a^20*b^27*c^45*d^18 -
648353352496064059760705536*a^21*b^26*c^44*d^19 + 1080394184249474617790431232*a^22*b^25*c^43*d^20 - 15247253
39928630029153468416*a^23*b^24*c^42*d^21 + 1834102420924176937716285440*a^24*b^23*c^41*d^22 - 1888062742223171
008426147840*a^25*b^22*c^40*d^23 + 1666588213584359199850102784*a^26*b^21*c^39*d^24 - 126156245380001477937646
7968*a^27*b^20*c^38*d^25 + 817528072151542384572760064*a^28*b^19*c^37*d^26 - 451847698934426396681830400*a^29*
b^18*c^36*d^27 + 211721890947778234390937600*a^30*b^17*c^35*d^28 - 83366248780838000977248256*a^31*b^16*c^34*d
^29 + 27241266624044306322685952*a^32*b^15*c^33*d^30 - 7257515800860571589410816*a^33*b^14*c^32*d^31 + 1536699
518639901947985920*a^34*b^13*c^31*d^32 - 248859486128715197317120*a^35*b^12*c^30*d^33 + 2896164204217252393779
2*a^36*b^11*c^29*d^34 - 2157438443758953693184*a^37*b^10*c^28*d^35 + 77302354662372933632*a^38*b^9*c^27*d^36)*
1i)*1i))*(-b^15/(16*a^19*d^12 + 16*a^7*b^12*c^12 - 192*a^8*b^11*c^11*d + 1056*a^9*b^10*c^10*d^2 - 3520*a^10*b^
9*c^9*d^3 + 7920*a^11*b^8*c^8*d^4 - 12672*a^12*b^7*c^7*d^5 + 14784*a^13*b^6*c^6*d^6 - 12672*a^14*b^5*c^5*d^7 +
7920*a^15*b^4*c^4*d^8 - 3520*a^16*b^3*c^3*d^9 + 1056*a^17*b^2*c^2*d^10 - 192*a^18*b*c*d^11))^(1/4) - (2/(3*a*
c) + (x^2*(121*a^2*d^3 + 64*b^2*c^2*d - 209*a*b*c*d^2))/(48*a*c*(b^2*c^3 + a^2*c*d^2 - 2*a*b*c^2*d)) + (d^2*x^
4*(77*a^2*d^2 + 32*b^2*c^2 - 133*a*b*c*d))/(48*a*c^2*(b^2*c^3 + a^2*c*d^2 - 2*a*b*c^2*d)))/(c^2*x^(3/2) + d^2*
x^(11/2) + 2*c*d*x^(7/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(5/2)/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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