∫ x3∕2 ___________________________

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

Optimal. Leaf size=601 \[ -\frac {b^{3/4} (7 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{3/4} (7 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^3}-\frac {b^{3/4} (7 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{3/4} (7 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} (a d+7 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} (b c-a d)^3}-\frac {d^{3/4} (a d+7 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} (b c-a d)^3}+\frac {d^{3/4} (a d+7 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} (b c-a d)^3}-\frac {d^{3/4} (a d+7 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{3/4} (b c-a d)^3}-\frac {d \sqrt {x}}{\left (c+d x^2\right ) (b c-a d)^2}-\frac {\sqrt {x}}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]

[Out]

-1/8*b^(3/4)*(7*a*d+b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/8*b^(3/4)*(7

*a*d+b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/8*d^(3/4)*(a*d+7*b*c)*arcta

n(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(3/4)/(-a*d+b*c)^3*2^(1/2)-1/8*d^(3/4)*(a*d+7*b*c)*arctan(1+d^(1/4)*2^(

1/2)*x^(1/2)/c^(1/4))/c^(3/4)/(-a*d+b*c)^3*2^(1/2)-1/16*b^(3/4)*(7*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/

4)*2^(1/2)*x^(1/2))/a^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/16*b^(3/4)*(7*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)

*2^(1/2)*x^(1/2))/a^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/16*d^(3/4)*(a*d+7*b*c)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2

^(1/2)*x^(1/2))/c^(3/4)/(-a*d+b*c)^3*2^(1/2)-1/16*d^(3/4)*(a*d+7*b*c)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(

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

d*x^2+c)

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Rubi [A] time = 0.69, antiderivative size = 601, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 471, 527, 522, 211, 1165, 628, 1162, 617, 204} \[ -\frac {b^{3/4} (7 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{3/4} (7 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^3}-\frac {b^{3/4} (7 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{3/4} (7 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} (a d+7 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} (b c-a d)^3}-\frac {d^{3/4} (a d+7 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} (b c-a d)^3}+\frac {d^{3/4} (a d+7 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} (b c-a d)^3}-\frac {d^{3/4} (a d+7 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{3/4} (b c-a d)^3}-\frac {d \sqrt {x}}{\left (c+d x^2\right ) (b c-a d)^2}-\frac {\sqrt {x}}{2 \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((d*Sqrt[x])/((b*c - a*d)^2*(c + d*x^2))) - Sqrt[x]/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (b^(3/4)*(b*c +

7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (b^(3/4)*(b*c + 7*a

*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(7*b*c + a*d)*

ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) - (d^(3/4)*(7*b*c + a*d)*ArcT

an[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)*(b*c - a*d)^3) - (b^(3/4)*(b*c + 7*a*d)*Log[Sqrt

[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (b^(3/4)*(b*c + 7*a*d)

*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(7*b

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

3/4)*(7*b*c + a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(3/4)*(b*c - a*d)^

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /

; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di

st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[

{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b

]]))

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno

minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*

x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege

rQ[p]

Rule 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -

1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -

a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)

+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n

, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f

)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a

, b, c, d, e, f, n}, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[

((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d

)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)

*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub

st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;

FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +

c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S

imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},

x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

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

(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /

; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {x^{3/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^4}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {c-7 d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)}\\ &=-\frac {d \sqrt {x}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {4 c (b c+a d)-24 b c d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{8 c (b c-a d)^2}\\ &=-\frac {d \sqrt {x}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {(d (7 b c+a d)) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^3}+\frac {(b (b c+7 a d)) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {(d (7 b c+a d)) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {c} (b c-a d)^3}-\frac {(d (7 b c+a d)) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {c} (b c-a d)^3}+\frac {(b (b c+7 a d)) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {a} (b c-a d)^3}+\frac {(b (b c+7 a d)) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {a} (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (\sqrt {d} (7 b c+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 \sqrt {c} (b c-a d)^3}-\frac {\left (\sqrt {d} (7 b c+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 \sqrt {c} (b c-a d)^3}+\frac {\left (d^{3/4} (7 b c+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^{3/4} (b c-a d)^3}+\frac {\left (d^{3/4} (7 b c+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^{3/4} (b c-a d)^3}+\frac {\left (\sqrt {b} (b c+7 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 \sqrt {a} (b c-a d)^3}+\frac {\left (\sqrt {b} (b c+7 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 \sqrt {a} (b c-a d)^3}-\frac {\left (b^{3/4} (b c+7 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^{3/4} (b c-a d)^3}-\frac {\left (b^{3/4} (b c+7 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^{3/4} (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{3/4} (b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{3/4} (b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} (7 b c+a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} (b c-a d)^3}-\frac {d^{3/4} (7 b c+a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} (b c-a d)^3}-\frac {\left (d^{3/4} (7 b c+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^{3/4} (b c-a d)^3}+\frac {\left (d^{3/4} (7 b c+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^{3/4} (b c-a d)^3}+\frac {\left (b^{3/4} (b c+7 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^{3/4} (b c-a d)^3}-\frac {\left (b^{3/4} (b c+7 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^{3/4} (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{(b c-a d)^2 \left (c+d x^2\right )}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{3/4} (b c+7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{3/4} (b c+7 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} (7 b c+a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} (b c-a d)^3}-\frac {d^{3/4} (7 b c+a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} (b c-a d)^3}-\frac {b^{3/4} (b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{3/4} (b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} (7 b c+a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} (b c-a d)^3}-\frac {d^{3/4} (7 b c+a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A] time = 1.03, size = 575, normalized size = 0.96 \[ \frac {1}{16} \left (\frac {\sqrt {2} b^{3/4} (7 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{3/4} (a d-b c)^3}+\frac {\sqrt {2} b^{3/4} (7 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{3/4} (b c-a d)^3}+\frac {2 \sqrt {2} b^{3/4} (7 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{3/4} (a d-b c)^3}-\frac {2 \sqrt {2} b^{3/4} (7 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{3/4} (a d-b c)^3}+\frac {\sqrt {2} d^{3/4} (a d+7 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{3/4} (b c-a d)^3}+\frac {\sqrt {2} d^{3/4} (a d+7 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{3/4} (a d-b c)^3}+\frac {2 \sqrt {2} d^{3/4} (a d+7 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{3/4} (b c-a d)^3}-\frac {2 \sqrt {2} d^{3/4} (a d+7 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{3/4} (b c-a d)^3}-\frac {8 b \sqrt {x}}{\left (a+b x^2\right ) (b c-a d)^2}-\frac {8 d \sqrt {x}}{\left (c+d x^2\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-8*b*Sqrt[x])/((b*c - a*d)^2*(a + b*x^2)) - (8*d*Sqrt[x])/((b*c - a*d)^2*(c + d*x^2)) + (2*Sqrt[2]*b^(3/4)*(

b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*(-(b*c) + a*d)^3) - (2*Sqrt[2]*b^(3/4)*(b

*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*d^(3/4)*(7*

b*c + a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(3/4)*(b*c - a*d)^3) - (2*Sqrt[2]*d^(3/4)*(7*b*c

+ a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(3/4)*(b*c - a*d)^3) + (Sqrt[2]*b^(3/4)*(b*c + 7*a*d)

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

c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*(b*c - a*d)^3) + (Sqrt[2]*d^(3

/4)*(7*b*c + a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(3/4)*(b*c - a*d)^3) + (Sqrt[

2]*d^(3/4)*(7*b*c + a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(3/4)*(-(b*c) + a*d)^3

))/16

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fricas [B] time = 130.29, size = 5474, normalized size = 9.11 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(b^7*c^4 + 28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3*b^4*c*d^3 +

2401*a^4*b^3*d^4)/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*

b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^

12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12))^(1/4)*arctan(-((a^2*b^9*c^9 - 9*a^3*b^8

*c^8*d + 36*a^4*b^7*c^7*d^2 - 84*a^5*b^6*c^6*d^3 + 126*a^6*b^5*c^5*d^4 - 126*a^7*b^4*c^4*d^5 + 84*a^8*b^3*c^3*

d^6 - 36*a^9*b^2*c^2*d^7 + 9*a^10*b*c*d^8 - a^11*d^9)*sqrt((b^4*c^2 + 14*a*b^3*c*d + 49*a^2*b^2*d^2)*x + (a^2*

b^6*c^6 - 6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6*b^2*c^2*d^4 - 6*a^7*b*c*d^5 + a^8

*d^6)*sqrt(-(b^7*c^4 + 28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/(a^3*b^12

*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^

7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2

*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)))*(-(b^7*c^4 + 28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3*b^4*c

*d^3 + 2401*a^4*b^3*d^4)/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 49

5*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 -

220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12))^(3/4) - (a^2*b^11*c^10 - 2*a^3*b^

10*c^9*d - 27*a^4*b^9*c^8*d^2 + 168*a^5*b^8*c^7*d^3 - 462*a^6*b^7*c^6*d^4 + 756*a^7*b^6*c^5*d^5 - 798*a^8*b^5*

c^4*d^6 + 552*a^9*b^4*c^3*d^7 - 243*a^10*b^3*c^2*d^8 + 62*a^11*b^2*c*d^9 - 7*a^12*b*d^10)*sqrt(x)*(-(b^7*c^4 +

28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/(a^3*b^12*c^12 - 12*a^4*b^11*c^

11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^

6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*

c*d^11 + a^15*d^12))^(3/4))/(b^7*c^4 + 28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3*b^4*c*d^3 + 2401*a^4*b^

3*d^4)) - 4*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 -

a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*

a^3*b*c*d^6 + a^4*d^7)/(b^12*c^15 - 12*a*b^11*c^14*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495*a^4*b

^8*c^11*d^4 - 792*a^5*b^7*c^10*d^5 + 924*a^6*b^6*c^9*d^6 - 792*a^7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 220*a^9

*b^3*c^6*d^9 + 66*a^10*b^2*c^5*d^10 - 12*a^11*b*c^4*d^11 + a^12*c^3*d^12))^(1/4)*arctan(-((b^9*c^11 - 9*a*b^8*

c^10*d + 36*a^2*b^7*c^9*d^2 - 84*a^3*b^6*c^8*d^3 + 126*a^4*b^5*c^7*d^4 - 126*a^5*b^4*c^6*d^5 + 84*a^6*b^3*c^5*

d^6 - 36*a^7*b^2*c^4*d^7 + 9*a^8*b*c^3*d^8 - a^9*c^2*d^9)*sqrt((49*b^2*c^2*d^2 + 14*a*b*c*d^3 + a^2*d^4)*x + (

b^6*c^8 - 6*a*b^5*c^7*d + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a^5*b*c^3*d^5 + a^6

*c^2*d^6)*sqrt(-(2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^4*d^7)/(b^12

*c^15 - 12*a*b^11*c^14*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495*a^4*b^8*c^11*d^4 - 792*a^5*b^7*c^

10*d^5 + 924*a^6*b^6*c^9*d^6 - 792*a^7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 + 66*a^10*b^2*c

^5*d^10 - 12*a^11*b*c^4*d^11 + a^12*c^3*d^12)))*(-(2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5

+ 28*a^3*b*c*d^6 + a^4*d^7)/(b^12*c^15 - 12*a*b^11*c^14*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495

*a^4*b^8*c^11*d^4 - 792*a^5*b^7*c^10*d^5 + 924*a^6*b^6*c^9*d^6 - 792*a^7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 2

20*a^9*b^3*c^6*d^9 + 66*a^10*b^2*c^5*d^10 - 12*a^11*b*c^4*d^11 + a^12*c^3*d^12))^(3/4) - (7*b^10*c^12*d - 62*a

*b^9*c^11*d^2 + 243*a^2*b^8*c^10*d^3 - 552*a^3*b^7*c^9*d^4 + 798*a^4*b^6*c^8*d^5 - 756*a^5*b^5*c^7*d^6 + 462*a

^6*b^4*c^6*d^7 - 168*a^7*b^3*c^5*d^8 + 27*a^8*b^2*c^4*d^9 + 2*a^9*b*c^3*d^10 - a^10*c^2*d^11)*sqrt(x)*(-(2401*

b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^4*d^7)/(b^12*c^15 - 12*a*b^11*c^14

*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495*a^4*b^8*c^11*d^4 - 792*a^5*b^7*c^10*d^5 + 924*a^6*b^6*c

^9*d^6 - 792*a^7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 + 66*a^10*b^2*c^5*d^10 - 12*a^11*b*c^

4*d^11 + a^12*c^3*d^12))^(3/4))/(2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6

+ a^4*d^7)) - (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3

- a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(b^7*c^4 + 28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3*b^4*c

*d^3 + 2401*a^4*b^3*d^4)/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 49

5*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 -

220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12))^(1/4)*log((b^2*c + 7*a*b*d)*sqrt(

x) + (a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*(-(b^7*c^4 + 28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2

+ 1372*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6

*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^

11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12))^(1/4)) + (a*b^2*

c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b

*c*d^2 + a^3*d^3)*x^2)*(-(b^7*c^4 + 28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3*b^4*c*d^3 + 2401*a^4*b^3*d

^4)/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 7

92*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9

+ 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12))^(1/4)*log((b^2*c + 7*a*b*d)*sqrt(x) - (a*b^3*c^3 - 3*a

^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*(-(b^7*c^4 + 28*a*b^6*c^3*d + 294*a^2*b^5*c^2*d^2 + 1372*a^3*b^4*c*d^3

+ 2401*a^4*b^3*d^4)/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^

7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*

a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12))^(1/4)) + (a*b^2*c^3 - 2*a^2*b*c^2*d +

a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2

)*(-(2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^4*d^7)/(b^12*c^15 - 12*a

*b^11*c^14*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495*a^4*b^8*c^11*d^4 - 792*a^5*b^7*c^10*d^5 + 924

*a^6*b^6*c^9*d^6 - 792*a^7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 + 66*a^10*b^2*c^5*d^10 - 12

*a^11*b*c^4*d^11 + a^12*c^3*d^12))^(1/4)*log((7*b*c*d + a*d^2)*sqrt(x) + (b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^

2*d^2 - a^3*c*d^3)*(-(2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^4*d^7)/

(b^12*c^15 - 12*a*b^11*c^14*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495*a^4*b^8*c^11*d^4 - 792*a^5*b

^7*c^10*d^5 + 924*a^6*b^6*c^9*d^6 - 792*a^7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 + 66*a^10*

b^2*c^5*d^10 - 12*a^11*b*c^4*d^11 + a^12*c^3*d^12))^(1/4)) - (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2

*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*(-(2401*b^4*c^4*d^3

+ 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^4*d^7)/(b^12*c^15 - 12*a*b^11*c^14*d + 66*a^2

*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495*a^4*b^8*c^11*d^4 - 792*a^5*b^7*c^10*d^5 + 924*a^6*b^6*c^9*d^6 - 79

2*a^7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 + 66*a^10*b^2*c^5*d^10 - 12*a^11*b*c^4*d^11 + a^

12*c^3*d^12))^(1/4)*log((7*b*c*d + a*d^2)*sqrt(x) - (b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*(-

(2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^4*d^7)/(b^12*c^15 - 12*a*b^1

1*c^14*d + 66*a^2*b^10*c^13*d^2 - 220*a^3*b^9*c^12*d^3 + 495*a^4*b^8*c^11*d^4 - 792*a^5*b^7*c^10*d^5 + 924*a^6

*b^6*c^9*d^6 - 792*a^7*b^5*c^8*d^7 + 495*a^8*b^4*c^7*d^8 - 220*a^9*b^3*c^6*d^9 + 66*a^10*b^2*c^5*d^10 - 12*a^1

1*b*c^4*d^11 + a^12*c^3*d^12))^(1/4)) + 4*(2*b*d*x^2 + b*c + a*d)*sqrt(x))/(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*

d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)

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giac [B] time = 1.73, size = 904, normalized size = 1.50 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((a*b^3)^(1/4)*b*c + 7*(a*b^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4)

)/(sqrt(2)*a*b^3*c^3 - 3*sqrt(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)*a^4*d^3) + 1/4*((a*b^3)^(1/4)

*b*c + 7*(a*b^3)^(1/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^3*

c^3 - 3*sqrt(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)*a^4*d^3) - 1/4*(7*(c*d^3)^(1/4)*b*c + (c*d^3)^

(1/4)*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^4 - 3*sqrt(2)*a*b^

2*c^3*d + 3*sqrt(2)*a^2*b*c^2*d^2 - sqrt(2)*a^3*c*d^3) - 1/4*(7*(c*d^3)^(1/4)*b*c + (c*d^3)^(1/4)*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^4 - 3*sqrt(2)*a*b^2*c^3*d + 3*sqrt(

2)*a^2*b*c^2*d^2 - sqrt(2)*a^3*c*d^3) + 1/8*((a*b^3)^(1/4)*b*c + 7*(a*b^3)^(1/4)*a*d)*log(sqrt(2)*sqrt(x)*(a/b

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

3) - 1/8*((a*b^3)^(1/4)*b*c + 7*(a*b^3)^(1/4)*a*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*

a*b^3*c^3 - 3*sqrt(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)*a^4*d^3) - 1/8*(7*(c*d^3)^(1/4)*b*c + (c

*d^3)^(1/4)*a*d)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^4 - 3*sqrt(2)*a*b^2*c^3*d + 3

*sqrt(2)*a^2*b*c^2*d^2 - sqrt(2)*a^3*c*d^3) + 1/8*(7*(c*d^3)^(1/4)*b*c + (c*d^3)^(1/4)*a*d)*log(-sqrt(2)*sqrt(

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

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

a*b*c*d + a^2*d^2))

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maple [A] time = 0.02, size = 770, normalized size = 1.28 \[ -\frac {a b d \sqrt {x}}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )}-\frac {a \,d^{2} \sqrt {x}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )}+\frac {b^{2} c \sqrt {x}}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )}+\frac {b c d \sqrt {x}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3} c}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3} c}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{2} \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{3} c}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3} a}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3} a}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (a d -b c \right )^{3} a}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3}}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3}}+\frac {7 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3}}+\frac {7 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3}}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b d \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (a d -b c \right )^{3}}+\frac {7 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b d \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{3}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b/(a*d-b*c)^3*x^(1/2)/(b*x^2+a)*a*d+1/2*b^2/(a*d-b*c)^3*x^(1/2)/(b*x^2+a)*c-7/8*b/(a*d-b*c)^3*(a/b)^(1/4)

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

)^(1/4)*x^(1/2)-1)*c-7/16*b/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-

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

2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))*c-7/8*b/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*a

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

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

*(c/d)^(1/4)/c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a+7/8*d/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^

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

*a+7/8*d/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b+1/16*d^2/(a*d-b*c)^3*(c/d)^(1

/4)/c*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a+7/

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

1/2)+(c/d)^(1/2)))*b

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maxima [A] time = 2.58, size = 620, normalized size = 1.03 \[ \frac {{\left (\frac {2 \, \sqrt {2} {\left (b c + 7 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b c + 7 \, a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b c + 7 \, a d\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b c + 7 \, a d\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} b}{16 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {2 \, b d x^{\frac {5}{2}} + {\left (b c + a d\right )} \sqrt {x}}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (7 \, b c d + a d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (7 \, b c d + a d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (7 \, b c d + a d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (7 \, b c d + a d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{16 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*sqrt(2)*(b*c + 7*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sq

rt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(b*c + 7*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4)

- 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(b*c + 7*a*d)*log(sqrt(

2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b*c + 7*a*d)*log(-sqrt(2)*a^(1/

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

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

*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2) - 1/16*(2*sqrt(2)*(7*b*c*d + a*

d^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(c)*sqrt(sqr

t(c)*sqrt(d))) + 2*sqrt(2)*(7*b*c*d + a*d^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(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(7*b*c*d + a*d^2)*log(sqrt(2)*c^(1/4)*d^(1/4

)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(7*b*c*d + a*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sq

rt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)

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mupad [B] time = 3.79, size = 34586, normalized size = 57.55 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan(-(((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 -

108544*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8

648704*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^1

3 - 11106304*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4

*d^17 + 337920*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*

d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*

d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11))

+ ((-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12

+ 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^

7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^

4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4)*(8192*a^2*b^17*c^

14*d^5 - 2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 24

55552*a^5*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11

+ 10070016*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^1

5 + 59392*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^

5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7*

c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3

*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d

^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 -

901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(3/4)*1i - (((2197*a*b^10*c^4*d^7)

/2 - (7*b^11*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c

^2*d^9)*1i)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3

*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3

+ 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336

*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*

b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*

c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4) - (x^(1/2)*(1225*a^6*b^7*d^13 + 1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^

8 + 18186*a^5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3*d^10 + 75975*a^4*b^9*c^2*d^11))/(8*(a^1

2*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 +

924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 -

12*a*b^11*c^11*d - 12*a^11*b*c*d^11)))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d

^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 -

901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 324403

2*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^1

4*b*c*d^11))^(1/4) + ((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b

^5*c*d^20 - 108544*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*

c^12*d^9 + 8648704*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*

b^12*c^8*d^13 - 11106304*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*

a^13*b^8*c^4*d^17 + 337920*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^

2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*

a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^1

1*b*c*d^11)) - ((-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(40

96*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3

+ 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 202

7520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4)*(8192

*a^2*b^17*c^14*d^5 - 2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c

^12*d^7 + 2455552*a^5*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^

11*c^8*d^11 + 10070016*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a^11*b^8*c^5*d^14 - 606208*a^12

*b^7*c^4*d^15 + 59392*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 5

6*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d

^7))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^1

2 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a

^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b

^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(3/4)*1i + (((2197*a*b

^10*c^4*d^7)/2 - (7*b^11*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 91

45*a^3*b^8*c^2*d^9)*1i)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56

*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a

^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^1

1*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 +

3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 2703

36*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4) - (x^(1/2)*(1225*a^6*b^7*d^13 + 1225*b^13*c^6*d^7 + 18186*a

*b^12*c^5*d^8 + 18186*a^5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3*d^10 + 75975*a^4*b^9*c^2*d^

11))/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^

7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^

2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a

^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^1

0*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*

d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10

- 49152*a^14*b*c*d^11))^(1/4))/(((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 +

4096*a^16*b^5*c*d^20 - 108544*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^17*c^13*d^8 - 32256

00*a^5*b^16*c^12*d^9 + 8648704*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^8*b^13*c^9*d^12 +

5294080*a^9*b^12*c^8*d^13 - 11106304*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^

16 + 153600*a^13*b^8*c^4*d^17 + 337920*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19)*1i)/(8*(a^12*d^12 + b^12*

c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^

6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^1

1*d - 12*a^11*b*c*d^11)) + ((-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^

6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*

b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c

^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))

^(1/4)*(8192*a^2*b^17*c^14*d^5 - 2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^16*c^13*d^6 - 60620

8*a^4*b^15*c^12*d^7 + 2455552*a^5*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b^12*c^9*d^10 - 118

94784*a^8*b^11*c^8*d^11 + 10070016*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a^11*b^8*c^5*d^14 -

606208*a^12*b^7*c^4*d^15 + 59392*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6

*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d -

8*a^7*b*c*d^7))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4

096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3

+ 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 20

27520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(3/4)*1i -

(((2197*a*b^10*c^4*d^7)/2 - (7*b^11*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/2 + 9145*a^2*b^9*

c^3*d^8 + 9145*a^3*b^8*c^2*d^9)*1i)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*

c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7*c^4 + 2401*a^4*b^3*

d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*

a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^

7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^

3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4)*1i - (x^(1/2)*(1225*a^6*b^7*d^13 + 1225*b^13*c^

6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3*d^10 + 75975

*a^4*b^9*c^2*d^11)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^

8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^

3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3

*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*

d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 37

84704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336

*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4) - ((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096

*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 - 108544*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^1

7*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8648704*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^

8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^13 - 11106304*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 32256

00*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4*d^17 + 337920*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19)*1i)/(8*

(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^

5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^

10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)) - ((-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5

*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10

*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 -

3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 491

52*a^14*b*c*d^11))^(1/4)*(8192*a^2*b^17*c^14*d^5 - 2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^1

6*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 2455552*a^5*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b

^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11 + 10070016*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a

^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^15 + 59392*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^

8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6

- 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 +

28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 9011

20*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^1

0*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c

*d^11))^(3/4)*1i + (((2197*a*b^10*c^4*d^7)/2 - (7*b^11*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)

/2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c^2*d^9)*1i)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5

*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7*c

^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*

b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^

4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 -

901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4)*1i - (x^(1/2)*(1225*a^6*b^7*d

^13 + 1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^1

0*c^3*d^10 + 75975*a^4*b^9*c^2*d^11)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^

3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^

8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)))*(-(b^7*c^4 + 2401*a^4*

b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49

152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^

8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^

3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4)))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*

b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d

+ 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 378

4704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*

a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4) - atan(-(((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4

+ 4096*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 - 108544*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a

^4*b^17*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8648704*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294

080*a^8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^13 - 11106304*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 -

3225600*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4*d^17 + 337920*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19))/

(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7

*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2

*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)) + ((-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*

b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c

^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6

- 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 -

49152*a^14*b*c*d^11))^(1/4)*(8192*a^2*b^17*c^14*d^5 - 2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*

b^16*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 2455552*a^5*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^

7*b^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11 + 10070016*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 245555

2*a^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^15 + 59392*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 +

b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*

d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2

+ 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 9

01120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*

a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*

b*c*d^11))^(3/4) - ((2197*a*b^10*c^4*d^7)/2 - (7*b^11*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/

2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c^2*d^9)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3

+ 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7*c^4 +

2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*

c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3

244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 90112

0*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4)*1i + (x^(1/2)*(1225*a^6*b^7*d^13 +

1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3

*d^10 + 75975*a^4*b^9*c^2*d^11)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 4

95*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 2

20*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)))*(-(b^7*c^4 + 2401*a^4*b^3*d

^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a

^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7

*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3

*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4) + ((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^

17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 - 108544*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 1

53600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8648704*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11

+ 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^13 - 11106304*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6

*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4*d^17 + 337920*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*

d^19))/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*

b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*

b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)) - ((-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 2

94*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5

*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*

c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*

d^10 - 49152*a^14*b*c*d^11))^(1/4)*(8192*a^2*b^17*c^14*d^5 - 2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 593

92*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 2455552*a^5*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 1007

0016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11 + 10070016*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 +

2455552*a^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^15 + 59392*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^

8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b

^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*

c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*

d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3

244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 4915

2*a^14*b*c*d^11))^(3/4) + ((2197*a*b^10*c^4*d^7)/2 - (7*b^11*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c

*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c^2*d^9)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c

^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7

*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^

3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*

d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8

- 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4)*1i + (x^(1/2)*(1225*a^6*b^7

*d^13 + 1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b

^10*c^3*d^10 + 75975*a^4*b^9*c^2*d^11)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*

d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*

d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)))*(-(b^7*c^4 + 2401*a^

4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 -

49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*

a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*

b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4))/(((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*

b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 - 108544*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*

d^7 + 153600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8648704*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^

10*d^11 + 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^13 - 11106304*a^10*b^11*c^7*d^14 + 8648704*a^11*b

^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4*d^17 + 337920*a^14*b^7*c^3*d^18 - 108544*a^15*b

^6*c^2*d^19))/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 7

92*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 6

6*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)) + ((-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*

d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270

336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a

^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b

^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4)*(8192*a^2*b^17*c^14*d^5 - 2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^

4 + 59392*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 2455552*a^5*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9

+ 10070016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11 + 10070016*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6

*d^13 + 2455552*a^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^15 + 59392*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^1

7))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 2

8*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a

^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^1

0*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*

d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10

- 49152*a^14*b*c*d^11))^(3/4) - ((2197*a*b^10*c^4*d^7)/2 - (7*b^11*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^

4*b^7*c*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c^2*d^9)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^

3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))

*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 +

4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b

^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c

^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4) + (x^(1/2)*(1225*a^6

*b^7*d^13 + 1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a

^3*b^10*c^3*d^10 + 75975*a^4*b^9*c^2*d^11))/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9

*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4

*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)))*(-(b^7*c^4 + 2401*a

^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 -

49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032

*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12

*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4) - ((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048

*b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 - 108544*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14

*d^7 + 153600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8648704*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c

^10*d^11 + 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^13 - 11106304*a^10*b^11*c^7*d^14 + 8648704*a^11*

b^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4*d^17 + 337920*a^14*b^7*c^3*d^18 - 108544*a^15*

b^6*c^2*d^19))/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 -

792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 +

66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)) - ((-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c

*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 27

0336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*

a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*

b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4)*(8192*a^2*b^17*c^14*d^5 - 2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d

^4 + 59392*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 2455552*a^5*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^

9 + 10070016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11 + 10070016*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^

6*d^13 + 2455552*a^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^15 + 59392*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^

17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 +

28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*

a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^

10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6

*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^1

0 - 49152*a^14*b*c*d^11))^(3/4) + ((2197*a*b^10*c^4*d^7)/2 - (7*b^11*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a

^4*b^7*c*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c^2*d^9)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a

^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7)

)*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 +

4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*

b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*

c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4) + (x^(1/2)*(1225*a^

6*b^7*d^13 + 1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*

a^3*b^10*c^3*d^10 + 75975*a^4*b^9*c^2*d^11))/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^

9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^

4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)))*(-(b^7*c^4 + 2401*

a^4*b^3*d^4 + 1372*a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12

- 49152*a^4*b^11*c^11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 324403

2*a^8*b^7*c^7*d^5 + 3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^1

2*b^3*c^3*d^9 + 270336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4)))*(-(b^7*c^4 + 2401*a^4*b^3*d^4 + 1372*

a^3*b^4*c*d^3 + 294*a^2*b^5*c^2*d^2 + 28*a*b^6*c^3*d)/(4096*a^15*d^12 + 4096*a^3*b^12*c^12 - 49152*a^4*b^11*c^

11*d + 270336*a^5*b^10*c^10*d^2 - 901120*a^6*b^9*c^9*d^3 + 2027520*a^7*b^8*c^8*d^4 - 3244032*a^8*b^7*c^7*d^5 +

3784704*a^9*b^6*c^6*d^6 - 3244032*a^10*b^5*c^5*d^7 + 2027520*a^11*b^4*c^4*d^8 - 901120*a^12*b^3*c^3*d^9 + 270

336*a^13*b^2*c^2*d^10 - 49152*a^14*b*c*d^11))^(1/4)*2i - ((x^(1/2)*(a*d + b*c))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*

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

(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 - 108544*a^2*b^19*c^15*

d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8648704*a^6*b^15*c^11*

d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^13 - 11106304*a^10*b^1

1*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4*d^17 + 337920*a^14*b

^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19))/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^

3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^

8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)) + ((-(a^4*d^7 + 2401*b^

4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 -

49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 324403

2*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*

b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)*(8192*a^2*b^17*c^14*d^5 - 2048*a^15*b^4*c

*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 2455552*a^5*b^14*c^11*d^8

- 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11 + 10070016*a^9*b^10*c^7*

d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^15 + 59392*a^13*b^6*c^3*d

^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d

^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 +

1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*

b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c

^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9

+ 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(3/4) - ((2197*a*b^10*c^4*d^7)/2 - (7*b^11*c^5*d^6)/2 - (7

*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c^2*d^9)/(a^8*d^8 + b^8*c^8 +

28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*

b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*

b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*

b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5

*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))

^(1/4)*1i + (x^(1/2)*(1225*a^6*b^7*d^13 + 1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^5*b^8*c*d^12 + 75

975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3*d^10 + 75975*a^4*b^9*c^2*d^11)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^

2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*

a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^1

1*b*c*d^11)))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096

*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 +

2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 202

7520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4) + ((((x

^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 - 108544*a^2*b^1

9*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8648704*a^6*b^1

5*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^13 - 11106304*a

^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4*d^17 + 337920

*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19))/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9

*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4

*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)) - ((-(a^4*d^7 +

2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*

d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 -

3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 9011

20*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)*(8192*a^2*b^17*c^14*d^5 - 2048*a^1

5*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 2455552*a^5*b^14*c^

11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11 + 10070016*a^9*b^

10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^15 + 59392*a^13*b^

6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^

4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b^4*c^

4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 4915

2*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^

5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*

c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(3/4) + ((2197*a*b^10*c^4*d^7)/2 - (7*b^11*c^5*d^6)

/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c^2*d^9)/(a^8*d^8 + b^

8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6

- 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 +

28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 9011

20*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*

a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c

^14*d))^(1/4)*1i + (x^(1/2)*(1225*a^6*b^7*d^13 + 1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^5*b^8*c*d^

12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3*d^10 + 75975*a^4*b^9*c^2*d^11)*1i)/(8*(a^12*d^12 + b^12*c^12

+ 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6

- 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d -

12*a^11*b*c*d^11)))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6

)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^1

2*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^

7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4))

/(((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 - 108544*

a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8648704*

a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^13 - 111

06304*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4*d^17 +

337920*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19))/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*

a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*

a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)) + ((-(a^4

*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^

12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^1

1*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8

- 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)*(8192*a^2*b^17*c^14*d^5 - 2

048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 2455552*a^5*

b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11 + 10070016

*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^15 + 59392*

a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70

*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401

*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12

- 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 324

4032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a

^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(3/4) - ((2197*a*b^10*c^4*d^7)/2 - (7*b^11*c

^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c^2*d^9)/(a^8*d

^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*

c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2

*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2

- 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3

244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a

*b^11*c^14*d))^(1/4) + (x^(1/2)*(1225*a^6*b^7*d^13 + 1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^5*b^8*

c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3*d^10 + 75975*a^4*b^9*c^2*d^11))/(8*(a^12*d^12 + b^12*c^12

+ 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^

6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d

- 12*a^11*b*c*d^11)))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^

6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^

12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d

^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)

- ((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 - 108544

*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8648704

*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^13 - 11

106304*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4*d^17

+ 337920*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19))/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220

*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495

*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)) - ((-(a^

4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a

^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^

11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^

8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)*(8192*a^2*b^17*c^14*d^5 -

2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 2455552*a^5

*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11 + 1007001

6*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^15 + 59392

*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 7

0*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 240

1*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^1

2 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 32

44032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*

a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(3/4) + ((2197*a*b^10*c^4*d^7)/2 - (7*b^11*

c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c^2*d^9)/(a^8*

d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2

*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^

2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^

2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 -

3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*

a*b^11*c^14*d))^(1/4) + (x^(1/2)*(1225*a^6*b^7*d^13 + 1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^5*b^8

*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3*d^10 + 75975*a^4*b^9*c^2*d^11))/(8*(a^12*d^12 + b^12*c^1

2 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d

^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d

- 12*a^11*b*c*d^11)))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d

^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c

^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*

d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4

)))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15

+ 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a

^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b

^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)*2i + 2*atan(-((((

(x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 - 108544*a^2*b

^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8648704*a^6*b

^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^13 - 11106304

*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4*d^17 + 3379

20*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^

3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^

8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)) + ((-(a^4*d

^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12

*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*

d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 -

901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)*(8192*a^2*b^17*c^14*d^5 - 204

8*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 2455552*a^5*b^

14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11 + 10070016*a

^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^15 + 59392*a^

13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a

^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b

^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 -

49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 32440

32*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9

*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(3/4)*1i - (((2197*a*b^10*c^4*d^7)/2 - (7*b^11

*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c^2*d^9)*1i)/

(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^

6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b

^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^

13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d

^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 4

9152*a*b^11*c^14*d))^(1/4) - (x^(1/2)*(1225*a^6*b^7*d^13 + 1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^

5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3*d^10 + 75975*a^4*b^9*c^2*d^11))/(8*(a^12*d^12 + b^1

2*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*

c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c

^11*d - 12*a^11*b*c*d^11)))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*

b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*

b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5

*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))

^(1/4) + ((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b^5*c*d^20 -

108544*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*c^12*d^9 + 8

648704*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*b^12*c^8*d^1

3 - 11106304*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*a^13*b^8*c^4

*d^17 + 337920*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*

d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*

d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11))

- ((-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15

+ 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a

^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b

^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)*(8192*a^2*b^17*c^

14*d^5 - 2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c^12*d^7 + 24

55552*a^5*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^11*c^8*d^11

+ 10070016*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a^11*b^8*c^5*d^14 - 606208*a^12*b^7*c^4*d^1

5 + 59392*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^

5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*

d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^1

2*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11

*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8

- 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(3/4)*1i + (((2197*a*b^10*c^4*d^7)

/2 - (7*b^11*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 9145*a^3*b^8*c

^2*d^9)*1i)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3

*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4

+ 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336

*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a

^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*

c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4) - (x^(1/2)*(1225*a^6*b^7*d^13 + 1225*b^13*c^6*d^7 + 18186*a*b^12*c^5*d^

8 + 18186*a^5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3*d^10 + 75975*a^4*b^9*c^2*d^11))/(8*(a^1

2*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 +

924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 -

12*a*b^11*c^11*d - 12*a^11*b*c*d^11)))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d

^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 -

901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 324

4032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b

^11*c^14*d))^(1/4))/(((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096*a*b^20*c^16*d^5 + 4096*a^16*b

^5*c*d^20 - 108544*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^17*c^13*d^8 - 3225600*a^5*b^16*

c^12*d^9 + 8648704*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^8*b^13*c^9*d^12 + 5294080*a^9*

b^12*c^8*d^13 - 11106304*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 3225600*a^12*b^9*c^5*d^16 + 153600*

a^13*b^8*c^4*d^17 + 337920*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^

2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*

a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^1

1*b*c*d^11)) + ((-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(40

96*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3

+ 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2

027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)*(8192

*a^2*b^17*c^14*d^5 - 2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^16*c^13*d^6 - 606208*a^4*b^15*c

^12*d^7 + 2455552*a^5*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b^12*c^9*d^10 - 11894784*a^8*b^

11*c^8*d^11 + 10070016*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a^11*b^8*c^5*d^14 - 606208*a^12

*b^7*c^4*d^15 + 59392*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 5

6*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d

^7))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^1

5 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*

a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*

b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(3/4)*1i - (((2197*a*b

^10*c^4*d^7)/2 - (7*b^11*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/2 + 9145*a^2*b^9*c^3*d^8 + 91

45*a^3*b^8*c^2*d^9)*1i)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56

*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a

*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d

^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5

+ 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 2703

36*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)*1i - (x^(1/2)*(1225*a^6*b^7*d^13 + 1225*b^13*c^6*d^7 + 1818

6*a*b^12*c^5*d^8 + 18186*a^5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3*d^10 + 75975*a^4*b^9*c^2

*d^11)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*

a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a

^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 +

294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a

^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6

*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^

5*d^10 - 49152*a*b^11*c^14*d))^(1/4) - ((((x^(1/2)*(2048*a^17*b^4*d^21 + 2048*b^21*c^17*d^4 + 4096*a*b^20*c^16

*d^5 + 4096*a^16*b^5*c*d^20 - 108544*a^2*b^19*c^15*d^6 + 337920*a^3*b^18*c^14*d^7 + 153600*a^4*b^17*c^13*d^8 -

3225600*a^5*b^16*c^12*d^9 + 8648704*a^6*b^15*c^11*d^10 - 11106304*a^7*b^14*c^10*d^11 + 5294080*a^8*b^13*c^9*d

^12 + 5294080*a^9*b^12*c^8*d^13 - 11106304*a^10*b^11*c^7*d^14 + 8648704*a^11*b^10*c^6*d^15 - 3225600*a^12*b^9*

c^5*d^16 + 153600*a^13*b^8*c^4*d^17 + 337920*a^14*b^7*c^3*d^18 - 108544*a^15*b^6*c^2*d^19)*1i)/(8*(a^12*d^12 +

b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*

b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^

11*c^11*d - 12*a^11*b*c*d^11)) - ((-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 2

8*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 90112

0*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a

^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^

14*d))^(1/4)*(8192*a^2*b^17*c^14*d^5 - 2048*a^15*b^4*c*d^18 - 2048*a*b^18*c^15*d^4 + 59392*a^3*b^16*c^13*d^6 -

606208*a^4*b^15*c^12*d^7 + 2455552*a^5*b^14*c^11*d^8 - 6037504*a^6*b^13*c^10*d^9 + 10070016*a^7*b^12*c^9*d^10

- 11894784*a^8*b^11*c^8*d^11 + 10070016*a^9*b^10*c^7*d^12 - 6037504*a^10*b^9*c^6*d^13 + 2455552*a^11*b^8*c^5*

d^14 - 606208*a^12*b^7*c^4*d^15 + 59392*a^13*b^6*c^3*d^16 + 8192*a^14*b^5*c^2*d^17))/(a^8*d^8 + b^8*c^8 + 28*a

^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c

^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d

^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c

^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*

d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(3/4

)*1i + (((2197*a*b^10*c^4*d^7)/2 - (7*b^11*c^5*d^6)/2 - (7*a^5*b^6*d^11)/2 + (2197*a^4*b^7*c*d^10)/2 + 9145*a^

2*b^9*c^3*d^8 + 9145*a^3*b^8*c^2*d^9)*1i)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^

4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))*(-(a^4*d^7 + 2401*b^

4*c^4*d^3 + 1372*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 -

49152*a^11*b*c^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 324403

2*a^5*b^7*c^10*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*

b^3*c^6*d^9 + 270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)*1i - (x^(1/2)*(1225*a^6*b^7*d^13 + 1225*b

^13*c^6*d^7 + 18186*a*b^12*c^5*d^8 + 18186*a^5*b^8*c*d^12 + 75975*a^2*b^11*c^4*d^9 + 71372*a^3*b^10*c^3*d^10 +

75975*a^4*b^9*c^2*d^11)*1i)/(8*(a^12*d^12 + b^12*c^12 + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*

b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*

b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a*b^11*c^11*d - 12*a^11*b*c*d^11)))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 13

72*a*b^3*c^3*d^4 + 294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c

^4*d^11 + 270336*a^2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10

*d^5 + 3784704*a^6*b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 +

270336*a^10*b^2*c^5*d^10 - 49152*a*b^11*c^14*d))^(1/4)))*(-(a^4*d^7 + 2401*b^4*c^4*d^3 + 1372*a*b^3*c^3*d^4 +

294*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6)/(4096*b^12*c^15 + 4096*a^12*c^3*d^12 - 49152*a^11*b*c^4*d^11 + 270336*a^

2*b^10*c^13*d^2 - 901120*a^3*b^9*c^12*d^3 + 2027520*a^4*b^8*c^11*d^4 - 3244032*a^5*b^7*c^10*d^5 + 3784704*a^6*

b^6*c^9*d^6 - 3244032*a^7*b^5*c^8*d^7 + 2027520*a^8*b^4*c^7*d^8 - 901120*a^9*b^3*c^6*d^9 + 270336*a^10*b^2*c^5

*d^10 - 49152*a*b^11*c^14*d))^(1/4)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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