\(\int \frac {1}{x^4 (a+b x^2)^2 (c+d x^2)^3} \, dx\) [712]

Optimal result

Integrand size = 22, antiderivative size = 377 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {20 b^3 c^3-24 a b^2 c^2 d+75 a^2 b c d^2-35 a^3 d^3}{24 a^2 c^3 (b c-a d)^3 x^3}+\frac {20 b^4 c^4-24 a b^3 c^3 d-24 a^2 b^2 c^2 d^2+75 a^3 b c d^3-35 a^4 d^4}{8 a^3 c^4 (b c-a d)^3 x}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (4 b^2 c^2+15 a b c d-7 a^2 d^2\right )}{8 a c^2 (b c-a d)^3 x^3 \left (c+d x^2\right )}+\frac {b^{9/2} (5 b c-11 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (b c-a d)^4}+\frac {d^{7/2} \left (99 b^2 c^2-110 a b c d+35 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} (b c-a d)^4} \] Output:

-1/24*(-35*a^3*d^3+75*a^2*b*c*d^2-24*a*b^2*c^2*d+20*b^3*c^3)/a^2/c^3/(-a*d 
+b*c)^3/x^3+1/8*(-35*a^4*d^4+75*a^3*b*c*d^3-24*a^2*b^2*c^2*d^2-24*a*b^3*c^ 
3*d+20*b^4*c^4)/a^3/c^4/(-a*d+b*c)^3/x+1/4*d*(a*d+2*b*c)/a/c/(-a*d+b*c)^2/ 
x^3/(d*x^2+c)^2+1/2*b/a/(-a*d+b*c)/x^3/(b*x^2+a)/(d*x^2+c)^2+1/8*d*(-7*a^2 
*d^2+15*a*b*c*d+4*b^2*c^2)/a/c^2/(-a*d+b*c)^3/x^3/(d*x^2+c)+1/2*b^(9/2)*(- 
11*a*d+5*b*c)*arctan(b^(1/2)*x/a^(1/2))/a^(7/2)/(-a*d+b*c)^4+1/8*d^(7/2)*( 
35*a^2*d^2-110*a*b*c*d+99*b^2*c^2)*arctan(d^(1/2)*x/c^(1/2))/c^(9/2)/(-a*d 
+b*c)^4
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {1}{24} \left (-\frac {8}{a^2 c^3 x^3}+\frac {48 b c+72 a d}{a^3 c^4 x}-\frac {12 b^5 x}{a^3 (-b c+a d)^3 \left (a+b x^2\right )}+\frac {6 d^4 x}{c^3 (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {3 d^4 (19 b c-11 a d) x}{c^4 (b c-a d)^3 \left (c+d x^2\right )}+\frac {12 b^{9/2} (5 b c-11 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2} (b c-a d)^4}+\frac {3 d^{7/2} \left (99 b^2 c^2-110 a b c d+35 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{9/2} (b c-a d)^4}\right ) \] Input:

Integrate[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^3),x]
 

Output:

(-8/(a^2*c^3*x^3) + (48*b*c + 72*a*d)/(a^3*c^4*x) - (12*b^5*x)/(a^3*(-(b*c 
) + a*d)^3*(a + b*x^2)) + (6*d^4*x)/(c^3*(b*c - a*d)^2*(c + d*x^2)^2) + (3 
*d^4*(19*b*c - 11*a*d)*x)/(c^4*(b*c - a*d)^3*(c + d*x^2)) + (12*b^(9/2)*(5 
*b*c - 11*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*(b*c - a*d)^4) + (3*d 
^(7/2)*(99*b^2*c^2 - 110*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]] 
)/(c^(9/2)*(b*c - a*d)^4))/24
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {374, 25, 441, 27, 441, 445, 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.

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

\(\Big \downarrow \) 374

\(\displaystyle \frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\int -\frac {9 b d x^2+5 b c-2 a d}{x^4 \left (b x^2+a\right ) \left (d x^2+c\right )^3}dx}{2 a (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {9 b d x^2+5 b c-2 a d}{x^4 \left (b x^2+a\right ) \left (d x^2+c\right )^3}dx}{2 a (b c-a d)}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 441

\(\displaystyle \frac {\frac {\int \frac {2 \left (10 b^2 c^2-8 a b d c+7 a^2 d^2+7 b d (2 b c+a d) x^2\right )}{x^4 \left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{4 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^3 \left (c+d x^2\right )^2 (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {10 b^2 c^2-8 a b d c+7 a^2 d^2+7 b d (2 b c+a d) x^2}{x^4 \left (b x^2+a\right ) \left (d x^2+c\right )^2}dx}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^3 \left (c+d x^2\right )^2 (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 441

\(\displaystyle \frac {\frac {\frac {\int \frac {20 b^3 c^3-24 a b^2 d c^2+75 a^2 b d^2 c-35 a^3 d^3+5 b d \left (4 b^2 c^2+15 a b d c-7 a^2 d^2\right ) x^2}{x^4 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{2 c (b c-a d)}+\frac {d \left (-7 a^2 d^2+15 a b c d+4 b^2 c^2\right )}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^3 \left (c+d x^2\right )^2 (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {3 \left (20 b^4 c^4-24 a b^3 d c^3-24 a^2 b^2 d^2 c^2+75 a^3 b d^3 c-35 a^4 d^4+b d \left (20 b^3 c^3-24 a b^2 d c^2+75 a^2 b d^2 c-35 a^3 d^3\right ) x^2\right )}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{3 a c}-\frac {-35 a^3 d^3+75 a^2 b c d^2-24 a b^2 c^2 d+20 b^3 c^3}{3 a c x^3}}{2 c (b c-a d)}+\frac {d \left (-7 a^2 d^2+15 a b c d+4 b^2 c^2\right )}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^3 \left (c+d x^2\right )^2 (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {20 b^4 c^4-24 a b^3 d c^3-24 a^2 b^2 d^2 c^2+75 a^3 b d^3 c-35 a^4 d^4+b d \left (20 b^3 c^3-24 a b^2 d c^2+75 a^2 b d^2 c-35 a^3 d^3\right ) x^2}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}dx}{a c}-\frac {-35 a^3 d^3+75 a^2 b c d^2-24 a b^2 c^2 d+20 b^3 c^3}{3 a c x^3}}{2 c (b c-a d)}+\frac {d \left (-7 a^2 d^2+15 a b c d+4 b^2 c^2\right )}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^3 \left (c+d x^2\right )^2 (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {\frac {-\frac {-\frac {\int \frac {20 b^5 c^5-24 a b^4 d c^4-24 a^2 b^3 d^2 c^3-24 a^3 b^2 d^3 c^2+75 a^4 b d^4 c-35 a^5 d^5+b d \left (20 b^4 c^4-24 a b^3 d c^3-24 a^2 b^2 d^2 c^2+75 a^3 b d^3 c-35 a^4 d^4\right ) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )}dx}{a c}-\frac {-35 a^4 d^4+75 a^3 b c d^3-24 a^2 b^2 c^2 d^2-24 a b^3 c^3 d+20 b^4 c^4}{a c x}}{a c}-\frac {-35 a^3 d^3+75 a^2 b c d^2-24 a b^2 c^2 d+20 b^3 c^3}{3 a c x^3}}{2 c (b c-a d)}+\frac {d \left (-7 a^2 d^2+15 a b c d+4 b^2 c^2\right )}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^3 \left (c+d x^2\right )^2 (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {\frac {-\frac {-\frac {\frac {a^3 d^4 \left (35 a^2 d^2-110 a b c d+99 b^2 c^2\right ) \int \frac {1}{d x^2+c}dx}{b c-a d}+\frac {4 b^5 c^4 (5 b c-11 a d) \int \frac {1}{b x^2+a}dx}{b c-a d}}{a c}-\frac {-35 a^4 d^4+75 a^3 b c d^3-24 a^2 b^2 c^2 d^2-24 a b^3 c^3 d+20 b^4 c^4}{a c x}}{a c}-\frac {-35 a^3 d^3+75 a^2 b c d^2-24 a b^2 c^2 d+20 b^3 c^3}{3 a c x^3}}{2 c (b c-a d)}+\frac {d \left (-7 a^2 d^2+15 a b c d+4 b^2 c^2\right )}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^3 \left (c+d x^2\right )^2 (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {d \left (-7 a^2 d^2+15 a b c d+4 b^2 c^2\right )}{2 c x^3 \left (c+d x^2\right ) (b c-a d)}+\frac {-\frac {-35 a^3 d^3+75 a^2 b c d^2-24 a b^2 c^2 d+20 b^3 c^3}{3 a c x^3}-\frac {-\frac {\frac {a^3 d^{7/2} \left (35 a^2 d^2-110 a b c d+99 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)}+\frac {4 b^{9/2} c^4 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (5 b c-11 a d)}{\sqrt {a} (b c-a d)}}{a c}-\frac {-35 a^4 d^4+75 a^3 b c d^3-24 a^2 b^2 c^2 d^2-24 a b^3 c^3 d+20 b^4 c^4}{a c x}}{a c}}{2 c (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^3 \left (c+d x^2\right )^2 (b c-a d)}}{2 a (b c-a d)}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\)

Input:

Int[1/(x^4*(a + b*x^2)^2*(c + d*x^2)^3),x]
 

Output:

b/(2*a*(b*c - a*d)*x^3*(a + b*x^2)*(c + d*x^2)^2) + ((d*(2*b*c + a*d))/(2* 
c*(b*c - a*d)*x^3*(c + d*x^2)^2) + ((d*(4*b^2*c^2 + 15*a*b*c*d - 7*a^2*d^2 
))/(2*c*(b*c - a*d)*x^3*(c + d*x^2)) + (-1/3*(20*b^3*c^3 - 24*a*b^2*c^2*d 
+ 75*a^2*b*c*d^2 - 35*a^3*d^3)/(a*c*x^3) - (-((20*b^4*c^4 - 24*a*b^3*c^3*d 
 - 24*a^2*b^2*c^2*d^2 + 75*a^3*b*c*d^3 - 35*a^4*d^4)/(a*c*x)) - ((4*b^(9/2 
)*c^4*(5*b*c - 11*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d)) 
+ (a^3*d^(7/2)*(99*b^2*c^2 - 110*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/ 
Sqrt[c]])/(Sqrt[c]*(b*c - a*d)))/(a*c))/(a*c))/(2*c*(b*c - a*d)))/(2*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 = 0.74 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.59

method	result	size
default	\(-\frac {b^{5} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (11 a d -5 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3} \left (a d -b c \right )^{4}}-\frac {1}{3 a^{2} c^{3} x^{3}}-\frac {-3 a d -2 b c}{x \,a^{3} c^{4}}+\frac {d^{4} \left (\frac {\left (\frac {11}{8} a^{2} d^{3}-\frac {15}{4} a c \,d^{2} b +\frac {19}{8} b^{2} c^{2} d \right ) x^{3}+\frac {c \left (13 a^{2} d^{2}-34 a b c d +21 b^{2} c^{2}\right ) x}{8}}{\left (x^{2} d +c \right )^{2}}+\frac {\left (35 a^{2} d^{2}-110 a b c d +99 b^{2} c^{2}\right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{8 \sqrt {c d}}\right )}{c^{4} \left (a d -b c \right )^{4}}\)	\(222\)
risch	\(\text {Expression too large to display}\)	\(4081\)

Input:

int(1/x^4/(b*x^2+a)^2/(d*x^2+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-b^5/a^3/(a*d-b*c)^4*((1/2*a*d-1/2*b*c)*x/(b*x^2+a)+1/2*(11*a*d-5*b*c)/(a* 
b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-1/3/a^2/c^3/x^3-(-3*a*d-2*b*c)/x/a^3/c^4 
+d^4/c^4/(a*d-b*c)^4*(((11/8*a^2*d^3-15/4*a*c*d^2*b+19/8*b^2*c^2*d)*x^3+1/ 
8*c*(13*a^2*d^2-34*a*b*c*d+21*b^2*c^2)*x)/(d*x^2+c)^2+1/8*(35*a^2*d^2-110* 
a*b*c*d+99*b^2*c^2)/(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. 1033 vs. \(2 (347) = 694\).

Time = 9.55 (sec) , antiderivative size = 4225, normalized size of antiderivative = 11.21 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/x**4/(b*x**2+a)**2/(d*x**2+c)**3,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (347) = 694\).

Time = 0.14 (sec) , antiderivative size = 738, normalized size of antiderivative = 1.96 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {{\left (5 \, b^{6} c - 11 \, a b^{5} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} \sqrt {a b}} + \frac {{\left (99 \, b^{2} c^{2} d^{4} - 110 \, a b c d^{5} + 35 \, a^{2} d^{6}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} \sqrt {c d}} - \frac {8 \, a^{2} b^{3} c^{6} - 24 \, a^{3} b^{2} c^{5} d + 24 \, a^{4} b c^{4} d^{2} - 8 \, a^{5} c^{3} d^{3} - 3 \, {\left (20 \, b^{5} c^{4} d^{2} - 24 \, a b^{4} c^{3} d^{3} - 24 \, a^{2} b^{3} c^{2} d^{4} + 75 \, a^{3} b^{2} c d^{5} - 35 \, a^{4} b d^{6}\right )} x^{8} - {\left (120 \, b^{5} c^{5} d - 104 \, a b^{4} c^{4} d^{2} - 192 \, a^{2} b^{3} c^{3} d^{3} + 303 \, a^{3} b^{2} c^{2} d^{4} + 50 \, a^{4} b c d^{5} - 105 \, a^{5} d^{6}\right )} x^{6} - {\left (60 \, b^{5} c^{6} + 8 \, a b^{4} c^{5} d - 176 \, a^{2} b^{3} c^{4} d^{2} + 319 \, a^{4} b c^{2} d^{4} - 175 \, a^{5} c d^{5}\right )} x^{4} - 8 \, {\left (5 \, a b^{4} c^{6} - 8 \, a^{2} b^{3} c^{5} d - 6 \, a^{3} b^{2} c^{4} d^{2} + 16 \, a^{4} b c^{3} d^{3} - 7 \, a^{5} c^{2} d^{4}\right )} x^{2}}{24 \, {\left ({\left (a^{3} b^{4} c^{7} d^{2} - 3 \, a^{4} b^{3} c^{6} d^{3} + 3 \, a^{5} b^{2} c^{5} d^{4} - a^{6} b c^{4} d^{5}\right )} x^{9} + {\left (2 \, a^{3} b^{4} c^{8} d - 5 \, a^{4} b^{3} c^{7} d^{2} + 3 \, a^{5} b^{2} c^{6} d^{3} + a^{6} b c^{5} d^{4} - a^{7} c^{4} d^{5}\right )} x^{7} + {\left (a^{3} b^{4} c^{9} - a^{4} b^{3} c^{8} d - 3 \, a^{5} b^{2} c^{7} d^{2} + 5 \, a^{6} b c^{6} d^{3} - 2 \, a^{7} c^{5} d^{4}\right )} x^{5} + {\left (a^{4} b^{3} c^{9} - 3 \, a^{5} b^{2} c^{8} d + 3 \, a^{6} b c^{7} d^{2} - a^{7} c^{6} d^{3}\right )} x^{3}\right )}} \] Input:

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")
 

Output:

1/2*(5*b^6*c - 11*a*b^5*d)*arctan(b*x/sqrt(a*b))/((a^3*b^4*c^4 - 4*a^4*b^3 
*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*sqrt(a*b)) + 1/8*(99 
*b^2*c^2*d^4 - 110*a*b*c*d^5 + 35*a^2*d^6)*arctan(d*x/sqrt(c*d))/((b^4*c^8 
 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4)*sqrt 
(c*d)) - 1/24*(8*a^2*b^3*c^6 - 24*a^3*b^2*c^5*d + 24*a^4*b*c^4*d^2 - 8*a^5 
*c^3*d^3 - 3*(20*b^5*c^4*d^2 - 24*a*b^4*c^3*d^3 - 24*a^2*b^3*c^2*d^4 + 75* 
a^3*b^2*c*d^5 - 35*a^4*b*d^6)*x^8 - (120*b^5*c^5*d - 104*a*b^4*c^4*d^2 - 1 
92*a^2*b^3*c^3*d^3 + 303*a^3*b^2*c^2*d^4 + 50*a^4*b*c*d^5 - 105*a^5*d^6)*x 
^6 - (60*b^5*c^6 + 8*a*b^4*c^5*d - 176*a^2*b^3*c^4*d^2 + 319*a^4*b*c^2*d^4 
 - 175*a^5*c*d^5)*x^4 - 8*(5*a*b^4*c^6 - 8*a^2*b^3*c^5*d - 6*a^3*b^2*c^4*d 
^2 + 16*a^4*b*c^3*d^3 - 7*a^5*c^2*d^4)*x^2)/((a^3*b^4*c^7*d^2 - 3*a^4*b^3* 
c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^9 + (2*a^3*b^4*c^8*d - 5*a^ 
4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^7 + (a^ 
3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^ 
5*d^4)*x^5 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^ 
3)*x^3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {b^{5} x}{2 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} {\left (b x^{2} + a\right )}} + \frac {{\left (5 \, b^{6} c - 11 \, a b^{5} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} \sqrt {a b}} + \frac {{\left (99 \, b^{2} c^{2} d^{4} - 110 \, a b c d^{5} + 35 \, a^{2} d^{6}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} \sqrt {c d}} + \frac {19 \, b c d^{5} x^{3} - 11 \, a d^{6} x^{3} + 21 \, b c^{2} d^{4} x - 13 \, a c d^{5} x}{8 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} + \frac {6 \, b c x^{2} + 9 \, a d x^{2} - a c}{3 \, a^{3} c^{4} x^{3}} \] Input:

integrate(1/x^4/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")
 

Output:

1/2*b^5*x/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*(b*x^ 
2 + a)) + 1/2*(5*b^6*c - 11*a*b^5*d)*arctan(b*x/sqrt(a*b))/((a^3*b^4*c^4 - 
 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*sqrt(a*b)) 
 + 1/8*(99*b^2*c^2*d^4 - 110*a*b*c*d^5 + 35*a^2*d^6)*arctan(d*x/sqrt(c*d)) 
/((b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d^2 - 4*a^3*b*c^5*d^3 + a^4*c^4 
*d^4)*sqrt(c*d)) + 1/8*(19*b*c*d^5*x^3 - 11*a*d^6*x^3 + 21*b*c^2*d^4*x - 1 
3*a*c*d^5*x)/((b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3)*(d 
*x^2 + c)^2) + 1/3*(6*b*c*x^2 + 9*a*d*x^2 - a*c)/(a^3*c^4*x^3)
 

Mupad [B] (verification not implemented)

Time = 2.92 (sec) , antiderivative size = 1161, normalized size of antiderivative = 3.08 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(x^4*(a + b*x^2)^2*(c + d*x^2)^3),x)
 

Output:

((x^2*(7*a*d + 5*b*c))/(3*a^2*c^2) - 1/(3*a*c) + (x^8*(35*a^4*b*d^6 - 20*b 
^5*c^4*d^2 + 24*a*b^4*c^3*d^3 - 75*a^3*b^2*c*d^5 + 24*a^2*b^3*c^2*d^4))/(8 
*a^3*c^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (x^4*(60*b 
^5*c^5 - 175*a^5*d^5 - 176*a^2*b^3*c^3*d^2 + 8*a*b^4*c^4*d + 319*a^4*b*c*d 
^4))/(24*a^3*c^3*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (d*x^6*(10 
5*a^5*d^5 - 120*b^5*c^5 + 192*a^2*b^3*c^3*d^2 - 303*a^3*b^2*c^2*d^3 + 104* 
a*b^4*c^4*d - 50*a^4*b*c*d^4))/(24*a^3*c^4*(a*d - b*c)*(a^2*d^2 + b^2*c^2 
- 2*a*b*c*d)))/(x^5*(b*c^2 + 2*a*c*d) + x^7*(a*d^2 + 2*b*c*d) + a*c^2*x^3 
+ b*d^2*x^9) + (atan((b^3*c^11*x*(-a^7*b^9)^(3/2)*400i + a^18*b*d^11*x*(-a 
^7*b^9)^(1/2)*1225i + a^14*b^5*c^4*d^7*x*(-a^7*b^9)^(1/2)*9801i - a^15*b^4 
*c^3*d^8*x*(-a^7*b^9)^(1/2)*21780i + a^16*b^3*c^2*d^9*x*(-a^7*b^9)^(1/2)*1 
9030i - a*b^2*c^10*d*x*(-a^7*b^9)^(3/2)*1760i + a^2*b*c^9*d^2*x*(-a^7*b^9) 
^(3/2)*1936i - a^17*b^2*c*d^10*x*(-a^7*b^9)^(1/2)*7700i)/(400*a^11*b^16*c^ 
11 - 1225*a^22*b^5*d^11 - 1760*a^12*b^15*c^10*d + 7700*a^21*b^6*c*d^10 + 1 
936*a^13*b^14*c^9*d^2 - 9801*a^18*b^9*c^4*d^7 + 21780*a^19*b^8*c^3*d^8 - 1 
9030*a^20*b^7*c^2*d^9))*(11*a*d - 5*b*c)*(-a^7*b^9)^(1/2)*1i)/(2*(a^11*d^4 
 + a^7*b^4*c^4 - 4*a^8*b^3*c^3*d + 6*a^9*b^2*c^2*d^2 - 4*a^10*b*c*d^3)) - 
(atan((a^11*d^5*x*(-c^9*d^7)^(3/2)*1225i + b^11*c^20*d*x*(-c^9*d^7)^(1/2)* 
400i - a^8*b^3*c^3*d^2*x*(-c^9*d^7)^(3/2)*21780i + a^9*b^2*c^2*d^3*x*(-c^9 
*d^7)^(3/2)*19030i + a^2*b^9*c^18*d^3*x*(-c^9*d^7)^(1/2)*1936i - a^10*b...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1490, normalized size of antiderivative = 3.95 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/x^4/(b*x^2+a)^2/(d*x^2+c)^3,x)
 

Output:

( - 132*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**4*c**7*d*x** 
3 - 264*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**4*c**6*d**2* 
x**5 - 132*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**4*c**5*d* 
*3*x**7 + 60*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**5*c**8*x** 
3 - 12*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**5*c**7*d*x**5 - 
204*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**5*c**6*d**2*x**7 - 
132*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**5*c**5*d**3*x**9 + 
60*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**6*c**8*x**5 + 120*sqrt 
(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**6*c**7*d*x**7 + 60*sqrt(b)*sq 
rt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**6*c**6*d**2*x**9 + 105*sqrt(d)*sqrt 
(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**7*c**2*d**5*x**3 + 210*sqrt(d)*sqrt(c 
)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**7*c*d**6*x**5 + 105*sqrt(d)*sqrt(c)*ata 
n((d*x)/(sqrt(d)*sqrt(c)))*a**7*d**7*x**7 - 330*sqrt(d)*sqrt(c)*atan((d*x) 
/(sqrt(d)*sqrt(c)))*a**6*b*c**3*d**4*x**3 - 555*sqrt(d)*sqrt(c)*atan((d*x) 
/(sqrt(d)*sqrt(c)))*a**6*b*c**2*d**5*x**5 - 120*sqrt(d)*sqrt(c)*atan((d*x) 
/(sqrt(d)*sqrt(c)))*a**6*b*c*d**6*x**7 + 105*sqrt(d)*sqrt(c)*atan((d*x)/(s 
qrt(d)*sqrt(c)))*a**6*b*d**7*x**9 + 297*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d 
)*sqrt(c)))*a**5*b**2*c**4*d**3*x**3 + 264*sqrt(d)*sqrt(c)*atan((d*x)/(sqr 
t(d)*sqrt(c)))*a**5*b**2*c**3*d**4*x**5 - 363*sqrt(d)*sqrt(c)*atan((d*x)/( 
sqrt(d)*sqrt(c)))*a**5*b**2*c**2*d**5*x**7 - 330*sqrt(d)*sqrt(c)*atan((...