∫ 1_______ x5∕2(a+bx2)(c+dx2)2 dx

3.478 \(\int \frac {1}{x^{5/2} (a+b x^2) (c+d x^2)^2} \, dx\)

Optimal. Leaf size=570 \[ \frac {b^{11/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^2}-\frac {b^{11/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^2}+\frac {b^{11/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^2}-\frac {b^{11/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{7/4} (b c-a d)^2}-\frac {d^{7/4} (11 b c-7 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} (b c-a d)^2}+\frac {d^{7/4} (11 b c-7 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} (b c-a d)^2}-\frac {d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} (b c-a d)^2}+\frac {d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{11/4} (b c-a d)^2}-\frac {4 b c-7 a d}{6 a c^2 x^{3/2} (b c-a d)}-\frac {d}{2 c x^{3/2} \left (c+d x^2\right ) (b c-a d)} \]

[Out]

1/6*(7*a*d-4*b*c)/a/c^2/(-a*d+b*c)/x^(3/2)-1/2*d/c/(-a*d+b*c)/x^(3/2)/(d*x^2+c)+1/2*b^(11/4)*arctan(1-b^(1/4)*

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

^(7/4)/(-a*d+b*c)^2*2^(1/2)-1/8*d^(7/4)*(-7*a*d+11*b*c)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/(-a

*d+b*c)^2*2^(1/2)+1/8*d^(7/4)*(-7*a*d+11*b*c)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/(-a*d+b*c)^2*

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

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

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

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

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Rubi [A] time = 0.82, antiderivative size = 570, 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, 472, 583, 522, 211, 1165, 628, 1162, 617, 204} \[ \frac {b^{11/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^2}-\frac {b^{11/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^2}+\frac {b^{11/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^2}-\frac {b^{11/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{7/4} (b c-a d)^2}-\frac {d^{7/4} (11 b c-7 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} (b c-a d)^2}+\frac {d^{7/4} (11 b c-7 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} (b c-a d)^2}-\frac {d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} (b c-a d)^2}+\frac {d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{11/4} (b c-a d)^2}-\frac {4 b c-7 a d}{6 a c^2 x^{3/2} (b c-a d)}-\frac {d}{2 c x^{3/2} \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

-(4*b*c - 7*a*d)/(6*a*c^2*(b*c - a*d)*x^(3/2)) - d/(2*c*(b*c - a*d)*x^(3/2)*(c + d*x^2)) + (b^(11/4)*ArcTan[1

- (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (b^(11/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*

Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (d^(7/4)*(11*b*c - 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqr

t[x])/c^(1/4)])/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^2) + (d^(7/4)*(11*b*c - 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqr

t[x])/c^(1/4)])/(4*Sqrt[2]*c^(11/4)*(b*c - a*d)^2) + (b^(11/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +

Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (b^(11/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqr

t[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (d^(7/4)*(11*b*c - 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*S

qrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*(b*c - a*d)^2) + (d^(7/4)*(11*b*c - 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1

/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*(b*c - a*d)^2)

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 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 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 583

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

x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*

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

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

IGtQ[n, 0] && LtQ[m, -1]

Rule 617

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

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

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

Rule 628

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

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

Rule 1162

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

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

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

Rule 1165

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

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

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

Rubi steps

\begin {align*} \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d}{2 c (b c-a d) x^{3/2} \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {4 b c-7 a d-7 b d x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{2 c (b c-a d)}\\ &=-\frac {4 b c-7 a d}{6 a c^2 (b c-a d) x^{3/2}}-\frac {d}{2 c (b c-a d) x^{3/2} \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {3 \left (4 b^2 c^2+4 a b c d-7 a^2 d^2\right )+3 b d (4 b c-7 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{6 a c^2 (b c-a d)}\\ &=-\frac {4 b c-7 a d}{6 a c^2 (b c-a d) x^{3/2}}-\frac {d}{2 c (b c-a d) x^{3/2} \left (c+d x^2\right )}-\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^2}+\frac {\left (d^2 (11 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c^2 (b c-a d)^2}\\ &=-\frac {4 b c-7 a d}{6 a c^2 (b c-a d) x^{3/2}}-\frac {d}{2 c (b c-a d) x^{3/2} \left (c+d x^2\right )}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} (b c-a d)^2}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} (b c-a d)^2}+\frac {\left (d^2 (11 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^{5/2} (b c-a d)^2}+\frac {\left (d^2 (11 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^{5/2} (b c-a d)^2}\\ &=-\frac {4 b c-7 a d}{6 a c^2 (b c-a d) x^{3/2}}-\frac {d}{2 c (b c-a d) x^{3/2} \left (c+d x^2\right )}-\frac {b^{5/2} \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 a^{3/2} (b c-a d)^2}-\frac {b^{5/2} \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 a^{3/2} (b c-a d)^2}+\frac {b^{11/4} \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} a^{7/4} (b c-a d)^2}+\frac {b^{11/4} \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} a^{7/4} (b c-a d)^2}+\frac {\left (d^{3/2} (11 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 c^{5/2} (b c-a d)^2}+\frac {\left (d^{3/2} (11 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 c^{5/2} (b c-a d)^2}-\frac {\left (d^{7/4} (11 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{11/4} (b c-a d)^2}-\frac {\left (d^{7/4} (11 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{11/4} (b c-a d)^2}\\ &=-\frac {4 b c-7 a d}{6 a c^2 (b c-a d) x^{3/2}}-\frac {d}{2 c (b c-a d) x^{3/2} \left (c+d x^2\right )}+\frac {b^{11/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^2}-\frac {b^{11/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^2}-\frac {d^{7/4} (11 b c-7 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} (b c-a d)^2}+\frac {d^{7/4} (11 b c-7 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} (b c-a d)^2}-\frac {b^{11/4} \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} a^{7/4} (b c-a d)^2}+\frac {b^{11/4} \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} a^{7/4} (b c-a d)^2}+\frac {\left (d^{7/4} (11 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} (b c-a d)^2}-\frac {\left (d^{7/4} (11 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} (b c-a d)^2}\\ &=-\frac {4 b c-7 a d}{6 a c^2 (b c-a d) x^{3/2}}-\frac {d}{2 c (b c-a d) x^{3/2} \left (c+d x^2\right )}+\frac {b^{11/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^2}-\frac {b^{11/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^2}-\frac {d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} (b c-a d)^2}+\frac {d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} (b c-a d)^2}+\frac {b^{11/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^2}-\frac {b^{11/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^2}-\frac {d^{7/4} (11 b c-7 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} (b c-a d)^2}+\frac {d^{7/4} (11 b c-7 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} (b c-a d)^2}\\ \end {align*}

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Mathematica [A] time = 6.15, size = 602, normalized size = 1.06 \[ \frac {b^{11/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^2}-\frac {b^{11/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^2}-\frac {b^{11/4} \tan ^{-1}\left (\frac {2 \sqrt [4]{b} \sqrt {x}-\sqrt {2} \sqrt [4]{a}}{\sqrt {2} \sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^2}-\frac {b^{11/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a}+2 \sqrt [4]{b} \sqrt {x}}{\sqrt {2} \sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^2}-\frac {d^{7/4} (11 b c-7 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} (a d-b c)^2}+\frac {d^{7/4} (11 b c-7 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} (a d-b c)^2}+\frac {d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (\frac {2 \sqrt [4]{d} \sqrt {x}-\sqrt {2} \sqrt [4]{c}}{\sqrt {2} \sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} (a d-b c)^2}+\frac {d^{7/4} (11 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c}+2 \sqrt [4]{d} \sqrt {x}}{\sqrt {2} \sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} (a d-b c)^2}+\frac {d^2 \sqrt {x}}{2 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac {2}{3 a c^2 x^{3/2}} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

-2/(3*a*c^2*x^(3/2)) + (d^2*Sqrt[x])/(2*c^2*(b*c - a*d)*(c + d*x^2)) - (b^(11/4)*ArcTan[(-(Sqrt[2]*a^(1/4)) +

2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (b^(11/4)*ArcTan[(Sqrt[2]*a^(1/4) + 2

*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^2) + (d^(7/4)*(11*b*c - 7*a*d)*ArcTan[(-(Sq

rt[2]*c^(1/4)) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(11/4)*(-(b*c) + a*d)^2) + (d^(7/4)*(11*b

*c - 7*a*d)*ArcTan[(Sqrt[2]*c^(1/4) + 2*d^(1/4)*Sqrt[x])/(Sqrt[2]*c^(1/4))])/(4*Sqrt[2]*c^(11/4)*(-(b*c) + a*d

)^2) + (b^(11/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^2)

- (b^(11/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^2) - (

d^(7/4)*(11*b*c - 7*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(11/4)*(-(b*

c) + a*d)^2) + (d^(7/4)*(11*b*c - 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2

]*c^(11/4)*(-(b*c) + a*d)^2)

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fricas [B] time = 111.97, size = 3439, normalized size = 6.03 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

-1/24*(12*((a*b*c^3*d - a^2*c^2*d^2)*x^4 + (a*b*c^4 - a^2*c^3*d)*x^2)*(-(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d

^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*b^6*c^17*

d^2 - 56*a^3*b^5*c^16*d^3 + 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*d^5 + 28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12*d^7

+ a^8*c^11*d^8))^(1/4)*arctan(((b^6*c^14 - 6*a*b^5*c^13*d + 15*a^2*b^4*c^12*d^2 - 20*a^3*b^3*c^11*d^3 + 15*a^

4*b^2*c^10*d^4 - 6*a^5*b*c^9*d^5 + a^6*c^8*d^6)*sqrt((121*b^2*c^2*d^4 - 154*a*b*c*d^5 + 49*a^2*d^6)*x + (b^4*c

^10 - 4*a*b^3*c^9*d + 6*a^2*b^2*c^8*d^2 - 4*a^3*b*c^7*d^3 + a^4*c^6*d^4)*sqrt(-(14641*b^4*c^4*d^7 - 37268*a*b^

3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*b^

6*c^17*d^2 - 56*a^3*b^5*c^16*d^3 + 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*d^5 + 28*a^6*b^2*c^13*d^6 - 8*a^7*b*c

^12*d^7 + a^8*c^11*d^8)))*(-(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d

^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*b^6*c^17*d^2 - 56*a^3*b^5*c^16*d^3 + 70*a^4*b^4*c^15*

d^4 - 56*a^5*b^3*c^14*d^5 + 28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12*d^7 + a^8*c^11*d^8))^(3/4) + (11*b^7*c^15*d^2 -

73*a*b^6*c^14*d^3 + 207*a^2*b^5*c^13*d^4 - 325*a^3*b^4*c^12*d^5 + 305*a^4*b^3*c^11*d^6 - 171*a^5*b^2*c^10*d^7

+ 53*a^6*b*c^9*d^8 - 7*a^7*c^8*d^9)*sqrt(x)*(-(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^

9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*b^6*c^17*d^2 - 56*a^3*b^5*c^16*d^3

+ 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*d^5 + 28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12*d^7 + a^8*c^11*d^8))^(3/4))/

(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^11)) + 48*(

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

2*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b*c*d^7 + a^15*d^8))^(1/4)*((a*b*c^3*d - a^2*c^2*d^2)*x^4 + (a*b*

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

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

0*b^5*c^5*d^3 + 70*a^11*b^4*c^4*d^4 - 56*a^12*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b*c*d^7 + a^15*d^8))^

(3/4)*sqrt(b^6*x + (a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4)*sqrt(-b^11/(a

^7*b^8*c^8 - 8*a^8*b^7*c^7*d + 28*a^9*b^6*c^6*d^2 - 56*a^10*b^5*c^5*d^3 + 70*a^11*b^4*c^4*d^4 - 56*a^12*b^3*c^

3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b*c*d^7 + a^15*d^8))) - (a^5*b^9*c^6 - 6*a^6*b^8*c^5*d + 15*a^7*b^7*c^4*d

^2 - 20*a^8*b^6*c^3*d^3 + 15*a^9*b^5*c^2*d^4 - 6*a^10*b^4*c*d^5 + a^11*b^3*d^6)*(-b^11/(a^7*b^8*c^8 - 8*a^8*b^

7*c^7*d + 28*a^9*b^6*c^6*d^2 - 56*a^10*b^5*c^5*d^3 + 70*a^11*b^4*c^4*d^4 - 56*a^12*b^3*c^3*d^5 + 28*a^13*b^2*c

^2*d^6 - 8*a^14*b*c*d^7 + a^15*d^8))^(3/4)*sqrt(x))/b^11) + 12*(-b^11/(a^7*b^8*c^8 - 8*a^8*b^7*c^7*d + 28*a^9*

b^6*c^6*d^2 - 56*a^10*b^5*c^5*d^3 + 70*a^11*b^4*c^4*d^4 - 56*a^12*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b

*c*d^7 + a^15*d^8))^(1/4)*((a*b*c^3*d - a^2*c^2*d^2)*x^4 + (a*b*c^4 - a^2*c^3*d)*x^2)*log(b^3*sqrt(x) + (-b^11

/(a^7*b^8*c^8 - 8*a^8*b^7*c^7*d + 28*a^9*b^6*c^6*d^2 - 56*a^10*b^5*c^5*d^3 + 70*a^11*b^4*c^4*d^4 - 56*a^12*b^3

*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b*c*d^7 + a^15*d^8))^(1/4)*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)) - 12

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

^12*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b*c*d^7 + a^15*d^8))^(1/4)*((a*b*c^3*d - a^2*c^2*d^2)*x^4 + (a*

b*c^4 - a^2*c^3*d)*x^2)*log(b^3*sqrt(x) - (-b^11/(a^7*b^8*c^8 - 8*a^8*b^7*c^7*d + 28*a^9*b^6*c^6*d^2 - 56*a^10

*b^5*c^5*d^3 + 70*a^11*b^4*c^4*d^4 - 56*a^12*b^3*c^3*d^5 + 28*a^13*b^2*c^2*d^6 - 8*a^14*b*c*d^7 + a^15*d^8))^(

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

(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^11)/(b^8*c^

19 - 8*a*b^7*c^18*d + 28*a^2*b^6*c^17*d^2 - 56*a^3*b^5*c^16*d^3 + 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*d^5 +

28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12*d^7 + a^8*c^11*d^8))^(1/4)*log(-(11*b*c*d^2 - 7*a*d^3)*sqrt(x) + (b^2*c^5 -

2*a*b*c^4*d + a^2*c^3*d^2)*(-(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c

*d^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*b^6*c^17*d^2 - 56*a^3*b^5*c^16*d^3 + 70*a^4*b^4*c^1

5*d^4 - 56*a^5*b^3*c^14*d^5 + 28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12*d^7 + a^8*c^11*d^8))^(1/4)) - 3*((a*b*c^3*d -

a^2*c^2*d^2)*x^4 + (a*b*c^4 - a^2*c^3*d)*x^2)*(-(14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*

d^9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*b^6*c^17*d^2 - 56*a^3*b^5*c^16*d

^3 + 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*d^5 + 28*a^6*b^2*c^13*d^6 - 8*a^7*b*c^12*d^7 + a^8*c^11*d^8))^(1/4)

*log(-(11*b*c*d^2 - 7*a*d^3)*sqrt(x) - (b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*(-(14641*b^4*c^4*d^7 - 37268*a*b^

3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10 + 2401*a^4*d^11)/(b^8*c^19 - 8*a*b^7*c^18*d + 28*a^2*b^

6*c^17*d^2 - 56*a^3*b^5*c^16*d^3 + 70*a^4*b^4*c^15*d^4 - 56*a^5*b^3*c^14*d^5 + 28*a^6*b^2*c^13*d^6 - 8*a^7*b*c

^12*d^7 + a^8*c^11*d^8))^(1/4)) + 4*(4*b*c^2 - 4*a*c*d + (4*b*c*d - 7*a*d^2)*x^2)*sqrt(x))/((a*b*c^3*d - a^2*c

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

________________________________________________________________________________________

giac [A] time = 1.15, size = 718, normalized size = 1.26 \[ -\frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}\right )}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{2} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}\right )}} + \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{5} - 2 \, \sqrt {2} a b c^{4} d + \sqrt {2} a^{2} c^{3} d^{2}\right )}} + \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{5} - 2 \, \sqrt {2} a b c^{4} d + \sqrt {2} a^{2} c^{3} d^{2}\right )}} + \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{5} - 2 \, \sqrt {2} a b c^{4} d + \sqrt {2} a^{2} c^{3} d^{2}\right )}} - \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{5} - 2 \, \sqrt {2} a b c^{4} d + \sqrt {2} a^{2} c^{3} d^{2}\right )}} + \frac {d^{2} \sqrt {x}}{2 \, {\left (b c^{3} - a c^{2} d\right )} {\left (d x^{2} + c\right )}} - \frac {2}{3 \, a c^{2} x^{\frac {3}{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="giac")

[Out]

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

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

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

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

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

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

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

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

*b^2*c^5 - 2*sqrt(2)*a*b*c^4*d + sqrt(2)*a^2*c^3*d^2) + 1/8*(11*(c*d^3)^(1/4)*b*c*d - 7*(c*d^3)^(1/4)*a*d^2)*l

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

- 1/8*(11*(c*d^3)^(1/4)*b*c*d - 7*(c*d^3)^(1/4)*a*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt

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

/3/(a*c^2*x^(3/2))

________________________________________________________________________________________

maple [A] time = 0.02, size = 588, normalized size = 1.03 \[ -\frac {a \,d^{3} \sqrt {x}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right ) c^{2}}+\frac {b \,d^{2} \sqrt {x}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right ) c}-\frac {7 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{2} c^{3}}-\frac {7 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{2} c^{3}}-\frac {7 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{3} \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{2} c^{3}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right )^{2} a^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right )^{2} a^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{3} \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (a d -b c \right )^{2} a^{2}}+\frac {11 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{2} c^{2}}+\frac {11 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{2} c^{2}}+\frac {11 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b \,d^{2} \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{2} c^{2}}-\frac {2}{3 a \,c^{2} x^{\frac {3}{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^2,x)

[Out]

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

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

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

(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a+11/8*d^2/c^2/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)

-1)*b-7/16*d^3/c^3/(a*d-b*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+11/16*d^2/c^2/(a*d-b*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)))*b-7/8*d^3/c^3/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*a

rctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a+11/8*d^2/c^2/(a*d-b*c)^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)

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

________________________________________________________________________________________

maxima [A] time = 2.49, size = 539, normalized size = 0.95 \[ -\frac {\frac {2 \, \sqrt {2} b^{3} \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^{3} \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 {11}{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 {11}{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}}}}{4 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} - \frac {4 \, b c^{2} - 4 \, a c d + {\left (4 \, b c d - 7 \, a d^{2}\right )} x^{2}}{6 \, {\left ({\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{\frac {7}{2}} + {\left (a b c^{4} - a^{2} c^{3} d\right )} x^{\frac {3}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (11 \, b c d^{2} - 7 \, a d^{3}\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 (11 \, b c d^{2} - 7 \, a d^{3}\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 (11 \, b c d^{2} - 7 \, a d^{3}\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 (11 \, b c d^{2} - 7 \, a d^{3}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{16 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

-1/4*(2*sqrt(2)*b^3*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^3*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^(11/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x

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

)/a^(3/4))/(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2) - 1/6*(4*b*c^2 - 4*a*c*d + (4*b*c*d - 7*a*d^2)*x^2)/((a*b*c^3*d

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

qrt(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)*(11*b*c*d^2 - 7*a*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c

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

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

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

________________________________________________________________________________________

mupad [B] time = 6.58, size = 27743, normalized size = 48.67 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^2),x)

[Out]

2*atan(((-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3

+ 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(1/4)*(x^(1/2)*(158

59712*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^10 + 600711168*a^11*b^22*c^29*d^11 - 2168807424*a^12*b^21

*c^28*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 14037065728*a^14*b^19*c^26*d^14 + 22648012800*a^15*b^18*c^25*d^1

5 - 26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^17*b^16*c^23*d^17 - 13398917120*a^18*b^15*c^22*d^18 + 5629

976576*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^20*d^20 + 261316608*a^21*b^12*c^19*d^21 - 19668992*a^22*b^

11*c^18*d^22) - (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*

c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(1/4)*((-b^

11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*

b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(3/4)*((-b^11/(16*a^15*d^8 + 16

*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^

12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(1/4)*(67108864*a^13*b^25*c^46*d^4 - 1140850688*a^1

4*b^24*c^45*d^5 + 9126805504*a^15*b^23*c^44*d^6 - 45818576896*a^16*b^22*c^43*d^7 + 162973876224*a^17*b^21*c^42

*d^8 - 442364854272*a^18*b^20*c^41*d^9 + 972004786176*a^19*b^19*c^40*d^10 - 1824220250112*a^20*b^18*c^39*d^11

+ 3052916441088*a^21*b^17*c^38*d^12 - 4642121449472*a^22*b^16*c^37*d^13 + 6347693228032*a^23*b^15*c^36*d^14 -

7600917708800*a^24*b^14*c^35*d^15 + 7756643827712*a^25*b^13*c^34*d^16 - 6603814207488*a^26*b^12*c^33*d^17 + 46

13600182272*a^27*b^11*c^32*d^18 - 2604562120704*a^28*b^10*c^31*d^19 + 1167090253824*a^29*b^9*c^30*d^20 - 40506

9103104*a^30*b^8*c^29*d^21 + 104958263296*a^31*b^7*c^28*d^22 - 19109249024*a^32*b^6*c^27*d^23 + 2181038080*a^3

3*b^5*c^26*d^24 - 117440512*a^34*b^4*c^25*d^25)*1i + x^(1/2)*(33554432*a^11*b^26*c^44*d^4 - 503316480*a^12*b^2

5*c^43*d^5 + 3523215360*a^13*b^24*c^42*d^6 - 15267266560*a^14*b^23*c^41*d^7 + 45801799680*a^15*b^22*c^40*d^8 -

100510203904*a^16*b^21*c^39*d^9 + 163810639872*a^17*b^20*c^38*d^10 - 184331272192*a^18*b^19*c^37*d^11 + 65011

712000*a^19*b^18*c^36*d^12 + 336173465600*a^20*b^17*c^35*d^13 - 1148861808640*a^21*b^16*c^34*d^14 + 2334365057

024*a^22*b^15*c^33*d^15 - 3542660153344*a^23*b^14*c^32*d^16 + 4221965434880*a^24*b^13*c^31*d^17 - 400906256384

0*a^25*b^12*c^30*d^18 + 3039679217664*a^26*b^11*c^29*d^19 - 1830545260544*a^27*b^10*c^28*d^20 + 864890650624*a

^28*b^9*c^27*d^21 - 313859768320*a^29*b^8*c^26*d^22 + 84473282560*a^30*b^7*c^25*d^23 - 15888023552*a^31*b^6*c^

24*d^24 + 1864368128*a^32*b^5*c^23*d^25 - 102760448*a^33*b^4*c^22*d^26))*1i + 11534336*a^9*b^25*c^35*d^7 - 111

149056*a^10*b^24*c^34*d^8 + 481296384*a^11*b^23*c^33*d^9 - 1233125376*a^12*b^22*c^32*d^10 + 1830010880*a^13*b^

21*c^31*d^11 + 391331840*a^14*b^20*c^30*d^12 - 12820119552*a^15*b^19*c^29*d^13 + 46592393216*a^16*b^18*c^28*d^

14 - 104394047488*a^17*b^17*c^27*d^15 + 165297111040*a^18*b^16*c^26*d^16 - 192702906368*a^19*b^15*c^25*d^17 +

167824392192*a^20*b^14*c^24*d^18 - 109211664384*a^21*b^13*c^23*d^19 + 52444708864*a^22*b^12*c^22*d^20 - 180622

13120*a^23*b^11*c^21*d^21 + 4224417792*a^24*b^10*c^20*d^22 - 601309184*a^25*b^9*c^19*d^23 + 39337984*a^26*b^8*

c^18*d^24)*1i) + (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5

*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(1/4)*(x^(

1/2)*(15859712*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^10 + 600711168*a^11*b^22*c^29*d^11 - 2168807424*

a^12*b^21*c^28*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 14037065728*a^14*b^19*c^26*d^14 + 22648012800*a^15*b^18

*c^25*d^15 - 26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^17*b^16*c^23*d^17 - 13398917120*a^18*b^15*c^22*d^

18 + 5629976576*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^20*d^20 + 261316608*a^21*b^12*c^19*d^21 - 1966899

2*a^22*b^11*c^18*d^22) + (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*

a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(1

/4)*((-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1

120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(3/4)*((-b^11/(16*a^15

*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4

- 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(1/4)*(67108864*a^13*b^25*c^46*d^4 - 11408

50688*a^14*b^24*c^45*d^5 + 9126805504*a^15*b^23*c^44*d^6 - 45818576896*a^16*b^22*c^43*d^7 + 162973876224*a^17*

b^21*c^42*d^8 - 442364854272*a^18*b^20*c^41*d^9 + 972004786176*a^19*b^19*c^40*d^10 - 1824220250112*a^20*b^18*c

^39*d^11 + 3052916441088*a^21*b^17*c^38*d^12 - 4642121449472*a^22*b^16*c^37*d^13 + 6347693228032*a^23*b^15*c^3

6*d^14 - 7600917708800*a^24*b^14*c^35*d^15 + 7756643827712*a^25*b^13*c^34*d^16 - 6603814207488*a^26*b^12*c^33*

d^17 + 4613600182272*a^27*b^11*c^32*d^18 - 2604562120704*a^28*b^10*c^31*d^19 + 1167090253824*a^29*b^9*c^30*d^2

0 - 405069103104*a^30*b^8*c^29*d^21 + 104958263296*a^31*b^7*c^28*d^22 - 19109249024*a^32*b^6*c^27*d^23 + 21810

38080*a^33*b^5*c^26*d^24 - 117440512*a^34*b^4*c^25*d^25)*1i - x^(1/2)*(33554432*a^11*b^26*c^44*d^4 - 503316480

*a^12*b^25*c^43*d^5 + 3523215360*a^13*b^24*c^42*d^6 - 15267266560*a^14*b^23*c^41*d^7 + 45801799680*a^15*b^22*c

^40*d^8 - 100510203904*a^16*b^21*c^39*d^9 + 163810639872*a^17*b^20*c^38*d^10 - 184331272192*a^18*b^19*c^37*d^1

1 + 65011712000*a^19*b^18*c^36*d^12 + 336173465600*a^20*b^17*c^35*d^13 - 1148861808640*a^21*b^16*c^34*d^14 + 2

334365057024*a^22*b^15*c^33*d^15 - 3542660153344*a^23*b^14*c^32*d^16 + 4221965434880*a^24*b^13*c^31*d^17 - 400

9062563840*a^25*b^12*c^30*d^18 + 3039679217664*a^26*b^11*c^29*d^19 - 1830545260544*a^27*b^10*c^28*d^20 + 86489

0650624*a^28*b^9*c^27*d^21 - 313859768320*a^29*b^8*c^26*d^22 + 84473282560*a^30*b^7*c^25*d^23 - 15888023552*a^

31*b^6*c^24*d^24 + 1864368128*a^32*b^5*c^23*d^25 - 102760448*a^33*b^4*c^22*d^26))*1i + 11534336*a^9*b^25*c^35*

d^7 - 111149056*a^10*b^24*c^34*d^8 + 481296384*a^11*b^23*c^33*d^9 - 1233125376*a^12*b^22*c^32*d^10 + 183001088

0*a^13*b^21*c^31*d^11 + 391331840*a^14*b^20*c^30*d^12 - 12820119552*a^15*b^19*c^29*d^13 + 46592393216*a^16*b^1

8*c^28*d^14 - 104394047488*a^17*b^17*c^27*d^15 + 165297111040*a^18*b^16*c^26*d^16 - 192702906368*a^19*b^15*c^2

5*d^17 + 167824392192*a^20*b^14*c^24*d^18 - 109211664384*a^21*b^13*c^23*d^19 + 52444708864*a^22*b^12*c^22*d^20

- 18062213120*a^23*b^11*c^21*d^21 + 4224417792*a^24*b^10*c^20*d^22 - 601309184*a^25*b^9*c^19*d^23 + 39337984*

a^26*b^8*c^18*d^24)*1i))/((-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896

*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(

1/4)*(x^(1/2)*(15859712*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^10 + 600711168*a^11*b^22*c^29*d^11 - 21

68807424*a^12*b^21*c^28*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 14037065728*a^14*b^19*c^26*d^14 + 22648012800*

a^15*b^18*c^25*d^15 - 26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^17*b^16*c^23*d^17 - 13398917120*a^18*b^1

5*c^22*d^18 + 5629976576*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^20*d^20 + 261316608*a^21*b^12*c^19*d^21

- 19668992*a^22*b^11*c^18*d^22) - (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d

^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c

*d^7))^(1/4)*((-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^

5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(3/4)*((-b^11

/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^

4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(1/4)*(67108864*a^13*b^25*c^46*d^

4 - 1140850688*a^14*b^24*c^45*d^5 + 9126805504*a^15*b^23*c^44*d^6 - 45818576896*a^16*b^22*c^43*d^7 + 162973876

224*a^17*b^21*c^42*d^8 - 442364854272*a^18*b^20*c^41*d^9 + 972004786176*a^19*b^19*c^40*d^10 - 1824220250112*a^

20*b^18*c^39*d^11 + 3052916441088*a^21*b^17*c^38*d^12 - 4642121449472*a^22*b^16*c^37*d^13 + 6347693228032*a^23

*b^15*c^36*d^14 - 7600917708800*a^24*b^14*c^35*d^15 + 7756643827712*a^25*b^13*c^34*d^16 - 6603814207488*a^26*b

^12*c^33*d^17 + 4613600182272*a^27*b^11*c^32*d^18 - 2604562120704*a^28*b^10*c^31*d^19 + 1167090253824*a^29*b^9

*c^30*d^20 - 405069103104*a^30*b^8*c^29*d^21 + 104958263296*a^31*b^7*c^28*d^22 - 19109249024*a^32*b^6*c^27*d^2

3 + 2181038080*a^33*b^5*c^26*d^24 - 117440512*a^34*b^4*c^25*d^25)*1i + x^(1/2)*(33554432*a^11*b^26*c^44*d^4 -

503316480*a^12*b^25*c^43*d^5 + 3523215360*a^13*b^24*c^42*d^6 - 15267266560*a^14*b^23*c^41*d^7 + 45801799680*a^

15*b^22*c^40*d^8 - 100510203904*a^16*b^21*c^39*d^9 + 163810639872*a^17*b^20*c^38*d^10 - 184331272192*a^18*b^19

*c^37*d^11 + 65011712000*a^19*b^18*c^36*d^12 + 336173465600*a^20*b^17*c^35*d^13 - 1148861808640*a^21*b^16*c^34

*d^14 + 2334365057024*a^22*b^15*c^33*d^15 - 3542660153344*a^23*b^14*c^32*d^16 + 4221965434880*a^24*b^13*c^31*d

^17 - 4009062563840*a^25*b^12*c^30*d^18 + 3039679217664*a^26*b^11*c^29*d^19 - 1830545260544*a^27*b^10*c^28*d^2

0 + 864890650624*a^28*b^9*c^27*d^21 - 313859768320*a^29*b^8*c^26*d^22 + 84473282560*a^30*b^7*c^25*d^23 - 15888

023552*a^31*b^6*c^24*d^24 + 1864368128*a^32*b^5*c^23*d^25 - 102760448*a^33*b^4*c^22*d^26))*1i + 11534336*a^9*b

^25*c^35*d^7 - 111149056*a^10*b^24*c^34*d^8 + 481296384*a^11*b^23*c^33*d^9 - 1233125376*a^12*b^22*c^32*d^10 +

1830010880*a^13*b^21*c^31*d^11 + 391331840*a^14*b^20*c^30*d^12 - 12820119552*a^15*b^19*c^29*d^13 + 46592393216

*a^16*b^18*c^28*d^14 - 104394047488*a^17*b^17*c^27*d^15 + 165297111040*a^18*b^16*c^26*d^16 - 192702906368*a^19

*b^15*c^25*d^17 + 167824392192*a^20*b^14*c^24*d^18 - 109211664384*a^21*b^13*c^23*d^19 + 52444708864*a^22*b^12*

c^22*d^20 - 18062213120*a^23*b^11*c^21*d^21 + 4224417792*a^24*b^10*c^20*d^22 - 601309184*a^25*b^9*c^19*d^23 +

39337984*a^26*b^8*c^18*d^24)*1i)*1i - (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c

^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14

*b*c*d^7))^(1/4)*(x^(1/2)*(15859712*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^10 + 600711168*a^11*b^22*c^

29*d^11 - 2168807424*a^12*b^21*c^28*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 14037065728*a^14*b^19*c^26*d^14 +

22648012800*a^15*b^18*c^25*d^15 - 26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^17*b^16*c^23*d^17 - 13398917

120*a^18*b^15*c^22*d^18 + 5629976576*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^20*d^20 + 261316608*a^21*b^1

2*c^19*d^21 - 19668992*a^22*b^11*c^18*d^22) + (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a

^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 -

128*a^14*b*c*d^7))^(1/4)*((-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896

*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(

3/4)*((-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 +

1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(1/4)*(67108864*a^13*

b^25*c^46*d^4 - 1140850688*a^14*b^24*c^45*d^5 + 9126805504*a^15*b^23*c^44*d^6 - 45818576896*a^16*b^22*c^43*d^7

+ 162973876224*a^17*b^21*c^42*d^8 - 442364854272*a^18*b^20*c^41*d^9 + 972004786176*a^19*b^19*c^40*d^10 - 1824

220250112*a^20*b^18*c^39*d^11 + 3052916441088*a^21*b^17*c^38*d^12 - 4642121449472*a^22*b^16*c^37*d^13 + 634769

3228032*a^23*b^15*c^36*d^14 - 7600917708800*a^24*b^14*c^35*d^15 + 7756643827712*a^25*b^13*c^34*d^16 - 66038142

07488*a^26*b^12*c^33*d^17 + 4613600182272*a^27*b^11*c^32*d^18 - 2604562120704*a^28*b^10*c^31*d^19 + 1167090253

824*a^29*b^9*c^30*d^20 - 405069103104*a^30*b^8*c^29*d^21 + 104958263296*a^31*b^7*c^28*d^22 - 19109249024*a^32*

b^6*c^27*d^23 + 2181038080*a^33*b^5*c^26*d^24 - 117440512*a^34*b^4*c^25*d^25)*1i - x^(1/2)*(33554432*a^11*b^26

*c^44*d^4 - 503316480*a^12*b^25*c^43*d^5 + 3523215360*a^13*b^24*c^42*d^6 - 15267266560*a^14*b^23*c^41*d^7 + 45

801799680*a^15*b^22*c^40*d^8 - 100510203904*a^16*b^21*c^39*d^9 + 163810639872*a^17*b^20*c^38*d^10 - 1843312721

92*a^18*b^19*c^37*d^11 + 65011712000*a^19*b^18*c^36*d^12 + 336173465600*a^20*b^17*c^35*d^13 - 1148861808640*a^

21*b^16*c^34*d^14 + 2334365057024*a^22*b^15*c^33*d^15 - 3542660153344*a^23*b^14*c^32*d^16 + 4221965434880*a^24

*b^13*c^31*d^17 - 4009062563840*a^25*b^12*c^30*d^18 + 3039679217664*a^26*b^11*c^29*d^19 - 1830545260544*a^27*b

^10*c^28*d^20 + 864890650624*a^28*b^9*c^27*d^21 - 313859768320*a^29*b^8*c^26*d^22 + 84473282560*a^30*b^7*c^25*

d^23 - 15888023552*a^31*b^6*c^24*d^24 + 1864368128*a^32*b^5*c^23*d^25 - 102760448*a^33*b^4*c^22*d^26))*1i + 11

534336*a^9*b^25*c^35*d^7 - 111149056*a^10*b^24*c^34*d^8 + 481296384*a^11*b^23*c^33*d^9 - 1233125376*a^12*b^22*

c^32*d^10 + 1830010880*a^13*b^21*c^31*d^11 + 391331840*a^14*b^20*c^30*d^12 - 12820119552*a^15*b^19*c^29*d^13 +

46592393216*a^16*b^18*c^28*d^14 - 104394047488*a^17*b^17*c^27*d^15 + 165297111040*a^18*b^16*c^26*d^16 - 19270

2906368*a^19*b^15*c^25*d^17 + 167824392192*a^20*b^14*c^24*d^18 - 109211664384*a^21*b^13*c^23*d^19 + 5244470886

4*a^22*b^12*c^22*d^20 - 18062213120*a^23*b^11*c^21*d^21 + 4224417792*a^24*b^10*c^20*d^22 - 601309184*a^25*b^9*

c^19*d^23 + 39337984*a^26*b^8*c^18*d^24)*1i)*1i))*(-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 4

48*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^

6 - 128*a^14*b*c*d^7))^(1/4) - 2*atan(-((-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2

*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 + 114688*a^2*b^6*

c^17*d^2 - 229376*a^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688*a^6*b^2*c^13*d

^6 - 32768*a*b^7*c^18*d))^(1/4)*(x^(1/2)*(15859712*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^10 + 6007111

68*a^11*b^22*c^29*d^11 - 2168807424*a^12*b^21*c^28*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 14037065728*a^14*b^

19*c^26*d^14 + 22648012800*a^15*b^18*c^25*d^15 - 26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^17*b^16*c^23*

d^17 - 13398917120*a^18*b^15*c^22*d^18 + 5629976576*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^20*d^20 + 261

316608*a^21*b^12*c^19*d^21 - 19668992*a^22*b^11*c^18*d^22) - (-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^

3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*

d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 +

114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(1/4)*(((-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*

d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 +

114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688

*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(1/4)*(67108864*a^13*b^25*c^46*d^4 - 1140850688*a^14*b^24*c^45*d^5 +

9126805504*a^15*b^23*c^44*d^6 - 45818576896*a^16*b^22*c^43*d^7 + 162973876224*a^17*b^21*c^42*d^8 - 44236485427

2*a^18*b^20*c^41*d^9 + 972004786176*a^19*b^19*c^40*d^10 - 1824220250112*a^20*b^18*c^39*d^11 + 3052916441088*a^

21*b^17*c^38*d^12 - 4642121449472*a^22*b^16*c^37*d^13 + 6347693228032*a^23*b^15*c^36*d^14 - 7600917708800*a^24

*b^14*c^35*d^15 + 7756643827712*a^25*b^13*c^34*d^16 - 6603814207488*a^26*b^12*c^33*d^17 + 4613600182272*a^27*b

^11*c^32*d^18 - 2604562120704*a^28*b^10*c^31*d^19 + 1167090253824*a^29*b^9*c^30*d^20 - 405069103104*a^30*b^8*c

^29*d^21 + 104958263296*a^31*b^7*c^28*d^22 - 19109249024*a^32*b^6*c^27*d^23 + 2181038080*a^33*b^5*c^26*d^24 -

117440512*a^34*b^4*c^25*d^25)*1i + x^(1/2)*(33554432*a^11*b^26*c^44*d^4 - 503316480*a^12*b^25*c^43*d^5 + 35232

15360*a^13*b^24*c^42*d^6 - 15267266560*a^14*b^23*c^41*d^7 + 45801799680*a^15*b^22*c^40*d^8 - 100510203904*a^16

*b^21*c^39*d^9 + 163810639872*a^17*b^20*c^38*d^10 - 184331272192*a^18*b^19*c^37*d^11 + 65011712000*a^19*b^18*c

^36*d^12 + 336173465600*a^20*b^17*c^35*d^13 - 1148861808640*a^21*b^16*c^34*d^14 + 2334365057024*a^22*b^15*c^33

*d^15 - 3542660153344*a^23*b^14*c^32*d^16 + 4221965434880*a^24*b^13*c^31*d^17 - 4009062563840*a^25*b^12*c^30*d

^18 + 3039679217664*a^26*b^11*c^29*d^19 - 1830545260544*a^27*b^10*c^28*d^20 + 864890650624*a^28*b^9*c^27*d^21

- 313859768320*a^29*b^8*c^26*d^22 + 84473282560*a^30*b^7*c^25*d^23 - 15888023552*a^31*b^6*c^24*d^24 + 18643681

28*a^32*b^5*c^23*d^25 - 102760448*a^33*b^4*c^22*d^26))*(-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*

d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 +

114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688

*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(3/4)*1i + 11534336*a^9*b^25*c^35*d^7 - 111149056*a^10*b^24*c^34*d^8

+ 481296384*a^11*b^23*c^33*d^9 - 1233125376*a^12*b^22*c^32*d^10 + 1830010880*a^13*b^21*c^31*d^11 + 391331840*a

^14*b^20*c^30*d^12 - 12820119552*a^15*b^19*c^29*d^13 + 46592393216*a^16*b^18*c^28*d^14 - 104394047488*a^17*b^1

7*c^27*d^15 + 165297111040*a^18*b^16*c^26*d^16 - 192702906368*a^19*b^15*c^25*d^17 + 167824392192*a^20*b^14*c^2

4*d^18 - 109211664384*a^21*b^13*c^23*d^19 + 52444708864*a^22*b^12*c^22*d^20 - 18062213120*a^23*b^11*c^21*d^21

+ 4224417792*a^24*b^10*c^20*d^22 - 601309184*a^25*b^9*c^19*d^23 + 39337984*a^26*b^8*c^18*d^24)*1i) + (-(2401*a

^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)/(4096*b^8*c^19

+ 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3 + 286720*a^4*b

^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(1/4)*(x^(1/2)*(1585971

2*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^10 + 600711168*a^11*b^22*c^29*d^11 - 2168807424*a^12*b^21*c^2

8*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 14037065728*a^14*b^19*c^26*d^14 + 22648012800*a^15*b^18*c^25*d^15 -

26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^17*b^16*c^23*d^17 - 13398917120*a^18*b^15*c^22*d^18 + 56299765

76*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^20*d^20 + 261316608*a^21*b^12*c^19*d^21 - 19668992*a^22*b^11*c

^18*d^22) + (-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c

*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^

16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(1

/4)*(((-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)

/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3

+ 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(1/4)*(6

7108864*a^13*b^25*c^46*d^4 - 1140850688*a^14*b^24*c^45*d^5 + 9126805504*a^15*b^23*c^44*d^6 - 45818576896*a^16*

b^22*c^43*d^7 + 162973876224*a^17*b^21*c^42*d^8 - 442364854272*a^18*b^20*c^41*d^9 + 972004786176*a^19*b^19*c^4

0*d^10 - 1824220250112*a^20*b^18*c^39*d^11 + 3052916441088*a^21*b^17*c^38*d^12 - 4642121449472*a^22*b^16*c^37*

d^13 + 6347693228032*a^23*b^15*c^36*d^14 - 7600917708800*a^24*b^14*c^35*d^15 + 7756643827712*a^25*b^13*c^34*d^

16 - 6603814207488*a^26*b^12*c^33*d^17 + 4613600182272*a^27*b^11*c^32*d^18 - 2604562120704*a^28*b^10*c^31*d^19

+ 1167090253824*a^29*b^9*c^30*d^20 - 405069103104*a^30*b^8*c^29*d^21 + 104958263296*a^31*b^7*c^28*d^22 - 1910

9249024*a^32*b^6*c^27*d^23 + 2181038080*a^33*b^5*c^26*d^24 - 117440512*a^34*b^4*c^25*d^25)*1i - x^(1/2)*(33554

432*a^11*b^26*c^44*d^4 - 503316480*a^12*b^25*c^43*d^5 + 3523215360*a^13*b^24*c^42*d^6 - 15267266560*a^14*b^23*

c^41*d^7 + 45801799680*a^15*b^22*c^40*d^8 - 100510203904*a^16*b^21*c^39*d^9 + 163810639872*a^17*b^20*c^38*d^10

- 184331272192*a^18*b^19*c^37*d^11 + 65011712000*a^19*b^18*c^36*d^12 + 336173465600*a^20*b^17*c^35*d^13 - 114

8861808640*a^21*b^16*c^34*d^14 + 2334365057024*a^22*b^15*c^33*d^15 - 3542660153344*a^23*b^14*c^32*d^16 + 42219

65434880*a^24*b^13*c^31*d^17 - 4009062563840*a^25*b^12*c^30*d^18 + 3039679217664*a^26*b^11*c^29*d^19 - 1830545

260544*a^27*b^10*c^28*d^20 + 864890650624*a^28*b^9*c^27*d^21 - 313859768320*a^29*b^8*c^26*d^22 + 84473282560*a

^30*b^7*c^25*d^23 - 15888023552*a^31*b^6*c^24*d^24 + 1864368128*a^32*b^5*c^23*d^25 - 102760448*a^33*b^4*c^22*d

^26))*(-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)

/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3

+ 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(3/4)*1i

+ 11534336*a^9*b^25*c^35*d^7 - 111149056*a^10*b^24*c^34*d^8 + 481296384*a^11*b^23*c^33*d^9 - 1233125376*a^12*

b^22*c^32*d^10 + 1830010880*a^13*b^21*c^31*d^11 + 391331840*a^14*b^20*c^30*d^12 - 12820119552*a^15*b^19*c^29*d

^13 + 46592393216*a^16*b^18*c^28*d^14 - 104394047488*a^17*b^17*c^27*d^15 + 165297111040*a^18*b^16*c^26*d^16 -

192702906368*a^19*b^15*c^25*d^17 + 167824392192*a^20*b^14*c^24*d^18 - 109211664384*a^21*b^13*c^23*d^19 + 52444

708864*a^22*b^12*c^22*d^20 - 18062213120*a^23*b^11*c^21*d^21 + 4224417792*a^24*b^10*c^20*d^22 - 601309184*a^25

*b^9*c^19*d^23 + 39337984*a^26*b^8*c^18*d^24)*1i))/((-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8

+ 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 + 114

688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688*a^

6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(1/4)*(x^(1/2)*(15859712*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^

10 + 600711168*a^11*b^22*c^29*d^11 - 2168807424*a^12*b^21*c^28*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 1403706

5728*a^14*b^19*c^26*d^14 + 22648012800*a^15*b^18*c^25*d^15 - 26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^1

7*b^16*c^23*d^17 - 13398917120*a^18*b^15*c^22*d^18 + 5629976576*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^2

0*d^20 + 261316608*a^21*b^12*c^19*d^21 - 19668992*a^22*b^11*c^18*d^22) - (-(2401*a^4*d^11 + 14641*b^4*c^4*d^7

- 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768

*a^7*b*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3

*c^14*d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(1/4)*(((-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 3726

8*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b

*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*

d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(1/4)*(67108864*a^13*b^25*c^46*d^4 - 1140850688*a^14*b^24

*c^45*d^5 + 9126805504*a^15*b^23*c^44*d^6 - 45818576896*a^16*b^22*c^43*d^7 + 162973876224*a^17*b^21*c^42*d^8 -

442364854272*a^18*b^20*c^41*d^9 + 972004786176*a^19*b^19*c^40*d^10 - 1824220250112*a^20*b^18*c^39*d^11 + 3052

916441088*a^21*b^17*c^38*d^12 - 4642121449472*a^22*b^16*c^37*d^13 + 6347693228032*a^23*b^15*c^36*d^14 - 760091

7708800*a^24*b^14*c^35*d^15 + 7756643827712*a^25*b^13*c^34*d^16 - 6603814207488*a^26*b^12*c^33*d^17 + 46136001

82272*a^27*b^11*c^32*d^18 - 2604562120704*a^28*b^10*c^31*d^19 + 1167090253824*a^29*b^9*c^30*d^20 - 40506910310

4*a^30*b^8*c^29*d^21 + 104958263296*a^31*b^7*c^28*d^22 - 19109249024*a^32*b^6*c^27*d^23 + 2181038080*a^33*b^5*

c^26*d^24 - 117440512*a^34*b^4*c^25*d^25)*1i + x^(1/2)*(33554432*a^11*b^26*c^44*d^4 - 503316480*a^12*b^25*c^43

*d^5 + 3523215360*a^13*b^24*c^42*d^6 - 15267266560*a^14*b^23*c^41*d^7 + 45801799680*a^15*b^22*c^40*d^8 - 10051

0203904*a^16*b^21*c^39*d^9 + 163810639872*a^17*b^20*c^38*d^10 - 184331272192*a^18*b^19*c^37*d^11 + 65011712000

*a^19*b^18*c^36*d^12 + 336173465600*a^20*b^17*c^35*d^13 - 1148861808640*a^21*b^16*c^34*d^14 + 2334365057024*a^

22*b^15*c^33*d^15 - 3542660153344*a^23*b^14*c^32*d^16 + 4221965434880*a^24*b^13*c^31*d^17 - 4009062563840*a^25

*b^12*c^30*d^18 + 3039679217664*a^26*b^11*c^29*d^19 - 1830545260544*a^27*b^10*c^28*d^20 + 864890650624*a^28*b^

9*c^27*d^21 - 313859768320*a^29*b^8*c^26*d^22 + 84473282560*a^30*b^7*c^25*d^23 - 15888023552*a^31*b^6*c^24*d^2

4 + 1864368128*a^32*b^5*c^23*d^25 - 102760448*a^33*b^4*c^22*d^26))*(-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 3726

8*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b

*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*

d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(3/4)*1i + 11534336*a^9*b^25*c^35*d^7 - 111149056*a^10*b^

24*c^34*d^8 + 481296384*a^11*b^23*c^33*d^9 - 1233125376*a^12*b^22*c^32*d^10 + 1830010880*a^13*b^21*c^31*d^11 +

391331840*a^14*b^20*c^30*d^12 - 12820119552*a^15*b^19*c^29*d^13 + 46592393216*a^16*b^18*c^28*d^14 - 104394047

488*a^17*b^17*c^27*d^15 + 165297111040*a^18*b^16*c^26*d^16 - 192702906368*a^19*b^15*c^25*d^17 + 167824392192*a

^20*b^14*c^24*d^18 - 109211664384*a^21*b^13*c^23*d^19 + 52444708864*a^22*b^12*c^22*d^20 - 18062213120*a^23*b^1

1*c^21*d^21 + 4224417792*a^24*b^10*c^20*d^22 - 601309184*a^25*b^9*c^19*d^23 + 39337984*a^26*b^8*c^18*d^24)*1i)

*1i - (-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)

/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3

+ 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(1/4)*(x

^(1/2)*(15859712*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^10 + 600711168*a^11*b^22*c^29*d^11 - 216880742

4*a^12*b^21*c^28*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 14037065728*a^14*b^19*c^26*d^14 + 22648012800*a^15*b^

18*c^25*d^15 - 26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^17*b^16*c^23*d^17 - 13398917120*a^18*b^15*c^22*

d^18 + 5629976576*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^20*d^20 + 261316608*a^21*b^12*c^19*d^21 - 19668

992*a^22*b^11*c^18*d^22) + (-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9

- 15092*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 22

9376*a^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*

b^7*c^18*d))^(1/4)*(((-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 1509

2*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a

^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^

18*d))^(1/4)*(67108864*a^13*b^25*c^46*d^4 - 1140850688*a^14*b^24*c^45*d^5 + 9126805504*a^15*b^23*c^44*d^6 - 45

818576896*a^16*b^22*c^43*d^7 + 162973876224*a^17*b^21*c^42*d^8 - 442364854272*a^18*b^20*c^41*d^9 + 97200478617

6*a^19*b^19*c^40*d^10 - 1824220250112*a^20*b^18*c^39*d^11 + 3052916441088*a^21*b^17*c^38*d^12 - 4642121449472*

a^22*b^16*c^37*d^13 + 6347693228032*a^23*b^15*c^36*d^14 - 7600917708800*a^24*b^14*c^35*d^15 + 7756643827712*a^

25*b^13*c^34*d^16 - 6603814207488*a^26*b^12*c^33*d^17 + 4613600182272*a^27*b^11*c^32*d^18 - 2604562120704*a^28

*b^10*c^31*d^19 + 1167090253824*a^29*b^9*c^30*d^20 - 405069103104*a^30*b^8*c^29*d^21 + 104958263296*a^31*b^7*c

^28*d^22 - 19109249024*a^32*b^6*c^27*d^23 + 2181038080*a^33*b^5*c^26*d^24 - 117440512*a^34*b^4*c^25*d^25)*1i -

x^(1/2)*(33554432*a^11*b^26*c^44*d^4 - 503316480*a^12*b^25*c^43*d^5 + 3523215360*a^13*b^24*c^42*d^6 - 1526726

6560*a^14*b^23*c^41*d^7 + 45801799680*a^15*b^22*c^40*d^8 - 100510203904*a^16*b^21*c^39*d^9 + 163810639872*a^17

*b^20*c^38*d^10 - 184331272192*a^18*b^19*c^37*d^11 + 65011712000*a^19*b^18*c^36*d^12 + 336173465600*a^20*b^17*

c^35*d^13 - 1148861808640*a^21*b^16*c^34*d^14 + 2334365057024*a^22*b^15*c^33*d^15 - 3542660153344*a^23*b^14*c^

32*d^16 + 4221965434880*a^24*b^13*c^31*d^17 - 4009062563840*a^25*b^12*c^30*d^18 + 3039679217664*a^26*b^11*c^29

*d^19 - 1830545260544*a^27*b^10*c^28*d^20 + 864890650624*a^28*b^9*c^27*d^21 - 313859768320*a^29*b^8*c^26*d^22

+ 84473282560*a^30*b^7*c^25*d^23 - 15888023552*a^31*b^6*c^24*d^24 + 1864368128*a^32*b^5*c^23*d^25 - 102760448*

a^33*b^4*c^22*d^26))*(-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 37268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 1509

2*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^7*b*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a

^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^14*d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^

18*d))^(3/4)*1i + 11534336*a^9*b^25*c^35*d^7 - 111149056*a^10*b^24*c^34*d^8 + 481296384*a^11*b^23*c^33*d^9 - 1

233125376*a^12*b^22*c^32*d^10 + 1830010880*a^13*b^21*c^31*d^11 + 391331840*a^14*b^20*c^30*d^12 - 12820119552*a

^15*b^19*c^29*d^13 + 46592393216*a^16*b^18*c^28*d^14 - 104394047488*a^17*b^17*c^27*d^15 + 165297111040*a^18*b^

16*c^26*d^16 - 192702906368*a^19*b^15*c^25*d^17 + 167824392192*a^20*b^14*c^24*d^18 - 109211664384*a^21*b^13*c^

23*d^19 + 52444708864*a^22*b^12*c^22*d^20 - 18062213120*a^23*b^11*c^21*d^21 + 4224417792*a^24*b^10*c^20*d^22 -

601309184*a^25*b^9*c^19*d^23 + 39337984*a^26*b^8*c^18*d^24)*1i)*1i))*(-(2401*a^4*d^11 + 14641*b^4*c^4*d^7 - 3

7268*a*b^3*c^3*d^8 + 35574*a^2*b^2*c^2*d^9 - 15092*a^3*b*c*d^10)/(4096*b^8*c^19 + 4096*a^8*c^11*d^8 - 32768*a^

7*b*c^12*d^7 + 114688*a^2*b^6*c^17*d^2 - 229376*a^3*b^5*c^16*d^3 + 286720*a^4*b^4*c^15*d^4 - 229376*a^5*b^3*c^

14*d^5 + 114688*a^6*b^2*c^13*d^6 - 32768*a*b^7*c^18*d))^(1/4) - (2/(3*a*c) + (d*x^2*(7*a*d - 4*b*c))/(6*a*c^2*

(a*d - b*c)))/(c*x^(3/2) + d*x^(7/2)) - atan(((-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a

^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 -

128*a^14*b*c*d^7))^(1/4)*(x^(1/2)*(15859712*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^10 + 600711168*a^11

*b^22*c^29*d^11 - 2168807424*a^12*b^21*c^28*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 14037065728*a^14*b^19*c^26

*d^14 + 22648012800*a^15*b^18*c^25*d^15 - 26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^17*b^16*c^23*d^17 -

13398917120*a^18*b^15*c^22*d^18 + 5629976576*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^20*d^20 + 261316608*

a^21*b^12*c^19*d^21 - 19668992*a^22*b^11*c^18*d^22) + (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d

+ 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^

2*d^6 - 128*a^14*b*c*d^7))^(1/4)*(11534336*a^9*b^25*c^35*d^7 - (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*

b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^

13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(3/4)*((-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a^9*

b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 - 128

*a^14*b*c*d^7))^(1/4)*(67108864*a^13*b^25*c^46*d^4 - 1140850688*a^14*b^24*c^45*d^5 + 9126805504*a^15*b^23*c^44

*d^6 - 45818576896*a^16*b^22*c^43*d^7 + 162973876224*a^17*b^21*c^42*d^8 - 442364854272*a^18*b^20*c^41*d^9 + 97

2004786176*a^19*b^19*c^40*d^10 - 1824220250112*a^20*b^18*c^39*d^11 + 3052916441088*a^21*b^17*c^38*d^12 - 46421

21449472*a^22*b^16*c^37*d^13 + 6347693228032*a^23*b^15*c^36*d^14 - 7600917708800*a^24*b^14*c^35*d^15 + 7756643

827712*a^25*b^13*c^34*d^16 - 6603814207488*a^26*b^12*c^33*d^17 + 4613600182272*a^27*b^11*c^32*d^18 - 260456212

0704*a^28*b^10*c^31*d^19 + 1167090253824*a^29*b^9*c^30*d^20 - 405069103104*a^30*b^8*c^29*d^21 + 104958263296*a

^31*b^7*c^28*d^22 - 19109249024*a^32*b^6*c^27*d^23 + 2181038080*a^33*b^5*c^26*d^24 - 117440512*a^34*b^4*c^25*d

^25) - x^(1/2)*(33554432*a^11*b^26*c^44*d^4 - 503316480*a^12*b^25*c^43*d^5 + 3523215360*a^13*b^24*c^42*d^6 - 1

5267266560*a^14*b^23*c^41*d^7 + 45801799680*a^15*b^22*c^40*d^8 - 100510203904*a^16*b^21*c^39*d^9 + 16381063987

2*a^17*b^20*c^38*d^10 - 184331272192*a^18*b^19*c^37*d^11 + 65011712000*a^19*b^18*c^36*d^12 + 336173465600*a^20

*b^17*c^35*d^13 - 1148861808640*a^21*b^16*c^34*d^14 + 2334365057024*a^22*b^15*c^33*d^15 - 3542660153344*a^23*b

^14*c^32*d^16 + 4221965434880*a^24*b^13*c^31*d^17 - 4009062563840*a^25*b^12*c^30*d^18 + 3039679217664*a^26*b^1

1*c^29*d^19 - 1830545260544*a^27*b^10*c^28*d^20 + 864890650624*a^28*b^9*c^27*d^21 - 313859768320*a^29*b^8*c^26

*d^22 + 84473282560*a^30*b^7*c^25*d^23 - 15888023552*a^31*b^6*c^24*d^24 + 1864368128*a^32*b^5*c^23*d^25 - 1027

60448*a^33*b^4*c^22*d^26)) - 111149056*a^10*b^24*c^34*d^8 + 481296384*a^11*b^23*c^33*d^9 - 1233125376*a^12*b^2

2*c^32*d^10 + 1830010880*a^13*b^21*c^31*d^11 + 391331840*a^14*b^20*c^30*d^12 - 12820119552*a^15*b^19*c^29*d^13

+ 46592393216*a^16*b^18*c^28*d^14 - 104394047488*a^17*b^17*c^27*d^15 + 165297111040*a^18*b^16*c^26*d^16 - 192

702906368*a^19*b^15*c^25*d^17 + 167824392192*a^20*b^14*c^24*d^18 - 109211664384*a^21*b^13*c^23*d^19 + 52444708

864*a^22*b^12*c^22*d^20 - 18062213120*a^23*b^11*c^21*d^21 + 4224417792*a^24*b^10*c^20*d^22 - 601309184*a^25*b^

9*c^19*d^23 + 39337984*a^26*b^8*c^18*d^24))*1i + (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 44

8*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6

- 128*a^14*b*c*d^7))^(1/4)*(x^(1/2)*(15859712*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^10 + 600711168*a

^11*b^22*c^29*d^11 - 2168807424*a^12*b^21*c^28*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 14037065728*a^14*b^19*c

^26*d^14 + 22648012800*a^15*b^18*c^25*d^15 - 26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^17*b^16*c^23*d^17

- 13398917120*a^18*b^15*c^22*d^18 + 5629976576*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^20*d^20 + 2613166

08*a^21*b^12*c^19*d^21 - 19668992*a^22*b^11*c^18*d^22) - (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^

7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2

*c^2*d^6 - 128*a^14*b*c*d^7))^(1/4)*(11534336*a^9*b^25*c^35*d^7 - (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a

^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448

*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(3/4)*((-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 448*a

^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6 -

128*a^14*b*c*d^7))^(1/4)*(67108864*a^13*b^25*c^46*d^4 - 1140850688*a^14*b^24*c^45*d^5 + 9126805504*a^15*b^23*c

^44*d^6 - 45818576896*a^16*b^22*c^43*d^7 + 162973876224*a^17*b^21*c^42*d^8 - 442364854272*a^18*b^20*c^41*d^9 +

972004786176*a^19*b^19*c^40*d^10 - 1824220250112*a^20*b^18*c^39*d^11 + 3052916441088*a^21*b^17*c^38*d^12 - 46

42121449472*a^22*b^16*c^37*d^13 + 6347693228032*a^23*b^15*c^36*d^14 - 7600917708800*a^24*b^14*c^35*d^15 + 7756

643827712*a^25*b^13*c^34*d^16 - 6603814207488*a^26*b^12*c^33*d^17 + 4613600182272*a^27*b^11*c^32*d^18 - 260456

2120704*a^28*b^10*c^31*d^19 + 1167090253824*a^29*b^9*c^30*d^20 - 405069103104*a^30*b^8*c^29*d^21 + 10495826329

6*a^31*b^7*c^28*d^22 - 19109249024*a^32*b^6*c^27*d^23 + 2181038080*a^33*b^5*c^26*d^24 - 117440512*a^34*b^4*c^2

5*d^25) + x^(1/2)*(33554432*a^11*b^26*c^44*d^4 - 503316480*a^12*b^25*c^43*d^5 + 3523215360*a^13*b^24*c^42*d^6

- 15267266560*a^14*b^23*c^41*d^7 + 45801799680*a^15*b^22*c^40*d^8 - 100510203904*a^16*b^21*c^39*d^9 + 16381063

9872*a^17*b^20*c^38*d^10 - 184331272192*a^18*b^19*c^37*d^11 + 65011712000*a^19*b^18*c^36*d^12 + 336173465600*a

^20*b^17*c^35*d^13 - 1148861808640*a^21*b^16*c^34*d^14 + 2334365057024*a^22*b^15*c^33*d^15 - 3542660153344*a^2

3*b^14*c^32*d^16 + 4221965434880*a^24*b^13*c^31*d^17 - 4009062563840*a^25*b^12*c^30*d^18 + 3039679217664*a^26*

b^11*c^29*d^19 - 1830545260544*a^27*b^10*c^28*d^20 + 864890650624*a^28*b^9*c^27*d^21 - 313859768320*a^29*b^8*c

^26*d^22 + 84473282560*a^30*b^7*c^25*d^23 - 15888023552*a^31*b^6*c^24*d^24 + 1864368128*a^32*b^5*c^23*d^25 - 1

02760448*a^33*b^4*c^22*d^26)) - 111149056*a^10*b^24*c^34*d^8 + 481296384*a^11*b^23*c^33*d^9 - 1233125376*a^12*

b^22*c^32*d^10 + 1830010880*a^13*b^21*c^31*d^11 + 391331840*a^14*b^20*c^30*d^12 - 12820119552*a^15*b^19*c^29*d

^13 + 46592393216*a^16*b^18*c^28*d^14 - 104394047488*a^17*b^17*c^27*d^15 + 165297111040*a^18*b^16*c^26*d^16 -

192702906368*a^19*b^15*c^25*d^17 + 167824392192*a^20*b^14*c^24*d^18 - 109211664384*a^21*b^13*c^23*d^19 + 52444

708864*a^22*b^12*c^22*d^20 - 18062213120*a^23*b^11*c^21*d^21 + 4224417792*a^24*b^10*c^20*d^22 - 601309184*a^25

*b^9*c^19*d^23 + 39337984*a^26*b^8*c^18*d^24))*1i)/((-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d +

448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*

d^6 - 128*a^14*b*c*d^7))^(1/4)*(x^(1/2)*(15859712*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^10 + 60071116

8*a^11*b^22*c^29*d^11 - 2168807424*a^12*b^21*c^28*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 14037065728*a^14*b^1

9*c^26*d^14 + 22648012800*a^15*b^18*c^25*d^15 - 26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^17*b^16*c^23*d

^17 - 13398917120*a^18*b^15*c^22*d^18 + 5629976576*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^20*d^20 + 2613

16608*a^21*b^12*c^19*d^21 - 19668992*a^22*b^11*c^18*d^22) + (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7

*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*

b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(1/4)*(11534336*a^9*b^25*c^35*d^7 - (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 12

8*a^8*b^7*c^7*d + 448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 +

448*a^13*b^2*c^2*d^6 - 128*a^14*b*c*d^7))^(3/4)*((-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d + 44

8*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*d^6

- 128*a^14*b*c*d^7))^(1/4)*(67108864*a^13*b^25*c^46*d^4 - 1140850688*a^14*b^24*c^45*d^5 + 9126805504*a^15*b^2

3*c^44*d^6 - 45818576896*a^16*b^22*c^43*d^7 + 162973876224*a^17*b^21*c^42*d^8 - 442364854272*a^18*b^20*c^41*d^

9 + 972004786176*a^19*b^19*c^40*d^10 - 1824220250112*a^20*b^18*c^39*d^11 + 3052916441088*a^21*b^17*c^38*d^12 -

4642121449472*a^22*b^16*c^37*d^13 + 6347693228032*a^23*b^15*c^36*d^14 - 7600917708800*a^24*b^14*c^35*d^15 + 7

756643827712*a^25*b^13*c^34*d^16 - 6603814207488*a^26*b^12*c^33*d^17 + 4613600182272*a^27*b^11*c^32*d^18 - 260

4562120704*a^28*b^10*c^31*d^19 + 1167090253824*a^29*b^9*c^30*d^20 - 405069103104*a^30*b^8*c^29*d^21 + 10495826

3296*a^31*b^7*c^28*d^22 - 19109249024*a^32*b^6*c^27*d^23 + 2181038080*a^33*b^5*c^26*d^24 - 117440512*a^34*b^4*

c^25*d^25) - x^(1/2)*(33554432*a^11*b^26*c^44*d^4 - 503316480*a^12*b^25*c^43*d^5 + 3523215360*a^13*b^24*c^42*d

^6 - 15267266560*a^14*b^23*c^41*d^7 + 45801799680*a^15*b^22*c^40*d^8 - 100510203904*a^16*b^21*c^39*d^9 + 16381

0639872*a^17*b^20*c^38*d^10 - 184331272192*a^18*b^19*c^37*d^11 + 65011712000*a^19*b^18*c^36*d^12 + 33617346560

0*a^20*b^17*c^35*d^13 - 1148861808640*a^21*b^16*c^34*d^14 + 2334365057024*a^22*b^15*c^33*d^15 - 3542660153344*

a^23*b^14*c^32*d^16 + 4221965434880*a^24*b^13*c^31*d^17 - 4009062563840*a^25*b^12*c^30*d^18 + 3039679217664*a^

26*b^11*c^29*d^19 - 1830545260544*a^27*b^10*c^28*d^20 + 864890650624*a^28*b^9*c^27*d^21 - 313859768320*a^29*b^

8*c^26*d^22 + 84473282560*a^30*b^7*c^25*d^23 - 15888023552*a^31*b^6*c^24*d^24 + 1864368128*a^32*b^5*c^23*d^25

- 102760448*a^33*b^4*c^22*d^26)) - 111149056*a^10*b^24*c^34*d^8 + 481296384*a^11*b^23*c^33*d^9 - 1233125376*a^

12*b^22*c^32*d^10 + 1830010880*a^13*b^21*c^31*d^11 + 391331840*a^14*b^20*c^30*d^12 - 12820119552*a^15*b^19*c^2

9*d^13 + 46592393216*a^16*b^18*c^28*d^14 - 104394047488*a^17*b^17*c^27*d^15 + 165297111040*a^18*b^16*c^26*d^16

- 192702906368*a^19*b^15*c^25*d^17 + 167824392192*a^20*b^14*c^24*d^18 - 109211664384*a^21*b^13*c^23*d^19 + 52

444708864*a^22*b^12*c^22*d^20 - 18062213120*a^23*b^11*c^21*d^21 + 4224417792*a^24*b^10*c^20*d^22 - 601309184*a

^25*b^9*c^19*d^23 + 39337984*a^26*b^8*c^18*d^24)) - (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7*c^7*d +

448*a^9*b^6*c^6*d^2 - 896*a^10*b^5*c^5*d^3 + 1120*a^11*b^4*c^4*d^4 - 896*a^12*b^3*c^3*d^5 + 448*a^13*b^2*c^2*

d^6 - 128*a^14*b*c*d^7))^(1/4)*(x^(1/2)*(15859712*a^9*b^24*c^31*d^9 - 131203072*a^10*b^23*c^30*d^10 + 60071116

8*a^11*b^22*c^29*d^11 - 2168807424*a^12*b^21*c^28*d^12 + 6343680000*a^13*b^20*c^27*d^13 - 14037065728*a^14*b^1

9*c^26*d^14 + 22648012800*a^15*b^18*c^25*d^15 - 26429997056*a^16*b^17*c^24*d^16 + 22256009216*a^17*b^16*c^23*d

^17 - 13398917120*a^18*b^15*c^22*d^18 + 5629976576*a^19*b^14*c^21*d^19 - 1569906688*a^20*b^13*c^20*d^20 + 2613

16608*a^21*b^12*c^19*d^21 - 19668992*a^22*b^11*c^18*d^22) - (-b^11/(16*a^15*d^8 + 16*a^7*b^8*c^8 - 128*a^8*b^7