3.5.91 \(\int \frac {\sqrt {x}}{(a+b x^2)^2 (c+d x^2)^2} \, dx\) [491]

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

[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 = 0.57, 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 = {477, 483, 593, 598, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {b^{5/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-9 a d)}{4 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {b^{5/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-9 a d)}{4 \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) \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 {d^{5/4} (9 b c-a d) \text {ArcTan}\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) \text {ArcTan}\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^{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 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)} \end {gather*}

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 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 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 {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=2 \text {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 {\text {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 {\text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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)) \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 (b c-a d)^3}+\frac {(b (b c-9 a d)) \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 (b c-a d)^3}+\frac {\left (b^{5/4} (b c-9 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^{5/4} (b c-a d)^3}+\frac {\left (b^{5/4} (b c-9 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^{5/4} (b c-a d)^3}+\frac {(d (9 b c-a d)) \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 (b c-a d)^3}+\frac {(d (9 b c-a d)) \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 (b c-a d)^3}+\frac {\left (d^{5/4} (9 b c-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^{5/4} (b c-a d)^3}+\frac {\left (d^{5/4} (9 b c-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^{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 ) \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^{5/4} (b c-a d)^3}-\frac {\left (b^{5/4} (b c-9 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^{5/4} (b c-a d)^3}+\frac {\left (d^{5/4} (9 b c-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^{5/4} (b c-a d)^3}-\frac {\left (d^{5/4} (9 b c-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^{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*}

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Mathematica [A]
time = 1.42, size = 357, normalized size = 0.57 \begin {gather*} \frac {1}{8} \left (\frac {4 x^{3/2} \left (a^2 d^2+a b d^2 x^2+b^2 c \left (c+d x^2\right )\right )}{a c (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt {2} b^{5/4} (b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{5/4} (-b c+a d)^3}+\frac {\sqrt {2} d^{5/4} (-9 b c+a d) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{5/4} (b c-a d)^3}+\frac {\sqrt {2} b^{5/4} (b c-9 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{5/4} (-b c+a d)^3}+\frac {\sqrt {2} d^{5/4} (-9 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{5/4} (b c-a d)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.12, size = 317, normalized size = 0.51

method	result	size
derivativedivides	\(\frac {2 b^{2} \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 a \left (b \,x^{2}+a \right )}+\frac {\left (9 a d -b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 d^{2} \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 c \left (d \,x^{2}+c \right )}+\frac {\left (a d -9 b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}\)	\(317\)
default	\(\frac {2 b^{2} \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 a \left (b \,x^{2}+a \right )}+\frac {\left (9 a d -b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 d^{2} \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 c \left (d \,x^{2}+c \right )}+\frac {\left (a d -9 b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}\)	\(317\)

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,method=_RETURNVERBOSE)

[Out]

2*b^2/(a*d-b*c)^3*(1/4*(a*d-b*c)/a*x^(3/2)/(b*x^2+a)+1/32*(9*a*d-b*c)/a/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^2/(a*d-b*c)^3*(1/4*(a*d-b*c)/c*x^(3/2)/(d*x^2+c)+1/32*(a*
d-9*b*c)/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)))

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Maxima [A]
time = 0.54, size = 610, normalized size = 0.98 \begin {gather*} \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 )}} \end {gather*}

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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6028 vs. \(2 (472) = 944\).
time = 84.73, size = 6028, normalized size = 9.66 \begin {gather*} \text {Too large to display} \end {gather*}

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]

-1/8*(4*(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)*(-(b^9*c^4 - 36*a*b^8*c^3*d + 486*a^2*b^7*c^2*d^2 -
2916*a^3*b^6*c*d^3 + 6561*a^4*b^5*d^4)/(a^5*b^12*c^12 - 12*a^6*b^11*c^11*d + 66*a^7*b^10*c^10*d^2 - 220*a^8*b^
9*c^9*d^3 + 495*a^9*b^8*c^8*d^4 - 792*a^10*b^7*c^7*d^5 + 924*a^11*b^6*c^6*d^6 - 792*a^12*b^5*c^5*d^7 + 495*a^1
3*b^4*c^4*d^8 - 220*a^14*b^3*c^3*d^9 + 66*a^15*b^2*c^2*d^10 - 12*a^16*b*c*d^11 + a^17*d^12))^(1/4)*arctan(-((a
*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*sqrt((b^14*c^6 - 54*a*b^13*c^5*d + 1215*a^2*b^12*c^4*d^2
- 14580*a^3*b^11*c^3*d^3 + 98415*a^4*b^10*c^2*d^4 - 354294*a^5*b^9*c*d^5 + 531441*a^6*b^8*d^6)*x - (a^3*b^15*
c^10 - 42*a^4*b^14*c^9*d + 717*a^5*b^13*c^8*d^2 - 6392*a^6*b^12*c^7*d^3 + 32082*a^7*b^11*c^6*d^4 - 93372*a^8*b
^10*c^5*d^5 + 164242*a^9*b^9*c^4*d^6 - 177912*a^10*b^8*c^3*d^7 + 116397*a^11*b^7*c^2*d^8 - 42282*a^12*b^6*c*d^
9 + 6561*a^13*b^5*d^10)*sqrt(-(b^9*c^4 - 36*a*b^8*c^3*d + 486*a^2*b^7*c^2*d^2 - 2916*a^3*b^6*c*d^3 + 6561*a^4*
b^5*d^4)/(a^5*b^12*c^12 - 12*a^6*b^11*c^11*d + 66*a^7*b^10*c^10*d^2 - 220*a^8*b^9*c^9*d^3 + 495*a^9*b^8*c^8*d^
4 - 792*a^10*b^7*c^7*d^5 + 924*a^11*b^6*c^6*d^6 - 792*a^12*b^5*c^5*d^7 + 495*a^13*b^4*c^4*d^8 - 220*a^14*b^3*c
^3*d^9 + 66*a^15*b^2*c^2*d^10 - 12*a^16*b*c*d^11 + a^17*d^12)))*(-(b^9*c^4 - 36*a*b^8*c^3*d + 486*a^2*b^7*c^2*
d^2 - 2916*a^3*b^6*c*d^3 + 6561*a^4*b^5*d^4)/(a^5*b^12*c^12 - 12*a^6*b^11*c^11*d + 66*a^7*b^10*c^10*d^2 - 220*
a^8*b^9*c^9*d^3 + 495*a^9*b^8*c^8*d^4 - 792*a^10*b^7*c^7*d^5 + 924*a^11*b^6*c^6*d^6 - 792*a^12*b^5*c^5*d^7 + 4
95*a^13*b^4*c^4*d^8 - 220*a^14*b^3*c^3*d^9 + 66*a^15*b^2*c^2*d^10 - 12*a^16*b*c*d^11 + a^17*d^12))^(1/4) + (a*
b^10*c^6 - 30*a^2*b^9*c^5*d + 327*a^3*b^8*c^4*d^2 - 1540*a^4*b^7*c^3*d^3 + 2943*a^5*b^6*c^2*d^4 - 2430*a^6*b^5
*c*d^5 + 729*a^7*b^4*d^6)*sqrt(x)*(-(b^9*c^4 - 36*a*b^8*c^3*d + 486*a^2*b^7*c^2*d^2 - 2916*a^3*b^6*c*d^3 + 656
1*a^4*b^5*d^4)/(a^5*b^12*c^12 - 12*a^6*b^11*c^11*d + 66*a^7*b^10*c^10*d^2 - 220*a^8*b^9*c^9*d^3 + 495*a^9*b^8*
c^8*d^4 - 792*a^10*b^7*c^7*d^5 + 924*a^11*b^6*c^6*d^6 - 792*a^12*b^5*c^5*d^7 + 495*a^13*b^4*c^4*d^8 - 220*a^14
*b^3*c^3*d^9 + 66*a^15*b^2*c^2*d^10 - 12*a^16*b*c*d^11 + a^17*d^12))^(1/4))/(b^9*c^4 - 36*a*b^8*c^3*d + 486*a^
2*b^7*c^2*d^2 - 2916*a^3*b^6*c*d^3 + 6561*a^4*b^5*d^4)) + 4*(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)*
(-(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 + a^4*d^9)/(b^12*c^17 - 12*a*b
^11*c^16*d + 66*a^2*b^10*c^15*d^2 - 220*a^3*b^9*c^14*d^3 + 495*a^4*b^8*c^13*d^4 - 792*a^5*b^7*c^12*d^5 + 924*a
^6*b^6*c^11*d^6 - 792*a^7*b^5*c^10*d^7 + 495*a^8*b^4*c^9*d^8 - 220*a^9*b^3*c^8*d^9 + 66*a^10*b^2*c^7*d^10 - 12
*a^11*b*c^6*d^11 + a^12*c^5*d^12))^(1/4)*arctan(-((b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*sqrt
((531441*b^6*c^6*d^8 - 354294*a*b^5*c^5*d^9 + 98415*a^2*b^4*c^4*d^10 - 14580*a^3*b^3*c^3*d^11 + 1215*a^4*b^2*c
^2*d^12 - 54*a^5*b*c*d^13 + a^6*d^14)*x - (6561*b^10*c^13*d^5 - 42282*a*b^9*c^12*d^6 + 116397*a^2*b^8*c^11*d^7
- 177912*a^3*b^7*c^10*d^8 + 164242*a^4*b^6*c^9*d^9 - 93372*a^5*b^5*c^8*d^10 + 32082*a^6*b^4*c^7*d^11 - 6392*a
^7*b^3*c^6*d^12 + 717*a^8*b^2*c^5*d^13 - 42*a^9*b*c^4*d^14 + a^10*c^3*d^15)*sqrt(-(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 + a^4*d^9)/(b^12*c^17 - 12*a*b^11*c^16*d + 66*a^2*b^10*c^15*
d^2 - 220*a^3*b^9*c^14*d^3 + 495*a^4*b^8*c^13*d^4 - 792*a^5*b^7*c^12*d^5 + 924*a^6*b^6*c^11*d^6 - 792*a^7*b^5*
c^10*d^7 + 495*a^8*b^4*c^9*d^8 - 220*a^9*b^3*c^8*d^9 + 66*a^10*b^2*c^7*d^10 - 12*a^11*b*c^6*d^11 + a^12*c^5*d^
12)))*(-(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 + a^4*d^9)/(b^12*c^17 -
12*a*b^11*c^16*d + 66*a^2*b^10*c^15*d^2 - 220*a^3*b^9*c^14*d^3 + 495*a^4*b^8*c^13*d^4 - 792*a^5*b^7*c^12*d^5 +
924*a^6*b^6*c^11*d^6 - 792*a^7*b^5*c^10*d^7 + 495*a^8*b^4*c^9*d^8 - 220*a^9*b^3*c^8*d^9 + 66*a^10*b^2*c^7*d^1
0 - 12*a^11*b*c^6*d^11 + a^12*c^5*d^12))^(1/4) + (729*b^6*c^7*d^4 - 2430*a*b^5*c^6*d^5 + 2943*a^2*b^4*c^5*d^6
- 1540*a^3*b^3*c^4*d^7 + 327*a^4*b^2*c^3*d^8 - 30*a^5*b*c^2*d^9 + a^6*c*d^10)*sqrt(x)*(-(6561*b^4*c^4*d^5 - 29
16*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8 + a^4*d^9)/(b^12*c^17 - 12*a*b^11*c^16*d + 66*a^2*b^10
*c^15*d^2 - 220*a^3*b^9*c^14*d^3 + 495*a^4*b^8*c^13*d^4 - 792*a^5*b^7*c^12*d^5 + 924*a^6*b^6*c^11*d^6 - 792*a^
7*b^5*c^10*d^7 + 495*a^8*b^4*c^9*d^8 - 220*a^9*b^3*c^8*d^9 + 66*a^10*b^2*c^7*d^10 - 12*a^11*b*c^6*d^11 + a^12*
c^5*d^12))^(1/4))/(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 + a^4*d^9)) +
(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)*(-(b^9*c^4 - 36*a*b^8*c^3*d + 486*a^2*b^7*c^2*d^2 - 2916*a^3
*b^6*c*d^3 + 6561*a^4*b^5*d^4)/(a^5*b^12*c^12 -...

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 973 vs. \(2 (472) = 944\).
time = 1.64, size = 973, normalized size = 1.56 \begin {gather*} \frac {{\left (\left (a b^{3}\right )^{\frac {3}{4}} b c - 9 \, \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^{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}\right )}} + \frac {{\left (\left (a b^{3}\right )^{\frac {3}{4}} b c - 9 \, \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^{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}\right )}} + \frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \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^{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}\right )}} + \frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \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^{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}\right )}} - \frac {{\left (\left (a b^{3}\right )^{\frac {3}{4}} b c - 9 \, \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^{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}\right )}} + \frac {{\left (\left (a b^{3}\right )^{\frac {3}{4}} b c - 9 \, \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^{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}\right )}} - \frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \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^{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}\right )}} + \frac {{\left (9 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \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^{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}\right )}} + \frac {b^{2} c d x^{\frac {7}{2}} + a b d^{2} x^{\frac {7}{2}} + b^{2} c^{2} x^{\frac {3}{2}} + a^{2} d^{2} x^{\frac {3}{2}}}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )}} \end {gather*}

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))

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Mupad [B]
time = 2.45, size = 2500, normalized size = 4.01 \begin {gather*} \text {Too large to display} \end {gather*}

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^1...

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