3.3.91 \(\int \frac {x^2}{(a+b x^2)^2 (c+d x^2)} \, dx\) [291]

Optimal. Leaf size=104 \[ -\frac {x}{2 (b c-a d) \left (a+b x^2\right )}+\frac {(b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} (b c-a d)^2}-\frac {\sqrt {c} \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{(b c-a d)^2} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 104, 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 = {482, 536, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (a d+b c)}{2 \sqrt {a} \sqrt {b} (b c-a d)^2}-\frac {\sqrt {c} \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{(b c-a d)^2}-\frac {x}{2 \left (a+b x^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.10, size = 104, normalized size = 1.00 \begin {gather*} \frac {x}{2 (-b c+a d) \left (a+b x^2\right )}+\frac {(b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} (-b c+a d)^2}-\frac {\sqrt {c} \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{(b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.18, size = 85, normalized size = 0.82

method	result	size
default	\(\frac {\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (a d +b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{\left (a d -b c \right )^{2}}-\frac {d c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right )^{2} \sqrt {c d}}\)	\(85\)
risch	\(\frac {x}{2 \left (a d -b c \right ) \left (b \,x^{2}+a \right )}-\frac {\ln \left (a \,b^{2} x -\left (-a b \right )^{\frac {3}{2}}\right ) a d}{4 \sqrt {-a b}\, \left (a d -b c \right )^{2}}-\frac {\ln \left (a \,b^{2} x -\left (-a b \right )^{\frac {3}{2}}\right ) b c}{4 \sqrt {-a b}\, \left (a d -b c \right )^{2}}+\frac {\ln \left (-a \,b^{2} x -\left (-a b \right )^{\frac {3}{2}}\right ) a d}{4 \sqrt {-a b}\, \left (a d -b c \right )^{2}}+\frac {\ln \left (-a \,b^{2} x -\left (-a b \right )^{\frac {3}{2}}\right ) b c}{4 \sqrt {-a b}\, \left (a d -b c \right )^{2}}+\frac {\sqrt {-c d}\, \ln \left (\left (-4 \left (-c d \right )^{\frac {3}{2}} a b d -4 \left (-c d \right )^{\frac {3}{2}} b^{2} c -a^{2} \sqrt {-c d}\, d^{3}-2 \sqrt {-c d}\, a b c \,d^{2}-5 b^{2} c^{2} \sqrt {-c d}\, d \right ) x -a^{2} c \,d^{3}+2 a b \,c^{2} d^{2}-b^{2} c^{3} d \right )}{2 \left (a d -b c \right )^{2}}-\frac {\sqrt {-c d}\, \ln \left (\left (4 \left (-c d \right )^{\frac {3}{2}} a b d +4 \left (-c d \right )^{\frac {3}{2}} b^{2} c +a^{2} \sqrt {-c d}\, d^{3}+2 \sqrt {-c d}\, a b c \,d^{2}+5 b^{2} c^{2} \sqrt {-c d}\, d \right ) x -a^{2} c \,d^{3}+2 a b \,c^{2} d^{2}-b^{2} c^{3} d \right )}{2 \left (a d -b c \right )^{2}}\)	\(403\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.54, size = 119, normalized size = 1.14 \begin {gather*} -\frac {c d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {{\left (b c + a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b}} - \frac {x}{2 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.02, size = 704, normalized size = 6.77 \begin {gather*} \left [-\frac {{\left (a b c + a^{2} d + {\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, {\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, {\left (a b^{2} c - a^{2} b d\right )} x}{4 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}, \frac {{\left (a b c + a^{2} d + {\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) - {\left (a b^{2} c - a^{2} b d\right )} x}{2 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac {4 \, {\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + {\left (a b c + a^{2} d + {\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (a b^{2} c - a^{2} b d\right )} x}{4 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}, \frac {{\left (a b c + a^{2} d + {\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - 2 \, {\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - {\left (a b^{2} c - a^{2} b d\right )} x}{2 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]
time = 0.47, size = 110, normalized size = 1.06 \begin {gather*} -\frac {c d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {{\left (b c + a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b}} - \frac {x}{2 \, {\left (b x^{2} + a\right )} {\left (b c - a d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.49, size = 3153, normalized size = 30.32 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(2*(a + b*x^2)*(a*d - b*c)) + (atan((((-c*d)^(1/2)*(((-c*d)^(1/2)*((2*b^6*c^5*d^2 - 8*a*b^5*c^4*d^3 + 2*a^4*
b^2*c*d^6 + 12*a^2*b^4*c^3*d^4 - 8*a^3*b^3*c^2*d^5)/(2*(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)*(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*(a^2*b*d
^5 + 5*b^3*c^2*d^3 + 2*a*b^2*c*d^4))/(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)
- ((-c*d)^(1/2)*(((-c*d)^(1/2)*((2*b^6*c^5*d^2 - 8*a*b^5*c^4*d^3 + 2*a^4*b^2*c*d^6 + 12*a^2*b^4*c^3*d^4 - 8*a^
3*b^3*c^2*d^5)/(2*(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)*(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*(a^2*b*d^5 + 5*b^3*c^2*d^3 + 2*a*b^2*c*d^4))/
(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))/(((b^2*c^2*d^3)/2 + (a*b*c*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) + ((-c*d)^(1/2)*(((-c*d)^(1/2)*((2*b^6*c^5*d^2 - 8*a*b^5*c^
4*d^3 + 2*a^4*b^2*c*d^6 + 12*a^2*b^4*c^3*d^4 - 8*a^3*b^3*c^2*d^5)/(2*(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)*(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*(a^2*b*d^5 + 5*b^3*c^2*d^3 + 2*a*b^2*c*d^4))/(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) + ((-c*d)^(1/2)*(((-c*d)^(1/2)*((2*b^6*c^5*d^2 - 8*a*b^5*c^4*d^3 + 2*a^4*b^2*c*d^6 + 12*a^2*b^4*c^3
*d^4 - 8*a^3*b^3*c^2*d^5)/(2*(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)*(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*(a^2*b*d^5 + 5*b^3*c^2*d^3 + 2*a*b
^2*c*d^4))/(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)))*(-c*d)^(1/2)*1i)/(a^2*d^2 +
b^2*c^2 - 2*a*b*c*d) - (atan((((-a*b)^(1/2)*((x*(a^2*b*d^5 + 5*b^3*c^2*d^3 + 2*a*b^2*c*d^4))/(2*(a^2*d^2 + b^
2*c^2 - 2*a*b*c*d)) - ((-a*b)^(1/2)*((2*b^6*c^5*d^2 - 8*a*b^5*c^4*d^3 + 2*a^4*b^2*c*d^6 + 12*a^2*b^4*c^3*d^4 -
8*a^3*b^3*c^2*d^5)/(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)*(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*b^3*c^2 + a^3*b*d^2 - 2*a^2*b^2*c*d)))*(a*d + b*c))/(4*(a*b^3*c^2 + a^3*b*d
^2 - 2*a^2*b^2*c*d)))*(a*d + b*c)*1i)/(4*(a*b^3*c^2 + a^3*b*d^2 - 2*a^2*b^2*c*d)) + ((-a*b)^(1/2)*((x*(a^2*b*d
^5 + 5*b^3*c^2*d^3 + 2*a*b^2*c*d^4))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + ((-a*b)^(1/2)*((2*b^6*c^5*d^2 - 8*a
*b^5*c^4*d^3 + 2*a^4*b^2*c*d^6 + 12*a^2*b^4*c^3*d^4 - 8*a^3*b^3*c^2*d^5)/(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)*(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*b^3*c^2 + a^3*b*d^2 -
2*a^2*b^2*c*d)))*(a*d + b*c))/(4*(a*b^3*c^2 + a^3*b*d^2 - 2*a^2*b^2*c*d)))*(a*d + b*c)*1i)/(4*(a*b^3*c^2 + a^3
*b*d^2 - 2*a^2*b^2*c*d)))/(((b^2*c^2*d^3)/2 + (a*b*c*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) - ((-a*b)^(1/2)*((x*(a^2*b*d^5 + 5*b^3*c^2*d^3 + 2*a*b^2*c*d^4))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - ((-a
*b)^(1/2)*((2*b^6*c^5*d^2 - 8*a*b^5*c^4*d^3 + 2*a^4*b^2*c*d^6 + 12*a^2*b^4*c^3*d^4 - 8*a^3*b^3*c^2*d^5)/(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)*(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*b^3*c^2 + a^3*b*d^2 - 2*a^2*b^2*c*d)))*(a*d + b*c))/(4*(a*b^3*c^2 + a^3*b*d^2 - 2*a^2*b^2*c*d)))*(a*d
+ b*c))/(4*(a*b^3*c^2 + a^3*b*d^2 - 2*a^2*b^2*c*d)) + ((-a*b)^(1/2)*((x*(a^2*b*d^5 + 5*b^3*c^2*d^3 + 2*a*b^2*
c*d^4))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + ((-a*b)^(1/2)*((2*b^6*c^5*d^2 - 8*a*b^5*c^4*d^3 + 2*a^4*b^2*c*d^
6 + 12*a^2*b^4*c^3*d^4 - 8*a^3*b^3*c^2*d^5)/(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)*(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*b^3*c^2 + a^3*b*d^2 - 2*a^2*b^2*c*d)))*(a*d + b*c))
/(4*(a*b^3*c^2 + a^3*b*d^2 - 2*a^2*b^2*c*d)))*(a*d + b*c))/(4*(a*b^3*c^2 + a^3*b*d^2 - 2*a^2*b^2*c*d))))*(-a*b
)^(1/2)*(a*d + b*c)*1i)/(2*(a*b^3*c^2 + a^3*b*d^2 - 2*a^2*b^2*c*d))

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