∫ 1_______√ x (a+bx2)(c+dx2)3 dx

3.484 \(\int \frac {1}{\sqrt {x} (a+b x^2) (c+d x^2)^3} \, dx\)

Optimal. Leaf size=633 \[ -\frac {b^{11/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}-\frac {b^{11/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}+\frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d \sqrt {x} (15 b c-7 a d)}{16 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {d \sqrt {x}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

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

1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/64*d^(3/4)*(21*a^2*d^2-66*a*b*c*d+77*b^2*c^2)*arc

tan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/64*d^(3/4)*(21*a^2*d^2-66*a*b*c*d+77*b^

2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/4*b^(11/4)*ln(a^(1/2)+x*b^(1/

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

1/4)*2^(1/2)*x^(1/2))/a^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/128*d^(3/4)*(21*a^2*d^2-66*a*b*c*d+77*b^2*c^2)*ln(c^(1/2)

+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/128*d^(3/4)*(21*a^2*d^2-66*a*b*c*d

+77*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/4*d*x^(1/2)

/c/(-a*d+b*c)/(d*x^2+c)^2-1/16*d*(-7*a*d+15*b*c)*x^(1/2)/c^2/(-a*d+b*c)^2/(d*x^2+c)

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Rubi [A] time = 0.83, antiderivative size = 633, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 414, 527, 522, 211, 1165, 628, 1162, 617, 204} \[ \frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}+\frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {b^{11/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}-\frac {b^{11/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}-\frac {d \sqrt {x} (15 b c-7 a d)}{16 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {d \sqrt {x}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(d*Sqrt[x])/(4*c*(b*c - a*d)*(c + d*x^2)^2) - (d*(15*b*c - 7*a*d)*Sqrt[x])/(16*c^2*(b*c - a*d)^2*(c + d*x^2))

- (b^(11/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (b^(11/4)*ArcTan

[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d +

21*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) - (d^(3/4)*(77

*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(11/4)*(b*c -

a*d)^3) - (b^(11/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*(b*c - a*d

)^3) + (b^(11/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*(b*c - a*d)^3)

+ (d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])

/(64*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) - (d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c

^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(11/4)*(b*c - a*d)^3)

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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(

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

(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,

n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial

Q[a, b, c, d, n, p, q, x]

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 522

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

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

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

Rule 527

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

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

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

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

Rule 617

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

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

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

Rule 628

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

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

Rule 1162

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

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

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

Rule 1165

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

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

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {8 b c-7 a d-7 b d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 c (b c-a d)}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {32 b^2 c^2-45 a b c d+21 a^2 d^2-3 b d (15 b c-7 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^2}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}-\frac {\left (d \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a} (b c-a d)^3}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a} (b c-a d)^3}-\frac {\left (d \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{5/2} (b c-a d)^3}-\frac {\left (d \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{5/2} (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {b^{5/2} \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 {a} (b c-a d)^3}+\frac {b^{5/2} \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 {a} (b c-a d)^3}-\frac {b^{11/4} \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} a^{3/4} (b c-a d)^3}-\frac {b^{11/4} \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} a^{3/4} (b c-a d)^3}-\frac {\left (\sqrt {d} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{5/2} (b c-a d)^3}-\frac {\left (\sqrt {d} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{5/2} (b c-a d)^3}+\frac {\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}+\frac {\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b^{11/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}+\frac {b^{11/4} \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} a^{3/4} (b c-a d)^3}-\frac {b^{11/4} \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} a^{3/4} (b c-a d)^3}-\frac {\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}+\frac {\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b^{11/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {b^{11/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A] time = 0.85, size = 620, normalized size = 0.98 \[ \frac {1}{128} \left (\frac {32 \sqrt {2} b^{11/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{3/4} (a d-b c)^3}+\frac {32 \sqrt {2} b^{11/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{3/4} (b c-a d)^3}+\frac {64 \sqrt {2} b^{11/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{3/4} (a d-b c)^3}-\frac {64 \sqrt {2} b^{11/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{3/4} (a d-b c)^3}+\frac {\sqrt {2} d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{11/4} (b c-a d)^3}+\frac {\sqrt {2} d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{11/4} (a d-b c)^3}+\frac {2 \sqrt {2} d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{11/4} (b c-a d)^3}-\frac {2 \sqrt {2} d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{11/4} (b c-a d)^3}+\frac {8 d \sqrt {x} (7 a d-15 b c)}{c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {32 d \sqrt {x}}{c \left (c+d x^2\right )^2 (b c-a d)}\right ) \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-32*d*Sqrt[x])/(c*(b*c - a*d)*(c + d*x^2)^2) + (8*d*(-15*b*c + 7*a*d)*Sqrt[x])/(c^2*(b*c - a*d)^2*(c + d*x^2

)) + (64*Sqrt[2]*b^(11/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*(-(b*c) + a*d)^3) - (64*Sqrt

[2]*b^(11/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*d^(3/4)*(7

7*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(11/4)*(b*c - a*d)^3) -

(2*Sqrt[2]*d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(

11/4)*(b*c - a*d)^3) + (32*Sqrt[2]*b^(11/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/

4)*(-(b*c) + a*d)^3) + (32*Sqrt[2]*b^(11/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/

4)*(b*c - a*d)^3) + (Sqrt[2]*d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1

/4)*Sqrt[x] + Sqrt[d]*x])/(c^(11/4)*(b*c - a*d)^3) + (Sqrt[2]*d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*L

og[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(11/4)*(-(b*c) + a*d)^3))/128

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(d*x^2+c)^3/x^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [A] time = 1.46, size = 960, normalized size = 1.52 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(d*x^2+c)^3/x^(1/2),x, algorithm="giac")

[Out]

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

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

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

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

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

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

*a^4*d^3) - 1/32*(77*(c*d^3)^(1/4)*b^2*c^2 - 66*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*s

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

*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/32*(77*(c*d^3)^(1/4)*b^2*c^2 - 66*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/

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

b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/64*(77*(c*d^3)^(1/4)*b^2*c^2 - 66*(c*d^3)^(1/4)

*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^6 - 3*sqr

t(2)*a*b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) + 1/64*(77*(c*d^3)^(1/4)*b^2*c^2 - 66*(c*d^3

)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^6

- 3*sqrt(2)*a*b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/16*(15*b*c*d^2*x^(5/2) - 7*a*d^3

*x^(5/2) + 19*b*c^2*d*sqrt(x) - 11*a*c*d^2*sqrt(x))/((b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*(d*x^2 + c)^2)

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maple [A] time = 0.02, size = 882, normalized size = 1.39 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)/(d*x^2+c)^3/x^(1/2),x)

[Out]

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

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

2*x^(5/2)*a^2-11/8*d^3/(a*d-b*c)^3/(d*x^2+c)^2/c*x^(5/2)*a*b+15/16*d^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(5/2)*b^2+11/

16*d^3/(a*d-b*c)^3/(d*x^2+c)^2/c*x^(1/2)*a^2-15/8*d^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(1/2)*a*b+19/16*d/(a*d-b*c)^3/

(d*x^2+c)^2*c*x^(1/2)*b^2+21/64*d^3/(a*d-b*c)^3/c^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*

a^2-33/32*d^2/(a*d-b*c)^3/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b+77/64*d/(a*d-b*c)^

3/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+21/64*d^3/(a*d-b*c)^3/c^3*(c/d)^(1/4)*2^(1/2

)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2-33/32*d^2/(a*d-b*c)^3/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)

^(1/4)*x^(1/2)-1)*a*b+77/64*d/(a*d-b*c)^3/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+21/1

28*d^3/(a*d-b*c)^3/c^3*(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^2-33/64*d^2/(a*d-b*c)^3/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*b+77/128*d/(a*d-b*c)^3/c*(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^2

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maxima [A] time = 2.61, size = 675, normalized size = 1.07 \[ -\frac {{\left (15 \, b c d^{2} - 7 \, a d^{3}\right )} x^{\frac {5}{2}} + {\left (19 \, b c^{2} d - 11 \, a c d^{2}\right )} \sqrt {x}}{16 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} b^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {11}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {11}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}}{4 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\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 (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\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 (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\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 (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(d*x^2+c)^3/x^(1/2),x, algorithm="maxima")

[Out]

-1/16*((15*b*c*d^2 - 7*a*d^3)*x^(5/2) + (19*b*c^2*d - 11*a*c*d^2)*sqrt(x))/(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^

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

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

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

(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(11/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sq

rt(b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^(11/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/a^(3

/4))/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - 1/128*(2*sqrt(2)*(77*b^2*c^2*d - 66*a*b*c*d^2 + 21*

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

(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(77*b^2*c^2*d - 66*a*b*c*d^2 + 21*a^2*d^3)*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)*(77*b^2*c^2*d -

66*a*b*c*d^2 + 21*a^2*d^3)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqr

t(2)*(77*b^2*c^2*d - 66*a*b*c*d^2 + 21*a^2*d^3)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c

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

________________________________________________________________________________________

mupad [B] time = 4.41, size = 36997, normalized size = 58.45 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(a + b*x^2)*(c + d*x^2)^3),x)

[Out]

atan(((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^

9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a

^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*((((194481*a^8*b^

8*d^14)/2048 + 1232*b^16*c^8*d^6 - (34792593*a*b^15*c^7*d^7)/2048 - (2250423*a^7*b^9*c*d^13)/2048 + (86420247*

a^2*b^14*c^6*d^8)/2048 - (106888869*a^3*b^13*c^5*d^9)/2048 + (80271027*a^4*b^12*c^4*d^10)/2048 - (38915667*a^5

*b^11*c^3*d^11)/2048 + (12127941*a^6*b^10*c^2*d^12)/2048)/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b

^6*c^14*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*

c^15*d) + ((x^(1/2)*(16777216*b^23*c^23*d^4 - 201326592*a*b^22*c^22*d^5 + 1107296256*a^2*b^21*c^21*d^6 - 35938

46784*a^3*b^20*c^20*d^7 + 6972506112*a^4*b^19*c^19*d^8 - 4753588224*a^5*b^18*c^18*d^9 - 18397265920*a^6*b^17*c

^17*d^10 + 80192667648*a^7*b^16*c^16*d^11 - 181503787008*a^8*b^15*c^15*d^12 + 289980416000*a^9*b^14*c^14*d^13

- 352258621440*a^10*b^13*c^13*d^14 + 334222688256*a^11*b^12*c^12*d^15 - 249961119744*a^12*b^11*c^11*d^16 + 147

248775168*a^13*b^10*c^10*d^17 - 67718086656*a^14*b^9*c^9*d^18 + 23871029248*a^15*b^8*c^8*d^19 - 6245842944*a^1

6*b^7*c^7*d^20 + 1146224640*a^17*b^6*c^6*d^21 - 132120576*a^18*b^5*c^5*d^22 + 7225344*a^19*b^4*c^4*d^23))/(409

6*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*

c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 792*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9

*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d)) - ((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^

4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 +

14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^

2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*(8192*a*b^19*c^22*d^4 - 90112*a^2*b^18*c^21*d^5 + 430848*a^3*b^17*c^20*

d^6 - 1117952*a^4*b^16*c^19*d^7 + 1427968*a^5*b^15*c^18*d^8 + 456192*a^6*b^14*c^17*d^9 - 5803776*a^7*b^13*c^16

*d^10 + 12866304*a^8*b^12*c^15*d^11 - 17335296*a^9*b^11*c^14*d^12 + 16344064*a^10*b^10*c^13*d^13 - 11221760*a^

11*b^9*c^12*d^14 + 5637888*a^12*b^8*c^11*d^15 - 2033152*a^13*b^7*c^10*d^16 + 501248*a^14*b^6*c^9*d^17 - 76032*

a^15*b^5*c^8*d^18 + 5376*a^16*b^4*c^7*d^19))/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2 -

56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d))*(-b^

11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 79

20*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^

4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(3/4))*(-b^11/(16*a^15*d^12 + 16*

a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 1

2672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*

c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*1i + (x^(1/2)*(194481*a^8*b^11*d^15 + 41224337*b^

19*c^8*d^7 - 130932648*a*b^18*c^7*d^8 - 2444904*a^7*b^12*c*d^14 + 201081276*a^2*b^17*c^6*d^9 - 189998424*a^3*b

^16*c^5*d^10 + 119638278*a^4*b^15*c^4*d^11 - 51043608*a^5*b^14*c^3*d^12 + 14378364*a^6*b^13*c^2*d^13)*1i)/(409

6*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*

c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 792*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9

*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d))) - (-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^

4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 +

14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^

2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*((((194481*a^8*b^8*d^14)/2048 + 1232*b^16*c^8*d^6 - (34792593*a*b^15*c^

7*d^7)/2048 - (2250423*a^7*b^9*c*d^13)/2048 + (86420247*a^2*b^14*c^6*d^8)/2048 - (106888869*a^3*b^13*c^5*d^9)/

2048 + (80271027*a^4*b^12*c^4*d^10)/2048 - (38915667*a^5*b^11*c^3*d^11)/2048 + (12127941*a^6*b^10*c^2*d^12)/20

48)/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^

4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d) - ((x^(1/2)*(16777216*b^23*c^23*d^4 - 20132659

2*a*b^22*c^22*d^5 + 1107296256*a^2*b^21*c^21*d^6 - 3593846784*a^3*b^20*c^20*d^7 + 6972506112*a^4*b^19*c^19*d^8

- 4753588224*a^5*b^18*c^18*d^9 - 18397265920*a^6*b^17*c^17*d^10 + 80192667648*a^7*b^16*c^16*d^11 - 1815037870

08*a^8*b^15*c^15*d^12 + 289980416000*a^9*b^14*c^14*d^13 - 352258621440*a^10*b^13*c^13*d^14 + 334222688256*a^11

*b^12*c^12*d^15 - 249961119744*a^12*b^11*c^11*d^16 + 147248775168*a^13*b^10*c^10*d^17 - 67718086656*a^14*b^9*c

^9*d^18 + 23871029248*a^15*b^8*c^8*d^19 - 6245842944*a^16*b^7*c^7*d^20 + 1146224640*a^17*b^6*c^6*d^21 - 132120

576*a^18*b^5*c^5*d^22 + 7225344*a^19*b^4*c^4*d^23))/(4096*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66

*a^2*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6

- 792*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19

*d)) + ((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*

c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920

*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*(8192*a*b^19*c^

22*d^4 - 90112*a^2*b^18*c^21*d^5 + 430848*a^3*b^17*c^20*d^6 - 1117952*a^4*b^16*c^19*d^7 + 1427968*a^5*b^15*c^1

8*d^8 + 456192*a^6*b^14*c^17*d^9 - 5803776*a^7*b^13*c^16*d^10 + 12866304*a^8*b^12*c^15*d^11 - 17335296*a^9*b^1

1*c^14*d^12 + 16344064*a^10*b^10*c^13*d^13 - 11221760*a^11*b^9*c^12*d^14 + 5637888*a^12*b^8*c^11*d^15 - 203315

2*a^13*b^7*c^10*d^16 + 501248*a^14*b^6*c^9*d^17 - 76032*a^15*b^5*c^8*d^18 + 5376*a^16*b^4*c^7*d^19))/(b^8*c^16

+ a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^

3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d))*(-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^1

1*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9

*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10

- 192*a^14*b*c*d^11))^(3/4))*(-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^

10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a

^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))

^(1/4)*1i - (x^(1/2)*(194481*a^8*b^11*d^15 + 41224337*b^19*c^8*d^7 - 130932648*a*b^18*c^7*d^8 - 2444904*a^7*b^

12*c*d^14 + 201081276*a^2*b^17*c^6*d^9 - 189998424*a^3*b^16*c^5*d^10 + 119638278*a^4*b^15*c^4*d^11 - 51043608*

a^5*b^14*c^3*d^12 + 14378364*a^6*b^13*c^2*d^13)*1i)/(4096*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66

*a^2*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6

- 792*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19

*d))))/((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*

c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920

*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*((((194481*a^8*

b^8*d^14)/2048 + 1232*b^16*c^8*d^6 - (34792593*a*b^15*c^7*d^7)/2048 - (2250423*a^7*b^9*c*d^13)/2048 + (8642024

7*a^2*b^14*c^6*d^8)/2048 - (106888869*a^3*b^13*c^5*d^9)/2048 + (80271027*a^4*b^12*c^4*d^10)/2048 - (38915667*a

^5*b^11*c^3*d^11)/2048 + (12127941*a^6*b^10*c^2*d^12)/2048)/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2

*b^6*c^14*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^

7*c^15*d) + ((x^(1/2)*(16777216*b^23*c^23*d^4 - 201326592*a*b^22*c^22*d^5 + 1107296256*a^2*b^21*c^21*d^6 - 359

3846784*a^3*b^20*c^20*d^7 + 6972506112*a^4*b^19*c^19*d^8 - 4753588224*a^5*b^18*c^18*d^9 - 18397265920*a^6*b^17

*c^17*d^10 + 80192667648*a^7*b^16*c^16*d^11 - 181503787008*a^8*b^15*c^15*d^12 + 289980416000*a^9*b^14*c^14*d^1

3 - 352258621440*a^10*b^13*c^13*d^14 + 334222688256*a^11*b^12*c^12*d^15 - 249961119744*a^12*b^11*c^11*d^16 + 1

47248775168*a^13*b^10*c^10*d^17 - 67718086656*a^14*b^9*c^9*d^18 + 23871029248*a^15*b^8*c^8*d^19 - 6245842944*a

^16*b^7*c^7*d^20 + 1146224640*a^17*b^6*c^6*d^21 - 132120576*a^18*b^5*c^5*d^22 + 7225344*a^19*b^4*c^4*d^23))/(4

096*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^

8*c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 792*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a

^9*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d)) - ((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*

a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5

+ 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*

b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*(8192*a*b^19*c^22*d^4 - 90112*a^2*b^18*c^21*d^5 + 430848*a^3*b^17*c^2

0*d^6 - 1117952*a^4*b^16*c^19*d^7 + 1427968*a^5*b^15*c^18*d^8 + 456192*a^6*b^14*c^17*d^9 - 5803776*a^7*b^13*c^

16*d^10 + 12866304*a^8*b^12*c^15*d^11 - 17335296*a^9*b^11*c^14*d^12 + 16344064*a^10*b^10*c^13*d^13 - 11221760*

a^11*b^9*c^12*d^14 + 5637888*a^12*b^8*c^11*d^15 - 2033152*a^13*b^7*c^10*d^16 + 501248*a^14*b^6*c^9*d^17 - 7603

2*a^15*b^5*c^8*d^18 + 5376*a^16*b^4*c^7*d^19))/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2

- 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d))*(-

b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 +

7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*

c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(3/4))*(-b^11/(16*a^15*d^12 + 1

6*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 -

12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^

3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4) + (x^(1/2)*(194481*a^8*b^11*d^15 + 41224337*b^1

9*c^8*d^7 - 130932648*a*b^18*c^7*d^8 - 2444904*a^7*b^12*c*d^14 + 201081276*a^2*b^17*c^6*d^9 - 189998424*a^3*b^

16*c^5*d^10 + 119638278*a^4*b^15*c^4*d^11 - 51043608*a^5*b^14*c^3*d^12 + 14378364*a^6*b^13*c^2*d^13))/(4096*(b

^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*c^16

*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 792*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9*b^3

*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d))) + (-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^

11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 147

84*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^

2*d^10 - 192*a^14*b*c*d^11))^(1/4)*((((194481*a^8*b^8*d^14)/2048 + 1232*b^16*c^8*d^6 - (34792593*a*b^15*c^7*d^

7)/2048 - (2250423*a^7*b^9*c*d^13)/2048 + (86420247*a^2*b^14*c^6*d^8)/2048 - (106888869*a^3*b^13*c^5*d^9)/2048

+ (80271027*a^4*b^12*c^4*d^10)/2048 - (38915667*a^5*b^11*c^3*d^11)/2048 + (12127941*a^6*b^10*c^2*d^12)/2048)/

(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 -

56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d) - ((x^(1/2)*(16777216*b^23*c^23*d^4 - 201326592*a*

b^22*c^22*d^5 + 1107296256*a^2*b^21*c^21*d^6 - 3593846784*a^3*b^20*c^20*d^7 + 6972506112*a^4*b^19*c^19*d^8 - 4

753588224*a^5*b^18*c^18*d^9 - 18397265920*a^6*b^17*c^17*d^10 + 80192667648*a^7*b^16*c^16*d^11 - 181503787008*a

^8*b^15*c^15*d^12 + 289980416000*a^9*b^14*c^14*d^13 - 352258621440*a^10*b^13*c^13*d^14 + 334222688256*a^11*b^1

2*c^12*d^15 - 249961119744*a^12*b^11*c^11*d^16 + 147248775168*a^13*b^10*c^10*d^17 - 67718086656*a^14*b^9*c^9*d

^18 + 23871029248*a^15*b^8*c^8*d^19 - 6245842944*a^16*b^7*c^7*d^20 + 1146224640*a^17*b^6*c^6*d^21 - 132120576*

a^18*b^5*c^5*d^22 + 7225344*a^19*b^4*c^4*d^23))/(4096*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2

*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 7

92*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d))

+ ((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*

d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^1

1*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*(8192*a*b^19*c^22*d

^4 - 90112*a^2*b^18*c^21*d^5 + 430848*a^3*b^17*c^20*d^6 - 1117952*a^4*b^16*c^19*d^7 + 1427968*a^5*b^15*c^18*d^

8 + 456192*a^6*b^14*c^17*d^9 - 5803776*a^7*b^13*c^16*d^10 + 12866304*a^8*b^12*c^15*d^11 - 17335296*a^9*b^11*c^

14*d^12 + 16344064*a^10*b^10*c^13*d^13 - 11221760*a^11*b^9*c^12*d^14 + 5637888*a^12*b^8*c^11*d^15 - 2033152*a^

13*b^7*c^10*d^16 + 501248*a^14*b^6*c^9*d^17 - 76032*a^15*b^5*c^8*d^18 + 5376*a^16*b^4*c^7*d^19))/(b^8*c^16 + a

^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^

11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d))*(-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d

+ 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6

*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 1

92*a^14*b*c*d^11))^(3/4))*(-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d

^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*

b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/

4) - (x^(1/2)*(194481*a^8*b^11*d^15 + 41224337*b^19*c^8*d^7 - 130932648*a*b^18*c^7*d^8 - 2444904*a^7*b^12*c*d^

14 + 201081276*a^2*b^17*c^6*d^9 - 189998424*a^3*b^16*c^5*d^10 + 119638278*a^4*b^15*c^4*d^11 - 51043608*a^5*b^1

4*c^3*d^12 + 14378364*a^6*b^13*c^2*d^13))/(4096*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*b^10*

c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 792*a^7

*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d)))))*(-

b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 +

7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*

c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*2i + 2*atan(((-b^11/(16*a

^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b

^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 -

3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*(((((194481*a^8*b^8*d^14)/2048 + 12

32*b^16*c^8*d^6 - (34792593*a*b^15*c^7*d^7)/2048 - (2250423*a^7*b^9*c*d^13)/2048 + (86420247*a^2*b^14*c^6*d^8)

/2048 - (106888869*a^3*b^13*c^5*d^9)/2048 + (80271027*a^4*b^12*c^4*d^10)/2048 - (38915667*a^5*b^11*c^3*d^11)/2

048 + (12127941*a^6*b^10*c^2*d^12)/2048)*1i)/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2 -

56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d) - ((x

^(1/2)*(16777216*b^23*c^23*d^4 - 201326592*a*b^22*c^22*d^5 + 1107296256*a^2*b^21*c^21*d^6 - 3593846784*a^3*b^2

0*c^20*d^7 + 6972506112*a^4*b^19*c^19*d^8 - 4753588224*a^5*b^18*c^18*d^9 - 18397265920*a^6*b^17*c^17*d^10 + 80

192667648*a^7*b^16*c^16*d^11 - 181503787008*a^8*b^15*c^15*d^12 + 289980416000*a^9*b^14*c^14*d^13 - 35225862144

0*a^10*b^13*c^13*d^14 + 334222688256*a^11*b^12*c^12*d^15 - 249961119744*a^12*b^11*c^11*d^16 + 147248775168*a^1

3*b^10*c^10*d^17 - 67718086656*a^14*b^9*c^9*d^18 + 23871029248*a^15*b^8*c^8*d^19 - 6245842944*a^16*b^7*c^7*d^2

0 + 1146224640*a^17*b^6*c^6*d^21 - 132120576*a^18*b^5*c^5*d^22 + 7225344*a^19*b^4*c^4*d^23)*1i)/(4096*(b^12*c^

20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*c^16*d^4 -

792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 792*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9*b^3*c^11*

d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d)) + ((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^1

1*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9

*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10

- 192*a^14*b*c*d^11))^(1/4)*(8192*a*b^19*c^22*d^4 - 90112*a^2*b^18*c^21*d^5 + 430848*a^3*b^17*c^20*d^6 - 1117

952*a^4*b^16*c^19*d^7 + 1427968*a^5*b^15*c^18*d^8 + 456192*a^6*b^14*c^17*d^9 - 5803776*a^7*b^13*c^16*d^10 + 12

866304*a^8*b^12*c^15*d^11 - 17335296*a^9*b^11*c^14*d^12 + 16344064*a^10*b^10*c^13*d^13 - 11221760*a^11*b^9*c^1

2*d^14 + 5637888*a^12*b^8*c^11*d^15 - 2033152*a^13*b^7*c^10*d^16 + 501248*a^14*b^6*c^9*d^17 - 76032*a^15*b^5*c

^8*d^18 + 5376*a^16*b^4*c^7*d^19))/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2 - 56*a^3*b^

5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d))*(-b^11/(16*a^1

5*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8

*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 35

20*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(3/4)*1i)*(-b^11/(16*a^15*d^12 + 16*a^3*b^1

2*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^

8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9

+ 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4) + (x^(1/2)*(194481*a^8*b^11*d^15 + 41224337*b^19*c^8*d^7

- 130932648*a*b^18*c^7*d^8 - 2444904*a^7*b^12*c*d^14 + 201081276*a^2*b^17*c^6*d^9 - 189998424*a^3*b^16*c^5*d^

10 + 119638278*a^4*b^15*c^4*d^11 - 51043608*a^5*b^14*c^3*d^12 + 14378364*a^6*b^13*c^2*d^13))/(4096*(b^12*c^20

+ a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*c^16*d^4 - 79

2*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 792*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9*b^3*c^11*d^9

+ 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d))) - (-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d

+ 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^

6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 -

192*a^14*b*c*d^11))^(1/4)*(((((194481*a^8*b^8*d^14)/2048 + 1232*b^16*c^8*d^6 - (34792593*a*b^15*c^7*d^7)/2048

- (2250423*a^7*b^9*c*d^13)/2048 + (86420247*a^2*b^14*c^6*d^8)/2048 - (106888869*a^3*b^13*c^5*d^9)/2048 + (8027

1027*a^4*b^12*c^4*d^10)/2048 - (38915667*a^5*b^11*c^3*d^11)/2048 + (12127941*a^6*b^10*c^2*d^12)/2048)*1i)/(b^8

*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a

^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d) + ((x^(1/2)*(16777216*b^23*c^23*d^4 - 201326592*a*b^22

*c^22*d^5 + 1107296256*a^2*b^21*c^21*d^6 - 3593846784*a^3*b^20*c^20*d^7 + 6972506112*a^4*b^19*c^19*d^8 - 47535

88224*a^5*b^18*c^18*d^9 - 18397265920*a^6*b^17*c^17*d^10 + 80192667648*a^7*b^16*c^16*d^11 - 181503787008*a^8*b

^15*c^15*d^12 + 289980416000*a^9*b^14*c^14*d^13 - 352258621440*a^10*b^13*c^13*d^14 + 334222688256*a^11*b^12*c^

12*d^15 - 249961119744*a^12*b^11*c^11*d^16 + 147248775168*a^13*b^10*c^10*d^17 - 67718086656*a^14*b^9*c^9*d^18

+ 23871029248*a^15*b^8*c^8*d^19 - 6245842944*a^16*b^7*c^7*d^20 + 1146224640*a^17*b^6*c^6*d^21 - 132120576*a^18

*b^5*c^5*d^22 + 7225344*a^19*b^4*c^4*d^23)*1i)/(4096*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*

b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 79

2*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d))

- ((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d

^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11

*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*(8192*a*b^19*c^22*d^

4 - 90112*a^2*b^18*c^21*d^5 + 430848*a^3*b^17*c^20*d^6 - 1117952*a^4*b^16*c^19*d^7 + 1427968*a^5*b^15*c^18*d^8

+ 456192*a^6*b^14*c^17*d^9 - 5803776*a^7*b^13*c^16*d^10 + 12866304*a^8*b^12*c^15*d^11 - 17335296*a^9*b^11*c^1

4*d^12 + 16344064*a^10*b^10*c^13*d^13 - 11221760*a^11*b^9*c^12*d^14 + 5637888*a^12*b^8*c^11*d^15 - 2033152*a^1

3*b^7*c^10*d^16 + 501248*a^14*b^6*c^9*d^17 - 76032*a^15*b^5*c^8*d^18 + 5376*a^16*b^4*c^7*d^19))/(b^8*c^16 + a^

8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^1

1*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d))*(-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d +

1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*

c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 19

2*a^14*b*c*d^11))^(3/4)*1i)*(-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10

*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^1

0*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(

1/4) - (x^(1/2)*(194481*a^8*b^11*d^15 + 41224337*b^19*c^8*d^7 - 130932648*a*b^18*c^7*d^8 - 2444904*a^7*b^12*c*

d^14 + 201081276*a^2*b^17*c^6*d^9 - 189998424*a^3*b^16*c^5*d^10 + 119638278*a^4*b^15*c^4*d^11 - 51043608*a^5*b

^14*c^3*d^12 + 14378364*a^6*b^13*c^2*d^13))/(4096*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*b^1

0*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 792*a

^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d))))/(

(-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3

+ 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^

4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*(((((194481*a^8*b^8*d^1

4)/2048 + 1232*b^16*c^8*d^6 - (34792593*a*b^15*c^7*d^7)/2048 - (2250423*a^7*b^9*c*d^13)/2048 + (86420247*a^2*b

^14*c^6*d^8)/2048 - (106888869*a^3*b^13*c^5*d^9)/2048 + (80271027*a^4*b^12*c^4*d^10)/2048 - (38915667*a^5*b^11

*c^3*d^11)/2048 + (12127941*a^6*b^10*c^2*d^12)/2048)*1i)/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^

6*c^14*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c

^15*d) - ((x^(1/2)*(16777216*b^23*c^23*d^4 - 201326592*a*b^22*c^22*d^5 + 1107296256*a^2*b^21*c^21*d^6 - 359384

6784*a^3*b^20*c^20*d^7 + 6972506112*a^4*b^19*c^19*d^8 - 4753588224*a^5*b^18*c^18*d^9 - 18397265920*a^6*b^17*c^

17*d^10 + 80192667648*a^7*b^16*c^16*d^11 - 181503787008*a^8*b^15*c^15*d^12 + 289980416000*a^9*b^14*c^14*d^13 -

352258621440*a^10*b^13*c^13*d^14 + 334222688256*a^11*b^12*c^12*d^15 - 249961119744*a^12*b^11*c^11*d^16 + 1472

48775168*a^13*b^10*c^10*d^17 - 67718086656*a^14*b^9*c^9*d^18 + 23871029248*a^15*b^8*c^8*d^19 - 6245842944*a^16

*b^7*c^7*d^20 + 1146224640*a^17*b^6*c^6*d^21 - 132120576*a^18*b^5*c^5*d^22 + 7225344*a^19*b^4*c^4*d^23)*1i)/(4

096*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^

8*c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 792*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a

^9*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d)) + ((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*