∫ x3∕2 _________________________

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

Optimal. Leaf size=627 \[ -\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{a} b^{7/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\sqrt [4]{a} b^{7/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} (b c-a d)^3}+\frac {\sqrt {x} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac {\sqrt {x}}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

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

1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/(-a*d+b*c)^3*2^(1/2)-1/64*(-3*a^2*d^2+14*a*b*c*d+21*b^2*c^2)*arctan(1-d^(1/

4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(7/4)/d^(1/4)/(-a*d+b*c)^3*2^(1/2)+1/64*(-3*a^2*d^2+14*a*b*c*d+21*b^2*c^2)*arcta

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

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

(1/4)*2^(1/2)*x^(1/2))/(-a*d+b*c)^3*2^(1/2)-1/128*(-3*a^2*d^2+14*a*b*c*d+21*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1

/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(7/4)/d^(1/4)/(-a*d+b*c)^3*2^(1/2)+1/128*(-3*a^2*d^2+14*a*b*c*d+21*b^2*c^2)*ln(

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

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

________________________________________________________________________________________

Rubi [A] time = 0.71, antiderivative size = 627, 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} \[ -\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{a} b^{7/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\sqrt [4]{a} b^{7/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} (b c-a d)^3}+\frac {\sqrt {x} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac {\sqrt {x}}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Sqrt[x]/(4*(b*c - a*d)*(c + d*x^2)^2) + ((7*b*c + a*d)*Sqrt[x])/(16*c*(b*c - a*d)^2*(c + d*x^2)) + (a^(1/4)*b^

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

rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - ((21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (

Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*d^(1/4)*(b*c - a*d)^3) + ((21*b^2*c^2 + 14*a*b*c*d - 3*

a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*d^(1/4)*(b*c - a*d)^3) + (a^(1/4)*

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

4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*(b*c - a*d)^3) - ((21*b^2*c^2 + 14*a

*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(7/4)*d^(1/4)*(b

*c - a*d)^3) + ((21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*

x])/(64*Sqrt[2]*c^(7/4)*d^(1/4)*(b*c - a*d)^3)

Rule 204

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

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

Rule 211

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

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

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

]]))

Rule 466

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

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

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

rQ[p]

Rule 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 ) \left (c+d x^2\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {a-7 b x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)}\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {a (11 b c-3 a d)-3 b (7 b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^2}\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^3}\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\left (\sqrt {a} b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}-\frac {\left (\sqrt {a} b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2} (b c-a d)^3}\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\left (\sqrt {a} b^{3/2}\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 )}{2 (b c-a d)^3}-\frac {\left (\sqrt {a} b^{3/2}\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 )}{2 (b c-a d)^3}+\frac {\left (\sqrt [4]{a} b^{7/4}\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 )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (\sqrt [4]{a} b^{7/4}\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 )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{3/2} \sqrt {d} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{3/2} \sqrt {d} (b c-a d)^3}-\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{7/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\sqrt [4]{a} b^{7/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (\sqrt [4]{a} b^{7/4}\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 )}{\sqrt {2} (b c-a d)^3}+\frac {\left (\sqrt [4]{a} b^{7/4}\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 )}{\sqrt {2} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}\\ &=\frac {\sqrt {x}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(7 b c+a d) \sqrt {x}}{16 c (b c-a d)^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{a} b^{7/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\sqrt [4]{a} b^{7/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac {\left (21 b^2 c^2+14 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}\\ \end {align*}

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Mathematica [A] time = 0.73, size = 543, normalized size = 0.87 \[ \frac {-\frac {\sqrt {2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{7/4} \sqrt [4]{d}}+\frac {\sqrt {2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{7/4} \sqrt [4]{d}}-\frac {2 \sqrt {2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{7/4} \sqrt [4]{d}}+\frac {2 \sqrt {2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{7/4} \sqrt [4]{d}}+32 \sqrt {2} \sqrt [4]{a} b^{7/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-32 \sqrt {2} \sqrt [4]{a} b^{7/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+64 \sqrt {2} \sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )-64 \sqrt {2} \sqrt [4]{a} b^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )+\frac {32 \sqrt {x} (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac {8 \sqrt {x} (a d+7 b c) (b c-a d)}{c \left (c+d x^2\right )}}{128 (b c-a d)^3} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((32*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2)^2 + (8*(b*c - a*d)*(7*b*c + a*d)*Sqrt[x])/(c*(c + d*x^2)) + 64*Sqrt[2]

*a^(1/4)*b^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 64*Sqrt[2]*a^(1/4)*b^(7/4)*ArcTan[1 + (Sqrt[2

]*b^(1/4)*Sqrt[x])/a^(1/4)] - (2*Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqr

t[x])/c^(1/4)])/(c^(7/4)*d^(1/4)) + (2*Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/

4)*Sqrt[x])/c^(1/4)])/(c^(7/4)*d^(1/4)) + 32*Sqrt[2]*a^(1/4)*b^(7/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqr

t[x] + Sqrt[b]*x] - 32*Sqrt[2]*a^(1/4)*b^(7/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - (S

qrt[2]*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(7

/4)*d^(1/4)) + (Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] +

Sqrt[d]*x])/(c^(7/4)*d^(1/4)))/(128*(b*c - a*d)^3)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [A] time = 1.40, size = 946, normalized size = 1.51 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

+ sqrt(a/b))/(sqrt(2)*b^3*c^3 - 3*sqrt(2)*a*b^2*c^2*d + 3*sqrt(2)*a^2*b*c*d^2 - sqrt(2)*a^3*d^3) + 1/32*(21*(c

*d^3)^(1/4)*b^2*c^2 + 14*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1

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

*a^3*c^2*d^4) + 1/32*(21*(c*d^3)^(1/4)*b^2*c^2 + 14*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1

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

t(2)*a^2*b*c^3*d^3 - sqrt(2)*a^3*c^2*d^4) + 1/64*(21*(c*d^3)^(1/4)*b^2*c^2 + 14*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d

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

^2 + 3*sqrt(2)*a^2*b*c^3*d^3 - sqrt(2)*a^3*c^2*d^4) - 1/64*(21*(c*d^3)^(1/4)*b^2*c^2 + 14*(c*d^3)^(1/4)*a*b*c*

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

a*b^2*c^4*d^2 + 3*sqrt(2)*a^2*b*c^3*d^3 - sqrt(2)*a^3*c^2*d^4) + 1/16*(7*b*c*d*x^(5/2) + a*d^2*x^(5/2) + 11*b*

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

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maple [A] time = 0.02, size = 848, normalized size = 1.35 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

*a^2+3/8/(a*d-b*c)^3/(d*x^2+c)^2*d^2*x^(5/2)*a*b-7/16/(a*d-b*c)^3/(d*x^2+c)^2*d*c*x^(5/2)*b^2+7/8/(a*d-b*c)^3/

(d*x^2+c)^2*x^(1/2)*a*b*c*d-11/16/(a*d-b*c)^3/(d*x^2+c)^2*x^(1/2)*b^2*c^2-3/16/(a*d-b*c)^3/(d*x^2+c)^2*x^(1/2)

*a^2*d^2+3/64/(a*d-b*c)^3/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2*d^2-7/32/(a*d-b*c)

^3/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b*d-21/64/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arc

tan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+3/64/(a*d-b*c)^3/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(

1/2)-1)*a^2*d^2-7/32/(a*d-b*c)^3/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b*d-21/64/(a*d-

b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+3/128/(a*d-b*c)^3/c^2*(c/d)^(1/4)*2^(1/2)

*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a^2*d^2-7/64/(a*d

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

/d)^(1/2)))*a*b*d-21/128/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-(c/

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

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maxima [A] time = 2.61, size = 654, normalized size = 1.04 \[ -\frac {{\left (\frac {2 \, \sqrt {2} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {7}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {7}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}\right )} a}{4 \, {\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 {{\left (7 \, b c d + a d^{2}\right )} x^{\frac {5}{2}} + {\left (11 \, b c^{2} - 3 \, a c d\right )} \sqrt {x}}{16 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (21 \, b^{2} c^{2} + 14 \, a b c d - 3 \, a^{2} 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 (21 \, b^{2} c^{2} + 14 \, a b c d - 3 \, a^{2} 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 (21 \, b^{2} c^{2} + 14 \, a b c d - 3 \, a^{2} 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 (21 \, b^{2} c^{2} + 14 \, a b c d - 3 \, a^{2} 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}}}}{128 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(2)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(s

qrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*b^2*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x)

)/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(7/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x)

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

a^(3/4))*a/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 1/16*((7*b*c*d + a*d^2)*x^(5/2) + (11*b*c^2 -

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

^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3)*x^2) + 1/128*(2*sqrt(2)*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*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))

) + 2*sqrt(2)*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*s

qrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*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)*(21*b^2*c^2 + 14*a*b*c

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

b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)

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mupad [B] time = 4.25, size = 36160, normalized size = 57.67 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan(((((((81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*

b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/

2048 + (6885*a^7*b^9*c^2*d^8)/2048)*1i)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a

^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (((-(a*b^7)/

(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^

5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 +

1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^1

7*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^

13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a

^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*

b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^

3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (x^(1/2)*(167

77216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^1

6*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^13*d^10 - 133610

86464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 1989672960*a^12*b

^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 - 12740

1984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^21)*1i

)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c^13*d^3 + 495*a^

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

*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^

2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6

- 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^

11*d - 192*a^11*b*c*d^11))^(3/4)*1i)*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^

3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 +

7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11)

)^(1/4) - (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^

4 + 859068*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 1753

2*a^8*b^11*c^2*d^9))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^

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

8*b^4*c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*

b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 147

84*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d

^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) - (((((81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 -

(194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5

*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)*1i)/(b^8*c^12 + a^8*c^4*d^8

- 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^

6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (((-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3

*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 +

7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))

^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^

7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11

*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^

7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 -

8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6

*b^2*c^6*d^6 - 8*a*b^7*c^11*d) + (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 10302259

20*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^14

*d^9 + 14203486208*a^8*b^15*c^13*d^10 - 13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 5521

702912*a^11*b^12*c^10*d^13 + 1989672960*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9*

c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 - 127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a^

18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^21)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2

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

92*a^7*b^5*c^9*d^7 + 495*a^8*b^4*c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-

(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 -

12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^

3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4)*1i)*(-(a*b^7)/(16*a^12*d^12 + 1

6*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 1

4784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2

*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) + (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 19

4481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 859068*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^

6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c^2*d^9))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*

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

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

a*b^11*c^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920

*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^

8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4))/((((((81*a^

9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)/2048

- (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*a^7*b

^9*c^2*d^8)/2048)*1i)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 7

0*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (((-(a*b^7)/(16*a^12*d^12 + 16

*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14

784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*

d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 50611

2*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958

976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 -

459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768

*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70

*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (x^(1/2)*(16777216*a^2*b^21*c^1

9*d^4 - 194101248*a^3*b^20*c^18*d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^16*d^7 + 7335837696

*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^13*d^10 - 13361086464*a^9*b^14*c^1

2*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 1989672960*a^12*b^11*c^9*d^14 - 499

38432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 - 127401984*a^16*b^7*c^5*

d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^21)*1i)/(4096*(b^12*c^16

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

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

66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 -

3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^

5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*

c*d^11))^(3/4)*1i)*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 79

20*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*

d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*1i - (x^(1

/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 859068*a^4*

b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c^2*

d^9)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c^13*d^3 +

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

8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 +

1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c

^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*

b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) + (((((81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 - (194481*a^2

*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^

6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)*1i)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^

5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^

6 - 8*a*b^7*c^11*d) - (((-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3

+ 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4

*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(8192

*a^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^7 + 3384832*

a^6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11*d^11 + 1115

136*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^7*d^15 - 762

88*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5

*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6

- 8*a*b^7*c^11*d) + (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 1030225920*a^4*b^19*

c^17*d^6 - 3328573440*a^5*b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^14*d^9 + 14203

486208*a^8*b^15*c^13*d^10 - 13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 5521702912*a^11*

b^12*c^10*d^13 + 1989672960*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9*c^7*d^16 + 3

43080960*a^15*b^8*c^6*d^17 - 127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a^18*b^5*c^3*d

^20 + 147456*a^19*b^4*c^2*d^21)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d

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

^9*d^7 + 495*a^8*b^4*c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(16*

a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^

7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056

*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4)*1i)*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12

+ 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6

*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*

a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*1i + (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 194481*a^2*

b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 859068*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^6*b^13*c^

4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c^2*d^9)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*

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

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

*c^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b

^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 35

20*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)))*(-(a*b^7)/(16*a^

12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*

c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a

^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) - atan((((((81*a^9*b^7*d^10)/2048 - (1431*a^8

*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)

/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)/(b^8*c^12

+ a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c

^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (((-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*

d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*

b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a

^11*b*c*d^11))^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^

5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824

*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 2565

12*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 +

a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^

7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*

d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738087424

*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^13*d^10 - 13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^

11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 1989672960*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 48444

2112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 - 127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^1

9 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^21))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^

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

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

^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8

*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520

*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4))*(-(a*b^7)/(16*a^12*

d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7

*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10

*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*1i + (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*

c*d^10 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 859068*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6

- 14266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c^2*d^9)*1i)/(4096*(b^12*c^16 + a^12*c^4*d

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

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

c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*

c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*

a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4

) - ((((81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*

c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048

+ (6885*a^7*b^9*c^2*d^8)/2048)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^

9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (((-(a*b^7)/(16*a^12*

d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7

*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10

*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^

5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d

^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c

^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d

^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9

*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) + (x^(1/2)*(16777216*a^2

*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^16*d^7 + 7

335837696*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^13*d^10 - 13361086464*a^9

*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 1989672960*a^12*b^11*c^9*d

^14 - 49938432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 - 127401984*a^16

*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^21))/(4096*(b^1

2*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c^13*d^3 + 495*a^4*b^8*c^12*d

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

*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*

d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*

b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a

^11*b*c*d^11))^(3/4))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 +

7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c

^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*1i - (x

^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 859068*a

^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c

^2*d^9)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c^13*d^

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

*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12

+ 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^

6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192

*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4))/(((((81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 - (194481*a

^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*

d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*

d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6

- 8*a*b^7*c^11*d) - (((-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 +

7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c

^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(8192*a

^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^7 + 3384832*a^

6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11*d^11 + 111513

6*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^7*d^15 - 76288

*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d

^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 -

8*a*b^7*c^11*d) - (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 1030225920*a^4*b^19*c^

17*d^6 - 3328573440*a^5*b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^14*d^9 + 1420348

6208*a^8*b^15*c^13*d^10 - 13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 5521702912*a^11*b^

12*c^10*d^13 + 1989672960*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9*c^7*d^16 + 343

080960*a^15*b^8*c^6*d^17 - 127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a^18*b^5*c^3*d^2

0 + 147456*a^19*b^4*c^2*d^21))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 -

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

7 + 495*a^8*b^4*c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(16*a^12*

d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7

*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10

*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a

^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6

- 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c

^11*d - 192*a^11*b*c*d^11))^(1/4) + (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 194481*a^2*b^17*c^8*d^

3 + 518616*a^3*b^16*c^7*d^4 + 859068*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^6*b^13*c^4*d^7 - 871

92*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c^2*d^9))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*

b^10*c^14*d^2 - 220*a^3*b^9*c^13*d^3 + 495*a^4*b^8*c^12*d^4 - 792*a^5*b^7*c^11*d^5 + 924*a^6*b^6*c^10*d^6 - 79

2*a^7*b^5*c^9*d^7 + 495*a^8*b^4*c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(

a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 1

2672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3

*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) + ((((81*a^9*b^7*d^10)/2048 - (1

431*a^8*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c

^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)/(b^

8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^

5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (((-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^1

0*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 126

72*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d

- 192*a^11*b*c*d^11))^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 160

7168*a^5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 -

3277824*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14

+ 256512*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8

*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5

*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) + (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^2

0*c^18*d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 1173

8087424*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^13*d^10 - 13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*

b^13*c^11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 1989672960*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15

- 484442112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 - 127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*

c^4*d^19 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^21))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b

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

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

*b^11*c^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*

a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8

- 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(3/4))*(-(a*b^7)/(1

6*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*

b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 10

56*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) - (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b

^10*c*d^10 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 859068*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5

*d^6 - 14266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c^2*d^9))/(4096*(b^12*c^16 + a^12*c^4*

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

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

*c^6*d^10 - 12*a*b^11*c^15*d)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9

*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920

*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/

4)))*(-(a*b^7)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8

*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9

*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*2i - atan(((-(81*a^8*d^8

+ 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d

^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d

^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^

9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287

555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9

*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*

c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*

c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*

c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147

584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^

10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*

b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*

a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8

*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072

*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9

- 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(3/4)*(((-(81*a^8*d^8 + 194481*b^8*c^8 + 40748

4*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6

+ 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^

18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a

^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 +

8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*(8192*a^2*b

^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^7 + 3384832*a^6*b^

14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11*d^11 + 1115136*a^

10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^7*d^15 - 76288*a^1

4*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 +

28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a

*b^7*c^11*d) - (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 1030225920*a^4*b^19*c^17*d

^6 - 3328573440*a^5*b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^14*d^9 + 14203486208

*a^8*b^15*c^13*d^10 - 13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 5521702912*a^11*b^12*c

^10*d^13 + 1989672960*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9*c^7*d^16 + 3430809

60*a^15*b^8*c^6*d^17 - 127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a^18*b^5*c^3*d^20 +

147456*a^19*b^4*c^2*d^21))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*

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

495*a^8*b^4*c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d))) - ((81*a^9*b^7*d^10)/20

48 - (1431*a^8*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4

*b^12*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/20

48)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4

- 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d))*1i - (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c

*d^10 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 859068*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6

- 14266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c^2*d^9)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^

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

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

^6*d^10 - 12*a*b^11*c^15*d))) - (-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3

- 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7

)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 110

7296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^

14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 369098752

0*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^

6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*

b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201

326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*

d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*

a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481

*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316

*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201

326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^

4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^

7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(

3/4)*(((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4

- 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d +

16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3

- 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*

b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 110

7296256*a^10*b^2*c^9*d^11))^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6

- 1607168*a^5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d

^10 - 3277824*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^

8*d^14 + 256512*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18)

)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 -

56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) + (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a

^3*b^20*c^18*d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8

- 11738087424*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^13*d^10 - 13361086464*a^9*b^14*c^12*d^11 + 9861857280

*a^10*b^13*c^11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 1989672960*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8

*d^15 - 484442112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 - 127401984*a^16*b^7*c^5*d^18 + 27394048*a^1

7*b^6*c^4*d^19 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^21))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*

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

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

- 12*a*b^11*c^15*d))) - ((81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 -

(713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b

^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d

^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d))*1i +

(x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 85906

8*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^1

1*c^2*d^9)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c^13

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

c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d))))/((-(81*a^8*d^8 + 194481*b^8*c^8 +

407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^

2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^

11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721

920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*

d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*((-(81

*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b

^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^

12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 369098752

0*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7

- 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10

*b^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*

a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(167772

16*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a

^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 +

15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3

*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(3/4)*(((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 +

8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*

d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a

^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13

287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*

c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488

*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8 - 492518

4*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10*d^12 +

199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 120

32*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2

- 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (x^(1

/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573440*a^5*

b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^13*d^10

- 13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 198967296

0*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17

- 127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*

d^21))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c^13*d^3 + 4

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

- 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d))) - ((81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c

*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/2048

+ (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)/(b^8*c^12 + a^8*

c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5

+ 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d)) - (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 194481*a^2*b^17

*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 859068*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^6*b^13*c^4*d^

7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c^2*d^9))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 +

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

^6 - 792*a^7*b^5*c^9*d^7 + 495*a^8*b^4*c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d

))) + (-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 -

1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d +

16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3

- 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b

^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107

296256*a^10*b^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d

^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d

^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1

107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*

c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987

520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*

c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*

a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 2

01326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^1

5*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 830472192

0*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(3/4)*(((-(81*a^8*d^8 + 194

481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8

316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 -

201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16

*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072

*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11)

)^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d

^7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^1

1*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c

^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8

- 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^

6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) + (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 1030225

920*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^1

4*d^9 + 14203486208*a^8*b^15*c^13*d^10 - 13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 552

1702912*a^11*b^12*c^10*d^13 + 1989672960*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9

*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 - 127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a

^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^21))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b

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

*a^7*b^5*c^9*d^7 + 495*a^8*b^4*c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d))) - ((

81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)

/2048 - (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*

a^7*b^9*c^2*d^8)/2048)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 +

70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d)) + (x^(1/2)*(81*a^10*b^9*d^11 -

1512*a^9*b^10*c*d^10 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 859068*a^4*b^15*c^6*d^5 + 610344*a

^5*b^14*c^5*d^6 - 14266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c^2*d^9))/(4096*(b^12*c^16

+ a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c^13*d^3 + 495*a^4*b^8*c^12*d^4 - 79

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

66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))))*(-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^

3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 151

2*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c

^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 132875550

72*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^

9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*2i + 2*atan(((-(81*a^8*d^8 + 194481*b^

8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^

6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326

592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 +

8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b

^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4

)*((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 11

76*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 167

77216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3

690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*

c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296

256*a^10*b^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3

- 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)

/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107

296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^1

4*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520

*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(3/4)*(((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^

6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*

b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201

326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*

d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*

a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*(8192*a^2*b^18*c^18*d^4

- 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8

- 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10

*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^

16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*

c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d)

- (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573

440*a^5*b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^

13*d^10 - 13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 1

989672960*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*

c^6*d^17 - 127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*

b^4*c^2*d^21)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c

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

^4*c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*1i - (((81*a^9*b^7*d^10)/2048 -

(1431*a^8*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12

*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)*1

i)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 -

56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d)) + (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^1

0 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 859068*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14

266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c^2*d^9))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12

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

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

- 12*a*b^11*c^15*d))) - (-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 8594

6*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(1677

7216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256

*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6

+ 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b

^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 +

8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7

*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*

a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 1

3287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4

*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^

8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^

2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*

a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 830

4721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c

^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(3/4)*((

(-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*

a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 167772

16*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690

987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^1

3*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256

*a^10*b^2*c^9*d^11))^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607

168*a^5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3

277824*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14

+ 256512*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*

c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*

b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) + (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20

*c^18*d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738

087424*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^13*d^10 - 13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b

^13*c^11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 1989672960*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 -

484442112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 - 127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c

^4*d^19 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^21)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11

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

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

*a*b^11*c^15*d)))*1i - (((81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 -

(713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b

^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)*1i)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^

10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d))

- (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 8590

68*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^

11*c^2*d^9))/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c^13*d

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

8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d))))/((-(81*a^8*d^8 + 194481*b^8*c^8 + 40

7484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*

d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11

*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 830472192

0*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^

8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*((-(81*a

^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3

*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12

*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*

a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 -

13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b

^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^

4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216

*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2

*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15

502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c

^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(3/4)*(((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 82

32*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d

- 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^1

1*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 1328

7555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^

11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488*a

^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*a^5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*

a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 3277824*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10*d^12 + 19

9936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 256512*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032

*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 -

56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) - (x^(1/2

)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^18*d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^

18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 11738087424*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^13*d^10 -

13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*c^11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 1989672960*

a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 484442112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 -

127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d^19 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^

21)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c^13*d^3 +

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

- 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))*1i - (((81*a^9*b^7*d^10)/2048 - (1431*a^8*

b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (713097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/

2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)*1i)/(b^8*c^

12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^

3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d))*1i + (x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 194

481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 859068*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^6

*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^11*c^2*d^9)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^1

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

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

2*a*b^11*c^15*d))) + (-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^

4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216

*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2

*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15

502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c

^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 823

2*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d -

1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11

*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287

555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^1

1*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4)*((-(81*a^8*d^8 + 194481*b^8*c^8 +

407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^

2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^

11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721

920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*

d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(3/4)*(((-(8

1*a^8*d^8 + 194481*b^8*c^8 + 407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*

b^3*c^3*d^5 + 8316*a^6*b^2*c^2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a

^12*c^7*d^13 - 201326592*a*b^11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 36909875

20*a^3*b^9*c^16*d^4 + 8304721920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^

7 - 13287555072*a^7*b^5*c^12*d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^1

0*b^2*c^9*d^11))^(1/4)*(8192*a^2*b^18*c^18*d^4 - 95488*a^3*b^17*c^17*d^5 + 506112*a^4*b^16*c^16*d^6 - 1607168*

a^5*b^15*c^15*d^7 + 3384832*a^6*b^14*c^14*d^8 - 4925184*a^7*b^13*c^13*d^9 + 4958976*a^8*b^12*c^12*d^10 - 32778

24*a^9*b^11*c^11*d^11 + 1115136*a^10*b^10*c^10*d^12 + 199936*a^11*b^9*c^9*d^13 - 459008*a^12*b^8*c^8*d^14 + 25

6512*a^13*b^7*c^7*d^15 - 76288*a^14*b^6*c^6*d^16 + 12032*a^15*b^5*c^5*d^17 - 768*a^16*b^4*c^4*d^18))/(b^8*c^12

+ a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*

c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d) + (x^(1/2)*(16777216*a^2*b^21*c^19*d^4 - 194101248*a^3*b^20*c^1

8*d^5 + 1030225920*a^4*b^19*c^17*d^6 - 3328573440*a^5*b^18*c^16*d^7 + 7335837696*a^6*b^17*c^15*d^8 - 117380874

24*a^7*b^16*c^14*d^9 + 14203486208*a^8*b^15*c^13*d^10 - 13361086464*a^9*b^14*c^12*d^11 + 9861857280*a^10*b^13*

c^11*d^12 - 5521702912*a^11*b^12*c^10*d^13 + 1989672960*a^12*b^11*c^9*d^14 - 49938432*a^13*b^10*c^8*d^15 - 484

442112*a^14*b^9*c^7*d^16 + 343080960*a^15*b^8*c^6*d^17 - 127401984*a^16*b^7*c^5*d^18 + 27394048*a^17*b^6*c^4*d

^19 - 3145728*a^18*b^5*c^3*d^20 + 147456*a^19*b^4*c^2*d^21)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c

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

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

^11*c^15*d)))*1i - (((81*a^9*b^7*d^10)/2048 - (1431*a^8*b^8*c*d^9)/2048 - (194481*a^2*b^14*c^7*d^3)/2048 - (71

3097*a^3*b^13*c^6*d^4)/2048 - (432453*a^4*b^12*c^5*d^5)/2048 + (18067*a^5*b^11*c^4*d^6)/2048 + (5709*a^6*b^10*

c^3*d^7)/2048 + (6885*a^7*b^9*c^2*d^8)/2048)*1i)/(b^8*c^12 + a^8*c^4*d^8 - 8*a^7*b*c^5*d^7 + 28*a^2*b^6*c^10*d

^2 - 56*a^3*b^5*c^9*d^3 + 70*a^4*b^4*c^8*d^4 - 56*a^5*b^3*c^7*d^5 + 28*a^6*b^2*c^6*d^6 - 8*a*b^7*c^11*d))*1i -

(x^(1/2)*(81*a^10*b^9*d^11 - 1512*a^9*b^10*c*d^10 + 194481*a^2*b^17*c^8*d^3 + 518616*a^3*b^16*c^7*d^4 + 85906

8*a^4*b^15*c^6*d^5 + 610344*a^5*b^14*c^5*d^6 - 14266*a^6*b^13*c^4*d^7 - 87192*a^7*b^12*c^3*d^8 + 17532*a^8*b^1

1*c^2*d^9)*1i)/(4096*(b^12*c^16 + a^12*c^4*d^12 - 12*a^11*b*c^5*d^11 + 66*a^2*b^10*c^14*d^2 - 220*a^3*b^9*c^13

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

c^8*d^8 - 220*a^9*b^3*c^7*d^9 + 66*a^10*b^2*c^6*d^10 - 12*a*b^11*c^15*d)))))*(-(81*a^8*d^8 + 194481*b^8*c^8 +

407484*a^2*b^6*c^6*d^2 + 8232*a^3*b^5*c^5*d^3 - 85946*a^4*b^4*c^4*d^4 - 1176*a^5*b^3*c^3*d^5 + 8316*a^6*b^2*c^

2*d^6 + 518616*a*b^7*c^7*d - 1512*a^7*b*c*d^7)/(16777216*b^12*c^19*d + 16777216*a^12*c^7*d^13 - 201326592*a*b^

11*c^18*d^2 - 201326592*a^11*b*c^8*d^12 + 1107296256*a^2*b^10*c^17*d^3 - 3690987520*a^3*b^9*c^16*d^4 + 8304721

920*a^4*b^8*c^15*d^5 - 13287555072*a^5*b^7*c^14*d^6 + 15502147584*a^6*b^6*c^13*d^7 - 13287555072*a^7*b^5*c^12*

d^8 + 8304721920*a^8*b^4*c^11*d^9 - 3690987520*a^9*b^3*c^10*d^10 + 1107296256*a^10*b^2*c^9*d^11))^(1/4) - ((x^

(1/2)*(3*a*d - 11*b*c))/(16*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (d*x^(5/2)*(a*d + 7*b*c))/(16*(b^2*c^3 + a^2*c*

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

________________________________________________________________________________________

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

[Out]

Timed out

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