∫ 1________√ x (a+bx2)2(c+dx2)2 dx

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

Optimal. Leaf size=628 \[ -\frac {b^{7/4} (3 b c-11 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {b \sqrt {x}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {d \sqrt {x} (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2} \]

[Out]

-1/8*b^(7/4)*(-11*a*d+3*b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/(-a*d+b*c)^3*2^(1/2)+1/8*b^(7/4

)*(-11*a*d+3*b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/(-a*d+b*c)^3*2^(1/2)-1/8*d^(7/4)*(-3*a*d+1

1*b*c)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(7/4)/(-a*d+b*c)^3*2^(1/2)+1/8*d^(7/4)*(-3*a*d+11*b*c)*arct

an(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(7/4)/(-a*d+b*c)^3*2^(1/2)-1/16*b^(7/4)*(-11*a*d+3*b*c)*ln(a^(1/2)+x*b

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

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

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

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

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

________________________________________________________________________________________

Rubi [A] time = 0.86, antiderivative size = 628, 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} \[ -\frac {b^{7/4} (3 b c-11 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {b \sqrt {x}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {d \sqrt {x} (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

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

- (b^(7/4)*(3*b*c - 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*

c - a*d)^3) + (b^(7/4)*(3*b*c - 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]

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

)/(8*Sqrt[2]*c^(7/4)*(b*c - a*d)^3) + (d^(7/4)*(11*b*c - 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]

+ Sqrt[d]*x])/(8*Sqrt[2]*c^(7/4)*(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 )^2 \left (c+d x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-3 b c+4 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 )}{2 a (b c-a d)}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-4 \left (3 b^2 c^2-8 a b c d+3 a^2 d^2\right )-12 b d (b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{8 a c (b c-a d)^2}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^2 (3 b c-11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a (b c-a d)^3}+\frac {\left (d^2 (11 b c-3 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c (b c-a d)^3}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^2 (3 b c-11 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{3/2} (b c-a d)^3}+\frac {\left (b^2 (3 b c-11 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{3/2} (b c-a d)^3}+\frac {\left (d^2 (11 b c-3 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^{3/2} (b c-a d)^3}+\frac {\left (d^2 (11 b c-3 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^{3/2} (b c-a d)^3}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^{3/2} (3 b c-11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^{3/2} (b c-a d)^3}+\frac {\left (b^{3/2} (3 b c-11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^{3/2} (b c-a d)^3}-\frac {\left (b^{7/4} (3 b c-11 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {\left (b^{7/4} (3 b c-11 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {\left (d^{3/2} (11 b c-3 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 c^{3/2} (b c-a d)^3}+\frac {\left (d^{3/2} (11 b c-3 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 c^{3/2} (b c-a d)^3}-\frac {\left (d^{7/4} (11 b c-3 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} c^{7/4} (b c-a d)^3}-\frac {\left (d^{7/4} (11 b c-3 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} c^{7/4} (b c-a d)^3}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {\left (b^{7/4} (3 b c-11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {\left (b^{7/4} (3 b c-11 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {\left (d^{7/4} (11 b c-3 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} c^{7/4} (b c-a d)^3}-\frac {\left (d^{7/4} (11 b c-3 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} c^{7/4} (b c-a d)^3}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^3}-\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A] time = 6.21, size = 661, normalized size = 1.05 \[ \frac {b^{7/4} (11 a d-3 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{7/4} (11 a d-3 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{7/4} (11 a d-3 b c) \tan ^{-1}\left (\frac {2 \sqrt [4]{b} \sqrt {x}-\sqrt {2} \sqrt [4]{a}}{\sqrt {2} \sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{7/4} (11 a d-3 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a}+2 \sqrt [4]{b} \sqrt {x}}{\sqrt {2} \sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^2 \sqrt {x}}{2 a \left (a+b x^2\right ) (a d-b c)^2}+\frac {d^{7/4} (11 b c-3 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (a d-b c)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (a d-b c)^3}-\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac {2 \sqrt [4]{d} \sqrt {x}-\sqrt {2} \sqrt [4]{c}}{\sqrt {2} \sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (a d-b c)^3}-\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c}+2 \sqrt [4]{d} \sqrt {x}}{\sqrt {2} \sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (a d-b c)^3}+\frac {d^2 \sqrt {x}}{2 c \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

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

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

- Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(7/4)*(-3*b*c + 11*a*d)

*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) + (d^(7/4)*(11*

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

- (d^(7/4)*(11*b*c - 3*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(7/4)*(-(

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

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

[Out]

Timed out

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giac [B] time = 1.29, size = 977, normalized size = 1.56 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

- 3*sqrt(2)*a*b^2*c^4*d + 3*sqrt(2)*a^2*b*c^3*d^2 - sqrt(2)*a^3*c^2*d^3) + 1/8*(3*(a*b^3)^(1/4)*b^2*c - 11*(a

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

^2*d + 3*sqrt(2)*a^4*b*c*d^2 - sqrt(2)*a^5*d^3) - 1/8*(3*(a*b^3)^(1/4)*b^2*c - 11*(a*b^3)^(1/4)*a*b*d)*log(-sq

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

d^2 - sqrt(2)*a^5*d^3) + 1/8*(11*(c*d^3)^(1/4)*b*c*d - 3*(c*d^3)^(1/4)*a*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4)

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

/8*(11*(c*d^3)^(1/4)*b*c*d - 3*(c*d^3)^(1/4)*a*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)

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

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

+ a*d*x^2 + a*c))

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maple [A] time = 0.02, size = 808, normalized size = 1.29 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2/(a*d-b*c)^3*x^(1/2)/(b*x^2+a)*d-1/2*b^3/(a*d-b*c)^3/a*x^(1/2)/(b*x^2+a)*c+11/8*b^2/(a*d-b*c)^3/a*(a/b)

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

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

b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))*d-3/16*b^3/(a*d-b*c)^3/a^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)))*c+11/8*b^2/(a*d-b*c)^3/a*(

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

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

c)*b+3/16*d^3/(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-11/16*d^2/(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+3/8*d^3/(a*d-b*c)^3/c^2*(c/d)^(1/4)*2^(1/2)*arct

an(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a-11/8*d^2/(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+3/8*d^3/(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-11/8*d^2/(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

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maxima [A] time = 2.59, size = 678, normalized size = 1.08 \[ \frac {{\left (\frac {2 \, \sqrt {2} {\left (3 \, b c - 11 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (3 \, b c - 11 \, a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (3 \, b c - 11 \, a d\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (3 \, b c - 11 \, a d\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} b^{2}}{16 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} + \frac {{\left (b^{2} c d + a b d^{2}\right )} x^{\frac {5}{2}} + {\left (b^{2} c^{2} + a^{2} d^{2}\right )} \sqrt {x}}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{4} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (11 \, b c d^{2} - 3 \, a 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 (11 \, b c d^{2} - 3 \, a 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 (11 \, b c d^{2} - 3 \, a 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 (11 \, b c d^{2} - 3 \, a 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}}}}{16 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

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

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

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(sq

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

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

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mupad [B] time = 5.07, size = 37332, normalized size = 59.45 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

*d^2 + b^2*c^2 - 2*a*b*c*d)))/(a*c + x^2*(a*d + b*c) + b*d*x^4) - atan((((((x^(1/2)*(36864*a^2*b^23*c^21*d^4 -

712704*a^3*b^22*c^20*d^5 + 6172672*a^4*b^21*c^19*d^6 - 31899648*a^5*b^20*c^18*d^7 + 110432256*a^6*b^19*c^17*d

^8 - 271552512*a^7*b^18*c^16*d^9 + 487280640*a^8*b^17*c^15*d^10 - 635523072*a^9*b^16*c^14*d^11 + 562982912*a^1

0*b^15*c^13*d^12 - 227217408*a^11*b^14*c^12*d^13 - 227217408*a^12*b^13*c^11*d^14 + 562982912*a^13*b^12*c^10*d^

15 - 635523072*a^14*b^11*c^9*d^16 + 487280640*a^15*b^10*c^8*d^17 - 271552512*a^16*b^9*c^7*d^18 + 110432256*a^1

7*b^8*c^6*d^19 - 31899648*a^18*b^7*c^5*d^20 + 6172672*a^19*b^6*c^4*d^21 - 712704*a^20*b^5*c^3*d^22 + 36864*a^2

1*b^4*c^2*d^23))/(16*(a^4*b^12*c^16 + a^16*c^4*d^12 - 12*a^5*b^11*c^15*d - 12*a^15*b*c^5*d^11 + 66*a^6*b^10*c^

14*d^2 - 220*a^7*b^9*c^13*d^3 + 495*a^8*b^8*c^12*d^4 - 792*a^9*b^7*c^11*d^5 + 924*a^10*b^6*c^10*d^6 - 792*a^11

*b^5*c^9*d^7 + 495*a^12*b^4*c^8*d^8 - 220*a^13*b^3*c^7*d^9 + 66*a^14*b^2*c^6*d^10)) + ((-(81*b^11*c^4 + 14641*

a^4*b^7*d^4 - 15972*a^3*b^8*c*d^3 + 6534*a^2*b^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12 + 4096*a^7*b^12*

c^12 - 49152*a^8*b^11*c^11*d + 270336*a^9*b^10*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a^11*b^8*c^8*d^4 -

3244032*a^12*b^7*c^7*d^5 + 3784704*a^13*b^6*c^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^15*b^4*c^4*d^8 - 9

01120*a^16*b^3*c^3*d^9 + 270336*a^17*b^2*c^2*d^10 - 49152*a^18*b*c*d^11))^(1/4)*(6144*a^4*b^19*c^19*d^4 - 9011

2*a^5*b^18*c^18*d^5 + 585728*a^6*b^17*c^17*d^6 - 2230272*a^7*b^16*c^16*d^7 + 5490688*a^8*b^15*c^15*d^8 - 89661

44*a^9*b^14*c^14*d^9 + 9191424*a^10*b^13*c^13*d^10 - 3987456*a^11*b^12*c^12*d^11 - 3987456*a^12*b^11*c^11*d^12

+ 9191424*a^13*b^10*c^10*d^13 - 8966144*a^14*b^9*c^9*d^14 + 5490688*a^15*b^8*c^8*d^15 - 2230272*a^16*b^7*c^7*

d^16 + 585728*a^17*b^6*c^6*d^17 - 90112*a^18*b^5*c^5*d^18 + 6144*a^19*b^4*c^4*d^19))/(a^4*b^8*c^12 + a^12*c^4*

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

*a^9*b^3*c^7*d^5 + 28*a^10*b^2*c^6*d^6))*(-(81*b^11*c^4 + 14641*a^4*b^7*d^4 - 15972*a^3*b^8*c*d^3 + 6534*a^2*b

^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12 + 4096*a^7*b^12*c^12 - 49152*a^8*b^11*c^11*d + 270336*a^9*b^10

*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a^11*b^8*c^8*d^4 - 3244032*a^12*b^7*c^7*d^5 + 3784704*a^13*b^6*c

^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^15*b^4*c^4*d^8 - 901120*a^16*b^3*c^3*d^9 + 270336*a^17*b^2*c^2*d

^10 - 49152*a^18*b*c*d^11))^(3/4) + ((891*a^8*b^7*d^15)/2 + (891*b^15*c^8*d^7)/2 - 6210*a*b^14*c^7*d^8 - 6210*

a^7*b^8*c*d^14 + 31509*a^2*b^13*c^6*d^9 - 66138*a^3*b^12*c^5*d^10 + 60307*a^4*b^11*c^4*d^11 - 66138*a^5*b^10*c

^3*d^12 + 31509*a^6*b^9*c^2*d^13)/(a^4*b^8*c^12 + a^12*c^4*d^8 - 8*a^5*b^7*c^11*d - 8*a^11*b*c^5*d^7 + 28*a^6*

b^6*c^10*d^2 - 56*a^7*b^5*c^9*d^3 + 70*a^8*b^4*c^8*d^4 - 56*a^9*b^3*c^7*d^5 + 28*a^10*b^2*c^6*d^6))*(-(81*b^11

*c^4 + 14641*a^4*b^7*d^4 - 15972*a^3*b^8*c*d^3 + 6534*a^2*b^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12 + 4

096*a^7*b^12*c^12 - 49152*a^8*b^11*c^11*d + 270336*a^9*b^10*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a^11*

b^8*c^8*d^4 - 3244032*a^12*b^7*c^7*d^5 + 3784704*a^13*b^6*c^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^15*b^

4*c^4*d^8 - 901120*a^16*b^3*c^3*d^9 + 270336*a^17*b^2*c^2*d^10 - 49152*a^18*b*c*d^11))^(1/4)*1i + (x^(1/2)*(98

01*a^8*b^9*d^17 + 9801*b^17*c^8*d^9 - 149094*a*b^16*c^7*d^10 - 149094*a^7*b^10*c*d^16 + 1001520*a^2*b^15*c^6*d

^11 - 3484602*a^3*b^14*c^5*d^12 + 5769038*a^4*b^13*c^4*d^13 - 3484602*a^5*b^12*c^3*d^14 + 1001520*a^6*b^11*c^2

*d^15)*1i)/(16*(a^4*b^12*c^16 + a^16*c^4*d^12 - 12*a^5*b^11*c^15*d - 12*a^15*b*c^5*d^11 + 66*a^6*b^10*c^14*d^2

- 220*a^7*b^9*c^13*d^3 + 495*a^8*b^8*c^12*d^4 - 792*a^9*b^7*c^11*d^5 + 924*a^10*b^6*c^10*d^6 - 792*a^11*b^5*c

^9*d^7 + 495*a^12*b^4*c^8*d^8 - 220*a^13*b^3*c^7*d^9 + 66*a^14*b^2*c^6*d^10)))*(-(81*b^11*c^4 + 14641*a^4*b^7*

d^4 - 15972*a^3*b^8*c*d^3 + 6534*a^2*b^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12 + 4096*a^7*b^12*c^12 - 4

9152*a^8*b^11*c^11*d + 270336*a^9*b^10*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a^11*b^8*c^8*d^4 - 3244032

*a^12*b^7*c^7*d^5 + 3784704*a^13*b^6*c^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^15*b^4*c^4*d^8 - 901120*a^

16*b^3*c^3*d^9 + 270336*a^17*b^2*c^2*d^10 - 49152*a^18*b*c*d^11))^(1/4) + ((((x^(1/2)*(36864*a^2*b^23*c^21*d^4

- 712704*a^3*b^22*c^20*d^5 + 6172672*a^4*b^21*c^19*d^6 - 31899648*a^5*b^20*c^18*d^7 + 110432256*a^6*b^19*c^17

*d^8 - 271552512*a^7*b^18*c^16*d^9 + 487280640*a^8*b^17*c^15*d^10 - 635523072*a^9*b^16*c^14*d^11 + 562982912*a

^10*b^15*c^13*d^12 - 227217408*a^11*b^14*c^12*d^13 - 227217408*a^12*b^13*c^11*d^14 + 562982912*a^13*b^12*c^10*

d^15 - 635523072*a^14*b^11*c^9*d^16 + 487280640*a^15*b^10*c^8*d^17 - 271552512*a^16*b^9*c^7*d^18 + 110432256*a

^17*b^8*c^6*d^19 - 31899648*a^18*b^7*c^5*d^20 + 6172672*a^19*b^6*c^4*d^21 - 712704*a^20*b^5*c^3*d^22 + 36864*a

^21*b^4*c^2*d^23))/(16*(a^4*b^12*c^16 + a^16*c^4*d^12 - 12*a^5*b^11*c^15*d - 12*a^15*b*c^5*d^11 + 66*a^6*b^10*

c^14*d^2 - 220*a^7*b^9*c^13*d^3 + 495*a^8*b^8*c^12*d^4 - 792*a^9*b^7*c^11*d^5 + 924*a^10*b^6*c^10*d^6 - 792*a^

11*b^5*c^9*d^7 + 495*a^12*b^4*c^8*d^8 - 220*a^13*b^3*c^7*d^9 + 66*a^14*b^2*c^6*d^10)) - ((-(81*b^11*c^4 + 1464

1*a^4*b^7*d^4 - 15972*a^3*b^8*c*d^3 + 6534*a^2*b^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12 + 4096*a^7*b^1

2*c^12 - 49152*a^8*b^11*c^11*d + 270336*a^9*b^10*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a^11*b^8*c^8*d^4

- 3244032*a^12*b^7*c^7*d^5 + 3784704*a^13*b^6*c^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^15*b^4*c^4*d^8 -

901120*a^16*b^3*c^3*d^9 + 270336*a^17*b^2*c^2*d^10 - 49152*a^18*b*c*d^11))^(1/4)*(6144*a^4*b^19*c^19*d^4 - 90

112*a^5*b^18*c^18*d^5 + 585728*a^6*b^17*c^17*d^6 - 2230272*a^7*b^16*c^16*d^7 + 5490688*a^8*b^15*c^15*d^8 - 896

6144*a^9*b^14*c^14*d^9 + 9191424*a^10*b^13*c^13*d^10 - 3987456*a^11*b^12*c^12*d^11 - 3987456*a^12*b^11*c^11*d^

12 + 9191424*a^13*b^10*c^10*d^13 - 8966144*a^14*b^9*c^9*d^14 + 5490688*a^15*b^8*c^8*d^15 - 2230272*a^16*b^7*c^

7*d^16 + 585728*a^17*b^6*c^6*d^17 - 90112*a^18*b^5*c^5*d^18 + 6144*a^19*b^4*c^4*d^19))/(a^4*b^8*c^12 + a^12*c^

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

56*a^9*b^3*c^7*d^5 + 28*a^10*b^2*c^6*d^6))*(-(81*b^11*c^4 + 14641*a^4*b^7*d^4 - 15972*a^3*b^8*c*d^3 + 6534*a^2

*b^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12 + 4096*a^7*b^12*c^12 - 49152*a^8*b^11*c^11*d + 270336*a^9*b^

10*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a^11*b^8*c^8*d^4 - 3244032*a^12*b^7*c^7*d^5 + 3784704*a^13*b^6

*c^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^15*b^4*c^4*d^8 - 901120*a^16*b^3*c^3*d^9 + 270336*a^17*b^2*c^2

*d^10 - 49152*a^18*b*c*d^11))^(3/4) - ((891*a^8*b^7*d^15)/2 + (891*b^15*c^8*d^7)/2 - 6210*a*b^14*c^7*d^8 - 621

0*a^7*b^8*c*d^14 + 31509*a^2*b^13*c^6*d^9 - 66138*a^3*b^12*c^5*d^10 + 60307*a^4*b^11*c^4*d^11 - 66138*a^5*b^10

*c^3*d^12 + 31509*a^6*b^9*c^2*d^13)/(a^4*b^8*c^12 + a^12*c^4*d^8 - 8*a^5*b^7*c^11*d - 8*a^11*b*c^5*d^7 + 28*a^

6*b^6*c^10*d^2 - 56*a^7*b^5*c^9*d^3 + 70*a^8*b^4*c^8*d^4 - 56*a^9*b^3*c^7*d^5 + 28*a^10*b^2*c^6*d^6))*(-(81*b^

11*c^4 + 14641*a^4*b^7*d^4 - 15972*a^3*b^8*c*d^3 + 6534*a^2*b^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12 +

4096*a^7*b^12*c^12 - 49152*a^8*b^11*c^11*d + 270336*a^9*b^10*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a^1

1*b^8*c^8*d^4 - 3244032*a^12*b^7*c^7*d^5 + 3784704*a^13*b^6*c^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^15*

b^4*c^4*d^8 - 901120*a^16*b^3*c^3*d^9 + 270336*a^17*b^2*c^2*d^10 - 49152*a^18*b*c*d^11))^(1/4)*1i + (x^(1/2)*(

9801*a^8*b^9*d^17 + 9801*b^17*c^8*d^9 - 149094*a*b^16*c^7*d^10 - 149094*a^7*b^10*c*d^16 + 1001520*a^2*b^15*c^6

*d^11 - 3484602*a^3*b^14*c^5*d^12 + 5769038*a^4*b^13*c^4*d^13 - 3484602*a^5*b^12*c^3*d^14 + 1001520*a^6*b^11*c

^2*d^15)*1i)/(16*(a^4*b^12*c^16 + a^16*c^4*d^12 - 12*a^5*b^11*c^15*d - 12*a^15*b*c^5*d^11 + 66*a^6*b^10*c^14*d

^2 - 220*a^7*b^9*c^13*d^3 + 495*a^8*b^8*c^12*d^4 - 792*a^9*b^7*c^11*d^5 + 924*a^10*b^6*c^10*d^6 - 792*a^11*b^5

*c^9*d^7 + 495*a^12*b^4*c^8*d^8 - 220*a^13*b^3*c^7*d^9 + 66*a^14*b^2*c^6*d^10)))*(-(81*b^11*c^4 + 14641*a^4*b^

7*d^4 - 15972*a^3*b^8*c*d^3 + 6534*a^2*b^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12 + 4096*a^7*b^12*c^12 -

49152*a^8*b^11*c^11*d + 270336*a^9*b^10*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a^11*b^8*c^8*d^4 - 32440

32*a^12*b^7*c^7*d^5 + 3784704*a^13*b^6*c^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^15*b^4*c^4*d^8 - 901120*

a^16*b^3*c^3*d^9 + 270336*a^17*b^2*c^2*d^10 - 49152*a^18*b*c*d^11))^(1/4))/(((((x^(1/2)*(36864*a^2*b^23*c^21*d

^4 - 712704*a^3*b^22*c^20*d^5 + 6172672*a^4*b^21*c^19*d^6 - 31899648*a^5*b^20*c^18*d^7 + 110432256*a^6*b^19*c^

17*d^8 - 271552512*a^7*b^18*c^16*d^9 + 487280640*a^8*b^17*c^15*d^10 - 635523072*a^9*b^16*c^14*d^11 + 562982912

*a^10*b^15*c^13*d^12 - 227217408*a^11*b^14*c^12*d^13 - 227217408*a^12*b^13*c^11*d^14 + 562982912*a^13*b^12*c^1

0*d^15 - 635523072*a^14*b^11*c^9*d^16 + 487280640*a^15*b^10*c^8*d^17 - 271552512*a^16*b^9*c^7*d^18 + 110432256

*a^17*b^8*c^6*d^19 - 31899648*a^18*b^7*c^5*d^20 + 6172672*a^19*b^6*c^4*d^21 - 712704*a^20*b^5*c^3*d^22 + 36864

*a^21*b^4*c^2*d^23))/(16*(a^4*b^12*c^16 + a^16*c^4*d^12 - 12*a^5*b^11*c^15*d - 12*a^15*b*c^5*d^11 + 66*a^6*b^1

0*c^14*d^2 - 220*a^7*b^9*c^13*d^3 + 495*a^8*b^8*c^12*d^4 - 792*a^9*b^7*c^11*d^5 + 924*a^10*b^6*c^10*d^6 - 792*

a^11*b^5*c^9*d^7 + 495*a^12*b^4*c^8*d^8 - 220*a^13*b^3*c^7*d^9 + 66*a^14*b^2*c^6*d^10)) + ((-(81*b^11*c^4 + 14

641*a^4*b^7*d^4 - 15972*a^3*b^8*c*d^3 + 6534*a^2*b^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12 + 4096*a^7*b

^12*c^12 - 49152*a^8*b^11*c^11*d + 270336*a^9*b^10*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a^11*b^8*c^8*d

^4 - 3244032*a^12*b^7*c^7*d^5 + 3784704*a^13*b^6*c^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^15*b^4*c^4*d^8

- 901120*a^16*b^3*c^3*d^9 + 270336*a^17*b^2*c^2*d^10 - 49152*a^18*b*c*d^11))^(1/4)*(6144*a^4*b^19*c^19*d^4 -

90112*a^5*b^18*c^18*d^5 + 585728*a^6*b^17*c^17*d^6 - 2230272*a^7*b^16*c^16*d^7 + 5490688*a^8*b^15*c^15*d^8 - 8

966144*a^9*b^14*c^14*d^9 + 9191424*a^10*b^13*c^13*d^10 - 3987456*a^11*b^12*c^12*d^11 - 3987456*a^12*b^11*c^11*

d^12 + 9191424*a^13*b^10*c^10*d^13 - 8966144*a^14*b^9*c^9*d^14 + 5490688*a^15*b^8*c^8*d^15 - 2230272*a^16*b^7*

c^7*d^16 + 585728*a^17*b^6*c^6*d^17 - 90112*a^18*b^5*c^5*d^18 + 6144*a^19*b^4*c^4*d^19))/(a^4*b^8*c^12 + a^12*

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

- 56*a^9*b^3*c^7*d^5 + 28*a^10*b^2*c^6*d^6))*(-(81*b^11*c^4 + 14641*a^4*b^7*d^4 - 15972*a^3*b^8*c*d^3 + 6534*a

^2*b^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12 + 4096*a^7*b^12*c^12 - 49152*a^8*b^11*c^11*d + 270336*a^9*

b^10*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a^11*b^8*c^8*d^4 - 3244032*a^12*b^7*c^7*d^5 + 3784704*a^13*b

^6*c^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^15*b^4*c^4*d^8 - 901120*a^16*b^3*c^3*d^9 + 270336*a^17*b^2*c

^2*d^10 - 49152*a^18*b*c*d^11))^(3/4) + ((891*a^8*b^7*d^15)/2 + (891*b^15*c^8*d^7)/2 - 6210*a*b^14*c^7*d^8 - 6

210*a^7*b^8*c*d^14 + 31509*a^2*b^13*c^6*d^9 - 66138*a^3*b^12*c^5*d^10 + 60307*a^4*b^11*c^4*d^11 - 66138*a^5*b^

10*c^3*d^12 + 31509*a^6*b^9*c^2*d^13)/(a^4*b^8*c^12 + a^12*c^4*d^8 - 8*a^5*b^7*c^11*d - 8*a^11*b*c^5*d^7 + 28*

a^6*b^6*c^10*d^2 - 56*a^7*b^5*c^9*d^3 + 70*a^8*b^4*c^8*d^4 - 56*a^9*b^3*c^7*d^5 + 28*a^10*b^2*c^6*d^6))*(-(81*

b^11*c^4 + 14641*a^4*b^7*d^4 - 15972*a^3*b^8*c*d^3 + 6534*a^2*b^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12

+ 4096*a^7*b^12*c^12 - 49152*a^8*b^11*c^11*d + 270336*a^9*b^10*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a

^11*b^8*c^8*d^4 - 3244032*a^12*b^7*c^7*d^5 + 3784704*a^13*b^6*c^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^1

5*b^4*c^4*d^8 - 901120*a^16*b^3*c^3*d^9 + 270336*a^17*b^2*c^2*d^10 - 49152*a^18*b*c*d^11))^(1/4) + (x^(1/2)*(9

801*a^8*b^9*d^17 + 9801*b^17*c^8*d^9 - 149094*a*b^16*c^7*d^10 - 149094*a^7*b^10*c*d^16 + 1001520*a^2*b^15*c^6*

d^11 - 3484602*a^3*b^14*c^5*d^12 + 5769038*a^4*b^13*c^4*d^13 - 3484602*a^5*b^12*c^3*d^14 + 1001520*a^6*b^11*c^

2*d^15))/(16*(a^4*b^12*c^16 + a^16*c^4*d^12 - 12*a^5*b^11*c^15*d - 12*a^15*b*c^5*d^11 + 66*a^6*b^10*c^14*d^2 -

220*a^7*b^9*c^13*d^3 + 495*a^8*b^8*c^12*d^4 - 792*a^9*b^7*c^11*d^5 + 924*a^10*b^6*c^10*d^6 - 792*a^11*b^5*c^9

*d^7 + 495*a^12*b^4*c^8*d^8 - 220*a^13*b^3*c^7*d^9 + 66*a^14*b^2*c^6*d^10)))*(-(81*b^11*c^4 + 14641*a^4*b^7*d^

4 - 15972*a^3*b^8*c*d^3 + 6534*a^2*b^9*c^2*d^2 - 1188*a*b^10*c^3*d)/(4096*a^19*d^12 + 4096*a^7*b^12*c^12 - 491

52*a^8*b^11*c^11*d + 270336*a^9*b^10*c^10*d^2 - 901120*a^10*b^9*c^9*d^3 + 2027520*a^11*b^8*c^8*d^4 - 3244032*a

^12*b^7*c^7*d^5 + 3784704*a^13*b^6*c^6*d^6 - 3244032*a^14*b^5*c^5*d^7 + 2027520*a^15*b^4*c^4*d^8 - 901120*a^16

*b^3*c^3*d^9 + 270336*a^17*b^2*c^2*d^10 - 49152*a^18*b*c*d^11))^(1/4) - ((((x^(1/2)*(36864*a^2*b^23*c^21*d^4 -

712704*a^3*b^22*c^20*d^5 + 6172672*a^4*b^21*c^19*d^6 - 31899648*a^5*b^20*c^18*d^7 + 110432256*a^6*b^19*c^17*d

^8 - 271552512*a^7*b^18*c^16*d^9 + 487280640*a^8*b^17*c^15*d^10 - 635523072*a^9*b^16*c^14*d^11 + 562982912*a^1

0*b^15*c^13*d^12 - 227217408*a^11*b^14*c^12*d^13 - 227217408*a^12*b^13*c^11*d^14 + 562982912*a^13*b^12*c^10*d^

15 - 635523072*a^14*b^11*c^9*d^16 + 487280640*a^15*b^10*c^8*d^17 - 271552512*a^16*b^9*c^7*d^18 + 110432256*a^1

7*b^8*c^6*d^19 - 31899648*a^18*b^7*c^5*d^20 + 6172672*a^19*b^6*c^4*d^21 - 712704*a^20*b^5*c^3*d^22 + 36864*a^2

1*b^4*c^2*d^23))/(16*(a^4*b^12*c^16 + a^16*c^4*d^12 - 12*a^5*b^11*c^15*d - 12*a^15*b*c^5*d^11 + 66*a^6*b^10*c^

14*d^2 - 220*a^7*b^9*c^13*d^3 + 495*a^8*b^8*c^12*d^4 - 792*a^9*b^7*c^11*d^5 + 924*a^10*b^6*c^10*d^6 - 792*a^11

*b^5*c^9*d^7 + 495*a^12*b^4*c^8*d^8 - 220*a^13*b^3*c^7*d^9 + 66*a^14*b^2*c^6*d^10)) - ((-(81*b^11*c^4 + 14641*