∫ 1_______ x3∕2(a+bx2)(c+dx2)2 dx

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

Optimal. Leaf size=570 \[ -\frac {b^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}+\frac {b^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}+\frac {b^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^2}-\frac {b^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{5/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {4 b c-5 a d}{2 a c^2 \sqrt {x} (b c-a d)}-\frac {d}{2 c \sqrt {x} \left (c+d x^2\right ) (b c-a d)} \]

[Out]

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

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

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

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

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

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

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

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Rubi [A] time = 0.75, antiderivative size = 570, 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, 583, 584, 297, 1162, 617, 204, 1165, 628} \[ -\frac {b^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}+\frac {b^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}+\frac {b^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^2}-\frac {b^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{5/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {4 b c-5 a d}{2 a c^2 \sqrt {x} (b c-a d)}-\frac {d}{2 c \sqrt {x} \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

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

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

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

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

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

Rule 204

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

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

Rule 297

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

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

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

b]]))

Rule 466

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

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

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

rQ[p]

Rule 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 583

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

x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*

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

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

IGtQ[n, 0] && LtQ[m, -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 {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {4 b c-5 a d-5 b d x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{2 c (b c-a d)}\\ &=-\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (4 b^2 c^2+4 a b c d-5 a^2 d^2+b d (4 b c-5 a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{2 a c^2 (b c-a d)}\\ &=-\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}-\frac {\operatorname {Subst}\left (\int \left (\frac {4 b^3 c^2 x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {a d^2 (-9 b c+5 a d) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a c^2 (b c-a d)}\\ &=-\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}-\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^2}+\frac {\left (d^2 (9 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c^2 (b c-a d)^2}\\ &=-\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}+\frac {b^{5/2} \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^2}-\frac {b^{5/2} \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^2}-\frac {\left (d^{3/2} (9 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^2 (b c-a d)^2}+\frac {\left (d^{3/2} (9 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^2 (b c-a d)^2}\\ &=-\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}-\frac {b^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 a (b c-a d)^2}-\frac {b^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 a (b c-a d)^2}-\frac {b^{9/4} \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^{5/4} (b c-a d)^2}-\frac {b^{9/4} \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^{5/4} (b c-a d)^2}+\frac {(d (9 b c-5 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^2 (b c-a d)^2}+\frac {(d (9 b c-5 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^2 (b c-a d)^2}+\frac {\left (d^{5/4} (9 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}+\frac {\left (d^{5/4} (9 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}\\ &=-\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}-\frac {b^{9/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}+\frac {b^{9/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {b^{9/4} \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^{5/4} (b c-a d)^2}+\frac {b^{9/4} \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^{5/4} (b c-a d)^2}+\frac {\left (d^{5/4} (9 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {\left (d^{5/4} (9 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}\\ &=-\frac {4 b c-5 a d}{2 a c^2 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) \sqrt {x} \left (c+d x^2\right )}+\frac {b^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^2}-\frac {b^{9/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {b^{9/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}+\frac {b^{9/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^2}+\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}-\frac {d^{5/4} (9 b c-5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^2}\\ \end {align*}

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Mathematica [A] time = 2.17, size = 540, normalized size = 0.95 \[ \frac {1}{16} \left (-\frac {4 \sqrt {2} b^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{5/4} (b c-a d)^2}+\frac {4 \sqrt {2} b^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{5/4} (b c-a d)^2}+\frac {8 \sqrt {2} b^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{5/4} (b c-a d)^2}-\frac {8 \sqrt {2} b^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{5/4} (b c-a d)^2}+\frac {\sqrt {2} d^{5/4} (9 b c-5 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{9/4} (b c-a d)^2}+\frac {\sqrt {2} d^{5/4} (5 a d-9 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{9/4} (b c-a d)^2}+\frac {2 \sqrt {2} d^{5/4} (5 a d-9 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{9/4} (b c-a d)^2}+\frac {2 \sqrt {2} d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{9/4} (b c-a d)^2}+\frac {8 d^2 x^{3/2}}{c^2 \left (c+d x^2\right ) (b c-a d)}-\frac {32}{a c^2 \sqrt {x}}\right ) \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-32/(a*c^2*Sqrt[x]) + (8*d^2*x^(3/2))/(c^2*(b*c - a*d)*(c + d*x^2)) + (8*Sqrt[2]*b^(9/4)*ArcTan[1 - (Sqrt[2]*

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

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

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

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

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

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

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

+ Sqrt[d]*x])/(c^(9/4)*(b*c - a*d)^2))/16

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fricas [B] time = 39.93, size = 3630, normalized size = 6.37 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*((a*b*c^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 1

2150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a

^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9

*d^8))^(1/4)*arctan(((b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*sqrt((531441*b^6*c^6*d^8 - 1771470*a*b^5*c^5*d^9 +

2460375*a^2*b^4*c^4*d^10 - 1822500*a^3*b^3*c^3*d^11 + 759375*a^4*b^2*c^2*d^12 - 168750*a^5*b*c*d^13 + 15625*a^

6*d^14)*x - (6561*b^8*c^13*d^5 - 40824*a*b^7*c^12*d^6 + 109836*a^2*b^6*c^11*d^7 - 166824*a^3*b^5*c^10*d^8 + 15

6406*a^4*b^4*c^9*d^9 - 92680*a^5*b^3*c^8*d^10 + 33900*a^6*b^2*c^7*d^11 - 7000*a^7*b*c^6*d^12 + 625*a^8*c^5*d^1

3)*sqrt(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^

8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^

5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8)))*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*

a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^

5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8)

)^(1/4) + (729*b^5*c^7*d^4 - 2673*a*b^4*c^6*d^5 + 3834*a^2*b^3*c^5*d^6 - 2690*a^3*b^2*c^4*d^7 + 925*a^4*b*c^3*

d^8 - 125*a^5*c^2*d^9)*sqrt(x)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*

c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*

d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(1/4))/(6561*b^4*c^4*d^5 -

14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)) + 16*(-b^9/(a^5*b^8*c^8 - 8*a^6

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

c^2*d^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(1/4)*((a*b*c^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*arctan((

sqrt(b^14*x - (a^3*b^13*c^4 - 4*a^4*b^12*c^3*d + 6*a^5*b^11*c^2*d^2 - 4*a^6*b^10*c*d^3 + a^7*b^9*d^4)*sqrt(-b^

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

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

*c^6*d^2 - 56*a^8*b^5*c^5*d^3 + 70*a^9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b*c*d^

7 + a^13*d^8))^(1/4)*(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2) - (a*b^9*c^2 - 2*a^2*b^8*c*d + a^3*b^7*d^2)*(-b^9/(a^

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

^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(1/4)*sqrt(x))/b^9) - 4*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*

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

^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(1/4)*((a*b*c^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*log(b^7*sqrt(

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

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

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

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

^11*b^2*c^2*d^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(1/4)*((a*b*c^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*

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

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

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

^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c

^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^

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

og((b^6*c^13 - 6*a*b^5*c^12*d + 15*a^2*b^4*c^11*d^2 - 20*a^3*b^3*c^10*d^3 + 15*a^4*b^2*c^9*d^4 - 6*a^5*b*c^8*d

^5 + a^6*c^7*d^6)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a

^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*

b^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(3/4) - (729*b^3*c^3*d^4 - 1215*a*b^2*c^

2*d^5 + 675*a^2*b*c*d^6 - 125*a^3*d^7)*sqrt(x)) + ((a*b*c^3*d - a^2*c^2*d^2)*x^3 + (a*b*c^4 - a^2*c^3*d)*x)*(-

(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 -

8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^

6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(1/4)*log(-(b^6*c^13 - 6*a*b^5*c^12*d + 15*a^2*b^4*c^11*d^2

- 20*a^3*b^3*c^10*d^3 + 15*a^4*b^2*c^9*d^4 - 6*a^5*b*c^8*d^5 + a^6*c^7*d^6)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*

c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15

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

7 + a^8*c^9*d^8))^(3/4) - (729*b^3*c^3*d^4 - 1215*a*b^2*c^2*d^5 + 675*a^2*b*c*d^6 - 125*a^3*d^7)*sqrt(x)) - 4*

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

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giac [A] time = 1.17, size = 725, normalized size = 1.27 \[ \frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{5} d - 2 \, \sqrt {2} a b c^{4} d^{2} + \sqrt {2} a^{2} c^{3} d^{3}\right )}} + \frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} b^{2} c^{5} d - 2 \, \sqrt {2} a b c^{4} d^{2} + \sqrt {2} a^{2} c^{3} d^{3}\right )}} - \frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{5} d - 2 \, \sqrt {2} a b c^{4} d^{2} + \sqrt {2} a^{2} c^{3} d^{3}\right )}} + \frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{5} d - 2 \, \sqrt {2} a b c^{4} d^{2} + \sqrt {2} a^{2} c^{3} d^{3}\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} c^{2} - 2 \, \sqrt {2} a^{3} b c d + \sqrt {2} a^{4} d^{2}\right )}} - \frac {4 \, b c d x^{2} - 5 \, a d^{2} x^{2} + 4 \, b c^{2} - 4 \, a c d}{2 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} {\left (d x^{\frac {5}{2}} + c \sqrt {x}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

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

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

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

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

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

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

d)*(d*x^(5/2) + c*sqrt(x)))

________________________________________________________________________________________

maple [A] time = 0.02, size = 582, normalized size = 1.02 \[ -\frac {a \,d^{3} x^{\frac {3}{2}}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right ) c^{2}}+\frac {b \,d^{2} x^{\frac {3}{2}}}{2 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right ) c}-\frac {5 \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 )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2}}-\frac {5 \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 )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2}}-\frac {5 \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 )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2}}-\frac {\sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} a}-\frac {\sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} a}-\frac {\sqrt {2}\, b^{2} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} a}+\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 )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} c}+\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 )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} c}+\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 )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}} c}-\frac {2}{a \,c^{2} \sqrt {x}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

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

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

/x^(1/2)

________________________________________________________________________________________

maxima [A] time = 2.58, size = 494, normalized size = 0.87 \[ -\frac {b^{3} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac {{\left (9 \, b c d^{2} - 5 \, 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^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} - \frac {4 \, b c^{2} - 4 \, a c d + {\left (4 \, b c d - 5 \, a d^{2}\right )} x^{2}}{2 \, {\left ({\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{\frac {5}{2}} + {\left (a b c^{4} - a^{2} c^{3} d\right )} \sqrt {x}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

tan(-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))*s

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

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

d)*sqrt(x))

________________________________________________________________________________________

mupad [B] time = 4.09, size = 21370, normalized size = 37.49 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 11

20*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^

8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 89

6*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(3/4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*

c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*

d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*

d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500

742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200

*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^

23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26

*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*

c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52

428800*a^33*b^4*c^23*d^25) - 16777216*a^11*b^25*c^42*d^4 + 218103808*a^12*b^24*c^41*d^5 - 1308622848*a^13*b^23

*c^40*d^6 + 4798283776*a^14*b^22*c^39*d^7 - 11995709440*a^15*b^21*c^38*d^8 + 21783379968*a^16*b^20*c^37*d^9 -

31592546304*a^17*b^19*c^36*d^10 + 48013246464*a^18*b^18*c^35*d^11 - 103424196608*a^19*b^17*c^34*d^12 + 2539546

21440*a^20*b^16*c^33*d^13 - 531641663488*a^21*b^15*c^32*d^14 + 875046109184*a^22*b^14*c^31*d^15 - 112586548838

4*a^23*b^13*c^30*d^16 + 1138334629888*a^24*b^12*c^29*d^17 - 906425794560*a^25*b^11*c^28*d^18 + 566347431936*a^

26*b^10*c^27*d^19 - 274688114688*a^27*b^9*c^26*d^20 + 101363744768*a^28*b^8*c^25*d^21 - 27505197056*a^29*b^7*c

^24*d^22 + 5174722560*a^30*b^6*c^23*d^23 - 602931200*a^31*b^5*c^22*d^24 + 32768000*a^32*b^4*c^21*d^25) - x^(1/

2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a^1

4*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26

*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465

100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20))*1i + (-b^9/(16*a^13*

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

896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 -

128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 +

448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(3/4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7

*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^

2*d^6 - 128*a^12*b*c*d^7))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14

*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39

*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12

- 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2

585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 106

1108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 462212300

80*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*

d^25) + 16777216*a^11*b^25*c^42*d^4 - 218103808*a^12*b^24*c^41*d^5 + 1308622848*a^13*b^23*c^40*d^6 - 479828377

6*a^14*b^22*c^39*d^7 + 11995709440*a^15*b^21*c^38*d^8 - 21783379968*a^16*b^20*c^37*d^9 + 31592546304*a^17*b^19

*c^36*d^10 - 48013246464*a^18*b^18*c^35*d^11 + 103424196608*a^19*b^17*c^34*d^12 - 253954621440*a^20*b^16*c^33*

d^13 + 531641663488*a^21*b^15*c^32*d^14 - 875046109184*a^22*b^14*c^31*d^15 + 1125865488384*a^23*b^13*c^30*d^16

- 1138334629888*a^24*b^12*c^29*d^17 + 906425794560*a^25*b^11*c^28*d^18 - 566347431936*a^26*b^10*c^27*d^19 + 2

74688114688*a^27*b^9*c^26*d^20 - 101363744768*a^28*b^8*c^25*d^21 + 27505197056*a^29*b^7*c^24*d^22 - 5174722560

*a^30*b^6*c^23*d^23 + 602931200*a^31*b^5*c^22*d^24 - 32768000*a^32*b^4*c^21*d^25) - x^(1/2)*(32366592*a^12*b^2

1*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19*c^29*d^11 + 42

69711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14 - 7603863552*a^

18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800*a^21*b^12*c^22

*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20))*1i)/((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8

- 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5

+ 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 4

48*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6

- 128*a^12*b*c*d^7))^(3/4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*

d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*

d^7))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 1526

7266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a

^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b

^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^1

3*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*

c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22

- 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25) - 16777216*a^11

*b^25*c^42*d^4 + 218103808*a^12*b^24*c^41*d^5 - 1308622848*a^13*b^23*c^40*d^6 + 4798283776*a^14*b^22*c^39*d^7

- 11995709440*a^15*b^21*c^38*d^8 + 21783379968*a^16*b^20*c^37*d^9 - 31592546304*a^17*b^19*c^36*d^10 + 48013246

464*a^18*b^18*c^35*d^11 - 103424196608*a^19*b^17*c^34*d^12 + 253954621440*a^20*b^16*c^33*d^13 - 531641663488*a

^21*b^15*c^32*d^14 + 875046109184*a^22*b^14*c^31*d^15 - 1125865488384*a^23*b^13*c^30*d^16 + 1138334629888*a^24

*b^12*c^29*d^17 - 906425794560*a^25*b^11*c^28*d^18 + 566347431936*a^26*b^10*c^27*d^19 - 274688114688*a^27*b^9*

c^26*d^20 + 101363744768*a^28*b^8*c^25*d^21 - 27505197056*a^29*b^7*c^24*d^22 + 5174722560*a^30*b^6*c^23*d^23 -

602931200*a^31*b^5*c^22*d^24 + 32768000*a^32*b^4*c^21*d^25) - x^(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832

*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^

28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4

414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800*a^21*b^12*c^22*d^18 - 72704000*a^22

*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20)) - (-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 44

8*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 -

128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896

*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(3/

4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^

3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*(33554432*a^

12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5 + 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d

^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313

817825280*a^19*b^18*c^37*d^11 + 539177779200*a^20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 15893224

48896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030

720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560

*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^27*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c

^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428800*a^33*b^4*c^23*d^25) + 16777216*a^11*b^25*c^42*d^4 - 2181038

08*a^12*b^24*c^41*d^5 + 1308622848*a^13*b^23*c^40*d^6 - 4798283776*a^14*b^22*c^39*d^7 + 11995709440*a^15*b^21*

c^38*d^8 - 21783379968*a^16*b^20*c^37*d^9 + 31592546304*a^17*b^19*c^36*d^10 - 48013246464*a^18*b^18*c^35*d^11

+ 103424196608*a^19*b^17*c^34*d^12 - 253954621440*a^20*b^16*c^33*d^13 + 531641663488*a^21*b^15*c^32*d^14 - 875

046109184*a^22*b^14*c^31*d^15 + 1125865488384*a^23*b^13*c^30*d^16 - 1138334629888*a^24*b^12*c^29*d^17 + 906425

794560*a^25*b^11*c^28*d^18 - 566347431936*a^26*b^10*c^27*d^19 + 274688114688*a^27*b^9*c^26*d^20 - 101363744768

*a^28*b^8*c^25*d^21 + 27505197056*a^29*b^7*c^24*d^22 - 5174722560*a^30*b^6*c^23*d^23 + 602931200*a^31*b^5*c^22

*d^24 - 32768000*a^32*b^4*c^21*d^25) - x^(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 18

6867712*a^13*b^20*c^30*d^10 - 1422057472*a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^1

6*b^17*c^27*d^13 + 9165979648*a^17*b^16*c^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24

*d^16 - 1766236160*a^20*b^13*c^23*d^17 + 465100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 512000

0*a^23*b^10*c^20*d^20)) - 29859840*a^11*b^21*c^29*d^9 + 228925440*a^12*b^20*c^28*d^10 - 774144000*a^13*b^19*c^

27*d^11 + 1514700800*a^14*b^18*c^26*d^12 - 1888665600*a^15*b^17*c^25*d^13 + 1555415040*a^16*b^16*c^24*d^14 - 8

45578240*a^17*b^15*c^23*d^15 + 292454400*a^18*b^14*c^22*d^16 - 58368000*a^19*b^13*c^21*d^17 + 5120000*a^20*b^1

2*c^20*d^18))*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*

d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*2i + 2*ata

n(((-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8 - 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*

a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*((-b^9/(16*a^13*d^8 +

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

^10*b^3*c^3*d^5 + 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(3/4)*(x^(1/2)*(-b^9/(16*a^13*d^8 + 16*a^5*b^8*c^8

- 128*a^6*b^7*c^7*d + 448*a^7*b^6*c^6*d^2 - 896*a^8*b^5*c^5*d^3 + 1120*a^9*b^4*c^4*d^4 - 896*a^10*b^3*c^3*d^5

+ 448*a^11*b^2*c^2*d^6 - 128*a^12*b*c*d^7))^(1/4)*(33554432*a^12*b^25*c^44*d^4 - 503316480*a^13*b^24*c^43*d^5

+ 3523215360*a^14*b^23*c^42*d^6 - 15267266560*a^15*b^22*c^41*d^7 + 45971668992*a^16*b^21*c^40*d^8 - 103500742

656*a^17*b^20*c^39*d^9 + 188659793920*a^18*b^19*c^38*d^10 - 313817825280*a^19*b^18*c^37*d^11 + 539177779200*a^

20*b^17*c^36*d^12 - 959547703296*a^21*b^16*c^35*d^13 + 1589322448896*a^22*b^15*c^34*d^14 - 2241016627200*a^23*

b^14*c^33*d^15 + 2585348014080*a^24*b^13*c^32*d^16 - 2405664030720*a^25*b^12*c^31*d^17 + 1792662306816*a^26*b^

11*c^30*d^18 - 1061108580352*a^27*b^10*c^29*d^19 + 492369346560*a^28*b^9*c^28*d^20 - 175279964160*a^29*b^8*c^2

7*d^21 + 46221230080*a^30*b^7*c^26*d^22 - 8506048512*a^31*b^6*c^25*d^23 + 975175680*a^32*b^5*c^24*d^24 - 52428

800*a^33*b^4*c^23*d^25)*1i - 16777216*a^11*b^25*c^42*d^4 + 218103808*a^12*b^24*c^41*d^5 - 1308622848*a^13*b^23

*c^40*d^6 + 4798283776*a^14*b^22*c^39*d^7 - 11995709440*a^15*b^21*c^38*d^8 + 21783379968*a^16*b^20*c^37*d^9 -

31592546304*a^17*b^19*c^36*d^10 + 48013246464*a^18*b^18*c^35*d^11 - 103424196608*a^19*b^17*c^34*d^12 + 2539546

21440*a^20*b^16*c^33*d^13 - 531641663488*a^21*b^15*c^32*d^14 + 875046109184*a^22*b^14*c^31*d^15 - 112586548838

4*a^23*b^13*c^30*d^16 + 1138334629888*a^24*b^12*c^29*d^17 - 906425794560*a^25*b^11*c^28*d^18 + 566347431936*a^

26*b^10*c^27*d^19 - 274688114688*a^27*b^9*c^26*d^20 + 101363744768*a^28*b^8*c^25*d^21 - 27505197056*a^29*b^7*c

^24*d^22 + 5174722560*a^30*b^6*c^23*d^23 - 602931200*a^31*b^5*c^22*d^24 + 32768000*a^32*b^4*c^21*d^25)*1i + x^

(1/2)*(32366592*a^12*b^21*c^31*d^9 - 10616832*a^11*b^22*c^32*d^8 + 186867712*a^13*b^20*c^30*d^10 - 1422057472*

a^14*b^19*c^29*d^11 + 4269711360*a^15*b^18*c^28*d^12 - 7664386048*a^16*b^17*c^27*d^13 + 9165979648*a^17*b^16*c

^26*d^14 - 7603863552*a^18*b^15*c^25*d^15 + 4414717952*a^19*b^14*c^24*d^16 - 1766236160*a^20*b^13*c^23*d^17 +

465100800*a^21*b^12*c^22*d^18 - 72704000*a^22*b^11*c^21*d^19 + 5120000*a^23*b^10*c^20*d^20)) + (-b^9/(16*a^13*