\(\int \frac {1}{x^{3/2} (a+b x^2) (c+d x^2)^3} \, dx\) [485]

Optimal result

Integrand size = 24, antiderivative size = 681 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {32 b^2 c^2-85 a b c d+45 a^2 d^2}{16 a c^3 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b^{13/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{13/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^3}-\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} (b c-a d)^3}-\frac {b^{13/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {b^{13/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} (b c-a d)^3}-\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} (b c-a d)^3} \]

[Out]

1/2*b^(13/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(5/4)/(-a*d+b*c)^3*2^(1/2)-1/2*b^(13/4)*arctan(1+b^(1
/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(5/4)/(-a*d+b*c)^3*2^(1/2)-1/64*d^(5/4)*(45*a^2*d^2-130*a*b*c*d+117*b^2*c^2)*ar
ctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(13/4)/(-a*d+b*c)^3*2^(1/2)+1/64*d^(5/4)*(45*a^2*d^2-130*a*b*c*d+117
*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(13/4)/(-a*d+b*c)^3*2^(1/2)-1/4*b^(13/4)*ln(a^(1/2)+x*b^
(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(5/4)/(-a*d+b*c)^3*2^(1/2)+1/4*b^(13/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*
b^(1/4)*2^(1/2)*x^(1/2))/a^(5/4)/(-a*d+b*c)^3*2^(1/2)+1/128*d^(5/4)*(45*a^2*d^2-130*a*b*c*d+117*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^(13/4)/(-a*d+b*c)^3*2^(1/2)-1/128*d^(5/4)*(45*a^2*d^2-130*a
*b*c*d+117*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^(13/4)/(-a*d+b*c)^3*2^(1/2)+1/16*(
-45*a^2*d^2+85*a*b*c*d-32*b^2*c^2)/a/c^3/(-a*d+b*c)^2/x^(1/2)-1/4*d/c/(-a*d+b*c)/(d*x^2+c)^2/x^(1/2)-1/16*d*(-
9*a*d+17*b*c)/c^2/(-a*d+b*c)^2/(d*x^2+c)/x^(1/2)

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {477, 483, 593, 597, 598, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {b^{13/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{13/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{13/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {b^{13/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^3}-\frac {d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} (b c-a d)^3}-\frac {d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} (b c-a d)^3}-\frac {45 a^2 d^2-85 a b c d+32 b^2 c^2}{16 a c^3 \sqrt {x} (b c-a d)^2}-\frac {d (17 b c-9 a d)}{16 c^2 \sqrt {x} \left (c+d x^2\right ) (b c-a d)^2}-\frac {d}{4 c \sqrt {x} \left (c+d x^2\right )^2 (b c-a d)} \]

[In]

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

[Out]

-1/16*(32*b^2*c^2 - 85*a*b*c*d + 45*a^2*d^2)/(a*c^3*(b*c - a*d)^2*Sqrt[x]) - d/(4*c*(b*c - a*d)*Sqrt[x]*(c + d
*x^2)^2) - (d*(17*b*c - 9*a*d))/(16*c^2*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2)) + (b^(13/4)*ArcTan[1 - (Sqrt[2]*b^(
1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (b^(13/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/
4)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d
^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) + (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d
^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) - (b^(13/4)*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*(b*c - a*d)^3) + (b^(13/4)*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*(b*c - a*d)^3) + (d^(5/4)*(117*b^2*c^2 - 130*a
*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*(b*c - a
*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*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^(13/4)*(b*c - a*d)^3)

Rule 210

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

Rule 303

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

Rule 477

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 483

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 593

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 597

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 598

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 631

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 642

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 1176

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 1179

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*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{x^2 \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) \sqrt {x} \left (c+d x^2\right )^2}+\frac {\text {Subst}\left (\int \frac {8 b c-9 a d-9 b d x^4}{x^2 \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) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {32 b^2 c^2-85 a b c d+45 a^2 d^2-5 b d (17 b c-9 a d) x^4}{x^2 \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}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (32 b^3 c^3+32 a b^2 c^2 d-85 a^2 b c d^2+45 a^3 d^3+b d \left (32 b^2 c^2-85 a b c d+45 a^2 d^2\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 a c^3 (b c-a d)^2} \\ & = -\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {32 b^4 c^3 x^2}{(b c-a d) \left (a+b x^4\right )}+\frac {a d^2 \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right ) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{16 a c^3 (b c-a d)^2} \\ & = -\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}-\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^3}+\frac {\left (d^2 \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^3 (b c-a d)^3} \\ & = -\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b^{7/2} \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^3}-\frac {b^{7/2} \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^3}-\frac {\left (d^{3/2} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^3 (b c-a d)^3}+\frac {\left (d^{3/2} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^3 (b c-a d)^3} \\ & = -\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}-\frac {b^3 \text {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 (b c-a d)^3}-\frac {b^3 \text {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 (b c-a d)^3}-\frac {b^{13/4} \text {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^{5/4} (b c-a d)^3}-\frac {b^{13/4} \text {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^{5/4} (b c-a d)^3}+\frac {\left (d \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right )\right ) \text {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^3 (b c-a d)^3}+\frac {\left (d \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right )\right ) \text {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^3 (b c-a d)^3}+\frac {\left (d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right )\right ) \text {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^{13/4} (b c-a d)^3}+\frac {\left (d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right )\right ) \text {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^{13/4} (b c-a d)^3} \\ & = -\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}-\frac {b^{13/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {b^{13/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3}-\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3}-\frac {b^{13/4} \text {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^{5/4} (b c-a d)^3}+\frac {b^{13/4} \text {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^{5/4} (b c-a d)^3}+\frac {\left (d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right )\right ) \text {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^{13/4} (b c-a d)^3}-\frac {\left (d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right )\right ) \text {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^{13/4} (b c-a d)^3} \\ & = -\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b^{13/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{13/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^3}-\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3}+\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3}-\frac {b^{13/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {b^{13/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3}-\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.60 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {1}{64} \left (-\frac {4 \left (32 b^2 c^2 \left (c+d x^2\right )^2+a^2 d^2 \left (32 c^2+81 c d x^2+45 d^2 x^4\right )-a b c d \left (64 c^2+153 c d x^2+85 d^2 x^4\right )\right )}{a c^3 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )^2}-\frac {32 \sqrt {2} b^{13/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{5/4} (-b c+a d)^3}-\frac {\sqrt {2} d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{13/4} (b c-a d)^3}-\frac {32 \sqrt {2} b^{13/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{5/4} (-b c+a d)^3}-\frac {\sqrt {2} d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{13/4} (b c-a d)^3}\right ) \]

[In]

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

[Out]

((-4*(32*b^2*c^2*(c + d*x^2)^2 + a^2*d^2*(32*c^2 + 81*c*d*x^2 + 45*d^2*x^4) - a*b*c*d*(64*c^2 + 153*c*d*x^2 +
85*d^2*x^4)))/(a*c^3*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2)^2) - (32*Sqrt[2]*b^(13/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/
(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(5/4)*(-(b*c) + a*d)^3) - (Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d +
45*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(13/4)*(b*c - a*d)^3) - (32*S
qrt[2]*b^(13/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(5/4)*(-(b*c) + a*d)^3) -
(Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c]
+ Sqrt[d]*x)])/(c^(13/4)*(b*c - a*d)^3))/64

Maple [A] (verified)

Time = 2.80 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.51

method	result	size
derivativedivides	\(\frac {b^{3} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 d^{2} \left (\frac {\left (\frac {13}{32} a^{2} d^{3}-\frac {17}{16} a b c \,d^{2}+\frac {21}{32} b^{2} c^{2} d \right ) x^{\frac {7}{2}}+\frac {c \left (17 a^{2} d^{2}-42 a b c d +25 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (\frac {45}{32} a^{2} d^{2}-\frac {65}{16} a b c d +\frac {117}{32} b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3} c^{3}}-\frac {2}{a \,c^{3} \sqrt {x}}\)	\(348\)
default	\(\frac {b^{3} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 d^{2} \left (\frac {\left (\frac {13}{32} a^{2} d^{3}-\frac {17}{16} a b c \,d^{2}+\frac {21}{32} b^{2} c^{2} d \right ) x^{\frac {7}{2}}+\frac {c \left (17 a^{2} d^{2}-42 a b c d +25 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (\frac {45}{32} a^{2} d^{2}-\frac {65}{16} a b c d +\frac {117}{32} b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3} c^{3}}-\frac {2}{a \,c^{3} \sqrt {x}}\)	\(348\)
risch	\(-\frac {2}{a \,c^{3} \sqrt {x}}-\frac {-\frac {b^{3} c^{3} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {2 a \,d^{2} \left (\frac {\left (\frac {13}{32} a^{2} d^{3}-\frac {17}{16} a b c \,d^{2}+\frac {21}{32} b^{2} c^{2} d \right ) x^{\frac {7}{2}}+\frac {c \left (17 a^{2} d^{2}-42 a b c d +25 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (\frac {45}{32} a^{2} d^{2}-\frac {65}{16} a b c d +\frac {117}{32} b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}}{a \,c^{3}}\)	\(355\)

[In]

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

[Out]

1/4*b^3/a/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*(ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/
2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))-2*d^
2/(a*d-b*c)^3/c^3*(((13/32*a^2*d^3-17/16*a*b*c*d^2+21/32*b^2*c^2*d)*x^(7/2)+1/32*c*(17*a^2*d^2-42*a*b*c*d+25*b
^2*c^2)*x^(3/2))/(d*x^2+c)^2+1/8*(45/32*a^2*d^2-65/16*a*b*c*d+117/32*b^2*c^2)/d/(c/d)^(1/4)*2^(1/2)*(ln((x-(c/
d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4
)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))-2/a/c^3/x^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 117.23 (sec) , antiderivative size = 6173, normalized size of antiderivative = 9.06 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 668, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {b^{4} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{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 {{\left (117 \, b^{2} c^{2} d^{2} - 130 \, a b c d^{3} + 45 \, a^{2} d^{4}\right )} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \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 {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{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 )}} - \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} - 85 \, a b c d^{3} + 45 \, a^{2} d^{4}\right )} x^{4} + {\left (64 \, b^{2} c^{3} d - 153 \, a b c^{2} d^{2} + 81 \, a^{2} c d^{3}\right )} x^{2}}{16 \, {\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 {9}{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 {5}{2}} + {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} \sqrt {x}\right )}} \]

[In]

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

[Out]

-1/4*b^4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(s
qrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sq
rt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x
+ sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b
^(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*(117*b^2*c^2*d^2 - 130*a*b*c*d^3 + 45
*a^2*d^4)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(
sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/s
qrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*
x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*
d^(3/4)))/(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) - 1/16*(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 - 85*a*b*c*d^3 + 45*a^2*d^4)*x^4 + (64*b^2*c^3*d - 153*a*b*c^2*d^2 + 81*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^(9/2) + 2*(a*b^2*c^6*d - 2*a^2*b*c^5*d^2 + a^3*c^4*d^3
)*x^(5/2) + (a*b^2*c^7 - 2*a^2*b*c^6*d + a^3*c^5*d^2)*sqrt(x))

Giac [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 987, normalized size of antiderivative = 1.45 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-(a*b^3)^(3/4)*b*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*sq
rt(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)^(3/4)*b*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)^(3/4)*b*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)^(3/4)*b*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*(117*(c*d^3)^(3/4)*b^2*c^2 - 130*(c*d^3)^(3/4)*a*b*c*d + 45*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/
2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^7*d - 3*sqrt(2)*a*b^2*c^6*d^2 + 3*sqrt
(2)*a^2*b*c^5*d^3 - sqrt(2)*a^3*c^4*d^4) + 1/32*(117*(c*d^3)^(3/4)*b^2*c^2 - 130*(c*d^3)^(3/4)*a*b*c*d + 45*(c
*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^7*d - 3
*sqrt(2)*a*b^2*c^6*d^2 + 3*sqrt(2)*a^2*b*c^5*d^3 - sqrt(2)*a^3*c^4*d^4) - 1/64*(117*(c*d^3)^(3/4)*b^2*c^2 - 13
0*(c*d^3)^(3/4)*a*b*c*d + 45*(c*d^3)^(3/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*
b^3*c^7*d - 3*sqrt(2)*a*b^2*c^6*d^2 + 3*sqrt(2)*a^2*b*c^5*d^3 - sqrt(2)*a^3*c^4*d^4) + 1/64*(117*(c*d^3)^(3/4)
*b^2*c^2 - 130*(c*d^3)^(3/4)*a*b*c*d + 45*(c*d^3)^(3/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c
/d))/(sqrt(2)*b^3*c^7*d - 3*sqrt(2)*a*b^2*c^6*d^2 + 3*sqrt(2)*a^2*b*c^5*d^3 - sqrt(2)*a^3*c^4*d^4) + 1/16*(21*
b*c*d^3*x^(7/2) - 13*a*d^4*x^(7/2) + 25*b*c^2*d^2*x^(3/2) - 17*a*c*d^3*x^(3/2))/((b^2*c^5 - 2*a*b*c^4*d + a^2*
c^3*d^2)*(d*x^2 + c)^2) - 2/(a*c^3*sqrt(x))

Mupad [B] (verification not implemented)

Time = 18.45 (sec) , antiderivative size = 33717, normalized size of antiderivative = 49.51 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan((a^21*c^16*d^20*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 167635494
0*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247
981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^1
1*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 132
87555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c
^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*217432
7193600i + b^17*c^20*d^4*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 16763
54940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 +
247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592
*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 -
13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b
^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*91
8653239296i + a*b^20*c^36*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676
354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10
+ 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 20132659
2*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4
- 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*
b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*1
099511627776i + a*b^16*c^19*d^5*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6
+ 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3
*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 20
1326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^2
1*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 830472192
0*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(
1/4)*10239255576576i - a^2*b^19*c^35*d*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c
^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*
b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^
12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*
b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 83
04721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^2
4*d))^(5/4)*13194139533312i - a^20*b*c^17*d^19*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760
*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063
000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12
*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721
920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*
d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*
b^11*c^24*d))^(5/4)*38654705664000i - a^2*b^15*c^18*d^6*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 -
832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9
- 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 1677
7216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3
+ 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*
b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201
326592*a*b^11*c^24*d))^(1/4)*52740124835840i + a^3*b^14*c^17*d^7*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c
^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^
4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^
25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*
c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 1328755
5072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d
^10 - 201326592*a*b^11*c^24*d))^(1/4)*109076423639040i - a^4*b^13*c^16*d^8*x^(1/2)*(-(4100625*a^8*d^13 + 18738
8721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673
350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(167772
16*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 369098752
0*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6
- 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*
b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*130225943347200i + a^5*b^12*c^15*d^9*x^(1/2)*(-(4100625*a^8*d^
13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8
+ 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^1
2)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 -
3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^
6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 110729
6256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*99593312665600i - a^6*b^11*c^14*d^10*x^(1/2)*(-(4100
625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b
^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a
^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*
c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147
584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^
9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*50139168768000i + a^7*b^10*c^13*d^11*x^(1/
2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 198916
3800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 -
47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256
*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5
+ 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b
^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*16251715584000i - a^8*b^9*c^12*d
^12*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7
- 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^
2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1
107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*
c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987
520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*3105423360000i + a^9*b^
8*c^11*d^13*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6
*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^
6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*
d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*
a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 -
3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*268738560000i +
a^3*b^18*c^34*d^2*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*
a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 24798
1500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*
b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287
555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^1
7*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*72567767
433216i - a^4*b^17*c^33*d^3*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 16
76354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^1
0 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326
592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^
4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^
8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)
*241892558110720i + a^5*b^16*c^32*d^4*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^
7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b
^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^1
2 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b
^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 830
4721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24
*d))^(5/4)*558956707577856i - a^6*b^15*c^31*d^5*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 83283876
0*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 76506
3000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^1
2*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 830472
1920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18
*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a
*b^11*c^24*d))^(5/4)*1079857857429504i + a^7*b^14*c^30*d^6*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5
- 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*
d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 1
6777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d
^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a
^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 -
201326592*a*b^11*c^24*d))^(5/4)*2407458018426880i - a^8*b^13*c^29*d^7*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*
b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a
^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^
12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3
*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13
287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c
^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*6626241184530432i + a^9*b^12*c^28*d^8*x^(1/2)*(-(4100625*a^8*d^13 +
187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1
519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(
16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 369
0987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^
19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256
*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*17102710096527360i - a^10*b^11*c^27*d^9*x^(1/2)*(-(41006
25*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^
5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^
7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c
^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 155021475
84*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9
+ 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*35386201192005632i + a^11*b^10*c^26*d^10*x^
(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 198
9163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11
- 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296
256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d
^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^
9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*57009634950512640i - a^12*b^9
*c^25*d^11*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*
c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6
*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d
^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a
^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 -
3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*7185868551251558
4i + a^13*b^8*c^24*d^12*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 167635
4940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 +
247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*
a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 -
13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^
4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*713
86451710836736i - a^14*b^7*c^23*d^13*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7
*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^
3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12
- 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^
8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304
721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*
d))^(5/4)*56058600342159360i + a^15*b^6*c^22*d^14*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838
760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765
063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a
^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304
721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^
18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592
*a*b^11*c^24*d))^(5/4)*34693912593432576i - a^16*b^5*c^21*d^15*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8
*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*
c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25
+ 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^
22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 132875550
72*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^1
0 - 201326592*a*b^11*c^24*d))^(5/4)*16752399678963712i + a^17*b^4*c^20*d^16*x^(1/2)*(-(4100625*a^8*d^13 + 1873
88721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 151967
3350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777
216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 36909875
20*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^
6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10
*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*6190641306402816i - a^18*b^3*c^19*d^17*x^(1/2)*(-(4100625*a^8
*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*
d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*
d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^
2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6
*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 110
7296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*1693591504158720i + a^19*b^2*c^18*d^18*x^(1/2)*(-
(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*
a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385
000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*
b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 155
02147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^
16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*323711685099520i)/(373669453125*a^16*
d^21 + 1679412953088*b^16*c^16*d^5 - 559804317696*a*b^15*c^15*d^6 + 1440180338688*a^2*b^14*c^14*d^7 + 10694583
91040*a^3*b^13*c^13*d^8 + 1465657589760*a^4*b^12*c^12*d^9 - 298323546568125*a^5*b^11*c^11*d^10 + 1436096821015
175*a^6*b^10*c^10*d^11 - 3384298041916875*a^7*b^9*c^9*d^12 + 5036592389645625*a^8*b^8*c^8*d^13 - 5207312367681
250*a^9*b^7*c^7*d^14 + 3905254606443750*a^10*b^6*c^6*d^15 - 2160273093693750*a^11*b^5*c^5*d^16 + 8795071393312
50*a^12*b^4*c^4*d^17 - 257930235140625*a^13*b^3*c^3*d^18 + 51862552171875*a^14*b^2*c^2*d^19 - 6435418359375*a^
15*b*c*d^20))*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d
^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*
c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 +
1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^
7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 36909
87520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*2i + atan((a^6*b^20*c
^25*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*
b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7
+ 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*33554432i
+ a^14*b^8*d^13*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^
2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12
*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1
/4)*8201250i + a^26*c^5*d^20*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*
b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6
- 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b
*c*d^11))^(5/4)*66355200i + a^5*b^17*c^9*d^4*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^
11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a
^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d
^10 - 192*a^16*b*c*d^11))^(1/4)*28035072i + a^6*b^16*c^8*d^5*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 -
192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c
^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 105
6*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*312477282i - a^7*b^15*c^7*d^6*x^(1/2)*(-b^13/(16*a^17*d^12 + 1
6*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 -
12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*
b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*1609500880i + a^8*b^14*c^6*d^7*x^(1/2)*(-b^13
/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920
*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^
4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*3328748280i - a^9*b^13*c^5*
d^8*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*
b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7
+ 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*397418040
0i + a^10*b^12*c^4*d^9*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c
^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 1267
2*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^1
1))^(1/4)*3039346700i - a^11*b^11*c^3*d^10*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11
*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^1
1*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^1
0 - 192*a^16*b*c*d^11))^(1/4)*1530126000i + a^12*b^10*c^2*d^11*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12
- 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7
*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1
056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*495963000i + a^8*b^18*c^23*d^2*x^(1/2)*(-b^13/(16*a^17*d^12
+ 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^
4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^
14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*2214592512i - a^9*b^17*c^22*d^3*x^(1/2)*(-
b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 +
7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^
4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*7381975040i + a^10*b^16
*c^21*d^4*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 352
0*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^
5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*170
58004992i - a^11*b^15*c^20*d^5*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^
7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^
6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16
*b*c*d^11))^(5/4)*32954646528i + a^12*b^14*c^19*d^6*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*
b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 +
14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^
2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*73469788160i - a^13*b^13*c^18*d^7*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5
*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 1267
2*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c
^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*202216833024i + a^14*b^12*c^17*d^8*x^(1/2)*(-b^13/
(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*
a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4
*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*521933291520i - a^15*b^11*c^
16*d^9*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a
^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d
^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*107990
1159424i + a^16*b^10*c^15*d^10*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^
7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^
6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16
*b*c*d^11))^(5/4)*1739795988480i - a^17*b^9*c^14*d^11*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^
6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5
+ 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*
b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*2192953049088i + a^18*b^8*c^13*d^12*x^(1/2)*(-b^13/(16*a^17*d^12 + 16
*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 -
12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b
^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*2178541617152i - a^19*b^7*c^12*d^13*x^(1/2)*(-
b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 +
7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^
4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*1710772715520i + a^20*b
^6*c^11*d^14*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 -
3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5
*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*
1058774188032i - a^21*b^5*c^10*d^15*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 10
56*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c
^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192
*a^16*b*c*d^11))^(5/4)*511242665984i + a^22*b^4*c^9*d^16*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192
*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d
^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^
15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*188923379712i - a^23*b^3*c^8*d^17*x^(1/2)*(-b^13/(16*a^17*d^12 + 1
6*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 -
12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*
b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*51684311040i + a^24*b^2*c^7*d^18*x^(1/2)*(-b^
13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 79
20*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*
c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(5/4)*9878896640i - a^13*b^9*c*
d^12*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8
*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7
+ 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*94770000
i - a^7*b^19*c^24*d*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10
*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a
^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))
^(5/4)*402653184i - a^25*b*c^6*d^19*x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 10
56*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c
^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192
*a^16*b*c*d^11))^(5/4)*1179648000i)/(1048576*b^21*c^10 + 4100625*a^10*b^11*d^10 - 35083125*a^9*b^12*c*d^9 + 62
91456*a^2*b^19*c^8*d^2 + 10485760*a^3*b^18*c^7*d^3 + 15728640*a^4*b^17*c^6*d^4 - 165368625*a^5*b^16*c^5*d^5 +
300032725*a^6*b^15*c^4*d^6 - 264422250*a^7*b^14*c^3*d^7 + 130430250*a^8*b^13*c^2*d^8 + 3145728*a*b^20*c^9*d))*
(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3
+ 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*
b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*2i + 2*atan(((x^(1/2)
*(15412443824768679936*a^11*b^35*c^52*d^8 - 43988011059341426688*a^12*b^34*c^51*d^9 - 1887306913007783641088*a
^13*b^33*c^50*d^10 + 24240068121369040125952*a^14*b^32*c^49*d^11 - 155426886723407276146688*a^15*b^31*c^48*d^1
2 + 661969678817672344633344*a^16*b^30*c^47*d^13 - 2072522435259453904257024*a^17*b^29*c^46*d^14 + 50256206139
85914706722816*a^18*b^28*c^45*d^15 - 9739734806850605210927104*a^19*b^27*c^44*d^16 + 1539558713198738688022937
6*a^20*b^26*c^43*d^17 - 20118464109716534770794496*a^21*b^25*c^42*d^18 + 21925688980693704834547712*a^22*b^24*
c^41*d^19 - 20031833528060137877536768*a^23*b^23*c^40*d^20 + 15375655212110710153674752*a^24*b^22*c^39*d^21 -
9908539789204785922572288*a^25*b^21*c^38*d^22 + 5342151752610266235273216*a^26*b^20*c^37*d^23 - 23933617400483
38255872000*a^27*b^19*c^36*d^24 + 881440288329629213655040*a^28*b^18*c^35*d^25 - 262552769009553086873600*a^29
*b^17*c^34*d^26 + 61737289250332318105600*a^30*b^16*c^33*d^27 - 11040709176673173504000*a^31*b^15*c^32*d^28 +
1412353884520710144000*a^32*b^14*c^31*d^29 - 115221946643251200000*a^33*b^13*c^30*d^30 + 4508684868648960000*a
^34*b^12*c^29*d^31) - (-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 -
3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^
5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(3/4)
*(x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^
9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 +
7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*(1844674407
3709551616*a^12*b^38*c^68*d^4 - 479615345916448342016*a^13*b^37*c^67*d^5 + 5995191823955604275200*a^14*b^36*c^
66*d^6 - 47961534591644834201600*a^15*b^35*c^65*d^7 + 276025423003154095538176*a^16*b^34*c^64*d^8 - 1220386399
802376518107136*a^17*b^33*c^63*d^9 + 4341880678999181785825280*a^18*b^32*c^62*d^10 - 1296658354285206707372032
0*a^19*b^31*c^61*d^11 + 34096448785847177707520000*a^20*b^30*c^60*d^12 - 83405832293258492567879680*a^21*b^29*
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6127977833519707586560*a^26*b^24*c^54*d^18 - 6855830429358878767993323520*a^27*b^23*c^53*d^19 + 10053745593205
095687740456960*a^28*b^22*c^52*d^20 - 12966109559707844614920601600*a^29*b^21*c^51*d^21 + 14704312650164876038
740377600*a^30*b^20*c^50*d^22 - 14666481047173052905774120960*a^31*b^19*c^49*d^23 + 12864666662378251575193763
840*a^32*b^18*c^48*d^24 - 9915254214005035782929121280*a^33*b^17*c^47*d^25 + 6703228082495101562834124800*a^34
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44*d^28 - 910688569282163512962973696*a^37*b^13*c^43*d^29 + 349132643065901184834338816*a^38*b^12*c^42*d^30 -
113859137485172722840371200*a^39*b^11*c^41*d^31 + 31155202813126960689971200*a^40*b^10*c^40*d^32 - 70198492619
36667709669376*a^41*b^9*c^39*d^33 + 1268449805817592472928256*a^42*b^8*c^38*d^34 - 176741065216378693222400*a^
43*b^7*c^37*d^35 + 17829840996752341073920*a^44*b^6*c^36*d^36 - 1159226544085165670400*a^45*b^5*c^35*d^37 + 36
479156981701017600*a^46*b^4*c^34*d^38)*1i - 9223372036854775808*a^11*b^38*c^65*d^4 + 212137556847659843584*a^1
2*b^37*c^64*d^5 - 2333513125324258279424*a^13*b^36*c^63*d^6 + 16334591877269807955968*a^14*b^35*c^62*d^7 - 816
72959386349039779840*a^15*b^34*c^61*d^8 + 310808059650000835051520*a^16*b^33*c^60*d^9 - 9429431718604071292108
80*a^17*b^32*c^59*d^10 + 2411982412523930344488960*a^18*b^31*c^58*d^11 - 5753067372685321201254400*a^19*b^30*c
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31295543449898524428206080*a^22*b^27*c^54*d^15 - 365631810199400875032576000*a^23*b^26*c^53*d^16 + 88861501991
6519951743057920*a^24*b^25*c^52*d^17 - 1859065088581792734285660160*a^25*b^24*c^51*d^18 + 33497204972588690635
43685120*a^26*b^23*c^50*d^19 - 5220292063815211666322227200*a^27*b^22*c^49*d^20 + 7067608268064143449134202880
*a^28*b^21*c^48*d^21 - 8342222871228251802477527040*a^29*b^20*c^47*d^22 + 8605396720616721741816791040*a^30*b^
19*c^46*d^23 - 7767500088979055902405427200*a^31*b^18*c^45*d^24 + 6135496566696171932913500160*a^32*b^17*c^44*
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334575022384247271808040960*a^35*b^14*c^41*d^28 + 603343239457650202481000448*a^36*b^13*c^40*d^29 - 2339671236
41003163353350144*a^37*b^12*c^39*d^30 + 77049527429528415176228864*a^38*b^11*c^38*d^31 - 212587498504804503943
90528*a^39*b^10*c^37*d^32 + 4823899363819901975265280*a^40*b^9*c^36*d^33 - 876898617974708020183040*a^41*b^8*c
^35*d^34 + 122811796684756379238400*a^42*b^7*c^34*d^35 - 12444332416601319014400*a^43*b^6*c^33*d^36 + 81223122
9670686720000*a^44*b^5*c^32*d^37 - 25649407252758528000*a^45*b^4*c^31*d^38)*1i)*(-b^13/(16*a^17*d^12 + 16*a^5*
b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672
*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^
3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4) + (x^(1/2)*(15412443824768679936*a^11*b^35*c^52*d^8
- 43988011059341426688*a^12*b^34*c^51*d^9 - 1887306913007783641088*a^13*b^33*c^50*d^10 + 24240068121369040125
952*a^14*b^32*c^49*d^11 - 155426886723407276146688*a^15*b^31*c^48*d^12 + 661969678817672344633344*a^16*b^30*c^
47*d^13 - 2072522435259453904257024*a^17*b^29*c^46*d^14 + 5025620613985914706722816*a^18*b^28*c^45*d^15 - 9739
734806850605210927104*a^19*b^27*c^44*d^16 + 15395587131987386880229376*a^20*b^26*c^43*d^17 - 20118464109716534
770794496*a^21*b^25*c^42*d^18 + 21925688980693704834547712*a^22*b^24*c^41*d^19 - 20031833528060137877536768*a^
23*b^23*c^40*d^20 + 15375655212110710153674752*a^24*b^22*c^39*d^21 - 9908539789204785922572288*a^25*b^21*c^38*
d^22 + 5342151752610266235273216*a^26*b^20*c^37*d^23 - 2393361740048338255872000*a^27*b^19*c^36*d^24 + 8814402
88329629213655040*a^28*b^18*c^35*d^25 - 262552769009553086873600*a^29*b^17*c^34*d^26 + 61737289250332318105600
*a^30*b^16*c^33*d^27 - 11040709176673173504000*a^31*b^15*c^32*d^28 + 1412353884520710144000*a^32*b^14*c^31*d^2
9 - 115221946643251200000*a^33*b^13*c^30*d^30 + 4508684868648960000*a^34*b^12*c^29*d^31) - (-b^13/(16*a^17*d^1
2 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*
d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*
a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(3/4)*(x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b
^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*
a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3
*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*(18446744073709551616*a^12*b^38*c^68*d^4 - 479615345
916448342016*a^13*b^37*c^67*d^5 + 5995191823955604275200*a^14*b^36*c^66*d^6 - 47961534591644834201600*a^15*b^3
5*c^65*d^7 + 276025423003154095538176*a^16*b^34*c^64*d^8 - 1220386399802376518107136*a^17*b^33*c^63*d^9 + 4341
880678999181785825280*a^18*b^32*c^62*d^10 - 12966583542852067073720320*a^19*b^31*c^61*d^11 + 34096448785847177
707520000*a^20*b^30*c^60*d^12 - 83405832293258492567879680*a^21*b^29*c^59*d^13 + 198753207063509910306160640*a
^22*b^28*c^58*d^14 - 466650996519299420897935360*a^23*b^27*c^57*d^15 + 1052056219870198056039219200*a^24*b^26*
c^56*d^16 - 2194458800584304697435750400*a^25*b^25*c^55*d^17 + 4119286127977833519707586560*a^26*b^24*c^54*d^1
8 - 6855830429358878767993323520*a^27*b^23*c^53*d^19 + 10053745593205095687740456960*a^28*b^22*c^52*d^20 - 129
66109559707844614920601600*a^29*b^21*c^51*d^21 + 14704312650164876038740377600*a^30*b^20*c^50*d^22 - 146664810
47173052905774120960*a^31*b^19*c^49*d^23 + 12864666662378251575193763840*a^32*b^18*c^48*d^24 - 991525421400503
5782929121280*a^33*b^17*c^47*d^25 + 6703228082495101562834124800*a^34*b^16*c^46*d^26 - 39637238143982610587582
46400*a^35*b^15*c^45*d^27 + 2041552487767277748019527680*a^36*b^14*c^44*d^28 - 910688569282163512962973696*a^3
7*b^13*c^43*d^29 + 349132643065901184834338816*a^38*b^12*c^42*d^30 - 113859137485172722840371200*a^39*b^11*c^4
1*d^31 + 31155202813126960689971200*a^40*b^10*c^40*d^32 - 7019849261936667709669376*a^41*b^9*c^39*d^33 + 12684
49805817592472928256*a^42*b^8*c^38*d^34 - 176741065216378693222400*a^43*b^7*c^37*d^35 + 1782984099675234107392
0*a^44*b^6*c^36*d^36 - 1159226544085165670400*a^45*b^5*c^35*d^37 + 36479156981701017600*a^46*b^4*c^34*d^38)*1i
+ 9223372036854775808*a^11*b^38*c^65*d^4 - 212137556847659843584*a^12*b^37*c^64*d^5 + 2333513125324258279424*
a^13*b^36*c^63*d^6 - 16334591877269807955968*a^14*b^35*c^62*d^7 + 81672959386349039779840*a^15*b^34*c^61*d^8 -
310808059650000835051520*a^16*b^33*c^60*d^9 + 942943171860407129210880*a^17*b^32*c^59*d^10 - 2411982412523930
344488960*a^18*b^31*c^58*d^11 + 5753067372685321201254400*a^19*b^30*c^57*d^12 - 14786194741349386435952640*a^2
0*b^29*c^56*d^13 + 43374839389541821883351040*a^21*b^28*c^55*d^14 - 131295543449898524428206080*a^22*b^27*c^54
*d^15 + 365631810199400875032576000*a^23*b^26*c^53*d^16 - 888615019916519951743057920*a^24*b^25*c^52*d^17 + 18
59065088581792734285660160*a^25*b^24*c^51*d^18 - 3349720497258869063543685120*a^26*b^23*c^50*d^19 + 5220292063
815211666322227200*a^27*b^22*c^49*d^20 - 7067608268064143449134202880*a^28*b^21*c^48*d^21 + 834222287122825180
2477527040*a^29*b^20*c^47*d^22 - 8605396720616721741816791040*a^30*b^19*c^46*d^23 + 77675000889790559024054272
00*a^31*b^18*c^45*d^24 - 6135496566696171932913500160*a^32*b^17*c^44*d^25 + 4236422046382466798589050880*a^33*
b^16*c^43*d^26 - 2550980661067485441771438080*a^34*b^15*c^42*d^27 + 1334575022384247271808040960*a^35*b^14*c^4
1*d^28 - 603343239457650202481000448*a^36*b^13*c^40*d^29 + 233967123641003163353350144*a^37*b^12*c^39*d^30 - 7
7049527429528415176228864*a^38*b^11*c^38*d^31 + 21258749850480450394390528*a^39*b^10*c^37*d^32 - 4823899363819
901975265280*a^40*b^9*c^36*d^33 + 876898617974708020183040*a^41*b^8*c^35*d^34 - 122811796684756379238400*a^42*
b^7*c^34*d^35 + 12444332416601319014400*a^43*b^6*c^33*d^36 - 812231229670686720000*a^44*b^5*c^32*d^37 + 256494
07252758528000*a^45*b^4*c^31*d^38)*1i)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^
7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^
6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16
*b*c*d^11))^(1/4))/((x^(1/2)*(15412443824768679936*a^11*b^35*c^52*d^8 - 43988011059341426688*a^12*b^34*c^51*d^
9 - 1887306913007783641088*a^13*b^33*c^50*d^10 + 24240068121369040125952*a^14*b^32*c^49*d^11 - 155426886723407
276146688*a^15*b^31*c^48*d^12 + 661969678817672344633344*a^16*b^30*c^47*d^13 - 2072522435259453904257024*a^17*
b^29*c^46*d^14 + 5025620613985914706722816*a^18*b^28*c^45*d^15 - 9739734806850605210927104*a^19*b^27*c^44*d^16
+ 15395587131987386880229376*a^20*b^26*c^43*d^17 - 20118464109716534770794496*a^21*b^25*c^42*d^18 + 219256889
80693704834547712*a^22*b^24*c^41*d^19 - 20031833528060137877536768*a^23*b^23*c^40*d^20 + 153756552121107101536
74752*a^24*b^22*c^39*d^21 - 9908539789204785922572288*a^25*b^21*c^38*d^22 + 5342151752610266235273216*a^26*b^2
0*c^37*d^23 - 2393361740048338255872000*a^27*b^19*c^36*d^24 + 881440288329629213655040*a^28*b^18*c^35*d^25 - 2
62552769009553086873600*a^29*b^17*c^34*d^26 + 61737289250332318105600*a^30*b^16*c^33*d^27 - 110407091766731735
04000*a^31*b^15*c^32*d^28 + 1412353884520710144000*a^32*b^14*c^31*d^29 - 115221946643251200000*a^33*b^13*c^30*
d^30 + 4508684868648960000*a^34*b^12*c^29*d^31) - (-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*
d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11
*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10
- 192*a^16*b*c*d^11))^(3/4)*(x^(1/2)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7
*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6
- 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*
b*c*d^11))^(1/4)*(18446744073709551616*a^12*b^38*c^68*d^4 - 479615345916448342016*a^13*b^37*c^67*d^5 + 5995191
823955604275200*a^14*b^36*c^66*d^6 - 47961534591644834201600*a^15*b^35*c^65*d^7 + 276025423003154095538176*a^1
6*b^34*c^64*d^8 - 1220386399802376518107136*a^17*b^33*c^63*d^9 + 4341880678999181785825280*a^18*b^32*c^62*d^10
- 12966583542852067073720320*a^19*b^31*c^61*d^11 + 34096448785847177707520000*a^20*b^30*c^60*d^12 - 834058322
93258492567879680*a^21*b^29*c^59*d^13 + 198753207063509910306160640*a^22*b^28*c^58*d^14 - 46665099651929942089
7935360*a^23*b^27*c^57*d^15 + 1052056219870198056039219200*a^24*b^26*c^56*d^16 - 2194458800584304697435750400*
a^25*b^25*c^55*d^17 + 4119286127977833519707586560*a^26*b^24*c^54*d^18 - 6855830429358878767993323520*a^27*b^2
3*c^53*d^19 + 10053745593205095687740456960*a^28*b^22*c^52*d^20 - 12966109559707844614920601600*a^29*b^21*c^51
*d^21 + 14704312650164876038740377600*a^30*b^20*c^50*d^22 - 14666481047173052905774120960*a^31*b^19*c^49*d^23
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28082495101562834124800*a^34*b^16*c^46*d^26 - 3963723814398261058758246400*a^35*b^15*c^45*d^27 + 2041552487767
277748019527680*a^36*b^14*c^44*d^28 - 910688569282163512962973696*a^37*b^13*c^43*d^29 + 3491326430659011848343
38816*a^38*b^12*c^42*d^30 - 113859137485172722840371200*a^39*b^11*c^41*d^31 + 31155202813126960689971200*a^40*
b^10*c^40*d^32 - 7019849261936667709669376*a^41*b^9*c^39*d^33 + 1268449805817592472928256*a^42*b^8*c^38*d^34 -
176741065216378693222400*a^43*b^7*c^37*d^35 + 17829840996752341073920*a^44*b^6*c^36*d^36 - 115922654408516567
0400*a^45*b^5*c^35*d^37 + 36479156981701017600*a^46*b^4*c^34*d^38)*1i - 9223372036854775808*a^11*b^38*c^65*d^4
+ 212137556847659843584*a^12*b^37*c^64*d^5 - 2333513125324258279424*a^13*b^36*c^63*d^6 + 16334591877269807955
968*a^14*b^35*c^62*d^7 - 81672959386349039779840*a^15*b^34*c^61*d^8 + 310808059650000835051520*a^16*b^33*c^60*
d^9 - 942943171860407129210880*a^17*b^32*c^59*d^10 + 2411982412523930344488960*a^18*b^31*c^58*d^11 - 575306737
2685321201254400*a^19*b^30*c^57*d^12 + 14786194741349386435952640*a^20*b^29*c^56*d^13 - 4337483938954182188335
1040*a^21*b^28*c^55*d^14 + 131295543449898524428206080*a^22*b^27*c^54*d^15 - 365631810199400875032576000*a^23*
b^26*c^53*d^16 + 888615019916519951743057920*a^24*b^25*c^52*d^17 - 1859065088581792734285660160*a^25*b^24*c^51
*d^18 + 3349720497258869063543685120*a^26*b^23*c^50*d^19 - 5220292063815211666322227200*a^27*b^22*c^49*d^20 +
7067608268064143449134202880*a^28*b^21*c^48*d^21 - 8342222871228251802477527040*a^29*b^20*c^47*d^22 + 86053967
20616721741816791040*a^30*b^19*c^46*d^23 - 7767500088979055902405427200*a^31*b^18*c^45*d^24 + 6135496566696171
932913500160*a^32*b^17*c^44*d^25 - 4236422046382466798589050880*a^33*b^16*c^43*d^26 + 255098066106748544177143
8080*a^34*b^15*c^42*d^27 - 1334575022384247271808040960*a^35*b^14*c^41*d^28 + 603343239457650202481000448*a^36
*b^13*c^40*d^29 - 233967123641003163353350144*a^37*b^12*c^39*d^30 + 77049527429528415176228864*a^38*b^11*c^38*
d^31 - 21258749850480450394390528*a^39*b^10*c^37*d^32 + 4823899363819901975265280*a^40*b^9*c^36*d^33 - 8768986
17974708020183040*a^41*b^8*c^35*d^34 + 122811796684756379238400*a^42*b^7*c^34*d^35 - 12444332416601319014400*a
^43*b^6*c^33*d^36 + 812231229670686720000*a^44*b^5*c^32*d^37 - 25649407252758528000*a^45*b^4*c^31*d^38)*1i)*(-
b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 +
7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^
4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*1i - (x^(1/2)*(15412443
824768679936*a^11*b^35*c^52*d^8 - 43988011059341426688*a^12*b^34*c^51*d^9 - 1887306913007783641088*a^13*b^33*c
^50*d^10 + 24240068121369040125952*a^14*b^32*c^49*d^11 - 155426886723407276146688*a^15*b^31*c^48*d^12 + 661969
678817672344633344*a^16*b^30*c^47*d^13 - 2072522435259453904257024*a^17*b^29*c^46*d^14 + 502562061398591470672
2816*a^18*b^28*c^45*d^15 - 9739734806850605210927104*a^19*b^27*c^44*d^16 + 15395587131987386880229376*a^20*b^2
6*c^43*d^17 - 20118464109716534770794496*a^21*b^25*c^42*d^18 + 21925688980693704834547712*a^22*b^24*c^41*d^19
- 20031833528060137877536768*a^23*b^23*c^40*d^20 + 15375655212110710153674752*a^24*b^22*c^39*d^21 - 9908539789
204785922572288*a^25*b^21*c^38*d^22 + 5342151752610266235273216*a^26*b^20*c^37*d^23 - 239336174004833825587200
0*a^27*b^19*c^36*d^24 + 881440288329629213655040*a^28*b^18*c^35*d^25 - 262552769009553086873600*a^29*b^17*c^34
*d^26 + 61737289250332318105600*a^30*b^16*c^33*d^27 - 11040709176673173504000*a^31*b^15*c^32*d^28 + 1412353884
520710144000*a^32*b^14*c^31*d^29 - 115221946643251200000*a^33*b^13*c^30*d^30 + 4508684868648960000*a^34*b^12*c
^29*d^31) - (-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*
b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7
+ 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(3/4)*(x^(1/2)*
(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3
+ 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*
b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*(18446744073709551616
*a^12*b^38*c^68*d^4 - 479615345916448342016*a^13*b^37*c^67*d^5 + 5995191823955604275200*a^14*b^36*c^66*d^6 - 4
7961534591644834201600*a^15*b^35*c^65*d^7 + 276025423003154095538176*a^16*b^34*c^64*d^8 - 12203863998023765181
07136*a^17*b^33*c^63*d^9 + 4341880678999181785825280*a^18*b^32*c^62*d^10 - 12966583542852067073720320*a^19*b^3
1*c^61*d^11 + 34096448785847177707520000*a^20*b^30*c^60*d^12 - 83405832293258492567879680*a^21*b^29*c^59*d^13
+ 198753207063509910306160640*a^22*b^28*c^58*d^14 - 466650996519299420897935360*a^23*b^27*c^57*d^15 + 10520562
19870198056039219200*a^24*b^26*c^56*d^16 - 2194458800584304697435750400*a^25*b^25*c^55*d^17 + 4119286127977833
519707586560*a^26*b^24*c^54*d^18 - 6855830429358878767993323520*a^27*b^23*c^53*d^19 + 100537455932050956877404
56960*a^28*b^22*c^52*d^20 - 12966109559707844614920601600*a^29*b^21*c^51*d^21 + 14704312650164876038740377600*
a^30*b^20*c^50*d^22 - 14666481047173052905774120960*a^31*b^19*c^49*d^23 + 12864666662378251575193763840*a^32*b
^18*c^48*d^24 - 9915254214005035782929121280*a^33*b^17*c^47*d^25 + 6703228082495101562834124800*a^34*b^16*c^46
*d^26 - 3963723814398261058758246400*a^35*b^15*c^45*d^27 + 2041552487767277748019527680*a^36*b^14*c^44*d^28 -
910688569282163512962973696*a^37*b^13*c^43*d^29 + 349132643065901184834338816*a^38*b^12*c^42*d^30 - 1138591374
85172722840371200*a^39*b^11*c^41*d^31 + 31155202813126960689971200*a^40*b^10*c^40*d^32 - 701984926193666770966
9376*a^41*b^9*c^39*d^33 + 1268449805817592472928256*a^42*b^8*c^38*d^34 - 176741065216378693222400*a^43*b^7*c^3
7*d^35 + 17829840996752341073920*a^44*b^6*c^36*d^36 - 1159226544085165670400*a^45*b^5*c^35*d^37 + 364791569817
01017600*a^46*b^4*c^34*d^38)*1i + 9223372036854775808*a^11*b^38*c^65*d^4 - 212137556847659843584*a^12*b^37*c^6
4*d^5 + 2333513125324258279424*a^13*b^36*c^63*d^6 - 16334591877269807955968*a^14*b^35*c^62*d^7 + 8167295938634
9039779840*a^15*b^34*c^61*d^8 - 310808059650000835051520*a^16*b^33*c^60*d^9 + 942943171860407129210880*a^17*b^
32*c^59*d^10 - 2411982412523930344488960*a^18*b^31*c^58*d^11 + 5753067372685321201254400*a^19*b^30*c^57*d^12 -
14786194741349386435952640*a^20*b^29*c^56*d^13 + 43374839389541821883351040*a^21*b^28*c^55*d^14 - 13129554344
9898524428206080*a^22*b^27*c^54*d^15 + 365631810199400875032576000*a^23*b^26*c^53*d^16 - 888615019916519951743
057920*a^24*b^25*c^52*d^17 + 1859065088581792734285660160*a^25*b^24*c^51*d^18 - 3349720497258869063543685120*a
^26*b^23*c^50*d^19 + 5220292063815211666322227200*a^27*b^22*c^49*d^20 - 7067608268064143449134202880*a^28*b^21
*c^48*d^21 + 8342222871228251802477527040*a^29*b^20*c^47*d^22 - 8605396720616721741816791040*a^30*b^19*c^46*d^
23 + 7767500088979055902405427200*a^31*b^18*c^45*d^24 - 6135496566696171932913500160*a^32*b^17*c^44*d^25 + 423
6422046382466798589050880*a^33*b^16*c^43*d^26 - 2550980661067485441771438080*a^34*b^15*c^42*d^27 + 13345750223
84247271808040960*a^35*b^14*c^41*d^28 - 603343239457650202481000448*a^36*b^13*c^40*d^29 + 23396712364100316335
3350144*a^37*b^12*c^39*d^30 - 77049527429528415176228864*a^38*b^11*c^38*d^31 + 21258749850480450394390528*a^39
*b^10*c^37*d^32 - 4823899363819901975265280*a^40*b^9*c^36*d^33 + 876898617974708020183040*a^41*b^8*c^35*d^34 -
122811796684756379238400*a^42*b^7*c^34*d^35 + 12444332416601319014400*a^43*b^6*c^33*d^36 - 812231229670686720
000*a^44*b^5*c^32*d^37 + 25649407252758528000*a^45*b^4*c^31*d^38)*1i)*(-b^13/(16*a^17*d^12 + 16*a^5*b^12*c^12
- 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*b^8*c^8*d^4 - 12672*a^10*b^7*
c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8 - 3520*a^14*b^3*c^3*d^9 + 10
56*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4)*1i - 149684329919262228480*a^11*b^34*c^48*d^9 + 2374404370680
061624320*a^12*b^33*c^47*d^10 - 17808722627439624192000*a^13*b^32*c^46*d^11 + 83960295795175519682560*a^14*b^3
1*c^45*d^12 - 278998813302985850880000*a^15*b^30*c^44*d^13 + 694438771802419400540160*a^16*b^29*c^43*d^14 - 13
42951722708271932375040*a^17*b^28*c^42*d^15 + 2065391322938120916172800*a^18*b^27*c^41*d^16 - 2564218746215699
966853120*a^19*b^26*c^40*d^17 + 2593338871410901332787200*a^20*b^25*c^39*d^18 - 2146065846150812380692480*a^21
*b^24*c^38*d^19 + 1453625441569727022366720*a^22*b^23*c^37*d^20 - 802881124954933087436800*a^23*b^22*c^36*d^21
+ 358581985606139180482560*a^24*b^21*c^35*d^22 - 127660361818125316915200*a^25*b^20*c^34*d^23 + 3541741940575
0750412800*a^26*b^19*c^33*d^24 - 7386837561454362624000*a^27*b^18*c^32*d^25 + 1090533977896255488000*a^28*b^17
*c^31*d^26 - 101695892037304320000*a^29*b^16*c^30*d^27 + 4508684868648960000*a^30*b^15*c^29*d^28))*(-b^13/(16*
a^17*d^12 + 16*a^5*b^12*c^12 - 192*a^6*b^11*c^11*d + 1056*a^7*b^10*c^10*d^2 - 3520*a^8*b^9*c^9*d^3 + 7920*a^9*
b^8*c^8*d^4 - 12672*a^10*b^7*c^7*d^5 + 14784*a^11*b^6*c^6*d^6 - 12672*a^12*b^5*c^5*d^7 + 7920*a^13*b^4*c^4*d^8
- 3520*a^14*b^3*c^3*d^9 + 1056*a^15*b^2*c^2*d^10 - 192*a^16*b*c*d^11))^(1/4) - (2/(a*c) + (x^2*(81*a^2*d^3 +
64*b^2*c^2*d - 153*a*b*c*d^2))/(16*a*c*(b^2*c^3 + a^2*c*d^2 - 2*a*b*c^2*d)) + (d^2*x^4*(45*a^2*d^2 + 32*b^2*c^
2 - 85*a*b*c*d))/(16*a*c^2*(b^2*c^3 + a^2*c*d^2 - 2*a*b*c^2*d)))/(c^2*x^(1/2) + d^2*x^(9/2) + 2*c*d*x^(5/2)) +
2*atan(((-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 -
1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d
^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107
296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^2
0*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520
*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*(x^(1/2)*(1541244382476867
9936*a^11*b^35*c^52*d^8 - 43988011059341426688*a^12*b^34*c^51*d^9 - 1887306913007783641088*a^13*b^33*c^50*d^10
+ 24240068121369040125952*a^14*b^32*c^49*d^11 - 155426886723407276146688*a^15*b^31*c^48*d^12 + 66196967881767
2344633344*a^16*b^30*c^47*d^13 - 2072522435259453904257024*a^17*b^29*c^46*d^14 + 5025620613985914706722816*a^1
8*b^28*c^45*d^15 - 9739734806850605210927104*a^19*b^27*c^44*d^16 + 15395587131987386880229376*a^20*b^26*c^43*d
^17 - 20118464109716534770794496*a^21*b^25*c^42*d^18 + 21925688980693704834547712*a^22*b^24*c^41*d^19 - 200318
33528060137877536768*a^23*b^23*c^40*d^20 + 15375655212110710153674752*a^24*b^22*c^39*d^21 - 990853978920478592
2572288*a^25*b^21*c^38*d^22 + 5342151752610266235273216*a^26*b^20*c^37*d^23 - 2393361740048338255872000*a^27*b
^19*c^36*d^24 + 881440288329629213655040*a^28*b^18*c^35*d^25 - 262552769009553086873600*a^29*b^17*c^34*d^26 +
61737289250332318105600*a^30*b^16*c^33*d^27 - 11040709176673173504000*a^31*b^15*c^32*d^28 + 141235388452071014
4000*a^32*b^14*c^31*d^29 - 115221946643251200000*a^33*b^13*c^30*d^30 + 4508684868648960000*a^34*b^12*c^29*d^31
) - (-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 19891
63800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 -
47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 110729625
6*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5
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b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(3/4)*(x^(1/2)*(-(4100625*a^8*d^13 +
187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 15
19673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(1
6777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690
987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^1
9*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*
a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*(18446744073709551616*a^12*b^38*c^68*d^4 - 47961534591644
8342016*a^13*b^37*c^67*d^5 + 5995191823955604275200*a^14*b^36*c^66*d^6 - 47961534591644834201600*a^15*b^35*c^6
5*d^7 + 276025423003154095538176*a^16*b^34*c^64*d^8 - 1220386399802376518107136*a^17*b^33*c^63*d^9 + 434188067
8999181785825280*a^18*b^32*c^62*d^10 - 12966583542852067073720320*a^19*b^31*c^61*d^11 + 3409644878584717770752
0000*a^20*b^30*c^60*d^12 - 83405832293258492567879680*a^21*b^29*c^59*d^13 + 198753207063509910306160640*a^22*b
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855830429358878767993323520*a^27*b^23*c^53*d^19 + 10053745593205095687740456960*a^28*b^22*c^52*d^20 - 12966109
559707844614920601600*a^29*b^21*c^51*d^21 + 14704312650164876038740377600*a^30*b^20*c^50*d^22 - 14666481047173
052905774120960*a^31*b^19*c^49*d^23 + 12864666662378251575193763840*a^32*b^18*c^48*d^24 - 99152542140050357829
29121280*a^33*b^17*c^47*d^25 + 6703228082495101562834124800*a^34*b^16*c^46*d^26 - 3963723814398261058758246400
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3*c^43*d^29 + 349132643065901184834338816*a^38*b^12*c^42*d^30 - 113859137485172722840371200*a^39*b^11*c^41*d^3
1 + 31155202813126960689971200*a^40*b^10*c^40*d^32 - 7019849261936667709669376*a^41*b^9*c^39*d^33 + 1268449805
817592472928256*a^42*b^8*c^38*d^34 - 176741065216378693222400*a^43*b^7*c^37*d^35 + 17829840996752341073920*a^4
4*b^6*c^36*d^36 - 1159226544085165670400*a^45*b^5*c^35*d^37 + 36479156981701017600*a^46*b^4*c^34*d^38)*1i - 92
23372036854775808*a^11*b^38*c^65*d^4 + 212137556847659843584*a^12*b^37*c^64*d^5 - 2333513125324258279424*a^13*
b^36*c^63*d^6 + 16334591877269807955968*a^14*b^35*c^62*d^7 - 81672959386349039779840*a^15*b^34*c^61*d^8 + 3108
08059650000835051520*a^16*b^33*c^60*d^9 - 942943171860407129210880*a^17*b^32*c^59*d^10 + 241198241252393034448
8960*a^18*b^31*c^58*d^11 - 5753067372685321201254400*a^19*b^30*c^57*d^12 + 14786194741349386435952640*a^20*b^2
9*c^56*d^13 - 43374839389541821883351040*a^21*b^28*c^55*d^14 + 131295543449898524428206080*a^22*b^27*c^54*d^15
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088581792734285660160*a^25*b^24*c^51*d^18 + 3349720497258869063543685120*a^26*b^23*c^50*d^19 - 522029206381521
1666322227200*a^27*b^22*c^49*d^20 + 7067608268064143449134202880*a^28*b^21*c^48*d^21 - 83422228712282518024775
27040*a^29*b^20*c^47*d^22 + 8605396720616721741816791040*a^30*b^19*c^46*d^23 - 7767500088979055902405427200*a^
31*b^18*c^45*d^24 + 6135496566696171932913500160*a^32*b^17*c^44*d^25 - 4236422046382466798589050880*a^33*b^16*
c^43*d^26 + 2550980661067485441771438080*a^34*b^15*c^42*d^27 - 1334575022384247271808040960*a^35*b^14*c^41*d^2
8 + 603343239457650202481000448*a^36*b^13*c^40*d^29 - 233967123641003163353350144*a^37*b^12*c^39*d^30 + 770495
27429528415176228864*a^38*b^11*c^38*d^31 - 21258749850480450394390528*a^39*b^10*c^37*d^32 + 482389936381990197
5265280*a^40*b^9*c^36*d^33 - 876898617974708020183040*a^41*b^8*c^35*d^34 + 122811796684756379238400*a^42*b^7*c
^34*d^35 - 12444332416601319014400*a^43*b^6*c^33*d^36 + 812231229670686720000*a^44*b^5*c^32*d^37 - 25649407252
758528000*a^45*b^4*c^31*d^38)*1i) + (-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 16
76354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^1
0 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326
592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^
4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^
8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)
*(x^(1/2)*(15412443824768679936*a^11*b^35*c^52*d^8 - 43988011059341426688*a^12*b^34*c^51*d^9 - 188730691300778
3641088*a^13*b^33*c^50*d^10 + 24240068121369040125952*a^14*b^32*c^49*d^11 - 155426886723407276146688*a^15*b^31
*c^48*d^12 + 661969678817672344633344*a^16*b^30*c^47*d^13 - 2072522435259453904257024*a^17*b^29*c^46*d^14 + 50
25620613985914706722816*a^18*b^28*c^45*d^15 - 9739734806850605210927104*a^19*b^27*c^44*d^16 + 1539558713198738
6880229376*a^20*b^26*c^43*d^17 - 20118464109716534770794496*a^21*b^25*c^42*d^18 + 21925688980693704834547712*a
^22*b^24*c^41*d^19 - 20031833528060137877536768*a^23*b^23*c^40*d^20 + 15375655212110710153674752*a^24*b^22*c^3
9*d^21 - 9908539789204785922572288*a^25*b^21*c^38*d^22 + 5342151752610266235273216*a^26*b^20*c^37*d^23 - 23933
61740048338255872000*a^27*b^19*c^36*d^24 + 881440288329629213655040*a^28*b^18*c^35*d^25 - 26255276900955308687
3600*a^29*b^17*c^34*d^26 + 61737289250332318105600*a^30*b^16*c^33*d^27 - 11040709176673173504000*a^31*b^15*c^3
2*d^28 + 1412353884520710144000*a^32*b^14*c^31*d^29 - 115221946643251200000*a^33*b^13*c^30*d^30 + 450868486864
8960000*a^34*b^12*c^29*d^31) - (-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354
940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 2
47981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a
^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 1
3287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4
*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(3/4)*(x^(
1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989
163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11
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56*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^
5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9
*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*(18446744073709551616*a^12*b^3
8*c^68*d^4 - 479615345916448342016*a^13*b^37*c^67*d^5 + 5995191823955604275200*a^14*b^36*c^66*d^6 - 4796153459
1644834201600*a^15*b^35*c^65*d^7 + 276025423003154095538176*a^16*b^34*c^64*d^8 - 1220386399802376518107136*a^1
7*b^33*c^63*d^9 + 4341880678999181785825280*a^18*b^32*c^62*d^10 - 12966583542852067073720320*a^19*b^31*c^61*d^
11 + 34096448785847177707520000*a^20*b^30*c^60*d^12 - 83405832293258492567879680*a^21*b^29*c^59*d^13 + 1987532
07063509910306160640*a^22*b^28*c^58*d^14 - 466650996519299420897935360*a^23*b^27*c^57*d^15 + 10520562198701980
56039219200*a^24*b^26*c^56*d^16 - 2194458800584304697435750400*a^25*b^25*c^55*d^17 + 4119286127977833519707586
560*a^26*b^24*c^54*d^18 - 6855830429358878767993323520*a^27*b^23*c^53*d^19 + 10053745593205095687740456960*a^2
8*b^22*c^52*d^20 - 12966109559707844614920601600*a^29*b^21*c^51*d^21 + 14704312650164876038740377600*a^30*b^20
*c^50*d^22 - 14666481047173052905774120960*a^31*b^19*c^49*d^23 + 12864666662378251575193763840*a^32*b^18*c^48*
d^24 - 9915254214005035782929121280*a^33*b^17*c^47*d^25 + 6703228082495101562834124800*a^34*b^16*c^46*d^26 - 3
963723814398261058758246400*a^35*b^15*c^45*d^27 + 2041552487767277748019527680*a^36*b^14*c^44*d^28 - 910688569
282163512962973696*a^37*b^13*c^43*d^29 + 349132643065901184834338816*a^38*b^12*c^42*d^30 - 1138591374851727228
40371200*a^39*b^11*c^41*d^31 + 31155202813126960689971200*a^40*b^10*c^40*d^32 - 7019849261936667709669376*a^41
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17829840996752341073920*a^44*b^6*c^36*d^36 - 1159226544085165670400*a^45*b^5*c^35*d^37 + 36479156981701017600*
a^46*b^4*c^34*d^38)*1i + 9223372036854775808*a^11*b^38*c^65*d^4 - 212137556847659843584*a^12*b^37*c^64*d^5 + 2
333513125324258279424*a^13*b^36*c^63*d^6 - 16334591877269807955968*a^14*b^35*c^62*d^7 + 8167295938634903977984
0*a^15*b^34*c^61*d^8 - 310808059650000835051520*a^16*b^33*c^60*d^9 + 942943171860407129210880*a^17*b^32*c^59*d
^10 - 2411982412523930344488960*a^18*b^31*c^58*d^11 + 5753067372685321201254400*a^19*b^30*c^57*d^12 - 14786194
741349386435952640*a^20*b^29*c^56*d^13 + 43374839389541821883351040*a^21*b^28*c^55*d^14 - 13129554344989852442
8206080*a^22*b^27*c^54*d^15 + 365631810199400875032576000*a^23*b^26*c^53*d^16 - 888615019916519951743057920*a^
24*b^25*c^52*d^17 + 1859065088581792734285660160*a^25*b^24*c^51*d^18 - 3349720497258869063543685120*a^26*b^23*
c^50*d^19 + 5220292063815211666322227200*a^27*b^22*c^49*d^20 - 7067608268064143449134202880*a^28*b^21*c^48*d^2
1 + 8342222871228251802477527040*a^29*b^20*c^47*d^22 - 8605396720616721741816791040*a^30*b^19*c^46*d^23 + 7767
500088979055902405427200*a^31*b^18*c^45*d^24 - 6135496566696171932913500160*a^32*b^17*c^44*d^25 + 423642204638
2466798589050880*a^33*b^16*c^43*d^26 - 2550980661067485441771438080*a^34*b^15*c^42*d^27 + 13345750223842472718
08040960*a^35*b^14*c^41*d^28 - 603343239457650202481000448*a^36*b^13*c^40*d^29 + 233967123641003163353350144*a
^37*b^12*c^39*d^30 - 77049527429528415176228864*a^38*b^11*c^38*d^31 + 21258749850480450394390528*a^39*b^10*c^3
7*d^32 - 4823899363819901975265280*a^40*b^9*c^36*d^33 + 876898617974708020183040*a^41*b^8*c^35*d^34 - 12281179
6684756379238400*a^42*b^7*c^34*d^35 + 12444332416601319014400*a^43*b^6*c^33*d^36 - 812231229670686720000*a^44*
b^5*c^32*d^37 + 25649407252758528000*a^45*b^4*c^31*d^38)*1i))/((-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 8
32838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9
- 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777
216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 +
8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b
^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 2013
26592*a*b^11*c^24*d))^(1/4)*(x^(1/2)*(15412443824768679936*a^11*b^35*c^52*d^8 - 43988011059341426688*a^12*b^34
*c^51*d^9 - 1887306913007783641088*a^13*b^33*c^50*d^10 + 24240068121369040125952*a^14*b^32*c^49*d^11 - 1554268
86723407276146688*a^15*b^31*c^48*d^12 + 661969678817672344633344*a^16*b^30*c^47*d^13 - 20725224352594539042570
24*a^17*b^29*c^46*d^14 + 5025620613985914706722816*a^18*b^28*c^45*d^15 - 9739734806850605210927104*a^19*b^27*c
^44*d^16 + 15395587131987386880229376*a^20*b^26*c^43*d^17 - 20118464109716534770794496*a^21*b^25*c^42*d^18 + 2
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760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765
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13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8
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38)*1i)*1i - 149684329919262228480*a^11*b^34*c^48*d^9 + 2374404370680061624320*a^12*b^33*c^47*d^10 - 178087226
27439624192000*a^13*b^32*c^46*d^11 + 83960295795175519682560*a^14*b^31*c^45*d^12 - 278998813302985850880000*a^
15*b^30*c^44*d^13 + 694438771802419400540160*a^16*b^29*c^43*d^14 - 1342951722708271932375040*a^17*b^28*c^42*d^
15 + 2065391322938120916172800*a^18*b^27*c^41*d^16 - 2564218746215699966853120*a^19*b^26*c^40*d^17 + 259333887
1410901332787200*a^20*b^25*c^39*d^18 - 2146065846150812380692480*a^21*b^24*c^38*d^19 + 14536254415697270223667
20*a^22*b^23*c^37*d^20 - 802881124954933087436800*a^23*b^22*c^36*d^21 + 358581985606139180482560*a^24*b^21*c^3
5*d^22 - 127660361818125316915200*a^25*b^20*c^34*d^23 + 35417419405750750412800*a^26*b^19*c^33*d^24 - 73868375
61454362624000*a^27*b^18*c^32*d^25 + 1090533977896255488000*a^28*b^17*c^31*d^26 - 101695892037304320000*a^29*b
^16*c^30*d^27 + 4508684868648960000*a^30*b^15*c^29*d^28))*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838
760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765
063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a
^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304
721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^
18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592
*a*b^11*c^24*d))^(1/4)