∫ √ _x _____(a+bx2 )2 (c+dx2 )2 dx

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

Optimal. Leaf size=624 \[ \frac {b^{5/4} (b c-9 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{5/4} (b c-9 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{5/4} (b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {b^{5/4} (b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {d^{5/4} (9 b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^3}-\frac {d^{5/4} (9 b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^3}-\frac {d^{5/4} (9 b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^3}+\frac {d^{5/4} (9 b c-a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^3}+\frac {d x^{3/2} (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2}+\frac {b x^{3/2}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]

[Out]

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

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

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

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

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

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

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

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

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

________________________________________________________________________________________

Rubi [A] time = 0.86, antiderivative size = 624, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 472, 579, 584, 297, 1162, 617, 204, 1165, 628} \[ \frac {b^{5/4} (b c-9 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{5/4} (b c-9 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{5/4} (b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {b^{5/4} (b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {d^{5/4} (9 b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^3}-\frac {d^{5/4} (9 b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^3}-\frac {d^{5/4} (9 b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^3}+\frac {d^{5/4} (9 b c-a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^3}+\frac {d x^{3/2} (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2}+\frac {b x^{3/2}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

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

b*c - 9*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (

b^(5/4)*(b*c - 9*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(5/4)*(b*c - a*

d)^3) + (d^(5/4)*(9*b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(5/4)*

(b*c - a*d)^3) - (d^(5/4)*(9*b*c - a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]

*c^(5/4)*(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 297

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

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

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

b]]))

Rule 466

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

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

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

rQ[p]

Rule 472

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

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

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

p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p

, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 579

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

_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +

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

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

e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 584

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

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

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

Rule 617

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

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

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

Rule 628

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

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

Rule 1162

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

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

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

Rule 1165

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

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

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

Rubi steps

\begin {align*} \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (-b c+4 a d-5 b d x^4\right )}{\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) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (-4 \left (b^2 c^2-8 a b c d+a^2 d^2\right )-4 b d (b c+a d) x^4\right )}{\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) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \left (-\frac {4 b^2 c (b c-9 a d) x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {4 a d^2 (-9 b c+a d) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{8 a c (b c-a d)^2}\\ &=\frac {d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^2 (b c-9 a d)\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a (b c-a d)^3}+\frac {\left (d^2 (9 b c-a d)\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c (b c-a d)^3}\\ &=\frac {d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (b^{3/2} (b c-9 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 (b c-a d)^3}+\frac {\left (b^{3/2} (b c-9 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 (b c-a d)^3}-\frac {\left (d^{3/2} (9 b c-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 (b c-a d)^3}+\frac {\left (d^{3/2} (9 b c-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 (b c-a d)^3}\\ &=\frac {d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {(b (b c-9 a d)) \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 (b c-a d)^3}+\frac {(b (b c-9 a d)) \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 (b c-a d)^3}+\frac {\left (b^{5/4} (b c-9 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^{5/4} (b c-a d)^3}+\frac {\left (b^{5/4} (b c-9 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^{5/4} (b c-a d)^3}+\frac {(d (9 b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 c (b c-a d)^3}+\frac {(d (9 b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 c (b c-a d)^3}+\frac {\left (d^{5/4} (9 b c-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^{5/4} (b c-a d)^3}+\frac {\left (d^{5/4} (9 b c-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^{5/4} (b c-a d)^3}\\ &=\frac {d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{5/4} (b c-9 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{5/4} (b c-9 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {d^{5/4} (9 b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^3}-\frac {d^{5/4} (9 b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^3}+\frac {\left (b^{5/4} (b c-9 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^{5/4} (b c-a d)^3}-\frac {\left (b^{5/4} (b c-9 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^{5/4} (b c-a d)^3}+\frac {\left (d^{5/4} (9 b c-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^{5/4} (b c-a d)^3}-\frac {\left (d^{5/4} (9 b c-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^{5/4} (b c-a d)^3}\\ &=\frac {d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{5/4} (b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {b^{5/4} (b c-9 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^3}-\frac {d^{5/4} (9 b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^3}+\frac {d^{5/4} (9 b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^3}+\frac {b^{5/4} (b c-9 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{5/4} (b c-9 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {d^{5/4} (9 b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^3}-\frac {d^{5/4} (9 b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A] time = 2.09, size = 589, normalized size = 0.94 \[ \frac {1}{16} \left (\frac {\sqrt {2} b^{5/4} (9 a d-b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{5/4} (a d-b c)^3}+\frac {\sqrt {2} b^{5/4} (b c-9 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{5/4} (a d-b c)^3}+\frac {2 \sqrt {2} b^{5/4} (b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{5/4} (a d-b c)^3}+\frac {2 \sqrt {2} b^{5/4} (9 a d-b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{5/4} (a d-b c)^3}+\frac {8 b^2 x^{3/2}}{a \left (a+b x^2\right ) (b c-a d)^2}+\frac {\sqrt {2} d^{5/4} (9 b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{5/4} (b c-a d)^3}+\frac {\sqrt {2} d^{5/4} (a d-9 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{5/4} (b c-a d)^3}+\frac {2 \sqrt {2} d^{5/4} (a d-9 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{5/4} (b c-a d)^3}+\frac {2 \sqrt {2} d^{5/4} (9 b c-a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{5/4} (b c-a d)^3}+\frac {8 d^2 x^{3/2}}{c \left (c+d x^2\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

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

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

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

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

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

)*(b*c - a*d)^3))/16

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

giac [B] time = 1.64, size = 973, normalized size = 1.56 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((a*b^3)^(3/4)*b*c - 9*(a*b^3)^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4)

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

^(3/4)*b*c - 9*(a*b^3)^(3/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*

a^2*b^4*c^3 - 3*sqrt(2)*a^3*b^3*c^2*d + 3*sqrt(2)*a^4*b^2*c*d^2 - sqrt(2)*a^5*b*d^3) + 1/4*(9*(c*d^3)^(3/4)*b*

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

3*sqrt(2)*a*b^2*c^4*d^2 + 3*sqrt(2)*a^2*b*c^3*d^3 - sqrt(2)*a^3*c^2*d^4) + 1/4*(9*(c*d^3)^(3/4)*b*c - (c*d^3)^

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

*b^2*c^4*d^2 + 3*sqrt(2)*a^2*b*c^3*d^3 - sqrt(2)*a^3*c^2*d^4) - 1/8*((a*b^3)^(3/4)*b*c - 9*(a*b^3)^(3/4)*a*d)*

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

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

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

d^3) - 1/8*(9*(c*d^3)^(3/4)*b*c - (c*d^3)^(3/4)*a*d)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)

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

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

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

^(3/2) + a^2*d^2*x^(3/2))/((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))

________________________________________________________________________________________

maple [A] time = 0.02, size = 778, normalized size = 1.25 \[ \frac {a \,d^{3} x^{\frac {3}{2}}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right ) c}-\frac {b^{3} c \,x^{\frac {3}{2}}}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right ) a}+\frac {b^{2} d \,x^{\frac {3}{2}}}{2 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )}-\frac {b \,d^{2} x^{\frac {3}{2}}}{2 \left (a d -b c \right )^{3} \left (d \,x^{2}+c \right )}+\frac {\sqrt {2}\, a \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}} c}+\frac {\sqrt {2}\, a \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}} c}+\frac {\sqrt {2}\, a \,d^{2} \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}} c}-\frac {\sqrt {2}\, b^{2} c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}} a}-\frac {\sqrt {2}\, b^{2} c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}} a}-\frac {\sqrt {2}\, b^{2} c \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}} a}+\frac {9 \sqrt {2}\, b d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {9 \sqrt {2}\, b d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {9 \sqrt {2}\, b d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {9 \sqrt {2}\, b d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {9 \sqrt {2}\, b d \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {9 \sqrt {2}\, b d \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (a d -b c \right )^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

-9/16*d/(a*d-b*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)))*b

________________________________________________________________________________________

maxima [A] time = 2.48, size = 610, normalized size = 0.98 \[ \frac {{\left (b^{3} c - 9 \, a b^{2} d\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{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 (9 \, b c d^{2} - a d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{16 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}} + \frac {{\left (b^{2} c d + a b d^{2}\right )} x^{\frac {7}{2}} + {\left (b^{2} c^{2} + a^{2} d^{2}\right )} x^{\frac {3}{2}}}{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 )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(b^3*c - 9*a*b^2*d)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt

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

t(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqr

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

t(a))/(a^(1/4)*b^(3/4)))/(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3) + 1/16*(9*b*c*d^2 - a*d^3)*(2

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

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

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

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

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

*x^(3/2))/(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)

________________________________________________________________________________________

mupad [B] time = 4.56, size = 32506, normalized size = 52.09 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan((((((32*a^19*b^4*d^23 + 32*b^23*c^19*d^4 - 1216*a*b^22*c^18*d^5 - 1216*a^18*b^5*c*d^22 + 19040*a^2*b^21

*c^17*d^6 - 161664*a^3*b^20*c^16*d^7 + 837408*a^4*b^19*c^15*d^8 - 2842656*a^5*b^18*c^14*d^9 + 6564768*a^6*b^17

*c^13*d^10 - 10331040*a^7*b^16*c^12*d^11 + 10374112*a^8*b^15*c^11*d^12 - 4458784*a^9*b^14*c^10*d^13 - 4458784*

a^10*b^13*c^9*d^14 + 10374112*a^11*b^12*c^8*d^15 - 10331040*a^12*b^11*c^7*d^16 + 6564768*a^13*b^10*c^6*d^17 -

2842656*a^14*b^9*c^5*d^18 + 837408*a^15*b^8*c^4*d^19 - 161664*a^16*b^7*c^3*d^20 + 19040*a^17*b^6*c^2*d^21)*1i)

/(a^2*b^14*c^16 + a^16*c^2*d^14 - 14*a^3*b^13*c^15*d - 14*a^15*b*c^3*d^13 + 91*a^4*b^12*c^14*d^2 - 364*a^5*b^1

1*c^13*d^3 + 1001*a^6*b^10*c^12*d^4 - 2002*a^7*b^9*c^11*d^5 + 3003*a^8*b^8*c^10*d^6 - 3432*a^9*b^7*c^9*d^7 + 3

003*a^10*b^6*c^8*d^8 - 2002*a^11*b^5*c^7*d^9 + 1001*a^12*b^4*c^6*d^10 - 364*a^13*b^3*c^5*d^11 + 91*a^14*b^2*c^

4*d^12) - (x^(1/2)*(-(a^4*d^9 + 6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8)/

(4096*b^12*c^17 + 4096*a^12*c^5*d^12 - 49152*a^11*b*c^6*d^11 + 270336*a^2*b^10*c^15*d^2 - 901120*a^3*b^9*c^14*

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

7 + 2027520*a^8*b^4*c^9*d^8 - 901120*a^9*b^3*c^8*d^9 + 270336*a^10*b^2*c^7*d^10 - 49152*a*b^11*c^16*d))^(1/4)*

(4096*a*b^22*c^19*d^4 + 4096*a^19*b^4*c*d^22 - 122880*a^2*b^21*c^18*d^5 + 1486848*a^3*b^20*c^17*d^6 - 9748480*

a^4*b^19*c^16*d^7 + 40476672*a^5*b^18*c^15*d^8 - 116785152*a^6*b^17*c^14*d^9 + 249192448*a^7*b^16*c^13*d^10 -

412041216*a^8*b^15*c^12*d^11 + 547700736*a^9*b^14*c^11*d^12 - 600326144*a^10*b^13*c^10*d^13 + 547700736*a^11*b

^12*c^9*d^14 - 412041216*a^12*b^11*c^8*d^15 + 249192448*a^13*b^10*c^7*d^16 - 116785152*a^14*b^9*c^6*d^17 + 404

76672*a^15*b^8*c^5*d^18 - 9748480*a^16*b^7*c^4*d^19 + 1486848*a^17*b^6*c^3*d^20 - 122880*a^18*b^5*c^2*d^21))/(

16*(a^2*b^12*c^14 + a^14*c^2*d^12 - 12*a^3*b^11*c^13*d - 12*a^13*b*c^3*d^11 + 66*a^4*b^10*c^12*d^2 - 220*a^5*b

^9*c^11*d^3 + 495*a^6*b^8*c^10*d^4 - 792*a^7*b^7*c^9*d^5 + 924*a^8*b^6*c^8*d^6 - 792*a^9*b^5*c^7*d^7 + 495*a^1

0*b^4*c^6*d^8 - 220*a^11*b^3*c^5*d^9 + 66*a^12*b^2*c^4*d^10)))*(-(a^4*d^9 + 6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*

d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8)/(4096*b^12*c^17 + 4096*a^12*c^5*d^12 - 49152*a^11*b*c^6*d^11 + 270

336*a^2*b^10*c^15*d^2 - 901120*a^3*b^9*c^14*d^3 + 2027520*a^4*b^8*c^13*d^4 - 3244032*a^5*b^7*c^12*d^5 + 378470

4*a^6*b^6*c^11*d^6 - 3244032*a^7*b^5*c^10*d^7 + 2027520*a^8*b^4*c^9*d^8 - 901120*a^9*b^3*c^8*d^9 + 270336*a^10

*b^2*c^7*d^10 - 49152*a*b^11*c^16*d))^(3/4) - (x^(1/2)*(81*a^7*b^8*d^15 + 81*b^15*c^7*d^8 + 3627*a*b^14*c^6*d^

9 + 3627*a^6*b^9*c*d^14 - 80999*a^2*b^13*c^5*d^10 + 339435*a^3*b^12*c^4*d^11 + 339435*a^4*b^11*c^3*d^12 - 8099

9*a^5*b^10*c^2*d^13))/(16*(a^2*b^12*c^14 + a^14*c^2*d^12 - 12*a^3*b^11*c^13*d - 12*a^13*b*c^3*d^11 + 66*a^4*b^

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

9*b^5*c^7*d^7 + 495*a^10*b^4*c^6*d^8 - 220*a^11*b^3*c^5*d^9 + 66*a^12*b^2*c^4*d^10)))*(-(a^4*d^9 + 6561*b^4*c^

4*d^5 - 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8)/(4096*b^12*c^17 + 4096*a^12*c^5*d^12 - 4915

2*a^11*b*c^6*d^11 + 270336*a^2*b^10*c^15*d^2 - 901120*a^3*b^9*c^14*d^3 + 2027520*a^4*b^8*c^13*d^4 - 3244032*a^

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

3*c^8*d^9 + 270336*a^10*b^2*c^7*d^10 - 49152*a*b^11*c^16*d))^(1/4) - ((((32*a^19*b^4*d^23 + 32*b^23*c^19*d^4 -

1216*a*b^22*c^18*d^5 - 1216*a^18*b^5*c*d^22 + 19040*a^2*b^21*c^17*d^6 - 161664*a^3*b^20*c^16*d^7 + 837408*a^4

*b^19*c^15*d^8 - 2842656*a^5*b^18*c^14*d^9 + 6564768*a^6*b^17*c^13*d^10 - 10331040*a^7*b^16*c^12*d^11 + 103741

12*a^8*b^15*c^11*d^12 - 4458784*a^9*b^14*c^10*d^13 - 4458784*a^10*b^13*c^9*d^14 + 10374112*a^11*b^12*c^8*d^15

- 10331040*a^12*b^11*c^7*d^16 + 6564768*a^13*b^10*c^6*d^17 - 2842656*a^14*b^9*c^5*d^18 + 837408*a^15*b^8*c^4*d

^19 - 161664*a^16*b^7*c^3*d^20 + 19040*a^17*b^6*c^2*d^21)*1i)/(a^2*b^14*c^16 + a^16*c^2*d^14 - 14*a^3*b^13*c^1

5*d - 14*a^15*b*c^3*d^13 + 91*a^4*b^12*c^14*d^2 - 364*a^5*b^11*c^13*d^3 + 1001*a^6*b^10*c^12*d^4 - 2002*a^7*b^

9*c^11*d^5 + 3003*a^8*b^8*c^10*d^6 - 3432*a^9*b^7*c^9*d^7 + 3003*a^10*b^6*c^8*d^8 - 2002*a^11*b^5*c^7*d^9 + 10

01*a^12*b^4*c^6*d^10 - 364*a^13*b^3*c^5*d^11 + 91*a^14*b^2*c^4*d^12) + (x^(1/2)*(-(a^4*d^9 + 6561*b^4*c^4*d^5

- 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8)/(4096*b^12*c^17 + 4096*a^12*c^5*d^12 - 49152*a^11

*b*c^6*d^11 + 270336*a^2*b^10*c^15*d^2 - 901120*a^3*b^9*c^14*d^3 + 2027520*a^4*b^8*c^13*d^4 - 3244032*a^5*b^7*

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

d^9 + 270336*a^10*b^2*c^7*d^10 - 49152*a*b^11*c^16*d))^(1/4)*(4096*a*b^22*c^19*d^4 + 4096*a^19*b^4*c*d^22 - 12

2880*a^2*b^21*c^18*d^5 + 1486848*a^3*b^20*c^17*d^6 - 9748480*a^4*b^19*c^16*d^7 + 40476672*a^5*b^18*c^15*d^8 -

116785152*a^6*b^17*c^14*d^9 + 249192448*a^7*b^16*c^13*d^10 - 412041216*a^8*b^15*c^12*d^11 + 547700736*a^9*b^14

*c^11*d^12 - 600326144*a^10*b^13*c^10*d^13 + 547700736*a^11*b^12*c^9*d^14 - 412041216*a^12*b^11*c^8*d^15 + 249

192448*a^13*b^10*c^7*d^16 - 116785152*a^14*b^9*c^6*d^17 + 40476672*a^15*b^8*c^5*d^18 - 9748480*a^16*b^7*c^4*d^

19 + 1486848*a^17*b^6*c^3*d^20 - 122880*a^18*b^5*c^2*d^21))/(16*(a^2*b^12*c^14 + a^14*c^2*d^12 - 12*a^3*b^11*c

^13*d - 12*a^13*b*c^3*d^11 + 66*a^4*b^10*c^12*d^2 - 220*a^5*b^9*c^11*d^3 + 495*a^6*b^8*c^10*d^4 - 792*a^7*b^7*

c^9*d^5 + 924*a^8*b^6*c^8*d^6 - 792*a^9*b^5*c^7*d^7 + 495*a^10*b^4*c^6*d^8 - 220*a^11*b^3*c^5*d^9 + 66*a^12*b^

2*c^4*d^10)))*(-(a^4*d^9 + 6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8)/(4096

*b^12*c^17 + 4096*a^12*c^5*d^12 - 49152*a^11*b*c^6*d^11 + 270336*a^2*b^10*c^15*d^2 - 901120*a^3*b^9*c^14*d^3 +

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

027520*a^8*b^4*c^9*d^8 - 901120*a^9*b^3*c^8*d^9 + 270336*a^10*b^2*c^7*d^10 - 49152*a*b^11*c^16*d))^(3/4) + (x^

(1/2)*(81*a^7*b^8*d^15 + 81*b^15*c^7*d^8 + 3627*a*b^14*c^6*d^9 + 3627*a^6*b^9*c*d^14 - 80999*a^2*b^13*c^5*d^10

+ 339435*a^3*b^12*c^4*d^11 + 339435*a^4*b^11*c^3*d^12 - 80999*a^5*b^10*c^2*d^13))/(16*(a^2*b^12*c^14 + a^14*c

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

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

^3*c^5*d^9 + 66*a^12*b^2*c^4*d^10)))*(-(a^4*d^9 + 6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7

- 36*a^3*b*c*d^8)/(4096*b^12*c^17 + 4096*a^12*c^5*d^12 - 49152*a^11*b*c^6*d^11 + 270336*a^2*b^10*c^15*d^2 - 90

1120*a^3*b^9*c^14*d^3 + 2027520*a^4*b^8*c^13*d^4 - 3244032*a^5*b^7*c^12*d^5 + 3784704*a^6*b^6*c^11*d^6 - 32440

32*a^7*b^5*c^10*d^7 + 2027520*a^8*b^4*c^9*d^8 - 901120*a^9*b^3*c^8*d^9 + 270336*a^10*b^2*c^7*d^10 - 49152*a*b^

11*c^16*d))^(1/4))/(((((32*a^19*b^4*d^23 + 32*b^23*c^19*d^4 - 1216*a*b^22*c^18*d^5 - 1216*a^18*b^5*c*d^22 + 19

040*a^2*b^21*c^17*d^6 - 161664*a^3*b^20*c^16*d^7 + 837408*a^4*b^19*c^15*d^8 - 2842656*a^5*b^18*c^14*d^9 + 6564

768*a^6*b^17*c^13*d^10 - 10331040*a^7*b^16*c^12*d^11 + 10374112*a^8*b^15*c^11*d^12 - 4458784*a^9*b^14*c^10*d^1

3 - 4458784*a^10*b^13*c^9*d^14 + 10374112*a^11*b^12*c^8*d^15 - 10331040*a^12*b^11*c^7*d^16 + 6564768*a^13*b^10

*c^6*d^17 - 2842656*a^14*b^9*c^5*d^18 + 837408*a^15*b^8*c^4*d^19 - 161664*a^16*b^7*c^3*d^20 + 19040*a^17*b^6*c

^2*d^21)*1i)/(a^2*b^14*c^16 + a^16*c^2*d^14 - 14*a^3*b^13*c^15*d - 14*a^15*b*c^3*d^13 + 91*a^4*b^12*c^14*d^2 -

364*a^5*b^11*c^13*d^3 + 1001*a^6*b^10*c^12*d^4 - 2002*a^7*b^9*c^11*d^5 + 3003*a^8*b^8*c^10*d^6 - 3432*a^9*b^7

*c^9*d^7 + 3003*a^10*b^6*c^8*d^8 - 2002*a^11*b^5*c^7*d^9 + 1001*a^12*b^4*c^6*d^10 - 364*a^13*b^3*c^5*d^11 + 91

*a^14*b^2*c^4*d^12) - (x^(1/2)*(-(a^4*d^9 + 6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a

^3*b*c*d^8)/(4096*b^12*c^17 + 4096*a^12*c^5*d^12 - 49152*a^11*b*c^6*d^11 + 270336*a^2*b^10*c^15*d^2 - 901120*a

^3*b^9*c^14*d^3 + 2027520*a^4*b^8*c^13*d^4 - 3244032*a^5*b^7*c^12*d^5 + 3784704*a^6*b^6*c^11*d^6 - 3244032*a^7

*b^5*c^10*d^7 + 2027520*a^8*b^4*c^9*d^8 - 901120*a^9*b^3*c^8*d^9 + 270336*a^10*b^2*c^7*d^10 - 49152*a*b^11*c^1

6*d))^(1/4)*(4096*a*b^22*c^19*d^4 + 4096*a^19*b^4*c*d^22 - 122880*a^2*b^21*c^18*d^5 + 1486848*a^3*b^20*c^17*d^

6 - 9748480*a^4*b^19*c^16*d^7 + 40476672*a^5*b^18*c^15*d^8 - 116785152*a^6*b^17*c^14*d^9 + 249192448*a^7*b^16*

c^13*d^10 - 412041216*a^8*b^15*c^12*d^11 + 547700736*a^9*b^14*c^11*d^12 - 600326144*a^10*b^13*c^10*d^13 + 5477

00736*a^11*b^12*c^9*d^14 - 412041216*a^12*b^11*c^8*d^15 + 249192448*a^13*b^10*c^7*d^16 - 116785152*a^14*b^9*c^

6*d^17 + 40476672*a^15*b^8*c^5*d^18 - 9748480*a^16*b^7*c^4*d^19 + 1486848*a^17*b^6*c^3*d^20 - 122880*a^18*b^5*

c^2*d^21))/(16*(a^2*b^12*c^14 + a^14*c^2*d^12 - 12*a^3*b^11*c^13*d - 12*a^13*b*c^3*d^11 + 66*a^4*b^10*c^12*d^2

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

^7 + 495*a^10*b^4*c^6*d^8 - 220*a^11*b^3*c^5*d^9 + 66*a^12*b^2*c^4*d^10)))*(-(a^4*d^9 + 6561*b^4*c^4*d^5 - 291

6*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8)/(4096*b^12*c^17 + 4096*a^12*c^5*d^12 - 49152*a^11*b*c^

6*d^11 + 270336*a^2*b^10*c^15*d^2 - 901120*a^3*b^9*c^14*d^3 + 2027520*a^4*b^8*c^13*d^4 - 3244032*a^5*b^7*c^12*

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

270336*a^10*b^2*c^7*d^10 - 49152*a*b^11*c^16*d))^(3/4)*1i - (x^(1/2)*(81*a^7*b^8*d^15 + 81*b^15*c^7*d^8 + 362

7*a*b^14*c^6*d^9 + 3627*a^6*b^9*c*d^14 - 80999*a^2*b^13*c^5*d^10 + 339435*a^3*b^12*c^4*d^11 + 339435*a^4*b^11*

c^3*d^12 - 80999*a^5*b^10*c^2*d^13)*1i)/(16*(a^2*b^12*c^14 + a^14*c^2*d^12 - 12*a^3*b^11*c^13*d - 12*a^13*b*c^

3*d^11 + 66*a^4*b^10*c^12*d^2 - 220*a^5*b^9*c^11*d^3 + 495*a^6*b^8*c^10*d^4 - 792*a^7*b^7*c^9*d^5 + 924*a^8*b^

6*c^8*d^6 - 792*a^9*b^5*c^7*d^7 + 495*a^10*b^4*c^6*d^8 - 220*a^11*b^3*c^5*d^9 + 66*a^12*b^2*c^4*d^10)))*(-(a^4

*d^9 + 6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8)/(4096*b^12*c^17 + 4096*a^

12*c^5*d^12 - 49152*a^11*b*c^6*d^11 + 270336*a^2*b^10*c^15*d^2 - 901120*a^3*b^9*c^14*d^3 + 2027520*a^4*b^8*c^1

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

^8 - 901120*a^9*b^3*c^8*d^9 + 270336*a^10*b^2*c^7*d^10 - 49152*a*b^11*c^16*d))^(1/4) - ((3645*a^6*b^9*d^15)/32

+ (3645*b^15*c^6*d^9)/32 - (49815*a*b^14*c^5*d^10)/16 - (49815*a^5*b^10*c*d^14)/16 + (918675*a^2*b^13*c^4*d^1

1)/32 - (739025*a^3*b^12*c^3*d^12)/8 + (918675*a^4*b^11*c^2*d^13)/32)/(a^2*b^14*c^16 + a^16*c^2*d^14 - 14*a^3*

b^13*c^15*d - 14*a^15*b*c^3*d^13 + 91*a^4*b^12*c^14*d^2 - 364*a^5*b^11*c^13*d^3 + 1001*a^6*b^10*c^12*d^4 - 200

2*a^7*b^9*c^11*d^5 + 3003*a^8*b^8*c^10*d^6 - 3432*a^9*b^7*c^9*d^7 + 3003*a^10*b^6*c^8*d^8 - 2002*a^11*b^5*c^7*

d^9 + 1001*a^12*b^4*c^6*d^10 - 364*a^13*b^3*c^5*d^11 + 91*a^14*b^2*c^4*d^12) + ((((32*a^19*b^4*d^23 + 32*b^23*

c^19*d^4 - 1216*a*b^22*c^18*d^5 - 1216*a^18*b^5*c*d^22 + 19040*a^2*b^21*c^17*d^6 - 161664*a^3*b^20*c^16*d^7 +

837408*a^4*b^19*c^15*d^8 - 2842656*a^5*b^18*c^14*d^9 + 6564768*a^6*b^17*c^13*d^10 - 10331040*a^7*b^16*c^12*d^1

1 + 10374112*a^8*b^15*c^11*d^12 - 4458784*a^9*b^14*c^10*d^13 - 4458784*a^10*b^13*c^9*d^14 + 10374112*a^11*b^12

*c^8*d^15 - 10331040*a^12*b^11*c^7*d^16 + 6564768*a^13*b^10*c^6*d^17 - 2842656*a^14*b^9*c^5*d^18 + 837408*a^15

*b^8*c^4*d^19 - 161664*a^16*b^7*c^3*d^20 + 19040*a^17*b^6*c^2*d^21)*1i)/(a^2*b^14*c^16 + a^16*c^2*d^14 - 14*a^

3*b^13*c^15*d - 14*a^15*b*c^3*d^13 + 91*a^4*b^12*c^14*d^2 - 364*a^5*b^11*c^13*d^3 + 1001*a^6*b^10*c^12*d^4 - 2

002*a^7*b^9*c^11*d^5 + 3003*a^8*b^8*c^10*d^6 - 3432*a^9*b^7*c^9*d^7 + 3003*a^10*b^6*c^8*d^8 - 2002*a^11*b^5*c^

7*d^9 + 1001*a^12*b^4*c^6*d^10 - 364*a^13*b^3*c^5*d^11 + 91*a^14*b^2*c^4*d^12) + (x^(1/2)*(-(a^4*d^9 + 6561*b^

4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8)/(4096*b^12*c^17 + 4096*a^12*c^5*d^12 -

49152*a^11*b*c^6*d^11 + 270336*a^2*b^10*c^15*d^2 - 901120*a^3*b^9*c^14*d^3 + 2027520*a^4*b^8*c^13*d^4 - 324403

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

9*b^3*c^8*d^9 + 270336*a^10*b^2*c^7*d^10 - 49152*a*b^11*c^16*d))^(1/4)*(4096*a*b^22*c^19*d^4 + 4096*a^19*b^4*c

*d^22 - 122880*a^2*b^21*c^18*d^5 + 1486848*a^3*b^20*c^17*d^6 - 9748480*a^4*b^19*c^16*d^7 + 40476672*a^5*b^18*c

^15*d^8 - 116785152*a^6*b^17*c^14*d^9 + 249192448*a^7*b^16*c^13*d^10 - 412041216*a^8*b^15*c^12*d^11 + 54770073

6*a^9*b^14*c^11*d^12 - 600326144*a^10*b^13*c^10*d^13 + 547700736*a^11*b^12*c^9*d^14 - 412041216*a^12*b^11*c^8*

d^15 + 249192448*a^13*b^10*c^7*d^16 - 116785152*a^14*b^9*c^6*d^17 + 40476672*a^15*b^8*c^5*d^18 - 9748480*a^16*

b^7*c^4*d^19 + 1486848*a^17*b^6*c^3*d^20 - 122880*a^18*b^5*c^2*d^21))/(16*(a^2*b^12*c^14 + a^14*c^2*d^12 - 12*

a^3*b^11*c^13*d - 12*a^13*b*c^3*d^11 + 66*a^4*b^10*c^12*d^2 - 220*a^5*b^9*c^11*d^3 + 495*a^6*b^8*c^10*d^4 - 79

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

66*a^12*b^2*c^4*d^10)))*(-(a^4*d^9 + 6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*

d^8)/(4096*b^12*c^17 + 4096*a^12*c^5*d^12 - 49152*a^11*b*c^6*d^11 + 270336*a^2*b^10*c^15*d^2 - 901120*a^3*b^9*

c^14*d^3 + 2027520*a^4*b^8*c^13*d^4 - 3244032*a^5*b^7*c^12*d^5 + 3784704*a^6*b^6*c^11*d^6 - 3244032*a^7*b^5*c^

10*d^7 + 2027520*a^8*b^4*c^9*d^8 - 901120*a^9*b^3*c^8*d^9 + 270336*a^10*b^2*c^7*d^10 - 49152*a*b^11*c^16*d))^(

3/4)*1i + (x^(1/2)*(81*a^7*b^8*d^15 + 81*b^15*c^7*d^8 + 3627*a*b^14*c^6*d^9 + 3627*a^6*b^9*c*d^14 - 80999*a^2*

b^13*c^5*d^10 + 339435*a^3*b^12*c^4*d^11 + 339435*a^4*b^11*c^3*d^12 - 80999*a^5*b^10*c^2*d^13)*1i)/(16*(a^2*b^

12*c^14 + a^14*c^2*d^12 - 12*a^3*b^11*c^13*d - 12*a^13*b*c^3*d^11 + 66*a^4*b^10*c^12*d^2 - 220*a^5*b^9*c^11*d^

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

d^8 - 220*a^11*b^3*c^5*d^9 + 66*a^12*b^2*c^4*d^10)))*(-(a^4*d^9 + 6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 486*

a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8)/(4096*b^12*c^17 + 4096*a^12*c^5*d^12 - 49152*a^11*b*c^6*d^11 + 270336*a^2*b^

10*c^15*d^2 - 901120*a^3*b^9*c^14*d^3 + 2027520*a^4*b^8*c^13*d^4 - 3244032*a^5*b^7*c^12*d^5 + 3784704*a^6*b^6*

c^11*d^6 - 3244032*a^7*b^5*c^10*d^7 + 2027520*a^8*b^4*c^9*d^8 - 901120*a^9*b^3*c^8*d^9 + 270336*a^10*b^2*c^7*d

^10 - 49152*a*b^11*c^16*d))^(1/4)))*(-(a^4*d^9 + 6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 -

36*a^3*b*c*d^8)/(4096*b^12*c^17 + 4096*a^12*c^5*d^12 - 49152*a^11*b*c^6*d^11 + 270336*a^2*b^10*c^15*d^2 - 901

120*a^3*b^9*c^14*d^3 + 2027520*a^4*b^8*c^13*d^4 - 3244032*a^5*b^7*c^12*d^5 + 3784704*a^6*b^6*c^11*d^6 - 324403

2*a^7*b^5*c^10*d^7 + 2027520*a^8*b^4*c^9*d^8 - 901120*a^9*b^3*c^8*d^9 + 270336*a^10*b^2*c^7*d^10 - 49152*a*b^1

1*c^16*d))^(1/4) - atan(((((32*a^19*b^4*d^23 + 32*b^23*c^19*d^4 - 1216*a*b^22*c^18*d^5 - 1216*a^18*b^5*c*d^22

+ 19040*a^2*b^21*c^17*d^6 - 161664*a^3*b^20*c^16*d^7 + 837408*a^4*b^19*c^15*d^8 - 2842656*a^5*b^18*c^14*d^9 +

6564768*a^6*b^17*c^13*d^10 - 10331040*a^7*b^16*c^12*d^11 + 10374112*a^8*b^15*c^11*d^12 - 4458784*a^9*b^14*c^10

*d^13 - 4458784*a^10*b^13*c^9*d^14 + 10374112*a^11*b^12*c^8*d^15 - 10331040*a^12*b^11*c^7*d^16 + 6564768*a^13*

b^10*c^6*d^17 - 2842656*a^14*b^9*c^5*d^18 + 837408*a^15*b^8*c^4*d^19 - 161664*a^16*b^7*c^3*d^20 + 19040*a^17*b

^6*c^2*d^21)/(a^2*b^14*c^16 + a^16*c^2*d^14 - 14*a^3*b^13*c^15*d - 14*a^15*b*c^3*d^13 + 91*a^4*b^12*c^14*d^2 -