∫ x7∕2 _________________________

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

Optimal. Leaf size=532 \[ -\frac {a^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{b} (b c-a d)^2}+\frac {a^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{b} (b c-a d)^2}-\frac {a^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{b} (b c-a d)^2}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{b} (b c-a d)^2}-\frac {\sqrt [4]{c} (b c-5 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{5/4} (b c-a d)^2}+\frac {\sqrt [4]{c} (b c-5 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{5/4} (b c-a d)^2}-\frac {\sqrt [4]{c} (b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{5/4} (b c-a d)^2}+\frac {\sqrt [4]{c} (b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} d^{5/4} (b c-a d)^2}-\frac {c \sqrt {x}}{2 d \left (c+d x^2\right ) (b c-a d)} \]

[Out]

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

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

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

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

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

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

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

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

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Rubi [A] time = 0.53, antiderivative size = 532, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {466, 470, 522, 211, 1165, 628, 1162, 617, 204} \[ -\frac {a^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{b} (b c-a d)^2}+\frac {a^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{b} (b c-a d)^2}-\frac {a^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{b} (b c-a d)^2}+\frac {a^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{b} (b c-a d)^2}-\frac {\sqrt [4]{c} (b c-5 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{5/4} (b c-a d)^2}+\frac {\sqrt [4]{c} (b c-5 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} d^{5/4} (b c-a d)^2}-\frac {\sqrt [4]{c} (b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{5/4} (b c-a d)^2}+\frac {\sqrt [4]{c} (b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} d^{5/4} (b c-a d)^2}-\frac {c \sqrt {x}}{2 d \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

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

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

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

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

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

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Mathematica [A] time = 0.30, size = 523, normalized size = 0.98 \[ \frac {-4 \sqrt {2} a^{5/4} d^{5/4} \left (c+d x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+4 \sqrt {2} a^{5/4} d^{5/4} \left (c+d x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-8 \sqrt {2} a^{5/4} d^{5/4} \left (c+d x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )+8 \sqrt {2} a^{5/4} d^{5/4} \left (c+d x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \left (c+d x^2\right ) (b c-5 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \left (c+d x^2\right ) (b c-5 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )-2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \left (c+d x^2\right ) (b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )+2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \left (c+d x^2\right ) (b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )-8 \sqrt [4]{b} c \sqrt [4]{d} \sqrt {x} (b c-a d)}{16 \sqrt [4]{b} d^{5/4} \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

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

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

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

)

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fricas [B] time = 9.99, size = 3224, normalized size = 6.06 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*(b*c^2*d - a*c*d^2 + (b*c*d^2 - a*d^3)*x^2)*(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2*c^3*d^2 - 500*a^

3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 28*a^2*b^6*c^6*d^7 - 56*a^3*b^5*c^5*d^8 + 70*a^4

*b^4*c^4*d^9 - 56*a^5*b^3*c^3*d^10 + 28*a^6*b^2*c^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13))^(1/4)*arctan(((b^6*c^6*

d^4 - 6*a*b^5*c^5*d^5 + 15*a^2*b^4*c^4*d^6 - 20*a^3*b^3*c^3*d^7 + 15*a^4*b^2*c^2*d^8 - 6*a^5*b*c*d^9 + a^6*d^1

0)*sqrt((b^2*c^2 - 10*a*b*c*d + 25*a^2*d^2)*x + (b^4*c^4*d^2 - 4*a*b^3*c^3*d^3 + 6*a^2*b^2*c^2*d^4 - 4*a^3*b*c

*d^5 + a^4*d^6)*sqrt(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2*c^3*d^2 - 500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^

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

10 + 28*a^6*b^2*c^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13)))*(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2*c^3*d^2 - 50

0*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 28*a^2*b^6*c^6*d^7 - 56*a^3*b^5*c^5*d^8 + 70

*a^4*b^4*c^4*d^9 - 56*a^5*b^3*c^3*d^10 + 28*a^6*b^2*c^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13))^(3/4) + (b^7*c^7*d^

4 - 11*a*b^6*c^6*d^5 + 45*a^2*b^5*c^5*d^6 - 95*a^3*b^4*c^4*d^7 + 115*a^4*b^3*c^3*d^8 - 81*a^5*b^2*c^2*d^9 + 31

*a^6*b*c*d^10 - 5*a^7*d^11)*sqrt(x)*(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2*c^3*d^2 - 500*a^3*b*c^2*d^3 + 62

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

6*a^5*b^3*c^3*d^10 + 28*a^6*b^2*c^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13))^(3/4))/(b^4*c^5 - 20*a*b^3*c^4*d + 150*

a^2*b^2*c^3*d^2 - 500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)) - 16*(-a^5/(b^9*c^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2

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

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

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

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

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

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

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

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

*a^7*b^2*c*d^7 + a^8*b*d^8))^(3/4)*sqrt(x))/a^5) + (b*c^2*d - a*c*d^2 + (b*c*d^2 - a*d^3)*x^2)*(-(b^4*c^5 - 20

*a*b^3*c^4*d + 150*a^2*b^2*c^3*d^2 - 500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 28*a^

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

c*d^12 + a^8*d^13))^(1/4)*log(-(b*c - 5*a*d)*sqrt(x) + (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(-(b^4*c^5 - 20*a*b

^3*c^4*d + 150*a^2*b^2*c^3*d^2 - 500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 28*a^2*b^

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

12 + a^8*d^13))^(1/4)) - (b*c^2*d - a*c*d^2 + (b*c*d^2 - a*d^3)*x^2)*(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2

*c^3*d^2 - 500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 28*a^2*b^6*c^6*d^7 - 56*a^3*b^5

*c^5*d^8 + 70*a^4*b^4*c^4*d^9 - 56*a^5*b^3*c^3*d^10 + 28*a^6*b^2*c^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13))^(1/4)*

log(-(b*c - 5*a*d)*sqrt(x) - (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(-(b^4*c^5 - 20*a*b^3*c^4*d + 150*a^2*b^2*c^3

*d^2 - 500*a^3*b*c^2*d^3 + 625*a^4*c*d^4)/(b^8*c^8*d^5 - 8*a*b^7*c^7*d^6 + 28*a^2*b^6*c^6*d^7 - 56*a^3*b^5*c^5

*d^8 + 70*a^4*b^4*c^4*d^9 - 56*a^5*b^3*c^3*d^10 + 28*a^6*b^2*c^2*d^11 - 8*a^7*b*c*d^12 + a^8*d^13))^(1/4)) - 4

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

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

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

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

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

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

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

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

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

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giac [A] time = 0.98, size = 669, normalized size = 1.26 \[ \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \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^{3} c^{2} - 2 \, \sqrt {2} a b^{2} c d + \sqrt {2} a^{2} b d^{2}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \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^{3} c^{2} - 2 \, \sqrt {2} a b^{2} c d + \sqrt {2} a^{2} b d^{2}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \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^{3} c^{2} - 2 \, \sqrt {2} a b^{2} c d + \sqrt {2} a^{2} b d^{2}\right )}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} a \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^{3} c^{2} - 2 \, \sqrt {2} a b^{2} c d + \sqrt {2} a^{2} b d^{2}\right )}} + \frac {{\left (\left (c d^{3}\right )^{\frac {1}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{2} d^{2} - 2 \, \sqrt {2} a b c d^{3} + \sqrt {2} a^{2} d^{4}\right )}} + \frac {{\left (\left (c d^{3}\right )^{\frac {1}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{2} d^{2} - 2 \, \sqrt {2} a b c d^{3} + \sqrt {2} a^{2} d^{4}\right )}} + \frac {{\left (\left (c d^{3}\right )^{\frac {1}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{2} d^{2} - 2 \, \sqrt {2} a b c d^{3} + \sqrt {2} a^{2} d^{4}\right )}} - \frac {{\left (\left (c d^{3}\right )^{\frac {1}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{2} d^{2} - 2 \, \sqrt {2} a b c d^{3} + \sqrt {2} a^{2} d^{4}\right )}} - \frac {c \sqrt {x}}{2 \, {\left (b c d - a d^{2}\right )} {\left (d x^{2} + c\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

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

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

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

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

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

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

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maple [A] time = 0.02, size = 533, normalized size = 1.00 \[ \frac {a c \sqrt {x}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}-\frac {b \,c^{2} \sqrt {x}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right ) d}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \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}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \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}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \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}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \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}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \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}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \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}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \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} d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \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} d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \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} d} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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

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

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

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

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

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

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maxima [A] time = 2.47, size = 468, normalized size = 0.88 \[ \frac {{\left (\frac {2 \, \sqrt {2} {\left (b c - 5 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b c - 5 \, a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b c - 5 \, a d\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b c - 5 \, a d\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}\right )} c}{16 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}} - \frac {c \sqrt {x}}{2 \, {\left (b c^{2} d - a c d^{2} + {\left (b c d^{2} - a d^{3}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} a^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}}}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

- 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(b*c - 5*a*d)*log(sqrt(

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

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

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

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

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

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

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

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mupad [B] time = 2.48, size = 21485, normalized size = 40.39 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

*a^4*b^5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*((2*(a^3*b^8*c^7 - 19*a

^4*b^7*c^6*d + 131*a^5*b^6*c^5*d^2 - 369*a^6*b^5*c^4*d^3 + 256*a^7*b^4*c^3*d^4 + 320*a^8*b^3*c^2*d^5))/(a^3*d^

4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) + ((2*(-a^5/(16*b^9*c^8 + 16*a^8*b*d^8 - 128*a^7*b^2*c*d^7 +

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

128*a*b^8*c^7*d))^(1/4)*(5120*a^3*b^12*c^10*d^5 - 40960*a^4*b^11*c^9*d^6 + 143360*a^5*b^10*c^8*d^7 - 286720*a

^6*b^9*c^7*d^8 + 358400*a^7*b^8*c^6*d^9 - 286720*a^8*b^7*c^5*d^10 + 143360*a^9*b^6*c^4*d^11 - 40960*a^10*b^5*c

^3*d^12 + 5120*a^11*b^4*c^2*d^13))/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) + (x^(1/2)*(256*a^3

*b^14*c^12*d^4 - 512*a^4*b^13*c^11*d^5 + 1280*a^5*b^12*c^10*d^6 - 22528*a^6*b^11*c^9*d^7 + 111104*a^7*b^10*c^8

*d^8 - 265216*a^8*b^9*c^7*d^9 + 369152*a^9*b^8*c^6*d^10 - 317440*a^10*b^7*c^5*d^11 + 167168*a^11*b^6*c^4*d^12

- 49664*a^12*b^5*c^3*d^13 + 6400*a^13*b^4*c^2*d^14))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d

^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6))*(-a^5/(16*b^9*c^8 + 16*a^8*b*d^8 - 128*a^7*b^2*

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

^2*d^6 - 128*a*b^8*c^7*d))^(3/4)) + (x^(1/2)*(a^4*b^9*c^8 - 20*a^5*b^8*c^7*d + 150*a^6*b^7*c^6*d^2 - 500*a^7*b

^6*c^5*d^3 + 641*a^8*b^5*c^4*d^4 - 160*a^9*b^4*c^3*d^5 + 400*a^10*b^3*c^2*d^6))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5

*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6))*(-a^5/(16*b^9*c^8 +

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

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

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

3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*((2*(a^3*b^8*c^7 - 19*a^4*b^7*c^6*d + 131*a^5*b^6*c^5*d^2 - 369*a^6*b^5*c^

4*d^3 + 256*a^7*b^4*c^3*d^4 + 320*a^8*b^3*c^2*d^5))/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) +

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

4*b^5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*(5120*a^3*b^12*c^10*d^5 -

40960*a^4*b^11*c^9*d^6 + 143360*a^5*b^10*c^8*d^7 - 286720*a^6*b^9*c^7*d^8 + 358400*a^7*b^8*c^6*d^9 - 286720*a^

8*b^7*c^5*d^10 + 143360*a^9*b^6*c^4*d^11 - 40960*a^10*b^5*c^3*d^12 + 5120*a^11*b^4*c^2*d^13))/(a^3*d^4 - b^3*c

^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) - (x^(1/2)*(256*a^3*b^14*c^12*d^4 - 512*a^4*b^13*c^11*d^5 + 1280*a^5*b

^12*c^10*d^6 - 22528*a^6*b^11*c^9*d^7 + 111104*a^7*b^10*c^8*d^8 - 265216*a^8*b^9*c^7*d^9 + 369152*a^9*b^8*c^6*

d^10 - 317440*a^10*b^7*c^5*d^11 + 167168*a^11*b^6*c^4*d^12 - 49664*a^12*b^5*c^3*d^13 + 6400*a^13*b^4*c^2*d^14)

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

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

1120*a^4*b^5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(3/4)) - (x^(1/2)*(a^4*b

^9*c^8 - 20*a^5*b^8*c^7*d + 150*a^6*b^7*c^6*d^2 - 500*a^7*b^6*c^5*d^3 + 641*a^8*b^5*c^4*d^4 - 160*a^9*b^4*c^3*

d^5 + 400*a^10*b^3*c^2*d^6))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4

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

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

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

3 + 1120*a^4*b^5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*((2*(a^3*b^8*c^

7 - 19*a^4*b^7*c^6*d + 131*a^5*b^6*c^5*d^2 - 369*a^6*b^5*c^4*d^3 + 256*a^7*b^4*c^3*d^4 + 320*a^8*b^3*c^2*d^5))

/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) + ((2*(-a^5/(16*b^9*c^8 + 16*a^8*b*d^8 - 128*a^7*b^2*

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

^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*(5120*a^3*b^12*c^10*d^5 - 40960*a^4*b^11*c^9*d^6 + 143360*a^5*b^10*c^8*d^7 -

286720*a^6*b^9*c^7*d^8 + 358400*a^7*b^8*c^6*d^9 - 286720*a^8*b^7*c^5*d^10 + 143360*a^9*b^6*c^4*d^11 - 40960*a^

10*b^5*c^3*d^12 + 5120*a^11*b^4*c^2*d^13))/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) + (x^(1/2)*

(256*a^3*b^14*c^12*d^4 - 512*a^4*b^13*c^11*d^5 + 1280*a^5*b^12*c^10*d^6 - 22528*a^6*b^11*c^9*d^7 + 111104*a^7*

b^10*c^8*d^8 - 265216*a^8*b^9*c^7*d^9 + 369152*a^9*b^8*c^6*d^10 - 317440*a^10*b^7*c^5*d^11 + 167168*a^11*b^6*c

^4*d^12 - 49664*a^12*b^5*c^3*d^13 + 6400*a^13*b^4*c^2*d^14))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b

^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6))*(-a^5/(16*b^9*c^8 + 16*a^8*b*d^8 - 128*

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

^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(3/4)) + (x^(1/2)*(a^4*b^9*c^8 - 20*a^5*b^8*c^7*d + 150*a^6*b^7*c^6*d^2 - 5

00*a^7*b^6*c^5*d^3 + 641*a^8*b^5*c^4*d^4 - 160*a^9*b^4*c^3*d^5 + 400*a^10*b^3*c^2*d^6))/(a^6*d^7 + b^6*c^6*d -

6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6))*(-a^5/(16*b^

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

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

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

^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*((2*(a^3*b^8*c^7 - 19*a^4*b^7*c^6*d + 131*a^5*b^6*c^5*d^2 - 369*a^6*b

^5*c^4*d^3 + 256*a^7*b^4*c^3*d^4 + 320*a^8*b^3*c^2*d^5))/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^

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

20*a^4*b^5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*(5120*a^3*b^12*c^10*d

^5 - 40960*a^4*b^11*c^9*d^6 + 143360*a^5*b^10*c^8*d^7 - 286720*a^6*b^9*c^7*d^8 + 358400*a^7*b^8*c^6*d^9 - 2867

20*a^8*b^7*c^5*d^10 + 143360*a^9*b^6*c^4*d^11 - 40960*a^10*b^5*c^3*d^12 + 5120*a^11*b^4*c^2*d^13))/(a^3*d^4 -

b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) - (x^(1/2)*(256*a^3*b^14*c^12*d^4 - 512*a^4*b^13*c^11*d^5 + 1280*

a^5*b^12*c^10*d^6 - 22528*a^6*b^11*c^9*d^7 + 111104*a^7*b^10*c^8*d^8 - 265216*a^8*b^9*c^7*d^9 + 369152*a^9*b^8

*c^6*d^10 - 317440*a^10*b^7*c^5*d^11 + 167168*a^11*b^6*c^4*d^12 - 49664*a^12*b^5*c^3*d^13 + 6400*a^13*b^4*c^2*

d^14))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 -

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

d^3 + 1120*a^4*b^5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(3/4)) - (x^(1/2)*(

a^4*b^9*c^8 - 20*a^5*b^8*c^7*d + 150*a^6*b^7*c^6*d^2 - 500*a^7*b^6*c^5*d^3 + 641*a^8*b^5*c^4*d^4 - 160*a^9*b^4

*c^3*d^5 + 400*a^10*b^3*c^2*d^6))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3

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

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

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

^3 + 1120*a^4*b^5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*2i + 2*atan(((

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

5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*((2*(a^3*b^8*c^7 - 19*a^4*b^7*

c^6*d + 131*a^5*b^6*c^5*d^2 - 369*a^6*b^5*c^4*d^3 + 256*a^7*b^4*c^3*d^4 + 320*a^8*b^3*c^2*d^5))/(a^3*d^4 - b^3

*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) - (((-a^5/(16*b^9*c^8 + 16*a^8*b*d^8 - 128*a^7*b^2*c*d^7 + 448*a^2*b

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

8*c^7*d))^(1/4)*(5120*a^3*b^12*c^10*d^5 - 40960*a^4*b^11*c^9*d^6 + 143360*a^5*b^10*c^8*d^7 - 286720*a^6*b^9*c^

7*d^8 + 358400*a^7*b^8*c^6*d^9 - 286720*a^8*b^7*c^5*d^10 + 143360*a^9*b^6*c^4*d^11 - 40960*a^10*b^5*c^3*d^12 +

5120*a^11*b^4*c^2*d^13)*2i)/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) + (x^(1/2)*(256*a^3*b^14*

c^12*d^4 - 512*a^4*b^13*c^11*d^5 + 1280*a^5*b^12*c^10*d^6 - 22528*a^6*b^11*c^9*d^7 + 111104*a^7*b^10*c^8*d^8 -

265216*a^8*b^9*c^7*d^9 + 369152*a^9*b^8*c^6*d^10 - 317440*a^10*b^7*c^5*d^11 + 167168*a^11*b^6*c^4*d^12 - 4966

4*a^12*b^5*c^3*d^13 + 6400*a^13*b^4*c^2*d^14))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 2

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

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

- 128*a*b^8*c^7*d))^(3/4)*1i)*1i + (x^(1/2)*(a^4*b^9*c^8 - 20*a^5*b^8*c^7*d + 150*a^6*b^7*c^6*d^2 - 500*a^7*b

^6*c^5*d^3 + 641*a^8*b^5*c^4*d^4 - 160*a^9*b^4*c^3*d^5 + 400*a^10*b^3*c^2*d^6))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5

*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6))*(-a^5/(16*b^9*c^8 +

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

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

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

^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*((2*(a^3*b^8*c^7 - 19*a^4*b^7*c^6*d + 131*a^5*b^6*c^5*d^2 - 369*a^6*b^5*c^4*d

^3 + 256*a^7*b^4*c^3*d^4 + 320*a^8*b^3*c^2*d^5))/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) - (((

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

*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*(5120*a^3*b^12*c^10*d^5 - 40960

*a^4*b^11*c^9*d^6 + 143360*a^5*b^10*c^8*d^7 - 286720*a^6*b^9*c^7*d^8 + 358400*a^7*b^8*c^6*d^9 - 286720*a^8*b^7

*c^5*d^10 + 143360*a^9*b^6*c^4*d^11 - 40960*a^10*b^5*c^3*d^12 + 5120*a^11*b^4*c^2*d^13)*2i)/(a^3*d^4 - b^3*c^3

*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) - (x^(1/2)*(256*a^3*b^14*c^12*d^4 - 512*a^4*b^13*c^11*d^5 + 1280*a^5*b^1

2*c^10*d^6 - 22528*a^6*b^11*c^9*d^7 + 111104*a^7*b^10*c^8*d^8 - 265216*a^8*b^9*c^7*d^9 + 369152*a^9*b^8*c^6*d^

10 - 317440*a^10*b^7*c^5*d^11 + 167168*a^11*b^6*c^4*d^12 - 49664*a^12*b^5*c^3*d^13 + 6400*a^13*b^4*c^2*d^14))/

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

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

120*a^4*b^5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(3/4)*1i)*1i - (x^(1/2)*(a

^4*b^9*c^8 - 20*a^5*b^8*c^7*d + 150*a^6*b^7*c^6*d^2 - 500*a^7*b^6*c^5*d^3 + 641*a^8*b^5*c^4*d^4 - 160*a^9*b^4*

c^3*d^5 + 400*a^10*b^3*c^2*d^6))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*

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

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

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

^3 + 1120*a^4*b^5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*((2*(a^3*b^8*c

^7 - 19*a^4*b^7*c^6*d + 131*a^5*b^6*c^5*d^2 - 369*a^6*b^5*c^4*d^3 + 256*a^7*b^4*c^3*d^4 + 320*a^8*b^3*c^2*d^5)

)/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) - (((-a^5/(16*b^9*c^8 + 16*a^8*b*d^8 - 128*a^7*b^2*c

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

2*d^6 - 128*a*b^8*c^7*d))^(1/4)*(5120*a^3*b^12*c^10*d^5 - 40960*a^4*b^11*c^9*d^6 + 143360*a^5*b^10*c^8*d^7 - 2

86720*a^6*b^9*c^7*d^8 + 358400*a^7*b^8*c^6*d^9 - 286720*a^8*b^7*c^5*d^10 + 143360*a^9*b^6*c^4*d^11 - 40960*a^1

0*b^5*c^3*d^12 + 5120*a^11*b^4*c^2*d^13)*2i)/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) + (x^(1/2

)*(256*a^3*b^14*c^12*d^4 - 512*a^4*b^13*c^11*d^5 + 1280*a^5*b^12*c^10*d^6 - 22528*a^6*b^11*c^9*d^7 + 111104*a^

7*b^10*c^8*d^8 - 265216*a^8*b^9*c^7*d^9 + 369152*a^9*b^8*c^6*d^10 - 317440*a^10*b^7*c^5*d^11 + 167168*a^11*b^6

*c^4*d^12 - 49664*a^12*b^5*c^3*d^13 + 6400*a^13*b^4*c^2*d^14))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2

*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6))*(-a^5/(16*b^9*c^8 + 16*a^8*b*d^8 - 12

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

*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(3/4)*1i)*1i + (x^(1/2)*(a^4*b^9*c^8 - 20*a^5*b^8*c^7*d + 150*a^6*b^7*c^6

*d^2 - 500*a^7*b^6*c^5*d^3 + 641*a^8*b^5*c^4*d^4 - 160*a^9*b^4*c^3*d^5 + 400*a^10*b^3*c^2*d^6))/(a^6*d^7 + b^6

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

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

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

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

d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*((2*(a^3*b^8*c^7 - 19*a^4*b^7*c^6*d + 131*a^5*b^6*c^5*d^2

- 369*a^6*b^5*c^4*d^3 + 256*a^7*b^4*c^3*d^4 + 320*a^8*b^3*c^2*d^5))/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3

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

*d^3 + 1120*a^4*b^5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(1/4)*(5120*a^3*b^

12*c^10*d^5 - 40960*a^4*b^11*c^9*d^6 + 143360*a^5*b^10*c^8*d^7 - 286720*a^6*b^9*c^7*d^8 + 358400*a^7*b^8*c^6*d

^9 - 286720*a^8*b^7*c^5*d^10 + 143360*a^9*b^6*c^4*d^11 - 40960*a^10*b^5*c^3*d^12 + 5120*a^11*b^4*c^2*d^13)*2i)

/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) - (x^(1/2)*(256*a^3*b^14*c^12*d^4 - 512*a^4*b^13*c^11

*d^5 + 1280*a^5*b^12*c^10*d^6 - 22528*a^6*b^11*c^9*d^7 + 111104*a^7*b^10*c^8*d^8 - 265216*a^8*b^9*c^7*d^9 + 36

9152*a^9*b^8*c^6*d^10 - 317440*a^10*b^7*c^5*d^11 + 167168*a^11*b^6*c^4*d^12 - 49664*a^12*b^5*c^3*d^13 + 6400*a

^13*b^4*c^2*d^14))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b

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

a^3*b^6*c^5*d^3 + 1120*a^4*b^5*c^4*d^4 - 896*a^5*b^4*c^3*d^5 + 448*a^6*b^3*c^2*d^6 - 128*a*b^8*c^7*d))^(3/4)*1

i)*1i - (x^(1/2)*(a^4*b^9*c^8 - 20*a^5*b^8*c^7*d + 150*a^6*b^7*c^6*d^2 - 500*a^7*b^6*c^5*d^3 + 641*a^8*b^5*c^4

*d^4 - 160*a^9*b^4*c^3*d^5 + 400*a^10*b^3*c^2*d^6))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^

3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6))*(-a^5/(16*b^9*c^8 + 16*a^8*b*d^8 - 128*a^7*b^2*c

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

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

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

^(1/4) + atan((((((2*(-(b^4*c^5 + 625*a^4*c*d^4 - 500*a^3*b*c^2*d^3 + 150*a^2*b^2*c^3*d^2 - 20*a*b^3*c^4*d)/(4

096*a^8*d^13 + 4096*b^8*c^8*d^5 - 32768*a*b^7*c^7*d^6 + 114688*a^2*b^6*c^6*d^7 - 229376*a^3*b^5*c^5*d^8 + 2867

20*a^4*b^4*c^4*d^9 - 229376*a^5*b^3*c^3*d^10 + 114688*a^6*b^2*c^2*d^11 - 32768*a^7*b*c*d^12))^(1/4)*(5120*a^3*

b^12*c^10*d^5 - 40960*a^4*b^11*c^9*d^6 + 143360*a^5*b^10*c^8*d^7 - 286720*a^6*b^9*c^7*d^8 + 358400*a^7*b^8*c^6

*d^9 - 286720*a^8*b^7*c^5*d^10 + 143360*a^9*b^6*c^4*d^11 - 40960*a^10*b^5*c^3*d^12 + 5120*a^11*b^4*c^2*d^13))/

(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) + (x^(1/2)*(256*a^3*b^14*c^12*d^4 - 512*a^4*b^13*c^11*

d^5 + 1280*a^5*b^12*c^10*d^6 - 22528*a^6*b^11*c^9*d^7 + 111104*a^7*b^10*c^8*d^8 - 265216*a^8*b^9*c^7*d^9 + 369

152*a^9*b^8*c^6*d^10 - 317440*a^10*b^7*c^5*d^11 + 167168*a^11*b^6*c^4*d^12 - 49664*a^12*b^5*c^3*d^13 + 6400*a^

13*b^4*c^2*d^14))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^

2*c^2*d^5 - 6*a^5*b*c*d^6))*(-(b^4*c^5 + 625*a^4*c*d^4 - 500*a^3*b*c^2*d^3 + 150*a^2*b^2*c^3*d^2 - 20*a*b^3*c^

4*d)/(4096*a^8*d^13 + 4096*b^8*c^8*d^5 - 32768*a*b^7*c^7*d^6 + 114688*a^2*b^6*c^6*d^7 - 229376*a^3*b^5*c^5*d^8

+ 286720*a^4*b^4*c^4*d^9 - 229376*a^5*b^3*c^3*d^10 + 114688*a^6*b^2*c^2*d^11 - 32768*a^7*b*c*d^12))^(3/4) + (

2*(a^3*b^8*c^7 - 19*a^4*b^7*c^6*d + 131*a^5*b^6*c^5*d^2 - 369*a^6*b^5*c^4*d^3 + 256*a^7*b^4*c^3*d^4 + 320*a^8*

b^3*c^2*d^5))/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3))*(-(b^4*c^5 + 625*a^4*c*d^4 - 500*a^3*b*

c^2*d^3 + 150*a^2*b^2*c^3*d^2 - 20*a*b^3*c^4*d)/(4096*a^8*d^13 + 4096*b^8*c^8*d^5 - 32768*a*b^7*c^7*d^6 + 1146

88*a^2*b^6*c^6*d^7 - 229376*a^3*b^5*c^5*d^8 + 286720*a^4*b^4*c^4*d^9 - 229376*a^5*b^3*c^3*d^10 + 114688*a^6*b^

2*c^2*d^11 - 32768*a^7*b*c*d^12))^(1/4) + (x^(1/2)*(a^4*b^9*c^8 - 20*a^5*b^8*c^7*d + 150*a^6*b^7*c^6*d^2 - 500

*a^7*b^6*c^5*d^3 + 641*a^8*b^5*c^4*d^4 - 160*a^9*b^4*c^3*d^5 + 400*a^10*b^3*c^2*d^6))/(a^6*d^7 + b^6*c^6*d - 6

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

25*a^4*c*d^4 - 500*a^3*b*c^2*d^3 + 150*a^2*b^2*c^3*d^2 - 20*a*b^3*c^4*d)/(4096*a^8*d^13 + 4096*b^8*c^8*d^5 - 3

2768*a*b^7*c^7*d^6 + 114688*a^2*b^6*c^6*d^7 - 229376*a^3*b^5*c^5*d^8 + 286720*a^4*b^4*c^4*d^9 - 229376*a^5*b^3

*c^3*d^10 + 114688*a^6*b^2*c^2*d^11 - 32768*a^7*b*c*d^12))^(1/4)*1i - ((((2*(-(b^4*c^5 + 625*a^4*c*d^4 - 500*a

^3*b*c^2*d^3 + 150*a^2*b^2*c^3*d^2 - 20*a*b^3*c^4*d)/(4096*a^8*d^13 + 4096*b^8*c^8*d^5 - 32768*a*b^7*c^7*d^6 +

114688*a^2*b^6*c^6*d^7 - 229376*a^3*b^5*c^5*d^8 + 286720*a^4*b^4*c^4*d^9 - 229376*a^5*b^3*c^3*d^10 + 114688*a

^6*b^2*c^2*d^11 - 32768*a^7*b*c*d^12))^(1/4)*(5120*a^3*b^12*c^10*d^5 - 40960*a^4*b^11*c^9*d^6 + 143360*a^5*b^1

0*c^8*d^7 - 286720*a^6*b^9*c^7*d^8 + 358400*a^7*b^8*c^6*d^9 - 286720*a^8*b^7*c^5*d^10 + 143360*a^9*b^6*c^4*d^1

1 - 40960*a^10*b^5*c^3*d^12 + 5120*a^11*b^4*c^2*d^13))/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)

- (x^(1/2)*(256*a^3*b^14*c^12*d^4 - 512*a^4*b^13*c^11*d^5 + 1280*a^5*b^12*c^10*d^6 - 22528*a^6*b^11*c^9*d^7 +

111104*a^7*b^10*c^8*d^8 - 265216*a^8*b^9*c^7*d^9 + 369152*a^9*b^8*c^6*d^10 - 317440*a^10*b^7*c^5*d^11 + 16716

8*a^11*b^6*c^4*d^12 - 49664*a^12*b^5*c^3*d^13 + 6400*a^13*b^4*c^2*d^14))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^

2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6))*(-(b^4*c^5 + 625*a^4*c*d^4

- 500*a^3*b*c^2*d^3 + 150*a^2*b^2*c^3*d^2 - 20*a*b^3*c^4*d)/(4096*a^8*d^13 + 4096*b^8*c^8*d^5 - 32768*a*b^7*c^

7*d^6 + 114688*a^2*b^6*c^6*d^7 - 229376*a^3*b^5*c^5*d^8 + 286720*a^4*b^4*c^4*d^9 - 229376*a^5*b^3*c^3*d^10 + 1

14688*a^6*b^2*c^2*d^11 - 32768*a^7*b*c*d^12))^(3/4) + (2*(a^3*b^8*c^7 - 19*a^4*b^7*c^6*d + 131*a^5*b^6*c^5*d^2

- 369*a^6*b^5*c^4*d^3 + 256*a^7*b^4*c^3*d^4 + 320*a^8*b^3*c^2*d^5))/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 -

3*a^2*b*c*d^3))*(-(b^4*c^5 + 625*a^4*c*d^4 - 500*a^3*b*c^2*d^3 + 150*a^2*b^2*c^3*d^2 - 20*a*b^3*c^4*d)/(4096*a

^8*d^13 + 4096*b^8*c^8*d^5 - 32768*a*b^7*c^7*d^6 + 114688*a^2*b^6*c^6*d^7 - 229376*a^3*b^5*c^5*d^8 + 286720*a^

4*b^4*c^4*d^9 - 229376*a^5*b^3*c^3*d^10 + 114688*a^6*b^2*c^2*d^11 - 32768*a^7*b*c*d^12))^(1/4) - (x^(1/2)*(a^4

*b^9*c^8 - 20*a^5*b^8*c^7*d + 150*a^6*b^7*c^6*d^2 - 500*a^7*b^6*c^5*d^3 + 641*a^8*b^5*c^4*d^4 - 160*a^9*b^4*c^

3*d^5 + 400*a^10*b^3*c^2*d^6))/(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^

4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6))*(-(b^4*c^5 + 625*a^4*c*d^4 - 500*a^3*b*c^2*d^3 + 150*a^2*b^2*c^3*d^2

- 20*a*b^3*c^4*d)/(4096*a^8*d^13 + 4096*b^8*c^8*d^5 - 32768*a*b^7*c^7*d^6 + 114688*a^2*b^6*c^6*d^7 - 229376*a^

3*b^5*c^5*d^8 + 286720*a^4*b^4*c^4*d^9 - 229376*a^5*b^3*c^3*d^10 + 114688*a^6*b^2*c^2*d^11 - 32768*a^7*b*c*d^1

2))^(1/4)*1i)/(((((2*(-(b^4*c^5 + 625*a^4*c*d^4 - 500*a^3*b*c^2*d^3 + 150*a^2*b^2*c^3*d^2 - 20*a*b^3*c^4*d)/(4