3.3.42 \(\int \frac {x^4}{(a+b x^2) (c+d x^2)^2} \, dx\) [242]

Optimal. Leaf size=108 \[ -\frac {c x}{2 d (b c-a d) \left (c+d x^2\right )}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (b c-a d)^2}+\frac {\sqrt {c} (b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 d^{3/2} (b c-a d)^2} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {481, 536, 211} \begin {gather*} \frac {a^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (b c-a d)^2}+\frac {\sqrt {c} (b c-3 a d) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 d^{3/2} (b c-a d)^2}-\frac {c x}{2 d \left (c+d x^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/((a + b*x^2)*(c + d*x^2)^2),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx &=-\frac {c x}{2 d (b c-a d) \left (c+d x^2\right )}+\frac {\int \frac {a c+(b c-2 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 d (b c-a d)}\\ &=-\frac {c x}{2 d (b c-a d) \left (c+d x^2\right )}+\frac {a^2 \int \frac {1}{a+b x^2} \, dx}{(b c-a d)^2}+\frac {(c (b c-3 a d)) \int \frac {1}{c+d x^2} \, dx}{2 d (b c-a d)^2}\\ &=-\frac {c x}{2 d (b c-a d) \left (c+d x^2\right )}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (b c-a d)^2}+\frac {\sqrt {c} (b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 d^{3/2} (b c-a d)^2}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 108, normalized size = 1.00 \begin {gather*} \frac {c x}{2 d (-b c+a d) \left (c+d x^2\right )}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (-b c+a d)^2}+\frac {\sqrt {c} (b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 d^{3/2} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/((a + b*x^2)*(c + d*x^2)^2),x]

[Out]

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

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Maple [A]
time = 0.16, size = 95, normalized size = 0.88

method	result	size
default	\(\frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right )^{2} \sqrt {a b}}-\frac {c \left (-\frac {\left (a d -b c \right ) x}{2 d \left (d \,x^{2}+c \right )}+\frac {\left (3 a d -b c \right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 d \sqrt {c d}}\right )}{\left (a d -b c \right )^{2}}\)	\(95\)
risch	\(\frac {c x}{2 d \left (a d -b c \right ) \left (d \,x^{2}+c \right )}+\frac {3 \sqrt {-c d}\, \ln \left (\left (-9 \left (-c d \right )^{\frac {3}{2}} a^{3} b \,d^{3}-3 \left (-c d \right )^{\frac {3}{2}} a^{2} b^{2} c \,d^{2}+5 \left (-c d \right )^{\frac {3}{2}} a \,b^{3} c^{2} d -\left (-c d \right )^{\frac {3}{2}} b^{4} c^{3}-4 a^{4} \sqrt {-c d}\, d^{5}-9 \sqrt {-c d}\, a^{2} b^{2} c^{2} d^{3}+6 \sqrt {-c d}\, a \,b^{3} c^{3} d^{2}-b^{4} c^{4} \sqrt {-c d}\, d \right ) x -4 a^{4} c \,d^{5}+9 b \,c^{2} d^{4} a^{3}-6 b^{2} c^{3} d^{3} a^{2}+a \,b^{3} c^{4} d^{2}\right ) a}{4 d \left (a d -b c \right )^{2}}-\frac {\sqrt {-c d}\, \ln \left (\left (-9 \left (-c d \right )^{\frac {3}{2}} a^{3} b \,d^{3}-3 \left (-c d \right )^{\frac {3}{2}} a^{2} b^{2} c \,d^{2}+5 \left (-c d \right )^{\frac {3}{2}} a \,b^{3} c^{2} d -\left (-c d \right )^{\frac {3}{2}} b^{4} c^{3}-4 a^{4} \sqrt {-c d}\, d^{5}-9 \sqrt {-c d}\, a^{2} b^{2} c^{2} d^{3}+6 \sqrt {-c d}\, a \,b^{3} c^{3} d^{2}-b^{4} c^{4} \sqrt {-c d}\, d \right ) x -4 a^{4} c \,d^{5}+9 b \,c^{2} d^{4} a^{3}-6 b^{2} c^{3} d^{3} a^{2}+a \,b^{3} c^{4} d^{2}\right ) b c}{4 d^{2} \left (a d -b c \right )^{2}}-\frac {3 \sqrt {-c d}\, \ln \left (\left (9 \left (-c d \right )^{\frac {3}{2}} a^{3} b \,d^{3}+3 \left (-c d \right )^{\frac {3}{2}} a^{2} b^{2} c \,d^{2}-5 \left (-c d \right )^{\frac {3}{2}} a \,b^{3} c^{2} d +\left (-c d \right )^{\frac {3}{2}} b^{4} c^{3}+4 a^{4} \sqrt {-c d}\, d^{5}+9 \sqrt {-c d}\, a^{2} b^{2} c^{2} d^{3}-6 \sqrt {-c d}\, a \,b^{3} c^{3} d^{2}+b^{4} c^{4} \sqrt {-c d}\, d \right ) x -4 a^{4} c \,d^{5}+9 b \,c^{2} d^{4} a^{3}-6 b^{2} c^{3} d^{3} a^{2}+a \,b^{3} c^{4} d^{2}\right ) a}{4 d \left (a d -b c \right )^{2}}+\frac {\sqrt {-c d}\, \ln \left (\left (9 \left (-c d \right )^{\frac {3}{2}} a^{3} b \,d^{3}+3 \left (-c d \right )^{\frac {3}{2}} a^{2} b^{2} c \,d^{2}-5 \left (-c d \right )^{\frac {3}{2}} a \,b^{3} c^{2} d +\left (-c d \right )^{\frac {3}{2}} b^{4} c^{3}+4 a^{4} \sqrt {-c d}\, d^{5}+9 \sqrt {-c d}\, a^{2} b^{2} c^{2} d^{3}-6 \sqrt {-c d}\, a \,b^{3} c^{3} d^{2}+b^{4} c^{4} \sqrt {-c d}\, d \right ) x -4 a^{4} c \,d^{5}+9 b \,c^{2} d^{4} a^{3}-6 b^{2} c^{3} d^{3} a^{2}+a \,b^{3} c^{4} d^{2}\right ) b c}{4 d^{2} \left (a d -b c \right )^{2}}+\frac {\sqrt {-a b}\, a \ln \left (\left (-4 \left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}-4 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}-4 a^{4} \sqrt {-a b}\, d^{4} b -9 \sqrt {-a b}\, a^{2} b^{3} c^{2} d^{2}+6 \sqrt {-a b}\, a \,b^{4} c^{3} d -b^{5} c^{4} \sqrt {-a b}\right ) x -4 a^{4} b^{2} c \,d^{3}+9 a^{3} b^{3} c^{2} d^{2}-6 a^{2} b^{4} c^{3} d +a \,b^{5} c^{4}\right )}{2 b \left (a d -b c \right )^{2}}-\frac {\sqrt {-a b}\, a \ln \left (\left (4 \left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}+4 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}+4 a^{4} \sqrt {-a b}\, d^{4} b +9 \sqrt {-a b}\, a^{2} b^{3} c^{2} d^{2}-6 \sqrt {-a b}\, a \,b^{4} c^{3} d +b^{5} c^{4} \sqrt {-a b}\right ) x -4 a^{4} b^{2} c \,d^{3}+9 a^{3} b^{3} c^{2} d^{2}-6 a^{2} b^{4} c^{3} d +a \,b^{5} c^{4}\right )}{2 b \left (a d -b c \right )^{2}}\)	\(1173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^2+a)/(d*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.52, size = 132, normalized size = 1.22 \begin {gather*} \frac {a^{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 {c x}{2 \, {\left (b c^{2} d - a c d^{2} + {\left (b c d^{2} - a d^{3}\right )} x^{2}\right )}} + \frac {{\left (b c^{2} - 3 \, a c d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {c d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.18, size = 718, normalized size = 6.65 \begin {gather*} \left [\frac {2 \, {\left (a d^{2} x^{2} + a c d\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - {\left (b c^{2} - 3 \, a c d + {\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} - 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - 2 \, {\left (b c^{2} - a c d\right )} x}{4 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac {4 \, {\left (a d^{2} x^{2} + a c d\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - {\left (b c^{2} - 3 \, a c d + {\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} - 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - 2 \, {\left (b c^{2} - a c d\right )} x}{4 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac {{\left (b c^{2} - 3 \, a c d + {\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) + {\left (a d^{2} x^{2} + a c d\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - {\left (b c^{2} - a c d\right )} x}{2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac {2 \, {\left (a d^{2} x^{2} + a c d\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + {\left (b c^{2} - 3 \, a c d + {\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) - {\left (b c^{2} - a c d\right )} x}{2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]
time = 1.17, size = 121, normalized size = 1.12 \begin {gather*} \frac {a^{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 (b c^{2} - 3 \, a c d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {c d}} - \frac {c x}{2 \, {\left (b c d - a d^{2}\right )} {\left (d x^{2} + c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.57, size = 2500, normalized size = 23.15 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((a + b*x^2)*(c + d*x^2)^2),x)

[Out]

(c*x)/(2*d*(c + d*x^2)*(a*d - b*c)) - (atan(((((x*(b^5*c^4 + 4*a^4*b*d^4 + 9*a^2*b^3*c^2*d^2 - 6*a*b^4*c^3*d))
/(2*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2)) - ((-c*d^3)^(1/2)*(3*a*d - b*c)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 -
8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*
c*d^3) - (x*(-c*d^3)^(1/2)*(3*a*d - b*c)*(16*a^5*b^2*d^8 + 16*b^7*c^5*d^3 - 48*a*b^6*c^4*d^4 - 48*a^4*b^3*c*d^
7 + 32*a^2*b^5*c^3*d^5 + 32*a^3*b^4*c^2*d^6))/(8*(a^2*d^5 + b^2*c^2*d^3 - 2*a*b*c*d^4)*(a^2*d^3 + b^2*c^2*d -
2*a*b*c*d^2))))/(4*(a^2*d^5 + b^2*c^2*d^3 - 2*a*b*c*d^4)))*(-c*d^3)^(1/2)*(3*a*d - b*c)*1i)/(4*(a^2*d^5 + b^2*
c^2*d^3 - 2*a*b*c*d^4)) + (((x*(b^5*c^4 + 4*a^4*b*d^4 + 9*a^2*b^3*c^2*d^2 - 6*a*b^4*c^3*d))/(2*(a^2*d^3 + b^2*
c^2*d - 2*a*b*c*d^2)) + ((-c*d^3)^(1/2)*(3*a*d - b*c)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3
+ 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) + (x*(-c*d^3
)^(1/2)*(3*a*d - b*c)*(16*a^5*b^2*d^8 + 16*b^7*c^5*d^3 - 48*a*b^6*c^4*d^4 - 48*a^4*b^3*c*d^7 + 32*a^2*b^5*c^3*
d^5 + 32*a^3*b^4*c^2*d^6))/(8*(a^2*d^5 + b^2*c^2*d^3 - 2*a*b*c*d^4)*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))))/(4*
(a^2*d^5 + b^2*c^2*d^3 - 2*a*b*c*d^4)))*(-c*d^3)^(1/2)*(3*a*d - b*c)*1i)/(4*(a^2*d^5 + b^2*c^2*d^3 - 2*a*b*c*d
^4)))/(((a^2*b^3*c^3)/2 - (5*a^3*b^2*c^2*d)/2 + 3*a^4*b*c*d^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) - (((x*(b^5*c^4 + 4*a^4*b*d^4 + 9*a^2*b^3*c^2*d^2 - 6*a*b^4*c^3*d))/(2*(a^2*d^3 + b^2*c^2*d - 2*a*b*c
*d^2)) - ((-c*d^3)^(1/2)*(3*a*d - b*c)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^
3*d^4 - 8*a^4*b^3*c^2*d^5)/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) - (x*(-c*d^3)^(1/2)*(3*a*d
- b*c)*(16*a^5*b^2*d^8 + 16*b^7*c^5*d^3 - 48*a*b^6*c^4*d^4 - 48*a^4*b^3*c*d^7 + 32*a^2*b^5*c^3*d^5 + 32*a^3*b^
4*c^2*d^6))/(8*(a^2*d^5 + b^2*c^2*d^3 - 2*a*b*c*d^4)*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))))/(4*(a^2*d^5 + b^2*
c^2*d^3 - 2*a*b*c*d^4)))*(-c*d^3)^(1/2)*(3*a*d - b*c))/(4*(a^2*d^5 + b^2*c^2*d^3 - 2*a*b*c*d^4)) + (((x*(b^5*c
^4 + 4*a^4*b*d^4 + 9*a^2*b^3*c^2*d^2 - 6*a*b^4*c^3*d))/(2*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2)) + ((-c*d^3)^(1/
2)*(3*a*d - b*c)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*
d^5)/(a^3*d^4 - b^3*c^3*d + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) + (x*(-c*d^3)^(1/2)*(3*a*d - b*c)*(16*a^5*b^2*d^8
+ 16*b^7*c^5*d^3 - 48*a*b^6*c^4*d^4 - 48*a^4*b^3*c*d^7 + 32*a^2*b^5*c^3*d^5 + 32*a^3*b^4*c^2*d^6))/(8*(a^2*d^
5 + b^2*c^2*d^3 - 2*a*b*c*d^4)*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))))/(4*(a^2*d^5 + b^2*c^2*d^3 - 2*a*b*c*d^4)
))*(-c*d^3)^(1/2)*(3*a*d - b*c))/(4*(a^2*d^5 + b^2*c^2*d^3 - 2*a*b*c*d^4))))*(-c*d^3)^(1/2)*(3*a*d - b*c)*1i)/
(2*(a^2*d^5 + b^2*c^2*d^3 - 2*a*b*c*d^4)) - (atan(-(((((-a^3*b)^(1/2)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*
a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(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)) - (x*(-a^3*b)^(1/2)*(16*a^5*b^2*d^8 + 16*b^7*c^5*d^3 - 48*a*b^6*c^4*d^4 - 48*a^4*b^3*c*d^7 + 32*a^2*b^
5*c^3*d^5 + 32*a^3*b^4*c^2*d^6))/(8*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))))
/(2*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)) - (x*(b^5*c^4 + 4*a^4*b*d^4 + 9*a^2*b^3*c^2*d^2 - 6*a*b^4*c^3*d))/(4*
(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2)))*(-a^3*b)^(1/2)*1i)/(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d) - ((((-a^3*b)^(1/
2)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(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)) + (x*(-a^3*b)^(1/2)*(16*a^5*b^2*d^8 + 16*b^7*c^5*d^3 - 48*a
*b^6*c^4*d^4 - 48*a^4*b^3*c*d^7 + 32*a^2*b^5*c^3*d^5 + 32*a^3*b^4*c^2*d^6))/(8*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*
c*d)*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))))/(2*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)) + (x*(b^5*c^4 + 4*a^4*b*d^
4 + 9*a^2*b^3*c^2*d^2 - 6*a*b^4*c^3*d))/(4*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2)))*(-a^3*b)^(1/2)*1i)/(b^3*c^2 +
a^2*b*d^2 - 2*a*b^2*c*d))/(((a^2*b^3*c^3)/2 - (5*a^3*b^2*c^2*d)/2 + 3*a^4*b*c*d^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) + ((((-a^3*b)^(1/2)*((2*a*b^6*c^5*d^2 + 2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12
*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(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)) - (x*(-a^3*b
)^(1/2)*(16*a^5*b^2*d^8 + 16*b^7*c^5*d^3 - 48*a*b^6*c^4*d^4 - 48*a^4*b^3*c*d^7 + 32*a^2*b^5*c^3*d^5 + 32*a^3*b
^4*c^2*d^6))/(8*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)*(a^2*d^3 + b^2*c^2*d - 2*a*b*c*d^2))))/(2*(b^3*c^2 + a^2*b
*d^2 - 2*a*b^2*c*d)) - (x*(b^5*c^4 + 4*a^4*b*d^4 + 9*a^2*b^3*c^2*d^2 - 6*a*b^4*c^3*d))/(4*(a^2*d^3 + b^2*c^2*d
- 2*a*b*c*d^2)))*(-a^3*b)^(1/2))/(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d) + ((((-a^3*b)^(1/2)*((2*a*b^6*c^5*d^2 +
2*a^5*b^2*c*d^6 - 8*a^2*b^5*c^4*d^3 + 12*a^3*b^4*c^3*d^4 - 8*a^4*b^3*c^2*d^5)/(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)) + (x*(-a^3*b)^(1/2)*(16*a^5*b^2*d^8 + 16*b^7*c^5*d^3 - 48*a*b^6*c^4*d^4 - 48*a^4*b
^3*c*d^7 + 32*a^2*b^5*c^3*d^5 + 32*a^3*b^4*c^2*...

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