\(\int \frac {x^4}{(a+b x^2)^2 (c+d x^2)^2} \, dx\) [301]
Optimal result
Integrand size = 22, antiderivative size = 162 \[
\int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {(b c+a d) x}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\sqrt {a} (3 b c+a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {b} (b c-a d)^3}+\frac {\sqrt {c} (b c+3 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {d} (b c-a d)^3}
\]
[Out]
1/2*(a*d+b*c)*x/b/(-a*d+b*c)^2/(d*x^2+c)+1/2*a*x/b/(-a*d+b*c)/(b*x^2+a)/(d*x^2+c)-1/2*(a*d+3*b*c)*arctan(x*b^(
1/2)/a^(1/2))*a^(1/2)/(-a*d+b*c)^3/b^(1/2)+1/2*(3*a*d+b*c)*arctan(x*d^(1/2)/c^(1/2))*c^(1/2)/(-a*d+b*c)^3/d^(1
/2)
Rubi [A] (verified)
Time = 0.14 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {481, 541, 536, 211}
\[
\int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (a d+3 b c)}{2 \sqrt {b} (b c-a d)^3}+\frac {\sqrt {c} (3 a d+b c) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {d} (b c-a d)^3}+\frac {x (a d+b c)}{2 b \left (c+d x^2\right ) (b c-a d)^2}+\frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}
\]
[In]
Int[x^4/((a + b*x^2)^2*(c + d*x^2)^2),x]
[Out]
((b*c + a*d)*x)/(2*b*(b*c - a*d)^2*(c + d*x^2)) + (a*x)/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (Sqrt[a]*(
3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[b]*(b*c - a*d)^3) + (Sqrt[c]*(b*c + 3*a*d)*ArcTan[(Sqrt[d]*x
)/Sqrt[c]])/(2*Sqrt[d]*(b*c - a*d)^3)
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 481
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 536
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]
Rule 541
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Rubi steps \begin{align*}
\text {integral}& = \frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\int \frac {a c+(-2 b c-a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{2 b (b c-a d)} \\ & = \frac {(b c+a d) x}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\int \frac {4 a b c^2-2 b c (b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{4 b c (b c-a d)^2} \\ & = \frac {(b c+a d) x}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {(a (3 b c+a d)) \int \frac {1}{a+b x^2} \, dx}{2 (b c-a d)^3}+\frac {(c (b c+3 a d)) \int \frac {1}{c+d x^2} \, dx}{2 (b c-a d)^3} \\ & = \frac {(b c+a d) x}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\sqrt {a} (3 b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {b} (b c-a d)^3}+\frac {\sqrt {c} (b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {d} (b c-a d)^3} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.84
\[
\int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {\sqrt {a} (3 b c+a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (-b c+a d)^3}+\frac {\frac {(b c-a d) x \left (2 a c+b c x^2+a d x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt {c} (b c+3 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d}}}{(b c-a d)^3}\right )
\]
[In]
Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)^2),x]
[Out]
((Sqrt[a]*(3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(-(b*c) + a*d)^3) + (((b*c - a*d)*x*(2*a*c + b*c
*x^2 + a*d*x^2))/((a + b*x^2)*(c + d*x^2)) + (Sqrt[c]*(b*c + 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/Sqrt[d])/(b*c
- a*d)^3)/2
Maple [A] (verified)
Time = 2.78 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.72
| | |
| method | result | size |
| | |
| default |
\(\frac {a \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (a d +3 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a d -b c \right )^{3}}-\frac {c \left (\frac {\left (-\frac {a d}{2}+\frac {b c}{2}\right ) x}{d \,x^{2}+c}+\frac {\left (3 a d +b c \right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}}\right )}{\left (a d -b c \right )^{3}}\) |
\(117\) |
| risch |
\(\text {Expression too large to display}\) |
\(1816\) |
| | |
|
|
|
|
[In]
int(x^4/(b*x^2+a)^2/(d*x^2+c)^2,x,method=_RETURNVERBOSE)
[Out]
a/(a*d-b*c)^3*((1/2*a*d-1/2*b*c)*x/(b*x^2+a)+1/2*(a*d+3*b*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-c/(a*d-b*c)^
3*((-1/2*a*d+1/2*b*c)*x/(d*x^2+c)+1/2*(3*a*d+b*c)/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2)))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (138) = 276\).
Time = 0.36 (sec) , antiderivative size = 1407, normalized size of antiderivative = 8.69
\[
\int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
integrate(x^4/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="fricas")
[Out]
[1/4*(2*(b^2*c^2 - a^2*d^2)*x^3 - ((3*b^2*c*d + a*b*d^2)*x^4 + 3*a*b*c^2 + a^2*c*d + (3*b^2*c^2 + 4*a*b*c*d +
a^2*d^2)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - ((b^2*c*d + 3*a*b*d^2)*x^4 + a*b*c^
2 + 3*a^2*c*d + (b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x^2)*sqrt(-c/d)*log((d*x^2 - 2*d*x*sqrt(-c/d) - c)/(d*x^2 +
c)) + 4*(a*b*c^2 - a^2*c*d)*x)/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b
^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2), 1/4*
(2*(b^2*c^2 - a^2*d^2)*x^3 - 2*((3*b^2*c*d + a*b*d^2)*x^4 + 3*a*b*c^2 + a^2*c*d + (3*b^2*c^2 + 4*a*b*c*d + a^2
*d^2)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - ((b^2*c*d + 3*a*b*d^2)*x^4 + a*b*c^2 + 3*a^2*c*d + (b^2*c^2 + 4
*a*b*c*d + 3*a^2*d^2)*x^2)*sqrt(-c/d)*log((d*x^2 - 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) + 4*(a*b*c^2 - a^2*c*d)*
x)/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3
- a^3*b*d^4)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2), 1/4*(2*(b^2*c^2 - a^2*d^2)*x^3 +
2*((b^2*c*d + 3*a*b*d^2)*x^4 + a*b*c^2 + 3*a^2*c*d + (b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x^2)*sqrt(c/d)*arctan(
d*x*sqrt(c/d)/c) - ((3*b^2*c*d + a*b*d^2)*x^4 + 3*a*b*c^2 + a^2*c*d + (3*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^2)*s
qrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 4*(a*b*c^2 - a^2*c*d)*x)/(a*b^3*c^4 - 3*a^2*b^2*c^
3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (b^4*c^4
- 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2), 1/2*((b^2*c^2 - a^2*d^2)*x^3 - ((3*b^2*c*d + a*b*d^2)*x^4 +
3*a*b*c^2 + a^2*c*d + (3*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + ((b^2*c*d + 3
*a*b*d^2)*x^4 + a*b*c^2 + 3*a^2*c*d + (b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x^2)*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c)
+ 2*(a*b*c^2 - a^2*c*d)*x)/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*
c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2)]
Sympy [F(-1)]
Timed out. \[
\int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out}
\]
[In]
integrate(x**4/(b*x**2+a)**2/(d*x**2+c)**2,x)
[Out]
Timed out
Maxima [A] (verification not implemented)
none
Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.54
\[
\int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {{\left (3 \, a b c + a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b}} + \frac {{\left (b c^{2} + 3 \, a c d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d}} + \frac {{\left (b c + a d\right )} x^{3} + 2 \, a c x}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}
\]
[In]
integrate(x^4/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="maxima")
[Out]
-1/2*(3*a*b*c + a^2*d)*arctan(b*x/sqrt(a*b))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a*b)) +
1/2*(b*c^2 + 3*a*c*d)*arctan(d*x/sqrt(c*d))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c*d)) +
1/2*((b*c + a*d)*x^3 + 2*a*c*x)/(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d
^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)
Giac [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.22
\[
\int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {{\left (3 \, a b c + a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b}} + \frac {{\left (b c^{2} + 3 \, a c d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d}} + \frac {b c x^{3} + a d x^{3} + 2 \, a c x}{2 \, {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}
\]
[In]
integrate(x^4/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="giac")
[Out]
-1/2*(3*a*b*c + a^2*d)*arctan(b*x/sqrt(a*b))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a*b)) +
1/2*(b*c^2 + 3*a*c*d)*arctan(d*x/sqrt(c*d))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c*d)) +
1/2*(b*c*x^3 + a*d*x^3 + 2*a*c*x)/((b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2))
Mupad [B] (verification not implemented)
Time = 6.60 (sec) , antiderivative size = 5395, normalized size of antiderivative = 33.30
\[
\int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
int(x^4/((a + b*x^2)^2*(c + d*x^2)^2),x)
[Out]
((x^3*(a*d + b*c))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (a*c*x)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(a*c + x^2*(
a*d + b*c) + b*d*x^4) - (atan((((-c*d)^(1/2)*((x*(a^4*b*d^5 + b^5*c^4*d + 6*a*b^4*c^3*d^2 + 6*a^3*b^2*c*d^4 +
18*a^2*b^3*c^2*d^3))/(2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - ((-c*d)^(1/
2)*((4*a*b^8*c^7*d^2 + 4*a^7*b^2*c*d^8 - 24*a^2*b^7*c^6*d^3 + 60*a^3*b^6*c^5*d^4 - 80*a^4*b^5*c^4*d^5 + 60*a^5
*b^4*c^3*d^6 - 24*a^6*b^3*c^2*d^7)/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c
^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) - (x*(-c*d)^(1/2)*(3*a*d + b*c)*(16*a^7*b^2*d^9 + 16*b^9*c^7*d^2 - 80*
a*b^8*c^6*d^3 - 80*a^6*b^3*c*d^8 + 144*a^2*b^7*c^5*d^4 - 80*a^3*b^6*c^4*d^5 - 80*a^4*b^5*c^3*d^6 + 144*a^5*b^4
*c^2*d^7))/(8*(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 -
4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))*(3*a*d + b*c))/(4*(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)))*
(3*a*d + b*c)*1i)/(4*(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)) + ((-c*d)^(1/2)*((x*(a^4*b*d^5 +
b^5*c^4*d + 6*a*b^4*c^3*d^2 + 6*a^3*b^2*c*d^4 + 18*a^2*b^3*c^2*d^3))/(2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^
2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + ((-c*d)^(1/2)*((4*a*b^8*c^7*d^2 + 4*a^7*b^2*c*d^8 - 24*a^2*b^7*c^6*d^3 +
60*a^3*b^6*c^5*d^4 - 80*a^4*b^5*c^4*d^5 + 60*a^5*b^4*c^3*d^6 - 24*a^6*b^3*c^2*d^7)/(a^6*d^6 + b^6*c^6 + 15*a^
2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) + (x*(-c*d)^(1/2)*(3*
a*d + b*c)*(16*a^7*b^2*d^9 + 16*b^9*c^7*d^2 - 80*a*b^8*c^6*d^3 - 80*a^6*b^3*c*d^8 + 144*a^2*b^7*c^5*d^4 - 80*a
^3*b^6*c^4*d^5 - 80*a^4*b^5*c^3*d^6 + 144*a^5*b^4*c^2*d^7))/(8*(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*
b*c*d^3)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))*(3*a*d + b*c))/(4*(a^3*d^4
- b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)))*(3*a*d + b*c)*1i)/(4*(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 -
3*a^2*b*c*d^3)))/(((13*a^2*b^3*c^3*d^2)/4 + (13*a^3*b^2*c^2*d^3)/4 + (3*a*b^4*c^4*d)/4 + (3*a^4*b*c*d^4)/4)/(
a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d
^5) - ((-c*d)^(1/2)*((x*(a^4*b*d^5 + b^5*c^4*d + 6*a*b^4*c^3*d^2 + 6*a^3*b^2*c*d^4 + 18*a^2*b^3*c^2*d^3))/(2*(
a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - ((-c*d)^(1/2)*((4*a*b^8*c^7*d^2 + 4*
a^7*b^2*c*d^8 - 24*a^2*b^7*c^6*d^3 + 60*a^3*b^6*c^5*d^4 - 80*a^4*b^5*c^4*d^5 + 60*a^5*b^4*c^3*d^6 - 24*a^6*b^3
*c^2*d^7)/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d -
6*a^5*b*c*d^5) - (x*(-c*d)^(1/2)*(3*a*d + b*c)*(16*a^7*b^2*d^9 + 16*b^9*c^7*d^2 - 80*a*b^8*c^6*d^3 - 80*a^6*b^
3*c*d^8 + 144*a^2*b^7*c^5*d^4 - 80*a^3*b^6*c^4*d^5 - 80*a^4*b^5*c^3*d^6 + 144*a^5*b^4*c^2*d^7))/(8*(a^3*d^4 -
b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*
c*d^3)))*(3*a*d + b*c))/(4*(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)))*(3*a*d + b*c))/(4*(a^3*d^
4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)) + ((-c*d)^(1/2)*((x*(a^4*b*d^5 + b^5*c^4*d + 6*a*b^4*c^3*d^2
+ 6*a^3*b^2*c*d^4 + 18*a^2*b^3*c^2*d^3))/(2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*
c*d^3)) + ((-c*d)^(1/2)*((4*a*b^8*c^7*d^2 + 4*a^7*b^2*c*d^8 - 24*a^2*b^7*c^6*d^3 + 60*a^3*b^6*c^5*d^4 - 80*a^4
*b^5*c^4*d^5 + 60*a^5*b^4*c^3*d^6 - 24*a^6*b^3*c^2*d^7)/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c
^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) + (x*(-c*d)^(1/2)*(3*a*d + b*c)*(16*a^7*b^2*d^9 +
16*b^9*c^7*d^2 - 80*a*b^8*c^6*d^3 - 80*a^6*b^3*c*d^8 + 144*a^2*b^7*c^5*d^4 - 80*a^3*b^6*c^4*d^5 - 80*a^4*b^5*
c^3*d^6 + 144*a^5*b^4*c^2*d^7))/(8*(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)*(a^4*d^4 + b^4*c^4
+ 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))*(3*a*d + b*c))/(4*(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^
2 - 3*a^2*b*c*d^3)))*(3*a*d + b*c))/(4*(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3))))*(-c*d)^(1/2)
*(3*a*d + b*c)*1i)/(2*(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3)) - (atan((((-a*b)^(1/2)*((x*(a^4
*b*d^5 + b^5*c^4*d + 6*a*b^4*c^3*d^2 + 6*a^3*b^2*c*d^4 + 18*a^2*b^3*c^2*d^3))/(2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^
2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - ((-a*b)^(1/2)*((4*a*b^8*c^7*d^2 + 4*a^7*b^2*c*d^8 - 24*a^2*b^7*c
^6*d^3 + 60*a^3*b^6*c^5*d^4 - 80*a^4*b^5*c^4*d^5 + 60*a^5*b^4*c^3*d^6 - 24*a^6*b^3*c^2*d^7)/(a^6*d^6 + b^6*c^6
+ 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) - (x*(-a*b)^(
1/2)*(a*d + 3*b*c)*(16*a^7*b^2*d^9 + 16*b^9*c^7*d^2 - 80*a*b^8*c^6*d^3 - 80*a^6*b^3*c*d^8 + 144*a^2*b^7*c^5*d^
4 - 80*a^3*b^6*c^4*d^5 - 80*a^4*b^5*c^3*d^6 + 144*a^5*b^4*c^2*d^7))/(8*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2
- 3*a*b^3*c^2*d)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))*(a*d + 3*b*c))/(4*(
b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)))*(a*d + 3*b*c)*1i)/(4*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2
*c*d^2 - 3*a*b^3*c^2*d)) + ((-a*b)^(1/2)*((x*(a^4*b*d^5 + b^5*c^4*d + 6*a*b^4*c^3*d^2 + 6*a^3*b^2*c*d^4 + 18*a
^2*b^3*c^2*d^3))/(2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + ((-a*b)^(1/2)*(
(4*a*b^8*c^7*d^2 + 4*a^7*b^2*c*d^8 - 24*a^2*b^7*c^6*d^3 + 60*a^3*b^6*c^5*d^4 - 80*a^4*b^5*c^4*d^5 + 60*a^5*b^4
*c^3*d^6 - 24*a^6*b^3*c^2*d^7)/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d
^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) + (x*(-a*b)^(1/2)*(a*d + 3*b*c)*(16*a^7*b^2*d^9 + 16*b^9*c^7*d^2 - 80*a*b^
8*c^6*d^3 - 80*a^6*b^3*c*d^8 + 144*a^2*b^7*c^5*d^4 - 80*a^3*b^6*c^4*d^5 - 80*a^4*b^5*c^3*d^6 + 144*a^5*b^4*c^2
*d^7))/(8*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a
*b^3*c^3*d - 4*a^3*b*c*d^3)))*(a*d + 3*b*c))/(4*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)))*(a*d
+ 3*b*c)*1i)/(4*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)))/(((13*a^2*b^3*c^3*d^2)/4 + (13*a^3*
b^2*c^2*d^3)/4 + (3*a*b^4*c^4*d)/4 + (3*a^4*b*c*d^4)/4)/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c
^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) - ((-a*b)^(1/2)*((x*(a^4*b*d^5 + b^5*c^4*d + 6*a*
b^4*c^3*d^2 + 6*a^3*b^2*c*d^4 + 18*a^2*b^3*c^2*d^3))/(2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d
- 4*a^3*b*c*d^3)) - ((-a*b)^(1/2)*((4*a*b^8*c^7*d^2 + 4*a^7*b^2*c*d^8 - 24*a^2*b^7*c^6*d^3 + 60*a^3*b^6*c^5*d
^4 - 80*a^4*b^5*c^4*d^5 + 60*a^5*b^4*c^3*d^6 - 24*a^6*b^3*c^2*d^7)/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 2
0*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5) - (x*(-a*b)^(1/2)*(a*d + 3*b*c)*(16*a^
7*b^2*d^9 + 16*b^9*c^7*d^2 - 80*a*b^8*c^6*d^3 - 80*a^6*b^3*c*d^8 + 144*a^2*b^7*c^5*d^4 - 80*a^3*b^6*c^4*d^5 -
80*a^4*b^5*c^3*d^6 + 144*a^5*b^4*c^2*d^7))/(8*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)*(a^4*d^4
+ b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))*(a*d + 3*b*c))/(4*(b^4*c^3 - a^3*b*d^3 + 3*a
^2*b^2*c*d^2 - 3*a*b^3*c^2*d)))*(a*d + 3*b*c))/(4*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)) + (
(-a*b)^(1/2)*((x*(a^4*b*d^5 + b^5*c^4*d + 6*a*b^4*c^3*d^2 + 6*a^3*b^2*c*d^4 + 18*a^2*b^3*c^2*d^3))/(2*(a^4*d^4
+ b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + ((-a*b)^(1/2)*((4*a*b^8*c^7*d^2 + 4*a^7*b^2
*c*d^8 - 24*a^2*b^7*c^6*d^3 + 60*a^3*b^6*c^5*d^4 - 80*a^4*b^5*c^4*d^5 + 60*a^5*b^4*c^3*d^6 - 24*a^6*b^3*c^2*d^
7)/(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b
*c*d^5) + (x*(-a*b)^(1/2)*(a*d + 3*b*c)*(16*a^7*b^2*d^9 + 16*b^9*c^7*d^2 - 80*a*b^8*c^6*d^3 - 80*a^6*b^3*c*d^8
+ 144*a^2*b^7*c^5*d^4 - 80*a^3*b^6*c^4*d^5 - 80*a^4*b^5*c^3*d^6 + 144*a^5*b^4*c^2*d^7))/(8*(b^4*c^3 - a^3*b*d
^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))
)*(a*d + 3*b*c))/(4*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)))*(a*d + 3*b*c))/(4*(b^4*c^3 - a^3
*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d))))*(-a*b)^(1/2)*(a*d + 3*b*c)*1i)/(2*(b^4*c^3 - a^3*b*d^3 + 3*a^2*b^
2*c*d^2 - 3*a*b^3*c^2*d))