∫ x7∕2 ___________________________

3.5.78 \(\int \frac {x^{7/2}}{(a+b x^2)^2 (c+d x^2)^3} \, dx\)

Optimal. Leaf size=718 \[ -\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}-\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}+\frac {\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} (b c-a d)^4}+\frac {a \sqrt {x}}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac {\sqrt {x} (17 a d+7 b c)}{16 \left (c+d x^2\right ) (b c-a d)^3}+\frac {\sqrt {x} (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2} \]

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Rubi [A] time = 1.04, antiderivative size = 718, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 470, 527, 522, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}-\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}+\frac {\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} (b c-a d)^4}+\frac {a \sqrt {x}}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac {\sqrt {x} (17 a d+7 b c)}{16 \left (c+d x^2\right ) (b c-a d)^3}+\frac {\sqrt {x} (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

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

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

- (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*(b*c - a*d)^4) - (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[1 +

(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*(b*c - a*d)^4) - ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*ArcTan[

1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(3/4)*d^(1/4)*(b*c - a*d)^4) + ((21*b^2*c^2 + 70*a*b*c*d

+ 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(3/4)*d^(1/4)*(b*c - a*d)^4) + (a^(

1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*(b*c - a*d

)^4) - (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]

*(b*c - a*d)^4) - ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[

d]*x])/(64*Sqrt[2]*c^(3/4)*d^(1/4)*(b*c - a*d)^4) + ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[

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

Rule 204

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

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

Rule 211

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

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

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

]]))

Rule 466

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

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

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

rQ[p]

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2

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

*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +

(a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f

)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a

, b, c, d, e, f, n}, x]

Rule 527

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

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

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

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

Rule 617

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

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

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

Rule 628

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

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

Rule 1162

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

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

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

Rule 1165

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

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

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

Rubi steps

\begin {align*} \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^8}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {a c+(-4 b c-7 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )}{2 b (b c-a d)}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {12 a b c^2-28 b c (b c+2 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{16 b c (b c-a d)^2}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {4 a b c^2 (19 b c+5 a d)-12 b^2 c^2 (7 b c+17 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{64 b c^2 (b c-a d)^3}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}-\frac {(a b (5 b c+7 a d)) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 (b c-a d)^4}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\left (\sqrt {a} b (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^4}-\frac {\left (\sqrt {a} b (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {c} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {c} (b c-a d)^4}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\left (\sqrt {a} \sqrt {b} (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 (b c-a d)^4}-\frac {\left (\sqrt {a} \sqrt {b} (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 (b c-a d)^4}+\frac {\left (\sqrt [4]{a} b^{3/4} (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} (b c-a d)^4}+\frac {\left (\sqrt [4]{a} b^{3/4} (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {c} \sqrt {d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {c} \sqrt {d} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}-\frac {\left (\sqrt [4]{a} b^{3/4} (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}+\frac {\left (\sqrt [4]{a} b^{3/4} (5 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}\\ \end {align*}

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Mathematica [A] time = 1.41, size = 604, normalized size = 0.84 \begin {gather*} \frac {-\frac {\sqrt {2} \left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{3/4} \sqrt [4]{d}}+\frac {\sqrt {2} \left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{3/4} \sqrt [4]{d}}-\frac {2 \sqrt {2} \left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{3/4} \sqrt [4]{d}}+\frac {2 \sqrt {2} \left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{3/4} \sqrt [4]{d}}+8 \sqrt {2} \sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-8 \sqrt {2} \sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+16 \sqrt {2} \sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )-16 \sqrt {2} \sqrt [4]{a} b^{3/4} (7 a d+5 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )+\frac {32 c \sqrt {x} (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac {64 a b \sqrt {x} (b c-a d)}{a+b x^2}+\frac {8 \sqrt {x} (9 a d+7 b c) (b c-a d)}{c+d x^2}}{128 (b c-a d)^4} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

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

+ 9*a*d)*Sqrt[x])/(c + d*x^2) + 16*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x

])/a^(1/4)] - 16*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - (2*Sq

rt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(3/4)*d^(1/4)) +

(2*Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(3/4)*d^(1

/4)) + 8*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] -

8*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - (Sqrt[2

]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(3/4)*d

^(1/4)) + (Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[

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

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IntegrateAlgebraic [A] time = 2.43, size = 456, normalized size = 0.64 \begin {gather*} \frac {\left (7 \sqrt {2} a^{5/4} b^{3/4} d+5 \sqrt {2} \sqrt [4]{a} b^{7/4} c\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{8 (b c-a d)^4}+\frac {\left (-7 \sqrt {2} a^{5/4} b^{3/4} d-5 \sqrt {2} \sqrt [4]{a} b^{7/4} c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{8 (b c-a d)^4}+\frac {\sqrt {x} \left (5 a^2 c d+9 a^2 d^2 x^2+19 a b c^2+28 a b c d x^2+17 a b d^2 x^4+11 b^2 c^2 x^2+7 b^2 c d x^4\right )}{16 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)^3}-\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

IntegrateAlgebraic[x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(Sqrt[x]*(19*a*b*c^2 + 5*a^2*c*d + 11*b^2*c^2*x^2 + 28*a*b*c*d*x^2 + 9*a^2*d^2*x^2 + 7*b^2*c*d*x^4 + 17*a*b*d^

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

4)*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(8*(b*c - a*d)^4) - ((21*b^2*c^2 + 70*a

*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(32*Sqrt[2]*c^(3/4)*d^(1/

4)*(b*c - a*d)^4) + ((-5*Sqrt[2]*a^(1/4)*b^(7/4)*c - 7*Sqrt[2]*a^(5/4)*b^(3/4)*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(

1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(8*(b*c - a*d)^4) + ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B] time = 2.21, size = 1193, normalized size = 1.66

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")

[Out]

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

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

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

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

qrt(2)*a^4*d^4) + 1/32*(21*(c*d^3)^(1/4)*b^2*c^2 + 70*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*arctan(

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

rt(2)*a^2*b^2*c^3*d^3 - 4*sqrt(2)*a^3*b*c^2*d^4 + sqrt(2)*a^4*c*d^5) + 1/32*(21*(c*d^3)^(1/4)*b^2*c^2 + 70*(c*

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

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

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

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

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

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

64*(21*(c*d^3)^(1/4)*b^2*c^2 + 70*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(

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

*b*c^2*d^4 + sqrt(2)*a^4*c*d^5) - 1/64*(21*(c*d^3)^(1/4)*b^2*c^2 + 70*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*

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

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

*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x^2 + a)) + 1/16*(7*b*c*d*x^(5/2) + 9*a*d^2*x^(5/2) + 11*b*c^2*sqrt(x) + 5*a*

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

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maple [A] time = 0.03, size = 1066, normalized size = 1.48

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

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

^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*d-5/8*b^2/(a*d-b*c)^4*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/

(a/b)^(1/4)*x^(1/2)+1)*c-7/8*a*b/(a*d-b*c)^4*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d-5/8*b

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

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

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

a/b)^(1/2)))*c-9/16/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2)*a^2*d^3+1/8/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2)*a*b*c*d^2+7/16

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

^2+c)^2*x^(1/2)*a*b*c^2*d+11/16/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*b^2*c^3+5/64/(a*d-b*c)^4*(c/d)^(1/4)/c*2^(1/2)

*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2*d^2+35/32/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4

)*x^(1/2)+1)*a*b*d+21/64/(a*d-b*c)^4*(c/d)^(1/4)*c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+5/64/(a*d

-b*c)^4*(c/d)^(1/4)/c*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2*d^2+35/32/(a*d-b*c)^4*(c/d)^(1/4)*2^(1

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

1/4)*x^(1/2)-1)*b^2+5/128/(a*d-b*c)^4*(c/d)^(1/4)/c*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x-

(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a^2*d^2+35/64/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*ln((x+(c/d)^(1/4)*2^(1

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

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

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maxima [A] time = 2.65, size = 855, normalized size = 1.19 \begin {gather*} -\frac {{\left (\frac {2 \, \sqrt {2} {\left (5 \, b c + 7 \, a d\right )} \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} {\left (5 \, b c + 7 \, a d\right )} \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} {\left (5 \, b c + 7 \, a d\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b c + 7 \, a d\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} a b}{16 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} + \frac {{\left (7 \, b^{2} c d + 17 \, a b d^{2}\right )} x^{\frac {9}{2}} + {\left (11 \, b^{2} c^{2} + 28 \, a b c d + 9 \, a^{2} d^{2}\right )} x^{\frac {5}{2}} + {\left (19 \, a b c^{2} + 5 \, a^{2} c d\right )} \sqrt {x}}{16 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

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

*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(5*b*c + 7*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^

(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(5*b*c + 7*a*d)*lo

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

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

*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) + 1/16*((7*b^2*c*d + 17*a*b*d^2)*x^(9/2) + (11*b^2*c^2 + 28*a*b*c*d + 9*a^

2*d^2)*x^(5/2) + (19*a*b*c^2 + 5*a^2*c*d)*sqrt(x))/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^

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

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

^4*c*d^4)*x^2) + 1/128*(2*sqrt(2)*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^

(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(21*b^2*c^2 + 70

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

/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*log(sqrt(2)*c^(1/4)*d^(1/4)*s

qrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*log(-sqrt(2)*c

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

- 4*a^3*b*c*d^3 + a^4*d^4)

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mupad [B] time = 7.73, size = 48950, normalized size = 68.18

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/((a + b*x^2)^2*(c + d*x^2)^3),x)

[Out]

2*atan(-((((((1473515*a^9*b^7*c*d^10)/2048 - (4375*a^10*b^6*d^11)/8192 + (972405*a^2*b^14*c^8*d^3)/8192 + (382

4793*a^3*b^13*c^7*d^4)/2048 + (11560479*a^4*b^12*c^6*d^5)/1024 + (69456793*a^5*b^11*c^5*d^6)/2048 + (218830061

*a^6*b^10*c^4*d^7)/4096 + (84943363*a^7*b^9*c^3*d^8)/2048 + (6507125*a^8*b^8*c^2*d^9)/512)*1i)/(a^13*d^13 - b^

13*c^13 - 78*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6

*b^7*c^7*d^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*

a^11*b^2*c^2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c*d^12) + (-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*

c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d

^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^1

5*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 3053453

3120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9

*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 -

73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14

*b^2*c^5*d^15))^(3/4)*(((-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3

+ 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^

7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d

^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 7328287948

8*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^

11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 305

34533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4)*(1280*a^20*b^

4*c*d^22 + 10240*a^2*b^22*c^19*d^4 - 144640*a^3*b^21*c^18*d^5 + 922880*a^4*b^20*c^17*d^6 - 3450880*a^5*b^19*c^

16*d^7 + 8038400*a^6*b^18*c^15*d^8 - 10501120*a^7*b^17*c^14*d^9 + 465920*a^8*b^16*c^13*d^10 + 31016960*a^9*b^1

5*c^12*d^11 - 77608960*a^10*b^14*c^11*d^12 + 115315200*a^11*b^13*c^10*d^13 - 121172480*a^12*b^12*c^9*d^14 + 94

382080*a^13*b^11*c^8*d^15 - 54978560*a^14*b^10*c^7*d^16 + 23618560*a^15*b^9*c^6*d^17 - 7193600*a^16*b^8*c^5*d^

18 + 1423360*a^17*b^7*c^4*d^19 - 143360*a^18*b^6*c^3*d^20 - 1280*a^19*b^5*c^2*d^21))/(a^13*d^13 - b^13*c^13 -

78*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6*b^7*c^7*d

^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*a^11*b^2*c

^2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c*d^12) - (x^(1/2)*(6553600*a^23*b^4*d^25 + 78643200*a^22*b^5*c*d^24 +

419430400*a^2*b^25*c^21*d^4 - 5420875776*a^3*b^24*c^20*d^5 + 31284264960*a^4*b^23*c^19*d^6 - 104224784384*a^5*

b^22*c^18*d^7 + 210842419200*a^6*b^21*c^17*d^8 - 218396098560*a^7*b^20*c^16*d^9 - 105331556352*a^8*b^19*c^15*d

^10 + 910845542400*a^9*b^18*c^14*d^11 - 2125492912128*a^10*b^17*c^13*d^12 + 3520229539840*a^11*b^16*c^12*d^13

- 4783425454080*a^12*b^15*c^11*d^14 + 5470166188032*a^13*b^14*c^10*d^15 - 5154201927680*a^14*b^13*c^9*d^16 + 3

867903787008*a^15*b^12*c^8*d^17 - 2229880750080*a^16*b^11*c^7*d^18 + 945071063040*a^17*b^10*c^6*d^19 - 2738922

45504*a^18*b^9*c^5*d^20 + 45719224320*a^19*b^8*c^4*d^21 - 1490026496*a^20*b^7*c^3*d^22 - 810024960*a^21*b^6*c^

2*d^23)*1i)/(65536*(a^18*d^18 + b^18*c^18 + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14

*d^4 - 8568*a^5*b^13*c^13*d^5 + 18564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 -

48620*a^9*b^9*c^9*d^9 + 43758*a^10*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^1

3*b^5*c^5*d^13 + 3060*a^14*b^4*c^4*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a*b^17*c^17*d - 1

8*a^17*b*c*d^17)))*1i)*(-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 +

30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7

*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^

16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488

*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^1

1*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 3053

4533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4) + (x^(1/2)*(38

72225*a^12*b^7*d^13 + 120299550*a^11*b^8*c*d^12 + 4862025*a^2*b^17*c^10*d^3 + 78440670*a^3*b^16*c^9*d^4 + 5374

50669*a^4*b^15*c^8*d^5 + 2030593320*a^5*b^14*c^7*d^6 + 4617534530*a^6*b^13*c^6*d^7 + 6551813940*a^7*b^12*c^5*d

^8 + 5932052274*a^8*b^11*c^4*d^9 + 3440955560*a^9*b^10*c^3*d^10 + 1143306165*a^10*b^9*c^2*d^11))/(65536*(a^18*

d^18 + b^18*c^18 + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13

*d^5 + 18564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 4

3758*a^10*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^1

4*b^4*c^4*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a*b^17*c^17*d - 18*a^17*b*c*d^17)))*(-(625

*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7

301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19

*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17

*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 13435194

5728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*

c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9

395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4) - (((((1473515*a^9*b^7*c*d^10)/2048 - (4375

*a^10*b^6*d^11)/8192 + (972405*a^2*b^14*c^8*d^3)/8192 + (3824793*a^3*b^13*c^7*d^4)/2048 + (11560479*a^4*b^12*c

^6*d^5)/1024 + (69456793*a^5*b^11*c^5*d^6)/2048 + (218830061*a^6*b^10*c^4*d^7)/4096 + (84943363*a^7*b^9*c^3*d^

8)/2048 + (6507125*a^8*b^8*c^2*d^9)/512)*1i)/(a^13*d^13 - b^13*c^13 - 78*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10

*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6*b^7*c^7*d^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5

*c^5*d^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*a^11*b^2*c^2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c

*d^12) + (-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*

b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(167

77216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 201326592

0*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14

*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931

351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b

^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(3/4)*(((-(625*a^8*d^8 + 194481*b^

8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d

^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c

^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^

3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d

^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 13435194

5728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c

^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4)*(1280*a^20*b^4*c*d^22 + 10240*a^2*b^22*c^19*d^4 - 144640*a^3*b^

21*c^18*d^5 + 922880*a^4*b^20*c^17*d^6 - 3450880*a^5*b^19*c^16*d^7 + 8038400*a^6*b^18*c^15*d^8 - 10501120*a^7*

b^17*c^14*d^9 + 465920*a^8*b^16*c^13*d^10 + 31016960*a^9*b^15*c^12*d^11 - 77608960*a^10*b^14*c^11*d^12 + 11531

5200*a^11*b^13*c^10*d^13 - 121172480*a^12*b^12*c^9*d^14 + 94382080*a^13*b^11*c^8*d^15 - 54978560*a^14*b^10*c^7

*d^16 + 23618560*a^15*b^9*c^6*d^17 - 7193600*a^16*b^8*c^5*d^18 + 1423360*a^17*b^7*c^4*d^19 - 143360*a^18*b^6*c

^3*d^20 - 1280*a^19*b^5*c^2*d^21))/(a^13*d^13 - b^13*c^13 - 78*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10*d^3 - 715

*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6*b^7*c^7*d^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d^8 +

715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*a^11*b^2*c^2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c*d^12) + (

x^(1/2)*(6553600*a^23*b^4*d^25 + 78643200*a^22*b^5*c*d^24 + 419430400*a^2*b^25*c^21*d^4 - 5420875776*a^3*b^24*

c^20*d^5 + 31284264960*a^4*b^23*c^19*d^6 - 104224784384*a^5*b^22*c^18*d^7 + 210842419200*a^6*b^21*c^17*d^8 - 2

18396098560*a^7*b^20*c^16*d^9 - 105331556352*a^8*b^19*c^15*d^10 + 910845542400*a^9*b^18*c^14*d^11 - 2125492912

128*a^10*b^17*c^13*d^12 + 3520229539840*a^11*b^16*c^12*d^13 - 4783425454080*a^12*b^15*c^11*d^14 + 547016618803

2*a^13*b^14*c^10*d^15 - 5154201927680*a^14*b^13*c^9*d^16 + 3867903787008*a^15*b^12*c^8*d^17 - 2229880750080*a^

16*b^11*c^7*d^18 + 945071063040*a^17*b^10*c^6*d^19 - 273892245504*a^18*b^9*c^5*d^20 + 45719224320*a^19*b^8*c^4

*d^21 - 1490026496*a^20*b^7*c^3*d^22 - 810024960*a^21*b^6*c^2*d^23)*1i)/(65536*(a^18*d^18 + b^18*c^18 + 153*a^

2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*d^5 + 18564*a^6*b^12*c^1

2*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 43758*a^10*b^8*c^8*d^10 -

31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^14*b^4*c^4*d^14 - 816*a^15

*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a*b^17*c^17*d - 18*a^17*b*c*d^17)))*1i)*(-(625*a^8*d^8 + 194481*b^8

*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^

5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^

3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3

*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^

7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945

728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^

6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4) - (x^(1/2)*(3872225*a^12*b^7*d^13 + 120299550*a^11*b^8*c*d^12 +

4862025*a^2*b^17*c^10*d^3 + 78440670*a^3*b^16*c^9*d^4 + 537450669*a^4*b^15*c^8*d^5 + 2030593320*a^5*b^14*c^7*d

^6 + 4617534530*a^6*b^13*c^6*d^7 + 6551813940*a^7*b^12*c^5*d^8 + 5932052274*a^8*b^11*c^4*d^9 + 3440955560*a^9*

b^10*c^3*d^10 + 1143306165*a^10*b^9*c^2*d^11))/(65536*(a^18*d^18 + b^18*c^18 + 153*a^2*b^16*c^16*d^2 - 816*a^3

*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*d^5 + 18564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^

11*d^7 + 43758*a^8*b^10*c^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 43758*a^10*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 +

18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^14*b^4*c^4*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*

b^2*c^2*d^16 - 18*a*b^17*c^17*d - 18*a^17*b*c*d^17)))*(-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d

^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 +

2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^1

8*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*

a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12

*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282

879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*

c^5*d^15))^(1/4))/((((((1473515*a^9*b^7*c*d^10)/2048 - (4375*a^10*b^6*d^11)/8192 + (972405*a^2*b^14*c^8*d^3)/8

192 + (3824793*a^3*b^13*c^7*d^4)/2048 + (11560479*a^4*b^12*c^6*d^5)/1024 + (69456793*a^5*b^11*c^5*d^6)/2048 +

(218830061*a^6*b^10*c^4*d^7)/4096 + (84943363*a^7*b^9*c^3*d^8)/2048 + (6507125*a^8*b^8*c^2*d^9)/512)*1i)/(a^13

*d^13 - b^13*c^13 - 78*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5

- 1716*a^6*b^7*c^7*d^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*

d^10 + 78*a^11*b^2*c^2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c*d^12) + (-(625*a^8*d^8 + 194481*b^8*c^8 + 1315062

0*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6

*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 26843

5456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4

+ 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 1919313510

40*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c

^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 20132

65920*a^14*b^2*c^5*d^15))^(3/4)*(((-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^

5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d

+ 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^

15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 -

73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*

a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*

d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4)*(12

80*a^20*b^4*c*d^22 + 10240*a^2*b^22*c^19*d^4 - 144640*a^3*b^21*c^18*d^5 + 922880*a^4*b^20*c^17*d^6 - 3450880*a

^5*b^19*c^16*d^7 + 8038400*a^6*b^18*c^15*d^8 - 10501120*a^7*b^17*c^14*d^9 + 465920*a^8*b^16*c^13*d^10 + 310169

60*a^9*b^15*c^12*d^11 - 77608960*a^10*b^14*c^11*d^12 + 115315200*a^11*b^13*c^10*d^13 - 121172480*a^12*b^12*c^9

*d^14 + 94382080*a^13*b^11*c^8*d^15 - 54978560*a^14*b^10*c^7*d^16 + 23618560*a^15*b^9*c^6*d^17 - 7193600*a^16*

b^8*c^5*d^18 + 1423360*a^17*b^7*c^4*d^19 - 143360*a^18*b^6*c^3*d^20 - 1280*a^19*b^5*c^2*d^21))/(a^13*d^13 - b^

13*c^13 - 78*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6

*b^7*c^7*d^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*

a^11*b^2*c^2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c*d^12) - (x^(1/2)*(6553600*a^23*b^4*d^25 + 78643200*a^22*b^5

*c*d^24 + 419430400*a^2*b^25*c^21*d^4 - 5420875776*a^3*b^24*c^20*d^5 + 31284264960*a^4*b^23*c^19*d^6 - 1042247

84384*a^5*b^22*c^18*d^7 + 210842419200*a^6*b^21*c^17*d^8 - 218396098560*a^7*b^20*c^16*d^9 - 105331556352*a^8*b

^19*c^15*d^10 + 910845542400*a^9*b^18*c^14*d^11 - 2125492912128*a^10*b^17*c^13*d^12 + 3520229539840*a^11*b^16*

c^12*d^13 - 4783425454080*a^12*b^15*c^11*d^14 + 5470166188032*a^13*b^14*c^10*d^15 - 5154201927680*a^14*b^13*c^

9*d^16 + 3867903787008*a^15*b^12*c^8*d^17 - 2229880750080*a^16*b^11*c^7*d^18 + 945071063040*a^17*b^10*c^6*d^19

- 273892245504*a^18*b^9*c^5*d^20 + 45719224320*a^19*b^8*c^4*d^21 - 1490026496*a^20*b^7*c^3*d^22 - 810024960*a

^21*b^6*c^2*d^23)*1i)/(65536*(a^18*d^18 + b^18*c^18 + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4

*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*d^5 + 18564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c

^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 43758*a^10*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12

- 8568*a^13*b^5*c^5*d^13 + 3060*a^14*b^4*c^4*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a*b^17*

c^17*d - 18*a^17*b*c*d^17)))*1i)*(-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5

*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d +

35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^1

5*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 7

3282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a

^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d

^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4)*1i +

(x^(1/2)*(3872225*a^12*b^7*d^13 + 120299550*a^11*b^8*c*d^12 + 4862025*a^2*b^17*c^10*d^3 + 78440670*a^3*b^16*c

^9*d^4 + 537450669*a^4*b^15*c^8*d^5 + 2030593320*a^5*b^14*c^7*d^6 + 4617534530*a^6*b^13*c^6*d^7 + 6551813940*a

^7*b^12*c^5*d^8 + 5932052274*a^8*b^11*c^4*d^9 + 3440955560*a^9*b^10*c^3*d^10 + 1143306165*a^10*b^9*c^2*d^11)*1

i)/(65536*(a^18*d^18 + b^18*c^18 + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 85

68*a^5*b^13*c^13*d^5 + 18564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 48620*a^9

*b^9*c^9*d^9 + 43758*a^10*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5

*d^13 + 3060*a^14*b^4*c^4*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a*b^17*c^17*d - 18*a^17*b*

c*d^17)))*(-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4

*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16

777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 20132659

20*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^1

4*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 19193

1351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*

b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4) + (((((1473515*a^9*b^7*c*d^

10)/2048 - (4375*a^10*b^6*d^11)/8192 + (972405*a^2*b^14*c^8*d^3)/8192 + (3824793*a^3*b^13*c^7*d^4)/2048 + (115

60479*a^4*b^12*c^6*d^5)/1024 + (69456793*a^5*b^11*c^5*d^6)/2048 + (218830061*a^6*b^10*c^4*d^7)/4096 + (8494336

3*a^7*b^9*c^3*d^8)/2048 + (6507125*a^8*b^8*c^2*d^9)/512)*1i)/(a^13*d^13 - b^13*c^13 - 78*a^2*b^11*c^11*d^2 + 2

86*a^3*b^10*c^10*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6*b^7*c^7*d^6 + 1716*a^7*b^6*c^6*d^

7 - 1287*a^8*b^5*c^5*d^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*a^11*b^2*c^2*d^11 + 13*a*b^12*c^12

*d - 13*a^12*b*c*d^12) + (-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3

+ 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a

^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*

d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 732828794

88*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c

^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30

534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(3/4)*(((-(625*a^8

*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 73010

00*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d +

16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3

- 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728

*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10

*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 93952

40960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4)*(1280*a^20*b^4*c*d^22 + 10240*a^2*b^22*c^19*d^4

- 144640*a^3*b^21*c^18*d^5 + 922880*a^4*b^20*c^17*d^6 - 3450880*a^5*b^19*c^16*d^7 + 8038400*a^6*b^18*c^15*d^8

- 10501120*a^7*b^17*c^14*d^9 + 465920*a^8*b^16*c^13*d^10 + 31016960*a^9*b^15*c^12*d^11 - 77608960*a^10*b^14*c

^11*d^12 + 115315200*a^11*b^13*c^10*d^13 - 121172480*a^12*b^12*c^9*d^14 + 94382080*a^13*b^11*c^8*d^15 - 549785

60*a^14*b^10*c^7*d^16 + 23618560*a^15*b^9*c^6*d^17 - 7193600*a^16*b^8*c^5*d^18 + 1423360*a^17*b^7*c^4*d^19 - 1

43360*a^18*b^6*c^3*d^20 - 1280*a^19*b^5*c^2*d^21))/(a^13*d^13 - b^13*c^13 - 78*a^2*b^11*c^11*d^2 + 286*a^3*b^1

0*c^10*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6*b^7*c^7*d^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a

^8*b^5*c^5*d^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*a^11*b^2*c^2*d^11 + 13*a*b^12*c^12*d - 13*a^

12*b*c*d^12) + (x^(1/2)*(6553600*a^23*b^4*d^25 + 78643200*a^22*b^5*c*d^24 + 419430400*a^2*b^25*c^21*d^4 - 5420

875776*a^3*b^24*c^20*d^5 + 31284264960*a^4*b^23*c^19*d^6 - 104224784384*a^5*b^22*c^18*d^7 + 210842419200*a^6*b

^21*c^17*d^8 - 218396098560*a^7*b^20*c^16*d^9 - 105331556352*a^8*b^19*c^15*d^10 + 910845542400*a^9*b^18*c^14*d

^11 - 2125492912128*a^10*b^17*c^13*d^12 + 3520229539840*a^11*b^16*c^12*d^13 - 4783425454080*a^12*b^15*c^11*d^1

4 + 5470166188032*a^13*b^14*c^10*d^15 - 5154201927680*a^14*b^13*c^9*d^16 + 3867903787008*a^15*b^12*c^8*d^17 -

2229880750080*a^16*b^11*c^7*d^18 + 945071063040*a^17*b^10*c^6*d^19 - 273892245504*a^18*b^9*c^5*d^20 + 45719224

320*a^19*b^8*c^4*d^21 - 1490026496*a^20*b^7*c^3*d^22 - 810024960*a^21*b^6*c^2*d^23)*1i)/(65536*(a^18*d^18 + b^

18*c^18 + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*d^5 + 18

564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 43758*a^10

*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^14*b^4*c^4

*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a*b^17*c^17*d - 18*a^17*b*c*d^17)))*1i)*(-(625*a^8*

d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 730100

0*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d +

16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3

- 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*

a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*

d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 939524

0960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4)*1i - (x^(1/2)*(3872225*a^12*b^7*d^13 + 120299550

*a^11*b^8*c*d^12 + 4862025*a^2*b^17*c^10*d^3 + 78440670*a^3*b^16*c^9*d^4 + 537450669*a^4*b^15*c^8*d^5 + 203059

3320*a^5*b^14*c^7*d^6 + 4617534530*a^6*b^13*c^6*d^7 + 6551813940*a^7*b^12*c^5*d^8 + 5932052274*a^8*b^11*c^4*d^

9 + 3440955560*a^9*b^10*c^3*d^10 + 1143306165*a^10*b^9*c^2*d^11)*1i)/(65536*(a^18*d^18 + b^18*c^18 + 153*a^2*b

^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*d^5 + 18564*a^6*b^12*c^12*d

^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 43758*a^10*b^8*c^8*d^10 - 318

24*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^14*b^4*c^4*d^14 - 816*a^15*b^

3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a*b^17*c^17*d - 18*a^17*b*c*d^17)))*(-(625*a^8*d^8 + 194481*b^8*c^8 +

13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745

500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17

- 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c

^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191

931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^1

0*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14

+ 2013265920*a^14*b^2*c^5*d^15))^(1/4)))*(-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200

*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7

*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 26843

5456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15

*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922

769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b

^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1

/4) - atan(-(((((1473515*a^9*b^7*c*d^10)/2048 - (4375*a^10*b^6*d^11)/8192 + (972405*a^2*b^14*c^8*d^3)/8192 + (

3824793*a^3*b^13*c^7*d^4)/2048 + (11560479*a^4*b^12*c^6*d^5)/1024 + (69456793*a^5*b^11*c^5*d^6)/2048 + (218830

061*a^6*b^10*c^4*d^7)/4096 + (84943363*a^7*b^9*c^3*d^8)/2048 + (6507125*a^8*b^8*c^2*d^9)/512)/(a^13*d^13 - b^1

3*c^13 - 78*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6*

b^7*c^7*d^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*a

^11*b^2*c^2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c*d^12) + (-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c

^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^

6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15

*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533

120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*

c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 7

3282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*

b^2*c^5*d^15))^(3/4)*(((-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 +

30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7

*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^

16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488

*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^1

1*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 3053

4533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4)*(1280*a^20*b^4

*c*d^22 + 10240*a^2*b^22*c^19*d^4 - 144640*a^3*b^21*c^18*d^5 + 922880*a^4*b^20*c^17*d^6 - 3450880*a^5*b^19*c^1

6*d^7 + 8038400*a^6*b^18*c^15*d^8 - 10501120*a^7*b^17*c^14*d^9 + 465920*a^8*b^16*c^13*d^10 + 31016960*a^9*b^15

*c^12*d^11 - 77608960*a^10*b^14*c^11*d^12 + 115315200*a^11*b^13*c^10*d^13 - 121172480*a^12*b^12*c^9*d^14 + 943

82080*a^13*b^11*c^8*d^15 - 54978560*a^14*b^10*c^7*d^16 + 23618560*a^15*b^9*c^6*d^17 - 7193600*a^16*b^8*c^5*d^1

8 + 1423360*a^17*b^7*c^4*d^19 - 143360*a^18*b^6*c^3*d^20 - 1280*a^19*b^5*c^2*d^21))/(a^13*d^13 - b^13*c^13 - 7

8*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6*b^7*c^7*d^

6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*a^11*b^2*c^

2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c*d^12) - (x^(1/2)*(6553600*a^23*b^4*d^25 + 78643200*a^22*b^5*c*d^24 + 4

19430400*a^2*b^25*c^21*d^4 - 5420875776*a^3*b^24*c^20*d^5 + 31284264960*a^4*b^23*c^19*d^6 - 104224784384*a^5*b

^22*c^18*d^7 + 210842419200*a^6*b^21*c^17*d^8 - 218396098560*a^7*b^20*c^16*d^9 - 105331556352*a^8*b^19*c^15*d^

10 + 910845542400*a^9*b^18*c^14*d^11 - 2125492912128*a^10*b^17*c^13*d^12 + 3520229539840*a^11*b^16*c^12*d^13 -

4783425454080*a^12*b^15*c^11*d^14 + 5470166188032*a^13*b^14*c^10*d^15 - 5154201927680*a^14*b^13*c^9*d^16 + 38

67903787008*a^15*b^12*c^8*d^17 - 2229880750080*a^16*b^11*c^7*d^18 + 945071063040*a^17*b^10*c^6*d^19 - 27389224

5504*a^18*b^9*c^5*d^20 + 45719224320*a^19*b^8*c^4*d^21 - 1490026496*a^20*b^7*c^3*d^22 - 810024960*a^21*b^6*c^2

*d^23))/(65536*(a^18*d^18 + b^18*c^18 + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4

- 8568*a^5*b^13*c^13*d^5 + 18564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 4862

0*a^9*b^9*c^9*d^9 + 43758*a^10*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^

5*c^5*d^13 + 3060*a^14*b^4*c^4*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a*b^17*c^17*d - 18*a^

17*b*c*d^17))))*(-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 302501

50*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^

7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 20

13265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^

11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 -

191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120

*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4)*1i - (x^(1/2)*(387222

5*a^12*b^7*d^13 + 120299550*a^11*b^8*c*d^12 + 4862025*a^2*b^17*c^10*d^3 + 78440670*a^3*b^16*c^9*d^4 + 53745066

9*a^4*b^15*c^8*d^5 + 2030593320*a^5*b^14*c^7*d^6 + 4617534530*a^6*b^13*c^6*d^7 + 6551813940*a^7*b^12*c^5*d^8 +

5932052274*a^8*b^11*c^4*d^9 + 3440955560*a^9*b^10*c^3*d^10 + 1143306165*a^10*b^9*c^2*d^11)*1i)/(65536*(a^18*d

^18 + b^18*c^18 + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*

d^5 + 18564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 43

758*a^10*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^14

*b^4*c^4*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a*b^17*c^17*d - 18*a^17*b*c*d^17)))*(-(625*

a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 73

01000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*

d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*

d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945

728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c

^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 93

95240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4) - ((((1473515*a^9*b^7*c*d^10)/2048 - (4375*a

^10*b^6*d^11)/8192 + (972405*a^2*b^14*c^8*d^3)/8192 + (3824793*a^3*b^13*c^7*d^4)/2048 + (11560479*a^4*b^12*c^6

*d^5)/1024 + (69456793*a^5*b^11*c^5*d^6)/2048 + (218830061*a^6*b^10*c^4*d^7)/4096 + (84943363*a^7*b^9*c^3*d^8)

/2048 + (6507125*a^8*b^8*c^2*d^9)/512)/(a^13*d^13 - b^13*c^13 - 78*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10*d^3 -

715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6*b^7*c^7*d^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d

^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*a^11*b^2*c^2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c*d^12)

+ (-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^

4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*

b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*

b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 +

134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040

*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7

*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(3/4)*(((-(625*a^8*d^8 + 194481*b^8*c^8

+ 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 7

45500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^1

7 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13

*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 1

91931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a

^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^1

4 + 2013265920*a^14*b^2*c^5*d^15))^(1/4)*(1280*a^20*b^4*c*d^22 + 10240*a^2*b^22*c^19*d^4 - 144640*a^3*b^21*c^1

8*d^5 + 922880*a^4*b^20*c^17*d^6 - 3450880*a^5*b^19*c^16*d^7 + 8038400*a^6*b^18*c^15*d^8 - 10501120*a^7*b^17*c

^14*d^9 + 465920*a^8*b^16*c^13*d^10 + 31016960*a^9*b^15*c^12*d^11 - 77608960*a^10*b^14*c^11*d^12 + 115315200*a

^11*b^13*c^10*d^13 - 121172480*a^12*b^12*c^9*d^14 + 94382080*a^13*b^11*c^8*d^15 - 54978560*a^14*b^10*c^7*d^16

+ 23618560*a^15*b^9*c^6*d^17 - 7193600*a^16*b^8*c^5*d^18 + 1423360*a^17*b^7*c^4*d^19 - 143360*a^18*b^6*c^3*d^2

0 - 1280*a^19*b^5*c^2*d^21))/(a^13*d^13 - b^13*c^13 - 78*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10*d^3 - 715*a^4*b

^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6*b^7*c^7*d^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d^8 + 715*a

^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*a^11*b^2*c^2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c*d^12) + (x^(1/2

)*(6553600*a^23*b^4*d^25 + 78643200*a^22*b^5*c*d^24 + 419430400*a^2*b^25*c^21*d^4 - 5420875776*a^3*b^24*c^20*d

^5 + 31284264960*a^4*b^23*c^19*d^6 - 104224784384*a^5*b^22*c^18*d^7 + 210842419200*a^6*b^21*c^17*d^8 - 2183960

98560*a^7*b^20*c^16*d^9 - 105331556352*a^8*b^19*c^15*d^10 + 910845542400*a^9*b^18*c^14*d^11 - 2125492912128*a^

10*b^17*c^13*d^12 + 3520229539840*a^11*b^16*c^12*d^13 - 4783425454080*a^12*b^15*c^11*d^14 + 5470166188032*a^13

*b^14*c^10*d^15 - 5154201927680*a^14*b^13*c^9*d^16 + 3867903787008*a^15*b^12*c^8*d^17 - 2229880750080*a^16*b^1

1*c^7*d^18 + 945071063040*a^17*b^10*c^6*d^19 - 273892245504*a^18*b^9*c^5*d^20 + 45719224320*a^19*b^8*c^4*d^21

- 1490026496*a^20*b^7*c^3*d^22 - 810024960*a^21*b^6*c^2*d^23))/(65536*(a^18*d^18 + b^18*c^18 + 153*a^2*b^16*c^

16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*d^5 + 18564*a^6*b^12*c^12*d^6 - 3

1824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 43758*a^10*b^8*c^8*d^10 - 31824*a^1

1*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^14*b^4*c^4*d^14 - 816*a^15*b^3*c^3*

d^15 + 153*a^16*b^2*c^2*d^16 - 18*a*b^17*c^17*d - 18*a^17*b*c*d^17))))*(-(625*a^8*d^8 + 194481*b^8*c^8 + 13150

620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a

^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268

435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d

^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 19193135

1040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6

*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 201

3265920*a^14*b^2*c^5*d^15))^(1/4)*1i + (x^(1/2)*(3872225*a^12*b^7*d^13 + 120299550*a^11*b^8*c*d^12 + 4862025*a

^2*b^17*c^10*d^3 + 78440670*a^3*b^16*c^9*d^4 + 537450669*a^4*b^15*c^8*d^5 + 2030593320*a^5*b^14*c^7*d^6 + 4617

534530*a^6*b^13*c^6*d^7 + 6551813940*a^7*b^12*c^5*d^8 + 5932052274*a^8*b^11*c^4*d^9 + 3440955560*a^9*b^10*c^3*

d^10 + 1143306165*a^10*b^9*c^2*d^11)*1i)/(65536*(a^18*d^18 + b^18*c^18 + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*

c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*d^5 + 18564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7

+ 43758*a^8*b^10*c^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 43758*a^10*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564

*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^14*b^4*c^4*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^

2*d^16 - 18*a*b^17*c^17*d - 18*a^17*b*c*d^17)))*(-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 3

0664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 259308

0*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2

- 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^

12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 +

215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488

*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^

15))^(1/4))/(((((1473515*a^9*b^7*c*d^10)/2048 - (4375*a^10*b^6*d^11)/8192 + (972405*a^2*b^14*c^8*d^3)/8192 + (