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

Optimal. Leaf size=676 \[ -\frac {b^{9/4} (5 b c-13 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {-5 a^2 d^2+8 a b c d-5 b^2 c^2}{2 a^2 c^2 \sqrt {x} (b c-a d)^2}-\frac {d^{9/4} (13 b c-5 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {d^{9/4} (13 b c-5 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^3}-\frac {d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {b}{2 a \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {d (a d+b c)}{2 a c \sqrt {x} \left (c+d x^2\right ) (b c-a d)^2} \]

[Out]

1/8*b^(9/4)*(-13*a*d+5*b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/(-a*d+b*c)^3*2^(1/2)-1/8*b^(9/4)
*(-13*a*d+5*b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/(-a*d+b*c)^3*2^(1/2)+1/8*d^(9/4)*(-5*a*d+13
*b*c)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(9/4)/(-a*d+b*c)^3*2^(1/2)-1/8*d^(9/4)*(-5*a*d+13*b*c)*arcta
n(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(9/4)/(-a*d+b*c)^3*2^(1/2)-1/16*b^(9/4)*(-13*a*d+5*b*c)*ln(a^(1/2)+x*b^
(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/(-a*d+b*c)^3*2^(1/2)+1/16*b^(9/4)*(-13*a*d+5*b*c)*ln(a^(1/2)+x*
b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/(-a*d+b*c)^3*2^(1/2)-1/16*d^(9/4)*(-5*a*d+13*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)^3*2^(1/2)+1/16*d^(9/4)*(-5*a*d+13*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)^3*2^(1/2)+1/2*(-5*a^2*d^2+8*a*b*c*d-5*b^2*c^2)
/a^2/c^2/(-a*d+b*c)^2/x^(1/2)+1/2*d*(a*d+b*c)/a/c/(-a*d+b*c)^2/(d*x^2+c)/x^(1/2)+1/2*b/a/(-a*d+b*c)/(b*x^2+a)/
(d*x^2+c)/x^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.06, antiderivative size = 676, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {466, 472, 579, 583, 584, 297, 1162, 617, 204, 1165, 628} \[ -\frac {5 a^2 d^2-8 a b c d+5 b^2 c^2}{2 a^2 c^2 \sqrt {x} (b c-a d)^2}-\frac {b^{9/4} (5 b c-13 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} (13 b c-5 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {d^{9/4} (13 b c-5 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^3}-\frac {d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {b}{2 a \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {d (a d+b c)}{2 a c \sqrt {x} \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(5*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)/(2*a^2*c^2*(b*c - a*d)^2*Sqrt[x]) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*
Sqrt[x]*(c + d*x^2)) + b/(2*a*(b*c - a*d)*Sqrt[x]*(a + b*x^2)*(c + d*x^2)) + (b^(9/4)*(5*b*c - 13*a*d)*ArcTan[
1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (b^(9/4)*(5*b*c - 13*a*d)*ArcTan[1
+ (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(13*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)^3) - (d^(9/4)*(13*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)^3) - (b^(9/4)*(5*b*c - 13*a*d)*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (b^(9/4)*(5*b*c - 13*a*d)
*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (d^(9/4)*(13*
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)^3) + (
d^(9/4)*(13*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)^3)

Rule 204

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

Rule 297

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 466

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

Rule 472

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

Rule 579

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

Rule 583

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

Rule 584

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

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

Rule 1162

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

Rule 1165

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

Rubi steps

\begin {align*} \int \frac {1}{x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-5 b c+4 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 )}{2 a (b c-a d)}\\ &=\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-4 \left (5 b^2 c^2-8 a b c d+5 a^2 d^2\right )-20 b d (b c+a d) x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{8 a c (b c-a d)^2}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (-4 (b c+a d) \left (5 b^2 c^2-13 a b c d+5 a^2 d^2\right )-4 b d \left (5 b^2 c^2-8 a b c d+5 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 )}{8 a^2 c^2 (b c-a d)^2}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \left (-\frac {4 b^3 c^2 (5 b c-13 a d) x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {4 a^2 d^3 (-13 b c+5 a d) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{8 a^2 c^2 (b c-a d)^2}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (b^3 (5 b c-13 a d)\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2 (b c-a d)^3}-\frac {\left (d^3 (13 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c^2 (b c-a d)^3}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^{5/2} (5 b c-13 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 (b c-a d)^3}-\frac {\left (b^{5/2} (5 b c-13 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 (b c-a d)^3}+\frac {\left (d^{5/2} (13 b c-5 a d)\right ) \operatorname {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)^3}-\frac {\left (d^{5/2} (13 b c-5 a d)\right ) \operatorname {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)^3}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (b^2 (5 b c-13 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^2 (b c-a d)^3}-\frac {\left (b^2 (5 b c-13 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^2 (b c-a d)^3}-\frac {\left (b^{9/4} (5 b c-13 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {\left (b^{9/4} (5 b c-13 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {\left (d^2 (13 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 c^2 (b c-a d)^3}-\frac {\left (d^2 (13 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 c^2 (b c-a d)^3}-\frac {\left (d^{9/4} (13 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^3}-\frac {\left (d^{9/4} (13 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^3}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} (13 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)^3}+\frac {d^{9/4} (13 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)^3}-\frac {\left (b^{9/4} (5 b c-13 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {\left (b^{9/4} (5 b c-13 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {\left (d^{9/4} (13 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {\left (d^{9/4} (13 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^3}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {d^{9/4} (13 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)^3}-\frac {d^{9/4} (13 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)^3}-\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} (13 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)^3}+\frac {d^{9/4} (13 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)^3}\\ \end {align*}

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Mathematica [A]  time = 6.16, size = 674, normalized size = 1.00 \[ \frac {b^{9/4} (13 a d-5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{9/4} (13 a d-5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (13 a d-5 b c) \tan ^{-1}\left (\frac {2 \sqrt [4]{b} \sqrt {x}-\sqrt {2} \sqrt [4]{a}}{\sqrt {2} \sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (13 a d-5 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a}+2 \sqrt [4]{b} \sqrt {x}}{\sqrt {2} \sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^3 x^{3/2}}{2 a^2 \left (a+b x^2\right ) (a d-b c)^2}-\frac {2}{a^2 c^2 \sqrt {x}}+\frac {d^{9/4} (13 b c-5 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (a d-b c)^3}-\frac {d^{9/4} (13 b c-5 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (a d-b c)^3}+\frac {d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (\frac {2 \sqrt [4]{d} \sqrt {x}-\sqrt {2} \sqrt [4]{c}}{\sqrt {2} \sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (a d-b c)^3}+\frac {d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c}+2 \sqrt [4]{d} \sqrt {x}}{\sqrt {2} \sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (a d-b c)^3}-\frac {d^3 x^{3/2}}{2 c^2 \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(a^2*c^2*Sqrt[x]) - (b^3*x^(3/2))/(2*a^2*(-(b*c) + a*d)^2*(a + b*x^2)) - (d^3*x^(3/2))/(2*c^2*(b*c - a*d)^2
*(c + d*x^2)) + (b^(9/4)*(-5*b*c + 13*a*d)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])
/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (b^(9/4)*(-5*b*c + 13*a*d)*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(
Sqrt[2]*a^(1/4))])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(13*b*c - 5*a*d)*ArcTan[(-(Sqrt[2]*c^(1/4)) +
2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(9/4)*(-(b*c) + a*d)^3) + (d^(9/4)*(13*b*c - 5*a*d)*ArcTan
[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(9/4)*(-(b*c) + a*d)^3) + (b^(9/4)*(-5
*b*c + 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) -
(b^(9/4)*(-5*b*c + 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*
c - a*d)^3) + (d^(9/4)*(13*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)^3) - (d^(9/4)*(13*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)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [A]  time = 1.56, size = 1035, normalized size = 1.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(5*(a*b^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(
1/4))/(sqrt(2)*a^3*b^3*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt(2)*a^6*d^3) - 1/4*(5*(a*b^
3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(
2)*a^3*b^3*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt(2)*a^6*d^3) - 1/4*(13*(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^3*c^6 -
3*sqrt(2)*a*b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/4*(13*(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^3*c^6 - 3*sqrt(2)*a*
b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) + 1/8*(5*(a*b^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*
log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^3*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^
5*b*c*d^2 - sqrt(2)*a^6*d^3) - 1/8*(5*(a*b^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/
4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^3*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt(2)*a^6*d^3)
+ 1/8*(13*(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
^3*c^6 - 3*sqrt(2)*a*b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/8*(13*(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^3*c^6 - 3*sqrt(2)*a*b^2*c^5*
d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/2*(5*b^3*c^2*d*x^4 - 8*a*b^2*c*d^2*x^4 + 5*a^2*b*d^3*x^
4 + 5*b^3*c^3*x^2 - 4*a*b^2*c^2*d*x^2 - 4*a^2*b*c*d^2*x^2 + 5*a^3*d^3*x^2 + 4*a*b^2*c^3 - 8*a^2*b*c^2*d + 4*a^
3*c*d^2)/((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*(b*d*x^(9/2) + b*c*x^(5/2) + a*d*x^(5/2) + a*c*sqrt(x)))

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maple [A]  time = 0.03, size = 825, normalized size = 1.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^3/a/(a*d-b*c)^3*x^(3/2)/(b*x^2+a)*d+1/2*b^4/a^2/(a*d-b*c)^3*x^(3/2)/(b*x^2+a)*c-13/16*b^2/a/(a*d-b*c)^3
/(a/b)^(1/4)*2^(1/2)*d*ln((x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/
2)))-13/8*b^2/a/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-13/8*b^2/a/(a*d-b*c)^3
/(a/b)^(1/4)*2^(1/2)*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+5/16*b^3/a^2/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*c*ln
((x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))+5/8*b^3/a^2/(a*d-b*c
)^3/(a/b)^(1/4)*2^(1/2)*c*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+5/8*b^3/a^2/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*c*
arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-1/2*d^4/c^2/(a*d-b*c)^3*x^(3/2)/(d*x^2+c)*a+1/2*d^3/c/(a*d-b*c)^3*x^(3/2
)/(d*x^2+c)*b-5/16*d^3/c^2/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*ln((x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x
+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))-5/8*d^3/c^2/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^
(1/4)*x^(1/2)+1)-5/8*d^3/c^2/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+13/16*d^2
/c/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*b*ln((x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*2^(1/2)*x^(
1/2)+(c/d)^(1/2)))+13/8*d^2/c/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+13/8*d^2
/c/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/a^2/c^2/x^(1/2)

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maxima [A]  time = 2.93, size = 694, normalized size = 1.03 \[ -\frac {{\left (5 \, b^{4} c - 13 \, a b^{3} d\right )} {\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 )}}{16 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}} - \frac {{\left (13 \, b c d^{3} - 5 \, a 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 )}}{16 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}} - \frac {4 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 4 \, a^{3} c d^{2} + {\left (5 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x^{4} + {\left (5 \, b^{3} c^{3} - 4 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} x^{2}}{2 \, {\left ({\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{\frac {9}{2}} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{\frac {5}{2}} + {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} \sqrt {x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(5*b^4*c - 13*a*b^3*d)*(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)))/(sqrt(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))/sqrt(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^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3) - 1/16*(13*b*c*d^3 - 5*
a*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)))/(sqr
t(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))/sqrt
(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^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3) - 1/2*(4*a*b^2*c^3 - 8*a^2*b*c^2*d + 4*a^3*c*
d^2 + (5*b^3*c^2*d - 8*a*b^2*c*d^2 + 5*a^2*b*d^3)*x^4 + (5*b^3*c^3 - 4*a*b^2*c^2*d - 4*a^2*b*c*d^2 + 5*a^3*d^3
)*x^2)/((a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^(9/2) + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3
*d^2 + a^5*c^2*d^3)*x^(5/2) + (a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*sqrt(x))

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mupad [B]  time = 12.38, size = 33548, normalized size = 49.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((((-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12
)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^
18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14
*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^
(3/4)*(x^(1/2)*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*
b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^
3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*
b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c
^20*d))^(1/4)*(52428800*a^23*b^38*c^57*d^4 - 1635778560*a^24*b^37*c^56*d^5 + 24482152448*a^25*b^36*c^55*d^6 -
234134437888*a^26*b^35*c^54*d^7 + 1607834009600*a^27*b^34*c^53*d^8 - 8446069964800*a^28*b^33*c^52*d^9 + 353031
82041088*a^29*b^32*c^51*d^10 - 120578363097088*a^30*b^31*c^50*d^11 + 342964201062400*a^31*b^30*c^49*d^12 - 823
887134720000*a^32*b^29*c^48*d^13 + 1690057100492800*a^33*b^28*c^47*d^14 - 2988135038320640*a^34*b^27*c^46*d^15
+ 4595616128696320*a^35*b^26*c^45*d^16 - 6215915829985280*a^36*b^25*c^44*d^17 + 7509830061260800*a^37*b^24*c^
43*d^18 - 8292025971507200*a^38*b^23*c^42*d^19 + 8624070071418880*a^39*b^22*c^41*d^20 - 8700497871503360*a^40*
b^21*c^40*d^21 + 8624070071418880*a^41*b^20*c^39*d^22 - 8292025971507200*a^42*b^19*c^38*d^23 + 750983006126080
0*a^43*b^18*c^37*d^24 - 6215915829985280*a^44*b^17*c^36*d^25 + 4595616128696320*a^45*b^16*c^35*d^26 - 29881350
38320640*a^46*b^15*c^34*d^27 + 1690057100492800*a^47*b^14*c^33*d^28 - 823887134720000*a^48*b^13*c^32*d^29 + 34
2964201062400*a^49*b^12*c^31*d^30 - 120578363097088*a^50*b^11*c^30*d^31 + 35303182041088*a^51*b^10*c^29*d^32 -
8446069964800*a^52*b^9*c^28*d^33 + 1607834009600*a^53*b^8*c^27*d^34 - 234134437888*a^54*b^7*c^26*d^35 + 24482
152448*a^55*b^6*c^25*d^36 - 1635778560*a^56*b^5*c^24*d^37 + 52428800*a^57*b^4*c^23*d^38) - 32768000*a^21*b^38*
c^55*d^4 + 1009254400*a^22*b^37*c^54*d^5 - 14833418240*a^23*b^36*c^53*d^6 + 138556735488*a^24*b^35*c^52*d^7 -
924185001984*a^25*b^34*c^51*d^8 + 4688465362944*a^26*b^33*c^50*d^9 - 18812623126528*a^27*b^32*c^49*d^10 + 6129
5191654400*a^28*b^31*c^48*d^11 - 165189260410880*a^29*b^30*c^47*d^12 + 373165003898880*a^30*b^29*c^46*d^13 - 7
13540118773760*a^31*b^28*c^45*d^14 + 1163349301657600*a^32*b^27*c^44*d^15 - 1627141704253440*a^33*b^26*c^43*d^
16 + 1966197351383040*a^34*b^25*c^42*d^17 - 2079216623943680*a^35*b^24*c^41*d^18 + 1981073955225600*a^36*b^23*
c^40*d^19 - 1807512431493120*a^37*b^22*c^39*d^20 + 1724885956034560*a^38*b^21*c^38*d^21 - 1807512431493120*a^3
9*b^20*c^37*d^22 + 1981073955225600*a^40*b^19*c^36*d^23 - 2079216623943680*a^41*b^18*c^35*d^24 + 1966197351383
040*a^42*b^17*c^34*d^25 - 1627141704253440*a^43*b^16*c^33*d^26 + 1163349301657600*a^44*b^15*c^32*d^27 - 713540
118773760*a^45*b^14*c^31*d^28 + 373165003898880*a^46*b^13*c^30*d^29 - 165189260410880*a^47*b^12*c^29*d^30 + 61
295191654400*a^48*b^11*c^28*d^31 - 18812623126528*a^49*b^10*c^27*d^32 + 4688465362944*a^50*b^9*c^26*d^33 - 924
185001984*a^51*b^8*c^25*d^34 + 138556735488*a^52*b^7*c^24*d^35 - 14833418240*a^53*b^6*c^23*d^36 + 1009254400*a
^54*b^5*c^22*d^37 - 32768000*a^55*b^4*c^21*d^38) + x^(1/2)*(54080000*a^20*b^33*c^43*d^10 - 1361152000*a^21*b^3
2*c^42*d^11 + 16011852800*a^22*b^31*c^41*d^12 - 116736734720*a^23*b^30*c^40*d^13 + 589861462528*a^24*b^29*c^39
*d^14 - 2187899577344*a^25*b^28*c^38*d^15 + 6149347117056*a^26*b^27*c^37*d^16 - 13298820601344*a^27*b^26*c^36*
d^17 + 22133436343296*a^28*b^25*c^35*d^18 - 27715689750528*a^29*b^24*c^34*d^19 + 24077503776768*a^30*b^23*c^33
*d^20 - 9645706816512*a^31*b^22*c^32*d^21 - 9645706816512*a^32*b^21*c^31*d^22 + 24077503776768*a^33*b^20*c^30*
d^23 - 27715689750528*a^34*b^19*c^29*d^24 + 22133436343296*a^35*b^18*c^28*d^25 - 13298820601344*a^36*b^17*c^27
*d^26 + 6149347117056*a^37*b^16*c^26*d^27 - 2187899577344*a^38*b^15*c^25*d^28 + 589861462528*a^39*b^14*c^24*d^
29 - 116736734720*a^40*b^13*c^23*d^30 + 16011852800*a^41*b^12*c^22*d^31 - 1361152000*a^42*b^11*c^21*d^32 + 540
80000*a^43*b^10*c^20*d^33))*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^1
1 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^
2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 -
3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 4
9152*a*b^11*c^20*d))^(1/4)*1i + ((-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^
2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^
19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*
d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^1
0 - 49152*a*b^11*c^20*d))^(3/4)*(x^(1/2)*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^
2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2
*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b
^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*
c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4)*(52428800*a^23*b^38*c^57*d^4 - 1635778560*a^24*b^37*c^56*d^5 + 2448215
2448*a^25*b^36*c^55*d^6 - 234134437888*a^26*b^35*c^54*d^7 + 1607834009600*a^27*b^34*c^53*d^8 - 8446069964800*a
^28*b^33*c^52*d^9 + 35303182041088*a^29*b^32*c^51*d^10 - 120578363097088*a^30*b^31*c^50*d^11 + 342964201062400
*a^31*b^30*c^49*d^12 - 823887134720000*a^32*b^29*c^48*d^13 + 1690057100492800*a^33*b^28*c^47*d^14 - 2988135038
320640*a^34*b^27*c^46*d^15 + 4595616128696320*a^35*b^26*c^45*d^16 - 6215915829985280*a^36*b^25*c^44*d^17 + 750
9830061260800*a^37*b^24*c^43*d^18 - 8292025971507200*a^38*b^23*c^42*d^19 + 8624070071418880*a^39*b^22*c^41*d^2
0 - 8700497871503360*a^40*b^21*c^40*d^21 + 8624070071418880*a^41*b^20*c^39*d^22 - 8292025971507200*a^42*b^19*c
^38*d^23 + 7509830061260800*a^43*b^18*c^37*d^24 - 6215915829985280*a^44*b^17*c^36*d^25 + 4595616128696320*a^45
*b^16*c^35*d^26 - 2988135038320640*a^46*b^15*c^34*d^27 + 1690057100492800*a^47*b^14*c^33*d^28 - 82388713472000
0*a^48*b^13*c^32*d^29 + 342964201062400*a^49*b^12*c^31*d^30 - 120578363097088*a^50*b^11*c^30*d^31 + 3530318204
1088*a^51*b^10*c^29*d^32 - 8446069964800*a^52*b^9*c^28*d^33 + 1607834009600*a^53*b^8*c^27*d^34 - 234134437888*
a^54*b^7*c^26*d^35 + 24482152448*a^55*b^6*c^25*d^36 - 1635778560*a^56*b^5*c^24*d^37 + 52428800*a^57*b^4*c^23*d
^38) + 32768000*a^21*b^38*c^55*d^4 - 1009254400*a^22*b^37*c^54*d^5 + 14833418240*a^23*b^36*c^53*d^6 - 13855673
5488*a^24*b^35*c^52*d^7 + 924185001984*a^25*b^34*c^51*d^8 - 4688465362944*a^26*b^33*c^50*d^9 + 18812623126528*
a^27*b^32*c^49*d^10 - 61295191654400*a^28*b^31*c^48*d^11 + 165189260410880*a^29*b^30*c^47*d^12 - 3731650038988
80*a^30*b^29*c^46*d^13 + 713540118773760*a^31*b^28*c^45*d^14 - 1163349301657600*a^32*b^27*c^44*d^15 + 16271417
04253440*a^33*b^26*c^43*d^16 - 1966197351383040*a^34*b^25*c^42*d^17 + 2079216623943680*a^35*b^24*c^41*d^18 - 1
981073955225600*a^36*b^23*c^40*d^19 + 1807512431493120*a^37*b^22*c^39*d^20 - 1724885956034560*a^38*b^21*c^38*d
^21 + 1807512431493120*a^39*b^20*c^37*d^22 - 1981073955225600*a^40*b^19*c^36*d^23 + 2079216623943680*a^41*b^18
*c^35*d^24 - 1966197351383040*a^42*b^17*c^34*d^25 + 1627141704253440*a^43*b^16*c^33*d^26 - 1163349301657600*a^
44*b^15*c^32*d^27 + 713540118773760*a^45*b^14*c^31*d^28 - 373165003898880*a^46*b^13*c^30*d^29 + 16518926041088
0*a^47*b^12*c^29*d^30 - 61295191654400*a^48*b^11*c^28*d^31 + 18812623126528*a^49*b^10*c^27*d^32 - 468846536294
4*a^50*b^9*c^26*d^33 + 924185001984*a^51*b^8*c^25*d^34 - 138556735488*a^52*b^7*c^24*d^35 + 14833418240*a^53*b^
6*c^23*d^36 - 1009254400*a^54*b^5*c^22*d^37 + 32768000*a^55*b^4*c^21*d^38) + x^(1/2)*(54080000*a^20*b^33*c^43*
d^10 - 1361152000*a^21*b^32*c^42*d^11 + 16011852800*a^22*b^31*c^41*d^12 - 116736734720*a^23*b^30*c^40*d^13 + 5
89861462528*a^24*b^29*c^39*d^14 - 2187899577344*a^25*b^28*c^38*d^15 + 6149347117056*a^26*b^27*c^37*d^16 - 1329
8820601344*a^27*b^26*c^36*d^17 + 22133436343296*a^28*b^25*c^35*d^18 - 27715689750528*a^29*b^24*c^34*d^19 + 240
77503776768*a^30*b^23*c^33*d^20 - 9645706816512*a^31*b^22*c^32*d^21 - 9645706816512*a^32*b^21*c^31*d^22 + 2407
7503776768*a^33*b^20*c^30*d^23 - 27715689750528*a^34*b^19*c^29*d^24 + 22133436343296*a^35*b^18*c^28*d^25 - 132
98820601344*a^36*b^17*c^27*d^26 + 6149347117056*a^37*b^16*c^26*d^27 - 2187899577344*a^38*b^15*c^25*d^28 + 5898
61462528*a^39*b^14*c^24*d^29 - 116736734720*a^40*b^13*c^23*d^30 + 16011852800*a^41*b^12*c^22*d^31 - 1361152000
*a^42*b^11*c^21*d^32 + 54080000*a^43*b^10*c^20*d^33))*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^
10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11
+ 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 +
3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270
336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4)*1i)/(((-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c
^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10
*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d
^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9
+ 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(3/4)*(x^(1/2)*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940
*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^1
1*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^
7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c
^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4)*(52428800*a^23*b^38*c^57*d^4 - 1635778560*a^
24*b^37*c^56*d^5 + 24482152448*a^25*b^36*c^55*d^6 - 234134437888*a^26*b^35*c^54*d^7 + 1607834009600*a^27*b^34*
c^53*d^8 - 8446069964800*a^28*b^33*c^52*d^9 + 35303182041088*a^29*b^32*c^51*d^10 - 120578363097088*a^30*b^31*c
^50*d^11 + 342964201062400*a^31*b^30*c^49*d^12 - 823887134720000*a^32*b^29*c^48*d^13 + 1690057100492800*a^33*b
^28*c^47*d^14 - 2988135038320640*a^34*b^27*c^46*d^15 + 4595616128696320*a^35*b^26*c^45*d^16 - 6215915829985280
*a^36*b^25*c^44*d^17 + 7509830061260800*a^37*b^24*c^43*d^18 - 8292025971507200*a^38*b^23*c^42*d^19 + 862407007
1418880*a^39*b^22*c^41*d^20 - 8700497871503360*a^40*b^21*c^40*d^21 + 8624070071418880*a^41*b^20*c^39*d^22 - 82
92025971507200*a^42*b^19*c^38*d^23 + 7509830061260800*a^43*b^18*c^37*d^24 - 6215915829985280*a^44*b^17*c^36*d^
25 + 4595616128696320*a^45*b^16*c^35*d^26 - 2988135038320640*a^46*b^15*c^34*d^27 + 1690057100492800*a^47*b^14*
c^33*d^28 - 823887134720000*a^48*b^13*c^32*d^29 + 342964201062400*a^49*b^12*c^31*d^30 - 120578363097088*a^50*b
^11*c^30*d^31 + 35303182041088*a^51*b^10*c^29*d^32 - 8446069964800*a^52*b^9*c^28*d^33 + 1607834009600*a^53*b^8
*c^27*d^34 - 234134437888*a^54*b^7*c^26*d^35 + 24482152448*a^55*b^6*c^25*d^36 - 1635778560*a^56*b^5*c^24*d^37
+ 52428800*a^57*b^4*c^23*d^38) - 32768000*a^21*b^38*c^55*d^4 + 1009254400*a^22*b^37*c^54*d^5 - 14833418240*a^2
3*b^36*c^53*d^6 + 138556735488*a^24*b^35*c^52*d^7 - 924185001984*a^25*b^34*c^51*d^8 + 4688465362944*a^26*b^33*
c^50*d^9 - 18812623126528*a^27*b^32*c^49*d^10 + 61295191654400*a^28*b^31*c^48*d^11 - 165189260410880*a^29*b^30
*c^47*d^12 + 373165003898880*a^30*b^29*c^46*d^13 - 713540118773760*a^31*b^28*c^45*d^14 + 1163349301657600*a^32
*b^27*c^44*d^15 - 1627141704253440*a^33*b^26*c^43*d^16 + 1966197351383040*a^34*b^25*c^42*d^17 - 20792166239436
80*a^35*b^24*c^41*d^18 + 1981073955225600*a^36*b^23*c^40*d^19 - 1807512431493120*a^37*b^22*c^39*d^20 + 1724885
956034560*a^38*b^21*c^38*d^21 - 1807512431493120*a^39*b^20*c^37*d^22 + 1981073955225600*a^40*b^19*c^36*d^23 -
2079216623943680*a^41*b^18*c^35*d^24 + 1966197351383040*a^42*b^17*c^34*d^25 - 1627141704253440*a^43*b^16*c^33*
d^26 + 1163349301657600*a^44*b^15*c^32*d^27 - 713540118773760*a^45*b^14*c^31*d^28 + 373165003898880*a^46*b^13*
c^30*d^29 - 165189260410880*a^47*b^12*c^29*d^30 + 61295191654400*a^48*b^11*c^28*d^31 - 18812623126528*a^49*b^1
0*c^27*d^32 + 4688465362944*a^50*b^9*c^26*d^33 - 924185001984*a^51*b^8*c^25*d^34 + 138556735488*a^52*b^7*c^24*
d^35 - 14833418240*a^53*b^6*c^23*d^36 + 1009254400*a^54*b^5*c^22*d^37 - 32768000*a^55*b^4*c^21*d^38) + x^(1/2)
*(54080000*a^20*b^33*c^43*d^10 - 1361152000*a^21*b^32*c^42*d^11 + 16011852800*a^22*b^31*c^41*d^12 - 1167367347
20*a^23*b^30*c^40*d^13 + 589861462528*a^24*b^29*c^39*d^14 - 2187899577344*a^25*b^28*c^38*d^15 + 6149347117056*
a^26*b^27*c^37*d^16 - 13298820601344*a^27*b^26*c^36*d^17 + 22133436343296*a^28*b^25*c^35*d^18 - 27715689750528
*a^29*b^24*c^34*d^19 + 24077503776768*a^30*b^23*c^33*d^20 - 9645706816512*a^31*b^22*c^32*d^21 - 9645706816512*
a^32*b^21*c^31*d^22 + 24077503776768*a^33*b^20*c^30*d^23 - 27715689750528*a^34*b^19*c^29*d^24 + 22133436343296
*a^35*b^18*c^28*d^25 - 13298820601344*a^36*b^17*c^27*d^26 + 6149347117056*a^37*b^16*c^26*d^27 - 2187899577344*
a^38*b^15*c^25*d^28 + 589861462528*a^39*b^14*c^24*d^29 - 116736734720*a^40*b^13*c^23*d^30 + 16011852800*a^41*b
^12*c^22*d^31 - 1361152000*a^42*b^11*c^21*d^32 + 54080000*a^43*b^10*c^20*d^33))*(-(625*a^4*d^13 + 28561*b^4*c^
4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^1
2 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3
244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901
120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4) - ((-(625*a^4*d^13 + 28561*b^4*
c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d
^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 -
3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 9
01120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(3/4)*(x^(1/2)*(-(625*a^4*d^13 + 28
561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^
12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^
17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13
*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4)*(52428800*a^23*b^38*c
^57*d^4 - 1635778560*a^24*b^37*c^56*d^5 + 24482152448*a^25*b^36*c^55*d^6 - 234134437888*a^26*b^35*c^54*d^7 + 1
607834009600*a^27*b^34*c^53*d^8 - 8446069964800*a^28*b^33*c^52*d^9 + 35303182041088*a^29*b^32*c^51*d^10 - 1205
78363097088*a^30*b^31*c^50*d^11 + 342964201062400*a^31*b^30*c^49*d^12 - 823887134720000*a^32*b^29*c^48*d^13 +
1690057100492800*a^33*b^28*c^47*d^14 - 2988135038320640*a^34*b^27*c^46*d^15 + 4595616128696320*a^35*b^26*c^45*
d^16 - 6215915829985280*a^36*b^25*c^44*d^17 + 7509830061260800*a^37*b^24*c^43*d^18 - 8292025971507200*a^38*b^2
3*c^42*d^19 + 8624070071418880*a^39*b^22*c^41*d^20 - 8700497871503360*a^40*b^21*c^40*d^21 + 8624070071418880*a
^41*b^20*c^39*d^22 - 8292025971507200*a^42*b^19*c^38*d^23 + 7509830061260800*a^43*b^18*c^37*d^24 - 62159158299
85280*a^44*b^17*c^36*d^25 + 4595616128696320*a^45*b^16*c^35*d^26 - 2988135038320640*a^46*b^15*c^34*d^27 + 1690
057100492800*a^47*b^14*c^33*d^28 - 823887134720000*a^48*b^13*c^32*d^29 + 342964201062400*a^49*b^12*c^31*d^30 -
120578363097088*a^50*b^11*c^30*d^31 + 35303182041088*a^51*b^10*c^29*d^32 - 8446069964800*a^52*b^9*c^28*d^33 +
1607834009600*a^53*b^8*c^27*d^34 - 234134437888*a^54*b^7*c^26*d^35 + 24482152448*a^55*b^6*c^25*d^36 - 1635778
560*a^56*b^5*c^24*d^37 + 52428800*a^57*b^4*c^23*d^38) + 32768000*a^21*b^38*c^55*d^4 - 1009254400*a^22*b^37*c^5
4*d^5 + 14833418240*a^23*b^36*c^53*d^6 - 138556735488*a^24*b^35*c^52*d^7 + 924185001984*a^25*b^34*c^51*d^8 - 4
688465362944*a^26*b^33*c^50*d^9 + 18812623126528*a^27*b^32*c^49*d^10 - 61295191654400*a^28*b^31*c^48*d^11 + 16
5189260410880*a^29*b^30*c^47*d^12 - 373165003898880*a^30*b^29*c^46*d^13 + 713540118773760*a^31*b^28*c^45*d^14
- 1163349301657600*a^32*b^27*c^44*d^15 + 1627141704253440*a^33*b^26*c^43*d^16 - 1966197351383040*a^34*b^25*c^4
2*d^17 + 2079216623943680*a^35*b^24*c^41*d^18 - 1981073955225600*a^36*b^23*c^40*d^19 + 1807512431493120*a^37*b
^22*c^39*d^20 - 1724885956034560*a^38*b^21*c^38*d^21 + 1807512431493120*a^39*b^20*c^37*d^22 - 1981073955225600
*a^40*b^19*c^36*d^23 + 2079216623943680*a^41*b^18*c^35*d^24 - 1966197351383040*a^42*b^17*c^34*d^25 + 162714170
4253440*a^43*b^16*c^33*d^26 - 1163349301657600*a^44*b^15*c^32*d^27 + 713540118773760*a^45*b^14*c^31*d^28 - 373
165003898880*a^46*b^13*c^30*d^29 + 165189260410880*a^47*b^12*c^29*d^30 - 61295191654400*a^48*b^11*c^28*d^31 +
18812623126528*a^49*b^10*c^27*d^32 - 4688465362944*a^50*b^9*c^26*d^33 + 924185001984*a^51*b^8*c^25*d^34 - 1385
56735488*a^52*b^7*c^24*d^35 + 14833418240*a^53*b^6*c^23*d^36 - 1009254400*a^54*b^5*c^22*d^37 + 32768000*a^55*b
^4*c^21*d^38) + x^(1/2)*(54080000*a^20*b^33*c^43*d^10 - 1361152000*a^21*b^32*c^42*d^11 + 16011852800*a^22*b^31
*c^41*d^12 - 116736734720*a^23*b^30*c^40*d^13 + 589861462528*a^24*b^29*c^39*d^14 - 2187899577344*a^25*b^28*c^3
8*d^15 + 6149347117056*a^26*b^27*c^37*d^16 - 13298820601344*a^27*b^26*c^36*d^17 + 22133436343296*a^28*b^25*c^3
5*d^18 - 27715689750528*a^29*b^24*c^34*d^19 + 24077503776768*a^30*b^23*c^33*d^20 - 9645706816512*a^31*b^22*c^3
2*d^21 - 9645706816512*a^32*b^21*c^31*d^22 + 24077503776768*a^33*b^20*c^30*d^23 - 27715689750528*a^34*b^19*c^2
9*d^24 + 22133436343296*a^35*b^18*c^28*d^25 - 13298820601344*a^36*b^17*c^27*d^26 + 6149347117056*a^37*b^16*c^2
6*d^27 - 2187899577344*a^38*b^15*c^25*d^28 + 589861462528*a^39*b^14*c^24*d^29 - 116736734720*a^40*b^13*c^23*d^
30 + 16011852800*a^41*b^12*c^22*d^31 - 1361152000*a^42*b^11*c^21*d^32 + 54080000*a^43*b^10*c^20*d^33))*(-(625*
a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c
^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 20275
20*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520
*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4) - 175760
000*a^20*b^31*c^39*d^12 + 3507088000*a^21*b^30*c^38*d^13 - 32026300800*a^22*b^29*c^37*d^14 + 177335474560*a^23
*b^28*c^36*d^15 - 664045775360*a^24*b^27*c^35*d^16 + 1770849815040*a^25*b^26*c^34*d^17 - 3431196106240*a^26*b^
25*c^33*d^18 + 4778178444800*a^27*b^24*c^32*d^19 - 4440824728320*a^28*b^23*c^31*d^20 + 1838397848320*a^29*b^22
*c^30*d^21 + 1838397848320*a^30*b^21*c^29*d^22 - 4440824728320*a^31*b^20*c^28*d^23 + 4778178444800*a^32*b^19*c
^27*d^24 - 3431196106240*a^33*b^18*c^26*d^25 + 1770849815040*a^34*b^17*c^25*d^26 - 664045775360*a^35*b^16*c^24
*d^27 + 177335474560*a^36*b^15*c^23*d^28 - 32026300800*a^37*b^14*c^22*d^29 + 3507088000*a^38*b^13*c^21*d^30 -
175760000*a^39*b^12*c^20*d^31))*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2
*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^1
9*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d
^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10
- 49152*a*b^11*c^20*d))^(1/4)*2i + 2*atan((((-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 2535
0*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336
*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a
^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*
b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(3/4)*(x^(1/2)*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^1
0 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11
+ 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3
784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 2703
36*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4)*(52428800*a^23*b^38*c^57*d^4 - 1635778560*a^24*b^37*c^56*d
^5 + 24482152448*a^25*b^36*c^55*d^6 - 234134437888*a^26*b^35*c^54*d^7 + 1607834009600*a^27*b^34*c^53*d^8 - 844
6069964800*a^28*b^33*c^52*d^9 + 35303182041088*a^29*b^32*c^51*d^10 - 120578363097088*a^30*b^31*c^50*d^11 + 342
964201062400*a^31*b^30*c^49*d^12 - 823887134720000*a^32*b^29*c^48*d^13 + 1690057100492800*a^33*b^28*c^47*d^14
- 2988135038320640*a^34*b^27*c^46*d^15 + 4595616128696320*a^35*b^26*c^45*d^16 - 6215915829985280*a^36*b^25*c^4
4*d^17 + 7509830061260800*a^37*b^24*c^43*d^18 - 8292025971507200*a^38*b^23*c^42*d^19 + 8624070071418880*a^39*b
^22*c^41*d^20 - 8700497871503360*a^40*b^21*c^40*d^21 + 8624070071418880*a^41*b^20*c^39*d^22 - 8292025971507200
*a^42*b^19*c^38*d^23 + 7509830061260800*a^43*b^18*c^37*d^24 - 6215915829985280*a^44*b^17*c^36*d^25 + 459561612
8696320*a^45*b^16*c^35*d^26 - 2988135038320640*a^46*b^15*c^34*d^27 + 1690057100492800*a^47*b^14*c^33*d^28 - 82
3887134720000*a^48*b^13*c^32*d^29 + 342964201062400*a^49*b^12*c^31*d^30 - 120578363097088*a^50*b^11*c^30*d^31
+ 35303182041088*a^51*b^10*c^29*d^32 - 8446069964800*a^52*b^9*c^28*d^33 + 1607834009600*a^53*b^8*c^27*d^34 - 2
34134437888*a^54*b^7*c^26*d^35 + 24482152448*a^55*b^6*c^25*d^36 - 1635778560*a^56*b^5*c^24*d^37 + 52428800*a^5
7*b^4*c^23*d^38)*1i - 32768000*a^21*b^38*c^55*d^4 + 1009254400*a^22*b^37*c^54*d^5 - 14833418240*a^23*b^36*c^53
*d^6 + 138556735488*a^24*b^35*c^52*d^7 - 924185001984*a^25*b^34*c^51*d^8 + 4688465362944*a^26*b^33*c^50*d^9 -
18812623126528*a^27*b^32*c^49*d^10 + 61295191654400*a^28*b^31*c^48*d^11 - 165189260410880*a^29*b^30*c^47*d^12
+ 373165003898880*a^30*b^29*c^46*d^13 - 713540118773760*a^31*b^28*c^45*d^14 + 1163349301657600*a^32*b^27*c^44*
d^15 - 1627141704253440*a^33*b^26*c^43*d^16 + 1966197351383040*a^34*b^25*c^42*d^17 - 2079216623943680*a^35*b^2
4*c^41*d^18 + 1981073955225600*a^36*b^23*c^40*d^19 - 1807512431493120*a^37*b^22*c^39*d^20 + 1724885956034560*a
^38*b^21*c^38*d^21 - 1807512431493120*a^39*b^20*c^37*d^22 + 1981073955225600*a^40*b^19*c^36*d^23 - 20792166239
43680*a^41*b^18*c^35*d^24 + 1966197351383040*a^42*b^17*c^34*d^25 - 1627141704253440*a^43*b^16*c^33*d^26 + 1163
349301657600*a^44*b^15*c^32*d^27 - 713540118773760*a^45*b^14*c^31*d^28 + 373165003898880*a^46*b^13*c^30*d^29 -
165189260410880*a^47*b^12*c^29*d^30 + 61295191654400*a^48*b^11*c^28*d^31 - 18812623126528*a^49*b^10*c^27*d^32
+ 4688465362944*a^50*b^9*c^26*d^33 - 924185001984*a^51*b^8*c^25*d^34 + 138556735488*a^52*b^7*c^24*d^35 - 1483
3418240*a^53*b^6*c^23*d^36 + 1009254400*a^54*b^5*c^22*d^37 - 32768000*a^55*b^4*c^21*d^38)*1i - x^(1/2)*(540800
00*a^20*b^33*c^43*d^10 - 1361152000*a^21*b^32*c^42*d^11 + 16011852800*a^22*b^31*c^41*d^12 - 116736734720*a^23*
b^30*c^40*d^13 + 589861462528*a^24*b^29*c^39*d^14 - 2187899577344*a^25*b^28*c^38*d^15 + 6149347117056*a^26*b^2
7*c^37*d^16 - 13298820601344*a^27*b^26*c^36*d^17 + 22133436343296*a^28*b^25*c^35*d^18 - 27715689750528*a^29*b^
24*c^34*d^19 + 24077503776768*a^30*b^23*c^33*d^20 - 9645706816512*a^31*b^22*c^32*d^21 - 9645706816512*a^32*b^2
1*c^31*d^22 + 24077503776768*a^33*b^20*c^30*d^23 - 27715689750528*a^34*b^19*c^29*d^24 + 22133436343296*a^35*b^
18*c^28*d^25 - 13298820601344*a^36*b^17*c^27*d^26 + 6149347117056*a^37*b^16*c^26*d^27 - 2187899577344*a^38*b^1
5*c^25*d^28 + 589861462528*a^39*b^14*c^24*d^29 - 116736734720*a^40*b^13*c^23*d^30 + 16011852800*a^41*b^12*c^22
*d^31 - 1361152000*a^42*b^11*c^21*d^32 + 54080000*a^43*b^10*c^20*d^33))*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 -
43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 4915
2*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a
^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*
b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4) + ((-(625*a^4*d^13 + 28561*b^4*c^4*d^9
- 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49
152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032
*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^
9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(3/4)*(x^(1/2)*(-(625*a^4*d^13 + 28561*b^4*
c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d
^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 -
3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 9
01120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4)*(52428800*a^23*b^38*c^57*d^4
- 1635778560*a^24*b^37*c^56*d^5 + 24482152448*a^25*b^36*c^55*d^6 - 234134437888*a^26*b^35*c^54*d^7 + 160783400
9600*a^27*b^34*c^53*d^8 - 8446069964800*a^28*b^33*c^52*d^9 + 35303182041088*a^29*b^32*c^51*d^10 - 120578363097
088*a^30*b^31*c^50*d^11 + 342964201062400*a^31*b^30*c^49*d^12 - 823887134720000*a^32*b^29*c^48*d^13 + 16900571
00492800*a^33*b^28*c^47*d^14 - 2988135038320640*a^34*b^27*c^46*d^15 + 4595616128696320*a^35*b^26*c^45*d^16 - 6
215915829985280*a^36*b^25*c^44*d^17 + 7509830061260800*a^37*b^24*c^43*d^18 - 8292025971507200*a^38*b^23*c^42*d
^19 + 8624070071418880*a^39*b^22*c^41*d^20 - 8700497871503360*a^40*b^21*c^40*d^21 + 8624070071418880*a^41*b^20
*c^39*d^22 - 8292025971507200*a^42*b^19*c^38*d^23 + 7509830061260800*a^43*b^18*c^37*d^24 - 6215915829985280*a^
44*b^17*c^36*d^25 + 4595616128696320*a^45*b^16*c^35*d^26 - 2988135038320640*a^46*b^15*c^34*d^27 + 169005710049
2800*a^47*b^14*c^33*d^28 - 823887134720000*a^48*b^13*c^32*d^29 + 342964201062400*a^49*b^12*c^31*d^30 - 1205783
63097088*a^50*b^11*c^30*d^31 + 35303182041088*a^51*b^10*c^29*d^32 - 8446069964800*a^52*b^9*c^28*d^33 + 1607834
009600*a^53*b^8*c^27*d^34 - 234134437888*a^54*b^7*c^26*d^35 + 24482152448*a^55*b^6*c^25*d^36 - 1635778560*a^56
*b^5*c^24*d^37 + 52428800*a^57*b^4*c^23*d^38)*1i + 32768000*a^21*b^38*c^55*d^4 - 1009254400*a^22*b^37*c^54*d^5
+ 14833418240*a^23*b^36*c^53*d^6 - 138556735488*a^24*b^35*c^52*d^7 + 924185001984*a^25*b^34*c^51*d^8 - 468846
5362944*a^26*b^33*c^50*d^9 + 18812623126528*a^27*b^32*c^49*d^10 - 61295191654400*a^28*b^31*c^48*d^11 + 1651892
60410880*a^29*b^30*c^47*d^12 - 373165003898880*a^30*b^29*c^46*d^13 + 713540118773760*a^31*b^28*c^45*d^14 - 116
3349301657600*a^32*b^27*c^44*d^15 + 1627141704253440*a^33*b^26*c^43*d^16 - 1966197351383040*a^34*b^25*c^42*d^1
7 + 2079216623943680*a^35*b^24*c^41*d^18 - 1981073955225600*a^36*b^23*c^40*d^19 + 1807512431493120*a^37*b^22*c
^39*d^20 - 1724885956034560*a^38*b^21*c^38*d^21 + 1807512431493120*a^39*b^20*c^37*d^22 - 1981073955225600*a^40
*b^19*c^36*d^23 + 2079216623943680*a^41*b^18*c^35*d^24 - 1966197351383040*a^42*b^17*c^34*d^25 + 16271417042534
40*a^43*b^16*c^33*d^26 - 1163349301657600*a^44*b^15*c^32*d^27 + 713540118773760*a^45*b^14*c^31*d^28 - 37316500
3898880*a^46*b^13*c^30*d^29 + 165189260410880*a^47*b^12*c^29*d^30 - 61295191654400*a^48*b^11*c^28*d^31 + 18812
623126528*a^49*b^10*c^27*d^32 - 4688465362944*a^50*b^9*c^26*d^33 + 924185001984*a^51*b^8*c^25*d^34 - 138556735
488*a^52*b^7*c^24*d^35 + 14833418240*a^53*b^6*c^23*d^36 - 1009254400*a^54*b^5*c^22*d^37 + 32768000*a^55*b^4*c^
21*d^38)*1i - x^(1/2)*(54080000*a^20*b^33*c^43*d^10 - 1361152000*a^21*b^32*c^42*d^11 + 16011852800*a^22*b^31*c
^41*d^12 - 116736734720*a^23*b^30*c^40*d^13 + 589861462528*a^24*b^29*c^39*d^14 - 2187899577344*a^25*b^28*c^38*
d^15 + 6149347117056*a^26*b^27*c^37*d^16 - 13298820601344*a^27*b^26*c^36*d^17 + 22133436343296*a^28*b^25*c^35*
d^18 - 27715689750528*a^29*b^24*c^34*d^19 + 24077503776768*a^30*b^23*c^33*d^20 - 9645706816512*a^31*b^22*c^32*
d^21 - 9645706816512*a^32*b^21*c^31*d^22 + 24077503776768*a^33*b^20*c^30*d^23 - 27715689750528*a^34*b^19*c^29*
d^24 + 22133436343296*a^35*b^18*c^28*d^25 - 13298820601344*a^36*b^17*c^27*d^26 + 6149347117056*a^37*b^16*c^26*
d^27 - 2187899577344*a^38*b^15*c^25*d^28 + 589861462528*a^39*b^14*c^24*d^29 - 116736734720*a^40*b^13*c^23*d^30
+ 16011852800*a^41*b^12*c^22*d^31 - 1361152000*a^42*b^11*c^21*d^32 + 54080000*a^43*b^10*c^20*d^33))*(-(625*a^
4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^2
1 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520
*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a
^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4))/(((-(625*
a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c
^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 20275
20*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520
*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(3/4)*(x^(1/2)
*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(409
6*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3
+ 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 +
2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4)*
(52428800*a^23*b^38*c^57*d^4 - 1635778560*a^24*b^37*c^56*d^5 + 24482152448*a^25*b^36*c^55*d^6 - 234134437888*a
^26*b^35*c^54*d^7 + 1607834009600*a^27*b^34*c^53*d^8 - 8446069964800*a^28*b^33*c^52*d^9 + 35303182041088*a^29*
b^32*c^51*d^10 - 120578363097088*a^30*b^31*c^50*d^11 + 342964201062400*a^31*b^30*c^49*d^12 - 823887134720000*a
^32*b^29*c^48*d^13 + 1690057100492800*a^33*b^28*c^47*d^14 - 2988135038320640*a^34*b^27*c^46*d^15 + 45956161286
96320*a^35*b^26*c^45*d^16 - 6215915829985280*a^36*b^25*c^44*d^17 + 7509830061260800*a^37*b^24*c^43*d^18 - 8292
025971507200*a^38*b^23*c^42*d^19 + 8624070071418880*a^39*b^22*c^41*d^20 - 8700497871503360*a^40*b^21*c^40*d^21
+ 8624070071418880*a^41*b^20*c^39*d^22 - 8292025971507200*a^42*b^19*c^38*d^23 + 7509830061260800*a^43*b^18*c^
37*d^24 - 6215915829985280*a^44*b^17*c^36*d^25 + 4595616128696320*a^45*b^16*c^35*d^26 - 2988135038320640*a^46*
b^15*c^34*d^27 + 1690057100492800*a^47*b^14*c^33*d^28 - 823887134720000*a^48*b^13*c^32*d^29 + 342964201062400*
a^49*b^12*c^31*d^30 - 120578363097088*a^50*b^11*c^30*d^31 + 35303182041088*a^51*b^10*c^29*d^32 - 8446069964800
*a^52*b^9*c^28*d^33 + 1607834009600*a^53*b^8*c^27*d^34 - 234134437888*a^54*b^7*c^26*d^35 + 24482152448*a^55*b^
6*c^25*d^36 - 1635778560*a^56*b^5*c^24*d^37 + 52428800*a^57*b^4*c^23*d^38)*1i - 32768000*a^21*b^38*c^55*d^4 +
1009254400*a^22*b^37*c^54*d^5 - 14833418240*a^23*b^36*c^53*d^6 + 138556735488*a^24*b^35*c^52*d^7 - 92418500198
4*a^25*b^34*c^51*d^8 + 4688465362944*a^26*b^33*c^50*d^9 - 18812623126528*a^27*b^32*c^49*d^10 + 61295191654400*
a^28*b^31*c^48*d^11 - 165189260410880*a^29*b^30*c^47*d^12 + 373165003898880*a^30*b^29*c^46*d^13 - 713540118773
760*a^31*b^28*c^45*d^14 + 1163349301657600*a^32*b^27*c^44*d^15 - 1627141704253440*a^33*b^26*c^43*d^16 + 196619
7351383040*a^34*b^25*c^42*d^17 - 2079216623943680*a^35*b^24*c^41*d^18 + 1981073955225600*a^36*b^23*c^40*d^19 -
1807512431493120*a^37*b^22*c^39*d^20 + 1724885956034560*a^38*b^21*c^38*d^21 - 1807512431493120*a^39*b^20*c^37
*d^22 + 1981073955225600*a^40*b^19*c^36*d^23 - 2079216623943680*a^41*b^18*c^35*d^24 + 1966197351383040*a^42*b^
17*c^34*d^25 - 1627141704253440*a^43*b^16*c^33*d^26 + 1163349301657600*a^44*b^15*c^32*d^27 - 713540118773760*a
^45*b^14*c^31*d^28 + 373165003898880*a^46*b^13*c^30*d^29 - 165189260410880*a^47*b^12*c^29*d^30 + 6129519165440
0*a^48*b^11*c^28*d^31 - 18812623126528*a^49*b^10*c^27*d^32 + 4688465362944*a^50*b^9*c^26*d^33 - 924185001984*a
^51*b^8*c^25*d^34 + 138556735488*a^52*b^7*c^24*d^35 - 14833418240*a^53*b^6*c^23*d^36 + 1009254400*a^54*b^5*c^2
2*d^37 - 32768000*a^55*b^4*c^21*d^38)*1i - x^(1/2)*(54080000*a^20*b^33*c^43*d^10 - 1361152000*a^21*b^32*c^42*d
^11 + 16011852800*a^22*b^31*c^41*d^12 - 116736734720*a^23*b^30*c^40*d^13 + 589861462528*a^24*b^29*c^39*d^14 -
2187899577344*a^25*b^28*c^38*d^15 + 6149347117056*a^26*b^27*c^37*d^16 - 13298820601344*a^27*b^26*c^36*d^17 + 2
2133436343296*a^28*b^25*c^35*d^18 - 27715689750528*a^29*b^24*c^34*d^19 + 24077503776768*a^30*b^23*c^33*d^20 -
9645706816512*a^31*b^22*c^32*d^21 - 9645706816512*a^32*b^21*c^31*d^22 + 24077503776768*a^33*b^20*c^30*d^23 - 2
7715689750528*a^34*b^19*c^29*d^24 + 22133436343296*a^35*b^18*c^28*d^25 - 13298820601344*a^36*b^17*c^27*d^26 +
6149347117056*a^37*b^16*c^26*d^27 - 2187899577344*a^38*b^15*c^25*d^28 + 589861462528*a^39*b^14*c^24*d^29 - 116
736734720*a^40*b^13*c^23*d^30 + 16011852800*a^41*b^12*c^22*d^31 - 1361152000*a^42*b^11*c^21*d^32 + 54080000*a^
43*b^10*c^20*d^33))*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500
*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 9011
20*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032
*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b
^11*c^20*d))^(1/4)*1i - ((-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 -
6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 -
901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 32
44032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 4915
2*a*b^11*c^20*d))^(3/4)*(x^(1/2)*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^
2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^
19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*
d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^1
0 - 49152*a*b^11*c^20*d))^(1/4)*(52428800*a^23*b^38*c^57*d^4 - 1635778560*a^24*b^37*c^56*d^5 + 24482152448*a^2
5*b^36*c^55*d^6 - 234134437888*a^26*b^35*c^54*d^7 + 1607834009600*a^27*b^34*c^53*d^8 - 8446069964800*a^28*b^33
*c^52*d^9 + 35303182041088*a^29*b^32*c^51*d^10 - 120578363097088*a^30*b^31*c^50*d^11 + 342964201062400*a^31*b^
30*c^49*d^12 - 823887134720000*a^32*b^29*c^48*d^13 + 1690057100492800*a^33*b^28*c^47*d^14 - 2988135038320640*a
^34*b^27*c^46*d^15 + 4595616128696320*a^35*b^26*c^45*d^16 - 6215915829985280*a^36*b^25*c^44*d^17 + 75098300612
60800*a^37*b^24*c^43*d^18 - 8292025971507200*a^38*b^23*c^42*d^19 + 8624070071418880*a^39*b^22*c^41*d^20 - 8700
497871503360*a^40*b^21*c^40*d^21 + 8624070071418880*a^41*b^20*c^39*d^22 - 8292025971507200*a^42*b^19*c^38*d^23
+ 7509830061260800*a^43*b^18*c^37*d^24 - 6215915829985280*a^44*b^17*c^36*d^25 + 4595616128696320*a^45*b^16*c^
35*d^26 - 2988135038320640*a^46*b^15*c^34*d^27 + 1690057100492800*a^47*b^14*c^33*d^28 - 823887134720000*a^48*b
^13*c^32*d^29 + 342964201062400*a^49*b^12*c^31*d^30 - 120578363097088*a^50*b^11*c^30*d^31 + 35303182041088*a^5
1*b^10*c^29*d^32 - 8446069964800*a^52*b^9*c^28*d^33 + 1607834009600*a^53*b^8*c^27*d^34 - 234134437888*a^54*b^7
*c^26*d^35 + 24482152448*a^55*b^6*c^25*d^36 - 1635778560*a^56*b^5*c^24*d^37 + 52428800*a^57*b^4*c^23*d^38)*1i
+ 32768000*a^21*b^38*c^55*d^4 - 1009254400*a^22*b^37*c^54*d^5 + 14833418240*a^23*b^36*c^53*d^6 - 138556735488*
a^24*b^35*c^52*d^7 + 924185001984*a^25*b^34*c^51*d^8 - 4688465362944*a^26*b^33*c^50*d^9 + 18812623126528*a^27*
b^32*c^49*d^10 - 61295191654400*a^28*b^31*c^48*d^11 + 165189260410880*a^29*b^30*c^47*d^12 - 373165003898880*a^
30*b^29*c^46*d^13 + 713540118773760*a^31*b^28*c^45*d^14 - 1163349301657600*a^32*b^27*c^44*d^15 + 1627141704253
440*a^33*b^26*c^43*d^16 - 1966197351383040*a^34*b^25*c^42*d^17 + 2079216623943680*a^35*b^24*c^41*d^18 - 198107
3955225600*a^36*b^23*c^40*d^19 + 1807512431493120*a^37*b^22*c^39*d^20 - 1724885956034560*a^38*b^21*c^38*d^21 +
1807512431493120*a^39*b^20*c^37*d^22 - 1981073955225600*a^40*b^19*c^36*d^23 + 2079216623943680*a^41*b^18*c^35
*d^24 - 1966197351383040*a^42*b^17*c^34*d^25 + 1627141704253440*a^43*b^16*c^33*d^26 - 1163349301657600*a^44*b^
15*c^32*d^27 + 713540118773760*a^45*b^14*c^31*d^28 - 373165003898880*a^46*b^13*c^30*d^29 + 165189260410880*a^4
7*b^12*c^29*d^30 - 61295191654400*a^48*b^11*c^28*d^31 + 18812623126528*a^49*b^10*c^27*d^32 - 4688465362944*a^5
0*b^9*c^26*d^33 + 924185001984*a^51*b^8*c^25*d^34 - 138556735488*a^52*b^7*c^24*d^35 + 14833418240*a^53*b^6*c^2
3*d^36 - 1009254400*a^54*b^5*c^22*d^37 + 32768000*a^55*b^4*c^21*d^38)*1i - x^(1/2)*(54080000*a^20*b^33*c^43*d^
10 - 1361152000*a^21*b^32*c^42*d^11 + 16011852800*a^22*b^31*c^41*d^12 - 116736734720*a^23*b^30*c^40*d^13 + 589
861462528*a^24*b^29*c^39*d^14 - 2187899577344*a^25*b^28*c^38*d^15 + 6149347117056*a^26*b^27*c^37*d^16 - 132988
20601344*a^27*b^26*c^36*d^17 + 22133436343296*a^28*b^25*c^35*d^18 - 27715689750528*a^29*b^24*c^34*d^19 + 24077
503776768*a^30*b^23*c^33*d^20 - 9645706816512*a^31*b^22*c^32*d^21 - 9645706816512*a^32*b^21*c^31*d^22 + 240775
03776768*a^33*b^20*c^30*d^23 - 27715689750528*a^34*b^19*c^29*d^24 + 22133436343296*a^35*b^18*c^28*d^25 - 13298
820601344*a^36*b^17*c^27*d^26 + 6149347117056*a^37*b^16*c^26*d^27 - 2187899577344*a^38*b^15*c^25*d^28 + 589861
462528*a^39*b^14*c^24*d^29 - 116736734720*a^40*b^13*c^23*d^30 + 16011852800*a^41*b^12*c^22*d^31 - 1361152000*a
^42*b^11*c^21*d^32 + 54080000*a^43*b^10*c^20*d^33))*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10
+ 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 +
270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 37
84704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 27033
6*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4)*1i + 175760000*a^20*b^31*c^39*d^12 - 3507088000*a^21*b^30*c
^38*d^13 + 32026300800*a^22*b^29*c^37*d^14 - 177335474560*a^23*b^28*c^36*d^15 + 664045775360*a^24*b^27*c^35*d^
16 - 1770849815040*a^25*b^26*c^34*d^17 + 3431196106240*a^26*b^25*c^33*d^18 - 4778178444800*a^27*b^24*c^32*d^19
+ 4440824728320*a^28*b^23*c^31*d^20 - 1838397848320*a^29*b^22*c^30*d^21 - 1838397848320*a^30*b^21*c^29*d^22 +
4440824728320*a^31*b^20*c^28*d^23 - 4778178444800*a^32*b^19*c^27*d^24 + 3431196106240*a^33*b^18*c^26*d^25 - 1
770849815040*a^34*b^17*c^25*d^26 + 664045775360*a^35*b^16*c^24*d^27 - 177335474560*a^36*b^15*c^23*d^28 + 32026
300800*a^37*b^14*c^22*d^29 - 3507088000*a^38*b^13*c^21*d^30 + 175760000*a^39*b^12*c^20*d^31))*(-(625*a^4*d^13
+ 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 409
6*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^
8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*
c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^(1/4) + 2*atan((819200*
a^11*b^20*c^23*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6
500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 -
901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 324
4032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*
a^20*b*c*d^11))^(5/4) + 845000*a^17*b^6*d^15*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^
3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d
+ 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 +
3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270
336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4) + 819200*a^31*c^3*d^20*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4
*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*
c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4
- 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 -
901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) - 9438000*a^16*b^7*c*d^14*x^
(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)
/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c
^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5
*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(
1/4) - 14090240*a^12*b^19*c^22*d*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^
2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^1
1*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15
*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2
*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) - 14090240*a^30*b*c^4*d^19*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4
- 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 4
9152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 32440
32*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*
a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) + 845000*a^8*b^15*c^9*d^6*x^(1/2)*(-
(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a
^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 +
2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2
027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4) - 9
438000*a^9*b^14*c^8*d^7*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^
2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^
10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*
d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10
- 49152*a^20*b*c*d^11))^(1/4) + 41158200*a^10*b^13*c^7*d^8*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 4394
0*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^
10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14
*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^
3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4) - 81358680*a^11*b^12*c^6*d^9*x^(1/2)*(-(625
*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*
d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 202
7520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 20275
20*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4) + 50890
632*a^12*b^11*c^5*d^10*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2
*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^1
0*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d
^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10
- 49152*a^20*b*c*d^11))^(1/4) + 50890632*a^13*b^10*c^4*d^11*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 4394
0*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^
10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14
*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^
3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4) - 81358680*a^14*b^9*c^3*d^12*x^(1/2)*(-(625
*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*
d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 202
7520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 20275
20*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4) + 41158
200*a^15*b^8*c^2*d^13*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*
d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10
*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^
6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 -
49152*a^20*b*c*d^11))^(1/4) + 110723072*a^13*b^18*c^21*d^2*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 4394
0*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^
10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14
*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^
3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) - 527826944*a^14*b^17*c^20*d^3*x^(1/2)*(-(6
25*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^2
1*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2
027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 202
7520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) + 170
8163072*a^15*b^16*c^19*d^4*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11
*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10
*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c
^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d
^10 - 49152*a^20*b*c*d^11))^(5/4) - 3975741440*a^16*b^15*c^18*d^5*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4
- 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49
152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 324403
2*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a
^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) + 6877478912*a^17*b^14*c^17*d^6*x^(1/
2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4
096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*
d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^
7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4
) - 9041543168*a^18*b^13*c^16*d^7*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a
^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^
11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^1
5*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^
2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) + 9313648640*a^19*b^12*c^15*d^8*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b
^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^
12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 -
3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 9
01120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) - 8184070144*a^20*b^11*c^14*d^
9*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^
3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b
^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5
*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11
))^(5/4) + 7464878080*a^21*b^10*c^13*d^10*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 +
25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 2
70336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 378
4704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336
*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) - 8184070144*a^22*b^9*c^12*d^11*x^(1/2)*(-(625*b^13*c^4 + 285
61*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9
*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c
^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4
*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) + 9313648640*a^23*b^8*
c^11*d^12*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a
*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 90112
0*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*
a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*
b*c*d^11))^(5/4) - 9041543168*a^24*b^7*c^10*d^13*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*
c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^1
1*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^
5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 +
270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) + 6877478912*a^25*b^6*c^9*d^14*x^(1/2)*(-(625*b^13*c^4
+ 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 40
96*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13
*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b
^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) - 3975741440*a^2
6*b^5*c^8*d^15*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6
500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 -
901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 324
4032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*
a^20*b*c*d^11))^(5/4) + 1708163072*a^27*b^4*c^7*d^16*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b
^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11
*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^
7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d
^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) - 527826944*a^28*b^3*c^6*d^17*x^(1/2)*(-(625*b^13*
c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 +
4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a
^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^1
7*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4) + 110723072*a
^29*b^2*c^5*d^18*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 -
6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2
- 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3
244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 4915
2*a^20*b*c*d^11))^(5/4))/(78125*b^21*c^13 - 1373125*a^13*b^8*d^13 + 11745500*a^12*b^9*c*d^12 + 7293750*a^2*b^1
9*c^11*d^2 - 22537500*a^3*b^18*c^10*d^3 + 34273125*a^4*b^17*c^9*d^4 - 17203200*a^5*b^16*c^8*d^5 - 8028160*a^6*
b^15*c^7*d^6 - 950272*a^7*b^14*c^6*d^7 + 4030464*a^8*b^13*c^5*d^8 + 3343923*a^9*b^12*c^4*d^9 + 30329580*a^10*b
^11*c^3*d^10 - 33424950*a^11*b^10*c^2*d^11 - 1187500*a*b^20*c^12*d))*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 439
40*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a
^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^1
4*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b
^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4) + atan((a^11*b^20*c^23*x^(1/2)*(-(625*b^13
*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12
+ 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*
a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^
17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*819200i + a^
17*b^6*d^15*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500
*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901
120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 324403
2*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^2
0*b*c*d^11))^(1/4)*845000i + a^31*c^3*d^20*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3
+ 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d +
270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 37
84704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 27033
6*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*819200i - a^16*b^7*c*d^14*x^(1/2)*(-(625*b^13*c^4 + 28561*a^
4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12
*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^
4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8
- 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4)*9438000i - a^12*b^19*c^22*d
*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3
*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^
9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*
c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11)
)^(5/4)*14090240i - a^30*b*c^4*d^19*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350
*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*
a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a
^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*
b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*14090240i + a^8*b^15*c^9*d^6*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^
9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^1
2 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 -
3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 90
1120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4)*845000i - a^9*b^14*c^8*d^7*x^(1
/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(
4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9
*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d
^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/
4)*9438000i + a^10*b^13*c^7*d^8*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2
*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11
*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*
b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*
c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4)*41158200i - a^11*b^12*c^6*d^9*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d
^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 -
49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 324
4032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 90112
0*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4)*81358680i + a^12*b^11*c^5*d^10*x^(
1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/
(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^
9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*
d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1
/4)*50890632i + a^13*b^10*c^4*d^11*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*
a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a
^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^
15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b
^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4)*50890632i - a^14*b^9*c^3*d^12*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^
9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^1
2 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 -
3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 90
1120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4)*81358680i + a^15*b^8*c^2*d^13*x
^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d
)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*
c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^
5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^
(1/4)*41158200i + a^13*b^18*c^21*d^2*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 2535
0*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336
*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*
a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19
*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*110723072i - a^14*b^17*c^20*d^3*x^(1/2)*(-(625*b^13*c^4 + 28561*a^
4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12
*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^
4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8
- 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*527826944i + a^15*b^16*c^19
*d^4*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12
*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^1
2*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*
b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d
^11))^(5/4)*1708163072i - a^16*b^15*c^18*d^5*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^
3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d
+ 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 +
3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270
336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*3975741440i + a^17*b^14*c^17*d^6*x^(1/2)*(-(625*b^13*c^4 +
28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096
*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b
^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4
*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*6877478912i - a^18
*b^13*c^16*d^7*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6
500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 -
901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 324
4032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*
a^20*b*c*d^11))^(5/4)*9041543168i + a^19*b^12*c^15*d^8*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3
*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^
11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*
c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3
*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*9313648640i - a^20*b^11*c^14*d^9*x^(1/2)*(-(625*
b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d
^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027
520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 202752
0*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*81840701
44i + a^21*b^10*c^13*d^10*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*
c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*
c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^
6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^
10 - 49152*a^20*b*c*d^11))^(5/4)*7464878080i - a^22*b^9*c^12*d^11*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4
- 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49
152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 324403
2*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a
^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*8184070144i + a^23*b^8*c^11*d^12*x^(1
/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(
4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9
*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d
^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/
4)*9313648640i - a^24*b^7*c^10*d^13*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350
*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*
a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a
^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*
b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*9041543168i + a^25*b^6*c^9*d^14*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4
*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*
c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4
- 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 -
901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*6877478912i - a^26*b^5*c^8*d
^15*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*
c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12
*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b
^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^
11))^(5/4)*3975741440i + a^27*b^4*c^7*d^16*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3
+ 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d +
270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 37
84704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 27033
6*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*1708163072i - a^28*b^3*c^6*d^17*x^(1/2)*(-(625*b^13*c^4 + 28
561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^
9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*
c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^
4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(5/4)*527826944i + a^29*b^2
*c^5*d^18*x^(1/2)*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 - 43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a
*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 49152*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 90112
0*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*
a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*
b*c*d^11))^(5/4)*110723072i)/(78125*b^21*c^13 - 1373125*a^13*b^8*d^13 + 11745500*a^12*b^9*c*d^12 + 7293750*a^2
*b^19*c^11*d^2 - 22537500*a^3*b^18*c^10*d^3 + 34273125*a^4*b^17*c^9*d^4 - 17203200*a^5*b^16*c^8*d^5 - 8028160*
a^6*b^15*c^7*d^6 - 950272*a^7*b^14*c^6*d^7 + 4030464*a^8*b^13*c^5*d^8 + 3343923*a^9*b^12*c^4*d^9 + 30329580*a^
10*b^11*c^3*d^10 - 33424950*a^11*b^10*c^2*d^11 - 1187500*a*b^20*c^12*d))*(-(625*b^13*c^4 + 28561*a^4*b^9*d^4 -
43940*a^3*b^10*c*d^3 + 25350*a^2*b^11*c^2*d^2 - 6500*a*b^12*c^3*d)/(4096*a^21*d^12 + 4096*a^9*b^12*c^12 - 491
52*a^10*b^11*c^11*d + 270336*a^11*b^10*c^10*d^2 - 901120*a^12*b^9*c^9*d^3 + 2027520*a^13*b^8*c^8*d^4 - 3244032
*a^14*b^7*c^7*d^5 + 3784704*a^15*b^6*c^6*d^6 - 3244032*a^16*b^5*c^5*d^7 + 2027520*a^17*b^4*c^4*d^8 - 901120*a^
18*b^3*c^3*d^9 + 270336*a^19*b^2*c^2*d^10 - 49152*a^20*b*c*d^11))^(1/4)*2i - (2/(a*c) + (x^2*(5*a^3*d^3 + 5*b^
3*c^3 - 4*a*b^2*c^2*d - 4*a^2*b*c*d^2))/(2*a^2*c^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (b*d*x^4*(5*a^2*d^2 + 5*
b^2*c^2 - 8*a*b*c*d))/(2*a^2*c^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x^(5/2)*(a*d + b*c) + a*c*x^(1/2) + b*d*x^
(9/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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