\(\int \frac {\sqrt {x}}{(a+b x^2) (c+d x^2)^2} \, dx\) [475]

Optimal result

Integrand size = 24, antiderivative size = 536 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {b^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}+\frac {b^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}+\frac {\sqrt [4]{d} (5 b c-a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}-\frac {\sqrt [4]{d} (5 b c-a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}+\frac {b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {\sqrt [4]{d} (5 b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^2}+\frac {\sqrt [4]{d} (5 b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 483, 598, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {b^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}+\frac {b^{5/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}+\frac {\sqrt [4]{d} (5 b c-a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}-\frac {\sqrt [4]{d} (5 b c-a d) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}+\frac {b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {\sqrt [4]{d} (5 b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^2}+\frac {\sqrt [4]{d} (5 b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^2}-\frac {d x^{3/2}}{2 c \left (c+d x^2\right ) (b c-a d)} \]

[In]

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

[Out]

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

Rule 210

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

Rule 303

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 477

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 483

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 598

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 631

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 642

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 1176

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 1179

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

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {\frac {4 d (-b c+a d) x^{3/2}}{c \left (c+d x^2\right )}-\frac {4 \sqrt {2} b^{5/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a}}+\frac {\sqrt {2} \sqrt [4]{d} (5 b c-a d) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{5/4}}-\frac {4 \sqrt {2} b^{5/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a}}+\frac {\sqrt {2} \sqrt [4]{d} (5 b c-a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{5/4}}}{8 (b c-a d)^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.84 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.50

method	result	size
derivativedivides	\(\frac {2 d \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 c \left (d \,x^{2}+c \right )}+\frac {\left (a d -5 b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{2}}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(270\)
default	\(\frac {2 d \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 c \left (d \,x^{2}+c \right )}+\frac {\left (a d -5 b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{2}}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(270\)

[In]

int(x^(1/2)/(b*x^2+a)/(d*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

2*d/(a*d-b*c)^2*(1/4*(a*d-b*c)/c*x^(3/2)/(d*x^2+c)+1/32*(a*d-5*b*c)/c/d/(c/d)^(1/4)*2^(1/2)*(ln((x-(c/d)^(1/4)
*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2
)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))+1/4*b/(a*d-b*c)^2/(a/b)^(1/4)*2^(1/2)*(ln((x-(a/b)^(1/4)*x^(1/2
)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*
arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.04 (sec) , antiderivative size = 3495, normalized size of antiderivative = 6.52 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/8*(4*d*x^(3/2) - 4*(-b^5/(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^3 + 70*a^5*b^
4*c^4*d^4 - 56*a^6*b^3*c^3*d^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(1/4)*(b*c^3 - a*c^2*d + (b*c^
2*d - a*c*d^2)*x^2)*log(b^4*sqrt(x) + (a*b^6*c^6 - 6*a^2*b^5*c^5*d + 15*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 +
15*a^5*b^2*c^2*d^4 - 6*a^6*b*c*d^5 + a^7*d^6)*(-b^5/(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^
4*b^5*c^5*d^3 + 70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*c^3*d^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(3/4)
) + 4*(-b^5/(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^3 + 70*a^5*b^4*c^4*d^4 - 56*a
^6*b^3*c^3*d^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(1/4)*(b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x
^2)*log(b^4*sqrt(x) - (a*b^6*c^6 - 6*a^2*b^5*c^5*d + 15*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^5*b^2*c^2*
d^4 - 6*a^6*b*c*d^5 + a^7*d^6)*(-b^5/(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^3 +
70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*c^3*d^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(3/4)) - 4*(-b^5/(a*b
^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^3 + 70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*c^3*d^5 +
28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(1/4)*(I*b*c^3 - I*a*c^2*d + I*(b*c^2*d - a*c*d^2)*x^2)*log(b^
4*sqrt(x) - (I*a*b^6*c^6 - 6*I*a^2*b^5*c^5*d + 15*I*a^3*b^4*c^4*d^2 - 20*I*a^4*b^3*c^3*d^3 + 15*I*a^5*b^2*c^2*
d^4 - 6*I*a^6*b*c*d^5 + I*a^7*d^6)*(-b^5/(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^
3 + 70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*c^3*d^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(3/4)) - 4*(-b^5/
(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^3 + 70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*c^3*d
^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(1/4)*(-I*b*c^3 + I*a*c^2*d - I*(b*c^2*d - a*c*d^2)*x^2)*l
og(b^4*sqrt(x) - (-I*a*b^6*c^6 + 6*I*a^2*b^5*c^5*d - 15*I*a^3*b^4*c^4*d^2 + 20*I*a^4*b^3*c^3*d^3 - 15*I*a^5*b^
2*c^2*d^4 + 6*I*a^6*b*c*d^5 - I*a^7*d^6)*(-b^5/(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*
c^5*d^3 + 70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*c^3*d^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(3/4)) - (b
*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)*(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*
b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4
- 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(1/4)*log((b^6*c^10 - 6*a*b^5*c^9
*d + 15*a^2*b^4*c^8*d^2 - 20*a^3*b^3*c^7*d^3 + 15*a^4*b^2*c^6*d^4 - 6*a^5*b*c^5*d^5 + a^6*c^4*d^6)*(-(625*b^4*
c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^
2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*
c^6*d^7 + a^8*c^5*d^8))^(3/4) - (125*b^3*c^3*d - 75*a*b^2*c^2*d^2 + 15*a^2*b*c*d^3 - a^3*d^4)*sqrt(x)) + (b*c^
3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)*(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c
*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 -
56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(1/4)*log(-(b^6*c^10 - 6*a*b^5*c^9*d
+ 15*a^2*b^4*c^8*d^2 - 20*a^3*b^3*c^7*d^3 + 15*a^4*b^2*c^6*d^4 - 6*a^5*b*c^5*d^5 + a^6*c^4*d^6)*(-(625*b^4*c^
4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*
b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^
6*d^7 + a^8*c^5*d^8))^(3/4) - (125*b^3*c^3*d - 75*a*b^2*c^2*d^2 + 15*a^2*b*c*d^3 - a^3*d^4)*sqrt(x)) - (I*b*c^
3 - I*a*c^2*d + I*(b*c^2*d - a*c*d^2)*x^2)*(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3
*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^
4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(1/4)*log(-(I*b^6*c^10 - 6*I*a*b
^5*c^9*d + 15*I*a^2*b^4*c^8*d^2 - 20*I*a^3*b^3*c^7*d^3 + 15*I*a^4*b^2*c^6*d^4 - 6*I*a^5*b*c^5*d^5 + I*a^6*c^4*
d^6)*(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^
7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^
7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(3/4) - (125*b^3*c^3*d - 75*a*b^2*c^2*d^2 + 15*a^2*b*c*d^3 - a^3*d^4)*
sqrt(x)) - (-I*b*c^3 + I*a*c^2*d - I*(b*c^2*d - a*c*d^2)*x^2)*(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b
^2*c^2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3
+ 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(1/4)*log(-(-
I*b^6*c^10 + 6*I*a*b^5*c^9*d - 15*I*a^2*b^4*c^8*d^2 + 20*I*a^3*b^3*c^7*d^3 - 15*I*a^4*b^2*c^6*d^4 + 6*I*a^5*b*
c^5*d^5 - I*a^6*c^4*d^6)*(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5
)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8
*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(3/4) - (125*b^3*c^3*d - 75*a*b^2*c^2*d^2 + 15*a^2
*b*c*d^3 - a^3*d^4)*sqrt(x)))/(b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 450, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {d x^{\frac {3}{2}}}{2 \, {\left (b c^{3} - a c^{2} d + {\left (b c^{2} d - a c d^{2}\right )} x^{2}\right )}} - \frac {{\left (5 \, b c d - a d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{16 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \left (c d^{3}\right )^{\frac {3}{4}} a d\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^{4} d^{2} - 2 \, \sqrt {2} a b c^{3} d^{3} + \sqrt {2} a^{2} c^{2} d^{4}\right )}} - \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \left (c d^{3}\right )^{\frac {3}{4}} a d\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^{4} d^{2} - 2 \, \sqrt {2} a b c^{3} d^{3} + \sqrt {2} a^{2} c^{2} d^{4}\right )}} + \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \left (c d^{3}\right )^{\frac {3}{4}} a d\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^{4} d^{2} - 2 \, \sqrt {2} a b c^{3} d^{3} + \sqrt {2} a^{2} c^{2} d^{4}\right )}} - \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \left (c d^{3}\right )^{\frac {3}{4}} a d\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^{4} d^{2} - 2 \, \sqrt {2} a b c^{3} d^{3} + \sqrt {2} a^{2} c^{2} d^{4}\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \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 b^{3} c^{2} - 2 \, \sqrt {2} a^{2} b^{2} c d + \sqrt {2} a^{3} b d^{2}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \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 b^{3} c^{2} - 2 \, \sqrt {2} a^{2} b^{2} c d + \sqrt {2} a^{3} b d^{2}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \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 b^{3} c^{2} - 2 \, \sqrt {2} a^{2} b^{2} c d + \sqrt {2} a^{3} b d^{2}\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \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 b^{3} c^{2} - 2 \, \sqrt {2} a^{2} b^{2} c d + \sqrt {2} a^{3} b d^{2}\right )}} - \frac {d x^{\frac {3}{2}}}{2 \, {\left (b c^{2} - a c d\right )} {\left (d x^{2} + c\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 7.47 (sec) , antiderivative size = 19453, normalized size of antiderivative = 36.29 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2*atan(((-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 112
0*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*((-b^5/(16*a^9*d^8 + 1
6*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b
^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(3/4)*(((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^
12*b^5*c*d^15 + 14336*a^2*b^15*c^11*d^5 - 39008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^
8*d^8 - 150784*a^6*b^11*c^7*d^9 + 219968*a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12
- 32000*a^10*b^7*c^3*d^13 + 6432*a^11*b^6*c^2*d^14)*1i)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*
c^7*d^2 - 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) + (x^(1/2)*(-b^5/(16*a
^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 -
896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^
16 - 32768*a^2*b^15*c^12*d^5 + 121088*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 -
661504*a^6*b^11*c^8*d^9 + 713216*a^7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 1372
16*a^10*b^7*c^4*d^13 + 34048*a^11*b^6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3
*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) - (x^(1/2)*(4*a^4*b^9*c*
d^8 - 625*a*b^12*c^4*d^5 - a^5*b^8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*d^7))/(b^6*c^8 + a^6*c^2*d^6 -
6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) - (-b^5/(16*
a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4
- 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*((-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*
a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a
^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(3/4)*(((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*b^5*c*d^15 + 14
336*a^2*b^15*c^11*d^5 - 39008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^8 - 150784*a^6
*b^11*c^7*d^9 + 219968*a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 32000*a^10*b^7*c
^3*d^13 + 6432*a^11*b^6*c^2*d^14)*1i)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b
^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) - (x^(1/2)*(-b^5/(16*a^9*d^8 + 16*a*b^8*
c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d
^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 32768*a^2*b^1
5*c^12*d^5 + 121088*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504*a^6*b^11*c
^8*d^9 + 713216*a^7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10*b^7*c^4*d^
13 + 34048*a^11*b^6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*
c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) + (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c
^4*d^5 - a^5*b^8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*d^7))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 +
15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)))/((-b^5/(16*a^9*d^8 + 16*a*b^8
*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*
d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*((-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 44
8*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 1
28*a^8*b*c*d^7))^(3/4)*(((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*b^5*c*d^15 + 14336*a^2*b^15*c^11*
d^5 - 39008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^8 - 150784*a^6*b^11*c^7*d^9 + 21
9968*a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 32000*a^10*b^7*c^3*d^13 + 6432*a^1
1*b^6*c^2*d^14)*1i)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3 + 35*a^
4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) + (x^(1/2)*(-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*
c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c
^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 32768*a^2*b^15*c^12*d^5 + 12108
8*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504*a^6*b^11*c^8*d^9 + 713216*a^
7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10*b^7*c^4*d^13 + 34048*a^11*b^
6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b
^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*1i - (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*b^
8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*d^7)*1i)/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*
c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) + (-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a
^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^
7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*((-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^
6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d
^7))^(3/4)*(((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*b^5*c*d^15 + 14336*a^2*b^15*c^11*d^5 - 39008*
a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^8 - 150784*a^6*b^11*c^7*d^9 + 219968*a^7*b^1
0*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 32000*a^10*b^7*c^3*d^13 + 6432*a^11*b^6*c^2*d^
14)*1i)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^
4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) - (x^(1/2)*(-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*
a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128
*a^8*b*c*d^7))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 32768*a^2*b^15*c^12*d^5 + 121088*a^3*b^14*c
^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504*a^6*b^11*c^8*d^9 + 713216*a^7*b^10*c^7*d
^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10*b^7*c^4*d^13 + 34048*a^11*b^6*c^3*d^14 -
4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 +
15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*1i + (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*b^8*d^9 + 100*
a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*d^7)*1i)/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20
*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) - (5*a^4*b^9*d^8 - 625*a*b^12*c^3*d^5 - 75*a^3*b^10*c*
d^7 + 375*a^2*b^11*c^2*d^6)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3
+ 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d)))*(-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c
^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^
2*d^6 - 128*a^8*b*c*d^7))^(1/4) - atan(((-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6
*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^
7))^(1/4)*((-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 +
1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(3/4)*((32*a^13*b^4*d^16
- 2048*a*b^16*c^12*d^4 - 704*a^12*b^5*c*d^15 + 14336*a^2*b^15*c^11*d^5 - 39008*a^3*b^14*c^10*d^6 + 41280*a^4*b
^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^8 - 150784*a^6*b^11*c^7*d^9 + 219968*a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5
*d^11 + 96320*a^9*b^8*c^4*d^12 - 32000*a^10*b^7*c^3*d^13 + 6432*a^11*b^6*c^2*d^14)/(b^7*c^9 - a^7*c^2*d^7 + 7*
a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^
8*d) + (x^(1/2)*(-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d
^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*(4096*a*b^16*c
^13*d^4 + 256*a^13*b^4*c*d^16 - 32768*a^2*b^15*c^12*d^5 + 121088*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7
+ 486656*a^5*b^12*c^9*d^8 - 661504*a^6*b^11*c^8*d^9 + 713216*a^7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344
576*a^9*b^8*c^5*d^12 - 137216*a^10*b^7*c^4*d^13 + 34048*a^11*b^6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8
+ a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d
))*1i - (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*b^8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*
d^7)*1i)/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d
^4 - 6*a*b^5*c^7*d)) - (-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^
5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*((-b^5/
(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*
d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(3/4)*((32*a^13*b^4*d^16 - 2048*a*b^16*c^1
2*d^4 - 704*a^12*b^5*c*d^15 + 14336*a^2*b^15*c^11*d^5 - 39008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 296
00*a^5*b^12*c^8*d^8 - 150784*a^6*b^11*c^7*d^9 + 219968*a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9
*b^8*c^4*d^12 - 32000*a^10*b^7*c^3*d^13 + 6432*a^11*b^6*c^2*d^14)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 2
1*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) - (x^(1/2)*(
-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4
*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^1
3*b^4*c*d^16 - 32768*a^2*b^15*c^12*d^5 + 121088*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12
*c^9*d^8 - 661504*a^6*b^11*c^8*d^9 + 713216*a^7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d
^12 - 137216*a^10*b^7*c^4*d^13 + 34048*a^11*b^6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6
*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*1i + (x^(1/2)*
(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*b^8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*d^7)*1i)/(b^6*c^8
+ a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*
d)))/((-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*
a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*((-b^5/(16*a^9*d^8 + 16*
a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3
*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(3/4)*((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*
b^5*c*d^15 + 14336*a^2*b^15*c^11*d^5 - 39008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d
^8 - 150784*a^6*b^11*c^7*d^9 + 219968*a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 3
2000*a^10*b^7*c^3*d^13 + 6432*a^11*b^6*c^2*d^14)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2
- 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) + (x^(1/2)*(-b^5/(16*a^9*d^8
+ 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^
6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 32
768*a^2*b^15*c^12*d^5 + 121088*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504
*a^6*b^11*c^8*d^9 + 713216*a^7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10
*b^7*c^4*d^13 + 34048*a^11*b^6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 +
15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) - (x^(1/2)*(4*a^4*b^9*c*d^8 - 6
25*a*b^12*c^4*d^5 - a^5*b^8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*d^7))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*
b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) + (-b^5/(16*a^9*d^8
+ 16*a*b^8*c^8 - 128*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a
^6*b^3*c^3*d^5 + 448*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*((-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^2*b^7
*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7*b^2*
c^2*d^6 - 128*a^8*b*c*d^7))^(3/4)*((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*b^5*c*d^15 + 14336*a^2*
b^15*c^11*d^5 - 39008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^8 - 150784*a^6*b^11*c^
7*d^9 + 219968*a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 32000*a^10*b^7*c^3*d^13
+ 6432*a^11*b^6*c^2*d^14)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3 +
35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) - (x^(1/2)*(-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 128*a^
2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 448*a^7
*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 32768*a^2*b^15*c^12*d^5 +
121088*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504*a^6*b^11*c^8*d^9 + 713
216*a^7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10*b^7*c^4*d^13 + 34048*a
^11*b^6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20
*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) + (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5
*b^8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*d^7))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*
c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) + (5*a^4*b^9*d^8 - 625*a*b^12*c^3*d^5 - 75
*a^3*b^10*c*d^7 + 375*a^2*b^11*c^2*d^6)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3
*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d)))*(-b^5/(16*a^9*d^8 + 16*a*b^8*c^8 - 1
28*a^2*b^7*c^7*d + 448*a^3*b^6*c^6*d^2 - 896*a^4*b^5*c^5*d^3 + 1120*a^5*b^4*c^4*d^4 - 896*a^6*b^3*c^3*d^5 + 44
8*a^7*b^2*c^2*d^6 - 128*a^8*b*c*d^7))^(1/4)*2i - atan(((((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*b
^5*c*d^15 + 14336*a^2*b^15*c^11*d^5 - 39008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^
8 - 150784*a^6*b^11*c^7*d^9 + 219968*a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 32
000*a^10*b^7*c^3*d^13 + 6432*a^11*b^6*c^2*d^14)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2
- 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) + (x^(1/2)*(-(a^4*d^5 + 625*b^
4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*
a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^
8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 3276
8*a^2*b^15*c^12*d^5 + 121088*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504*a
^6*b^11*c^8*d^9 + 713216*a^7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10*b
^7*c^4*d^13 + 34048*a^11*b^6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15
*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*
a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7
+ 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688
*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(3/4)*1i - (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*b^8*d^
9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*d^7)*1i)/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*
d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2
+ 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*
b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*
d^6 - 32768*a*b^7*c^12*d))^(1/4) - (((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*b^5*c*d^15 + 14336*a^
2*b^15*c^11*d^5 - 39008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^8 - 150784*a^6*b^11*
c^7*d^9 + 219968*a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 32000*a^10*b^7*c^3*d^1
3 + 6432*a^11*b^6*c^2*d^14)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3
+ 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) - (x^(1/2)*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*
c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 1146
88*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b
^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 32768*a^2*b^15*c^12*d^5
+ 121088*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504*a^6*b^11*c^8*d^9 + 71
3216*a^7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10*b^7*c^4*d^13 + 34048*
a^11*b^6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 2
0*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*
a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^1
1*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 3
2768*a*b^7*c^12*d))^(3/4)*1i + (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*b^8*d^9 + 100*a^2*b^11*c^3
*d^6 + 10*a^3*b^10*c^2*d^7)*1i)/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5
*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d
^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 22937
6*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^
12*d))^(1/4))/((((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*b^5*c*d^15 + 14336*a^2*b^15*c^11*d^5 - 39
008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^8 - 150784*a^6*b^11*c^7*d^9 + 219968*a^7
*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 32000*a^10*b^7*c^3*d^13 + 6432*a^11*b^6*c^
2*d^14)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^
4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) + (x^(1/2)*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^
2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2
- 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a
*b^7*c^12*d))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 32768*a^2*b^15*c^12*d^5 + 121088*a^3*b^14*c^
11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504*a^6*b^11*c^8*d^9 + 713216*a^7*b^10*c^7*d^
10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10*b^7*c^4*d^13 + 34048*a^11*b^6*c^3*d^14 -
4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 +
15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20
*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b
^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^
(3/4) - (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*b^8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*
d^7))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4
- 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096
*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 28672
0*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(1/4) + (((32*a^13*
b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*b^5*c*d^15 + 14336*a^2*b^15*c^11*d^5 - 39008*a^3*b^14*c^10*d^6 + 41
280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^8 - 150784*a^6*b^11*c^7*d^9 + 219968*a^7*b^10*c^6*d^10 - 183424*a^
8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 32000*a^10*b^7*c^3*d^13 + 6432*a^11*b^6*c^2*d^14)/(b^7*c^9 - a^7*c^2
*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7
*a*b^6*c^8*d) - (x^(1/2)*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4
)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3
+ 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(1/4)*(4096*
a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 32768*a^2*b^15*c^12*d^5 + 121088*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c
^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504*a^6*b^11*c^8*d^9 + 713216*a^7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^
11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10*b^7*c^4*d^13 + 34048*a^11*b^6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(
b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b
^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^
13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b
^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(3/4) + (x^(1/2)*(4*a^4*b^
9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*b^8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*d^7))/(b^6*c^8 + a^6*c^2*d
^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d
^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d
^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376
*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(1/4) + (5*a^4*b^9*d^8 - 625*a*b^12*c^3*d^5 -
75*a^3*b^10*c*d^7 + 375*a^2*b^11*c^2*d^6)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*
a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d)))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*
a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7
+ 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688
*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(1/4)*2i + 2*atan((((((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a
^12*b^5*c*d^15 + 14336*a^2*b^15*c^11*d^5 - 39008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c
^8*d^8 - 150784*a^6*b^11*c^7*d^9 + 219968*a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12
- 32000*a^10*b^7*c^3*d^13 + 6432*a^11*b^6*c^2*d^14)*1i)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5
*c^7*d^2 - 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) + (x^(1/2)*(-(a^4*d^5
+ 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8
- 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a
^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^
16 - 32768*a^2*b^15*c^12*d^5 + 121088*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 -
661504*a^6*b^11*c^8*d^9 + 713216*a^7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 1372
16*a^10*b^7*c^4*d^13 + 34048*a^11*b^6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3
*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4
*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b
*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5
+ 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(3/4) - (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*
b^8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*d^7))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c
^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*
d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a
^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c
^7*d^6 - 32768*a*b^7*c^12*d))^(1/4) - ((((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*b^5*c*d^15 + 1433
6*a^2*b^15*c^11*d^5 - 39008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^8 - 150784*a^6*b
^11*c^7*d^9 + 219968*a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 32000*a^10*b^7*c^3
*d^13 + 6432*a^11*b^6*c^2*d^14)*1i)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b^4
*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) - (x^(1/2)*(-(a^4*d^5 + 625*b^4*c^4*d - 50
0*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^
7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 1146
88*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 32768*a^2*b^15*c
^12*d^5 + 121088*a^3*b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504*a^6*b^11*c^8*
d^9 + 713216*a^7*b^10*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10*b^7*c^4*d^13
+ 34048*a^11*b^6*c^3*d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6
*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^
2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2
*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7
*d^6 - 32768*a*b^7*c^12*d))^(3/4) + (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*b^8*d^9 + 100*a^2*b^1
1*c^3*d^6 + 10*a^3*b^10*c^2*d^7))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c
^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2
*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229
376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*
c^12*d))^(1/4))/(((((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*b^5*c*d^15 + 14336*a^2*b^15*c^11*d^5 -
39008*a^3*b^14*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^8 - 150784*a^6*b^11*c^7*d^9 + 219968*
a^7*b^10*c^6*d^10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 32000*a^10*b^7*c^3*d^13 + 6432*a^11*b^6
*c^2*d^14)*1i)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3
*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) + (x^(1/2)*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150
*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^
11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 -
32768*a*b^7*c^12*d))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 32768*a^2*b^15*c^12*d^5 + 121088*a^3*
b^14*c^11*d^6 - 283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504*a^6*b^11*c^8*d^9 + 713216*a^7*b^10
*c^7*d^10 - 584704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10*b^7*c^4*d^13 + 34048*a^11*b^6*c^3*
d^14 - 4608*a^12*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5
*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d
^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 22937
6*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^
12*d))^(3/4)*1i - (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*b^8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3
*b^10*c^2*d^7)*1i)/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4
*b^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b
*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^1
0*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(1/4)
+ ((((32*a^13*b^4*d^16 - 2048*a*b^16*c^12*d^4 - 704*a^12*b^5*c*d^15 + 14336*a^2*b^15*c^11*d^5 - 39008*a^3*b^14
*c^10*d^6 + 41280*a^4*b^13*c^9*d^7 + 29600*a^5*b^12*c^8*d^8 - 150784*a^6*b^11*c^7*d^9 + 219968*a^7*b^10*c^6*d^
10 - 183424*a^8*b^9*c^5*d^11 + 96320*a^9*b^8*c^4*d^12 - 32000*a^10*b^7*c^3*d^13 + 6432*a^11*b^6*c^2*d^14)*1i)/
(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 + 21*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a
^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d) - (x^(1/2)*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^
3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376
*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^1
2*d))^(1/4)*(4096*a*b^16*c^13*d^4 + 256*a^13*b^4*c*d^16 - 32768*a^2*b^15*c^12*d^5 + 121088*a^3*b^14*c^11*d^6 -
283136*a^4*b^13*c^10*d^7 + 486656*a^5*b^12*c^9*d^8 - 661504*a^6*b^11*c^8*d^9 + 713216*a^7*b^10*c^7*d^10 - 584
704*a^8*b^9*c^6*d^11 + 344576*a^9*b^8*c^5*d^12 - 137216*a^10*b^7*c^4*d^13 + 34048*a^11*b^6*c^3*d^14 - 4608*a^1
2*b^5*c^2*d^15))/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b
^2*c^4*d^4 - 6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c
*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*
d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(3/4)*1i
+ (x^(1/2)*(4*a^4*b^9*c*d^8 - 625*a*b^12*c^4*d^5 - a^5*b^8*d^9 + 100*a^2*b^11*c^3*d^6 + 10*a^3*b^10*c^2*d^7)*
1i)/(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 -
6*a*b^5*c^7*d))*(-(a^4*d^5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b
^8*c^13 + 4096*a^8*c^5*d^8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*
a^4*b^4*c^9*d^4 - 229376*a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(1/4) - (5*a^4*b^9*d^
8 - 625*a*b^12*c^3*d^5 - 75*a^3*b^10*c*d^7 + 375*a^2*b^11*c^2*d^6)/(b^7*c^9 - a^7*c^2*d^7 + 7*a^6*b*c^3*d^6 +
21*a^2*b^5*c^7*d^2 - 35*a^3*b^4*c^6*d^3 + 35*a^4*b^3*c^5*d^4 - 21*a^5*b^2*c^4*d^5 - 7*a*b^6*c^8*d)))*(-(a^4*d^
5 + 625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4)/(4096*b^8*c^13 + 4096*a^8*c^5*d^
8 - 32768*a^7*b*c^6*d^7 + 114688*a^2*b^6*c^11*d^2 - 229376*a^3*b^5*c^10*d^3 + 286720*a^4*b^4*c^9*d^4 - 229376*
a^5*b^3*c^8*d^5 + 114688*a^6*b^2*c^7*d^6 - 32768*a*b^7*c^12*d))^(1/4) + (d*x^(3/2))/(2*c*(c + d*x^2)*(a*d - b*
c))