3.5.93 \(\int \frac {1}{x^{3/2} (a+b x^2)^2 (c+d x^2)^2} \, dx\) [493]

Optimal. Leaf size=676 \[ \frac {-5 b^2 c^2+8 a b c d-5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^3}-\frac {d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^3}-\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} (13 b c-5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {d^{9/4} (13 b c-5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} (b c-a d)^3} \]

[Out]

1/8*b^(9/4)*(-13*a*d+5*b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/(-a*d+b*c)^3*2^(1/2)-1/8*b^(9/4)
*(-13*a*d+5*b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/(-a*d+b*c)^3*2^(1/2)+1/8*d^(9/4)*(-5*a*d+13
*b*c)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(9/4)/(-a*d+b*c)^3*2^(1/2)-1/8*d^(9/4)*(-5*a*d+13*b*c)*arcta
n(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(9/4)/(-a*d+b*c)^3*2^(1/2)-1/16*b^(9/4)*(-13*a*d+5*b*c)*ln(a^(1/2)+x*b^
(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/(-a*d+b*c)^3*2^(1/2)+1/16*b^(9/4)*(-13*a*d+5*b*c)*ln(a^(1/2)+x*
b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/(-a*d+b*c)^3*2^(1/2)-1/16*d^(9/4)*(-5*a*d+13*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)^3*2^(1/2)+1/16*d^(9/4)*(-5*a*d+13*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)^3*2^(1/2)+1/2*(-5*a^2*d^2+8*a*b*c*d-5*b^2*c^2)
/a^2/c^2/(-a*d+b*c)^2/x^(1/2)+1/2*d*(a*d+b*c)/a/c/(-a*d+b*c)^2/(d*x^2+c)/x^(1/2)+1/2*b/a/(-a*d+b*c)/(b*x^2+a)/
(d*x^2+c)/x^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.71, antiderivative size = 676, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {477, 483, 593, 597, 598, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {b^{9/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (5 b c-13 a d)}{4 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{9/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (5 b c-13 a d)}{4 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{9/4} (5 b c-13 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {5 a^2 d^2-8 a b c d+5 b^2 c^2}{2 a^2 c^2 \sqrt {x} (b c-a d)^2}+\frac {d^{9/4} (13 b c-5 a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^3}-\frac {d^{9/4} (13 b c-5 a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{9/4} (b c-a d)^3}-\frac {d^{9/4} (13 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)^3}+\frac {d^{9/4} (13 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)^3}+\frac {b}{2 a \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {d (a d+b c)}{2 a c \sqrt {x} \left (c+d x^2\right ) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(5*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)/(a^2*c^2*(b*c - a*d)^2*Sqrt[x]) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^
2*Sqrt[x]*(c + d*x^2)) + b/(2*a*(b*c - a*d)*Sqrt[x]*(a + b*x^2)*(c + d*x^2)) + (b^(9/4)*(5*b*c - 13*a*d)*ArcTa
n[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (b^(9/4)*(5*b*c - 13*a*d)*ArcTan
[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(13*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)^3) - (d^(9/4)*(13*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)^3) - (b^(9/4)*(5*b*c - 13*a*d)*Log[Sqrt[
a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (b^(9/4)*(5*b*c - 13*a*
d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (d^(9/4)*(1
3*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)^3) +
(d^(9/4)*(13*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)^3)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 303

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 477

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 483

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 593

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 597

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 598

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 631

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 642

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 1176

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 1179

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 )^2 \left (c+d x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {-5 b c+4 a d-9 b d x^4}{x^2 \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)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {-4 \left (5 b^2 c^2-8 a b c d+5 a^2 d^2\right )-20 b d (b c+a d) x^4}{x^2 \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 {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (-4 (b c+a d) \left (5 b^2 c^2-13 a b c d+5 a^2 d^2\right )-4 b d \left (5 b^2 c^2-8 a b c d+5 a^2 d^2\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{8 a^2 c^2 (b c-a d)^2}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \left (-\frac {4 b^3 c^2 (5 b c-13 a d) x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {4 a^2 d^3 (-13 b c+5 a d) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{8 a^2 c^2 (b c-a d)^2}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (b^3 (5 b c-13 a d)\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2 (b c-a d)^3}-\frac {\left (d^3 (13 b c-5 a d)\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c^2 (b c-a d)^3}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^{5/2} (5 b c-13 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 (b c-a d)^3}-\frac {\left (b^{5/2} (5 b c-13 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 (b c-a d)^3}+\frac {\left (d^{5/2} (13 b c-5 a d)\right ) \text {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)^3}-\frac {\left (d^{5/2} (13 b c-5 a d)\right ) \text {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)^3}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (b^2 (5 b c-13 a d)\right ) \text {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^2 (b c-a d)^3}-\frac {\left (b^2 (5 b c-13 a d)\right ) \text {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^2 (b c-a d)^3}-\frac {\left (b^{9/4} (5 b c-13 a d)\right ) \text {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^{9/4} (b c-a d)^3}-\frac {\left (b^{9/4} (5 b c-13 a d)\right ) \text {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^{9/4} (b c-a d)^3}-\frac {\left (d^2 (13 b c-5 a d)\right ) \text {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)^3}-\frac {\left (d^2 (13 b c-5 a d)\right ) \text {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)^3}-\frac {\left (d^{9/4} (13 b c-5 a d)\right ) \text {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)^3}-\frac {\left (d^{9/4} (13 b c-5 a d)\right ) \text {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)^3}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} (13 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)^3}+\frac {d^{9/4} (13 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)^3}-\frac {\left (b^{9/4} (5 b c-13 a d)\right ) \text {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^{9/4} (b c-a d)^3}+\frac {\left (b^{9/4} (5 b c-13 a d)\right ) \text {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^{9/4} (b c-a d)^3}-\frac {\left (d^{9/4} (13 b c-5 a d)\right ) \text {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)^3}+\frac {\left (d^{9/4} (13 b c-5 a d)\right ) \text {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)^3}\\ &=-\frac {5 b^2 c^2-8 a b c d+5 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{9/4} (5 b c-13 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {d^{9/4} (13 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)^3}-\frac {d^{9/4} (13 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)^3}-\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{9/4} (5 b c-13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} (13 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)^3}+\frac {d^{9/4} (13 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)^3}\\ \end {align*}

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Mathematica [A]
time = 1.20, size = 421, normalized size = 0.62 \begin {gather*} \frac {1}{8} \left (-\frac {4 \left (5 b^3 c^2 x^2 \left (c+d x^2\right )+a^3 d^2 \left (4 c+5 d x^2\right )+4 a b^2 c \left (c^2-c d x^2-2 d^2 x^4\right )+a^2 b d \left (-8 c^2-4 c d x^2+5 d^2 x^4\right )\right )}{a^2 c^2 (b c-a d)^2 \sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt {2} b^{9/4} (-5 b c+13 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{9/4} (-b c+a d)^3}+\frac {\sqrt {2} d^{9/4} (13 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{9/4} (b c-a d)^3}+\frac {\sqrt {2} b^{9/4} (-5 b c+13 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{9/4} (-b c+a d)^3}+\frac {\sqrt {2} d^{9/4} (13 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{9/4} (b c-a d)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*(5*b^3*c^2*x^2*(c + d*x^2) + a^3*d^2*(4*c + 5*d*x^2) + 4*a*b^2*c*(c^2 - c*d*x^2 - 2*d^2*x^4) + a^2*b*d*(-
8*c^2 - 4*c*d*x^2 + 5*d^2*x^4)))/(a^2*c^2*(b*c - a*d)^2*Sqrt[x]*(a + b*x^2)*(c + d*x^2)) + (Sqrt[2]*b^(9/4)*(-
5*b*c + 13*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(9/4)*(-(b*c) + a*d)^3) +
(Sqrt[2]*d^(9/4)*(13*b*c - 5*a*d)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(9/4)*(b
*c - a*d)^3) + (Sqrt[2]*b^(9/4)*(-5*b*c + 13*a*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]
*x)])/(a^(9/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(9/4)*(13*b*c - 5*a*d)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])
/(Sqrt[c] + Sqrt[d]*x)])/(c^(9/4)*(b*c - a*d)^3))/8

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Maple [A]
time = 0.19, size = 323, normalized size = 0.48 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b^3/a^2/(a*d-b*c)^3*((1/4*a*d-1/4*b*c)*x^(3/2)/(b*x^2+a)+1/8*(13/4*a*d-5/4*b*c)/b/(a/b)^(1/4)*2^(1/2)*(ln((
x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)
^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))-2*d^3/c^2/(a*d-b*c)^3*((1/4*a*d-1/4*b*c)*x^(3/2)/(
d*x^2+c)+1/8*(5/4*a*d-13/4*b*c)/d/(c/d)^(1/4)*2^(1/2)*(ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)
^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1
/2)-1)))-2/a^2/c^2/x^(1/2)

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Maxima [A]
time = 0.56, size = 694, normalized size = 1.03 \begin {gather*} -\frac {{\left (5 \, b^{4} c - 13 \, a b^{3} 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^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}} - \frac {{\left (13 \, b c d^{3} - 5 \, a d^{4}\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^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}} - \frac {4 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 4 \, a^{3} c d^{2} + {\left (5 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x^{4} + {\left (5 \, b^{3} c^{3} - 4 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} x^{2}}{2 \, {\left ({\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{\frac {9}{2}} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{\frac {5}{2}} + {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} \sqrt {x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(5*b^4*c - 13*a*b^3*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
*sqrt(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)
*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^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3) - 1/16*(13*b*c*d^3 - 5*
a*d^4)*(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)))/(sqr
t(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^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3) - 1/2*(4*a*b^2*c^3 - 8*a^2*b*c^2*d + 4*a^3*c*
d^2 + (5*b^3*c^2*d - 8*a*b^2*c*d^2 + 5*a^2*b*d^3)*x^4 + (5*b^3*c^3 - 4*a*b^2*c^2*d - 4*a^2*b*c*d^2 + 5*a^3*d^3
)*x^2)/((a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^(9/2) + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3
*d^2 + a^5*c^2*d^3)*x^(5/2) + (a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*sqrt(x))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6207 vs. \(2 (520) = 1040\).
time = 270.75, size = 6207, normalized size = 9.18 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*((a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^5 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2
+ a^5*c^2*d^3)*x^3 + (a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*x)*(-(625*b^13*c^4 - 6500*a*b^12*c^3*d + 253
50*a^2*b^11*c^2*d^2 - 43940*a^3*b^10*c*d^3 + 28561*a^4*b^9*d^4)/(a^9*b^12*c^12 - 12*a^10*b^11*c^11*d + 66*a^11
*b^10*c^10*d^2 - 220*a^12*b^9*c^9*d^3 + 495*a^13*b^8*c^8*d^4 - 792*a^14*b^7*c^7*d^5 + 924*a^15*b^6*c^6*d^6 - 7
92*a^16*b^5*c^5*d^7 + 495*a^17*b^4*c^4*d^8 - 220*a^18*b^3*c^3*d^9 + 66*a^19*b^2*c^2*d^10 - 12*a^20*b*c*d^11 +
a^21*d^12))^(1/4)*arctan(-((a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*sqrt((15625*b^20*c^6 - 24
3750*a*b^19*c^5*d + 1584375*a^2*b^18*c^4*d^2 - 5492500*a^3*b^17*c^3*d^3 + 10710375*a^4*b^16*c^2*d^4 - 11138790
*a^5*b^15*c*d^5 + 4826809*a^6*b^14*d^6)*x - (625*a^5*b^19*c^10 - 10250*a^6*b^18*c^9*d + 73725*a^7*b^17*c^8*d^2
- 306040*a^8*b^16*c^7*d^3 + 811826*a^9*b^15*c^6*d^4 - 1438716*a^10*b^14*c^5*d^5 + 1727090*a^11*b^13*c^4*d^6 -
1388920*a^12*b^12*c^3*d^7 + 717405*a^13*b^11*c^2*d^8 - 215306*a^14*b^10*c*d^9 + 28561*a^15*b^9*d^10)*sqrt(-(6
25*b^13*c^4 - 6500*a*b^12*c^3*d + 25350*a^2*b^11*c^2*d^2 - 43940*a^3*b^10*c*d^3 + 28561*a^4*b^9*d^4)/(a^9*b^12
*c^12 - 12*a^10*b^11*c^11*d + 66*a^11*b^10*c^10*d^2 - 220*a^12*b^9*c^9*d^3 + 495*a^13*b^8*c^8*d^4 - 792*a^14*b
^7*c^7*d^5 + 924*a^15*b^6*c^6*d^6 - 792*a^16*b^5*c^5*d^7 + 495*a^17*b^4*c^4*d^8 - 220*a^18*b^3*c^3*d^9 + 66*a^
19*b^2*c^2*d^10 - 12*a^20*b*c*d^11 + a^21*d^12)))*(-(625*b^13*c^4 - 6500*a*b^12*c^3*d + 25350*a^2*b^11*c^2*d^2
- 43940*a^3*b^10*c*d^3 + 28561*a^4*b^9*d^4)/(a^9*b^12*c^12 - 12*a^10*b^11*c^11*d + 66*a^11*b^10*c^10*d^2 - 22
0*a^12*b^9*c^9*d^3 + 495*a^13*b^8*c^8*d^4 - 792*a^14*b^7*c^7*d^5 + 924*a^15*b^6*c^6*d^6 - 792*a^16*b^5*c^5*d^7
+ 495*a^17*b^4*c^4*d^8 - 220*a^18*b^3*c^3*d^9 + 66*a^19*b^2*c^2*d^10 - 12*a^20*b*c*d^11 + a^21*d^12))^(1/4) +
(125*a^2*b^13*c^6 - 1350*a^3*b^12*c^5*d + 5835*a^4*b^11*c^4*d^2 - 12852*a^5*b^10*c^3*d^3 + 15171*a^6*b^9*c^2*
d^4 - 9126*a^7*b^8*c*d^5 + 2197*a^8*b^7*d^6)*sqrt(x)*(-(625*b^13*c^4 - 6500*a*b^12*c^3*d + 25350*a^2*b^11*c^2*
d^2 - 43940*a^3*b^10*c*d^3 + 28561*a^4*b^9*d^4)/(a^9*b^12*c^12 - 12*a^10*b^11*c^11*d + 66*a^11*b^10*c^10*d^2 -
220*a^12*b^9*c^9*d^3 + 495*a^13*b^8*c^8*d^4 - 792*a^14*b^7*c^7*d^5 + 924*a^15*b^6*c^6*d^6 - 792*a^16*b^5*c^5*
d^7 + 495*a^17*b^4*c^4*d^8 - 220*a^18*b^3*c^3*d^9 + 66*a^19*b^2*c^2*d^10 - 12*a^20*b*c*d^11 + a^21*d^12))^(1/4
))/(625*b^13*c^4 - 6500*a*b^12*c^3*d + 25350*a^2*b^11*c^2*d^2 - 43940*a^3*b^10*c*d^3 + 28561*a^4*b^9*d^4)) + 4
*((a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x^5 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5
*c^2*d^3)*x^3 + (a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*x)*(-(28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 2
5350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12 + 625*a^4*d^13)/(b^12*c^21 - 12*a*b^11*c^20*d + 66*a^2*b^10*c^19*d^2
- 220*a^3*b^9*c^18*d^3 + 495*a^4*b^8*c^17*d^4 - 792*a^5*b^7*c^16*d^5 + 924*a^6*b^6*c^15*d^6 - 792*a^7*b^5*c^1
4*d^7 + 495*a^8*b^4*c^13*d^8 - 220*a^9*b^3*c^12*d^9 + 66*a^10*b^2*c^11*d^10 - 12*a^11*b*c^10*d^11 + a^12*c^9*d
^12))^(1/4)*arctan(-((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*sqrt((4826809*b^6*c^6*d^14 - 11
138790*a*b^5*c^5*d^15 + 10710375*a^2*b^4*c^4*d^16 - 5492500*a^3*b^3*c^3*d^17 + 1584375*a^4*b^2*c^2*d^18 - 2437
50*a^5*b*c*d^19 + 15625*a^6*d^20)*x - (28561*b^10*c^15*d^9 - 215306*a*b^9*c^14*d^10 + 717405*a^2*b^8*c^13*d^11
- 1388920*a^3*b^7*c^12*d^12 + 1727090*a^4*b^6*c^11*d^13 - 1438716*a^5*b^5*c^10*d^14 + 811826*a^6*b^4*c^9*d^15
- 306040*a^7*b^3*c^8*d^16 + 73725*a^8*b^2*c^7*d^17 - 10250*a^9*b*c^6*d^18 + 625*a^10*c^5*d^19)*sqrt(-(28561*b
^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12 + 625*a^4*d^13)/(b^12*c^21 - 12
*a*b^11*c^20*d + 66*a^2*b^10*c^19*d^2 - 220*a^3*b^9*c^18*d^3 + 495*a^4*b^8*c^17*d^4 - 792*a^5*b^7*c^16*d^5 + 9
24*a^6*b^6*c^15*d^6 - 792*a^7*b^5*c^14*d^7 + 495*a^8*b^4*c^13*d^8 - 220*a^9*b^3*c^12*d^9 + 66*a^10*b^2*c^11*d^
10 - 12*a^11*b*c^10*d^11 + a^12*c^9*d^12)))*(-(28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^
11 - 6500*a^3*b*c*d^12 + 625*a^4*d^13)/(b^12*c^21 - 12*a*b^11*c^20*d + 66*a^2*b^10*c^19*d^2 - 220*a^3*b^9*c^18
*d^3 + 495*a^4*b^8*c^17*d^4 - 792*a^5*b^7*c^16*d^5 + 924*a^6*b^6*c^15*d^6 - 792*a^7*b^5*c^14*d^7 + 495*a^8*b^4
*c^13*d^8 - 220*a^9*b^3*c^12*d^9 + 66*a^10*b^2*c^11*d^10 - 12*a^11*b*c^10*d^11 + a^12*c^9*d^12))^(1/4) + (2197
*b^6*c^8*d^7 - 9126*a*b^5*c^7*d^8 + 15171*a^2*b^4*c^6*d^9 - 12852*a^3*b^3*c^5*d^10 + 5835*a^4*b^2*c^4*d^11 - 1
350*a^5*b*c^3*d^12 + 125*a^6*c^2*d^13)*sqrt(x)*(-(28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2
*d^11 - 6500*a^3*b*c*d^12 + 625*a^4*d^13)/(b^12*c^21 - 12*a*b^11*c^20*d + 66*a^2*b^10*c^19*d^2 - 220*a^3*b^9*c
^18*d^3 + 495*a^4*b^8*c^17*d^4 - 792*a^5*b^7*c^16*d^5 + 924*a^6*b^6*c^15*d^6 - 792*a^7*b^5*c^14*d^7 + 495*a^8*
b^4*c^13*d^8 - 220*a^9*b^3*c^12*d^9 + 66*a^10*b^2*c^11*d^10 - 12*a^11*b*c^10*d^11 + a^12*c^9*d^12))^(1/4))/(28
561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 1.30, size = 1035, normalized size = 1.53 \begin {gather*} -\frac {{\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b c - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \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 )}{4 \, {\left (\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}\right )}} - \frac {{\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b c - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \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 )}{4 \, {\left (\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}\right )}} - \frac {{\left (13 \, \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^{3} c^{6} - 3 \, \sqrt {2} a b^{2} c^{5} d + 3 \, \sqrt {2} a^{2} b c^{4} d^{2} - \sqrt {2} a^{3} c^{3} d^{3}\right )}} - \frac {{\left (13 \, \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^{3} c^{6} - 3 \, \sqrt {2} a b^{2} c^{5} d + 3 \, \sqrt {2} a^{2} b c^{4} d^{2} - \sqrt {2} a^{3} c^{3} d^{3}\right )}} + \frac {{\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b c - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}\right )}} - \frac {{\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b c - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}\right )}} + \frac {{\left (13 \, \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^{3} c^{6} - 3 \, \sqrt {2} a b^{2} c^{5} d + 3 \, \sqrt {2} a^{2} b c^{4} d^{2} - \sqrt {2} a^{3} c^{3} d^{3}\right )}} - \frac {{\left (13 \, \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^{3} c^{6} - 3 \, \sqrt {2} a b^{2} c^{5} d + 3 \, \sqrt {2} a^{2} b c^{4} d^{2} - \sqrt {2} a^{3} c^{3} d^{3}\right )}} - \frac {5 \, b^{3} c^{2} d x^{4} - 8 \, a b^{2} c d^{2} x^{4} + 5 \, a^{2} b d^{3} x^{4} + 5 \, b^{3} c^{3} x^{2} - 4 \, a b^{2} c^{2} d x^{2} - 4 \, a^{2} b c d^{2} x^{2} + 5 \, a^{3} d^{3} x^{2} + 4 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 4 \, a^{3} c d^{2}}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left (b d x^{\frac {9}{2}} + b c x^{\frac {5}{2}} + a d x^{\frac {5}{2}} + a c \sqrt {x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(5*(a*b^3)^(3/4)*b*c - 13*(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^3*b^3*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt(2)*a^6*d^3) - 1/4*(5*(a*b^
3)^(3/4)*b*c - 13*(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^3*b^3*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt(2)*a^6*d^3) - 1/4*(13*(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^3*c^6 -
3*sqrt(2)*a*b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/4*(13*(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^3*c^6 - 3*sqrt(2)*a*
b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) + 1/8*(5*(a*b^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*
log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^3*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^
5*b*c*d^2 - sqrt(2)*a^6*d^3) - 1/8*(5*(a*b^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/
4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^3*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt(2)*a^6*d^3)
+ 1/8*(13*(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
^3*c^6 - 3*sqrt(2)*a*b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/8*(13*(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^3*c^6 - 3*sqrt(2)*a*b^2*c^5*
d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/2*(5*b^3*c^2*d*x^4 - 8*a*b^2*c*d^2*x^4 + 5*a^2*b*d^3*x^
4 + 5*b^3*c^3*x^2 - 4*a*b^2*c^2*d*x^2 - 4*a^2*b*c*d^2*x^2 + 5*a^3*d^3*x^2 + 4*a*b^2*c^3 - 8*a^2*b*c^2*d + 4*a^
3*c*d^2)/((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*(b*d*x^(9/2) + b*c*x^(5/2) + a*d*x^(5/2) + a*c*sqrt(x)))

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Mupad [B]
time = 5.96, size = 2500, normalized size = 3.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((((-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*b*c*d^12
)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^3*b^9*c^
18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*b^5*c^14
*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c^20*d))^
(3/4)*(x^(1/2)*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^11 - 6500*a^3*
b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^2 - 901120*a^
3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 - 3244032*a^7*
b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 49152*a*b^11*c
^20*d))^(1/4)*(52428800*a^23*b^38*c^57*d^4 - 1635778560*a^24*b^37*c^56*d^5 + 24482152448*a^25*b^36*c^55*d^6 -
234134437888*a^26*b^35*c^54*d^7 + 1607834009600*a^27*b^34*c^53*d^8 - 8446069964800*a^28*b^33*c^52*d^9 + 353031
82041088*a^29*b^32*c^51*d^10 - 120578363097088*a^30*b^31*c^50*d^11 + 342964201062400*a^31*b^30*c^49*d^12 - 823
887134720000*a^32*b^29*c^48*d^13 + 1690057100492800*a^33*b^28*c^47*d^14 - 2988135038320640*a^34*b^27*c^46*d^15
+ 4595616128696320*a^35*b^26*c^45*d^16 - 6215915829985280*a^36*b^25*c^44*d^17 + 7509830061260800*a^37*b^24*c^
43*d^18 - 8292025971507200*a^38*b^23*c^42*d^19 + 8624070071418880*a^39*b^22*c^41*d^20 - 8700497871503360*a^40*
b^21*c^40*d^21 + 8624070071418880*a^41*b^20*c^39*d^22 - 8292025971507200*a^42*b^19*c^38*d^23 + 750983006126080
0*a^43*b^18*c^37*d^24 - 6215915829985280*a^44*b^17*c^36*d^25 + 4595616128696320*a^45*b^16*c^35*d^26 - 29881350
38320640*a^46*b^15*c^34*d^27 + 1690057100492800*a^47*b^14*c^33*d^28 - 823887134720000*a^48*b^13*c^32*d^29 + 34
2964201062400*a^49*b^12*c^31*d^30 - 120578363097088*a^50*b^11*c^30*d^31 + 35303182041088*a^51*b^10*c^29*d^32 -
8446069964800*a^52*b^9*c^28*d^33 + 1607834009600*a^53*b^8*c^27*d^34 - 234134437888*a^54*b^7*c^26*d^35 + 24482
152448*a^55*b^6*c^25*d^36 - 1635778560*a^56*b^5*c^24*d^37 + 52428800*a^57*b^4*c^23*d^38) - 32768000*a^21*b^38*
c^55*d^4 + 1009254400*a^22*b^37*c^54*d^5 - 14833418240*a^23*b^36*c^53*d^6 + 138556735488*a^24*b^35*c^52*d^7 -
924185001984*a^25*b^34*c^51*d^8 + 4688465362944*a^26*b^33*c^50*d^9 - 18812623126528*a^27*b^32*c^49*d^10 + 6129
5191654400*a^28*b^31*c^48*d^11 - 165189260410880*a^29*b^30*c^47*d^12 + 373165003898880*a^30*b^29*c^46*d^13 - 7
13540118773760*a^31*b^28*c^45*d^14 + 1163349301657600*a^32*b^27*c^44*d^15 - 1627141704253440*a^33*b^26*c^43*d^
16 + 1966197351383040*a^34*b^25*c^42*d^17 - 2079216623943680*a^35*b^24*c^41*d^18 + 1981073955225600*a^36*b^23*
c^40*d^19 - 1807512431493120*a^37*b^22*c^39*d^20 + 1724885956034560*a^38*b^21*c^38*d^21 - 1807512431493120*a^3
9*b^20*c^37*d^22 + 1981073955225600*a^40*b^19*c^36*d^23 - 2079216623943680*a^41*b^18*c^35*d^24 + 1966197351383
040*a^42*b^17*c^34*d^25 - 1627141704253440*a^43*b^16*c^33*d^26 + 1163349301657600*a^44*b^15*c^32*d^27 - 713540
118773760*a^45*b^14*c^31*d^28 + 373165003898880*a^46*b^13*c^30*d^29 - 165189260410880*a^47*b^12*c^29*d^30 + 61
295191654400*a^48*b^11*c^28*d^31 - 18812623126528*a^49*b^10*c^27*d^32 + 4688465362944*a^50*b^9*c^26*d^33 - 924
185001984*a^51*b^8*c^25*d^34 + 138556735488*a^52*b^7*c^24*d^35 - 14833418240*a^53*b^6*c^23*d^36 + 1009254400*a
^54*b^5*c^22*d^37 - 32768000*a^55*b^4*c^21*d^38) + x^(1/2)*(54080000*a^20*b^33*c^43*d^10 - 1361152000*a^21*b^3
2*c^42*d^11 + 16011852800*a^22*b^31*c^41*d^12 - 116736734720*a^23*b^30*c^40*d^13 + 589861462528*a^24*b^29*c^39
*d^14 - 2187899577344*a^25*b^28*c^38*d^15 + 6149347117056*a^26*b^27*c^37*d^16 - 13298820601344*a^27*b^26*c^36*
d^17 + 22133436343296*a^28*b^25*c^35*d^18 - 27715689750528*a^29*b^24*c^34*d^19 + 24077503776768*a^30*b^23*c^33
*d^20 - 9645706816512*a^31*b^22*c^32*d^21 - 9645706816512*a^32*b^21*c^31*d^22 + 24077503776768*a^33*b^20*c^30*
d^23 - 27715689750528*a^34*b^19*c^29*d^24 + 22133436343296*a^35*b^18*c^28*d^25 - 13298820601344*a^36*b^17*c^27
*d^26 + 6149347117056*a^37*b^16*c^26*d^27 - 2187899577344*a^38*b^15*c^25*d^28 + 589861462528*a^39*b^14*c^24*d^
29 - 116736734720*a^40*b^13*c^23*d^30 + 16011852800*a^41*b^12*c^22*d^31 - 1361152000*a^42*b^11*c^21*d^32 + 540
80000*a^43*b^10*c^20*d^33))*(-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^2*d^1
1 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^19*d^
2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*b^8*c^17*d^4 - 3244032*a^5*b^7*c^16*d^5 + 3784704*a^6*b^6*c^15*d^6 -
3244032*a^7*b^5*c^14*d^7 + 2027520*a^8*b^4*c^13*d^8 - 901120*a^9*b^3*c^12*d^9 + 270336*a^10*b^2*c^11*d^10 - 4
9152*a*b^11*c^20*d))^(1/4)*1i + ((-(625*a^4*d^13 + 28561*b^4*c^4*d^9 - 43940*a*b^3*c^3*d^10 + 25350*a^2*b^2*c^
2*d^11 - 6500*a^3*b*c*d^12)/(4096*b^12*c^21 + 4096*a^12*c^9*d^12 - 49152*a^11*b*c^10*d^11 + 270336*a^2*b^10*c^
19*d^2 - 901120*a^3*b^9*c^18*d^3 + 2027520*a^4*...

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