\(\int \frac {x^{3/2}}{(a+b x^2) (c+d x^2)^2} \, dx\) [474]

Optimal result

Integrand size = 24, antiderivative size = 528 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {\sqrt {x}}{2 (b c-a d) \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^2}-\frac {\sqrt [4]{a} b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^2}-\frac {(3 b c+a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^2}+\frac {(3 b c+a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^2}+\frac {\sqrt [4]{a} b^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^2}-\frac {\sqrt [4]{a} b^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^2}-\frac {(3 b c+a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^2}+\frac {(3 b c+a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 482, 536, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {\sqrt [4]{a} b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^2}-\frac {\sqrt [4]{a} b^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} (b c-a d)^2}-\frac {(a d+3 b c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^2}+\frac {(a d+3 b c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^2}+\frac {\sqrt [4]{a} b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^2}-\frac {\sqrt [4]{a} b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^2}-\frac {(a d+3 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^2}+\frac {(a d+3 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^2}+\frac {\sqrt {x}}{2 \left (c+d x^2\right ) (b c-a d)} \]

[In]

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

[Out]

Sqrt[x]/(2*(b*c - a*d)*(c + d*x^2)) + (a^(1/4)*b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]
*(b*c - a*d)^2) - (a^(1/4)*b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^2) - ((
3*b*c + a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)*d^(1/4)*(b*c - a*d)^2) + ((3*b*
c + a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)*d^(1/4)*(b*c - a*d)^2) + (a^(1/4)*b
^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*(b*c - a*d)^2) - (a^(1/4)*b^(3/4
)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*(b*c - a*d)^2) - ((3*b*c + a*d)*Log[S
qrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(3/4)*d^(1/4)*(b*c - a*d)^2) + ((3*b*c + a
*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(3/4)*d^(1/4)*(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 217

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 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 482

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

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

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.51 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {\frac {4 (b c-a d) \sqrt {x}}{c+d x^2}+4 \sqrt {2} \sqrt [4]{a} b^{3/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-\frac {\sqrt {2} (3 b c+a d) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{3/4} \sqrt [4]{d}}-4 \sqrt {2} \sqrt [4]{a} b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )+\frac {\sqrt {2} (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{3/4} \sqrt [4]{d}}}{8 (b c-a d)^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.74 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.50

method	result	size
derivativedivides	\(\frac {\frac {2 \left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (a d +3 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c}}{\left (a d -b c \right )^{2}}-\frac {b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2}}\)	\(263\)
default	\(\frac {\frac {2 \left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (a d +3 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c}}{\left (a d -b c \right )^{2}}-\frac {b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2}}\)	\(263\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.26 (sec) , antiderivative size = 2823, normalized size of antiderivative = 5.35 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*((b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d
^3 + a^4*d^4)/(b^8*c^11*d - 8*a*b^7*c^10*d^2 + 28*a^2*b^6*c^9*d^3 - 56*a^3*b^5*c^8*d^4 + 70*a^4*b^4*c^7*d^5 -
56*a^5*b^3*c^6*d^6 + 28*a^6*b^2*c^5*d^7 - 8*a^7*b*c^4*d^8 + a^8*c^3*d^9))^(1/4)*log((3*b*c + a*d)*sqrt(x) + (b
^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*
d^4)/(b^8*c^11*d - 8*a*b^7*c^10*d^2 + 28*a^2*b^6*c^9*d^3 - 56*a^3*b^5*c^8*d^4 + 70*a^4*b^4*c^7*d^5 - 56*a^5*b^
3*c^6*d^6 + 28*a^6*b^2*c^5*d^7 - 8*a^7*b*c^4*d^8 + a^8*c^3*d^9))^(1/4)) - (b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2
)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(b^8*c^11*d - 8*a*b^7*c^10*
d^2 + 28*a^2*b^6*c^9*d^3 - 56*a^3*b^5*c^8*d^4 + 70*a^4*b^4*c^7*d^5 - 56*a^5*b^3*c^6*d^6 + 28*a^6*b^2*c^5*d^7 -
8*a^7*b*c^4*d^8 + a^8*c^3*d^9))^(1/4)*log((3*b*c + a*d)*sqrt(x) - (b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*(-(81*b
^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(b^8*c^11*d - 8*a*b^7*c^10*d^2 + 28*
a^2*b^6*c^9*d^3 - 56*a^3*b^5*c^8*d^4 + 70*a^4*b^4*c^7*d^5 - 56*a^5*b^3*c^6*d^6 + 28*a^6*b^2*c^5*d^7 - 8*a^7*b*
c^4*d^8 + a^8*c^3*d^9))^(1/4)) + (-I*b*c^2 + I*a*c*d - I*(b*c*d - a*d^2)*x^2)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d
+ 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(b^8*c^11*d - 8*a*b^7*c^10*d^2 + 28*a^2*b^6*c^9*d^3 - 56*a^3*
b^5*c^8*d^4 + 70*a^4*b^4*c^7*d^5 - 56*a^5*b^3*c^6*d^6 + 28*a^6*b^2*c^5*d^7 - 8*a^7*b*c^4*d^8 + a^8*c^3*d^9))^(
1/4)*log((3*b*c + a*d)*sqrt(x) - (I*b^2*c^3 - 2*I*a*b*c^2*d + I*a^2*c*d^2)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 5
4*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(b^8*c^11*d - 8*a*b^7*c^10*d^2 + 28*a^2*b^6*c^9*d^3 - 56*a^3*b^5
*c^8*d^4 + 70*a^4*b^4*c^7*d^5 - 56*a^5*b^3*c^6*d^6 + 28*a^6*b^2*c^5*d^7 - 8*a^7*b*c^4*d^8 + a^8*c^3*d^9))^(1/4
)) + (I*b*c^2 - I*a*c*d + I*(b*c*d - a*d^2)*x^2)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3
*b*c*d^3 + a^4*d^4)/(b^8*c^11*d - 8*a*b^7*c^10*d^2 + 28*a^2*b^6*c^9*d^3 - 56*a^3*b^5*c^8*d^4 + 70*a^4*b^4*c^7*
d^5 - 56*a^5*b^3*c^6*d^6 + 28*a^6*b^2*c^5*d^7 - 8*a^7*b*c^4*d^8 + a^8*c^3*d^9))^(1/4)*log((3*b*c + a*d)*sqrt(x
) - (-I*b^2*c^3 + 2*I*a*b*c^2*d - I*a^2*c*d^2)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b
*c*d^3 + a^4*d^4)/(b^8*c^11*d - 8*a*b^7*c^10*d^2 + 28*a^2*b^6*c^9*d^3 - 56*a^3*b^5*c^8*d^4 + 70*a^4*b^4*c^7*d^
5 - 56*a^5*b^3*c^6*d^6 + 28*a^6*b^2*c^5*d^7 - 8*a^7*b*c^4*d^8 + a^8*c^3*d^9))^(1/4)) - 4*(-a*b^3/(b^8*c^8 - 8*
a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c
^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8))^(1/4)*(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)*log(b*sqrt(x) + (b^2*c^2 - 2*a*
b*c*d + a^2*d^2)*(-a*b^3/(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d
^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8))^(1/4)) + 4*(-a*b^3/(b^8*c^8 - 8*a*b^7
*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^
6 - 8*a^7*b*c*d^7 + a^8*d^8))^(1/4)*(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)*log(b*sqrt(x) - (b^2*c^2 - 2*a*b*c*d
+ a^2*d^2)*(-a*b^3/(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 -
56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8))^(1/4)) + 4*(-a*b^3/(b^8*c^8 - 8*a*b^7*c^7*
d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8
*a^7*b*c*d^7 + a^8*d^8))^(1/4)*(I*b*c^2 - I*a*c*d + I*(b*c*d - a*d^2)*x^2)*log(b*sqrt(x) - (I*b^2*c^2 - 2*I*a*
b*c*d + I*a^2*d^2)*(-a*b^3/(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4
*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8))^(1/4)) + 4*(-a*b^3/(b^8*c^8 - 8*a*b
^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*
d^6 - 8*a^7*b*c*d^7 + a^8*d^8))^(1/4)*(-I*b*c^2 + I*a*c*d - I*(b*c*d - a*d^2)*x^2)*log(b*sqrt(x) - (-I*b^2*c^2
+ 2*I*a*b*c*d - I*a^2*d^2)*(-a*b^3/(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^
4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8))^(1/4)) + 4*sqrt(x))/(b*c^2
- a*c*d + (b*c*d - a*d^2)*x^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.87 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {{\left (\frac {2 \, \sqrt {2} b \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 \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 {3}{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 {3}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}\right )} a}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, b c + a d\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 (3 \, b c + a d\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 (3 \, b c + a d\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 (3 \, b c + a d\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^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac {\sqrt {x}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}} \]

[In]

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

[Out]

-1/4*(2*sqrt(2)*b*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqr
t(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*b*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sq
rt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(3/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + s
qrt(b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^(3/4)*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*b*c*d + a^2*d^2) + 1/16*(2*sqrt(2)*(3*b*c + a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(
1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(3*b*c + a*d)*arc
tan(-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)*sq
rt(d))) + sqrt(2)*(3*b*c + a*d)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) -
sqrt(2)*(3*b*c + a*d)*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^2
- 2*a*b*c*d + a^2*d^2) + 1/2*sqrt(x)/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.24 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b c + \left (c d^{3}\right )^{\frac {1}{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^{3} d - 2 \, \sqrt {2} a b c^{2} d^{2} + \sqrt {2} a^{2} c d^{3}\right )}} + \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b c + \left (c d^{3}\right )^{\frac {1}{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^{3} d - 2 \, \sqrt {2} a b c^{2} d^{2} + \sqrt {2} a^{2} c d^{3}\right )}} + \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b c + \left (c d^{3}\right )^{\frac {1}{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^{3} d - 2 \, \sqrt {2} a b c^{2} d^{2} + \sqrt {2} a^{2} c d^{3}\right )}} - \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b c + \left (c d^{3}\right )^{\frac {1}{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^{3} d - 2 \, \sqrt {2} a b c^{2} d^{2} + \sqrt {2} a^{2} c d^{3}\right )}} - \frac {\left (a b^{3}\right )^{\frac {1}{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} b^{2} c^{2} - 2 \, \sqrt {2} a b c d + \sqrt {2} a^{2} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {1}{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} b^{2} c^{2} - 2 \, \sqrt {2} a b c d + \sqrt {2} a^{2} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {1}{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} b^{2} c^{2} - 2 \, \sqrt {2} a b c d + \sqrt {2} a^{2} d^{2}\right )}} + \frac {\left (a b^{3}\right )^{\frac {1}{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} b^{2} c^{2} - 2 \, \sqrt {2} a b c d + \sqrt {2} a^{2} d^{2}\right )}} + \frac {\sqrt {x}}{2 \, {\left (d x^{2} + c\right )} {\left (b c - a d\right )}} \]

[In]

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

[Out]

1/4*(3*(c*d^3)^(1/4)*b*c + (c*d^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4)
)/(sqrt(2)*b^2*c^3*d - 2*sqrt(2)*a*b*c^2*d^2 + sqrt(2)*a^2*c*d^3) + 1/4*(3*(c*d^3)^(1/4)*b*c + (c*d^3)^(1/4)*a
*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^3*d - 2*sqrt(2)*a*b*c^2*
d^2 + sqrt(2)*a^2*c*d^3) + 1/8*(3*(c*d^3)^(1/4)*b*c + (c*d^3)^(1/4)*a*d)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x +
sqrt(c/d))/(sqrt(2)*b^2*c^3*d - 2*sqrt(2)*a*b*c^2*d^2 + sqrt(2)*a^2*c*d^3) - 1/8*(3*(c*d^3)^(1/4)*b*c + (c*d^
3)^(1/4)*a*d)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^3*d - 2*sqrt(2)*a*b*c^2*d^2 + s
qrt(2)*a^2*c*d^3) - (a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b
^2*c^2 - 2*sqrt(2)*a*b*c*d + sqrt(2)*a^2*d^2) - (a*b^3)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqr
t(x))/(a/b)^(1/4))/(sqrt(2)*b^2*c^2 - 2*sqrt(2)*a*b*c*d + sqrt(2)*a^2*d^2) - 1/2*(a*b^3)^(1/4)*log(sqrt(2)*sqr
t(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^2*c^2 - 2*sqrt(2)*a*b*c*d + sqrt(2)*a^2*d^2) + 1/2*(a*b^3)^(1/4)*
log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^2*c^2 - 2*sqrt(2)*a*b*c*d + sqrt(2)*a^2*d^2) + 1/
2*sqrt(x)/((d*x^2 + c)*(b*c - a*d))

Mupad [B] (verification not implemented)

Time = 7.73 (sec) , antiderivative size = 20689, normalized size of antiderivative = 39.18 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- atan(((((2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 +
3*a*b^2*c^2*d - 3*a^2*b*c*d^2) + ((x^(1/2)*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4
- 30464*a^3*b^14*c^10*d^5 + 97792*a^4*b^13*c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 14694
4*a^7*b^10*c^6*d^9 + 78848*a^8*b^9*c^5*d^10 - 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6
*c^2*d^13))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d
- 6*a^5*b*c*d^5) + (2*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^
4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*(1024*a^
11*b^4*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 31744*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^
7*d^7 + 229376*a^6*b^9*c^6*d^8 - 157696*a^7*b^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 409
6*a^10*b^5*c^2*d^12))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8
+ 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6
- 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(3/4))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a
^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*
b*c*d^7))^(1/4) + (x^(1/2)*(17*a^6*b^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 +
198*a^4*b^9*c^2*d^5))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*
b^5*c^5*d - 6*a^5*b*c*d^5))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1
120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*1i
- (((2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a*
b^2*c^2*d - 3*a^2*b*c*d^2) - ((x^(1/2)*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 304
64*a^3*b^14*c^10*d^5 + 97792*a^4*b^13*c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7
*b^10*c^6*d^9 + 78848*a^8*b^9*c^5*d^10 - 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*
d^13))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a
^5*b*c*d^5) - (2*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4
*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*(1024*a^11*b^
4*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 31744*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7
+ 229376*a^6*b^9*c^6*d^8 - 157696*a^7*b^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^1
0*b^5*c^2*d^12))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448
*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 12
8*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(3/4))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^
5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d
^7))^(1/4) - (x^(1/2)*(17*a^6*b^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a
^4*b^9*c^2*d^5))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c
^5*d - 6*a^5*b*c*d^5))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a
^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*1i)/(((
(2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c
^2*d - 3*a^2*b*c*d^2) + ((x^(1/2)*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 30464*a^
3*b^14*c^10*d^5 + 97792*a^4*b^13*c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7*b^10
*c^6*d^9 + 78848*a^8*b^9*c^5*d^10 - 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*d^13)
)/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*
c*d^5) + (2*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*
d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*(1024*a^11*b^4*c*d
^13 + 4096*a^2*b^13*c^10*d^4 - 31744*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7 + 22
9376*a^6*b^9*c^6*d^8 - 157696*a^7*b^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10*b^5
*c^2*d^12))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*
b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b
^7*c^7*d - 128*a^7*b*c*d^7))^(3/4))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5
*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^
(1/4) + (x^(1/2)*(17*a^6*b^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a^4*b^
9*c^2*d^5))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d
- 6*a^5*b*c*d^5))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^
4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4) + (((2*(51*a
^4*b^7*c*d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3
*a^2*b*c*d^2) - ((x^(1/2)*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 30464*a^3*b^14*c
^10*d^5 + 97792*a^4*b^13*c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7*b^10*c^6*d^9
+ 78848*a^8*b^9*c^5*d^10 - 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*d^13))/(a^6*d
^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) -
(2*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 89
6*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*(1024*a^11*b^4*c*d^13 + 40
96*a^2*b^13*c^10*d^4 - 31744*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7 + 229376*a^6
*b^9*c^6*d^8 - 157696*a^7*b^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10*b^5*c^2*d^1
2))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*
d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d
- 128*a^7*b*c*d^7))^(3/4))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1
120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4) -
(x^(1/2)*(17*a^6*b^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a^4*b^9*c^2*d^
5))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*
b*c*d^5))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^
4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)))*(-(a*b^3)/(16*a^8*
d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 44
8*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*2i - 2*atan(((((2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^
6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) - ((x^(1/2)
*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 30464*a^3*b^14*c^10*d^5 + 97792*a^4*b^13*
c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7*b^10*c^6*d^9 + 78848*a^8*b^9*c^5*d^10
- 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*d^13))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4
*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) + ((-(a*b^3)/(16*a^8*d^8 +
16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6
*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*(1024*a^11*b^4*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 3174
4*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7 + 229376*a^6*b^9*c^6*d^8 - 157696*a^7*b
^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10*b^5*c^2*d^12)*2i)/(a^3*d^3 - b^3*c^3 +
3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^
3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(3/
4)*1i)*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 -
896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*1i + (x^(1/2)*(17*a^6*b
^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a^4*b^9*c^2*d^5))/(a^6*d^6 + b^6
*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))*(-(a*b^3
)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^
3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4) - (((2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^6
+ 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) + ((x^(1/2)*
(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 30464*a^3*b^14*c^10*d^5 + 97792*a^4*b^13*c
^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7*b^10*c^6*d^9 + 78848*a^8*b^9*c^5*d^10
- 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*d^13))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*
c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) - ((-(a*b^3)/(16*a^8*d^8 +
16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*
b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*(1024*a^11*b^4*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 31744
*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7 + 229376*a^6*b^9*c^6*d^8 - 157696*a^7*b^
8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10*b^5*c^2*d^12)*2i)/(a^3*d^3 - b^3*c^3 +
3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3
+ 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(3/4
)*1i)*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 -
896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*1i - (x^(1/2)*(17*a^6*b^
7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a^4*b^9*c^2*d^5))/(a^6*d^6 + b^6*
c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))*(-(a*b^3)
/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3
*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4))/((((2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^6
+ 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) - ((x^(1/2)*(
256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 30464*a^3*b^14*c^10*d^5 + 97792*a^4*b^13*c^
9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7*b^10*c^6*d^9 + 78848*a^8*b^9*c^5*d^10 -
36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*d^13))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c
^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) + ((-(a*b^3)/(16*a^8*d^8 + 1
6*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b
^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*(1024*a^11*b^4*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 31744*
a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7 + 229376*a^6*b^9*c^6*d^8 - 157696*a^7*b^8
*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10*b^5*c^2*d^12)*2i)/(a^3*d^3 - b^3*c^3 + 3
*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3
+ 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(3/4)
*1i)*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 8
96*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*1i + (x^(1/2)*(17*a^6*b^7
*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a^4*b^9*c^2*d^5))/(a^6*d^6 + b^6*c
^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))*(-(a*b^3)/
(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*
d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*1i + (((2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^
6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) + ((x^(1/2)
*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 30464*a^3*b^14*c^10*d^5 + 97792*a^4*b^13*
c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7*b^10*c^6*d^9 + 78848*a^8*b^9*c^5*d^10
- 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*d^13))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4
*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) - ((-(a*b^3)/(16*a^8*d^8 +
16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6
*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*(1024*a^11*b^4*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 3174
4*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7 + 229376*a^6*b^9*c^6*d^8 - 157696*a^7*b
^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10*b^5*c^2*d^12)*2i)/(a^3*d^3 - b^3*c^3 +
3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^
3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(3/
4)*1i)*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 -
896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*1i - (x^(1/2)*(17*a^6*b
^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a^4*b^9*c^2*d^5))/(a^6*d^6 + b^6
*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))*(-(a*b^3
)/(16*a^8*d^8 + 16*b^8*c^8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^
3*d^5 + 448*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*1i))*(-(a*b^3)/(16*a^8*d^8 + 16*b^8*c^
8 + 448*a^2*b^6*c^6*d^2 - 896*a^3*b^5*c^5*d^3 + 1120*a^4*b^4*c^4*d^4 - 896*a^5*b^3*c^3*d^5 + 448*a^6*b^2*c^2*d
^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4) - atan(-(((((x^(1/2)*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 +
4096*a^2*b^15*c^11*d^4 - 30464*a^3*b^14*c^10*d^5 + 97792*a^4*b^13*c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*
a^6*b^11*c^7*d^8 - 146944*a^7*b^10*c^6*d^9 + 78848*a^8*b^9*c^5*d^10 - 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*
c^3*d^12 - 2816*a^11*b^6*c^2*d^13))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*
c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) - (2*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d +
12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2
*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d
^7))^(1/4)*(1024*a^11*b^4*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 31744*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 -
200704*a^5*b^10*c^7*d^7 + 229376*a^6*b^9*c^6*d^8 - 157696*a^7*b^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9
*b^6*c^3*d^11 - 4096*a^10*b^5*c^2*d^12))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*
b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a
*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5
- 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(3/4) - (2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c
^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 +
54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*
d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*
a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4) + (x^(1/2)*(17*a^6*b^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*
c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a^4*b^9*c^2*d^5))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^
3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 10
8*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d
^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 11468
8*a^6*b^2*c^5*d^7))^(1/4)*1i + ((((x^(1/2)*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 -
30464*a^3*b^14*c^10*d^5 + 97792*a^4*b^13*c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944
*a^7*b^10*c^6*d^9 + 78848*a^8*b^9*c^5*d^10 - 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*
c^2*d^13))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d -
6*a^5*b*c*d^5) + (2*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^
8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3
*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*(1024*a^11*b^4
*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 31744*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7
+ 229376*a^6*b^9*c^6*d^8 - 157696*a^7*b^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10
*b^5*c^2*d^12))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*
d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7
*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6
+ 114688*a^6*b^2*c^5*d^7))^(3/4) + (2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*
d^4))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*
a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8
+ 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*
a^6*b^2*c^5*d^7))^(1/4) + (x^(1/2)*(17*a^6*b^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^
3*d^4 + 198*a^4*b^9*c^2*d^5))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^
4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*
c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d
^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)
*1i)/(((((x^(1/2)*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 30464*a^3*b^14*c^10*d^5
+ 97792*a^4*b^13*c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7*b^10*c^6*d^9 + 78848
*a^8*b^9*c^5*d^10 - 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*d^13))/(a^6*d^6 + b^6
*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) - (2*(-(a
^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d
^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4
*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*(1024*a^11*b^4*c*d^13 + 4096*a^2*b^13*c
^10*d^4 - 31744*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7 + 229376*a^6*b^9*c^6*d^8
- 157696*a^7*b^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10*b^5*c^2*d^12))/(a^3*d^3
- b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 1
2*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b
^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7
))^(3/4) - (2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3
+ 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*
d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3
- 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4) +
(x^(1/2)*(17*a^6*b^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a^4*b^9*c^2*d
^5))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5
*b*c*d^5))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d +
4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d
^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4) - ((((x^(1/2)*(256*a^13*
b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 30464*a^3*b^14*c^10*d^5 + 97792*a^4*b^13*c^9*d^6 - 1
76896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7*b^10*c^6*d^9 + 78848*a^8*b^9*c^5*d^10 - 36352*a^
9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*d^13))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 -
20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) + (2*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2
*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 -
32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^
3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*(1024*a^11*b^4*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 31744*a^3*b^12*c^9
*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7 + 229376*a^6*b^9*c^6*d^8 - 157696*a^7*b^8*c^5*d^9 + 5
7344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10*b^5*c^2*d^12))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d -
3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^1
1*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*
c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(3/4) + (2*(51*a^4*b^7*c*
d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*
d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096
*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 +
286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4) + (x^(1/2)*(17*a^6*b^7*d^7 +
108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a^4*b^9*c^2*d^5))/(a^6*d^6 + b^6*c^6 + 15
*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))*(-(a^4*d^4 + 81*b
^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*
b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5
- 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108
*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^
8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688
*a^6*b^2*c^5*d^7))^(1/4)*2i - 2*atan(-(((((x^(1/2)*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^
11*d^4 - 30464*a^3*b^14*c^10*d^5 + 97792*a^4*b^13*c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8
- 146944*a^7*b^10*c^6*d^9 + 78848*a^8*b^9*c^5*d^10 - 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a
^11*b^6*c^2*d^13))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5
*c^5*d - 6*a^5*b*c*d^5) - ((-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4
096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 2293
76*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*(1024*a^
11*b^4*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 31744*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^
7*d^7 + 229376*a^6*b^9*c^6*d^8 - 157696*a^7*b^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 409
6*a^10*b^5*c^2*d^12)*2i)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2
*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 -
32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^
3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(3/4)*1i + (2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c^3*d^3 + 189*
a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^
2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a
^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d
^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*1i - (x^(1/2)*(17*a^6*b^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 +
108*a^3*b^10*c^3*d^4 + 198*a^4*b^9*c^2*d^5))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15
*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^
3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 11468
8*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*
c^5*d^7))^(1/4) + ((((x^(1/2)*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 30464*a^3*b^
14*c^10*d^5 + 97792*a^4*b^13*c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7*b^10*c^6
*d^9 + 78848*a^8*b^9*c^5*d^10 - 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*d^13))/(a
^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^
5) + ((-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096
*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 +
286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*(1024*a^11*b^4*c*d^13 + 4096*
a^2*b^13*c^10*d^4 - 31744*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7 + 229376*a^6*b^
9*c^6*d^8 - 157696*a^7*b^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10*b^5*c^2*d^12)*
2i)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*
b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 +
114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^
6*b^2*c^5*d^7))^(3/4)*1i - (2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^
3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3
*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688
*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c
^5*d^7))^(1/4)*1i - (x^(1/2)*(17*a^6*b^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4
+ 198*a^4*b^9*c^2*d^5))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*
a*b^5*c^5*d - 6*a^5*b*c*d^5))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)
/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 2
29376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4))/((((
(x^(1/2)*(256*a^13*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 30464*a^3*b^14*c^10*d^5 + 97792*a
^4*b^13*c^9*d^6 - 176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7*b^10*c^6*d^9 + 78848*a^8*b^9*
c^5*d^10 - 36352*a^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*d^13))/(a^6*d^6 + b^6*c^6 + 15
*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) - ((-(a^4*d^4 + 81
*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*
a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^
5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*(1024*a^11*b^4*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 3
1744*a^3*b^12*c^9*d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7 + 229376*a^6*b^9*c^6*d^8 - 157696*a^
7*b^8*c^5*d^9 + 57344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10*b^5*c^2*d^12)*2i)/(a^3*d^3 - b^3*c^
3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*
c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d
^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(3/4)
*1i + (2*(51*a^4*b^7*c*d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a
*b^2*c^2*d - 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/
(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 22
9376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*1i - (
x^(1/2)*(17*a^6*b^7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a^4*b^9*c^2*d^5
))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b
*c*d^5))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4
096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4
+ 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*1i - ((((x^(1/2)*(256*a^13
*b^4*d^15 - 512*a^12*b^5*c*d^14 + 4096*a^2*b^15*c^11*d^4 - 30464*a^3*b^14*c^10*d^5 + 97792*a^4*b^13*c^9*d^6 -
176896*a^5*b^12*c^8*d^7 + 198656*a^6*b^11*c^7*d^8 - 146944*a^7*b^10*c^6*d^9 + 78848*a^8*b^9*c^5*d^10 - 36352*a
^9*b^8*c^4*d^11 + 14336*a^10*b^7*c^3*d^12 - 2816*a^11*b^6*c^2*d^13))/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 -
20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) + ((-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*
b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 3
2768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3
*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*(1024*a^11*b^4*c*d^13 + 4096*a^2*b^13*c^10*d^4 - 31744*a^3*b^12*c^9*
d^5 + 106496*a^4*b^11*c^8*d^6 - 200704*a^5*b^10*c^7*d^7 + 229376*a^6*b^9*c^6*d^8 - 157696*a^7*b^8*c^5*d^9 + 57
344*a^8*b^7*c^4*d^10 - 4096*a^9*b^6*c^3*d^11 - 4096*a^10*b^5*c^2*d^12)*2i)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d
- 3*a^2*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c
^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^
5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(3/4)*1i - (2*(51*a^4*b
^7*c*d^5 - a^5*b^6*d^6 + 81*a^2*b^9*c^3*d^3 + 189*a^3*b^8*c^2*d^4))/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2
*b*c*d^2))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d +
4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d
^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*1i - (x^(1/2)*(17*a^6*b^
7*d^7 + 108*a^5*b^8*c*d^6 + 81*a^2*b^11*c^4*d^3 + 108*a^3*b^10*c^3*d^4 + 198*a^4*b^9*c^2*d^5))/(a^6*d^6 + b^6*
c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))*(-(a^4*d^
4 + 81*b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 -
32768*a*b^7*c^10*d^2 - 32768*a^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*
c^7*d^5 - 229376*a^5*b^3*c^6*d^6 + 114688*a^6*b^2*c^5*d^7))^(1/4)*1i))*(-(a^4*d^4 + 81*b^4*c^4 + 54*a^2*b^2*c^
2*d^2 + 108*a*b^3*c^3*d + 12*a^3*b*c*d^3)/(4096*b^8*c^11*d + 4096*a^8*c^3*d^9 - 32768*a*b^7*c^10*d^2 - 32768*a
^7*b*c^4*d^8 + 114688*a^2*b^6*c^9*d^3 - 229376*a^3*b^5*c^8*d^4 + 286720*a^4*b^4*c^7*d^5 - 229376*a^5*b^3*c^6*d
^6 + 114688*a^6*b^2*c^5*d^7))^(1/4) - x^(1/2)/(2*(c + d*x^2)*(a*d - b*c))