\(\int \frac {1}{x^{3/2} (a+b x^2) (c+d x^2)^2} \, dx\) [477]

Optimal result

Integrand size = 24, antiderivative size = 570 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}+\frac {b^{9/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^2}-\frac {b^{9/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {b^{9/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}+\frac {b^{9/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2} \]

[Out]

1/2*b^(9/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(5/4)/(-a*d+b*c)^2*2^(1/2)-1/2*b^(9/4)*arctan(1+b^(1/4
)*2^(1/2)*x^(1/2)/a^(1/4))/a^(5/4)/(-a*d+b*c)^2*2^(1/2)-1/8*d^(5/4)*(-5*a*d+9*b*c)*arctan(1-d^(1/4)*2^(1/2)*x^
(1/2)/c^(1/4))/c^(9/4)/(-a*d+b*c)^2*2^(1/2)+1/8*d^(5/4)*(-5*a*d+9*b*c)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4
))/c^(9/4)/(-a*d+b*c)^2*2^(1/2)-1/4*b^(9/4)*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)^2*2^(1/2)+1/4*b^(9/4)*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)^2*2^(1/2
)+1/16*d^(5/4)*(-5*a*d+9*b*c)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(9/4)/(-a*d+b*c)^2*2^(1/
2)-1/16*d^(5/4)*(-5*a*d+9*b*c)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(9/4)/(-a*d+b*c)^2*2^(1
/2)+1/2*(5*a*d-4*b*c)/a/c^2/(-a*d+b*c)/x^(1/2)-1/2*d/c/(-a*d+b*c)/(d*x^2+c)/x^(1/2)

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {477, 483, 597, 598, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b^{9/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^2}-\frac {b^{9/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{5/4} (b c-a d)^2}-\frac {b^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}+\frac {b^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {4 b c-5 a d}{2 a c^2 \sqrt {x} (b c-a d)}-\frac {d}{2 c \sqrt {x} \left (c+d x^2\right ) (b c-a d)} \]

[In]

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

[Out]

-1/2*(4*b*c - 5*a*d)/(a*c^2*(b*c - a*d)*Sqrt[x]) - d/(2*c*(b*c - a*d)*Sqrt[x]*(c + d*x^2)) + (b^(9/4)*ArcTan[1
- (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^2) - (b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*
Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^2) - (d^(5/4)*(9*b*c - 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt
[x])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^2) + (d^(5/4)*(9*b*c - 5*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x
])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^2) - (b^(9/4)*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)^2) + (b^(9/4)*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)^2) + (d^(5/4)*(9*b*c - 5*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
+ Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)*(b*c - a*d)^2) - (d^(5/4)*(9*b*c - 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/
4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)*(b*c - a*d)^2)

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 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 )^2} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {4 b c-5 a d-5 b d x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{2 c (b c-a d)} \\ & = -\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (4 b^2 c^2+4 a b c d-5 a^2 d^2+b d (4 b c-5 a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{2 a c^2 (b c-a d)} \\ & = -\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {4 b^3 c^2 x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {a d^2 (-9 b c+5 a d) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a c^2 (b c-a d)} \\ & = -\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}-\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^2}+\frac {\left (d^2 (9 b c-5 a d)\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c^2 (b c-a d)^2} \\ & = -\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}+\frac {b^{5/2} \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)^2}-\frac {b^{5/2} \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)^2}-\frac {\left (d^{3/2} (9 b c-5 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^2 (b c-a d)^2}+\frac {\left (d^{3/2} (9 b c-5 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^2 (b c-a d)^2} \\ & = -\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}-\frac {b^2 \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)^2}-\frac {b^2 \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)^2}-\frac {b^{9/4} \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)^2}-\frac {b^{9/4} \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)^2}+\frac {(d (9 b c-5 a d)) \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 )}{8 c^2 (b c-a d)^2}+\frac {(d (9 b c-5 a d)) \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 )}{8 c^2 (b c-a d)^2}+\frac {\left (d^{5/4} (9 b c-5 a d)\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 )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}+\frac {\left (d^{5/4} (9 b c-5 a d)\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 )}{8 \sqrt {2} c^{9/4} (b c-a d)^2} \\ & = -\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}-\frac {b^{9/4} \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)^2}+\frac {b^{9/4} \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)^2}+\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {b^{9/4} \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)^2}+\frac {b^{9/4} \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)^2}+\frac {\left (d^{5/4} (9 b c-5 a d)\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 )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {\left (d^{5/4} (9 b c-5 a d)\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 )}{4 \sqrt {2} c^{9/4} (b c-a d)^2} \\ & = -\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}+\frac {b^{9/4} \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)^2}-\frac {b^{9/4} \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)^2}-\frac {d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {b^{9/4} \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)^2}+\frac {b^{9/4} \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)^2}+\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {1}{8} \left (\frac {16 b c \left (c+d x^2\right )-4 a d \left (4 c+5 d x^2\right )}{a c^2 (-b c+a d) \sqrt {x} \left (c+d x^2\right )}+\frac {4 \sqrt {2} b^{9/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{5/4} (b c-a d)^2}+\frac {\sqrt {2} d^{5/4} (-9 b c+5 a d) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{9/4} (b c-a d)^2}+\frac {4 \sqrt {2} b^{9/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{5/4} (b c-a d)^2}+\frac {\sqrt {2} d^{5/4} (-9 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{9/4} (b c-a d)^2}\right ) \]

[In]

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

[Out]

((16*b*c*(c + d*x^2) - 4*a*d*(4*c + 5*d*x^2))/(a*c^2*(-(b*c) + a*d)*Sqrt[x]*(c + d*x^2)) + (4*Sqrt[2]*b^(9/4)*
ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(5/4)*(b*c - a*d)^2) + (Sqrt[2]*d^(5/4)*(-
9*b*c + 5*a*d)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(9/4)*(b*c - a*d)^2) + (4*S
qrt[2]*b^(9/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(5/4)*(b*c - a*d)^2) + (Sq
rt[2]*d^(5/4)*(-9*b*c + 5*a*d)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(9/4)*(b*c
- a*d)^2))/8

Maple [A] (verified)

Time = 2.80 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.50

method	result	size
derivativedivides	\(-\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 d^{2} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {5 a d}{4}-\frac {9 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{2} c^{2}}-\frac {2}{a \,c^{2} \sqrt {x}}\)	\(286\)
default	\(-\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 d^{2} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {5 a d}{4}-\frac {9 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{2} c^{2}}-\frac {2}{a \,c^{2} \sqrt {x}}\)	\(286\)
risch	\(-\frac {2}{a \,c^{2} \sqrt {x}}-\frac {\frac {b^{2} c^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {2 a \,d^{2} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {5 a d}{4}-\frac {9 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{2}}}{a \,c^{2}}\)	\(293\)

[In]

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

[Out]

-1/4*b^2/a/(a*d-b*c)^2/(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)^2/c^2*((1/4*a*d-1/4*b*c)*x^(3/2)/(d*x^2+c)+1/8*(5/4*a*d-9/4*b*c)/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^2/x^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 12.26 (sec) , antiderivative size = 3677, normalized size of antiderivative = 6.45 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/8*(4*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*d^2 - 56*a^8*b^5*c^5*d^3 + 70*a^9*b^4*c^4*d^4 -
56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(1/4)*((a*b*c^3*d - a^2*c^2*d^2)*x^3 +
(a*b*c^4 - a^2*c^3*d)*x)*log(b^7*sqrt(x) + (a^4*b^6*c^6 - 6*a^5*b^5*c^5*d + 15*a^6*b^4*c^4*d^2 - 20*a^7*b^3*c
^3*d^3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5 + a^10*d^6)*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*
d^2 - 56*a^8*b^5*c^5*d^3 + 70*a^9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b*c*d^7 + a
^13*d^8))^(3/4)) - 4*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*d^2 - 56*a^8*b^5*c^5*d^3 + 70*a^9*b
^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(1/4)*((a*b*c^3*d - a^2*c
^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*log(b^7*sqrt(x) - (a^4*b^6*c^6 - 6*a^5*b^5*c^5*d + 15*a^6*b^4*c^4*d^2 -
20*a^7*b^3*c^3*d^3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5 + a^10*d^6)*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28
*a^7*b^6*c^6*d^2 - 56*a^8*b^5*c^5*d^3 + 70*a^9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a^1
2*b*c*d^7 + a^13*d^8))^(3/4)) - 4*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*d^2 - 56*a^8*b^5*c^5*d
^3 + 70*a^9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(1/4)*(-I*(a
*b*c^3*d - a^2*c^2*d^2)*x^3 - I*(a*b*c^4 - a^2*c^3*d)*x)*log(b^7*sqrt(x) - (I*a^4*b^6*c^6 - 6*I*a^5*b^5*c^5*d
+ 15*I*a^6*b^4*c^4*d^2 - 20*I*a^7*b^3*c^3*d^3 + 15*I*a^8*b^2*c^2*d^4 - 6*I*a^9*b*c*d^5 + I*a^10*d^6)*(-b^9/(a^
5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*d^2 - 56*a^8*b^5*c^5*d^3 + 70*a^9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d
^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(3/4)) - 4*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^
7*b^6*c^6*d^2 - 56*a^8*b^5*c^5*d^3 + 70*a^9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b
*c*d^7 + a^13*d^8))^(1/4)*(I*(a*b*c^3*d - a^2*c^2*d^2)*x^3 + I*(a*b*c^4 - a^2*c^3*d)*x)*log(b^7*sqrt(x) - (-I*
a^4*b^6*c^6 + 6*I*a^5*b^5*c^5*d - 15*I*a^6*b^4*c^4*d^2 + 20*I*a^7*b^3*c^3*d^3 - 15*I*a^8*b^2*c^2*d^4 + 6*I*a^9
*b*c*d^5 - I*a^10*d^6)*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*d^2 - 56*a^8*b^5*c^5*d^3 + 70*a^9
*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(3/4)) + ((a*b*c^3*d -
a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7
- 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*
a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(1/4)*log((b^6
*c^13 - 6*a*b^5*c^12*d + 15*a^2*b^4*c^11*d^2 - 20*a^3*b^3*c^10*d^3 + 15*a^4*b^2*c^9*d^4 - 6*a^5*b*c^8*d^5 + a^
6*c^7*d^6)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)
/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^1
2*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(3/4) - (729*b^3*c^3*d^4 - 1215*a*b^2*c^2*d^5 +
675*a^2*b*c*d^6 - 125*a^3*d^7)*sqrt(x)) - ((a*b*c^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*(-(6561*b
^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7
*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^2*c
^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(1/4)*log(-(b^6*c^13 - 6*a*b^5*c^12*d + 15*a^2*b^4*c^11*d^2 - 20*a^
3*b^3*c^10*d^3 + 15*a^4*b^2*c^9*d^4 - 6*a^5*b*c^8*d^5 + a^6*c^7*d^6)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6
+ 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 -
56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8
*c^9*d^8))^(3/4) - (729*b^3*c^3*d^4 - 1215*a*b^2*c^2*d^5 + 675*a^2*b*c*d^6 - 125*a^3*d^7)*sqrt(x)) - (-I*(a*b*
c^3*d - a^2*c^2*d^2)*x^3 - I*(a*b*c^4 - a^2*c^3*d)*x)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^
2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14
*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(1/4
)*log(-(I*b^6*c^13 - 6*I*a*b^5*c^12*d + 15*I*a^2*b^4*c^11*d^2 - 20*I*a^3*b^3*c^10*d^3 + 15*I*a^4*b^2*c^9*d^4 -
6*I*a^5*b*c^8*d^5 + I*a^6*c^7*d^6)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a
^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*
c^13*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(3/4) - (729*b^3*c^3*d
^4 - 1215*a*b^2*c^2*d^5 + 675*a^2*b*c*d^6 - 125*a^3*d^7)*sqrt(x)) - (I*(a*b*c^3*d - a^2*c^2*d^2)*x^3 + I*(a*b*
c^4 - a^2*c^3*d)*x)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625
*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^
5*b^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(1/4)*log(-(-I*b^6*c^13 + 6*I*a*b^5*c^
12*d - 15*I*a^2*b^4*c^11*d^2 + 20*I*a^3*b^3*c^10*d^3 - 15*I*a^4*b^2*c^9*d^4 + 6*I*a^5*b*c^8*d^5 - I*a^6*c^7*d^
6)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^
17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5 +
28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(3/4) - (729*b^3*c^3*d^4 - 1215*a*b^2*c^2*d^5 + 675*a^2
*b*c*d^6 - 125*a^3*d^7)*sqrt(x)) + 4*(4*b*c^2 - 4*a*c*d + (4*b*c*d - 5*a*d^2)*x^2)*sqrt(x))/((a*b*c^3*d - a^2*
c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 494, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {b^{3} {\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^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac {{\left (9 \, b c d^{2} - 5 \, a d^{3}\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 )}}{16 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} - \frac {4 \, b c^{2} - 4 \, a c d + {\left (4 \, b c d - 5 \, a d^{2}\right )} x^{2}}{2 \, {\left ({\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{\frac {5}{2}} + {\left (a b c^{4} - a^{2} c^{3} d\right )} \sqrt {x}\right )}} \]

[In]

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

[Out]

-1/4*b^3*(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^2*c^2 - 2*a^2*b*c*d + a^3*d^2) + 1/16*(9*b*c*d^2 - 5*a*d^3)*(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)*arc
tan(-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))*s
qrt(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^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2
) - 1/2*(4*b*c^2 - 4*a*c*d + (4*b*c*d - 5*a*d^2)*x^2)/((a*b*c^3*d - a^2*c^2*d^2)*x^(5/2) + (a*b*c^4 - a^2*c^3*
d)*sqrt(x))

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{5} d - 2 \, \sqrt {2} a b c^{4} d^{2} + \sqrt {2} a^{2} c^{3} d^{3}\right )}} + \frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{5} d - 2 \, \sqrt {2} a b c^{4} d^{2} + \sqrt {2} a^{2} c^{3} d^{3}\right )}} - \frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{5} d - 2 \, \sqrt {2} a b c^{4} d^{2} + \sqrt {2} a^{2} c^{3} d^{3}\right )}} + \frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{5} d - 2 \, \sqrt {2} a b c^{4} d^{2} + \sqrt {2} a^{2} c^{3} d^{3}\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}\right )}} - \frac {4 \, b c d x^{2} - 5 \, a d^{2} x^{2} + 4 \, b c^{2} - 4 \, a c d}{2 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} {\left (d x^{\frac {5}{2}} + c \sqrt {x}\right )}} \]

[In]

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

[Out]

1/4*(9*(c*d^3)^(3/4)*b*c - 5*(c*d^3)^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/
4))/(sqrt(2)*b^2*c^5*d - 2*sqrt(2)*a*b*c^4*d^2 + sqrt(2)*a^2*c^3*d^3) + 1/4*(9*(c*d^3)^(3/4)*b*c - 5*(c*d^3)^(
3/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^5*d - 2*sqrt(2)*a*
b*c^4*d^2 + sqrt(2)*a^2*c^3*d^3) - 1/8*(9*(c*d^3)^(3/4)*b*c - 5*(c*d^3)^(3/4)*a*d)*log(sqrt(2)*sqrt(x)*(c/d)^(
1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^5*d - 2*sqrt(2)*a*b*c^4*d^2 + sqrt(2)*a^2*c^3*d^3) + 1/8*(9*(c*d^3)^(3/4)
*b*c - 5*(c*d^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^5*d - 2*sqrt(2)*a
*b*c^4*d^2 + sqrt(2)*a^2*c^3*d^3) - (a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(
1/4))/(sqrt(2)*a^2*b^2*c^2 - 2*sqrt(2)*a^3*b*c*d + sqrt(2)*a^4*d^2) - (a*b^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(
2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^2*c^2 - 2*sqrt(2)*a^3*b*c*d + sqrt(2)*a^4*d^2) + 1/2*(
a*b^3)^(3/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^2*c^2 - 2*sqrt(2)*a^3*b*c*d + sqr
t(2)*a^4*d^2) - 1/2*(a*b^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^2*c^2 - 2*s
qrt(2)*a^3*b*c*d + sqrt(2)*a^4*d^2) - 1/2*(4*b*c*d*x^2 - 5*a*d^2*x^2 + 4*b*c^2 - 4*a*c*d)/((a*b*c^3 - a^2*c^2*
d)*(d*x^(5/2) + c*sqrt(x)))

Mupad [B] (verification not implemented)

Time = 9.23 (sec) , antiderivative size = 21370, normalized size of antiderivative = 37.49 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 11
20*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^
8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 89
6*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(3/4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*
c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*
d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*
d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500
742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200
*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^
23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26
*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*
c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52
428800*a^33*b^4*c^23*d^25) - 16777216*a^11*b^25*c^42*d^4 + 218103808*a^12*b^24*c^41*d^5 - 1308622848*a^13*b^23
*c^40*d^6 + 4798283776*a^14*b^22*c^39*d^7 - 11995709440*a^15*b^21*c^38*d^8 + 21783379968*a^16*b^20*c^37*d^9 -
31592546304*a^17*b^19*c^36*d^10 + 48013246464*a^18*b^18*c^35*d^11 - 103424196608*a^19*b^17*c^34*d^12 + 2539546
21440*a^20*b^16*c^33*d^13 - 531641663488*a^21*b^15*c^32*d^14 + 875046109184*a^22*b^14*c^31*d^15 - 112586548838
4*a^23*b^13*c^30*d^16 + 1138334629888*a^24*b^12*c^29*d^17 - 906425794560*a^25*b^11*c^28*d^18 + 566347431936*a^
26*b^10*c^27*d^19 - 274688114688*a^27*b^9*c^26*d^20 + 101363744768*a^28*b^8*c^25*d^21 - 27505197056*a^29*b^7*c
^24*d^22 + 5174722560*a^30*b^6*c^23*d^23 - 602931200*a^31*b^5*c^22*d^24 + 32768000*a^32*b^4*c^21*d^25) - x^(1/
2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a^1
4*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26
*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465
100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20))*1i + (-b^9/(16*a^13*
d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 -
896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 -
128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 +
448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(3/4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7
*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^
2*d^6 - 128*a^12*b*c*d^7))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14
*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39
*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12
- 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2
585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 106
1108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 462212300
80*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*
d^25) + 16777216*a^11*b^25*c^42*d^4 - 218103808*a^12*b^24*c^41*d^5 + 1308622848*a^13*b^23*c^40*d^6 - 479828377
6*a^14*b^22*c^39*d^7 + 11995709440*a^15*b^21*c^38*d^8 - 21783379968*a^16*b^20*c^37*d^9 + 31592546304*a^17*b^19
*c^36*d^10 - 48013246464*a^18*b^18*c^35*d^11 + 103424196608*a^19*b^17*c^34*d^12 - 253954621440*a^20*b^16*c^33*
d^13 + 531641663488*a^21*b^15*c^32*d^14 - 875046109184*a^22*b^14*c^31*d^15 + 1125865488384*a^23*b^13*c^30*d^16
- 1138334629888*a^24*b^12*c^29*d^17 + 906425794560*a^25*b^11*c^28*d^18 - 566347431936*a^26*b^10*c^27*d^19 + 2
74688114688*a^27*b^9*c^26*d^20 - 101363744768*a^28*b^8*c^25*d^21 + 27505197056*a^29*b^7*c^24*d^22 - 5174722560
*a^30*b^6*c^23*d^23 + 602931200*a^31*b^5*c^22*d^24 - 32768000*a^32*b^4*c^21*d^25) - x^(1/2)*(32366592*a^12*b^2
1*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19*c^29*d^11 + 42
69711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14 - 7603863552*a^
18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800*a^21*b^12*c^22
*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20))*1i)/((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8
- 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5
+ 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 4
48*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6
- 128*a^12*b*c*d^7))^(3/4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*
d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*
d^7))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 1526
7266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a
^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b
^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^1
3*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*
c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22
- 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25) - 16777216*a^11
*b^25*c^42*d^4 + 218103808*a^12*b^24*c^41*d^5 - 1308622848*a^13*b^23*c^40*d^6 + 4798283776*a^14*b^22*c^39*d^7
- 11995709440*a^15*b^21*c^38*d^8 + 21783379968*a^16*b^20*c^37*d^9 - 31592546304*a^17*b^19*c^36*d^10 + 48013246
464*a^18*b^18*c^35*d^11 - 103424196608*a^19*b^17*c^34*d^12 + 253954621440*a^20*b^16*c^33*d^13 - 531641663488*a
^21*b^15*c^32*d^14 + 875046109184*a^22*b^14*c^31*d^15 - 1125865488384*a^23*b^13*c^30*d^16 + 1138334629888*a^24
*b^12*c^29*d^17 - 906425794560*a^25*b^11*c^28*d^18 + 566347431936*a^26*b^10*c^27*d^19 - 274688114688*a^27*b^9*
c^26*d^20 + 101363744768*a^28*b^8*c^25*d^21 - 27505197056*a^29*b^7*c^24*d^22 + 5174722560*a^30*b^6*c^23*d^23 -
602931200*a^31*b^5*c^22*d^24 + 32768000*a^32*b^4*c^21*d^25) - x^(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832
*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^
28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4
414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800*a^21*b^12*c^22*d^18 - 72704000*a^22
*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20)) - (-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 44
8*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 -
128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896
*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(3/
4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^
3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*(33554432*a^
12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d
^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313
817825280*a^19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 15893224
48896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030
720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560
*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c
^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25) + 16777216*a^11*b^25*c^42*d^4 - 2181038
08*a^12*b^24*c^41*d^5 + 1308622848*a^13*b^23*c^40*d^6 - 4798283776*a^14*b^22*c^39*d^7 + 11995709440*a^15*b^21*
c^38*d^8 - 21783379968*a^16*b^20*c^37*d^9 + 31592546304*a^17*b^19*c^36*d^10 - 48013246464*a^18*b^18*c^35*d^11
+ 103424196608*a^19*b^17*c^34*d^12 - 253954621440*a^20*b^16*c^33*d^13 + 531641663488*a^21*b^15*c^32*d^14 - 875
046109184*a^22*b^14*c^31*d^15 + 1125865488384*a^23*b^13*c^30*d^16 - 1138334629888*a^24*b^12*c^29*d^17 + 906425
794560*a^25*b^11*c^28*d^18 - 566347431936*a^26*b^10*c^27*d^19 + 274688114688*a^27*b^9*c^26*d^20 - 101363744768
*a^28*b^8*c^25*d^21 + 27505197056*a^29*b^7*c^24*d^22 - 5174722560*a^30*b^6*c^23*d^23 + 602931200*a^31*b^5*c^22
*d^24 - 32768000*a^32*b^4*c^21*d^25) - x^(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 18
6867712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^1
6*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24
*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 512000
0*a^23*b^10*c^20*d^20)) - 29859840*a^11*b^21*c^29*d^9 + 228925440*a^12*b^20*c^28*d^10 - 774144000*a^13*b^19*c^
27*d^11 + 1514700800*a^14*b^18*c^26*d^12 - 1888665600*a^15*b^17*c^25*d^13 + 1555415040*a^16*b^16*c^24*d^14 - 8
45578240*a^17*b^15*c^23*d^15 + 292454400*a^18*b^14*c^22*d^16 - 58368000*a^19*b^13*c^21*d^17 + 5120000*a^20*b^1
2*c^20*d^18))*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*
d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*2i + 2*ata
n(((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*
a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^8 +
16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a
^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(3/4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8
- 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5
+ 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5
+ 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742
656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200*a^
20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*
b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^
11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^2
7*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428
800*a^33*b^4*c^23*d^25)*1i - 16777216*a^11*b^25*c^42*d^4 + 218103808*a^12*b^24*c^41*d^5 - 1308622848*a^13*b^23
*c^40*d^6 + 4798283776*a^14*b^22*c^39*d^7 - 11995709440*a^15*b^21*c^38*d^8 + 21783379968*a^16*b^20*c^37*d^9 -
31592546304*a^17*b^19*c^36*d^10 + 48013246464*a^18*b^18*c^35*d^11 - 103424196608*a^19*b^17*c^34*d^12 + 2539546
21440*a^20*b^16*c^33*d^13 - 531641663488*a^21*b^15*c^32*d^14 + 875046109184*a^22*b^14*c^31*d^15 - 112586548838
4*a^23*b^13*c^30*d^16 + 1138334629888*a^24*b^12*c^29*d^17 - 906425794560*a^25*b^11*c^28*d^18 + 566347431936*a^
26*b^10*c^27*d^19 - 274688114688*a^27*b^9*c^26*d^20 + 101363744768*a^28*b^8*c^25*d^21 - 27505197056*a^29*b^7*c
^24*d^22 + 5174722560*a^30*b^6*c^23*d^23 - 602931200*a^31*b^5*c^22*d^24 + 32768000*a^32*b^4*c^21*d^25)*1i + x^
(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*
a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c
^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 +
465100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20)) + (-b^9/(16*a^13*
d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 -
896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 -
128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 +
448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(3/4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7
*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^
2*d^6 - 128*a^12*b*c*d^7))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14
*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39
*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12
- 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2
585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 106
1108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 462212300
80*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*
d^25)*1i + 16777216*a^11*b^25*c^42*d^4 - 218103808*a^12*b^24*c^41*d^5 + 1308622848*a^13*b^23*c^40*d^6 - 479828
3776*a^14*b^22*c^39*d^7 + 11995709440*a^15*b^21*c^38*d^8 - 21783379968*a^16*b^20*c^37*d^9 + 31592546304*a^17*b
^19*c^36*d^10 - 48013246464*a^18*b^18*c^35*d^11 + 103424196608*a^19*b^17*c^34*d^12 - 253954621440*a^20*b^16*c^
33*d^13 + 531641663488*a^21*b^15*c^32*d^14 - 875046109184*a^22*b^14*c^31*d^15 + 1125865488384*a^23*b^13*c^30*d
^16 - 1138334629888*a^24*b^12*c^29*d^17 + 906425794560*a^25*b^11*c^28*d^18 - 566347431936*a^26*b^10*c^27*d^19
+ 274688114688*a^27*b^9*c^26*d^20 - 101363744768*a^28*b^8*c^25*d^21 + 27505197056*a^29*b^7*c^24*d^22 - 5174722
560*a^30*b^6*c^23*d^23 + 602931200*a^31*b^5*c^22*d^24 - 32768000*a^32*b^4*c^21*d^25)*1i + x^(1/2)*(32366592*a^
12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19*c^29*d^1
1 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14 - 7603863
552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800*a^21*b^1
2*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20)))/((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c
^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d
^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d
+ 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d
^6 - 128*a^12*b*c*d^7))^(3/4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c
^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b
*c*d^7))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 1
5267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 18865979392
0*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^2
1*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*
b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^
10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d
^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25)*1i - 1677721
6*a^11*b^25*c^42*d^4 + 218103808*a^12*b^24*c^41*d^5 - 1308622848*a^13*b^23*c^40*d^6 + 4798283776*a^14*b^22*c^3
9*d^7 - 11995709440*a^15*b^21*c^38*d^8 + 21783379968*a^16*b^20*c^37*d^9 - 31592546304*a^17*b^19*c^36*d^10 + 48
013246464*a^18*b^18*c^35*d^11 - 103424196608*a^19*b^17*c^34*d^12 + 253954621440*a^20*b^16*c^33*d^13 - 53164166
3488*a^21*b^15*c^32*d^14 + 875046109184*a^22*b^14*c^31*d^15 - 1125865488384*a^23*b^13*c^30*d^16 + 113833462988
8*a^24*b^12*c^29*d^17 - 906425794560*a^25*b^11*c^28*d^18 + 566347431936*a^26*b^10*c^27*d^19 - 274688114688*a^2
7*b^9*c^26*d^20 + 101363744768*a^28*b^8*c^25*d^21 - 27505197056*a^29*b^7*c^24*d^22 + 5174722560*a^30*b^6*c^23*
d^23 - 602931200*a^31*b^5*c^22*d^24 + 32768000*a^32*b^4*c^21*d^25)*1i + x^(1/2)*(32366592*a^12*b^21*c^31*d^9 -
10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19*c^29*d^11 + 4269711360*a^1
5*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14 - 7603863552*a^18*b^15*c^25
*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800*a^21*b^12*c^22*d^18 - 7270
4000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20))*1i - (-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^
7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b
^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c
^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b
*c*d^7))^(3/4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^
8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*
(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15
*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^3
8*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^1
3 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16
- 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 +
492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 850604851
2*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25)*1i + 16777216*a^11*b^25*c^4
2*d^4 - 218103808*a^12*b^24*c^41*d^5 + 1308622848*a^13*b^23*c^40*d^6 - 4798283776*a^14*b^22*c^39*d^7 + 1199570
9440*a^15*b^21*c^38*d^8 - 21783379968*a^16*b^20*c^37*d^9 + 31592546304*a^17*b^19*c^36*d^10 - 48013246464*a^18*
b^18*c^35*d^11 + 103424196608*a^19*b^17*c^34*d^12 - 253954621440*a^20*b^16*c^33*d^13 + 531641663488*a^21*b^15*
c^32*d^14 - 875046109184*a^22*b^14*c^31*d^15 + 1125865488384*a^23*b^13*c^30*d^16 - 1138334629888*a^24*b^12*c^2
9*d^17 + 906425794560*a^25*b^11*c^28*d^18 - 566347431936*a^26*b^10*c^27*d^19 + 274688114688*a^27*b^9*c^26*d^20
- 101363744768*a^28*b^8*c^25*d^21 + 27505197056*a^29*b^7*c^24*d^22 - 5174722560*a^30*b^6*c^23*d^23 + 60293120
0*a^31*b^5*c^22*d^24 - 32768000*a^32*b^4*c^21*d^25)*1i + x^(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*
b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^1
2 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717
952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*
c^21*d^19 + 5120000*a^23*b^10*c^20*d^20))*1i + 29859840*a^11*b^21*c^29*d^9 - 228925440*a^12*b^20*c^28*d^10 + 7
74144000*a^13*b^19*c^27*d^11 - 1514700800*a^14*b^18*c^26*d^12 + 1888665600*a^15*b^17*c^25*d^13 - 1555415040*a^
16*b^16*c^24*d^14 + 845578240*a^17*b^15*c^23*d^15 - 292454400*a^18*b^14*c^22*d^16 + 58368000*a^19*b^13*c^21*d^
17 - 5120000*a^20*b^12*c^20*d^18))*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d
^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d
^7))^(1/4) - (2/(a*c) + (d*x^2*(5*a*d - 4*b*c))/(2*a*c^2*(a*d - b*c)))/(c*x^(1/2) + d*x^(5/2)) + atan(((-(625*
a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 +
4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c
^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(x^(1/2)*(32366592*a^
12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19*c^29*d^1
1 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14 - 7603863
552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800*a^21*b^1
2*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20) - (-(625*a^4*d^9 + 6561*b^4*c^4*d^5
- 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^
7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^
12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(3/4)*(x^(1/2)*(-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 145
80*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c
^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^
5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^
43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103
500742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779
200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200
*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a
^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b
^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 -
52428800*a^33*b^4*c^23*d^25) - 16777216*a^11*b^25*c^42*d^4 + 218103808*a^12*b^24*c^41*d^5 - 1308622848*a^13*b
^23*c^40*d^6 + 4798283776*a^14*b^22*c^39*d^7 - 11995709440*a^15*b^21*c^38*d^8 + 21783379968*a^16*b^20*c^37*d^9
- 31592546304*a^17*b^19*c^36*d^10 + 48013246464*a^18*b^18*c^35*d^11 - 103424196608*a^19*b^17*c^34*d^12 + 2539
54621440*a^20*b^16*c^33*d^13 - 531641663488*a^21*b^15*c^32*d^14 + 875046109184*a^22*b^14*c^31*d^15 - 112586548
8384*a^23*b^13*c^30*d^16 + 1138334629888*a^24*b^12*c^29*d^17 - 906425794560*a^25*b^11*c^28*d^18 + 566347431936
*a^26*b^10*c^27*d^19 - 274688114688*a^27*b^9*c^26*d^20 + 101363744768*a^28*b^8*c^25*d^21 - 27505197056*a^29*b^
7*c^24*d^22 + 5174722560*a^30*b^6*c^23*d^23 - 602931200*a^31*b^5*c^22*d^24 + 32768000*a^32*b^4*c^21*d^25))*1i
+ (-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^
8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*
a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(x^(1/2)*(32
366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19
*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14
- 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800
*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20) - (-(625*a^4*d^9 + 6561*b^4
*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 -
32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 - 229376*a
^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(3/4)*(x^(1/2)*(-(625*a^4*d^9 + 6561*b^4*c^4*
d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 3276
8*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^
3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^1
3*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*
d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 +
539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241
016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 179266
2306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 1752799641
60*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^
24*d^24 - 52428800*a^33*b^4*c^23*d^25) + 16777216*a^11*b^25*c^42*d^4 - 218103808*a^12*b^24*c^41*d^5 + 13086228
48*a^13*b^23*c^40*d^6 - 4798283776*a^14*b^22*c^39*d^7 + 11995709440*a^15*b^21*c^38*d^8 - 21783379968*a^16*b^20
*c^37*d^9 + 31592546304*a^17*b^19*c^36*d^10 - 48013246464*a^18*b^18*c^35*d^11 + 103424196608*a^19*b^17*c^34*d^
12 - 253954621440*a^20*b^16*c^33*d^13 + 531641663488*a^21*b^15*c^32*d^14 - 875046109184*a^22*b^14*c^31*d^15 +
1125865488384*a^23*b^13*c^30*d^16 - 1138334629888*a^24*b^12*c^29*d^17 + 906425794560*a^25*b^11*c^28*d^18 - 566
347431936*a^26*b^10*c^27*d^19 + 274688114688*a^27*b^9*c^26*d^20 - 101363744768*a^28*b^8*c^25*d^21 + 2750519705
6*a^29*b^7*c^24*d^22 - 5174722560*a^30*b^6*c^23*d^23 + 602931200*a^31*b^5*c^22*d^24 - 32768000*a^32*b^4*c^21*d
^25))*1i)/((-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)
/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3
+ 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(x^
(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*
a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c
^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 +
465100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20) - (-(625*a^4*d^9 +
6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*
c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 -
229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(3/4)*(x^(1/2)*(-(625*a^4*d^9 + 6561
*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d
^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 - 2293
76*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 5033
16480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b
^21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^3
7*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^
14 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17
+ 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 1
75279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^
32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25) - 16777216*a^11*b^25*c^42*d^4 + 218103808*a^12*b^24*c^41*d^5 -
1308622848*a^13*b^23*c^40*d^6 + 4798283776*a^14*b^22*c^39*d^7 - 11995709440*a^15*b^21*c^38*d^8 + 21783379968*
a^16*b^20*c^37*d^9 - 31592546304*a^17*b^19*c^36*d^10 + 48013246464*a^18*b^18*c^35*d^11 - 103424196608*a^19*b^1
7*c^34*d^12 + 253954621440*a^20*b^16*c^33*d^13 - 531641663488*a^21*b^15*c^32*d^14 + 875046109184*a^22*b^14*c^3
1*d^15 - 1125865488384*a^23*b^13*c^30*d^16 + 1138334629888*a^24*b^12*c^29*d^17 - 906425794560*a^25*b^11*c^28*d
^18 + 566347431936*a^26*b^10*c^27*d^19 - 274688114688*a^27*b^9*c^26*d^20 + 101363744768*a^28*b^8*c^25*d^21 - 2
7505197056*a^29*b^7*c^24*d^22 + 5174722560*a^30*b^6*c^23*d^23 - 602931200*a^31*b^5*c^22*d^24 + 32768000*a^32*b
^4*c^21*d^25)) - (-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*
c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^1
4*d^3 + 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/
4)*(x^(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 14220
57472*a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*
b^16*c^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d
^17 + 465100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20) - (-(625*a^4
*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 409
6*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13
*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(3/4)*(x^(1/2)*(-(625*a^4*d^9
+ 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8
*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4
- 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(33554432*a^12*b^25*c^44*d^4
- 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*
a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^
18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c
^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^3
1*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^
20 - 175279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175
680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25) + 16777216*a^11*b^25*c^42*d^4 - 218103808*a^12*b^24*c^41
*d^5 + 1308622848*a^13*b^23*c^40*d^6 - 4798283776*a^14*b^22*c^39*d^7 + 11995709440*a^15*b^21*c^38*d^8 - 217833
79968*a^16*b^20*c^37*d^9 + 31592546304*a^17*b^19*c^36*d^10 - 48013246464*a^18*b^18*c^35*d^11 + 103424196608*a^
19*b^17*c^34*d^12 - 253954621440*a^20*b^16*c^33*d^13 + 531641663488*a^21*b^15*c^32*d^14 - 875046109184*a^22*b^
14*c^31*d^15 + 1125865488384*a^23*b^13*c^30*d^16 - 1138334629888*a^24*b^12*c^29*d^17 + 906425794560*a^25*b^11*
c^28*d^18 - 566347431936*a^26*b^10*c^27*d^19 + 274688114688*a^27*b^9*c^26*d^20 - 101363744768*a^28*b^8*c^25*d^
21 + 27505197056*a^29*b^7*c^24*d^22 - 5174722560*a^30*b^6*c^23*d^23 + 602931200*a^31*b^5*c^22*d^24 - 32768000*
a^32*b^4*c^21*d^25)) + 29859840*a^11*b^21*c^29*d^9 - 228925440*a^12*b^20*c^28*d^10 + 774144000*a^13*b^19*c^27*
d^11 - 1514700800*a^14*b^18*c^26*d^12 + 1888665600*a^15*b^17*c^25*d^13 - 1555415040*a^16*b^16*c^24*d^14 + 8455
78240*a^17*b^15*c^23*d^15 - 292454400*a^18*b^14*c^22*d^16 + 58368000*a^19*b^13*c^21*d^17 - 5120000*a^20*b^12*c
^20*d^18))*(-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)
/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3
+ 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*2i
+ 2*atan(((-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/
(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 +
286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(x^(
1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a
^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^
26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 4
65100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20) + (-(625*a^4*d^9 +
6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c
^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 -
229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(3/4)*(x^(1/2)*(-(625*a^4*d^9 + 6561*
b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^
8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 - 22937
6*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 50331
6480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^
21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37
*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^1
4 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17
+ 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 17
5279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^3
2*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25)*1i - 16777216*a^11*b^25*c^42*d^4 + 218103808*a^12*b^24*c^41*d^5
- 1308622848*a^13*b^23*c^40*d^6 + 4798283776*a^14*b^22*c^39*d^7 - 11995709440*a^15*b^21*c^38*d^8 + 2178337996
8*a^16*b^20*c^37*d^9 - 31592546304*a^17*b^19*c^36*d^10 + 48013246464*a^18*b^18*c^35*d^11 - 103424196608*a^19*b
^17*c^34*d^12 + 253954621440*a^20*b^16*c^33*d^13 - 531641663488*a^21*b^15*c^32*d^14 + 875046109184*a^22*b^14*c
^31*d^15 - 1125865488384*a^23*b^13*c^30*d^16 + 1138334629888*a^24*b^12*c^29*d^17 - 906425794560*a^25*b^11*c^28
*d^18 + 566347431936*a^26*b^10*c^27*d^19 - 274688114688*a^27*b^9*c^26*d^20 + 101363744768*a^28*b^8*c^25*d^21 -
27505197056*a^29*b^7*c^24*d^22 + 5174722560*a^30*b^6*c^23*d^23 - 602931200*a^31*b^5*c^22*d^24 + 32768000*a^32
*b^4*c^21*d^25)*1i) + (-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a
^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^
5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d)
)^(1/4)*(x^(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 -
1422057472*a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*
a^17*b^16*c^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c
^23*d^17 + 465100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20) + (-(62
5*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17
+ 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4
*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(3/4)*(x^(1/2)*(-(625*a^4
*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 409
6*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13
*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(33554432*a^12*b^25*c^44
*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 4597166
8992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^
19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b
^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^1
2*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^
28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 9
75175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25)*1i + 16777216*a^11*b^25*c^42*d^4 - 218103808*a^12*b
^24*c^41*d^5 + 1308622848*a^13*b^23*c^40*d^6 - 4798283776*a^14*b^22*c^39*d^7 + 11995709440*a^15*b^21*c^38*d^8
- 21783379968*a^16*b^20*c^37*d^9 + 31592546304*a^17*b^19*c^36*d^10 - 48013246464*a^18*b^18*c^35*d^11 + 1034241
96608*a^19*b^17*c^34*d^12 - 253954621440*a^20*b^16*c^33*d^13 + 531641663488*a^21*b^15*c^32*d^14 - 875046109184
*a^22*b^14*c^31*d^15 + 1125865488384*a^23*b^13*c^30*d^16 - 1138334629888*a^24*b^12*c^29*d^17 + 906425794560*a^
25*b^11*c^28*d^18 - 566347431936*a^26*b^10*c^27*d^19 + 274688114688*a^27*b^9*c^26*d^20 - 101363744768*a^28*b^8
*c^25*d^21 + 27505197056*a^29*b^7*c^24*d^22 - 5174722560*a^30*b^6*c^23*d^23 + 602931200*a^31*b^5*c^22*d^24 - 3
2768000*a^32*b^4*c^21*d^25)*1i))/((-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*
d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 2
29376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a
*b^7*c^16*d))^(1/4)*(x^(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*
c^30*d^10 - 1422057472*a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 +
9165979648*a^17*b^16*c^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160
*a^20*b^13*c^23*d^17 + 465100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d
^20) + (-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(40
96*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 28
6720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(3/4)*(x^(1/2
)*(-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^
8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*
a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(33554432*a^
12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d
^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313
817825280*a^19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 15893224
48896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030
720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560
*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c
^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25)*1i - 16777216*a^11*b^25*c^42*d^4 + 2181
03808*a^12*b^24*c^41*d^5 - 1308622848*a^13*b^23*c^40*d^6 + 4798283776*a^14*b^22*c^39*d^7 - 11995709440*a^15*b^
21*c^38*d^8 + 21783379968*a^16*b^20*c^37*d^9 - 31592546304*a^17*b^19*c^36*d^10 + 48013246464*a^18*b^18*c^35*d^
11 - 103424196608*a^19*b^17*c^34*d^12 + 253954621440*a^20*b^16*c^33*d^13 - 531641663488*a^21*b^15*c^32*d^14 +
875046109184*a^22*b^14*c^31*d^15 - 1125865488384*a^23*b^13*c^30*d^16 + 1138334629888*a^24*b^12*c^29*d^17 - 906
425794560*a^25*b^11*c^28*d^18 + 566347431936*a^26*b^10*c^27*d^19 - 274688114688*a^27*b^9*c^26*d^20 + 101363744
768*a^28*b^8*c^25*d^21 - 27505197056*a^29*b^7*c^24*d^22 + 5174722560*a^30*b^6*c^23*d^23 - 602931200*a^31*b^5*c
^22*d^24 + 32768000*a^32*b^4*c^21*d^25)*1i)*1i - (-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 121
50*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b
^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^1
1*d^6 - 32768*a*b^7*c^16*d))^(1/4)*(x^(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 18686
7712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b
^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^
16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a
^23*b^10*c^20*d^20) + (-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a
^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^
5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d)
)^(3/4)*(x^(1/2)*(-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*
c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^2*b^6*c^15*d^2 - 229376*a^3*b^5*c^1
4*d^3 + 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*c^11*d^6 - 32768*a*b^7*c^16*d))^(1/
4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a
^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*
c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*
d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^
16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19
+ 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 850604
8512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25)*1i + 16777216*a^11*b^25*
c^42*d^4 - 218103808*a^12*b^24*c^41*d^5 + 1308622848*a^13*b^23*c^40*d^6 - 4798283776*a^14*b^22*c^39*d^7 + 1199
5709440*a^15*b^21*c^38*d^8 - 21783379968*a^16*b^20*c^37*d^9 + 31592546304*a^17*b^19*c^36*d^10 - 48013246464*a^
18*b^18*c^35*d^11 + 103424196608*a^19*b^17*c^34*d^12 - 253954621440*a^20*b^16*c^33*d^13 + 531641663488*a^21*b^
15*c^32*d^14 - 875046109184*a^22*b^14*c^31*d^15 + 1125865488384*a^23*b^13*c^30*d^16 - 1138334629888*a^24*b^12*
c^29*d^17 + 906425794560*a^25*b^11*c^28*d^18 - 566347431936*a^26*b^10*c^27*d^19 + 274688114688*a^27*b^9*c^26*d
^20 - 101363744768*a^28*b^8*c^25*d^21 + 27505197056*a^29*b^7*c^24*d^22 - 5174722560*a^30*b^6*c^23*d^23 + 60293
1200*a^31*b^5*c^22*d^24 - 32768000*a^32*b^4*c^21*d^25)*1i)*1i + 29859840*a^11*b^21*c^29*d^9 - 228925440*a^12*b
^20*c^28*d^10 + 774144000*a^13*b^19*c^27*d^11 - 1514700800*a^14*b^18*c^26*d^12 + 1888665600*a^15*b^17*c^25*d^1
3 - 1555415040*a^16*b^16*c^24*d^14 + 845578240*a^17*b^15*c^23*d^15 - 292454400*a^18*b^14*c^22*d^16 + 58368000*
a^19*b^13*c^21*d^17 - 5120000*a^20*b^12*c^20*d^18))*(-(625*a^4*d^9 + 6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 +
12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8)/(4096*b^8*c^17 + 4096*a^8*c^9*d^8 - 32768*a^7*b*c^10*d^7 + 114688*a^
2*b^6*c^15*d^2 - 229376*a^3*b^5*c^14*d^3 + 286720*a^4*b^4*c^13*d^4 - 229376*a^5*b^3*c^12*d^5 + 114688*a^6*b^2*
c^11*d^6 - 32768*a*b^7*c^16*d))^(1/4)