\(\int \frac {x^2}{(a+b x^2) (c+d x^2)^2} \, dx\) [244]
Optimal result
Integrand size = 22, antiderivative size = 104 \[
\int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {x}{2 (b c-a d) \left (c+d x^2\right )}-\frac {\sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^2}+\frac {(b c+a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d} (b c-a d)^2}
\]
[Out]
1/2*x/(-a*d+b*c)/(d*x^2+c)-arctan(x*b^(1/2)/a^(1/2))*a^(1/2)*b^(1/2)/(-a*d+b*c)^2+1/2*(a*d+b*c)*arctan(x*d^(1/
2)/c^(1/2))/(-a*d+b*c)^2/c^(1/2)/d^(1/2)
Rubi [A] (verified)
Time = 0.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of
steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {482, 536, 211}
\[
\int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {\sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^2}+\frac {(a d+b c) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d} (b c-a d)^2}+\frac {x}{2 \left (c+d x^2\right ) (b c-a d)}
\]
[In]
Int[x^2/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
x/(2*(b*c - a*d)*(c + d*x^2)) - (Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^2 + ((b*c + a*d)*Arc
Tan[(Sqrt[d]*x)/Sqrt[c]])/(2*Sqrt[c]*Sqrt[d]*(b*c - a*d)^2)
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 482
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 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] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && 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]
Rubi steps \begin{align*}
\text {integral}& = \frac {x}{2 (b c-a d) \left (c+d x^2\right )}-\frac {\int \frac {a-b x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 (b c-a d)} \\ & = \frac {x}{2 (b c-a d) \left (c+d x^2\right )}-\frac {(a b) \int \frac {1}{a+b x^2} \, dx}{(b c-a d)^2}+\frac {(b c+a d) \int \frac {1}{c+d x^2} \, dx}{2 (b c-a d)^2} \\ & = \frac {x}{2 (b c-a d) \left (c+d x^2\right )}-\frac {\sqrt {a} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^2}+\frac {(b c+a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d} (b c-a d)^2} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87
\[
\int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {\frac {(b c-a d) x}{c+d x^2}-2 \sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+\frac {(b c+a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}}{2 (b c-a d)^2}
\]
[In]
Integrate[x^2/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
(((b*c - a*d)*x)/(c + d*x^2) - 2*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + ((b*c + a*d)*ArcTan[(Sqrt[d]*x)
/Sqrt[c]])/(Sqrt[c]*Sqrt[d]))/(2*(b*c - a*d)^2)
Maple [A] (verified)
Time = 2.68 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.82
| | |
method | result | size |
| | |
default |
\(-\frac {a b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right )^{2} \sqrt {a b}}+\frac {\frac {\left (-\frac {a d}{2}+\frac {b c}{2}\right ) x}{d \,x^{2}+c}+\frac {\left (a d +b c \right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}}}{\left (a d -b c \right )^{2}}\) |
\(85\) |
risch |
\(-\frac {x}{2 \left (a d -b c \right ) \left (d \,x^{2}+c \right )}+\frac {\sqrt {-a b}\, \ln \left (\left (-4 \left (-a b \right )^{\frac {3}{2}} a \,d^{2}-4 \left (-a b \right )^{\frac {3}{2}} b c d -5 a^{2} \sqrt {-a b}\, d^{2} b -2 \sqrt {-a b}\, a \,b^{2} c d -b^{3} c^{2} \sqrt {-a b}\right ) x -a^{3} b \,d^{2}+2 a^{2} b^{2} c d -a \,b^{3} c^{2}\right )}{2 \left (a d -b c \right )^{2}}-\frac {\sqrt {-a b}\, \ln \left (\left (4 \left (-a b \right )^{\frac {3}{2}} a \,d^{2}+4 \left (-a b \right )^{\frac {3}{2}} b c d +5 a^{2} \sqrt {-a b}\, d^{2} b +2 \sqrt {-a b}\, a \,b^{2} c d +b^{3} c^{2} \sqrt {-a b}\right ) x -a^{3} b \,d^{2}+2 a^{2} b^{2} c d -a \,b^{3} c^{2}\right )}{2 \left (a d -b c \right )^{2}}-\frac {\ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a d}{4 \sqrt {-c d}\, \left (a d -b c \right )^{2}}-\frac {\ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b c}{4 \sqrt {-c d}\, \left (a d -b c \right )^{2}}+\frac {\ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a d}{4 \sqrt {-c d}\, \left (a d -b c \right )^{2}}+\frac {\ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b c}{4 \sqrt {-c d}\, \left (a d -b c \right )^{2}}\) |
\(403\) |
| | |
|
|
|
[In]
int(x^2/(b*x^2+a)/(d*x^2+c)^2,x,method=_RETURNVERBOSE)
[Out]
-a*b/(a*d-b*c)^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))+1/(a*d-b*c)^2*((-1/2*a*d+1/2*b*c)*x/(d*x^2+c)+1/2*(a*d+b*
c)/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2)))
Fricas [A] (verification not implemented)
none
Time = 0.29 (sec) , antiderivative size = 705, normalized size of antiderivative = 6.78
\[
\int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\left [\frac {2 \, {\left (c d^{2} x^{2} + c^{2} d\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - {\left (b c^{2} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x}{4 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}}, \frac {{\left (b c^{2} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + {\left (c d^{2} x^{2} + c^{2} d\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + {\left (b c^{2} d - a c d^{2}\right )} x}{2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}}, -\frac {4 \, {\left (c d^{2} x^{2} + c^{2} d\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (b c^{2} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) - 2 \, {\left (b c^{2} d - a c d^{2}\right )} x}{4 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}}, -\frac {2 \, {\left (c d^{2} x^{2} + c^{2} d\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (b c^{2} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - {\left (b c^{2} d - a c d^{2}\right )} x}{2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}}\right ]
\]
[In]
integrate(x^2/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="fricas")
[Out]
[1/4*(2*(c*d^2*x^2 + c^2*d)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - (b*c^2 + a*c*d + (b*c*d
+ a*d^2)*x^2)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) + 2*(b*c^2*d - a*c*d^2)*x)/(b^2*c^4*d
- 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^2), 1/2*((b*c^2 + a*c*d + (b*c*d +
a*d^2)*x^2)*sqrt(c*d)*arctan(sqrt(c*d)*x/c) + (c*d^2*x^2 + c^2*d)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)
/(b*x^2 + a)) + (b*c^2*d - a*c*d^2)*x)/(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^3*d^2 - 2*a*b*c^2*d^3
+ a^2*c*d^4)*x^2), -1/4*(4*(c*d^2*x^2 + c^2*d)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) + (b*c^2 + a*c*d + (b*c*d + a*
d^2)*x^2)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) - 2*(b*c^2*d - a*c*d^2)*x)/(b^2*c^4*d - 2*a
*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^2), -1/2*(2*(c*d^2*x^2 + c^2*d)*sqrt(a*
b)*arctan(sqrt(a*b)*x/a) - (b*c^2 + a*c*d + (b*c*d + a*d^2)*x^2)*sqrt(c*d)*arctan(sqrt(c*d)*x/c) - (b*c^2*d -
a*c*d^2)*x)/(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^2)]
Sympy [F(-1)]
Timed out. \[
\int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Timed out}
\]
[In]
integrate(x**2/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
Timed out
Maxima [A] (verification not implemented)
none
Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.14
\[
\int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {a b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b}} + \frac {{\left (b c + a d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {x}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}
\]
[In]
integrate(x^2/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="maxima")
[Out]
-a*b*arctan(b*x/sqrt(a*b))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(a*b)) + 1/2*(b*c + a*d)*arctan(d*x/sqrt(c*d))
/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c*d)) + 1/2*x/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)
Giac [A] (verification not implemented)
none
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.06
\[
\int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {a b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b}} + \frac {{\left (b c + a d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {x}{2 \, {\left (d x^{2} + c\right )} {\left (b c - a d\right )}}
\]
[In]
integrate(x^2/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="giac")
[Out]
-a*b*arctan(b*x/sqrt(a*b))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(a*b)) + 1/2*(b*c + a*d)*arctan(d*x/sqrt(c*d))
/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c*d)) + 1/2*x/((d*x^2 + c)*(b*c - a*d))
Mupad [B] (verification not implemented)
Time = 5.79 (sec) , antiderivative size = 3154, normalized size of antiderivative = 30.33
\[
\int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Too large to display}
\]
[In]
int(x^2/((a + b*x^2)*(c + d*x^2)^2),x)
[Out]
(atan(-(((-a*b)^(1/2)*(((-a*b)^(1/2)*((2*a^5*b^2*d^6 + 2*a*b^6*c^4*d^2 - 8*a^4*b^3*c*d^5 - 8*a^2*b^5*c^3*d^3 +
12*a^3*b^4*c^2*d^4)/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (x*(-a*b)^(1/2)*(16*a^5*b^2*d^7
+ 16*b^7*c^5*d^2 - 48*a*b^6*c^4*d^3 - 48*a^4*b^3*c*d^6 + 32*a^2*b^5*c^3*d^4 + 32*a^3*b^4*c^2*d^5))/(8*(a^2*d^
2 + b^2*c^2 - 2*a*b*c*d)^2)))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (x*(b^5*c^2*d + 5*a^2*b^3*d^3 + 2*a*b^4*c*
d^2))/(4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))*1i)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) - ((-a*b)^(1/2)*(((-a*b)^(1/2)*
((2*a^5*b^2*d^6 + 2*a*b^6*c^4*d^2 - 8*a^4*b^3*c*d^5 - 8*a^2*b^5*c^3*d^3 + 12*a^3*b^4*c^2*d^4)/(2*(a^3*d^3 - b^
3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x*(-a*b)^(1/2)*(16*a^5*b^2*d^7 + 16*b^7*c^5*d^2 - 48*a*b^6*c^4*d^3
- 48*a^4*b^3*c*d^6 + 32*a^2*b^5*c^3*d^4 + 32*a^3*b^4*c^2*d^5))/(8*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2)))/(2*(a^2
*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(b^5*c^2*d + 5*a^2*b^3*d^3 + 2*a*b^4*c*d^2))/(4*(a^2*d^2 + b^2*c^2 - 2*a*b*c
*d)))*1i)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(((a^2*b^3*d^2)/2 + (a*b^4*c*d)/2)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2
*d - 3*a^2*b*c*d^2) + ((-a*b)^(1/2)*(((-a*b)^(1/2)*((2*a^5*b^2*d^6 + 2*a*b^6*c^4*d^2 - 8*a^4*b^3*c*d^5 - 8*a^2
*b^5*c^3*d^3 + 12*a^3*b^4*c^2*d^4)/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (x*(-a*b)^(1/2)*(
16*a^5*b^2*d^7 + 16*b^7*c^5*d^2 - 48*a*b^6*c^4*d^3 - 48*a^4*b^3*c*d^6 + 32*a^2*b^5*c^3*d^4 + 32*a^3*b^4*c^2*d^
5))/(8*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2)))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (x*(b^5*c^2*d + 5*a^2*b^3*d^
3 + 2*a*b^4*c*d^2))/(4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) + ((-a*b)^(1/2)*(((-
a*b)^(1/2)*((2*a^5*b^2*d^6 + 2*a*b^6*c^4*d^2 - 8*a^4*b^3*c*d^5 - 8*a^2*b^5*c^3*d^3 + 12*a^3*b^4*c^2*d^4)/(2*(a
^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x*(-a*b)^(1/2)*(16*a^5*b^2*d^7 + 16*b^7*c^5*d^2 - 48*a*b
^6*c^4*d^3 - 48*a^4*b^3*c*d^6 + 32*a^2*b^5*c^3*d^4 + 32*a^3*b^4*c^2*d^5))/(8*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2
)))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(b^5*c^2*d + 5*a^2*b^3*d^3 + 2*a*b^4*c*d^2))/(4*(a^2*d^2 + b^2*c^
2 - 2*a*b*c*d))))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))*(-a*b)^(1/2)*1i)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) - x/(2*(c
+ d*x^2)*(a*d - b*c)) + (atan((((-c*d)^(1/2)*((x*(b^5*c^2*d + 5*a^2*b^3*d^3 + 2*a*b^4*c*d^2))/(2*(a^2*d^2 + b
^2*c^2 - 2*a*b*c*d)) - ((-c*d)^(1/2)*((2*a^5*b^2*d^6 + 2*a*b^6*c^4*d^2 - 8*a^4*b^3*c*d^5 - 8*a^2*b^5*c^3*d^3 +
12*a^3*b^4*c^2*d^4)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) - (x*(-c*d)^(1/2)*(a*d + b*c)*(16*a^5
*b^2*d^7 + 16*b^7*c^5*d^2 - 48*a*b^6*c^4*d^3 - 48*a^4*b^3*c*d^6 + 32*a^2*b^5*c^3*d^4 + 32*a^3*b^4*c^2*d^5))/(8
*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2*c*d^3 + b^2*c^3*d - 2*a*b*c^2*d^2)))*(a*d + b*c))/(4*(a^2*c*d^3 + b^2*c^
3*d - 2*a*b*c^2*d^2)))*(a*d + b*c)*1i)/(4*(a^2*c*d^3 + b^2*c^3*d - 2*a*b*c^2*d^2)) + ((-c*d)^(1/2)*((x*(b^5*c^
2*d + 5*a^2*b^3*d^3 + 2*a*b^4*c*d^2))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + ((-c*d)^(1/2)*((2*a^5*b^2*d^6 + 2*
a*b^6*c^4*d^2 - 8*a^4*b^3*c*d^5 - 8*a^2*b^5*c^3*d^3 + 12*a^3*b^4*c^2*d^4)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d -
3*a^2*b*c*d^2) + (x*(-c*d)^(1/2)*(a*d + b*c)*(16*a^5*b^2*d^7 + 16*b^7*c^5*d^2 - 48*a*b^6*c^4*d^3 - 48*a^4*b^3
*c*d^6 + 32*a^2*b^5*c^3*d^4 + 32*a^3*b^4*c^2*d^5))/(8*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2*c*d^3 + b^2*c^3*d -
2*a*b*c^2*d^2)))*(a*d + b*c))/(4*(a^2*c*d^3 + b^2*c^3*d - 2*a*b*c^2*d^2)))*(a*d + b*c)*1i)/(4*(a^2*c*d^3 + b^
2*c^3*d - 2*a*b*c^2*d^2)))/(((a^2*b^3*d^2)/2 + (a*b^4*c*d)/2)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d
^2) - ((-c*d)^(1/2)*((x*(b^5*c^2*d + 5*a^2*b^3*d^3 + 2*a*b^4*c*d^2))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - ((-
c*d)^(1/2)*((2*a^5*b^2*d^6 + 2*a*b^6*c^4*d^2 - 8*a^4*b^3*c*d^5 - 8*a^2*b^5*c^3*d^3 + 12*a^3*b^4*c^2*d^4)/(a^3*
d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) - (x*(-c*d)^(1/2)*(a*d + b*c)*(16*a^5*b^2*d^7 + 16*b^7*c^5*d^2
- 48*a*b^6*c^4*d^3 - 48*a^4*b^3*c*d^6 + 32*a^2*b^5*c^3*d^4 + 32*a^3*b^4*c^2*d^5))/(8*(a^2*d^2 + b^2*c^2 - 2*a*
b*c*d)*(a^2*c*d^3 + b^2*c^3*d - 2*a*b*c^2*d^2)))*(a*d + b*c))/(4*(a^2*c*d^3 + b^2*c^3*d - 2*a*b*c^2*d^2)))*(a*
d + b*c))/(4*(a^2*c*d^3 + b^2*c^3*d - 2*a*b*c^2*d^2)) + ((-c*d)^(1/2)*((x*(b^5*c^2*d + 5*a^2*b^3*d^3 + 2*a*b^4
*c*d^2))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + ((-c*d)^(1/2)*((2*a^5*b^2*d^6 + 2*a*b^6*c^4*d^2 - 8*a^4*b^3*c*d
^5 - 8*a^2*b^5*c^3*d^3 + 12*a^3*b^4*c^2*d^4)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) + (x*(-c*d)^(
1/2)*(a*d + b*c)*(16*a^5*b^2*d^7 + 16*b^7*c^5*d^2 - 48*a*b^6*c^4*d^3 - 48*a^4*b^3*c*d^6 + 32*a^2*b^5*c^3*d^4 +
32*a^3*b^4*c^2*d^5))/(8*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2*c*d^3 + b^2*c^3*d - 2*a*b*c^2*d^2)))*(a*d + b*c)
)/(4*(a^2*c*d^3 + b^2*c^3*d - 2*a*b*c^2*d^2)))*(a*d + b*c))/(4*(a^2*c*d^3 + b^2*c^3*d - 2*a*b*c^2*d^2))))*(-c*
d)^(1/2)*(a*d + b*c)*1i)/(2*(a^2*c*d^3 + b^2*c^3*d - 2*a*b*c^2*d^2))