∫ x5∕2 _________________________

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

Optimal. Leaf size=528 \[ -\frac {a^{3/4} \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^2}+\frac {a^{3/4} \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^2}+\frac {a^{3/4} \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^2}-\frac {a^{3/4} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} (b c-a d)^2}+\frac {(3 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)^2}-\frac {(3 a d+b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)^2}-\frac {(3 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)^2}+\frac {(3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)^2}+\frac {x^{3/2}}{2 \left (c+d x^2\right ) (b c-a d)} \]

[Out]

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

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

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

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

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

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Rubi [A] time = 0.59, antiderivative size = 528, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {466, 471, 584, 297, 1162, 617, 204, 1165, 628} \[ -\frac {a^{3/4} \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^2}+\frac {a^{3/4} \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^2}+\frac {a^{3/4} \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^2}-\frac {a^{3/4} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} (b c-a d)^2}+\frac {(3 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)^2}-\frac {(3 a d+b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)^2}-\frac {(3 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)^2}+\frac {(3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)^2}+\frac {x^{3/2}}{2 \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

^(1/4)*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^(3/4)*b^(1/4

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

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

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

Rule 204

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

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

Rule 297

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

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

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

b]]))

Rule 466

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

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

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

rQ[p]

Rule 471

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 584

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

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

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

Rule 617

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

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

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

Rule 628

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

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

Rule 1162

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

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

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

Rule 1165

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

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

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

Rubi steps

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

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Mathematica [A] time = 0.29, size = 522, normalized size = 0.99 \[ \frac {-4 \sqrt {2} a^{3/4} \sqrt [4]{b} \sqrt [4]{c} d^{3/4} \left (c+d x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+4 \sqrt {2} a^{3/4} \sqrt [4]{b} \sqrt [4]{c} d^{3/4} \left (c+d x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+8 \sqrt {2} a^{3/4} \sqrt [4]{b} \sqrt [4]{c} d^{3/4} \left (c+d x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )-8 \sqrt {2} a^{3/4} \sqrt [4]{b} \sqrt [4]{c} d^{3/4} \left (c+d x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )+8 \sqrt [4]{c} d^{3/4} x^{3/2} (b c-a d)+\sqrt {2} \left (c+d x^2\right ) (3 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )-\sqrt {2} \left (c+d x^2\right ) (3 a d+b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )-2 \sqrt {2} \left (c+d x^2\right ) (3 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )+2 \sqrt {2} \left (c+d x^2\right ) (3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{16 \sqrt [4]{c} d^{3/4} \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

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

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

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

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

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fricas [B] time = 14.96, size = 3393, normalized size = 6.43 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)*(-(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d

^3 + 81*a^4*d^4)/(b^8*c^9*d^3 - 8*a*b^7*c^8*d^4 + 28*a^2*b^6*c^7*d^5 - 56*a^3*b^5*c^6*d^6 + 70*a^4*b^4*c^5*d^7

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

c*d^2 + a^2*d^3)*sqrt((b^6*c^6 + 18*a*b^5*c^5*d + 135*a^2*b^4*c^4*d^2 + 540*a^3*b^3*c^3*d^3 + 1215*a^4*b^2*c^2

*d^4 + 1458*a^5*b*c*d^5 + 729*a^6*d^6)*x - (b^8*c^9*d + 8*a*b^7*c^8*d^2 + 12*a^2*b^6*c^7*d^3 - 40*a^3*b^5*c^6*

d^4 - 74*a^4*b^4*c^5*d^5 + 120*a^5*b^3*c^4*d^6 + 108*a^6*b^2*c^3*d^7 - 216*a^7*b*c^2*d^8 + 81*a^8*c*d^9)*sqrt(

-(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^8*c^9*d^3 - 8*a*b^7*c^8*d^4

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

a^7*b*c^2*d^10 + a^8*c*d^11)))*(-(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4

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

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

3*c^3*d^3 - 18*a^3*b^2*c^2*d^4 - 27*a^4*b*c*d^5 + 27*a^5*d^6)*sqrt(x)*(-(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2

*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^8*c^9*d^3 - 8*a*b^7*c^8*d^4 + 28*a^2*b^6*c^7*d^5 - 56*a^3*b^5*c^6*

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

^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)) - 16*(-a^3*b/(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)*arctan((sqrt(a^4*b^2*x - (a^3*b^5*c^4

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

8*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)))*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(-a^3*b/(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 5

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

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

*b/(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(a^2*b*sqrt(

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

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

2)*(-(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^8*c^9*d^3 - 8*a*b^7*c^8

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

- 8*a^7*b*c^2*d^10 + a^8*c*d^11))^(1/4)*log((b^6*c^7*d^2 - 6*a*b^5*c^6*d^3 + 15*a^2*b^4*c^5*d^4 - 20*a^3*b^3*c

^4*d^5 + 15*a^4*b^2*c^3*d^6 - 6*a^5*b*c^2*d^7 + a^6*c*d^8)*(-(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 +

108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^8*c^9*d^3 - 8*a*b^7*c^8*d^4 + 28*a^2*b^6*c^7*d^5 - 56*a^3*b^5*c^6*d^6 + 70*a^

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

*a*b^2*c^2*d + 27*a^2*b*c*d^2 + 27*a^3*d^3)*sqrt(x)) + (b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)*(-(b^4*c^4 + 12*a

*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^8*c^9*d^3 - 8*a*b^7*c^8*d^4 + 28*a^2*b^6*c^

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

a^8*c*d^11))^(1/4)*log(-(b^6*c^7*d^2 - 6*a*b^5*c^6*d^3 + 15*a^2*b^4*c^5*d^4 - 20*a^3*b^3*c^4*d^5 + 15*a^4*b^2

*c^3*d^6 - 6*a^5*b*c^2*d^7 + a^6*c*d^8)*(-(b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 8

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

a^5*b^3*c^4*d^8 + 28*a^6*b^2*c^3*d^9 - 8*a^7*b*c^2*d^10 + a^8*c*d^11))^(3/4) + (b^3*c^3 + 9*a*b^2*c^2*d + 27*a

^2*b*c*d^2 + 27*a^3*d^3)*sqrt(x)) - 4*x^(3/2))/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)

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giac [A] time = 0.93, size = 683, normalized size = 1.29 \[ \frac {{\left (\left (c d^{3}\right )^{\frac {3}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{3} d^{3} - 2 \, \sqrt {2} a b c^{2} d^{4} + \sqrt {2} a^{2} c d^{5}\right )}} + \frac {{\left (\left (c d^{3}\right )^{\frac {3}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{3} d^{3} - 2 \, \sqrt {2} a b c^{2} d^{4} + \sqrt {2} a^{2} c d^{5}\right )}} - \frac {{\left (\left (c d^{3}\right )^{\frac {3}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{3} d^{3} - 2 \, \sqrt {2} a b c^{2} d^{4} + \sqrt {2} a^{2} c d^{5}\right )}} + \frac {{\left (\left (c d^{3}\right )^{\frac {3}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{3} d^{3} - 2 \, \sqrt {2} a b c^{2} d^{4} + \sqrt {2} a^{2} c d^{5}\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} c^{2} - 2 \, \sqrt {2} a b^{3} c d + \sqrt {2} a^{2} b^{2} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} c^{2} - 2 \, \sqrt {2} a b^{3} c d + \sqrt {2} a^{2} b^{2} d^{2}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{4} c^{2} - 2 \, \sqrt {2} a b^{3} c d + \sqrt {2} a^{2} b^{2} d^{2}\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{4} c^{2} - 2 \, \sqrt {2} a b^{3} c d + \sqrt {2} a^{2} b^{2} d^{2}\right )}} + \frac {x^{\frac {3}{2}}}{2 \, {\left (d x^{2} + c\right )} {\left (b c - a d\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

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

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

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

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

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

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

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maple [A] time = 0.02, size = 528, normalized size = 1.00 \[ -\frac {a d \,x^{\frac {3}{2}}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}+\frac {b c \,x^{\frac {3}{2}}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}-\frac {\sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, a \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, a \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} d}+\frac {\sqrt {2}\, b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} d}+\frac {\sqrt {2}\, b c \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} d} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

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

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maxima [A] time = 2.45, size = 436, normalized size = 0.83 \[ -\frac {a b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac {{\left (b c + 3 \, a d\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{16 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac {x^{\frac {3}{2}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

t(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sq

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

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

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

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mupad [B] time = 2.48, size = 18673, normalized size = 35.37 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((864*a^13*b^4*c*d^13 - 32*a^3*b^14*c^11*d^3 + 1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*

a^6*b^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d

^10 + 10336*a^11*b^6*c^3*d^11 - 5184*a^12*b^5*c^2*d^12)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c

^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) + (x^(1/2)*(-(a^3*b)/(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 + 4

48*a^6*b^2*c^2*d^6 - 128*a*b^7*c^7*d - 128*a^7*b*c*d^7))^(1/4)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4

- 33280*a^4*b^13*c^10*d^5 + 111872*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*d^8 - 2652

16*a^8*b^9*c^6*d^9 + 197120*a^9*b^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 - 16896*a^12

*b^5*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^3*b)/(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 + (x^

(1/2)*(a^3*b^10*c^6*d + 144*a^8*b^5*c*d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4 + 17

7*a^7*b^6*c^2*d^5)*1i)/(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^3*b)/(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) -

(((864*a^13*b^4*c*d^13 - 32*a^3*b^14*c^11*d^3 + 1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b

^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 +

10336*a^11*b^6*c^3*d^11 - 5184*a^12*b^5*c^2*d^12)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^

3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) - (x^(1/2)*(-(a^3*b)/(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)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 332

80*a^4*b^13*c^10*d^5 + 111872*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*d^8 - 265216*a^

8*b^9*c^6*d^9 + 197120*a^9*b^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 - 16896*a^12*b^5*

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^3*b)/(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 - (x^(1/2)

*(a^3*b^10*c^6*d + 144*a^8*b^5*c*d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4 + 177*a^7

*b^6*c^2*d^5)*1i)/(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^3*b)/(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^4

*b^9*c^5*d + 108*a^8*b^5*c*d^5 + 13*a^5*b^8*c^4*d^2 + 63*a^6*b^7*c^3*d^3 + 135*a^7*b^6*c^2*d^4)/(a^7*d^7 - b^7

*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a

^6*b*c*d^6) + (((864*a^13*b^4*c*d^13 - 32*a^3*b^14*c^11*d^3 + 1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c^9*d^5

+ 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256*a^10*b

^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11 - 5184*a^12*b^5*c^2*d^12)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a

^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) + (x^(1/2)*(-(a^3*b)

/(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)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14*c

^11*d^4 - 33280*a^4*b^13*c^10*d^5 + 111872*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*d^

8 - 265216*a^8*b^9*c^6*d^9 + 197120*a^9*b^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 - 16

896*a^12*b^5*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^3*b)/(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)

+ (x^(1/2)*(a^3*b^10*c^6*d + 144*a^8*b^5*c*d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4

+ 177*a^7*b^6*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^3*b)/(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)

+ (((864*a^13*b^4*c*d^13 - 32*a^3*b^14*c^11*d^3 + 1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6

*b^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10

+ 10336*a^11*b^6*c^3*d^11 - 5184*a^12*b^5*c^2*d^12)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*

d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) - (x^(1/2)*(-(a^3*b)/(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)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 3

3280*a^4*b^13*c^10*d^5 + 111872*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*d^8 - 265216*

a^8*b^9*c^6*d^9 + 197120*a^9*b^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 - 16896*a^12*b^

5*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^3*b)/(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) - (x^(1/2)*

(a^3*b^10*c^6*d + 144*a^8*b^5*c*d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4 + 177*a^7*

b^6*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^3*b)/(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^3*b

)/(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)*2i - 2*atan((((((864*a^13*b^4*c*d^13 -

32*a^3*b^14*c^11*d^3 + 1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b

^10*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11

- 5184*a^12*b^5*c^2*d^12)*1i)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^

4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) + (x^(1/2)*(-(a^3*b)/(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)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 33280*a^4*b^13*c^10*d^5

+ 111872*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*d^8 - 265216*a^8*b^9*c^6*d^9 + 19712

0*a^9*b^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 - 16896*a^12*b^5*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

^3*b)/(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) + (x^(1/2)*(a^3*b^10*c^6*d + 144*a

^8*b^5*c*d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4 + 177*a^7*b^6*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^3*b)/(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) - ((((864*a^13*b^4*c*d^13 - 32*a^

3*b^14*c^11*d^3 + 1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b^10*c^

7*d^7 + 74368*a^8*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11 - 5184

*a^12*b^5*c^2*d^12)*1i)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21

*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) - (x^(1/2)*(-(a^3*b)/(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)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 33280*a^4*b^13*c^10*d^5 + 1118

72*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*d^8 - 265216*a^8*b^9*c^6*d^9 + 197120*a^9*

b^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 - 16896*a^12*b^5*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^3*b)/

(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) - (x^(1/2)*(a^3*b^10*c^6*d + 144*a^8*b^5

*c*d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4 + 177*a^7*b^6*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^3*b)

/(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))/(((((864*a^13*b^4*c*d^13 - 32*a^3*b^14

*c^11*d^3 + 1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7

+ 74368*a^8*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11 - 5184*a^12*

b^5*c^2*d^12)*1i)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b

^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) + (x^(1/2)*(-(a^3*b)/(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)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 33280*a^4*b^13*c^10*d^5 + 111872*a^5

*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*d^8 - 265216*a^8*b^9*c^6*d^9 + 197120*a^9*b^8*c^

5*d^10 - 120832*a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 - 16896*a^12*b^5*c^2*d^13))/(a^6*d^6 + b^6*c^6 + 1

5*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^3*b)/(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 + (x^(1/2)*(a^3*b^10*c^6*d + 144*a^8*b^5*c*

d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4 + 177*a^7*b^6*c^2*d^5)*1i)/(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^3*b)

/(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^4*b^9*c^5*d + 108*a^8*b^5*c*d^5 +

13*a^5*b^8*c^4*d^2 + 63*a^6*b^7*c^3*d^3 + 135*a^7*b^6*c^2*d^4)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^

3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) + ((((864*a^13*b^4*c*

d^13 - 32*a^3*b^14*c^11*d^3 + 1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816

*a^7*b^10*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3

*d^11 - 5184*a^12*b^5*c^2*d^12)*1i)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*

c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) - (x^(1/2)*(-(a^3*b)/(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)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 33280*a^4*b^13*c^1

0*d^5 + 111872*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*d^8 - 265216*a^8*b^9*c^6*d^9 +

197120*a^9*b^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 - 16896*a^12*b^5*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^3*b)/(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 - (x^(1/2)*(a^3*b^10*c^6*

d + 144*a^8*b^5*c*d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4 + 177*a^7*b^6*c^2*d^5)*1

i)/(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^3*b)/(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^3*b)/(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(((((864*a^13*b^4*c*d^13 - 32*a^3*b^14*c^11

*d^3 + 1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7 + 743

68*a^8*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11 - 5184*a^12*b^5*c

^2*d^12)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^

5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) + (x^(1/2)*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d +

108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 114688*a^2

*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d

^9))^(1/4)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 33280*a^4*b^13*c^10*d^5 + 111872*a^5*b^12*c^9*d^6

- 219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*d^8 - 265216*a^8*b^9*c^6*d^9 + 197120*a^9*b^8*c^5*d^10 - 12083

2*a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 - 16896*a^12*b^5*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))*(-(81*a^4*d^4 + b^4*c^4 + 54*a

^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 -

32768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*

b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(3/4)*1i + (x^(1/2)*(a^3*b^10*c^6*d + 144*a^8*b^5*c*d^6 + 12*a^4*b^9*c^

5*d^2 + 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4 + 177*a^7*b^6*c^2*d^5)*1i)/(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))*(-(81*a^4*d^4 + b^4*c^4 + 54

*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4

- 32768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^

5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(1/4) - (((864*a^13*b^4*c*d^13 - 32*a^3*b^14*c^11*d^3 + 1984*a^4*b^13

*c^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 -

37184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11 - 5184*a^12*b^5*c^2*d^12)/(a^7*d^7 -

b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d -

7*a^6*b*c*d^6) - (x^(1/2)*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(40

96*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 22937

6*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(1/4)*(2304*a^1

3*b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 33280*a^4*b^13*c^10*d^5 + 111872*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^

8*d^7 + 283136*a^7*b^10*c^7*d^8 - 265216*a^8*b^9*c^6*d^9 + 197120*a^9*b^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11

+ 56576*a^11*b^6*c^3*d^12 - 16896*a^12*b^5*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))*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*

a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^1

0 + 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688

*a^6*b^2*c^3*d^9))^(3/4)*1i - (x^(1/2)*(a^3*b^10*c^6*d + 144*a^8*b^5*c*d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c

^4*d^3 + 124*a^6*b^7*c^3*d^4 + 177*a^7*b^6*c^2*d^5)*1i)/(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))*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 1

2*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d

^10 + 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 1146

88*a^6*b^2*c^3*d^9))^(1/4))/((a^4*b^9*c^5*d + 108*a^8*b^5*c*d^5 + 13*a^5*b^8*c^4*d^2 + 63*a^6*b^7*c^3*d^3 + 13

5*a^7*b^6*c^2*d^4)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*

b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) + (((864*a^13*b^4*c*d^13 - 32*a^3*b^14*c^11*d^3 + 1984*a^4*b^13*c

^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 - 3

7184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11 - 5184*a^12*b^5*c^2*d^12)/(a^7*d^7 - b^

7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*

a^6*b*c*d^6) + (x^(1/2)*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096

*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229376*

a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(1/4)*(2304*a^13*

b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 33280*a^4*b^13*c^10*d^5 + 111872*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*

d^7 + 283136*a^7*b^10*c^7*d^8 - 265216*a^8*b^9*c^6*d^9 + 197120*a^9*b^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11 +

56576*a^11*b^6*c^3*d^12 - 16896*a^12*b^5*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))*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*

b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10

+ 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a

^6*b^2*c^3*d^9))^(3/4) + (x^(1/2)*(a^3*b^10*c^6*d + 144*a^8*b^5*c*d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*d^

3 + 124*a^6*b^7*c^3*d^4 + 177*a^7*b^6*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))*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*

c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 11

4688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b

^2*c^3*d^9))^(1/4) + (((864*a^13*b^4*c*d^13 - 32*a^3*b^14*c^11*d^3 + 1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c

^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256

*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11 - 5184*a^12*b^5*c^2*d^12)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2

+ 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) - (x^(1/2)*(-

(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9

*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a

^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(1/4)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14

*c^11*d^4 - 33280*a^4*b^13*c^10*d^5 + 111872*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*

d^8 - 265216*a^8*b^9*c^6*d^9 + 197120*a^9*b^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 -

16896*a^12*b^5*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))*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^

3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 -

229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(3/4) - (

x^(1/2)*(a^3*b^10*c^6*d + 144*a^8*b^5*c*d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4 +

177*a^7*b^6*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))*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(

4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229

376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(1/4)))*(-(81

*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^

3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*

b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(1/4)*2i - 2*atan((((((864*a^13*b^4*c*d^13 - 3

2*a^3*b^14*c^11*d^3 + 1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b^1

0*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11 -

5184*a^12*b^5*c^2*d^12)*1i)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4

+ 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) + (x^(1/2)*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2

+ 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^

2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 1

14688*a^6*b^2*c^3*d^9))^(1/4)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 33280*a^4*b^13*c^10*d^5 + 11187

2*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*d^8 - 265216*a^8*b^9*c^6*d^9 + 197120*a^9*b

^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 - 16896*a^12*b^5*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))*(-(81*a^4*d

^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32

768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^

5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(3/4) + (x^(1/2)*(a^3*b^10*c^6*d + 144*a^8*b^5*c*d^6

+ 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4 + 177*a^7*b^6*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))*(-(81*a^4*d^4 +

b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*

a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^

7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(1/4) - ((((864*a^13*b^4*c*d^13 - 32*a^3*b^14*c^11*d^3 +

1984*a^4*b^13*c^10*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7 + 74368*a^8

*b^9*c^6*d^8 - 37184*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11 - 5184*a^12*b^5*c^2*d^1

2)*1i)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5

+ 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) - (x^(1/2)*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 10

8*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 114688*a^2*b

^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9

))^(1/4)*(2304*a^13*b^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 33280*a^4*b^13*c^10*d^5 + 111872*a^5*b^12*c^9*d^6 -

219136*a^6*b^11*c^8*d^7 + 283136*a^7*b^10*c^7*d^8 - 265216*a^8*b^9*c^6*d^9 + 197120*a^9*b^8*c^5*d^10 - 120832*

a^10*b^7*c^4*d^11 + 56576*a^11*b^6*c^3*d^12 - 16896*a^12*b^5*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))*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2

*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 3

2768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^

3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(3/4) - (x^(1/2)*(a^3*b^10*c^6*d + 144*a^8*b^5*c*d^6 + 12*a^4*b^9*c^5*d^2

+ 54*a^5*b^8*c^4*d^3 + 124*a^6*b^7*c^3*d^4 + 177*a^7*b^6*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))*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2

*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768

*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^

4*d^8 + 114688*a^6*b^2*c^3*d^9))^(1/4))/(((((864*a^13*b^4*c*d^13 - 32*a^3*b^14*c^11*d^3 + 1984*a^4*b^13*c^10*d

^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 - 37184*

a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11 - 5184*a^12*b^5*c^2*d^12)*1i)/(a^7*d^7 - b^7

*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a

^6*b*c*d^6) + (x^(1/2)*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*

a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 + 114688*a^2*b^6*c^7*d^5 - 229376*a

^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^6*b^2*c^3*d^9))^(1/4)*(2304*a^13*b

^4*c*d^14 + 4352*a^3*b^14*c^11*d^4 - 33280*a^4*b^13*c^10*d^5 + 111872*a^5*b^12*c^9*d^6 - 219136*a^6*b^11*c^8*d

^7 + 283136*a^7*b^10*c^7*d^8 - 265216*a^8*b^9*c^6*d^9 + 197120*a^9*b^8*c^5*d^10 - 120832*a^10*b^7*c^4*d^11 + 5

6576*a^11*b^6*c^3*d^12 - 16896*a^12*b^5*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))*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a*b

^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10 +

114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*a^

6*b^2*c^3*d^9))^(3/4)*1i + (x^(1/2)*(a^3*b^10*c^6*d + 144*a^8*b^5*c*d^6 + 12*a^4*b^9*c^5*d^2 + 54*a^5*b^8*c^4*

d^3 + 124*a^6*b^7*c^3*d^4 + 177*a^7*b^6*c^2*d^5)*1i)/(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))*(-(81*a^4*d^4 + b^4*c^4 + 54*a^2*b^2*c^2*d^2 + 12*a

*b^3*c^3*d + 108*a^3*b*c*d^3)/(4096*a^8*c*d^11 + 4096*b^8*c^9*d^3 - 32768*a*b^7*c^8*d^4 - 32768*a^7*b*c^2*d^10

+ 114688*a^2*b^6*c^7*d^5 - 229376*a^3*b^5*c^6*d^6 + 286720*a^4*b^4*c^5*d^7 - 229376*a^5*b^3*c^4*d^8 + 114688*

a^6*b^2*c^3*d^9))^(1/4) - (a^4*b^9*c^5*d + 108*a^8*b^5*c*d^5 + 13*a^5*b^8*c^4*d^2 + 63*a^6*b^7*c^3*d^3 + 135*a

^7*b^6*c^2*d^4)/(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2

*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6) + ((((864*a^13*b^4*c*d^13 - 32*a^3*b^14*c^11*d^3 + 1984*a^4*b^13*c^1

0*d^4 - 13856*a^5*b^12*c^9*d^5 + 43264*a^6*b^11*c^8*d^6 - 74816*a^7*b^10*c^7*d^7 + 74368*a^8*b^9*c^6*d^8 - 371

84*a^9*b^8*c^5*d^9 + 256*a^10*b^7*c^4*d^10 + 10336*a^11*b^6*c^3*d^11 - 5184*a^12*b^5*c^2*d^12)*1i)/(a^7*d^7 -