\(\int \frac {1}{x^2 (a+b x^2)^2 (c+d x^2)^2} \, dx\) [701]

Optimal result

Integrand size = 22, antiderivative size = 218 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {3 b^2 c^2-4 a b c d+3 a^2 d^2}{2 a^2 c^2 (b c-a d)^2 x}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{5/2} (3 b c-7 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)^3}-\frac {d^{5/2} (7 b c-3 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{5/2} (b c-a d)^3} \] Output:

-1/2*(3*a^2*d^2-4*a*b*c*d+3*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/x+1/2*d*(a*d+b*c 
)/a/c/(-a*d+b*c)^2/x/(d*x^2+c)+1/2*b/a/(-a*d+b*c)/x/(b*x^2+a)/(d*x^2+c)-1/ 
2*b^(5/2)*(-7*a*d+3*b*c)*arctan(b^(1/2)*x/a^(1/2))/a^(5/2)/(-a*d+b*c)^3-1/ 
2*d^(5/2)*(-3*a*d+7*b*c)*arctan(d^(1/2)*x/c^(1/2))/c^(5/2)/(-a*d+b*c)^3
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {2}{a^2 c^2 x}-\frac {b^3 x}{a^2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d^3 x}{c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {b^{5/2} (3 b c-7 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (-b c+a d)^3}+\frac {d^{5/2} (-7 b c+3 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} (b c-a d)^3}\right ) \] Input:

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

Output:

(-2/(a^2*c^2*x) - (b^3*x)/(a^2*(b*c - a*d)^2*(a + b*x^2)) - (d^3*x)/(c^2*( 
b*c - a*d)^2*(c + d*x^2)) + (b^(5/2)*(3*b*c - 7*a*d)*ArcTan[(Sqrt[b]*x)/Sq 
rt[a]])/(a^(5/2)*(-(b*c) + a*d)^3) + (d^(5/2)*(-7*b*c + 3*a*d)*ArcTan[(Sqr 
t[d]*x)/Sqrt[c]])/(c^(5/2)*(b*c - a*d)^3))/2
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {374, 25, 441, 27, 445, 397, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 374

\(\displaystyle \frac {b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\int -\frac {5 b d x^2+3 b c-2 a d}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {5 b d x^2+3 b c-2 a d}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 a (b c-a d)}+\frac {b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 441

\(\displaystyle \frac {\frac {\int \frac {2 \left (3 b^2 c^2-4 a b d c+3 a^2 d^2+3 b d (b c+a d) x^2\right )}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{2 c (b c-a d)}+\frac {d (a d+b c)}{c x \left (c+d x^2\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {3 b^2 c^2-4 a b d c+3 a^2 d^2+3 b d (b c+a d) x^2}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{c (b c-a d)}+\frac {d (a d+b c)}{c x \left (c+d x^2\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {-\frac {\int \frac {b d \left (3 b^2 c^2-4 a b d c+3 a^2 d^2\right ) x^2+(b c+a d) \left (3 b^2 c^2-7 a b d c+3 a^2 d^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}dx}{a c}-\frac {\frac {3 b^2 c}{a}+\frac {3 a d^2}{c}-4 b d}{x}}{c (b c-a d)}+\frac {d (a d+b c)}{c x \left (c+d x^2\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {-\frac {\frac {a^2 d^3 (7 b c-3 a d) \int \frac {1}{d x^2+c}dx}{b c-a d}+\frac {b^3 c^2 (3 b c-7 a d) \int \frac {1}{b x^2+a}dx}{b c-a d}}{a c}-\frac {\frac {3 b^2 c}{a}+\frac {3 a d^2}{c}-4 b d}{x}}{c (b c-a d)}+\frac {d (a d+b c)}{c x \left (c+d x^2\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {-\frac {\frac {a^2 d^{5/2} (7 b c-3 a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)}+\frac {b^{5/2} c^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (3 b c-7 a d)}{\sqrt {a} (b c-a d)}}{a c}-\frac {\frac {3 b^2 c}{a}+\frac {3 a d^2}{c}-4 b d}{x}}{c (b c-a d)}+\frac {d (a d+b c)}{c x \left (c+d x^2\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 374

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
 + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - 
a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, 
c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, 
 c, d, e, m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 441

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a 
+ b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si 
mp[1/(a*2*(b*c - a*d)*(p + 1))   Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 
)^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m 
 + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, 
 x] && LtQ[p, -1]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.65

method	result	size
default	\(-\frac {b^{3} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (7 a d -3 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2} \left (a d -b c \right )^{3}}-\frac {1}{a^{2} c^{2} x}-\frac {d^{3} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{x^{2} d +c}+\frac {\left (3 a d -7 b c \right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{2 \sqrt {c d}}\right )}{c^{2} \left (a d -b c \right )^{3}}\)	\(141\)
risch	\(\text {Expression too large to display}\)	\(2040\)

Input:

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

Output:

-b^3/a^2/(a*d-b*c)^3*((1/2*a*d-1/2*b*c)*x/(b*x^2+a)+1/2*(7*a*d-3*b*c)/(a*b 
)^(1/2)*arctan(b*x/(a*b)^(1/2)))-1/a^2/c^2/x-d^3/c^2/(a*d-b*c)^3*((1/2*a*d 
-1/2*b*c)*x/(d*x^2+c)+1/2*(3*a*d-7*b*c)/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2) 
))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (192) = 384\).

Time = 1.12 (sec) , antiderivative size = 2113, normalized size of antiderivative = 9.69 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/4*(4*a*b^3*c^4 - 12*a^2*b^2*c^3*d + 12*a^3*b*c^2*d^2 - 4*a^4*c*d^3 + 2 
*(3*b^4*c^3*d - 7*a*b^3*c^2*d^2 + 7*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x^4 + 2*( 
3*b^4*c^4 - 5*a*b^3*c^3*d + 5*a^3*b*c*d^3 - 3*a^4*d^4)*x^2 - ((3*b^4*c^3*d 
 - 7*a*b^3*c^2*d^2)*x^5 + (3*b^4*c^4 - 4*a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2)* 
x^3 + (3*a*b^3*c^4 - 7*a^2*b^2*c^3*d)*x)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqr 
t(-b/a) - a)/(b*x^2 + a)) - ((7*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x^5 + (7*a^2* 
b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 3*a^4*d^4)*x^3 + (7*a^3*b*c^2*d^2 - 3*a^4*c* 
d^3)*x)*sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^2* 
b^4*c^5*d - 3*a^3*b^3*c^4*d^2 + 3*a^4*b^2*c^3*d^3 - a^5*b*c^2*d^4)*x^5 + ( 
a^2*b^4*c^6 - 2*a^3*b^3*c^5*d + 2*a^5*b*c^3*d^3 - a^6*c^2*d^4)*x^3 + (a^3* 
b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3)*x), -1/4*(4*a*b 
^3*c^4 - 12*a^2*b^2*c^3*d + 12*a^3*b*c^2*d^2 - 4*a^4*c*d^3 + 2*(3*b^4*c^3* 
d - 7*a*b^3*c^2*d^2 + 7*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x^4 + 2*(3*b^4*c^4 - 
5*a*b^3*c^3*d + 5*a^3*b*c*d^3 - 3*a^4*d^4)*x^2 + 2*((7*a^2*b^2*c*d^3 - 3*a 
^3*b*d^4)*x^5 + (7*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 3*a^4*d^4)*x^3 + (7*a 
^3*b*c^2*d^2 - 3*a^4*c*d^3)*x)*sqrt(d/c)*arctan(x*sqrt(d/c)) - ((3*b^4*c^3 
*d - 7*a*b^3*c^2*d^2)*x^5 + (3*b^4*c^4 - 4*a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 
)*x^3 + (3*a*b^3*c^4 - 7*a^2*b^2*c^3*d)*x)*sqrt(-b/a)*log((b*x^2 - 2*a*x*s 
qrt(-b/a) - a)/(b*x^2 + a)))/((a^2*b^4*c^5*d - 3*a^3*b^3*c^4*d^2 + 3*a^4*b 
^2*c^3*d^3 - a^5*b*c^2*d^4)*x^5 + (a^2*b^4*c^6 - 2*a^3*b^3*c^5*d + 2*a^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/x**2/(b*x**2+a)**2/(d*x**2+c)**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.73 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {{\left (3 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} - \frac {{\left (7 \, b c d^{3} - 3 \, a d^{4}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} \sqrt {c d}} - \frac {2 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + 2 \, a^{3} c d^{2} + {\left (3 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x^{4} + {\left (3 \, b^{3} c^{3} - 2 \, a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} x^{2}}{2 \, {\left ({\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{5} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{3} + {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x\right )}} \] Input:

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

Output:

-1/2*(3*b^4*c - 7*a*b^3*d)*arctan(b*x/sqrt(a*b))/((a^2*b^3*c^3 - 3*a^3*b^2 
*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*sqrt(a*b)) - 1/2*(7*b*c*d^3 - 3*a*d^4)*a 
rctan(d*x/sqrt(c*d))/((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2 
*d^3)*sqrt(c*d)) - 1/2*(2*a*b^2*c^3 - 4*a^2*b*c^2*d + 2*a^3*c*d^2 + (3*b^3 
*c^2*d - 4*a*b^2*c*d^2 + 3*a^2*b*d^3)*x^4 + (3*b^3*c^3 - 2*a*b^2*c^2*d - 2 
*a^2*b*c*d^2 + 3*a^3*d^3)*x^2)/((a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b 
*c^2*d^3)*x^5 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3 
)*x^3 + (a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=-\frac {{\left (3 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} - \frac {{\left (7 \, b c d^{3} - 3 \, a d^{4}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} \sqrt {c d}} - \frac {3 \, b^{3} c^{2} d x^{4} - 4 \, a b^{2} c d^{2} x^{4} + 3 \, a^{2} b d^{3} x^{4} + 3 \, b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{2} d x^{2} - 2 \, a^{2} b c d^{2} x^{2} + 3 \, a^{3} d^{3} x^{2} + 2 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + 2 \, a^{3} c d^{2}}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left (b d x^{5} + b c x^{3} + a d x^{3} + a c x\right )}} \] Input:

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

Output:

-1/2*(3*b^4*c - 7*a*b^3*d)*arctan(b*x/sqrt(a*b))/((a^2*b^3*c^3 - 3*a^3*b^2 
*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*sqrt(a*b)) - 1/2*(7*b*c*d^3 - 3*a*d^4)*a 
rctan(d*x/sqrt(c*d))/((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2 
*d^3)*sqrt(c*d)) - 1/2*(3*b^3*c^2*d*x^4 - 4*a*b^2*c*d^2*x^4 + 3*a^2*b*d^3* 
x^4 + 3*b^3*c^3*x^2 - 2*a*b^2*c^2*d*x^2 - 2*a^2*b*c*d^2*x^2 + 3*a^3*d^3*x^ 
2 + 2*a*b^2*c^3 - 4*a^2*b*c^2*d + 2*a^3*c*d^2)/((a^2*b^2*c^4 - 2*a^3*b*c^3 
*d + a^4*c^2*d^2)*(b*d*x^5 + b*c*x^3 + a*d*x^3 + a*c*x))
 

Mupad [B] (verification not implemented)

Time = 2.27 (sec) , antiderivative size = 3747, normalized size of antiderivative = 17.19 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

- (1/(a*c) + (x^2*(3*a^3*d^3 + 3*b^3*c^3 - 2*a*b^2*c^2*d - 2*a^2*b*c*d^2)) 
/(2*a^2*c^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (b*d*x^4*(3*a^2*d^2 + 3*b^2 
*c^2 - 4*a*b*c*d))/(2*a^2*c^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x^3*(a*d 
+ b*c) + a*c*x + b*d*x^5) - (atan((a^7*d^3*x*(-c^5*d^5)^(3/2)*9i + b^7*c^1 
2*d*x*(-c^5*d^5)^(1/2)*9i + a^2*b^5*c^10*d^3*x*(-c^5*d^5)^(1/2)*49i - a^6* 
b*c*d^2*x*(-c^5*d^5)^(3/2)*42i + a^5*b^2*c^2*d*x*(-c^5*d^5)^(3/2)*49i - a* 
b^6*c^11*d^2*x*(-c^5*d^5)^(1/2)*42i)/(9*a^7*c^8*d^10 - 9*b^7*c^15*d^3 + 42 
*a*b^6*c^14*d^4 - 42*a^6*b*c^9*d^9 - 49*a^2*b^5*c^13*d^5 + 49*a^5*b^2*c^10 
*d^8))*(3*a*d - 7*b*c)*(-c^5*d^5)^(1/2)*1i)/(2*(b^3*c^8 - a^3*c^5*d^3 + 3* 
a^2*b*c^6*d^2 - 3*a*b^2*c^7*d)) - (atan((((7*a*d - 3*b*c)*(x*(144*a^6*b^15 
*c^18*d^3 - 1536*a^7*b^14*c^17*d^4 + 6976*a^8*b^13*c^16*d^5 - 17664*a^9*b^ 
12*c^15*d^6 + 28144*a^10*b^11*c^14*d^7 - 32000*a^11*b^10*c^13*d^8 + 31872* 
a^12*b^9*c^12*d^9 - 32000*a^13*b^8*c^11*d^10 + 28144*a^14*b^7*c^10*d^11 - 
17664*a^15*b^6*c^9*d^12 + 6976*a^16*b^5*c^8*d^13 - 1536*a^17*b^4*c^7*d^14 
+ 144*a^18*b^3*c^6*d^15) - ((7*a*d - 3*b*c)*(-a^5*b^5)^(1/2)*(192*a^8*b^15 
*c^21*d^2 - 2176*a^9*b^14*c^20*d^3 + 10944*a^10*b^13*c^19*d^4 - 31808*a^11 
*b^12*c^18*d^5 + 57600*a^12*b^11*c^17*d^6 - 62784*a^13*b^10*c^16*d^7 + 280 
32*a^14*b^9*c^15*d^8 + 28032*a^15*b^8*c^14*d^9 - 62784*a^16*b^7*c^13*d^10 
+ 57600*a^17*b^6*c^12*d^11 - 31808*a^18*b^5*c^11*d^12 + 10944*a^19*b^4*c^1 
0*d^13 - 2176*a^20*b^3*c^9*d^14 + 192*a^21*b^2*c^8*d^15 - (x*(7*a*d - 3...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 722, normalized size of antiderivative = 3.31 \[ \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 7*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**2*c**4*d*x - 7 
*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**2*c**3*d**2*x**3 + 
3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**3*c**5*x - 4*sqrt(b)* 
sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**3*c**4*d*x**3 - 7*sqrt(b)*sqrt( 
a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**3*c**3*d**2*x**5 + 3*sqrt(b)*sqrt(a) 
*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**5*x**3 + 3*sqrt(b)*sqrt(a)*atan((b* 
x)/(sqrt(b)*sqrt(a)))*b**4*c**4*d*x**5 - 3*sqrt(d)*sqrt(c)*atan((d*x)/(sqr 
t(d)*sqrt(c)))*a**5*c*d**3*x - 3*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt( 
c)))*a**5*d**4*x**3 + 7*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4 
*b*c**2*d**2*x + 4*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4*b*c* 
d**3*x**3 - 3*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4*b*d**4*x* 
*5 + 7*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*b**2*c**2*d**2*x 
**3 + 7*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*b**2*c*d**3*x** 
5 - 2*a**5*c**2*d**3 - 3*a**5*c*d**4*x**2 + 6*a**4*b*c**3*d**2 + 5*a**4*b* 
c**2*d**3*x**2 - 3*a**4*b*c*d**4*x**4 - 6*a**3*b**2*c**4*d + 7*a**3*b**2*c 
**2*d**3*x**4 + 2*a**2*b**3*c**5 - 5*a**2*b**3*c**4*d*x**2 - 7*a**2*b**3*c 
**3*d**2*x**4 + 3*a*b**4*c**5*x**2 + 3*a*b**4*c**4*d*x**4)/(2*a**3*c**3*x* 
(a**4*c*d**3 + a**4*d**4*x**2 - 3*a**3*b*c**2*d**2 - 2*a**3*b*c*d**3*x**2 
+ a**3*b*d**4*x**4 + 3*a**2*b**2*c**3*d - 3*a**2*b**2*c*d**3*x**4 - a*b**3 
*c**4 + 2*a*b**3*c**3*d*x**2 + 3*a*b**3*c**2*d**2*x**4 - b**4*c**4*x**2...