3.1.25 \(\int \frac {1}{(a+b x^2) (c+d x^2)^2} \, dx\) [25]

Optimal. Leaf size=109 \[ -\frac {d x}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)^2}-\frac {\sqrt {d} (3 b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^2} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {425, 536, 211} \begin {gather*} \frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)^2}-\frac {\sqrt {d} (3 b c-a d) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^2}-\frac {d x}{2 c \left (c+d x^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)*(c + d*x^2)^2),x]

[Out]

-1/2*(d*x)/(c*(b*c - a*d)*(c + d*x^2)) + (b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d)^2) - (Sqrt
[d]*(3*b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*(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 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, 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*} \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx &=-\frac {d x}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {\int \frac {2 b c-a d-b d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=-\frac {d x}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {b^2 \int \frac {1}{a+b x^2} \, dx}{(b c-a d)^2}-\frac {(d (3 b c-a d)) \int \frac {1}{c+d x^2} \, dx}{2 c (b c-a d)^2}\\ &=-\frac {d x}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)^2}-\frac {\sqrt {d} (3 b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^2}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 95, normalized size = 0.87 \begin {gather*} \frac {\frac {d (-b c+a d) x}{c \left (c+d x^2\right )}+\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {\sqrt {d} (-3 b c+a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2}}}{2 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*(-(b*c) + a*d)*x)/(c*(c + d*x^2)) + (2*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (Sqrt[d]*(-3*b*c + a
*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(3/2))/(2*(b*c - a*d)^2)

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Maple [A]
time = 0.15, size = 93, normalized size = 0.85

method	result	size
default	\(\frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right )^{2} \sqrt {a b}}+\frac {d \left (\frac {\left (a d -b c \right ) x}{2 c \left (d \,x^{2}+c \right )}+\frac {\left (a d -3 b c \right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 c \sqrt {c d}}\right )}{\left (a d -b c \right )^{2}}\)	\(93\)
risch	\(\frac {d x}{2 c \left (a d -b c \right ) \left (d \,x^{2}+c \right )}+\frac {\sqrt {-a b}\, b \ln \left (\left (-4 \left (-a b \right )^{\frac {3}{2}} a b \,c^{2} d -4 \left (-a b \right )^{\frac {3}{2}} b^{2} c^{3}-\sqrt {-a b}\, a^{4} d^{3}+6 \sqrt {-a b}\, a^{3} b c \,d^{2}-13 \sqrt {-a b}\, a^{2} b^{2} c^{2} d \right ) x +a^{5} d^{3}-6 a^{4} b c \,d^{2}+9 a^{3} b^{2} c^{2} d -4 b^{3} a^{2} c^{3}\right )}{2 a \left (a d -b c \right )^{2}}-\frac {\sqrt {-a b}\, b \ln \left (\left (4 \left (-a b \right )^{\frac {3}{2}} a b \,c^{2} d +4 \left (-a b \right )^{\frac {3}{2}} b^{2} c^{3}+\sqrt {-a b}\, a^{4} d^{3}-6 \sqrt {-a b}\, a^{3} b c \,d^{2}+13 \sqrt {-a b}\, a^{2} b^{2} c^{2} d \right ) x +a^{5} d^{3}-6 a^{4} b c \,d^{2}+9 a^{3} b^{2} c^{2} d -4 b^{3} a^{2} c^{3}\right )}{2 a \left (a d -b c \right )^{2}}+\frac {\sqrt {-c d}\, \ln \left (\left (-\left (-c d \right )^{\frac {3}{2}} a^{3} d^{3}+5 \left (-c d \right )^{\frac {3}{2}} a^{2} b c \,d^{2}-3 \left (-c d \right )^{\frac {3}{2}} a \,b^{2} c^{2} d -9 \left (-c d \right )^{\frac {3}{2}} b^{3} c^{3}-\sqrt {-c d}\, a^{2} b \,c^{2} d^{3}+6 \sqrt {-c d}\, a \,b^{2} c^{3} d^{2}-13 \sqrt {-c d}\, b^{3} c^{4} d \right ) x -a^{3} c^{2} d^{4}+6 a^{2} b \,c^{3} d^{3}-9 a \,b^{2} d^{2} c^{4}+4 b^{3} c^{5} d \right ) a d}{4 c^{2} \left (a d -b c \right )^{2}}-\frac {3 \sqrt {-c d}\, \ln \left (\left (-\left (-c d \right )^{\frac {3}{2}} a^{3} d^{3}+5 \left (-c d \right )^{\frac {3}{2}} a^{2} b c \,d^{2}-3 \left (-c d \right )^{\frac {3}{2}} a \,b^{2} c^{2} d -9 \left (-c d \right )^{\frac {3}{2}} b^{3} c^{3}-\sqrt {-c d}\, a^{2} b \,c^{2} d^{3}+6 \sqrt {-c d}\, a \,b^{2} c^{3} d^{2}-13 \sqrt {-c d}\, b^{3} c^{4} d \right ) x -a^{3} c^{2} d^{4}+6 a^{2} b \,c^{3} d^{3}-9 a \,b^{2} d^{2} c^{4}+4 b^{3} c^{5} d \right ) b}{4 c \left (a d -b c \right )^{2}}-\frac {\sqrt {-c d}\, \ln \left (\left (\left (-c d \right )^{\frac {3}{2}} a^{3} d^{3}-5 \left (-c d \right )^{\frac {3}{2}} a^{2} b c \,d^{2}+3 \left (-c d \right )^{\frac {3}{2}} a \,b^{2} c^{2} d +9 \left (-c d \right )^{\frac {3}{2}} b^{3} c^{3}+\sqrt {-c d}\, a^{2} b \,c^{2} d^{3}-6 \sqrt {-c d}\, a \,b^{2} c^{3} d^{2}+13 \sqrt {-c d}\, b^{3} c^{4} d \right ) x -a^{3} c^{2} d^{4}+6 a^{2} b \,c^{3} d^{3}-9 a \,b^{2} d^{2} c^{4}+4 b^{3} c^{5} d \right ) a d}{4 c^{2} \left (a d -b c \right )^{2}}+\frac {3 \sqrt {-c d}\, \ln \left (\left (\left (-c d \right )^{\frac {3}{2}} a^{3} d^{3}-5 \left (-c d \right )^{\frac {3}{2}} a^{2} b c \,d^{2}+3 \left (-c d \right )^{\frac {3}{2}} a \,b^{2} c^{2} d +9 \left (-c d \right )^{\frac {3}{2}} b^{3} c^{3}+\sqrt {-c d}\, a^{2} b \,c^{2} d^{3}-6 \sqrt {-c d}\, a \,b^{2} c^{3} d^{2}+13 \sqrt {-c d}\, b^{3} c^{4} d \right ) x -a^{3} c^{2} d^{4}+6 a^{2} b \,c^{3} d^{3}-9 a \,b^{2} d^{2} c^{4}+4 b^{3} c^{5} d \right ) b}{4 c \left (a d -b c \right )^{2}}\)	\(1039\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 133, normalized size = 1.22 \begin {gather*} \frac {b^{2} \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 {d x}{2 \, {\left (b c^{3} - a c^{2} d + {\left (b c^{2} d - a c d^{2}\right )} x^{2}\right )}} - \frac {{\left (3 \, b c d - a d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {c d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.06, size = 711, normalized size = 6.52 \begin {gather*} \left [\frac {2 \, {\left (b c d x^{2} + b c^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - {\left (3 \, b c^{2} - a c d + {\left (3 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) - 2 \, {\left (b c d - a d^{2}\right )} x}{4 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, -\frac {{\left (3 \, b c^{2} - a c d + {\left (3 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) - {\left (b c d x^{2} + b c^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + {\left (b c d - a d^{2}\right )} x}{2 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, \frac {4 \, {\left (b c d x^{2} + b c^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - {\left (3 \, b c^{2} - a c d + {\left (3 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) - 2 \, {\left (b c d - a d^{2}\right )} x}{4 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, \frac {2 \, {\left (b c d x^{2} + b c^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - {\left (3 \, b c^{2} - a c d + {\left (3 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) - {\left (b c d - a d^{2}\right )} x}{2 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*(b*c*d*x^2 + b*c^2)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - (3*b*c^2 - a*c*d + (3
*b*c*d - a*d^2)*x^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) - 2*(b*c*d - a*d^2)*x)/(b^2*c^
4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x^2), -1/2*((3*b*c^2 - a*c*d + (3*b*c*
d - a*d^2)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) - (b*c*d*x^2 + b*c^2)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) -
a)/(b*x^2 + a)) + (b*c*d - a*d^2)*x)/(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*
c*d^3)*x^2), 1/4*(4*(b*c*d*x^2 + b*c^2)*sqrt(b/a)*arctan(x*sqrt(b/a)) - (3*b*c^2 - a*c*d + (3*b*c*d - a*d^2)*x
^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) - 2*(b*c*d - a*d^2)*x)/(b^2*c^4 - 2*a*b*c^3*d +
a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x^2), 1/2*(2*(b*c*d*x^2 + b*c^2)*sqrt(b/a)*arctan(x*sqr
t(b/a)) - (3*b*c^2 - a*c*d + (3*b*c*d - a*d^2)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) - (b*c*d - a*d^2)*x)/(b^2*c^
4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x^2)]

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

[Out]

Timed out

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Giac [A]
time = 1.43, size = 122, normalized size = 1.12 \begin {gather*} \frac {b^{2} \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 (3 \, b c d - a d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {c d}} - \frac {d x}{2 \, {\left (b c^{2} - a c d\right )} {\left (d x^{2} + c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.69, size = 2500, normalized size = 22.94 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^2)*(c + d*x^2)^2),x)

[Out]

(d*x)/(2*c*(c + d*x^2)*(a*d - b*c)) - (atan((((-c^3*d)^(1/2)*(a*d - 3*b*c)*((x*(a^2*b^3*d^5 + 13*b^5*c^2*d^3 -
6*a*b^4*c*d^4))/(2*(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)) - (((4*b^7*c^6*d^2 - 18*a*b^6*c^5*d^3 - 2*a^5*b^2*c
*d^7 + 32*a^2*b^5*c^4*d^4 - 28*a^3*b^4*c^3*d^5 + 12*a^4*b^3*c^2*d^6)/(b^3*c^5 - a^3*c^2*d^3 + 3*a^2*b*c^3*d^2
- 3*a*b^2*c^4*d) - (x*(-c^3*d)^(1/2)*(a*d - 3*b*c)*(16*b^7*c^7*d^2 - 48*a*b^6*c^6*d^3 + 32*a^2*b^5*c^5*d^4 + 3
2*a^3*b^4*c^4*d^5 - 48*a^4*b^3*c^3*d^6 + 16*a^5*b^2*c^2*d^7))/(8*(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)*(b^2*c^
5 + a^2*c^3*d^2 - 2*a*b*c^4*d)))*(-c^3*d)^(1/2)*(a*d - 3*b*c))/(4*(b^2*c^5 + a^2*c^3*d^2 - 2*a*b*c^4*d)))*1i)/
(4*(b^2*c^5 + a^2*c^3*d^2 - 2*a*b*c^4*d)) + ((-c^3*d)^(1/2)*(a*d - 3*b*c)*((x*(a^2*b^3*d^5 + 13*b^5*c^2*d^3 -
6*a*b^4*c*d^4))/(2*(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)) + (((4*b^7*c^6*d^2 - 18*a*b^6*c^5*d^3 - 2*a^5*b^2*c*
d^7 + 32*a^2*b^5*c^4*d^4 - 28*a^3*b^4*c^3*d^5 + 12*a^4*b^3*c^2*d^6)/(b^3*c^5 - a^3*c^2*d^3 + 3*a^2*b*c^3*d^2 -
3*a*b^2*c^4*d) + (x*(-c^3*d)^(1/2)*(a*d - 3*b*c)*(16*b^7*c^7*d^2 - 48*a*b^6*c^6*d^3 + 32*a^2*b^5*c^5*d^4 + 32
*a^3*b^4*c^4*d^5 - 48*a^4*b^3*c^3*d^6 + 16*a^5*b^2*c^2*d^7))/(8*(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)*(b^2*c^5
+ a^2*c^3*d^2 - 2*a*b*c^4*d)))*(-c^3*d)^(1/2)*(a*d - 3*b*c))/(4*(b^2*c^5 + a^2*c^3*d^2 - 2*a*b*c^4*d)))*1i)/(
4*(b^2*c^5 + a^2*c^3*d^2 - 2*a*b*c^4*d)))/(((a*b^4*d^4)/2 - (3*b^5*c*d^3)/2)/(b^3*c^5 - a^3*c^2*d^3 + 3*a^2*b*
c^3*d^2 - 3*a*b^2*c^4*d) + ((-c^3*d)^(1/2)*(a*d - 3*b*c)*((x*(a^2*b^3*d^5 + 13*b^5*c^2*d^3 - 6*a*b^4*c*d^4))/(
2*(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)) - (((4*b^7*c^6*d^2 - 18*a*b^6*c^5*d^3 - 2*a^5*b^2*c*d^7 + 32*a^2*b^5*
c^4*d^4 - 28*a^3*b^4*c^3*d^5 + 12*a^4*b^3*c^2*d^6)/(b^3*c^5 - a^3*c^2*d^3 + 3*a^2*b*c^3*d^2 - 3*a*b^2*c^4*d) -
(x*(-c^3*d)^(1/2)*(a*d - 3*b*c)*(16*b^7*c^7*d^2 - 48*a*b^6*c^6*d^3 + 32*a^2*b^5*c^5*d^4 + 32*a^3*b^4*c^4*d^5
- 48*a^4*b^3*c^3*d^6 + 16*a^5*b^2*c^2*d^7))/(8*(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)*(b^2*c^5 + a^2*c^3*d^2 -
2*a*b*c^4*d)))*(-c^3*d)^(1/2)*(a*d - 3*b*c))/(4*(b^2*c^5 + a^2*c^3*d^2 - 2*a*b*c^4*d))))/(4*(b^2*c^5 + a^2*c^3
*d^2 - 2*a*b*c^4*d)) - ((-c^3*d)^(1/2)*(a*d - 3*b*c)*((x*(a^2*b^3*d^5 + 13*b^5*c^2*d^3 - 6*a*b^4*c*d^4))/(2*(b
^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)) + (((4*b^7*c^6*d^2 - 18*a*b^6*c^5*d^3 - 2*a^5*b^2*c*d^7 + 32*a^2*b^5*c^4*
d^4 - 28*a^3*b^4*c^3*d^5 + 12*a^4*b^3*c^2*d^6)/(b^3*c^5 - a^3*c^2*d^3 + 3*a^2*b*c^3*d^2 - 3*a*b^2*c^4*d) + (x*
(-c^3*d)^(1/2)*(a*d - 3*b*c)*(16*b^7*c^7*d^2 - 48*a*b^6*c^6*d^3 + 32*a^2*b^5*c^5*d^4 + 32*a^3*b^4*c^4*d^5 - 48
*a^4*b^3*c^3*d^6 + 16*a^5*b^2*c^2*d^7))/(8*(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)*(b^2*c^5 + a^2*c^3*d^2 - 2*a*
b*c^4*d)))*(-c^3*d)^(1/2)*(a*d - 3*b*c))/(4*(b^2*c^5 + a^2*c^3*d^2 - 2*a*b*c^4*d))))/(4*(b^2*c^5 + a^2*c^3*d^2
- 2*a*b*c^4*d))))*(-c^3*d)^(1/2)*(a*d - 3*b*c)*1i)/(2*(b^2*c^5 + a^2*c^3*d^2 - 2*a*b*c^4*d)) - (atan((((-a*b^
3)^(1/2)*((((4*b^7*c^6*d^2 - 18*a*b^6*c^5*d^3 - 2*a^5*b^2*c*d^7 + 32*a^2*b^5*c^4*d^4 - 28*a^3*b^4*c^3*d^5 + 12
*a^4*b^3*c^2*d^6)/(2*(b^3*c^5 - a^3*c^2*d^3 + 3*a^2*b*c^3*d^2 - 3*a*b^2*c^4*d)) - (x*(-a*b^3)^(1/2)*(16*b^7*c^
7*d^2 - 48*a*b^6*c^6*d^3 + 32*a^2*b^5*c^5*d^4 + 32*a^3*b^4*c^4*d^5 - 48*a^4*b^3*c^3*d^6 + 16*a^5*b^2*c^2*d^7))
/(8*(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)*(a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d)))*(-a*b^3)^(1/2))/(2*(a^3*d^2 +
a*b^2*c^2 - 2*a^2*b*c*d)) - (x*(a^2*b^3*d^5 + 13*b^5*c^2*d^3 - 6*a*b^4*c*d^4))/(4*(b^2*c^4 + a^2*c^2*d^2 - 2*a
*b*c^3*d)))*1i)/(a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d) - ((-a*b^3)^(1/2)*((((4*b^7*c^6*d^2 - 18*a*b^6*c^5*d^3 - 2
*a^5*b^2*c*d^7 + 32*a^2*b^5*c^4*d^4 - 28*a^3*b^4*c^3*d^5 + 12*a^4*b^3*c^2*d^6)/(2*(b^3*c^5 - a^3*c^2*d^3 + 3*a
^2*b*c^3*d^2 - 3*a*b^2*c^4*d)) + (x*(-a*b^3)^(1/2)*(16*b^7*c^7*d^2 - 48*a*b^6*c^6*d^3 + 32*a^2*b^5*c^5*d^4 + 3
2*a^3*b^4*c^4*d^5 - 48*a^4*b^3*c^3*d^6 + 16*a^5*b^2*c^2*d^7))/(8*(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)*(a^3*d^
2 + a*b^2*c^2 - 2*a^2*b*c*d)))*(-a*b^3)^(1/2))/(2*(a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d)) + (x*(a^2*b^3*d^5 + 13*
b^5*c^2*d^3 - 6*a*b^4*c*d^4))/(4*(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)))*1i)/(a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*
d))/(((-a*b^3)^(1/2)*((((4*b^7*c^6*d^2 - 18*a*b^6*c^5*d^3 - 2*a^5*b^2*c*d^7 + 32*a^2*b^5*c^4*d^4 - 28*a^3*b^4*
c^3*d^5 + 12*a^4*b^3*c^2*d^6)/(2*(b^3*c^5 - a^3*c^2*d^3 + 3*a^2*b*c^3*d^2 - 3*a*b^2*c^4*d)) - (x*(-a*b^3)^(1/2
)*(16*b^7*c^7*d^2 - 48*a*b^6*c^6*d^3 + 32*a^2*b^5*c^5*d^4 + 32*a^3*b^4*c^4*d^5 - 48*a^4*b^3*c^3*d^6 + 16*a^5*b
^2*c^2*d^7))/(8*(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d)*(a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d)))*(-a*b^3)^(1/2))/(2
*(a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d)) - (x*(a^2*b^3*d^5 + 13*b^5*c^2*d^3 - 6*a*b^4*c*d^4))/(4*(b^2*c^4 + a^2*c
^2*d^2 - 2*a*b*c^3*d))))/(a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d) - ((a*b^4*d^4)/2 - (3*b^5*c*d^3)/2)/(b^3*c^5 - a^
3*c^2*d^3 + 3*a^2*b*c^3*d^2 - 3*a*b^2*c^4*d) + ((-a*b^3)^(1/2)*((((4*b^7*c^6*d^2 - 18*a*b^6*c^5*d^3 - 2*a^5*b^
2*c*d^7 + 32*a^2*b^5*c^4*d^4 - 28*a^3*b^4*c^3*d^5 + 12*a^4*b^3*c^2*d^6)/(2*(b^3*c^5 - a^3*c^2*d^3 + 3*a^2*b*c^
3*d^2 - 3*a*b^2*c^4*d)) + (x*(-a*b^3)^(1/2)*(16...

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