3.1.42 \(\int \frac {1}{(a+b x^2)^3 (c+d x^2)^3} \, dx\) [42]

Optimal. Leaf size=315 \[ \frac {d \left (3 b^2 c^2-13 a b c d-2 a^2 d^2\right ) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )^2}+\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac {b (3 b c-11 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {3 d (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x}{8 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {3 b^{5/2} \left (b^2 c^2-6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} (b c-a d)^5}-\frac {3 d^{5/2} \left (21 b^2 c^2-6 a b c d+a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} (b c-a d)^5} \]

[Out]

1/8*d*(-2*a^2*d^2-13*a*b*c*d+3*b^2*c^2)*x/a^2/c/(-a*d+b*c)^3/(d*x^2+c)^2+1/4*b*x/a/(-a*d+b*c)/(b*x^2+a)^2/(d*x
^2+c)^2+1/8*b*(-11*a*d+3*b*c)*x/a^2/(-a*d+b*c)^2/(b*x^2+a)/(d*x^2+c)^2+3/8*d*(a*d+b*c)*(a^2*d^2-6*a*b*c*d+b^2*
c^2)*x/a^2/c^2/(-a*d+b*c)^4/(d*x^2+c)+3/8*b^(5/2)*(21*a^2*d^2-6*a*b*c*d+b^2*c^2)*arctan(x*b^(1/2)/a^(1/2))/a^(
5/2)/(-a*d+b*c)^5-3/8*d^(5/2)*(a^2*d^2-6*a*b*c*d+21*b^2*c^2)*arctan(x*d^(1/2)/c^(1/2))/c^(5/2)/(-a*d+b*c)^5

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Rubi [A]
time = 0.31, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {425, 541, 536, 211} \begin {gather*} -\frac {3 d^{5/2} \left (a^2 d^2-6 a b c d+21 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} (b c-a d)^5}+\frac {3 d x (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{8 a^2 c^2 \left (c+d x^2\right ) (b c-a d)^4}+\frac {d x \left (-2 a^2 d^2-13 a b c d+3 b^2 c^2\right )}{8 a^2 c \left (c+d x^2\right )^2 (b c-a d)^3}+\frac {b x (3 b c-11 a d)}{8 a^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)^2}+\frac {3 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (21 a^2 d^2-6 a b c d+b^2 c^2\right )}{8 a^{5/2} (b c-a d)^5}+\frac {b x}{4 a \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)^3*(c + d*x^2)^3),x]

[Out]

(d*(3*b^2*c^2 - 13*a*b*c*d - 2*a^2*d^2)*x)/(8*a^2*c*(b*c - a*d)^3*(c + d*x^2)^2) + (b*x)/(4*a*(b*c - a*d)*(a +
b*x^2)^2*(c + d*x^2)^2) + (b*(3*b*c - 11*a*d)*x)/(8*a^2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)^2) + (3*d*(b*c
+ a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x)/(8*a^2*c^2*(b*c - a*d)^4*(c + d*x^2)) + (3*b^(5/2)*(b^2*c^2 - 6*a*b*
c*d + 21*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(b*c - a*d)^5) - (3*d^(5/2)*(21*b^2*c^2 - 6*a*b*c*d
+ a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*(b*c - a*d)^5)

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]

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*} \int \frac {1}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^3} \, dx &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}-\frac {\int \frac {-3 b c+4 a d-7 b d x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx}{4 a (b c-a d)}\\ &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac {b (3 b c-11 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {\int \frac {3 b^2 c^2-3 a b c d+8 a^2 d^2+5 b d (3 b c-11 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx}{8 a^2 (b c-a d)^2}\\ &=\frac {d \left (3 b^2 c^2-13 a b c d-2 a^2 d^2\right ) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )^2}+\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac {b (3 b c-11 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {\int \frac {12 \left (b^3 c^3-3 a b^2 c^2 d+8 a^2 b c d^2-2 a^3 d^3\right )+12 b d \left (3 b^2 c^2-13 a b c d-2 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{32 a^2 c (b c-a d)^3}\\ &=\frac {d \left (3 b^2 c^2-13 a b c d-2 a^2 d^2\right ) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )^2}+\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac {b (3 b c-11 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {3 d (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x}{8 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {\int \frac {24 \left (b^4 c^4-5 a b^3 c^3 d+16 a^2 b^2 c^2 d^2-5 a^3 b c d^3+a^4 d^4\right )+24 b d (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{64 a^2 c^2 (b c-a d)^4}\\ &=\frac {d \left (3 b^2 c^2-13 a b c d-2 a^2 d^2\right ) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )^2}+\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac {b (3 b c-11 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {3 d (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x}{8 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}-\frac {\left (3 d^3 \left (21 b^2 c^2-6 a b c d+a^2 d^2\right )\right ) \int \frac {1}{c+d x^2} \, dx}{8 c^2 (b c-a d)^5}+\frac {\left (3 b^3 \left (b^2 c^2-6 a b c d+21 a^2 d^2\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^2 (b c-a d)^5}\\ &=\frac {d \left (3 b^2 c^2-13 a b c d-2 a^2 d^2\right ) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )^2}+\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac {b (3 b c-11 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {3 d (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x}{8 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac {3 b^{5/2} \left (b^2 c^2-6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} (b c-a d)^5}-\frac {3 d^{5/2} \left (21 b^2 c^2-6 a b c d+a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} (b c-a d)^5}\\ \end {align*}

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Mathematica [A]
time = 0.61, size = 233, normalized size = 0.74 \begin {gather*} \frac {1}{8} \left (-\frac {3 b^{5/2} \left (b^2 c^2-6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (-b c+a d)^5}+\frac {(b c-a d) x \left (\frac {3 b^4 c}{a^2 \left (a+b x^2\right )}+\frac {3 a d^4}{c^2 \left (c+d x^2\right )}+\frac {b^3 \left (2 b c-17 a d-15 b d x^2\right )}{a \left (a+b x^2\right )^2}-\frac {d^3 \left (17 b c-2 a d+15 b d x^2\right )}{c \left (c+d x^2\right )^2}\right )-\frac {3 d^{5/2} \left (21 b^2 c^2-6 a b c d+a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2}}}{(b c-a d)^5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^2)^3*(c + d*x^2)^3),x]

[Out]

((-3*b^(5/2)*(b^2*c^2 - 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^5) + ((b*
c - a*d)*x*((3*b^4*c)/(a^2*(a + b*x^2)) + (3*a*d^4)/(c^2*(c + d*x^2)) + (b^3*(2*b*c - 17*a*d - 15*b*d*x^2))/(a
*(a + b*x^2)^2) - (d^3*(17*b*c - 2*a*d + 15*b*d*x^2))/(c*(c + d*x^2)^2)) - (3*d^(5/2)*(21*b^2*c^2 - 6*a*b*c*d
+ a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(5/2))/(b*c - a*d)^5)/8

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Maple [A]
time = 0.32, size = 257, normalized size = 0.82

method	result	size
default	\(-\frac {b^{3} \left (\frac {\frac {3 b \left (5 a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) x^{3}}{8 a^{2}}+\frac {\left (17 a^{2} d^{2}-22 a b c d +5 b^{2} c^{2}\right ) x}{8 a}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \left (21 a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}\right )}{\left (a d -b c \right )^{5}}+\frac {d^{3} \left (\frac {\frac {3 d \left (a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right ) x^{3}}{8 c^{2}}+\frac {\left (5 a^{2} d^{2}-22 a b c d +17 b^{2} c^{2}\right ) x}{8 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {3 \left (a^{2} d^{2}-6 a b c d +21 b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 c^{2} \sqrt {c d}}\right )}{\left (a d -b c \right )^{5}}\)	\(257\)
risch	\(\text {Expression too large to display}\)	\(5650\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)^3/(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

-b^3/(a*d-b*c)^5*((3/8*b*(5*a^2*d^2-6*a*b*c*d+b^2*c^2)/a^2*x^3+1/8*(17*a^2*d^2-22*a*b*c*d+5*b^2*c^2)/a*x)/(b*x
^2+a)^2+3/8*(21*a^2*d^2-6*a*b*c*d+b^2*c^2)/a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))+d^3/(a*d-b*c)^5*((3/8*d*(a
^2*d^2-6*a*b*c*d+5*b^2*c^2)/c^2*x^3+1/8*(5*a^2*d^2-22*a*b*c*d+17*b^2*c^2)/c*x)/(d*x^2+c)^2+3/8*(a^2*d^2-6*a*b*
c*d+21*b^2*c^2)/c^2/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 820 vs. \(2 (287) = 574\).
time = 0.58, size = 820, normalized size = 2.60 \begin {gather*} \frac {3 \, {\left (b^{5} c^{2} - 6 \, a b^{4} c d + 21 \, a^{2} b^{3} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{2} b^{5} c^{5} - 5 \, a^{3} b^{4} c^{4} d + 10 \, a^{4} b^{3} c^{3} d^{2} - 10 \, a^{5} b^{2} c^{2} d^{3} + 5 \, a^{6} b c d^{4} - a^{7} d^{5}\right )} \sqrt {a b}} - \frac {3 \, {\left (21 \, b^{2} c^{2} d^{3} - 6 \, a b c d^{4} + a^{2} d^{5}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{5} c^{7} - 5 \, a b^{4} c^{6} d + 10 \, a^{2} b^{3} c^{5} d^{2} - 10 \, a^{3} b^{2} c^{4} d^{3} + 5 \, a^{4} b c^{3} d^{4} - a^{5} c^{2} d^{5}\right )} \sqrt {c d}} + \frac {3 \, {\left (b^{5} c^{3} d^{2} - 5 \, a b^{4} c^{2} d^{3} - 5 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{7} + {\left (6 \, b^{5} c^{4} d - 25 \, a b^{4} c^{3} d^{2} - 34 \, a^{2} b^{3} c^{2} d^{3} - 25 \, a^{3} b^{2} c d^{4} + 6 \, a^{4} b d^{5}\right )} x^{5} + {\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 34 \, a^{2} b^{3} c^{3} d^{2} - 34 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5}\right )} x^{3} + {\left (5 \, a b^{4} c^{5} - 17 \, a^{2} b^{3} c^{4} d - 17 \, a^{4} b c^{2} d^{3} + 5 \, a^{5} c d^{4}\right )} x}{8 \, {\left (a^{4} b^{4} c^{8} - 4 \, a^{5} b^{3} c^{7} d + 6 \, a^{6} b^{2} c^{6} d^{2} - 4 \, a^{7} b c^{5} d^{3} + a^{8} c^{4} d^{4} + {\left (a^{2} b^{6} c^{6} d^{2} - 4 \, a^{3} b^{5} c^{5} d^{3} + 6 \, a^{4} b^{4} c^{4} d^{4} - 4 \, a^{5} b^{3} c^{3} d^{5} + a^{6} b^{2} c^{2} d^{6}\right )} x^{8} + 2 \, {\left (a^{2} b^{6} c^{7} d - 3 \, a^{3} b^{5} c^{6} d^{2} + 2 \, a^{4} b^{4} c^{5} d^{3} + 2 \, a^{5} b^{3} c^{4} d^{4} - 3 \, a^{6} b^{2} c^{3} d^{5} + a^{7} b c^{2} d^{6}\right )} x^{6} + {\left (a^{2} b^{6} c^{8} - 9 \, a^{4} b^{4} c^{6} d^{2} + 16 \, a^{5} b^{3} c^{5} d^{3} - 9 \, a^{6} b^{2} c^{4} d^{4} + a^{8} c^{2} d^{6}\right )} x^{4} + 2 \, {\left (a^{3} b^{5} c^{8} - 3 \, a^{4} b^{4} c^{7} d + 2 \, a^{5} b^{3} c^{6} d^{2} + 2 \, a^{6} b^{2} c^{5} d^{3} - 3 \, a^{7} b c^{4} d^{4} + a^{8} c^{3} d^{5}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^3/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

3/8*(b^5*c^2 - 6*a*b^4*c*d + 21*a^2*b^3*d^2)*arctan(b*x/sqrt(a*b))/((a^2*b^5*c^5 - 5*a^3*b^4*c^4*d + 10*a^4*b^
3*c^3*d^2 - 10*a^5*b^2*c^2*d^3 + 5*a^6*b*c*d^4 - a^7*d^5)*sqrt(a*b)) - 3/8*(21*b^2*c^2*d^3 - 6*a*b*c*d^4 + a^2
*d^5)*arctan(d*x/sqrt(c*d))/((b^5*c^7 - 5*a*b^4*c^6*d + 10*a^2*b^3*c^5*d^2 - 10*a^3*b^2*c^4*d^3 + 5*a^4*b*c^3*
d^4 - a^5*c^2*d^5)*sqrt(c*d)) + 1/8*(3*(b^5*c^3*d^2 - 5*a*b^4*c^2*d^3 - 5*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^7 + (
6*b^5*c^4*d - 25*a*b^4*c^3*d^2 - 34*a^2*b^3*c^2*d^3 - 25*a^3*b^2*c*d^4 + 6*a^4*b*d^5)*x^5 + (3*b^5*c^5 - 5*a*b
^4*c^4*d - 34*a^2*b^3*c^3*d^2 - 34*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 3*a^5*d^5)*x^3 + (5*a*b^4*c^5 - 17*a^2*b^
3*c^4*d - 17*a^4*b*c^2*d^3 + 5*a^5*c*d^4)*x)/(a^4*b^4*c^8 - 4*a^5*b^3*c^7*d + 6*a^6*b^2*c^6*d^2 - 4*a^7*b*c^5*
d^3 + a^8*c^4*d^4 + (a^2*b^6*c^6*d^2 - 4*a^3*b^5*c^5*d^3 + 6*a^4*b^4*c^4*d^4 - 4*a^5*b^3*c^3*d^5 + a^6*b^2*c^2
*d^6)*x^8 + 2*(a^2*b^6*c^7*d - 3*a^3*b^5*c^6*d^2 + 2*a^4*b^4*c^5*d^3 + 2*a^5*b^3*c^4*d^4 - 3*a^6*b^2*c^3*d^5 +
a^7*b*c^2*d^6)*x^6 + (a^2*b^6*c^8 - 9*a^4*b^4*c^6*d^2 + 16*a^5*b^3*c^5*d^3 - 9*a^6*b^2*c^4*d^4 + a^8*c^2*d^6)
*x^4 + 2*(a^3*b^5*c^8 - 3*a^4*b^4*c^7*d + 2*a^5*b^3*c^6*d^2 + 2*a^6*b^2*c^5*d^3 - 3*a^7*b*c^4*d^4 + a^8*c^3*d^
5)*x^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1242 vs. \(2 (287) = 574\).
time = 9.25, size = 5070, normalized size = 16.10 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^3/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

[1/16*(6*(b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 6*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^7 + 2*(6*b^6*c^5*d - 31*a*b^5*c^4*d
^2 - 9*a^2*b^4*c^3*d^3 + 9*a^3*b^3*c^2*d^4 + 31*a^4*b^2*c*d^5 - 6*a^5*b*d^6)*x^5 + 2*(3*b^6*c^6 - 8*a*b^5*c^5*
d - 29*a^2*b^4*c^4*d^2 + 29*a^4*b^2*c^2*d^4 + 8*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 - 3*(a^2*b^4*c^6 - 6*a^3*b^3*c^5*
d + 21*a^4*b^2*c^4*d^2 + (b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 21*a^2*b^4*c^2*d^4)*x^8 + 2*(b^6*c^5*d - 5*a*b^5*c^4
*d^2 + 15*a^2*b^4*c^3*d^3 + 21*a^3*b^3*c^2*d^4)*x^6 + (b^6*c^6 - 2*a*b^5*c^5*d - 2*a^2*b^4*c^4*d^2 + 78*a^3*b^
3*c^3*d^3 + 21*a^4*b^2*c^2*d^4)*x^4 + 2*(a*b^5*c^6 - 5*a^2*b^4*c^5*d + 15*a^3*b^3*c^4*d^2 + 21*a^4*b^2*c^3*d^3
)*x^2)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 3*(21*a^4*b^2*c^4*d^2 - 6*a^5*b*c^3*d^3 +
a^6*c^2*d^4 + (21*a^2*b^4*c^2*d^4 - 6*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^8 + 2*(21*a^2*b^4*c^3*d^3 + 15*a^3*b^3*c^
2*d^4 - 5*a^4*b^2*c*d^5 + a^5*b*d^6)*x^6 + (21*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 - 2*a^4*b^2*c^2*d^4 - 2*a^
5*b*c*d^5 + a^6*d^6)*x^4 + 2*(21*a^3*b^3*c^4*d^2 + 15*a^4*b^2*c^3*d^3 - 5*a^5*b*c^2*d^4 + a^6*c*d^5)*x^2)*sqrt
(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(5*a*b^5*c^6 - 22*a^2*b^4*c^5*d + 17*a^3*b^3*c^4*d^
2 - 17*a^4*b^2*c^3*d^3 + 22*a^5*b*c^2*d^4 - 5*a^6*c*d^5)*x)/(a^4*b^5*c^9 - 5*a^5*b^4*c^8*d + 10*a^6*b^3*c^7*d^
2 - 10*a^7*b^2*c^6*d^3 + 5*a^8*b*c^5*d^4 - a^9*c^4*d^5 + (a^2*b^7*c^7*d^2 - 5*a^3*b^6*c^6*d^3 + 10*a^4*b^5*c^5
*d^4 - 10*a^5*b^4*c^4*d^5 + 5*a^6*b^3*c^3*d^6 - a^7*b^2*c^2*d^7)*x^8 + 2*(a^2*b^7*c^8*d - 4*a^3*b^6*c^7*d^2 +
5*a^4*b^5*c^6*d^3 - 5*a^6*b^3*c^4*d^5 + 4*a^7*b^2*c^3*d^6 - a^8*b*c^2*d^7)*x^6 + (a^2*b^7*c^9 - a^3*b^6*c^8*d
- 9*a^4*b^5*c^7*d^2 + 25*a^5*b^4*c^6*d^3 - 25*a^6*b^3*c^5*d^4 + 9*a^7*b^2*c^4*d^5 + a^8*b*c^3*d^6 - a^9*c^2*d^
7)*x^4 + 2*(a^3*b^6*c^9 - 4*a^4*b^5*c^8*d + 5*a^5*b^4*c^7*d^2 - 5*a^7*b^2*c^5*d^4 + 4*a^8*b*c^4*d^5 - a^9*c^3*
d^6)*x^2), 1/16*(6*(b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 6*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^7 + 2*(6*b^6*c^5*d - 31*a
*b^5*c^4*d^2 - 9*a^2*b^4*c^3*d^3 + 9*a^3*b^3*c^2*d^4 + 31*a^4*b^2*c*d^5 - 6*a^5*b*d^6)*x^5 + 2*(3*b^6*c^6 - 8*
a*b^5*c^5*d - 29*a^2*b^4*c^4*d^2 + 29*a^4*b^2*c^2*d^4 + 8*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 - 6*(21*a^4*b^2*c^4*d^2
- 6*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (21*a^2*b^4*c^2*d^4 - 6*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^8 + 2*(21*a^2*b^4*c^
3*d^3 + 15*a^3*b^3*c^2*d^4 - 5*a^4*b^2*c*d^5 + a^5*b*d^6)*x^6 + (21*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 - 2*a
^4*b^2*c^2*d^4 - 2*a^5*b*c*d^5 + a^6*d^6)*x^4 + 2*(21*a^3*b^3*c^4*d^2 + 15*a^4*b^2*c^3*d^3 - 5*a^5*b*c^2*d^4 +
a^6*c*d^5)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) - 3*(a^2*b^4*c^6 - 6*a^3*b^3*c^5*d + 21*a^4*b^2*c^4*d^2 + (b^6*
c^4*d^2 - 6*a*b^5*c^3*d^3 + 21*a^2*b^4*c^2*d^4)*x^8 + 2*(b^6*c^5*d - 5*a*b^5*c^4*d^2 + 15*a^2*b^4*c^3*d^3 + 21
*a^3*b^3*c^2*d^4)*x^6 + (b^6*c^6 - 2*a*b^5*c^5*d - 2*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 + 21*a^4*b^2*c^2*d^4
)*x^4 + 2*(a*b^5*c^6 - 5*a^2*b^4*c^5*d + 15*a^3*b^3*c^4*d^2 + 21*a^4*b^2*c^3*d^3)*x^2)*sqrt(-b/a)*log((b*x^2 -
2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 2*(5*a*b^5*c^6 - 22*a^2*b^4*c^5*d + 17*a^3*b^3*c^4*d^2 - 17*a^4*b^2*c^3*
d^3 + 22*a^5*b*c^2*d^4 - 5*a^6*c*d^5)*x)/(a^4*b^5*c^9 - 5*a^5*b^4*c^8*d + 10*a^6*b^3*c^7*d^2 - 10*a^7*b^2*c^6*
d^3 + 5*a^8*b*c^5*d^4 - a^9*c^4*d^5 + (a^2*b^7*c^7*d^2 - 5*a^3*b^6*c^6*d^3 + 10*a^4*b^5*c^5*d^4 - 10*a^5*b^4*c
^4*d^5 + 5*a^6*b^3*c^3*d^6 - a^7*b^2*c^2*d^7)*x^8 + 2*(a^2*b^7*c^8*d - 4*a^3*b^6*c^7*d^2 + 5*a^4*b^5*c^6*d^3 -
5*a^6*b^3*c^4*d^5 + 4*a^7*b^2*c^3*d^6 - a^8*b*c^2*d^7)*x^6 + (a^2*b^7*c^9 - a^3*b^6*c^8*d - 9*a^4*b^5*c^7*d^2
+ 25*a^5*b^4*c^6*d^3 - 25*a^6*b^3*c^5*d^4 + 9*a^7*b^2*c^4*d^5 + a^8*b*c^3*d^6 - a^9*c^2*d^7)*x^4 + 2*(a^3*b^6
*c^9 - 4*a^4*b^5*c^8*d + 5*a^5*b^4*c^7*d^2 - 5*a^7*b^2*c^5*d^4 + 4*a^8*b*c^4*d^5 - a^9*c^3*d^6)*x^2), 1/16*(6*
(b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 6*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^7 + 2*(6*b^6*c^5*d - 31*a*b^5*c^4*d^2 - 9*a^
2*b^4*c^3*d^3 + 9*a^3*b^3*c^2*d^4 + 31*a^4*b^2*c*d^5 - 6*a^5*b*d^6)*x^5 + 2*(3*b^6*c^6 - 8*a*b^5*c^5*d - 29*a^
2*b^4*c^4*d^2 + 29*a^4*b^2*c^2*d^4 + 8*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 + 6*(a^2*b^4*c^6 - 6*a^3*b^3*c^5*d + 21*a^
4*b^2*c^4*d^2 + (b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 21*a^2*b^4*c^2*d^4)*x^8 + 2*(b^6*c^5*d - 5*a*b^5*c^4*d^2 + 15
*a^2*b^4*c^3*d^3 + 21*a^3*b^3*c^2*d^4)*x^6 + (b^6*c^6 - 2*a*b^5*c^5*d - 2*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3
+ 21*a^4*b^2*c^2*d^4)*x^4 + 2*(a*b^5*c^6 - 5*a^2*b^4*c^5*d + 15*a^3*b^3*c^4*d^2 + 21*a^4*b^2*c^3*d^3)*x^2)*sq
rt(b/a)*arctan(x*sqrt(b/a)) - 3*(21*a^4*b^2*c^4*d^2 - 6*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (21*a^2*b^4*c^2*d^4 - 6*
a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^8 + 2*(21*a^2*b^4*c^3*d^3 + 15*a^3*b^3*c^2*d^4 - 5*a^4*b^2*c*d^5 + a^5*b*d^6)*x
^6 + (21*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 - 2*a^4*b^2*c^2*d^4 - 2*a^5*b*c*d^5 + a^6*d^6)*x^4 + 2*(21*a^3*b
^3*c^4*d^2 + 15*a^4*b^2*c^3*d^3 - 5*a^5*b*c^2*d^4 + a^6*c*d^5)*x^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) -
c)/(d*x^2 + c)) + 2*(5*a*b^5*c^6 - 22*a^2*b^4*c^5*d + 17*a^3*b^3*c^4*d^2 - 17*a^4*b^2*c^3*d^3 + 22*a^5*b*c^2*
d^4 - 5*a^6*c*d^5)*x)/(a^4*b^5*c^9 - 5*a^5*b^4*...

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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)**3/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [A]
time = 1.74, size = 574, normalized size = 1.82 \begin {gather*} \frac {3 \, {\left (b^{5} c^{2} - 6 \, a b^{4} c d + 21 \, a^{2} b^{3} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{2} b^{5} c^{5} - 5 \, a^{3} b^{4} c^{4} d + 10 \, a^{4} b^{3} c^{3} d^{2} - 10 \, a^{5} b^{2} c^{2} d^{3} + 5 \, a^{6} b c d^{4} - a^{7} d^{5}\right )} \sqrt {a b}} - \frac {3 \, {\left (21 \, b^{2} c^{2} d^{3} - 6 \, a b c d^{4} + a^{2} d^{5}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{5} c^{7} - 5 \, a b^{4} c^{6} d + 10 \, a^{2} b^{3} c^{5} d^{2} - 10 \, a^{3} b^{2} c^{4} d^{3} + 5 \, a^{4} b c^{3} d^{4} - a^{5} c^{2} d^{5}\right )} \sqrt {c d}} + \frac {3 \, b^{5} c^{3} d^{2} x^{7} - 15 \, a b^{4} c^{2} d^{3} x^{7} - 15 \, a^{2} b^{3} c d^{4} x^{7} + 3 \, a^{3} b^{2} d^{5} x^{7} + 6 \, b^{5} c^{4} d x^{5} - 25 \, a b^{4} c^{3} d^{2} x^{5} - 34 \, a^{2} b^{3} c^{2} d^{3} x^{5} - 25 \, a^{3} b^{2} c d^{4} x^{5} + 6 \, a^{4} b d^{5} x^{5} + 3 \, b^{5} c^{5} x^{3} - 5 \, a b^{4} c^{4} d x^{3} - 34 \, a^{2} b^{3} c^{3} d^{2} x^{3} - 34 \, a^{3} b^{2} c^{2} d^{3} x^{3} - 5 \, a^{4} b c d^{4} x^{3} + 3 \, a^{5} d^{5} x^{3} + 5 \, a b^{4} c^{5} x - 17 \, a^{2} b^{3} c^{4} d x - 17 \, a^{4} b c^{2} d^{3} x + 5 \, a^{5} c d^{4} x}{8 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4}\right )} {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^3/(d*x^2+c)^3,x, algorithm="giac")

[Out]

3/8*(b^5*c^2 - 6*a*b^4*c*d + 21*a^2*b^3*d^2)*arctan(b*x/sqrt(a*b))/((a^2*b^5*c^5 - 5*a^3*b^4*c^4*d + 10*a^4*b^
3*c^3*d^2 - 10*a^5*b^2*c^2*d^3 + 5*a^6*b*c*d^4 - a^7*d^5)*sqrt(a*b)) - 3/8*(21*b^2*c^2*d^3 - 6*a*b*c*d^4 + a^2
*d^5)*arctan(d*x/sqrt(c*d))/((b^5*c^7 - 5*a*b^4*c^6*d + 10*a^2*b^3*c^5*d^2 - 10*a^3*b^2*c^4*d^3 + 5*a^4*b*c^3*
d^4 - a^5*c^2*d^5)*sqrt(c*d)) + 1/8*(3*b^5*c^3*d^2*x^7 - 15*a*b^4*c^2*d^3*x^7 - 15*a^2*b^3*c*d^4*x^7 + 3*a^3*b
^2*d^5*x^7 + 6*b^5*c^4*d*x^5 - 25*a*b^4*c^3*d^2*x^5 - 34*a^2*b^3*c^2*d^3*x^5 - 25*a^3*b^2*c*d^4*x^5 + 6*a^4*b*
d^5*x^5 + 3*b^5*c^5*x^3 - 5*a*b^4*c^4*d*x^3 - 34*a^2*b^3*c^3*d^2*x^3 - 34*a^3*b^2*c^2*d^3*x^3 - 5*a^4*b*c*d^4*
x^3 + 3*a^5*d^5*x^3 + 5*a*b^4*c^5*x - 17*a^2*b^3*c^4*d*x - 17*a^4*b*c^2*d^3*x + 5*a^5*c*d^4*x)/((a^2*b^4*c^6 -
4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4)*(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c)^2)

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Mupad [B]
time = 8.55, size = 2500, normalized size = 7.94 \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)^3*(c + d*x^2)^3),x)

[Out]

((x*(5*a^4*d^4 + 5*b^4*c^4 - 17*a*b^3*c^3*d - 17*a^3*b*c*d^3))/(8*a*c*(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)) - (x^3*(34*a^2*b^3*c^3*d^2 - 3*b^5*c^5 - 3*a^5*d^5 + 34*a^3*b^2*c^2*d^3 + 5*a
*b^4*c^4*d + 5*a^4*b*c*d^4))/(8*a^2*c^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
)) - (x^5*(25*a*b^4*c^3*d^2 - 6*b^5*c^4*d - 6*a^4*b*d^5 + 25*a^3*b^2*c*d^4 + 34*a^2*b^3*c^2*d^3))/(8*a^2*c^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)) + (3*b*d*x^7*(a^3*b*d^4 + b^4*c^3*d -
5*a*b^3*c^2*d^2 - 5*a^2*b^2*c*d^3))/(8*a^2*c^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)))/(x^4*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d) + x^2*(2*a*b*c^2 + 2*a^2*c*d) + x^6*(2*a*b*d^2 + 2*b^2*c*d) +
a^2*c^2 + b^2*d^2*x^8) - (atan(((((x*(9*a^8*b^3*d^11 + 9*b^11*c^8*d^3 - 108*a*b^10*c^7*d^4 - 108*a^7*b^4*c*d^1
0 + 702*a^2*b^9*c^6*d^5 - 2268*a^3*b^8*c^5*d^6 + 7938*a^4*b^7*c^4*d^7 - 2268*a^5*b^6*c^3*d^8 + 702*a^6*b^5*c^2
*d^9))/(32*(a^4*b^8*c^12 + a^12*c^4*d^8 - 8*a^5*b^7*c^11*d - 8*a^11*b*c^5*d^7 + 28*a^6*b^6*c^10*d^2 - 56*a^7*b
^5*c^9*d^3 + 70*a^8*b^4*c^8*d^4 - 56*a^9*b^3*c^7*d^5 + 28*a^10*b^2*c^6*d^6)) - (3*(((3*a^2*b^16*c^16*d^2)/2 -
(45*a^3*b^15*c^15*d^3)/2 + (333*a^4*b^14*c^14*d^4)/2 - 765*a^5*b^13*c^13*d^5 + (4743*a^6*b^12*c^12*d^6)/2 - (1
0371*a^7*b^11*c^11*d^7)/2 + (16425*a^8*b^10*c^10*d^8)/2 - 9558*a^9*b^9*c^9*d^9 + (16425*a^10*b^8*c^8*d^10)/2 -
(10371*a^11*b^7*c^7*d^11)/2 + (4743*a^12*b^6*c^6*d^12)/2 - 765*a^13*b^5*c^5*d^13 + (333*a^14*b^4*c^4*d^14)/2
- (45*a^15*b^3*c^3*d^15)/2 + (3*a^16*b^2*c^2*d^16)/2)/(a^4*b^12*c^16 + a^16*c^4*d^12 - 12*a^5*b^11*c^15*d - 12
*a^15*b*c^5*d^11 + 66*a^6*b^10*c^14*d^2 - 220*a^7*b^9*c^13*d^3 + 495*a^8*b^8*c^12*d^4 - 792*a^9*b^7*c^11*d^5 +
924*a^10*b^6*c^10*d^6 - 792*a^11*b^5*c^9*d^7 + 495*a^12*b^4*c^8*d^8 - 220*a^13*b^3*c^7*d^9 + 66*a^14*b^2*c^6*
d^10) - (3*x*(-a^5*b^5)^(1/2)*(21*a^2*d^2 + b^2*c^2 - 6*a*b*c*d)*(256*a^4*b^13*c^15*d^2 - 2304*a^5*b^12*c^14*d
^3 + 8960*a^6*b^11*c^13*d^4 - 19200*a^7*b^10*c^12*d^5 + 23040*a^8*b^9*c^11*d^6 - 10752*a^9*b^8*c^10*d^7 - 1075
2*a^10*b^7*c^9*d^8 + 23040*a^11*b^6*c^8*d^9 - 19200*a^12*b^5*c^7*d^10 + 8960*a^13*b^4*c^6*d^11 - 2304*a^14*b^3
*c^5*d^12 + 256*a^15*b^2*c^4*d^13))/(512*(a^10*d^5 - a^5*b^5*c^5 + 5*a^6*b^4*c^4*d - 10*a^7*b^3*c^3*d^2 + 10*a
^8*b^2*c^2*d^3 - 5*a^9*b*c*d^4)*(a^4*b^8*c^12 + a^12*c^4*d^8 - 8*a^5*b^7*c^11*d - 8*a^11*b*c^5*d^7 + 28*a^6*b^
6*c^10*d^2 - 56*a^7*b^5*c^9*d^3 + 70*a^8*b^4*c^8*d^4 - 56*a^9*b^3*c^7*d^5 + 28*a^10*b^2*c^6*d^6)))*(-a^5*b^5)^
(1/2)*(21*a^2*d^2 + b^2*c^2 - 6*a*b*c*d))/(16*(a^10*d^5 - a^5*b^5*c^5 + 5*a^6*b^4*c^4*d - 10*a^7*b^3*c^3*d^2 +
10*a^8*b^2*c^2*d^3 - 5*a^9*b*c*d^4)))*(-a^5*b^5)^(1/2)*(21*a^2*d^2 + b^2*c^2 - 6*a*b*c*d)*3i)/(16*(a^10*d^5 -
a^5*b^5*c^5 + 5*a^6*b^4*c^4*d - 10*a^7*b^3*c^3*d^2 + 10*a^8*b^2*c^2*d^3 - 5*a^9*b*c*d^4)) + (((x*(9*a^8*b^3*d
^11 + 9*b^11*c^8*d^3 - 108*a*b^10*c^7*d^4 - 108*a^7*b^4*c*d^10 + 702*a^2*b^9*c^6*d^5 - 2268*a^3*b^8*c^5*d^6 +
7938*a^4*b^7*c^4*d^7 - 2268*a^5*b^6*c^3*d^8 + 702*a^6*b^5*c^2*d^9))/(32*(a^4*b^8*c^12 + a^12*c^4*d^8 - 8*a^5*b
^7*c^11*d - 8*a^11*b*c^5*d^7 + 28*a^6*b^6*c^10*d^2 - 56*a^7*b^5*c^9*d^3 + 70*a^8*b^4*c^8*d^4 - 56*a^9*b^3*c^7*
d^5 + 28*a^10*b^2*c^6*d^6)) + (3*(((3*a^2*b^16*c^16*d^2)/2 - (45*a^3*b^15*c^15*d^3)/2 + (333*a^4*b^14*c^14*d^4
)/2 - 765*a^5*b^13*c^13*d^5 + (4743*a^6*b^12*c^12*d^6)/2 - (10371*a^7*b^11*c^11*d^7)/2 + (16425*a^8*b^10*c^10*
d^8)/2 - 9558*a^9*b^9*c^9*d^9 + (16425*a^10*b^8*c^8*d^10)/2 - (10371*a^11*b^7*c^7*d^11)/2 + (4743*a^12*b^6*c^6
*d^12)/2 - 765*a^13*b^5*c^5*d^13 + (333*a^14*b^4*c^4*d^14)/2 - (45*a^15*b^3*c^3*d^15)/2 + (3*a^16*b^2*c^2*d^16
)/2)/(a^4*b^12*c^16 + a^16*c^4*d^12 - 12*a^5*b^11*c^15*d - 12*a^15*b*c^5*d^11 + 66*a^6*b^10*c^14*d^2 - 220*a^7
*b^9*c^13*d^3 + 495*a^8*b^8*c^12*d^4 - 792*a^9*b^7*c^11*d^5 + 924*a^10*b^6*c^10*d^6 - 792*a^11*b^5*c^9*d^7 + 4
95*a^12*b^4*c^8*d^8 - 220*a^13*b^3*c^7*d^9 + 66*a^14*b^2*c^6*d^10) + (3*x*(-a^5*b^5)^(1/2)*(21*a^2*d^2 + b^2*c
^2 - 6*a*b*c*d)*(256*a^4*b^13*c^15*d^2 - 2304*a^5*b^12*c^14*d^3 + 8960*a^6*b^11*c^13*d^4 - 19200*a^7*b^10*c^12
*d^5 + 23040*a^8*b^9*c^11*d^6 - 10752*a^9*b^8*c^10*d^7 - 10752*a^10*b^7*c^9*d^8 + 23040*a^11*b^6*c^8*d^9 - 192
00*a^12*b^5*c^7*d^10 + 8960*a^13*b^4*c^6*d^11 - 2304*a^14*b^3*c^5*d^12 + 256*a^15*b^2*c^4*d^13))/(512*(a^10*d^
5 - a^5*b^5*c^5 + 5*a^6*b^4*c^4*d - 10*a^7*b^3*c^3*d^2 + 10*a^8*b^2*c^2*d^3 - 5*a^9*b*c*d^4)*(a^4*b^8*c^12 + a
^12*c^4*d^8 - 8*a^5*b^7*c^11*d - 8*a^11*b*c^5*d^7 + 28*a^6*b^6*c^10*d^2 - 56*a^7*b^5*c^9*d^3 + 70*a^8*b^4*c^8*
d^4 - 56*a^9*b^3*c^7*d^5 + 28*a^10*b^2*c^6*d^6)))*(-a^5*b^5)^(1/2)*(21*a^2*d^2 + b^2*c^2 - 6*a*b*c*d))/(16*(a^
10*d^5 - a^5*b^5*c^5 + 5*a^6*b^4*c^4*d - 10*a^7*b^3*c^3*d^2 + 10*a^8*b^2*c^2*d^3 - 5*a^9*b*c*d^4)))*(-a^5*b^5)
^(1/2)*(21*a^2*d^2 + b^2*c^2 - 6*a*b*c*d)*3i)/(16*(a^10*d^5 - a^5*b^5*c^5 + 5*a^6*b^4*c^4*d - 10*a^7*b^3*c^3*d
^2 + 10*a^8*b^2*c^2*d^3 - 5*a^9*b*c*d^4)))/(((567*a^7*b^5*d^12)/256 + (567*b^12*c^7*d^5)/256 - (6399*a*b^11*c^
6*d^6)/256 - (6399*a^6*b^6*c*d^11)/256 + (27891...

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