\(\int \frac {x^2}{(a+b x^2)^2 (c+d x^2)^3} \, dx\) [312]

Optimal result

Integrand size = 22, antiderivative size = 200 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {3 d x}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (11 b c+a d) x}{8 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {b^{3/2} (b c+5 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} (b c-a d)^4}-\frac {\sqrt {d} \left (15 b^2 c^2+10 a b c d-a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} (b c-a d)^4} \]

[Out]

-3/4*d*x/(-a*d+b*c)^2/(d*x^2+c)^2-1/2*x/(-a*d+b*c)/(b*x^2+a)/(d*x^2+c)^2-1/8*d*(a*d+11*b*c)*x/c/(-a*d+b*c)^3/(
d*x^2+c)+1/2*b^(3/2)*(5*a*d+b*c)*arctan(x*b^(1/2)/a^(1/2))/(-a*d+b*c)^4/a^(1/2)-1/8*(-a^2*d^2+10*a*b*c*d+15*b^
2*c^2)*arctan(x*d^(1/2)/c^(1/2))*d^(1/2)/c^(3/2)/(-a*d+b*c)^4

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {482, 541, 536, 211} \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {\sqrt {d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} (b c-a d)^4}+\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (5 a d+b c)}{2 \sqrt {a} (b c-a d)^4}-\frac {d x (a d+11 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^3}-\frac {3 d x}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac {x}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

[In]

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

[Out]

(-3*d*x)/(4*(b*c - a*d)^2*(c + d*x^2)^2) - x/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (d*(11*b*c + a*d)*x)/
(8*c*(b*c - a*d)^3*(c + d*x^2)) + (b^(3/2)*(b*c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*(b*c - a*d)^4
) - (Sqrt[d]*(15*b^2*c^2 + 10*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*(b*c - a*d)^4)

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]

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*} \text {integral}& = -\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {\int \frac {c-5 d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx}{2 (b c-a d)} \\ & = -\frac {3 d x}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {\int \frac {2 c (2 b c+a d)-18 b c d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{8 c (b c-a d)^2} \\ & = -\frac {3 d x}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (11 b c+a d) x}{8 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\int \frac {2 c \left (4 b^2 c^2+9 a b c d-a^2 d^2\right )-2 b c d (11 b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{16 c^2 (b c-a d)^3} \\ & = -\frac {3 d x}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (11 b c+a d) x}{8 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (b^2 (b c+5 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 (b c-a d)^4}-\frac {\left (d \left (15 b^2 c^2+10 a b c d-a^2 d^2\right )\right ) \int \frac {1}{c+d x^2} \, dx}{8 c (b c-a d)^4} \\ & = -\frac {3 d x}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (11 b c+a d) x}{8 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {b^{3/2} (b c+5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} (b c-a d)^4}-\frac {\sqrt {d} \left (15 b^2 c^2+10 a b c d-a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} (b c-a d)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {-\frac {4 b^2 (b c-a d) x}{a+b x^2}-\frac {2 d (b c-a d)^2 x}{\left (c+d x^2\right )^2}+\frac {d (-b c+a d) (7 b c+a d) x}{c \left (c+d x^2\right )}+\frac {4 b^{3/2} (b c+5 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {\sqrt {d} \left (-15 b^2 c^2-10 a b c d+a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2}}}{8 (b c-a d)^4} \]

[In]

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

[Out]

((-4*b^2*(b*c - a*d)*x)/(a + b*x^2) - (2*d*(b*c - a*d)^2*x)/(c + d*x^2)^2 + (d*(-(b*c) + a*d)*(7*b*c + a*d)*x)
/(c*(c + d*x^2)) + (4*b^(3/2)*(b*c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (Sqrt[d]*(-15*b^2*c^2 - 10*
a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(3/2))/(8*(b*c - a*d)^4)

Maple [A] (verified)

Time = 2.91 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {b^{2} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (5 a d +b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a d -b c \right )^{4}}+\frac {d \left (\frac {\frac {d \left (a^{2} d^{2}+6 a b c d -7 b^{2} c^{2}\right ) x^{3}}{8 c}+\left (\frac {5}{4} a b c d -\frac {9}{8} b^{2} c^{2}-\frac {1}{8} a^{2} d^{2}\right ) x}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (a^{2} d^{2}-10 a b c d -15 b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 c \sqrt {c d}}\right )}{\left (a d -b c \right )^{4}}\)	\(182\)
risch	\(\text {Expression too large to display}\)	\(2962\)

[In]

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

[Out]

b^2/(a*d-b*c)^4*((1/2*a*d-1/2*b*c)*x/(b*x^2+a)+1/2*(5*a*d+b*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))+d/(a*d-b*c
)^4*((1/8*d*(a^2*d^2+6*a*b*c*d-7*b^2*c^2)/c*x^3+(5/4*a*b*c*d-9/8*b^2*c^2-1/8*a^2*d^2)*x)/(d*x^2+c)^2+1/8*(a^2*
d^2-10*a*b*c*d-15*b^2*c^2)/c/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 698 vs. \(2 (174) = 348\).

Time = 1.18 (sec) , antiderivative size = 2891, normalized size of antiderivative = 14.46 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/16*(2*(11*b^3*c^2*d^2 - 10*a*b^2*c*d^3 - a^2*b*d^4)*x^5 + 2*(17*b^3*c^3*d - 11*a*b^2*c^2*d^2 - 5*a^2*b*c*d
^3 - a^3*d^4)*x^3 - 4*(a*b^2*c^4 + 5*a^2*b*c^3*d + (b^3*c^2*d^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d + 11*a*b^2
*c^2*d^2 + 5*a^2*b*c*d^3)*x^4 + (b^3*c^4 + 7*a*b^2*c^3*d + 10*a^2*b*c^2*d^2)*x^2)*sqrt(-b/a)*log((b*x^2 + 2*a*
x*sqrt(-b/a) - a)/(b*x^2 + a)) + (15*a*b^2*c^4 + 10*a^2*b*c^3*d - a^3*c^2*d^2 + (15*b^3*c^2*d^2 + 10*a*b^2*c*d
^3 - a^2*b*d^4)*x^6 + (30*b^3*c^3*d + 35*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 - a^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2
*c^3*d + 19*a^2*b*c^2*d^2 - 2*a^3*c*d^3)*x^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(
4*b^3*c^4 + 5*a*b^2*c^3*d - 10*a^2*b*c^2*d^2 + a^3*c*d^3)*x)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c^5*d^2
- 4*a^4*b*c^4*d^3 + a^5*c^3*d^4 + (b^5*c^5*d^2 - 4*a*b^4*c^4*d^3 + 6*a^2*b^3*c^3*d^4 - 4*a^3*b^2*c^2*d^5 + a^4
*b*c*d^6)*x^6 + (2*b^5*c^6*d - 7*a*b^4*c^5*d^2 + 8*a^2*b^3*c^4*d^3 - 2*a^3*b^2*c^3*d^4 - 2*a^4*b*c^2*d^5 + a^5
*c*d^6)*x^4 + (b^5*c^7 - 2*a*b^4*c^6*d - 2*a^2*b^3*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^3*d^4 + 2*a^5*c^2*d
^5)*x^2), -1/8*((11*b^3*c^2*d^2 - 10*a*b^2*c*d^3 - a^2*b*d^4)*x^5 + (17*b^3*c^3*d - 11*a*b^2*c^2*d^2 - 5*a^2*b
*c*d^3 - a^3*d^4)*x^3 + (15*a*b^2*c^4 + 10*a^2*b*c^3*d - a^3*c^2*d^2 + (15*b^3*c^2*d^2 + 10*a*b^2*c*d^3 - a^2*
b*d^4)*x^6 + (30*b^3*c^3*d + 35*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 - a^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2*c^3*d +
19*a^2*b*c^2*d^2 - 2*a^3*c*d^3)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) - 2*(a*b^2*c^4 + 5*a^2*b*c^3*d + (b^3*c^2*d
^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d + 11*a*b^2*c^2*d^2 + 5*a^2*b*c*d^3)*x^4 + (b^3*c^4 + 7*a*b^2*c^3*d + 10
*a^2*b*c^2*d^2)*x^2)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + (4*b^3*c^4 + 5*a*b^2*c^3*d -
10*a^2*b*c^2*d^2 + a^3*c*d^3)*x)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c^5*d^2 - 4*a^4*b*c^4*d^3 + a^5*c^3
*d^4 + (b^5*c^5*d^2 - 4*a*b^4*c^4*d^3 + 6*a^2*b^3*c^3*d^4 - 4*a^3*b^2*c^2*d^5 + a^4*b*c*d^6)*x^6 + (2*b^5*c^6*
d - 7*a*b^4*c^5*d^2 + 8*a^2*b^3*c^4*d^3 - 2*a^3*b^2*c^3*d^4 - 2*a^4*b*c^2*d^5 + a^5*c*d^6)*x^4 + (b^5*c^7 - 2*
a*b^4*c^6*d - 2*a^2*b^3*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^3*d^4 + 2*a^5*c^2*d^5)*x^2), -1/16*(2*(11*b^3*
c^2*d^2 - 10*a*b^2*c*d^3 - a^2*b*d^4)*x^5 + 2*(17*b^3*c^3*d - 11*a*b^2*c^2*d^2 - 5*a^2*b*c*d^3 - a^3*d^4)*x^3
- 8*(a*b^2*c^4 + 5*a^2*b*c^3*d + (b^3*c^2*d^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d + 11*a*b^2*c^2*d^2 + 5*a^2*b
*c*d^3)*x^4 + (b^3*c^4 + 7*a*b^2*c^3*d + 10*a^2*b*c^2*d^2)*x^2)*sqrt(b/a)*arctan(x*sqrt(b/a)) + (15*a*b^2*c^4
+ 10*a^2*b*c^3*d - a^3*c^2*d^2 + (15*b^3*c^2*d^2 + 10*a*b^2*c*d^3 - a^2*b*d^4)*x^6 + (30*b^3*c^3*d + 35*a*b^2*
c^2*d^2 + 8*a^2*b*c*d^3 - a^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2*c^3*d + 19*a^2*b*c^2*d^2 - 2*a^3*c*d^3)*x^2)*s
qrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(4*b^3*c^4 + 5*a*b^2*c^3*d - 10*a^2*b*c^2*d^2 +
a^3*c*d^3)*x)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c^5*d^2 - 4*a^4*b*c^4*d^3 + a^5*c^3*d^4 + (b^5*c^5*d^2
- 4*a*b^4*c^4*d^3 + 6*a^2*b^3*c^3*d^4 - 4*a^3*b^2*c^2*d^5 + a^4*b*c*d^6)*x^6 + (2*b^5*c^6*d - 7*a*b^4*c^5*d^2
+ 8*a^2*b^3*c^4*d^3 - 2*a^3*b^2*c^3*d^4 - 2*a^4*b*c^2*d^5 + a^5*c*d^6)*x^4 + (b^5*c^7 - 2*a*b^4*c^6*d - 2*a^2*
b^3*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^3*d^4 + 2*a^5*c^2*d^5)*x^2), -1/8*((11*b^3*c^2*d^2 - 10*a*b^2*c*d^
3 - a^2*b*d^4)*x^5 + (17*b^3*c^3*d - 11*a*b^2*c^2*d^2 - 5*a^2*b*c*d^3 - a^3*d^4)*x^3 - 4*(a*b^2*c^4 + 5*a^2*b*
c^3*d + (b^3*c^2*d^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d + 11*a*b^2*c^2*d^2 + 5*a^2*b*c*d^3)*x^4 + (b^3*c^4 +
7*a*b^2*c^3*d + 10*a^2*b*c^2*d^2)*x^2)*sqrt(b/a)*arctan(x*sqrt(b/a)) + (15*a*b^2*c^4 + 10*a^2*b*c^3*d - a^3*c^
2*d^2 + (15*b^3*c^2*d^2 + 10*a*b^2*c*d^3 - a^2*b*d^4)*x^6 + (30*b^3*c^3*d + 35*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 -
a^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2*c^3*d + 19*a^2*b*c^2*d^2 - 2*a^3*c*d^3)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/
c)) + (4*b^3*c^4 + 5*a*b^2*c^3*d - 10*a^2*b*c^2*d^2 + a^3*c*d^3)*x)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c
^5*d^2 - 4*a^4*b*c^4*d^3 + a^5*c^3*d^4 + (b^5*c^5*d^2 - 4*a*b^4*c^4*d^3 + 6*a^2*b^3*c^3*d^4 - 4*a^3*b^2*c^2*d^
5 + a^4*b*c*d^6)*x^6 + (2*b^5*c^6*d - 7*a*b^4*c^5*d^2 + 8*a^2*b^3*c^4*d^3 - 2*a^3*b^2*c^3*d^4 - 2*a^4*b*c^2*d^
5 + a^5*c*d^6)*x^4 + (b^5*c^7 - 2*a*b^4*c^6*d - 2*a^2*b^3*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^3*d^4 + 2*a^
5*c^2*d^5)*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(x**2/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (174) = 348\).

Time = 0.31 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.36 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {{\left (b^{3} c + 5 \, a b^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {a b}} - \frac {{\left (15 \, b^{2} c^{2} d + 10 \, a b c d^{2} - a^{2} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4}\right )} \sqrt {c d}} - \frac {{\left (11 \, b^{2} c d^{2} + a b d^{3}\right )} x^{5} + {\left (17 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + {\left (4 \, b^{2} c^{3} + 9 \, a b c^{2} d - a^{2} c d^{2}\right )} x}{8 \, {\left (a b^{3} c^{6} - 3 \, a^{2} b^{2} c^{5} d + 3 \, a^{3} b c^{4} d^{2} - a^{4} c^{3} d^{3} + {\left (b^{4} c^{4} d^{2} - 3 \, a b^{3} c^{3} d^{3} + 3 \, a^{2} b^{2} c^{2} d^{4} - a^{3} b c d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{5} d - 5 \, a b^{3} c^{4} d^{2} + 3 \, a^{2} b^{2} c^{3} d^{3} + a^{3} b c^{2} d^{4} - a^{4} c d^{5}\right )} x^{4} + {\left (b^{4} c^{6} - a b^{3} c^{5} d - 3 \, a^{2} b^{2} c^{4} d^{2} + 5 \, a^{3} b c^{3} d^{3} - 2 \, a^{4} c^{2} d^{4}\right )} x^{2}\right )}} \]

[In]

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

[Out]

1/2*(b^3*c + 5*a*b^2*d)*arctan(b*x/sqrt(a*b))/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 +
a^4*d^4)*sqrt(a*b)) - 1/8*(15*b^2*c^2*d + 10*a*b*c*d^2 - a^2*d^3)*arctan(d*x/sqrt(c*d))/((b^4*c^5 - 4*a*b^3*c^
4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4)*sqrt(c*d)) - 1/8*((11*b^2*c*d^2 + a*b*d^3)*x^5 + (17*b^
2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^3 + (4*b^2*c^3 + 9*a*b*c^2*d - a^2*c*d^2)*x)/(a*b^3*c^6 - 3*a^2*b^2*c^5*d +
3*a^3*b*c^4*d^2 - a^4*c^3*d^3 + (b^4*c^4*d^2 - 3*a*b^3*c^3*d^3 + 3*a^2*b^2*c^2*d^4 - a^3*b*c*d^5)*x^6 + (2*b^
4*c^5*d - 5*a*b^3*c^4*d^2 + 3*a^2*b^2*c^3*d^3 + a^3*b*c^2*d^4 - a^4*c*d^5)*x^4 + (b^4*c^6 - a*b^3*c^5*d - 3*a^
2*b^2*c^4*d^2 + 5*a^3*b*c^3*d^3 - 2*a^4*c^2*d^4)*x^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.58 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {b^{2} x}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (b x^{2} + a\right )}} + \frac {{\left (b^{3} c + 5 \, a b^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {a b}} - \frac {{\left (15 \, b^{2} c^{2} d + 10 \, a b c d^{2} - a^{2} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4}\right )} \sqrt {c d}} - \frac {7 \, b c d^{2} x^{3} + a d^{3} x^{3} + 9 \, b c^{2} d x - a c d^{2} x}{8 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} \]

[In]

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

[Out]

-1/2*b^2*x/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x^2 + a)) + 1/2*(b^3*c + 5*a*b^2*d)*arctan(
b*x/sqrt(a*b))/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(a*b)) - 1/8*(15*b
^2*c^2*d + 10*a*b*c*d^2 - a^2*d^3)*arctan(d*x/sqrt(c*d))/((b^4*c^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3
*b*c^2*d^3 + a^4*c*d^4)*sqrt(c*d)) - 1/8*(7*b*c*d^2*x^3 + a*d^3*x^3 + 9*b*c^2*d*x - a*c*d^2*x)/((b^3*c^4 - 3*a
*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*(d*x^2 + c)^2)

Mupad [B] (verification not implemented)

Time = 7.65 (sec) , antiderivative size = 7929, normalized size of antiderivative = 39.64 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

int(x^2/((a + b*x^2)^2*(c + d*x^2)^3),x)

[Out]

((x^5*(11*b^2*c*d^2 + a*b*d^3))/(8*c*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x*(4*b^2*c^2 - a^
2*d^2 + 9*a*b*c*d))/(8*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (d*x^3*(a^2*d^2 + 17*b^2*c^2 + 6*a*b*c*d
))/(8*c*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(a*c^2 + x^2*(b*c^2 + 2*a*c*d) + x^4*(a*d^2 + 2*b*c*d) +
b*d^2*x^6) + (atan((((-a*b^3)^(1/2)*(5*a*d + b*c)*((x*(a^4*b^3*d^7 + 241*b^7*c^4*d^3 + 460*a*b^6*c^3*d^4 - 20
*a^3*b^4*c*d^6 + 470*a^2*b^5*c^2*d^5))/(32*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*
a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) - (((2*b^12*c^11*d^2 - (23*a*b^11*c^10*d^3)/2 - (a^10*b
^2*c*d^12)/2 + (39*a^2*b^10*c^9*d^4)/2 + 18*a^3*b^9*c^8*d^5 - 126*a^4*b^8*c^7*d^6 + 231*a^5*b^7*c^6*d^7 - 231*
a^6*b^6*c^5*d^8 + 138*a^7*b^5*c^4*d^9 - 48*a^8*b^4*c^3*d^10 + (17*a^9*b^3*c^2*d^11)/2)/(b^9*c^11 - a^9*c^2*d^9
+ 9*a^8*b*c^3*d^8 + 36*a^2*b^7*c^9*d^2 - 84*a^3*b^6*c^8*d^3 + 126*a^4*b^5*c^7*d^4 - 126*a^5*b^4*c^6*d^5 + 84*
a^6*b^3*c^5*d^6 - 36*a^7*b^2*c^4*d^7 - 9*a*b^8*c^10*d) - (x*(-a*b^3)^(1/2)*(5*a*d + b*c)*(256*b^11*c^11*d^2 -
1792*a*b^10*c^10*d^3 + 5120*a^2*b^9*c^9*d^4 - 7168*a^3*b^8*c^8*d^5 + 3584*a^4*b^7*c^7*d^6 + 3584*a^5*b^6*c^6*d
^7 - 7168*a^6*b^5*c^5*d^8 + 5120*a^7*b^4*c^4*d^9 - 1792*a^8*b^3*c^3*d^10 + 256*a^9*b^2*c^2*d^11))/(128*(a^5*d^
4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5
+ 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)))*(-a*b^3)^(1/2)*(5*a*d + b*c)
)/(4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)))*1i)/(4*(a^5*d^4 + a*b^4*c^4
- 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)) + ((-a*b^3)^(1/2)*(5*a*d + b*c)*((x*(a^4*b^3*d^7 + 24
1*b^7*c^4*d^3 + 460*a*b^6*c^3*d^4 - 20*a^3*b^4*c*d^6 + 470*a^2*b^5*c^2*d^5))/(32*(b^6*c^8 + a^6*c^2*d^6 - 6*a^
5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) + (((2*b^12*c^11*
d^2 - (23*a*b^11*c^10*d^3)/2 - (a^10*b^2*c*d^12)/2 + (39*a^2*b^10*c^9*d^4)/2 + 18*a^3*b^9*c^8*d^5 - 126*a^4*b^
8*c^7*d^6 + 231*a^5*b^7*c^6*d^7 - 231*a^6*b^6*c^5*d^8 + 138*a^7*b^5*c^4*d^9 - 48*a^8*b^4*c^3*d^10 + (17*a^9*b^
3*c^2*d^11)/2)/(b^9*c^11 - a^9*c^2*d^9 + 9*a^8*b*c^3*d^8 + 36*a^2*b^7*c^9*d^2 - 84*a^3*b^6*c^8*d^3 + 126*a^4*b
^5*c^7*d^4 - 126*a^5*b^4*c^6*d^5 + 84*a^6*b^3*c^5*d^6 - 36*a^7*b^2*c^4*d^7 - 9*a*b^8*c^10*d) + (x*(-a*b^3)^(1/
2)*(5*a*d + b*c)*(256*b^11*c^11*d^2 - 1792*a*b^10*c^10*d^3 + 5120*a^2*b^9*c^9*d^4 - 7168*a^3*b^8*c^8*d^5 + 358
4*a^4*b^7*c^7*d^6 + 3584*a^5*b^6*c^6*d^7 - 7168*a^6*b^5*c^5*d^8 + 5120*a^7*b^4*c^4*d^9 - 1792*a^8*b^3*c^3*d^10
+ 256*a^9*b^2*c^2*d^11))/(128*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)*(b^
6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5
*c^7*d)))*(-a*b^3)^(1/2)*(5*a*d + b*c))/(4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*
b*c*d^3)))*1i)/(4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)))/(((165*b^8*c^4
*d^3)/64 - (5*a^4*b^4*d^7)/64 + (475*a*b^7*c^3*d^4)/32 - (3*a^3*b^5*c*d^6)/32 + (39*a^2*b^6*c^2*d^5)/4)/(b^9*c
^11 - a^9*c^2*d^9 + 9*a^8*b*c^3*d^8 + 36*a^2*b^7*c^9*d^2 - 84*a^3*b^6*c^8*d^3 + 126*a^4*b^5*c^7*d^4 - 126*a^5*
b^4*c^6*d^5 + 84*a^6*b^3*c^5*d^6 - 36*a^7*b^2*c^4*d^7 - 9*a*b^8*c^10*d) - ((-a*b^3)^(1/2)*(5*a*d + b*c)*((x*(a
^4*b^3*d^7 + 241*b^7*c^4*d^3 + 460*a*b^6*c^3*d^4 - 20*a^3*b^4*c*d^6 + 470*a^2*b^5*c^2*d^5))/(32*(b^6*c^8 + a^6
*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) -
(((2*b^12*c^11*d^2 - (23*a*b^11*c^10*d^3)/2 - (a^10*b^2*c*d^12)/2 + (39*a^2*b^10*c^9*d^4)/2 + 18*a^3*b^9*c^8*d
^5 - 126*a^4*b^8*c^7*d^6 + 231*a^5*b^7*c^6*d^7 - 231*a^6*b^6*c^5*d^8 + 138*a^7*b^5*c^4*d^9 - 48*a^8*b^4*c^3*d^
10 + (17*a^9*b^3*c^2*d^11)/2)/(b^9*c^11 - a^9*c^2*d^9 + 9*a^8*b*c^3*d^8 + 36*a^2*b^7*c^9*d^2 - 84*a^3*b^6*c^8*
d^3 + 126*a^4*b^5*c^7*d^4 - 126*a^5*b^4*c^6*d^5 + 84*a^6*b^3*c^5*d^6 - 36*a^7*b^2*c^4*d^7 - 9*a*b^8*c^10*d) -
(x*(-a*b^3)^(1/2)*(5*a*d + b*c)*(256*b^11*c^11*d^2 - 1792*a*b^10*c^10*d^3 + 5120*a^2*b^9*c^9*d^4 - 7168*a^3*b^
8*c^8*d^5 + 3584*a^4*b^7*c^7*d^6 + 3584*a^5*b^6*c^6*d^7 - 7168*a^6*b^5*c^5*d^8 + 5120*a^7*b^4*c^4*d^9 - 1792*a
^8*b^3*c^3*d^10 + 256*a^9*b^2*c^2*d^11))/(128*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a
^4*b*c*d^3)*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^
4*d^4 - 6*a*b^5*c^7*d)))*(-a*b^3)^(1/2)*(5*a*d + b*c))/(4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c
^2*d^2 - 4*a^4*b*c*d^3))))/(4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)) + (
(-a*b^3)^(1/2)*(5*a*d + b*c)*((x*(a^4*b^3*d^7 + 241*b^7*c^4*d^3 + 460*a*b^6*c^3*d^4 - 20*a^3*b^4*c*d^6 + 470*a
^2*b^5*c^2*d^5))/(32*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a
^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) + (((2*b^12*c^11*d^2 - (23*a*b^11*c^10*d^3)/2 - (a^10*b^2*c*d^12)/2 + (39*a^2
*b^10*c^9*d^4)/2 + 18*a^3*b^9*c^8*d^5 - 126*a^4*b^8*c^7*d^6 + 231*a^5*b^7*c^6*d^7 - 231*a^6*b^6*c^5*d^8 + 138*
a^7*b^5*c^4*d^9 - 48*a^8*b^4*c^3*d^10 + (17*a^9*b^3*c^2*d^11)/2)/(b^9*c^11 - a^9*c^2*d^9 + 9*a^8*b*c^3*d^8 + 3
6*a^2*b^7*c^9*d^2 - 84*a^3*b^6*c^8*d^3 + 126*a^4*b^5*c^7*d^4 - 126*a^5*b^4*c^6*d^5 + 84*a^6*b^3*c^5*d^6 - 36*a
^7*b^2*c^4*d^7 - 9*a*b^8*c^10*d) + (x*(-a*b^3)^(1/2)*(5*a*d + b*c)*(256*b^11*c^11*d^2 - 1792*a*b^10*c^10*d^3 +
5120*a^2*b^9*c^9*d^4 - 7168*a^3*b^8*c^8*d^5 + 3584*a^4*b^7*c^7*d^6 + 3584*a^5*b^6*c^6*d^7 - 7168*a^6*b^5*c^5*
d^8 + 5120*a^7*b^4*c^4*d^9 - 1792*a^8*b^3*c^3*d^10 + 256*a^9*b^2*c^2*d^11))/(128*(a^5*d^4 + a*b^4*c^4 - 4*a^2*
b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 -
20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)))*(-a*b^3)^(1/2)*(5*a*d + b*c))/(4*(a^5*d^4 + a*b^4*
c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3))))/(4*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^
3*b^2*c^2*d^2 - 4*a^4*b*c*d^3))))*(-a*b^3)^(1/2)*(5*a*d + b*c)*1i)/(2*(a^5*d^4 + a*b^4*c^4 - 4*a^2*b^3*c^3*d +
6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3)) + (atan(((((x*(a^4*b^3*d^7 + 241*b^7*c^4*d^3 + 460*a*b^6*c^3*d^4 - 20*a^3
*b^4*c*d^6 + 470*a^2*b^5*c^2*d^5))/(32*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*
b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) - (((2*b^12*c^11*d^2 - (23*a*b^11*c^10*d^3)/2 - (a^10*b^2*c
*d^12)/2 + (39*a^2*b^10*c^9*d^4)/2 + 18*a^3*b^9*c^8*d^5 - 126*a^4*b^8*c^7*d^6 + 231*a^5*b^7*c^6*d^7 - 231*a^6*
b^6*c^5*d^8 + 138*a^7*b^5*c^4*d^9 - 48*a^8*b^4*c^3*d^10 + (17*a^9*b^3*c^2*d^11)/2)/(b^9*c^11 - a^9*c^2*d^9 + 9
*a^8*b*c^3*d^8 + 36*a^2*b^7*c^9*d^2 - 84*a^3*b^6*c^8*d^3 + 126*a^4*b^5*c^7*d^4 - 126*a^5*b^4*c^6*d^5 + 84*a^6*
b^3*c^5*d^6 - 36*a^7*b^2*c^4*d^7 - 9*a*b^8*c^10*d) - (x*(-c^3*d)^(1/2)*(15*b^2*c^2 - a^2*d^2 + 10*a*b*c*d)*(25
6*b^11*c^11*d^2 - 1792*a*b^10*c^10*d^3 + 5120*a^2*b^9*c^9*d^4 - 7168*a^3*b^8*c^8*d^5 + 3584*a^4*b^7*c^7*d^6 +
3584*a^5*b^6*c^6*d^7 - 7168*a^6*b^5*c^5*d^8 + 5120*a^7*b^4*c^4*d^9 - 1792*a^8*b^3*c^3*d^10 + 256*a^9*b^2*c^2*d
^11))/(512*(b^4*c^7 + a^4*c^3*d^4 - 4*a^3*b*c^4*d^3 + 6*a^2*b^2*c^5*d^2 - 4*a*b^3*c^6*d)*(b^6*c^8 + a^6*c^2*d^
6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)))*(-c^3*d)
^(1/2)*(15*b^2*c^2 - a^2*d^2 + 10*a*b*c*d))/(16*(b^4*c^7 + a^4*c^3*d^4 - 4*a^3*b*c^4*d^3 + 6*a^2*b^2*c^5*d^2 -
4*a*b^3*c^6*d)))*(-c^3*d)^(1/2)*(15*b^2*c^2 - a^2*d^2 + 10*a*b*c*d)*1i)/(16*(b^4*c^7 + a^4*c^3*d^4 - 4*a^3*b*
c^4*d^3 + 6*a^2*b^2*c^5*d^2 - 4*a*b^3*c^6*d)) + (((x*(a^4*b^3*d^7 + 241*b^7*c^4*d^3 + 460*a*b^6*c^3*d^4 - 20*a
^3*b^4*c*d^6 + 470*a^2*b^5*c^2*d^5))/(32*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^
3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) + (((2*b^12*c^11*d^2 - (23*a*b^11*c^10*d^3)/2 - (a^10*b^2
*c*d^12)/2 + (39*a^2*b^10*c^9*d^4)/2 + 18*a^3*b^9*c^8*d^5 - 126*a^4*b^8*c^7*d^6 + 231*a^5*b^7*c^6*d^7 - 231*a^
6*b^6*c^5*d^8 + 138*a^7*b^5*c^4*d^9 - 48*a^8*b^4*c^3*d^10 + (17*a^9*b^3*c^2*d^11)/2)/(b^9*c^11 - a^9*c^2*d^9 +
9*a^8*b*c^3*d^8 + 36*a^2*b^7*c^9*d^2 - 84*a^3*b^6*c^8*d^3 + 126*a^4*b^5*c^7*d^4 - 126*a^5*b^4*c^6*d^5 + 84*a^
6*b^3*c^5*d^6 - 36*a^7*b^2*c^4*d^7 - 9*a*b^8*c^10*d) + (x*(-c^3*d)^(1/2)*(15*b^2*c^2 - a^2*d^2 + 10*a*b*c*d)*(
256*b^11*c^11*d^2 - 1792*a*b^10*c^10*d^3 + 5120*a^2*b^9*c^9*d^4 - 7168*a^3*b^8*c^8*d^5 + 3584*a^4*b^7*c^7*d^6
+ 3584*a^5*b^6*c^6*d^7 - 7168*a^6*b^5*c^5*d^8 + 5120*a^7*b^4*c^4*d^9 - 1792*a^8*b^3*c^3*d^10 + 256*a^9*b^2*c^2
*d^11))/(512*(b^4*c^7 + a^4*c^3*d^4 - 4*a^3*b*c^4*d^3 + 6*a^2*b^2*c^5*d^2 - 4*a*b^3*c^6*d)*(b^6*c^8 + a^6*c^2*
d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)))*(-c^3*
d)^(1/2)*(15*b^2*c^2 - a^2*d^2 + 10*a*b*c*d))/(16*(b^4*c^7 + a^4*c^3*d^4 - 4*a^3*b*c^4*d^3 + 6*a^2*b^2*c^5*d^2
- 4*a*b^3*c^6*d)))*(-c^3*d)^(1/2)*(15*b^2*c^2 - a^2*d^2 + 10*a*b*c*d)*1i)/(16*(b^4*c^7 + a^4*c^3*d^4 - 4*a^3*
b*c^4*d^3 + 6*a^2*b^2*c^5*d^2 - 4*a*b^3*c^6*d)))/(((165*b^8*c^4*d^3)/64 - (5*a^4*b^4*d^7)/64 + (475*a*b^7*c^3*
d^4)/32 - (3*a^3*b^5*c*d^6)/32 + (39*a^2*b^6*c^2*d^5)/4)/(b^9*c^11 - a^9*c^2*d^9 + 9*a^8*b*c^3*d^8 + 36*a^2*b^
7*c^9*d^2 - 84*a^3*b^6*c^8*d^3 + 126*a^4*b^5*c^7*d^4 - 126*a^5*b^4*c^6*d^5 + 84*a^6*b^3*c^5*d^6 - 36*a^7*b^2*c
^4*d^7 - 9*a*b^8*c^10*d) - (((x*(a^4*b^3*d^7 + 241*b^7*c^4*d^3 + 460*a*b^6*c^3*d^4 - 20*a^3*b^4*c*d^6 + 470*a^
2*b^5*c^2*d^5))/(32*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^
4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) - (((2*b^12*c^11*d^2 - (23*a*b^11*c^10*d^3)/2 - (a^10*b^2*c*d^12)/2 + (39*a^2*
b^10*c^9*d^4)/2 + 18*a^3*b^9*c^8*d^5 - 126*a^4*b^8*c^7*d^6 + 231*a^5*b^7*c^6*d^7 - 231*a^6*b^6*c^5*d^8 + 138*a
^7*b^5*c^4*d^9 - 48*a^8*b^4*c^3*d^10 + (17*a^9*b^3*c^2*d^11)/2)/(b^9*c^11 - a^9*c^2*d^9 + 9*a^8*b*c^3*d^8 + 36
*a^2*b^7*c^9*d^2 - 84*a^3*b^6*c^8*d^3 + 126*a^4*b^5*c^7*d^4 - 126*a^5*b^4*c^6*d^5 + 84*a^6*b^3*c^5*d^6 - 36*a^
7*b^2*c^4*d^7 - 9*a*b^8*c^10*d) - (x*(-c^3*d)^(1/2)*(15*b^2*c^2 - a^2*d^2 + 10*a*b*c*d)*(256*b^11*c^11*d^2 - 1
792*a*b^10*c^10*d^3 + 5120*a^2*b^9*c^9*d^4 - 7168*a^3*b^8*c^8*d^5 + 3584*a^4*b^7*c^7*d^6 + 3584*a^5*b^6*c^6*d^
7 - 7168*a^6*b^5*c^5*d^8 + 5120*a^7*b^4*c^4*d^9 - 1792*a^8*b^3*c^3*d^10 + 256*a^9*b^2*c^2*d^11))/(512*(b^4*c^7
+ a^4*c^3*d^4 - 4*a^3*b*c^4*d^3 + 6*a^2*b^2*c^5*d^2 - 4*a*b^3*c^6*d)*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5
+ 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)))*(-c^3*d)^(1/2)*(15*b^2*c^2
- a^2*d^2 + 10*a*b*c*d))/(16*(b^4*c^7 + a^4*c^3*d^4 - 4*a^3*b*c^4*d^3 + 6*a^2*b^2*c^5*d^2 - 4*a*b^3*c^6*d)))*(
-c^3*d)^(1/2)*(15*b^2*c^2 - a^2*d^2 + 10*a*b*c*d))/(16*(b^4*c^7 + a^4*c^3*d^4 - 4*a^3*b*c^4*d^3 + 6*a^2*b^2*c^
5*d^2 - 4*a*b^3*c^6*d)) + (((x*(a^4*b^3*d^7 + 241*b^7*c^4*d^3 + 460*a*b^6*c^3*d^4 - 20*a^3*b^4*c*d^6 + 470*a^2
*b^5*c^2*d^5))/(32*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4
*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) + (((2*b^12*c^11*d^2 - (23*a*b^11*c^10*d^3)/2 - (a^10*b^2*c*d^12)/2 + (39*a^2*b
^10*c^9*d^4)/2 + 18*a^3*b^9*c^8*d^5 - 126*a^4*b^8*c^7*d^6 + 231*a^5*b^7*c^6*d^7 - 231*a^6*b^6*c^5*d^8 + 138*a^
7*b^5*c^4*d^9 - 48*a^8*b^4*c^3*d^10 + (17*a^9*b^3*c^2*d^11)/2)/(b^9*c^11 - a^9*c^2*d^9 + 9*a^8*b*c^3*d^8 + 36*
a^2*b^7*c^9*d^2 - 84*a^3*b^6*c^8*d^3 + 126*a^4*b^5*c^7*d^4 - 126*a^5*b^4*c^6*d^5 + 84*a^6*b^3*c^5*d^6 - 36*a^7
*b^2*c^4*d^7 - 9*a*b^8*c^10*d) + (x*(-c^3*d)^(1/2)*(15*b^2*c^2 - a^2*d^2 + 10*a*b*c*d)*(256*b^11*c^11*d^2 - 17
92*a*b^10*c^10*d^3 + 5120*a^2*b^9*c^9*d^4 - 7168*a^3*b^8*c^8*d^5 + 3584*a^4*b^7*c^7*d^6 + 3584*a^5*b^6*c^6*d^7
- 7168*a^6*b^5*c^5*d^8 + 5120*a^7*b^4*c^4*d^9 - 1792*a^8*b^3*c^3*d^10 + 256*a^9*b^2*c^2*d^11))/(512*(b^4*c^7
+ a^4*c^3*d^4 - 4*a^3*b*c^4*d^3 + 6*a^2*b^2*c^5*d^2 - 4*a*b^3*c^6*d)*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5
+ 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)))*(-c^3*d)^(1/2)*(15*b^2*c^2 -
a^2*d^2 + 10*a*b*c*d))/(16*(b^4*c^7 + a^4*c^3*d^4 - 4*a^3*b*c^4*d^3 + 6*a^2*b^2*c^5*d^2 - 4*a*b^3*c^6*d)))*(-
c^3*d)^(1/2)*(15*b^2*c^2 - a^2*d^2 + 10*a*b*c*d))/(16*(b^4*c^7 + a^4*c^3*d^4 - 4*a^3*b*c^4*d^3 + 6*a^2*b^2*c^5
*d^2 - 4*a*b^3*c^6*d))))*(-c^3*d)^(1/2)*(15*b^2*c^2 - a^2*d^2 + 10*a*b*c*d)*1i)/(8*(b^4*c^7 + a^4*c^3*d^4 - 4*
a^3*b*c^4*d^3 + 6*a^2*b^2*c^5*d^2 - 4*a*b^3*c^6*d))