∫ √ _x _____(a+bx2 )(c+dx2 )3 dx

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

Optimal. Leaf size=633 \[ -\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^3}-\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {b^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {b^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {b^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {b^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {d x^{3/2} (13 b c-5 a d)}{16 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {d x^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

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

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

(1/2)/a^(1/4))/a^(1/4)/(-a*d+b*c)^3*2^(1/2)+1/64*d^(1/4)*(5*a^2*d^2-18*a*b*c*d+45*b^2*c^2)*arctan(1-d^(1/4)*2^

(1/2)*x^(1/2)/c^(1/4))/c^(9/4)/(-a*d+b*c)^3*2^(1/2)-1/64*d^(1/4)*(5*a^2*d^2-18*a*b*c*d+45*b^2*c^2)*arctan(1+d^

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

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

/a^(1/4)/(-a*d+b*c)^3*2^(1/2)-1/128*d^(1/4)*(5*a^2*d^2-18*a*b*c*d+45*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^(9/4)/(-a*d+b*c)^3*2^(1/2)+1/128*d^(1/4)*(5*a^2*d^2-18*a*b*c*d+45*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^(9/4)/(-a*d+b*c)^3*2^(1/2)

________________________________________________________________________________________

Rubi [A] time = 0.82, antiderivative size = 633, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 472, 579, 584, 297, 1162, 617, 204, 1165, 628} \[ -\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^3}-\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {b^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {b^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {b^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {b^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {d x^{3/2} (13 b c-5 a d)}{16 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {d x^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

+ (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*(b*c - a*d)^3) + (d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5

*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^3) - (d^(1/4)*(45*b^2

*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)^

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

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

/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt

[2]*c^(9/4)*(b*c - a*d)^3) + (d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1

/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(9/4)*(b*c - a*d)^3)

Rule 204

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

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

Rule 297

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

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

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

b]]))

Rule 466

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

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

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

rQ[p]

Rule 472

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

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

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

p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p

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

Rule 579

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x

_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +

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

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

e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy

mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,

f, g, m, p}, x] && IGtQ[n, 0]

Rule 617

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

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

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

Rule 628

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

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

Rule 1162

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

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

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

Rule 1165

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

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

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

Rubi steps

\begin {align*} \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d x^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (8 b c-5 a d-5 b d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 c (b c-a d)}\\ &=-\frac {d x^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (13 b c-5 a d) x^{3/2}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (32 b^2 c^2-13 a b c d+5 a^2 d^2-b d (13 b c-5 a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^2}\\ &=-\frac {d x^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (13 b c-5 a d) x^{3/2}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \left (\frac {32 b^3 c^2 x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {d \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right ) x^2}{(b c-a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^2}\\ &=-\frac {d x^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (13 b c-5 a d) x^{3/2}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}-\frac {\left (d \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^3}\\ &=-\frac {d x^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (13 b c-5 a d) x^{3/2}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b^{5/2} \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 {b^{5/2} \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 {d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^2 (b c-a d)^3}-\frac {\left (\sqrt {d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^2 (b c-a d)^3}\\ &=-\frac {d x^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (13 b c-5 a d) x^{3/2}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {b^2 \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 {b^2 \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 {b^{9/4} \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} \sqrt [4]{a} (b c-a d)^3}+\frac {b^{9/4} \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} \sqrt [4]{a} (b c-a d)^3}-\frac {\left (45 b^2 c^2-18 a b c d+5 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^2 (b c-a d)^3}-\frac {\left (45 b^2 c^2-18 a b c d+5 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^2 (b c-a d)^3}-\frac {\left (\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right )\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^{9/4} (b c-a d)^3}-\frac {\left (\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right )\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^{9/4} (b c-a d)^3}\\ &=-\frac {d x^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (13 b c-5 a d) x^{3/2}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {b^{9/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {b^{9/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 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^{9/4} (b c-a d)^3}+\frac {\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 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^{9/4} (b c-a d)^3}+\frac {b^{9/4} \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} \sqrt [4]{a} (b c-a d)^3}-\frac {b^{9/4} \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} \sqrt [4]{a} (b c-a d)^3}-\frac {\left (\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right )\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^{9/4} (b c-a d)^3}+\frac {\left (\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right )\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^{9/4} (b c-a d)^3}\\ &=-\frac {d x^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (13 b c-5 a d) x^{3/2}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {b^{9/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 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^{9/4} (b c-a d)^3}-\frac {\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 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^{9/4} (b c-a d)^3}+\frac {b^{9/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {b^{9/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 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^{9/4} (b c-a d)^3}+\frac {\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 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^{9/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A] time = 0.86, size = 620, normalized size = 0.98 \[ \frac {1}{128} \left (\frac {\sqrt {2} \sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{9/4} (a d-b c)^3}+\frac {\sqrt {2} \sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{9/4} (b c-a d)^3}+\frac {2 \sqrt {2} \sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{9/4} (b c-a d)^3}-\frac {2 \sqrt {2} \sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{9/4} (b c-a d)^3}+\frac {32 \sqrt {2} b^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{a} (b c-a d)^3}+\frac {32 \sqrt {2} b^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{a} (a d-b c)^3}+\frac {64 \sqrt {2} b^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} (a d-b c)^3}-\frac {64 \sqrt {2} b^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a} (a d-b c)^3}+\frac {8 d x^{3/2} (5 a d-13 b c)}{c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {32 d x^{3/2}}{c \left (c+d x^2\right )^2 (b c-a d)}\right ) \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-32*d*x^(3/2))/(c*(b*c - a*d)*(c + d*x^2)^2) + (8*d*(-13*b*c + 5*a*d)*x^(3/2))/(c^2*(b*c - a*d)^2*(c + d*x^2

)) + (64*Sqrt[2]*b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(-(b*c) + a*d)^3) - (64*Sqrt[

2]*b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*d^(1/4)*(45*

b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(9/4)*(b*c - a*d)^3) - (2*

Sqrt[2]*d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(9/4)*

(b*c - a*d)^3) + (32*Sqrt[2]*b^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(b*c

- a*d)^3) + (32*Sqrt[2]*b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(-(b*c)

+ a*d)^3) + (Sqrt[2]*d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[

x] + Sqrt[d]*x])/(c^(9/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[

c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(9/4)*(b*c - a*d)^3))/128

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

[Out]

Timed out

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giac [A] time = 1.61, size = 968, normalized size = 1.53 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(45*(c*d^3)^(3/4)*b^2*c^2 - 18*(c*d^3)^(3/4)*a*b*c*d + 5*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt

(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^6*d^2 - 3*sqrt(2)*a*b^2*c^5*d^3 + 3*sqrt(2)*a^2*b*c^4

*d^4 - sqrt(2)*a^3*c^3*d^5) - 1/32*(45*(c*d^3)^(3/4)*b^2*c^2 - 18*(c*d^3)^(3/4)*a*b*c*d + 5*(c*d^3)^(3/4)*a^2*

d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^6*d^2 - 3*sqrt(2)*a*b^2

*c^5*d^3 + 3*sqrt(2)*a^2*b*c^4*d^4 - sqrt(2)*a^3*c^3*d^5) + 1/64*(45*(c*d^3)^(3/4)*b^2*c^2 - 18*(c*d^3)^(3/4)*

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

qrt(2)*a*b^2*c^5*d^3 + 3*sqrt(2)*a^2*b*c^4*d^4 - sqrt(2)*a^3*c^3*d^5) - 1/64*(45*(c*d^3)^(3/4)*b^2*c^2 - 18*(c

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

c^6*d^2 - 3*sqrt(2)*a*b^2*c^5*d^3 + 3*sqrt(2)*a^2*b*c^4*d^4 - sqrt(2)*a^3*c^3*d^5) + (a*b^3)^(3/4)*arctan(1/2*

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

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

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

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

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

a*b^3*c^3 - 3*sqrt(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)*a^4*d^3) - 1/16*(13*b*c*d^2*x^(7/2) - 5*

a*d^3*x^(7/2) + 17*b*c^2*d*x^(3/2) - 9*a*c*d^2*x^(3/2))/((b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*(d*x^2 + c)^2)

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maple [A] time = 0.02, size = 855, normalized size = 1.35 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

/2)*a^2-9/8*d^3/(a*d-b*c)^3/(d*x^2+c)^2/c*x^(7/2)*a*b+13/16*d^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(7/2)*b^2+9/16*d^3/(

a*d-b*c)^3/(d*x^2+c)^2/c*x^(3/2)*a^2-13/8*d^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(3/2)*a*b+17/16*d/(a*d-b*c)^3/(d*x^2+c

)^2*c*x^(3/2)*b^2+5/64*d^2/(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^2-9/32*

d/(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+45/64/(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+5/128*d^2/(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^2-9/64*d/(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+45/

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+5/64*d^2/(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^2-9

/32*d/(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+45/64/(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

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maxima [A] time = 2.52, size = 594, normalized size = 0.94 \[ \frac {b^{3} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (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 (45 \, b^{2} c^{2} d - 18 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}} - \frac {{\left (13 \, b c d^{2} - 5 \, a d^{3}\right )} x^{\frac {7}{2}} + {\left (17 \, b c^{2} d - 9 \, a c d^{2}\right )} x^{\frac {3}{2}}}{16 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x^{2}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

(3/4)))/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - 1/128*(45*b^2*c^2*d - 18*a*b*c*d^2 + 5*a^2*d^3)*

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

c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c

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

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

(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3) - 1/16*((13*b*c*d^2 - 5*a*d^3)*x^(7/2) + (17*b*c^2*d

- 9*a*c*d^2)*x^(3/2))/(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*x^4

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

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mupad [B] time = 4.35, size = 32735, normalized size = 51.71 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan((((((2048*a*b^23*c^20*d^4 + (125*a^20*b^4*c*d^23)/16 - 22528*a^2*b^22*c^19*d^5 + (1711115*a^3*b^21*c^18

*d^6)/16 - (4294995*a^4*b^20*c^17*d^7)/16 + (565575*a^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15*d^9)/2 - (934

7799*a^7*b^17*c^14*d^10)/4 + (20337495*a^8*b^16*c^13*d^11)/4 - (14638795*a^9*b^15*c^12*d^12)/2 + (15550975*a^1

0*b^14*c^11*d^13)/2 - (50934983*a^11*b^13*c^10*d^14)/8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207335*a^13*b^11*

c^8*d^16)/2 + (1717635*a^14*b^10*c^7*d^17)/2 - (1110975*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*c^5*d^19)/4 -

(26949*a^17*b^7*c^4*d^20)/2 + (3745*a^18*b^6*c^3*d^21)/2 - (2725*a^19*b^5*c^2*d^22)/16)*1i)/(b^14*c^20 + a^14*

c^6*d^14 - 14*a^13*b*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a

^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6 - 3432*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^

9 + 1001*a^10*b^4*c^10*d^10 - 364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d) - (x^(1/2)*(-b^

9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*

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

- 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4)*(16777216*a*b^22*c^21*d^4 - 2013

26592*a^2*b^21*c^20*d^5 + 1140473856*a^3*b^20*c^19*d^6 - 4115660800*a^4*b^19*c^18*d^7 + 10825629696*a^5*b^18*c

^17*d^8 - 22493528064*a^6*b^17*c^16*d^9 + 38637076480*a^7*b^16*c^15*d^10 - 55691968512*a^8*b^15*c^14*d^11 + 66

935193600*a^9*b^14*c^13*d^12 - 66085978112*a^10*b^13*c^12*d^13 + 52807434240*a^11*b^12*c^11*d^14 - 33731641344

*a^12*b^11*c^10*d^15 + 17037131776*a^13*b^10*c^9*d^16 - 6723993600*a^14*b^9*c^8*d^17 + 2040201216*a^15*b^8*c^7

*d^18 - 463470592*a^16*b^7*c^6*d^19 + 75104256*a^17*b^6*c^5*d^20 - 7864320*a^18*b^5*c^4*d^21 + 409600*a^19*b^4

*c^3*d^22))/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^

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

10*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12

- 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^8*d^4 - 12672*a^6*b^7*

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

a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(3/4) - (x^(1/2)*(625*a^9*b^10*d^13 + 4100625*a*b^18*c^8*d^5 - 9000*a^

8*b^11*c*d^12 - 4487400*a^2*b^17*c^7*d^6 + 4100220*a^3*b^16*c^6*d^7 - 2444184*a^4*b^15*c^5*d^8 + 1099206*a^5*b

^14*c^4*d^9 - 334040*a^6*b^13*c^3*d^10 + 71100*a^7*b^12*c^2*d^11))/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*

b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a

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

2*a*b^11*c^17*d)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*

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

+ 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4) - ((((204

8*a*b^23*c^20*d^4 + (125*a^20*b^4*c*d^23)/16 - 22528*a^2*b^22*c^19*d^5 + (1711115*a^3*b^21*c^18*d^6)/16 - (429

4995*a^4*b^20*c^17*d^7)/16 + (565575*a^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15*d^9)/2 - (9347799*a^7*b^17*c

^14*d^10)/4 + (20337495*a^8*b^16*c^13*d^11)/4 - (14638795*a^9*b^15*c^12*d^12)/2 + (15550975*a^10*b^14*c^11*d^1

3)/2 - (50934983*a^11*b^13*c^10*d^14)/8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207335*a^13*b^11*c^8*d^16)/2 + (

1717635*a^14*b^10*c^7*d^17)/2 - (1110975*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*c^5*d^19)/4 - (26949*a^17*b^7

*c^4*d^20)/2 + (3745*a^18*b^6*c^3*d^21)/2 - (2725*a^19*b^5*c^2*d^22)/16)*1i)/(b^14*c^20 + a^14*c^6*d^14 - 14*a

^13*b*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5

+ 3003*a^6*b^8*c^14*d^6 - 3432*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b

^4*c^10*d^10 - 364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d) + (x^(1/2)*(-b^9/(16*a^13*d^12

+ 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^8*d^4

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

3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4)*(16777216*a*b^22*c^21*d^4 - 201326592*a^2*b^21*

c^20*d^5 + 1140473856*a^3*b^20*c^19*d^6 - 4115660800*a^4*b^19*c^18*d^7 + 10825629696*a^5*b^18*c^17*d^8 - 22493

528064*a^6*b^17*c^16*d^9 + 38637076480*a^7*b^16*c^15*d^10 - 55691968512*a^8*b^15*c^14*d^11 + 66935193600*a^9*b

^14*c^13*d^12 - 66085978112*a^10*b^13*c^12*d^13 + 52807434240*a^11*b^12*c^11*d^14 - 33731641344*a^12*b^11*c^10

*d^15 + 17037131776*a^13*b^10*c^9*d^16 - 6723993600*a^14*b^9*c^8*d^17 + 2040201216*a^15*b^8*c^7*d^18 - 4634705

92*a^16*b^7*c^6*d^19 + 75104256*a^17*b^6*c^5*d^20 - 7864320*a^18*b^5*c^4*d^21 + 409600*a^19*b^4*c^3*d^22))/(40

96*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8

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

9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11

*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^8*d^4 - 12672*a^6*b^7*c^7*d^5 + 14784

*a^7*b^6*c^6*d^6 - 12672*a^8*b^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^

10 - 192*a^12*b*c*d^11))^(3/4) + (x^(1/2)*(625*a^9*b^10*d^13 + 4100625*a*b^18*c^8*d^5 - 9000*a^8*b^11*c*d^12 -

4487400*a^2*b^17*c^7*d^6 + 4100220*a^3*b^16*c^6*d^7 - 2444184*a^4*b^15*c^5*d^8 + 1099206*a^5*b^14*c^4*d^9 - 3

34040*a^6*b^13*c^3*d^10 + 71100*a^7*b^12*c^2*d^11))/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66

*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a^6*b^6*c^12*d^6

- 792*a^7*b^5*c^11*d^7 + 495*a^8*b^4*c^10*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d

)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3

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

*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4))/(((((2048*a*b^23*c^20*d

^4 + (125*a^20*b^4*c*d^23)/16 - 22528*a^2*b^22*c^19*d^5 + (1711115*a^3*b^21*c^18*d^6)/16 - (4294995*a^4*b^20*c

^17*d^7)/16 + (565575*a^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15*d^9)/2 - (9347799*a^7*b^17*c^14*d^10)/4 + (

20337495*a^8*b^16*c^13*d^11)/4 - (14638795*a^9*b^15*c^12*d^12)/2 + (15550975*a^10*b^14*c^11*d^13)/2 - (5093498

3*a^11*b^13*c^10*d^14)/8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207335*a^13*b^11*c^8*d^16)/2 + (1717635*a^14*b^

10*c^7*d^17)/2 - (1110975*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*c^5*d^19)/4 - (26949*a^17*b^7*c^4*d^20)/2 +

(3745*a^18*b^6*c^3*d^21)/2 - (2725*a^19*b^5*c^2*d^22)/16)*1i)/(b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13

+ 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8

*c^14*d^6 - 3432*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 -

364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d) - (x^(1/2)*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^

12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^8*d^4 - 12672*a^6*b^

7*c^7*d^5 + 14784*a^7*b^6*c^6*d^6 - 12672*a^8*b^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 105

6*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4)*(16777216*a*b^22*c^21*d^4 - 201326592*a^2*b^21*c^20*d^5 + 1140

473856*a^3*b^20*c^19*d^6 - 4115660800*a^4*b^19*c^18*d^7 + 10825629696*a^5*b^18*c^17*d^8 - 22493528064*a^6*b^17

*c^16*d^9 + 38637076480*a^7*b^16*c^15*d^10 - 55691968512*a^8*b^15*c^14*d^11 + 66935193600*a^9*b^14*c^13*d^12 -

66085978112*a^10*b^13*c^12*d^13 + 52807434240*a^11*b^12*c^11*d^14 - 33731641344*a^12*b^11*c^10*d^15 + 1703713

1776*a^13*b^10*c^9*d^16 - 6723993600*a^14*b^9*c^8*d^17 + 2040201216*a^15*b^8*c^7*d^18 - 463470592*a^16*b^7*c^6

*d^19 + 75104256*a^17*b^6*c^5*d^20 - 7864320*a^18*b^5*c^4*d^21 + 409600*a^19*b^4*c^3*d^22))/(4096*(b^12*c^18 +

a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792

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

66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*

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

6 - 12672*a^8*b^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b

*c*d^11))^(3/4)*1i - (x^(1/2)*(625*a^9*b^10*d^13 + 4100625*a*b^18*c^8*d^5 - 9000*a^8*b^11*c*d^12 - 4487400*a^2

*b^17*c^7*d^6 + 4100220*a^3*b^16*c^6*d^7 - 2444184*a^4*b^15*c^5*d^8 + 1099206*a^5*b^14*c^4*d^9 - 334040*a^6*b^

13*c^3*d^10 + 71100*a^7*b^12*c^2*d^11)*1i)/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10

*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a^6*b^6*c^12*d^6 - 792*a^

7*b^5*c^11*d^7 + 495*a^8*b^4*c^10*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^9

/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a

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

- 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4) + ((((2048*a*b^23*c^20*d^4 + (125

*a^20*b^4*c*d^23)/16 - 22528*a^2*b^22*c^19*d^5 + (1711115*a^3*b^21*c^18*d^6)/16 - (4294995*a^4*b^20*c^17*d^7)/

16 + (565575*a^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15*d^9)/2 - (9347799*a^7*b^17*c^14*d^10)/4 + (20337495*

a^8*b^16*c^13*d^11)/4 - (14638795*a^9*b^15*c^12*d^12)/2 + (15550975*a^10*b^14*c^11*d^13)/2 - (50934983*a^11*b^

13*c^10*d^14)/8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207335*a^13*b^11*c^8*d^16)/2 + (1717635*a^14*b^10*c^7*d^

17)/2 - (1110975*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*c^5*d^19)/4 - (26949*a^17*b^7*c^4*d^20)/2 + (3745*a^1

8*b^6*c^3*d^21)/2 - (2725*a^19*b^5*c^2*d^22)/16)*1i)/(b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13 + 91*a^2*

b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6

- 3432*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 - 364*a^11*

b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d) + (x^(1/2)*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*

a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^8*d^4 - 12672*a^6*b^7*c^7*d^5

+ 14784*a^7*b^6*c^6*d^6 - 12672*a^8*b^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^

2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4)*(16777216*a*b^22*c^21*d^4 - 201326592*a^2*b^21*c^20*d^5 + 1140473856*a^

3*b^20*c^19*d^6 - 4115660800*a^4*b^19*c^18*d^7 + 10825629696*a^5*b^18*c^17*d^8 - 22493528064*a^6*b^17*c^16*d^9

+ 38637076480*a^7*b^16*c^15*d^10 - 55691968512*a^8*b^15*c^14*d^11 + 66935193600*a^9*b^14*c^13*d^12 - 66085978

112*a^10*b^13*c^12*d^13 + 52807434240*a^11*b^12*c^11*d^14 - 33731641344*a^12*b^11*c^10*d^15 + 17037131776*a^13

*b^10*c^9*d^16 - 6723993600*a^14*b^9*c^8*d^17 + 2040201216*a^15*b^8*c^7*d^18 - 463470592*a^16*b^7*c^6*d^19 + 7

5104256*a^17*b^6*c^5*d^20 - 7864320*a^18*b^5*c^4*d^21 + 409600*a^19*b^4*c^3*d^22))/(4096*(b^12*c^18 + a^12*c^6

*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*

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

b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*

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

*a^8*b^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))

^(3/4)*1i + (x^(1/2)*(625*a^9*b^10*d^13 + 4100625*a*b^18*c^8*d^5 - 9000*a^8*b^11*c*d^12 - 4487400*a^2*b^17*c^7

*d^6 + 4100220*a^3*b^16*c^6*d^7 - 2444184*a^4*b^15*c^5*d^8 + 1099206*a^5*b^14*c^4*d^9 - 334040*a^6*b^13*c^3*d^

10 + 71100*a^7*b^12*c^2*d^11)*1i)/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2

- 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a^6*b^6*c^12*d^6 - 792*a^7*b^5*c^1

1*d^7 + 495*a^8*b^4*c^10*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^9/(16*a^13

*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^

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

10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4) + ((625*a^8*b^12*d^12)/4096 - (4100625*a*b

^19*c^7*d^5)/4096 - (12375*a^7*b^13*c*d^11)/4096 + (5376375*a^2*b^18*c^6*d^6)/4096 - (3881925*a^3*b^17*c^5*d^7

)/4096 + (1726515*a^4*b^16*c^4*d^8)/4096 - (521235*a^5*b^15*c^3*d^9)/4096 + (101925*a^6*b^14*c^2*d^10)/4096)/(

b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*

c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6 - 3432*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 200

2*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 - 364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d

)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3

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

*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4) - atan(((((2048*a*b^23*c

^20*d^4 + (125*a^20*b^4*c*d^23)/16 - 22528*a^2*b^22*c^19*d^5 + (1711115*a^3*b^21*c^18*d^6)/16 - (4294995*a^4*b

^20*c^17*d^7)/16 + (565575*a^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15*d^9)/2 - (9347799*a^7*b^17*c^14*d^10)/

4 + (20337495*a^8*b^16*c^13*d^11)/4 - (14638795*a^9*b^15*c^12*d^12)/2 + (15550975*a^10*b^14*c^11*d^13)/2 - (50

934983*a^11*b^13*c^10*d^14)/8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207335*a^13*b^11*c^8*d^16)/2 + (1717635*a^

14*b^10*c^7*d^17)/2 - (1110975*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*c^5*d^19)/4 - (26949*a^17*b^7*c^4*d^20)

/2 + (3745*a^18*b^6*c^3*d^21)/2 - (2725*a^19*b^5*c^2*d^22)/16)/(b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13

+ 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^

8*c^14*d^6 - 3432*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 -

364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d) - (x^(1/2)*(-b^9/(16*a^13*d^12 + 16*a*b^12*c

^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^8*d^4 - 12672*a^6*b

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

56*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4)*(16777216*a*b^22*c^21*d^4 - 201326592*a^2*b^21*c^20*d^5 + 114

0473856*a^3*b^20*c^19*d^6 - 4115660800*a^4*b^19*c^18*d^7 + 10825629696*a^5*b^18*c^17*d^8 - 22493528064*a^6*b^1

7*c^16*d^9 + 38637076480*a^7*b^16*c^15*d^10 - 55691968512*a^8*b^15*c^14*d^11 + 66935193600*a^9*b^14*c^13*d^12

- 66085978112*a^10*b^13*c^12*d^13 + 52807434240*a^11*b^12*c^11*d^14 - 33731641344*a^12*b^11*c^10*d^15 + 170371

31776*a^13*b^10*c^9*d^16 - 6723993600*a^14*b^9*c^8*d^17 + 2040201216*a^15*b^8*c^7*d^18 - 463470592*a^16*b^7*c^

6*d^19 + 75104256*a^17*b^6*c^5*d^20 - 7864320*a^18*b^5*c^4*d^21 + 409600*a^19*b^4*c^3*d^22))/(4096*(b^12*c^18

+ a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 79

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

+ 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056

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

^6 - 12672*a^8*b^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*

b*c*d^11))^(3/4)*1i - (x^(1/2)*(625*a^9*b^10*d^13 + 4100625*a*b^18*c^8*d^5 - 9000*a^8*b^11*c*d^12 - 4487400*a^

2*b^17*c^7*d^6 + 4100220*a^3*b^16*c^6*d^7 - 2444184*a^4*b^15*c^5*d^8 + 1099206*a^5*b^14*c^4*d^9 - 334040*a^6*b

^13*c^3*d^10 + 71100*a^7*b^12*c^2*d^11)*1i)/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^1

0*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a^6*b^6*c^12*d^6 - 792*a

^7*b^5*c^11*d^7 + 495*a^8*b^4*c^10*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^

9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*

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

- 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4) - (((2048*a*b^23*c^20*d^4 + (125

*a^20*b^4*c*d^23)/16 - 22528*a^2*b^22*c^19*d^5 + (1711115*a^3*b^21*c^18*d^6)/16 - (4294995*a^4*b^20*c^17*d^7)/

16 + (565575*a^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15*d^9)/2 - (9347799*a^7*b^17*c^14*d^10)/4 + (20337495*

a^8*b^16*c^13*d^11)/4 - (14638795*a^9*b^15*c^12*d^12)/2 + (15550975*a^10*b^14*c^11*d^13)/2 - (50934983*a^11*b^

13*c^10*d^14)/8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207335*a^13*b^11*c^8*d^16)/2 + (1717635*a^14*b^10*c^7*d^

17)/2 - (1110975*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*c^5*d^19)/4 - (26949*a^17*b^7*c^4*d^20)/2 + (3745*a^1

8*b^6*c^3*d^21)/2 - (2725*a^19*b^5*c^2*d^22)/16)/(b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13 + 91*a^2*b^12

*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6 - 3

432*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 - 364*a^11*b^3*

c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d) + (x^(1/2)*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*

b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^8*d^4 - 12672*a^6*b^7*c^7*d^5 + 1

4784*a^7*b^6*c^6*d^6 - 12672*a^8*b^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^

2*d^10 - 192*a^12*b*c*d^11))^(1/4)*(16777216*a*b^22*c^21*d^4 - 201326592*a^2*b^21*c^20*d^5 + 1140473856*a^3*b^

20*c^19*d^6 - 4115660800*a^4*b^19*c^18*d^7 + 10825629696*a^5*b^18*c^17*d^8 - 22493528064*a^6*b^17*c^16*d^9 + 3

8637076480*a^7*b^16*c^15*d^10 - 55691968512*a^8*b^15*c^14*d^11 + 66935193600*a^9*b^14*c^13*d^12 - 66085978112*

a^10*b^13*c^12*d^13 + 52807434240*a^11*b^12*c^11*d^14 - 33731641344*a^12*b^11*c^10*d^15 + 17037131776*a^13*b^1

0*c^9*d^16 - 6723993600*a^14*b^9*c^8*d^17 + 2040201216*a^15*b^8*c^7*d^18 - 463470592*a^16*b^7*c^6*d^19 + 75104

256*a^17*b^6*c^5*d^20 - 7864320*a^18*b^5*c^4*d^21 + 409600*a^19*b^4*c^3*d^22))/(4096*(b^12*c^18 + a^12*c^6*d^1

2 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13

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

c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10

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

*b^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(3/

4)*1i + (x^(1/2)*(625*a^9*b^10*d^13 + 4100625*a*b^18*c^8*d^5 - 9000*a^8*b^11*c*d^12 - 4487400*a^2*b^17*c^7*d^6

+ 4100220*a^3*b^16*c^6*d^7 - 2444184*a^4*b^15*c^5*d^8 + 1099206*a^5*b^14*c^4*d^9 - 334040*a^6*b^13*c^3*d^10 +

71100*a^7*b^12*c^2*d^11)*1i)/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 2

20*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a^6*b^6*c^12*d^6 - 792*a^7*b^5*c^11*d^

7 + 495*a^8*b^4*c^10*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^9/(16*a^13*d^1

2 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^8*d^

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

^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4))/((((2048*a*b^23*c^20*d^4 + (125*a^20*b^4*c*d^

23)/16 - 22528*a^2*b^22*c^19*d^5 + (1711115*a^3*b^21*c^18*d^6)/16 - (4294995*a^4*b^20*c^17*d^7)/16 + (565575*a

^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15*d^9)/2 - (9347799*a^7*b^17*c^14*d^10)/4 + (20337495*a^8*b^16*c^13*

d^11)/4 - (14638795*a^9*b^15*c^12*d^12)/2 + (15550975*a^10*b^14*c^11*d^13)/2 - (50934983*a^11*b^13*c^10*d^14)/

8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207335*a^13*b^11*c^8*d^16)/2 + (1717635*a^14*b^10*c^7*d^17)/2 - (11109

75*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*c^5*d^19)/4 - (26949*a^17*b^7*c^4*d^20)/2 + (3745*a^18*b^6*c^3*d^21

)/2 - (2725*a^19*b^5*c^2*d^22)/16)/(b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 36

4*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6 - 3432*a^7*b^7*c^

13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 - 364*a^11*b^3*c^9*d^11 + 91*

a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d) - (x^(1/2)*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d +

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

^6*d^6 - 12672*a^8*b^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a

^12*b*c*d^11))^(1/4)*(16777216*a*b^22*c^21*d^4 - 201326592*a^2*b^21*c^20*d^5 + 1140473856*a^3*b^20*c^19*d^6 -

4115660800*a^4*b^19*c^18*d^7 + 10825629696*a^5*b^18*c^17*d^8 - 22493528064*a^6*b^17*c^16*d^9 + 38637076480*a^7

*b^16*c^15*d^10 - 55691968512*a^8*b^15*c^14*d^11 + 66935193600*a^9*b^14*c^13*d^12 - 66085978112*a^10*b^13*c^12

*d^13 + 52807434240*a^11*b^12*c^11*d^14 - 33731641344*a^12*b^11*c^10*d^15 + 17037131776*a^13*b^10*c^9*d^16 - 6

723993600*a^14*b^9*c^8*d^17 + 2040201216*a^15*b^8*c^7*d^18 - 463470592*a^16*b^7*c^6*d^19 + 75104256*a^17*b^6*c

^5*d^20 - 7864320*a^18*b^5*c^4*d^21 + 409600*a^19*b^4*c^3*d^22))/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*

c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a^6

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

a*b^11*c^17*d)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^

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

7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(3/4) - (x^(1/2)*

(625*a^9*b^10*d^13 + 4100625*a*b^18*c^8*d^5 - 9000*a^8*b^11*c*d^12 - 4487400*a^2*b^17*c^7*d^6 + 4100220*a^3*b^

16*c^6*d^7 - 2444184*a^4*b^15*c^5*d^8 + 1099206*a^5*b^14*c^4*d^9 - 334040*a^6*b^13*c^3*d^10 + 71100*a^7*b^12*c

^2*d^11))/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3

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

*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 -

192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^8*d^4 - 12672*a^6*b^7*c^

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

11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4) + (((2048*a*b^23*c^20*d^4 + (125*a^20*b^4*c*d^23)/16 - 22528*a^2*b

^22*c^19*d^5 + (1711115*a^3*b^21*c^18*d^6)/16 - (4294995*a^4*b^20*c^17*d^7)/16 + (565575*a^5*b^19*c^16*d^8)/2

+ (844557*a^6*b^18*c^15*d^9)/2 - (9347799*a^7*b^17*c^14*d^10)/4 + (20337495*a^8*b^16*c^13*d^11)/4 - (14638795*

a^9*b^15*c^12*d^12)/2 + (15550975*a^10*b^14*c^11*d^13)/2 - (50934983*a^11*b^13*c^10*d^14)/8 + (32835743*a^12*b

^12*c^9*d^15)/8 - (4207335*a^13*b^11*c^8*d^16)/2 + (1717635*a^14*b^10*c^7*d^17)/2 - (1110975*a^15*b^9*c^6*d^18

)/4 + (280623*a^16*b^8*c^5*d^19)/4 - (26949*a^17*b^7*c^4*d^20)/2 + (3745*a^18*b^6*c^3*d^21)/2 - (2725*a^19*b^5

*c^2*d^22)/16)/(b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3

+ 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6 - 3432*a^7*b^7*c^13*d^7 + 3003*a^8*b^

6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 - 364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 -

14*a*b^13*c^19*d) + (x^(1/2)*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d

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

^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4)

*(16777216*a*b^22*c^21*d^4 - 201326592*a^2*b^21*c^20*d^5 + 1140473856*a^3*b^20*c^19*d^6 - 4115660800*a^4*b^19*

c^18*d^7 + 10825629696*a^5*b^18*c^17*d^8 - 22493528064*a^6*b^17*c^16*d^9 + 38637076480*a^7*b^16*c^15*d^10 - 55

691968512*a^8*b^15*c^14*d^11 + 66935193600*a^9*b^14*c^13*d^12 - 66085978112*a^10*b^13*c^12*d^13 + 52807434240*

a^11*b^12*c^11*d^14 - 33731641344*a^12*b^11*c^10*d^15 + 17037131776*a^13*b^10*c^9*d^16 - 6723993600*a^14*b^9*c

^8*d^17 + 2040201216*a^15*b^8*c^7*d^18 - 463470592*a^16*b^7*c^6*d^19 + 75104256*a^17*b^6*c^5*d^20 - 7864320*a^

18*b^5*c^4*d^21 + 409600*a^19*b^4*c^3*d^22))/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^

10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a^6*b^6*c^12*d^6 - 792*

a^7*b^5*c^11*d^7 + 495*a^8*b^4*c^10*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b

^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920

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

8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(3/4) + (x^(1/2)*(625*a^9*b^10*d^13 +

4100625*a*b^18*c^8*d^5 - 9000*a^8*b^11*c*d^12 - 4487400*a^2*b^17*c^7*d^6 + 4100220*a^3*b^16*c^6*d^7 - 2444184

*a^4*b^15*c^5*d^8 + 1099206*a^5*b^14*c^4*d^9 - 334040*a^6*b^13*c^3*d^10 + 71100*a^7*b^12*c^2*d^11))/(4096*(b^1

2*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d

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

^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^11*c^11*d

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

6*c^6*d^6 - 12672*a^8*b^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^10 - 19

2*a^12*b*c*d^11))^(1/4) - ((625*a^8*b^12*d^12)/4096 - (4100625*a*b^19*c^7*d^5)/4096 - (12375*a^7*b^13*c*d^11)/

4096 + (5376375*a^2*b^18*c^6*d^6)/4096 - (3881925*a^3*b^17*c^5*d^7)/4096 + (1726515*a^4*b^16*c^4*d^8)/4096 - (

521235*a^5*b^15*c^3*d^9)/4096 + (101925*a^6*b^14*c^2*d^10)/4096)/(b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^

13 + 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*

b^8*c^14*d^6 - 3432*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10

- 364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d)))*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 1

92*a^2*b^11*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^8*d^4 - 12672*a^6*b^7*c^7*

d^5 + 14784*a^7*b^6*c^6*d^6 - 12672*a^8*b^5*c^5*d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11

*b^2*c^2*d^10 - 192*a^12*b*c*d^11))^(1/4)*2i + ((x^(3/2)*(9*a*d^2 - 17*b*c*d))/(16*(b^2*c^3 + a^2*c*d^2 - 2*a*

b*c^2*d)) + (d^2*x^(7/2)*(5*a*d - 13*b*c))/(16*c*(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) - atan(((-(625*a^8*d^9 + 4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^

3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8)/(

16777216*b^12*c^21 + 16777216*a^12*c^9*d^12 - 201326592*a^11*b*c^10*d^11 + 1107296256*a^2*b^10*c^19*d^2 - 3690

987520*a^3*b^9*c^18*d^3 + 8304721920*a^4*b^8*c^17*d^4 - 13287555072*a^5*b^7*c^16*d^5 + 15502147584*a^6*b^6*c^1

5*d^6 - 13287555072*a^7*b^5*c^14*d^7 + 8304721920*a^8*b^4*c^13*d^8 - 3690987520*a^9*b^3*c^12*d^9 + 1107296256*

a^10*b^2*c^11*d^10 - 201326592*a*b^11*c^20*d))^(1/4)*(((2048*a*b^23*c^20*d^4 + (125*a^20*b^4*c*d^23)/16 - 2252

8*a^2*b^22*c^19*d^5 + (1711115*a^3*b^21*c^18*d^6)/16 - (4294995*a^4*b^20*c^17*d^7)/16 + (565575*a^5*b^19*c^16*

d^8)/2 + (844557*a^6*b^18*c^15*d^9)/2 - (9347799*a^7*b^17*c^14*d^10)/4 + (20337495*a^8*b^16*c^13*d^11)/4 - (14

638795*a^9*b^15*c^12*d^12)/2 + (15550975*a^10*b^14*c^11*d^13)/2 - (50934983*a^11*b^13*c^10*d^14)/8 + (32835743

*a^12*b^12*c^9*d^15)/8 - (4207335*a^13*b^11*c^8*d^16)/2 + (1717635*a^14*b^10*c^7*d^17)/2 - (1110975*a^15*b^9*c

^6*d^18)/4 + (280623*a^16*b^8*c^5*d^19)/4 - (26949*a^17*b^7*c^4*d^20)/2 + (3745*a^18*b^6*c^3*d^21)/2 - (2725*a

^19*b^5*c^2*d^22)/16)/(b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^

17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6 - 3432*a^7*b^7*c^13*d^7 + 3003

*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 - 364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*

d^12 - 14*a*b^13*c^19*d) - (x^(1/2)*(-(625*a^8*d^9 + 4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b

^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^

7 - 9000*a^7*b*c*d^8)/(16777216*b^12*c^21 + 16777216*a^12*c^9*d^12 - 201326592*a^11*b*c^10*d^11 + 1107296256*a

^2*b^10*c^19*d^2 - 3690987520*a^3*b^9*c^18*d^3 + 8304721920*a^4*b^8*c^17*d^4 - 13287555072*a^5*b^7*c^16*d^5 +

15502147584*a^6*b^6*c^15*d^6 - 13287555072*a^7*b^5*c^14*d^7 + 8304721920*a^8*b^4*c^13*d^8 - 3690987520*a^9*b^3

*c^12*d^9 + 1107296256*a^10*b^2*c^11*d^10 - 201326592*a*b^11*c^20*d))^(1/4)*(16777216*a*b^22*c^21*d^4 - 201326

592*a^2*b^21*c^20*d^5 + 1140473856*a^3*b^20*c^19*d^6 - 4115660800*a^4*b^19*c^18*d^7 + 10825629696*a^5*b^18*c^1

7*d^8 - 22493528064*a^6*b^17*c^16*d^9 + 38637076480*a^7*b^16*c^15*d^10 - 55691968512*a^8*b^15*c^14*d^11 + 6693

5193600*a^9*b^14*c^13*d^12 - 66085978112*a^10*b^13*c^12*d^13 + 52807434240*a^11*b^12*c^11*d^14 - 33731641344*a

^12*b^11*c^10*d^15 + 17037131776*a^13*b^10*c^9*d^16 - 6723993600*a^14*b^9*c^8*d^17 + 2040201216*a^15*b^8*c^7*d

^18 - 463470592*a^16*b^7*c^6*d^19 + 75104256*a^17*b^6*c^5*d^20 - 7864320*a^18*b^5*c^4*d^21 + 409600*a^19*b^4*c

^3*d^22))/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3

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

*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-(625*a^8*d^9 + 4100625*b^8*c^8*d - 6

561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^

5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8)/(16777216*b^12*c^21 + 16777216*a^12*c^9*d^12 - 20132

6592*a^11*b*c^10*d^11 + 1107296256*a^2*b^10*c^19*d^2 - 3690987520*a^3*b^9*c^18*d^3 + 8304721920*a^4*b^8*c^17*d

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

^8*b^4*c^13*d^8 - 3690987520*a^9*b^3*c^12*d^9 + 1107296256*a^10*b^2*c^11*d^10 - 201326592*a*b^11*c^20*d))^(3/4

)*1i - (x^(1/2)*(625*a^9*b^10*d^13 + 4100625*a*b^18*c^8*d^5 - 9000*a^8*b^11*c*d^12 - 4487400*a^2*b^17*c^7*d^6

+ 4100220*a^3*b^16*c^6*d^7 - 2444184*a^4*b^15*c^5*d^8 + 1099206*a^5*b^14*c^4*d^9 - 334040*a^6*b^13*c^3*d^10 +

71100*a^7*b^12*c^2*d^11)*1i)/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 22

0*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a^6*b^6*c^12*d^6 - 792*a^7*b^5*c^11*d^7

+ 495*a^8*b^4*c^10*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d))) - (-(625*a^8*d^9 +

4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^

4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8)/(16777216*b^12*c^21 + 16777216*

a^12*c^9*d^12 - 201326592*a^11*b*c^10*d^11 + 1107296256*a^2*b^10*c^19*d^2 - 3690987520*a^3*b^9*c^18*d^3 + 8304

721920*a^4*b^8*c^17*d^4 - 13287555072*a^5*b^7*c^16*d^5 + 15502147584*a^6*b^6*c^15*d^6 - 13287555072*a^7*b^5*c^

14*d^7 + 8304721920*a^8*b^4*c^13*d^8 - 3690987520*a^9*b^3*c^12*d^9 + 1107296256*a^10*b^2*c^11*d^10 - 201326592

*a*b^11*c^20*d))^(1/4)*(((2048*a*b^23*c^20*d^4 + (125*a^20*b^4*c*d^23)/16 - 22528*a^2*b^22*c^19*d^5 + (1711115

*a^3*b^21*c^18*d^6)/16 - (4294995*a^4*b^20*c^17*d^7)/16 + (565575*a^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15

*d^9)/2 - (9347799*a^7*b^17*c^14*d^10)/4 + (20337495*a^8*b^16*c^13*d^11)/4 - (14638795*a^9*b^15*c^12*d^12)/2 +

(15550975*a^10*b^14*c^11*d^13)/2 - (50934983*a^11*b^13*c^10*d^14)/8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207

335*a^13*b^11*c^8*d^16)/2 + (1717635*a^14*b^10*c^7*d^17)/2 - (1110975*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*

c^5*d^19)/4 - (26949*a^17*b^7*c^4*d^20)/2 + (3745*a^18*b^6*c^3*d^21)/2 - (2725*a^19*b^5*c^2*d^22)/16)/(b^14*c^

20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^

4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6 - 3432*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b

^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 - 364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d) + (x^

(1/2)*(-(625*a^8*d^9 + 4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c

^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8)/(1677721

6*b^12*c^21 + 16777216*a^12*c^9*d^12 - 201326592*a^11*b*c^10*d^11 + 1107296256*a^2*b^10*c^19*d^2 - 3690987520*

a^3*b^9*c^18*d^3 + 8304721920*a^4*b^8*c^17*d^4 - 13287555072*a^5*b^7*c^16*d^5 + 15502147584*a^6*b^6*c^15*d^6 -

13287555072*a^7*b^5*c^14*d^7 + 8304721920*a^8*b^4*c^13*d^8 - 3690987520*a^9*b^3*c^12*d^9 + 1107296256*a^10*b^

2*c^11*d^10 - 201326592*a*b^11*c^20*d))^(1/4)*(16777216*a*b^22*c^21*d^4 - 201326592*a^2*b^21*c^20*d^5 + 114047

3856*a^3*b^20*c^19*d^6 - 4115660800*a^4*b^19*c^18*d^7 + 10825629696*a^5*b^18*c^17*d^8 - 22493528064*a^6*b^17*c

^16*d^9 + 38637076480*a^7*b^16*c^15*d^10 - 55691968512*a^8*b^15*c^14*d^11 + 66935193600*a^9*b^14*c^13*d^12 - 6

6085978112*a^10*b^13*c^12*d^13 + 52807434240*a^11*b^12*c^11*d^14 - 33731641344*a^12*b^11*c^10*d^15 + 170371317

76*a^13*b^10*c^9*d^16 - 6723993600*a^14*b^9*c^8*d^17 + 2040201216*a^15*b^8*c^7*d^18 - 463470592*a^16*b^7*c^6*d

^19 + 75104256*a^17*b^6*c^5*d^20 - 7864320*a^18*b^5*c^4*d^21 + 409600*a^19*b^4*c^3*d^22))/(4096*(b^12*c^18 + a

^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a

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

6*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-(625*a^8*d^9 + 4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100

*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*

c^2*d^7 - 9000*a^7*b*c*d^8)/(16777216*b^12*c^21 + 16777216*a^12*c^9*d^12 - 201326592*a^11*b*c^10*d^11 + 110729

6256*a^2*b^10*c^19*d^2 - 3690987520*a^3*b^9*c^18*d^3 + 8304721920*a^4*b^8*c^17*d^4 - 13287555072*a^5*b^7*c^16*

d^5 + 15502147584*a^6*b^6*c^15*d^6 - 13287555072*a^7*b^5*c^14*d^7 + 8304721920*a^8*b^4*c^13*d^8 - 3690987520*a

^9*b^3*c^12*d^9 + 1107296256*a^10*b^2*c^11*d^10 - 201326592*a*b^11*c^20*d))^(3/4)*1i + (x^(1/2)*(625*a^9*b^10*

d^13 + 4100625*a*b^18*c^8*d^5 - 9000*a^8*b^11*c*d^12 - 4487400*a^2*b^17*c^7*d^6 + 4100220*a^3*b^16*c^6*d^7 - 2

444184*a^4*b^15*c^5*d^8 + 1099206*a^5*b^14*c^4*d^9 - 334040*a^6*b^13*c^3*d^10 + 71100*a^7*b^12*c^2*d^11)*1i)/(

4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b

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

a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d))))/((-(625*a^8*d^9 + 4100625*b^8*c^8*d - 6561000*a*

b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3

*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8)/(16777216*b^12*c^21 + 16777216*a^12*c^9*d^12 - 201326592*a^11

*b*c^10*d^11 + 1107296256*a^2*b^10*c^19*d^2 - 3690987520*a^3*b^9*c^18*d^3 + 8304721920*a^4*b^8*c^17*d^4 - 1328

7555072*a^5*b^7*c^16*d^5 + 15502147584*a^6*b^6*c^15*d^6 - 13287555072*a^7*b^5*c^14*d^7 + 8304721920*a^8*b^4*c^

13*d^8 - 3690987520*a^9*b^3*c^12*d^9 + 1107296256*a^10*b^2*c^11*d^10 - 201326592*a*b^11*c^20*d))^(1/4)*(((2048

*a*b^23*c^20*d^4 + (125*a^20*b^4*c*d^23)/16 - 22528*a^2*b^22*c^19*d^5 + (1711115*a^3*b^21*c^18*d^6)/16 - (4294

995*a^4*b^20*c^17*d^7)/16 + (565575*a^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15*d^9)/2 - (9347799*a^7*b^17*c^

14*d^10)/4 + (20337495*a^8*b^16*c^13*d^11)/4 - (14638795*a^9*b^15*c^12*d^12)/2 + (15550975*a^10*b^14*c^11*d^13

)/2 - (50934983*a^11*b^13*c^10*d^14)/8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207335*a^13*b^11*c^8*d^16)/2 + (1

717635*a^14*b^10*c^7*d^17)/2 - (1110975*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*c^5*d^19)/4 - (26949*a^17*b^7*

c^4*d^20)/2 + (3745*a^18*b^6*c^3*d^21)/2 - (2725*a^19*b^5*c^2*d^22)/16)/(b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b

*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 30

03*a^6*b^8*c^14*d^6 - 3432*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^

10*d^10 - 364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d) - (x^(1/2)*(-(625*a^8*d^9 + 4100625

*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d

^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8)/(16777216*b^12*c^21 + 16777216*a^12*c^

9*d^12 - 201326592*a^11*b*c^10*d^11 + 1107296256*a^2*b^10*c^19*d^2 - 3690987520*a^3*b^9*c^18*d^3 + 8304721920*

a^4*b^8*c^17*d^4 - 13287555072*a^5*b^7*c^16*d^5 + 15502147584*a^6*b^6*c^15*d^6 - 13287555072*a^7*b^5*c^14*d^7

+ 8304721920*a^8*b^4*c^13*d^8 - 3690987520*a^9*b^3*c^12*d^9 + 1107296256*a^10*b^2*c^11*d^10 - 201326592*a*b^11

*c^20*d))^(1/4)*(16777216*a*b^22*c^21*d^4 - 201326592*a^2*b^21*c^20*d^5 + 1140473856*a^3*b^20*c^19*d^6 - 41156

60800*a^4*b^19*c^18*d^7 + 10825629696*a^5*b^18*c^17*d^8 - 22493528064*a^6*b^17*c^16*d^9 + 38637076480*a^7*b^16

*c^15*d^10 - 55691968512*a^8*b^15*c^14*d^11 + 66935193600*a^9*b^14*c^13*d^12 - 66085978112*a^10*b^13*c^12*d^13

+ 52807434240*a^11*b^12*c^11*d^14 - 33731641344*a^12*b^11*c^10*d^15 + 17037131776*a^13*b^10*c^9*d^16 - 672399

3600*a^14*b^9*c^8*d^17 + 2040201216*a^15*b^8*c^7*d^18 - 463470592*a^16*b^7*c^6*d^19 + 75104256*a^17*b^6*c^5*d^

20 - 7864320*a^18*b^5*c^4*d^21 + 409600*a^19*b^4*c^3*d^22))/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d

^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a^6*b^6*

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

1*c^17*d)))*(-(625*a^8*d^9 + 4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^3

*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8)/(1

6777216*b^12*c^21 + 16777216*a^12*c^9*d^12 - 201326592*a^11*b*c^10*d^11 + 1107296256*a^2*b^10*c^19*d^2 - 36909

87520*a^3*b^9*c^18*d^3 + 8304721920*a^4*b^8*c^17*d^4 - 13287555072*a^5*b^7*c^16*d^5 + 15502147584*a^6*b^6*c^15

*d^6 - 13287555072*a^7*b^5*c^14*d^7 + 8304721920*a^8*b^4*c^13*d^8 - 3690987520*a^9*b^3*c^12*d^9 + 1107296256*a

^10*b^2*c^11*d^10 - 201326592*a*b^11*c^20*d))^(3/4) - (x^(1/2)*(625*a^9*b^10*d^13 + 4100625*a*b^18*c^8*d^5 - 9

000*a^8*b^11*c*d^12 - 4487400*a^2*b^17*c^7*d^6 + 4100220*a^3*b^16*c^6*d^7 - 2444184*a^4*b^15*c^5*d^8 + 1099206

*a^5*b^14*c^4*d^9 - 334040*a^6*b^13*c^3*d^10 + 71100*a^7*b^12*c^2*d^11))/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12

*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 +

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

10 - 12*a*b^11*c^17*d))) + (-(625*a^8*d^9 + 4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^

3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*

a^7*b*c*d^8)/(16777216*b^12*c^21 + 16777216*a^12*c^9*d^12 - 201326592*a^11*b*c^10*d^11 + 1107296256*a^2*b^10*c

^19*d^2 - 3690987520*a^3*b^9*c^18*d^3 + 8304721920*a^4*b^8*c^17*d^4 - 13287555072*a^5*b^7*c^16*d^5 + 155021475

84*a^6*b^6*c^15*d^6 - 13287555072*a^7*b^5*c^14*d^7 + 8304721920*a^8*b^4*c^13*d^8 - 3690987520*a^9*b^3*c^12*d^9

+ 1107296256*a^10*b^2*c^11*d^10 - 201326592*a*b^11*c^20*d))^(1/4)*(((2048*a*b^23*c^20*d^4 + (125*a^20*b^4*c*d

^23)/16 - 22528*a^2*b^22*c^19*d^5 + (1711115*a^3*b^21*c^18*d^6)/16 - (4294995*a^4*b^20*c^17*d^7)/16 + (565575*

a^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15*d^9)/2 - (9347799*a^7*b^17*c^14*d^10)/4 + (20337495*a^8*b^16*c^13

*d^11)/4 - (14638795*a^9*b^15*c^12*d^12)/2 + (15550975*a^10*b^14*c^11*d^13)/2 - (50934983*a^11*b^13*c^10*d^14)

/8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207335*a^13*b^11*c^8*d^16)/2 + (1717635*a^14*b^10*c^7*d^17)/2 - (1110

975*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*c^5*d^19)/4 - (26949*a^17*b^7*c^4*d^20)/2 + (3745*a^18*b^6*c^3*d^2

1)/2 - (2725*a^19*b^5*c^2*d^22)/16)/(b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 3

64*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6 - 3432*a^7*b^7*c

^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 - 364*a^11*b^3*c^9*d^11 + 91

*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d) + (x^(1/2)*(-(625*a^8*d^9 + 4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 +

5759100*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*

a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8)/(16777216*b^12*c^21 + 16777216*a^12*c^9*d^12 - 201326592*a^11*b*c^10*d^11

+ 1107296256*a^2*b^10*c^19*d^2 - 3690987520*a^3*b^9*c^18*d^3 + 8304721920*a^4*b^8*c^17*d^4 - 13287555072*a^5*b

^7*c^16*d^5 + 15502147584*a^6*b^6*c^15*d^6 - 13287555072*a^7*b^5*c^14*d^7 + 8304721920*a^8*b^4*c^13*d^8 - 3690

987520*a^9*b^3*c^12*d^9 + 1107296256*a^10*b^2*c^11*d^10 - 201326592*a*b^11*c^20*d))^(1/4)*(16777216*a*b^22*c^2

1*d^4 - 201326592*a^2*b^21*c^20*d^5 + 1140473856*a^3*b^20*c^19*d^6 - 4115660800*a^4*b^19*c^18*d^7 + 1082562969

6*a^5*b^18*c^17*d^8 - 22493528064*a^6*b^17*c^16*d^9 + 38637076480*a^7*b^16*c^15*d^10 - 55691968512*a^8*b^15*c^

14*d^11 + 66935193600*a^9*b^14*c^13*d^12 - 66085978112*a^10*b^13*c^12*d^13 + 52807434240*a^11*b^12*c^11*d^14 -

33731641344*a^12*b^11*c^10*d^15 + 17037131776*a^13*b^10*c^9*d^16 - 6723993600*a^14*b^9*c^8*d^17 + 2040201216*

a^15*b^8*c^7*d^18 - 463470592*a^16*b^7*c^6*d^19 + 75104256*a^17*b^6*c^5*d^20 - 7864320*a^18*b^5*c^4*d^21 + 409

600*a^19*b^4*c^3*d^22))/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*d^2 - 220*a^3

*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a^6*b^6*c^12*d^6 - 792*a^7*b^5*c^11*d^7 + 49

5*a^8*b^4*c^10*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d)))*(-(625*a^8*d^9 + 4100625

*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d

^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8)/(16777216*b^12*c^21 + 16777216*a^12*c^

9*d^12 - 201326592*a^11*b*c^10*d^11 + 1107296256*a^2*b^10*c^19*d^2 - 3690987520*a^3*b^9*c^18*d^3 + 8304721920*

a^4*b^8*c^17*d^4 - 13287555072*a^5*b^7*c^16*d^5 + 15502147584*a^6*b^6*c^15*d^6 - 13287555072*a^7*b^5*c^14*d^7

+ 8304721920*a^8*b^4*c^13*d^8 - 3690987520*a^9*b^3*c^12*d^9 + 1107296256*a^10*b^2*c^11*d^10 - 201326592*a*b^11

*c^20*d))^(3/4) + (x^(1/2)*(625*a^9*b^10*d^13 + 4100625*a*b^18*c^8*d^5 - 9000*a^8*b^11*c*d^12 - 4487400*a^2*b^

17*c^7*d^6 + 4100220*a^3*b^16*c^6*d^7 - 2444184*a^4*b^15*c^5*d^8 + 1099206*a^5*b^14*c^4*d^9 - 334040*a^6*b^13*

c^3*d^10 + 71100*a^7*b^12*c^2*d^11))/(4096*(b^12*c^18 + a^12*c^6*d^12 - 12*a^11*b*c^7*d^11 + 66*a^2*b^10*c^16*

d^2 - 220*a^3*b^9*c^15*d^3 + 495*a^4*b^8*c^14*d^4 - 792*a^5*b^7*c^13*d^5 + 924*a^6*b^6*c^12*d^6 - 792*a^7*b^5*

c^11*d^7 + 495*a^8*b^4*c^10*d^8 - 220*a^9*b^3*c^9*d^9 + 66*a^10*b^2*c^8*d^10 - 12*a*b^11*c^17*d))) - ((625*a^8

*b^12*d^12)/4096 - (4100625*a*b^19*c^7*d^5)/4096 - (12375*a^7*b^13*c*d^11)/4096 + (5376375*a^2*b^18*c^6*d^6)/4

096 - (3881925*a^3*b^17*c^5*d^7)/4096 + (1726515*a^4*b^16*c^4*d^8)/4096 - (521235*a^5*b^15*c^3*d^9)/4096 + (10

1925*a^6*b^14*c^2*d^10)/4096)/(b^14*c^20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 364*a^3

*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6 - 3432*a^7*b^7*c^13*d^

7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 - 364*a^11*b^3*c^9*d^11 + 91*a^12*

b^2*c^8*d^12 - 14*a*b^13*c^19*d)))*(-(625*a^8*d^9 + 4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^

6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7

- 9000*a^7*b*c*d^8)/(16777216*b^12*c^21 + 16777216*a^12*c^9*d^12 - 201326592*a^11*b*c^10*d^11 + 1107296256*a^

2*b^10*c^19*d^2 - 3690987520*a^3*b^9*c^18*d^3 + 8304721920*a^4*b^8*c^17*d^4 - 13287555072*a^5*b^7*c^16*d^5 + 1

5502147584*a^6*b^6*c^15*d^6 - 13287555072*a^7*b^5*c^14*d^7 + 8304721920*a^8*b^4*c^13*d^8 - 3690987520*a^9*b^3*

c^12*d^9 + 1107296256*a^10*b^2*c^11*d^10 - 201326592*a*b^11*c^20*d))^(1/4)*2i + 2*atan(((-(625*a^8*d^9 + 41006

25*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4

*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8)/(16777216*b^12*c^21 + 16777216*a^12*

c^9*d^12 - 201326592*a^11*b*c^10*d^11 + 1107296256*a^2*b^10*c^19*d^2 - 3690987520*a^3*b^9*c^18*d^3 + 830472192

0*a^4*b^8*c^17*d^4 - 13287555072*a^5*b^7*c^16*d^5 + 15502147584*a^6*b^6*c^15*d^6 - 13287555072*a^7*b^5*c^14*d^

7 + 8304721920*a^8*b^4*c^13*d^8 - 3690987520*a^9*b^3*c^12*d^9 + 1107296256*a^10*b^2*c^11*d^10 - 201326592*a*b^

11*c^20*d))^(1/4)*((((2048*a*b^23*c^20*d^4 + (125*a^20*b^4*c*d^23)/16 - 22528*a^2*b^22*c^19*d^5 + (1711115*a^3

*b^21*c^18*d^6)/16 - (4294995*a^4*b^20*c^17*d^7)/16 + (565575*a^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15*d^9

)/2 - (9347799*a^7*b^17*c^14*d^10)/4 + (20337495*a^8*b^16*c^13*d^11)/4 - (14638795*a^9*b^15*c^12*d^12)/2 + (15

550975*a^10*b^14*c^11*d^13)/2 - (50934983*a^11*b^13*c^10*d^14)/8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207335*

a^13*b^11*c^8*d^16)/2 + (1717635*a^14*b^10*c^7*d^17)/2 - (1110975*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*c^5*

d^19)/4 - (26949*a^17*b^7*c^4*d^20)/2 + (3745*a^18*b^6*c^3*d^21)/2 - (2725*a^19*b^5*c^2*d^22)/16)*1i)/(b^14*c^

20 + a^14*c^6*d^14 - 14*a^13*b*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1001*a^4*b^10*c^16*d^

4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6 - 3432*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b

^5*c^11*d^9 + 1001*a^10*b^4*c^10*d^10 - 364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a*b^13*c^19*d) - (x^