\(\int \frac {1}{(a+b x^2) (c+d x^2)^2 (e+f x^2)^2} \, dx\) [256]

Optimal result

Integrand size = 28, antiderivative size = 270 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=-\frac {d^3 x}{2 c (b c-a d) (d e-c f)^2 \left (c+d x^2\right )}-\frac {f^3 x}{2 e (b e-a f) (d e-c f)^2 \left (e+f x^2\right )}+\frac {b^{7/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)^2 (b e-a f)^2}-\frac {d^{5/2} (b c (3 d e-7 c f)-a d (d e-5 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^2 (d e-c f)^3}-\frac {f^{5/2} (b e (7 d e-3 c f)-a f (5 d e-c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} (b e-a f)^2 (d e-c f)^3} \] Output:

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

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {d^3 x}{c (b c-a d) (d e-c f)^2 \left (c+d x^2\right )}-\frac {f^3 x}{e (b e-a f) (d e-c f)^2 \left (e+f x^2\right )}+\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)^2 (b e-a f)^2}-\frac {d^{5/2} (a d (d e-5 c f)+b c (-3 d e+7 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^2 (-d e+c f)^3}-\frac {f^{5/2} (b e (7 d e-3 c f)+a f (-5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{3/2} (b e-a f)^2 (d e-c f)^3}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.53, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {421, 316, 397, 218, 402, 25, 402, 27, 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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {b^2 \left (\frac {\frac {2 b^{3/2} e \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b e-a f)}-\frac {\sqrt {f} (3 b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}}{2 e (b e-a f)}-\frac {f x}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}-\frac {d \int \frac {b d x^2+2 b c-a d}{\left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{(b c-a d)^2}\)

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 397

\(\displaystyle \frac {b^2 \left (\frac {\frac {2 b^{3/2} e \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b e-a f)}-\frac {\sqrt {f} (3 b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}}{2 e (b e-a f)}-\frac {f x}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}-\frac {d \left (\frac {\frac {f x \left (2 b c^2 f-a d (c f+d e)\right )}{e \left (e+f x^2\right ) (d e-c f)}-\frac {\frac {c f \left (a d f (5 d e-c f)+b \left (2 c^2 f^2-9 c d e f+3 d^2 e^2\right )\right ) \int \frac {1}{f x^2+e}dx}{d e-c f}-\frac {d^2 e (b c (3 d e-7 c f)-a d (d e-5 c f)) \int \frac {1}{d x^2+c}dx}{d e-c f}}{e (d e-c f)}}{2 c (d e-c f)}+\frac {d x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right ) (d e-c f)}\right )}{(b c-a d)^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b^2 \left (\frac {\frac {2 b^{3/2} e \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b e-a f)}-\frac {\sqrt {f} (3 b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}}{2 e (b e-a f)}-\frac {f x}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{(b c-a d)^2}-\frac {d \left (\frac {\frac {f x \left (2 b c^2 f-a d (c f+d e)\right )}{e \left (e+f x^2\right ) (d e-c f)}-\frac {\frac {c \sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a d f (5 d e-c f)+b \left (2 c^2 f^2-9 c d e f+3 d^2 e^2\right )\right )}{\sqrt {e} (d e-c f)}-\frac {d^{3/2} e \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) (b c (3 d e-7 c f)-a d (d e-5 c f))}{\sqrt {c} (d e-c f)}}{e (d e-c f)}}{2 c (d e-c f)}+\frac {d x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right ) (d e-c f)}\right )}{(b c-a d)^2}\)

Input:

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

Output:

(b^2*(-1/2*(f*x)/(e*(b*e - a*f)*(e + f*x^2)) + ((2*b^(3/2)*e*ArcTan[(Sqrt[ 
b]*x)/Sqrt[a]])/(Sqrt[a]*(b*e - a*f)) - (Sqrt[f]*(3*b*e - a*f)*ArcTan[(Sqr 
t[f]*x)/Sqrt[e]])/(Sqrt[e]*(b*e - a*f)))/(2*e*(b*e - a*f))))/(b*c - a*d)^2 
 - (d*((d*(b*c - a*d)*x)/(2*c*(d*e - c*f)*(c + d*x^2)*(e + f*x^2)) + ((f*( 
2*b*c^2*f - a*d*(d*e + c*f))*x)/(e*(d*e - c*f)*(e + f*x^2)) - (-((d^(3/2)* 
e*(b*c*(3*d*e - 7*c*f) - a*d*(d*e - 5*c*f))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/( 
Sqrt[c]*(d*e - c*f))) + (c*Sqrt[f]*(a*d*f*(5*d*e - c*f) + b*(3*d^2*e^2 - 9 
*c*d*e*f + 2*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(d*e - c*f))) 
/(e*(d*e - c*f)))/(2*c*(d*e - c*f))))/(b*c - a*d)^2
 

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 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 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 402

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

rule 421

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

Maple [A] (verified)

Time = 1.72 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right )^{2} \left (a f -b e \right )^{2} \sqrt {a b}}+\frac {f^{3} \left (\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) x}{2 e \left (f \,x^{2}+e \right )}+\frac {\left (a c \,f^{2}-5 a d e f -3 b c e f +7 b d \,e^{2}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 e \sqrt {e f}}\right )}{\left (a f -b e \right )^{2} \left (c f -d e \right )^{3}}+\frac {d^{3} \left (\frac {\left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) x}{2 c \left (x^{2} d +c \right )}+\frac {\left (5 a c d f -a \,d^{2} e -7 b \,c^{2} f +3 b c d e \right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{2 c \sqrt {c d}}\right )}{\left (a d -b c \right )^{2} \left (c f -d e \right )^{3}}\)	\(261\)
risch	\(\text {Expression too large to display}\)	\(245072\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 795 vs. \(2 (238) = 476\).

Time = 0.14 (sec) , antiderivative size = 795, normalized size of antiderivative = 2.94 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{4} c^{2} e^{2} - 2 \, a b^{3} c d e^{2} + a^{2} b^{2} d^{2} e^{2} - 2 \, a b^{3} c^{2} e f + 4 \, a^{2} b^{2} c d e f - 2 \, a^{3} b d^{2} e f + a^{2} b^{2} c^{2} f^{2} - 2 \, a^{3} b c d f^{2} + a^{4} d^{2} f^{2}\right )} \sqrt {a b}} - \frac {{\left (3 \, b c d^{4} e - a d^{5} e - 7 \, b c^{2} d^{3} f + 5 \, a c d^{4} f\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{2} c^{3} d^{3} e^{3} - 2 \, a b c^{2} d^{4} e^{3} + a^{2} c d^{5} e^{3} - 3 \, b^{2} c^{4} d^{2} e^{2} f + 6 \, a b c^{3} d^{3} e^{2} f - 3 \, a^{2} c^{2} d^{4} e^{2} f + 3 \, b^{2} c^{5} d e f^{2} - 6 \, a b c^{4} d^{2} e f^{2} + 3 \, a^{2} c^{3} d^{3} e f^{2} - b^{2} c^{6} f^{3} + 2 \, a b c^{5} d f^{3} - a^{2} c^{4} d^{2} f^{3}\right )} \sqrt {c d}} - \frac {{\left (7 \, b d e^{2} f^{3} - 3 \, b c e f^{4} - 5 \, a d e f^{4} + a c f^{5}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \, {\left (b^{2} d^{3} e^{6} - 3 \, b^{2} c d^{2} e^{5} f - 2 \, a b d^{3} e^{5} f + 3 \, b^{2} c^{2} d e^{4} f^{2} + 6 \, a b c d^{2} e^{4} f^{2} + a^{2} d^{3} e^{4} f^{2} - b^{2} c^{3} e^{3} f^{3} - 6 \, a b c^{2} d e^{3} f^{3} - 3 \, a^{2} c d^{2} e^{3} f^{3} + 2 \, a b c^{3} e^{2} f^{4} + 3 \, a^{2} c^{2} d e^{2} f^{4} - a^{2} c^{3} e f^{5}\right )} \sqrt {e f}} - \frac {b d^{3} e^{2} f x^{3} - a d^{3} e f^{2} x^{3} + b c^{2} d f^{3} x^{3} - a c d^{2} f^{3} x^{3} + b d^{3} e^{3} x - a d^{3} e^{2} f x + b c^{3} f^{3} x - a c^{2} d f^{3} x}{2 \, {\left (b^{2} c^{2} d^{2} e^{4} - a b c d^{3} e^{4} - 2 \, b^{2} c^{3} d e^{3} f + a b c^{2} d^{2} e^{3} f + a^{2} c d^{3} e^{3} f + b^{2} c^{4} e^{2} f^{2} + a b c^{3} d e^{2} f^{2} - 2 \, a^{2} c^{2} d^{2} e^{2} f^{2} - a b c^{4} e f^{3} + a^{2} c^{3} d e f^{3}\right )} {\left (d f x^{4} + d e x^{2} + c f x^{2} + c e\right )}} \] Input:

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

Output:

b^4*arctan(b*x/sqrt(a*b))/((b^4*c^2*e^2 - 2*a*b^3*c*d*e^2 + a^2*b^2*d^2*e^ 
2 - 2*a*b^3*c^2*e*f + 4*a^2*b^2*c*d*e*f - 2*a^3*b*d^2*e*f + a^2*b^2*c^2*f^ 
2 - 2*a^3*b*c*d*f^2 + a^4*d^2*f^2)*sqrt(a*b)) - 1/2*(3*b*c*d^4*e - a*d^5*e 
 - 7*b*c^2*d^3*f + 5*a*c*d^4*f)*arctan(d*x/sqrt(c*d))/((b^2*c^3*d^3*e^3 - 
2*a*b*c^2*d^4*e^3 + a^2*c*d^5*e^3 - 3*b^2*c^4*d^2*e^2*f + 6*a*b*c^3*d^3*e^ 
2*f - 3*a^2*c^2*d^4*e^2*f + 3*b^2*c^5*d*e*f^2 - 6*a*b*c^4*d^2*e*f^2 + 3*a^ 
2*c^3*d^3*e*f^2 - b^2*c^6*f^3 + 2*a*b*c^5*d*f^3 - a^2*c^4*d^2*f^3)*sqrt(c* 
d)) - 1/2*(7*b*d*e^2*f^3 - 3*b*c*e*f^4 - 5*a*d*e*f^4 + a*c*f^5)*arctan(f*x 
/sqrt(e*f))/((b^2*d^3*e^6 - 3*b^2*c*d^2*e^5*f - 2*a*b*d^3*e^5*f + 3*b^2*c^ 
2*d*e^4*f^2 + 6*a*b*c*d^2*e^4*f^2 + a^2*d^3*e^4*f^2 - b^2*c^3*e^3*f^3 - 6* 
a*b*c^2*d*e^3*f^3 - 3*a^2*c*d^2*e^3*f^3 + 2*a*b*c^3*e^2*f^4 + 3*a^2*c^2*d* 
e^2*f^4 - a^2*c^3*e*f^5)*sqrt(e*f)) - 1/2*(b*d^3*e^2*f*x^3 - a*d^3*e*f^2*x 
^3 + b*c^2*d*f^3*x^3 - a*c*d^2*f^3*x^3 + b*d^3*e^3*x - a*d^3*e^2*f*x + b*c 
^3*f^3*x - a*c^2*d*f^3*x)/((b^2*c^2*d^2*e^4 - a*b*c*d^3*e^4 - 2*b^2*c^3*d* 
e^3*f + a*b*c^2*d^2*e^3*f + a^2*c*d^3*e^3*f + b^2*c^4*e^2*f^2 + a*b*c^3*d* 
e^2*f^2 - 2*a^2*c^2*d^2*e^2*f^2 - a*b*c^4*e*f^3 + a^2*c^3*d*e*f^3)*(d*f*x^ 
4 + d*e*x^2 + c*f*x^2 + c*e))
 

Mupad [B] (verification not implemented)

Time = 16.35 (sec) , antiderivative size = 123740, normalized size of antiderivative = 458.30 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log(root(315707392*a^9*b^8*c^13*d^10*e^13*f^10*z^6 + 229605376*a^11 
*b^6*c^12*d^11*e^12*f^11*z^6 + 229605376*a^7*b^10*c^14*d^9*e^14*f^9*z^6 + 
224387072*a^10*b^7*c^14*d^9*e^11*f^12*z^6 + 224387072*a^10*b^7*c^11*d^12*e 
^14*f^9*z^6 + 224387072*a^8*b^9*c^15*d^8*e^12*f^11*z^6 + 224387072*a^8*b^9 
*c^12*d^11*e^15*f^8*z^6 - 193077248*a^9*b^8*c^14*d^9*e^12*f^11*z^6 - 19307 
7248*a^9*b^8*c^12*d^11*e^14*f^9*z^6 - 143503360*a^11*b^6*c^13*d^10*e^11*f^ 
12*z^6 - 143503360*a^11*b^6*c^11*d^12*e^13*f^10*z^6 - 143503360*a^7*b^10*c 
^15*d^8*e^13*f^10*z^6 - 143503360*a^7*b^10*c^13*d^10*e^15*f^8*z^6 - 140894 
208*a^10*b^7*c^15*d^8*e^10*f^13*z^6 - 140894208*a^10*b^7*c^10*d^13*e^15*f^ 
8*z^6 - 140894208*a^8*b^9*c^16*d^7*e^11*f^12*z^6 - 140894208*a^8*b^9*c^11* 
d^12*e^16*f^7*z^6 + 114802688*a^12*b^5*c^13*d^10*e^10*f^13*z^6 + 114802688 
*a^12*b^5*c^10*d^13*e^13*f^10*z^6 - 114802688*a^10*b^7*c^13*d^10*e^12*f^11 
*z^6 - 114802688*a^10*b^7*c^12*d^11*e^13*f^10*z^6 - 114802688*a^8*b^9*c^14 
*d^9*e^13*f^10*z^6 - 114802688*a^8*b^9*c^13*d^10*e^14*f^9*z^6 + 114802688* 
a^6*b^11*c^16*d^7*e^13*f^10*z^6 + 114802688*a^6*b^11*c^13*d^10*e^16*f^7*z^ 
6 + 111992832*a^9*b^8*c^16*d^7*e^10*f^13*z^6 + 111992832*a^9*b^8*c^10*d^13 
*e^16*f^7*z^6 + 86102016*a^13*b^4*c^11*d^12*e^11*f^12*z^6 + 86102016*a^5*b 
^12*c^15*d^8*e^15*f^8*z^6 - 77529088*a^12*b^5*c^14*d^9*e^9*f^14*z^6 - 7752 
9088*a^12*b^5*c^9*d^14*e^14*f^9*z^6 - 77529088*a^6*b^11*c^17*d^6*e^12*f^11 
*z^6 - 77529088*a^6*b^11*c^12*d^11*e^17*f^6*z^6 + 77156352*a^11*b^6*c^1...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 4534, normalized size of antiderivative = 16.79 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**6*e**3*f**3 + 2*s 
qrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**6*e**2*f**4*x**2 - 6* 
sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**5*d*e**4*f**2 - 4*sq 
rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**5*d*e**3*f**3*x**2 + 2 
*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**5*d*e**2*f**4*x**4 
+ 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**4*d**2*e**5*f - 
6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**4*d**2*e**3*f**3*x 
**4 - 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**3*d**3*e**6 
+ 4*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**3*d**3*e**5*f*x* 
*2 + 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**3*d**3*e**4*f 
**2*x**4 - 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**2*d**4* 
e**6*x**2 - 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c**2*d**4 
*e**5*f*x**4 + 5*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4*c**2*d 
**3*e**3*f**3 + 5*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4*c**2* 
d**3*e**2*f**4*x**2 - sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4*c 
*d**4*e**4*f**2 + 4*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4*c*d 
**4*e**3*f**3*x**2 + 5*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4* 
c*d**4*e**2*f**4*x**4 - sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4 
*d**5*e**4*f**2*x**2 - sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**4* 
d**5*e**3*f**3*x**4 - 7*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a...