\(\int \frac {(a+b x^2)^2}{(c+d x^2)^2 (e+f x^2)^2} \, dx\) [239]

Optimal result

Integrand size = 28, antiderivative size = 258 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=-\frac {\left (4 a b c d e f-b^2 c e (d e+c f)-a^2 d f (d e+c f)\right ) x}{2 c d e (d e-c f)^2 \left (e+f x^2\right )}+\frac {(b c-a d)^2 x}{2 c d (d e-c f) \left (c+d x^2\right ) \left (e+f x^2\right )}-\frac {(b c-a d) (a d (d e-5 c f)+b c (3 d e+c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d} (d e-c f)^3}-\frac {(b e-a f) (a f (5 d e-c f)-b e (d e+3 c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} \sqrt {f} (d e-c f)^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {(b c-a d)^2 x}{c (d e-c f)^2 \left (c+d x^2\right )}+\frac {(b e-a f)^2 x}{e (d e-c f)^2 \left (e+f x^2\right )}+\frac {(b c-a d) (a d (d e-5 c f)+b c (3 d e+c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} \sqrt {d} (-d e+c f)^3}+\frac {(b e-a f) (a f (-5 d e+c f)+b e (d e+3 c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{3/2} \sqrt {f} (d e-c f)^3}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.60, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {425, 402, 25, 397, 218, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(b*(((b*e - a*f)*x)/(2*e*(d*e - c*f)*(e + f*x^2)) - ((2*Sqrt[d]*(b*c - a*d 
)*e*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(d*e - c*f)) + ((a*f*(3*d*e - c* 
f) - b*e*(d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]*(d*e - 
 c*f)))/(2*e*(d*e - c*f))))/d - ((b*c - a*d)*(-1/2*((b*c - a*d)*x)/(c*(d*e 
 - c*f)*(c + d*x^2)*(e + f*x^2)) + (-((f*(2*b*c*e - a*d*e - a*c*f)*x)/(e*( 
d*e - c*f)*(e + f*x^2))) + ((Sqrt[d]*e*(a*d*(d*e - 5*c*f) + b*c*(d*e + 3*c 
*f))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(d*e - c*f)) + (c*Sqrt[f]*(a*f* 
(5*d*e - c*f) - b*e*(3*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*( 
d*e - c*f)))/(e*(d*e - c*f)))/(2*c*(d*e - c*f))))/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 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 425

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ 
)^2)^(r_), x_Symbol] :> Simp[d/b   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 
 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b   Int[(a + b*x^2)^p*(c + d*x 
^2)^(q - 1)*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && ILt 
Q[p, 0] && GtQ[q, 0]
 

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.22

method	result	size
default	\(\frac {\frac {\left (a^{2} c \,f^{3}-a^{2} d e \,f^{2}-2 a b c e \,f^{2}+2 a b d \,e^{2} f +b^{2} c \,e^{2} f -b^{2} d \,e^{3}\right ) x}{2 e \left (f \,x^{2}+e \right )}+\frac {\left (a^{2} c \,f^{3}-5 a^{2} d e \,f^{2}+2 a b c e \,f^{2}+6 a b d \,e^{2} f -3 b^{2} c \,e^{2} f -b^{2} d \,e^{3}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 e \sqrt {e f}}}{\left (c f -d e \right )^{3}}+\frac {\frac {\left (a^{2} c f \,d^{2}-a^{2} d^{3} e -2 a b \,c^{2} d f +2 a b c \,d^{2} e +b^{2} c^{3} f -b^{2} c^{2} d e \right ) x}{2 c \left (x^{2} d +c \right )}+\frac {\left (5 a^{2} c f \,d^{2}-a^{2} d^{3} e -6 a b \,c^{2} d f -2 a b c \,d^{2} e +b^{2} c^{3} f +3 b^{2} c^{2} d e \right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{2 c \sqrt {c d}}}{\left (c f -d e \right )^{3}}\)	\(316\)
risch	\(\text {Expression too large to display}\)	\(1151\)

Input:

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

Output:

1/(c*f-d*e)^3*(1/2*(a^2*c*f^3-a^2*d*e*f^2-2*a*b*c*e*f^2+2*a*b*d*e^2*f+b^2* 
c*e^2*f-b^2*d*e^3)/e*x/(f*x^2+e)+1/2*(a^2*c*f^3-5*a^2*d*e*f^2+2*a*b*c*e*f^ 
2+6*a*b*d*e^2*f-3*b^2*c*e^2*f-b^2*d*e^3)/e/(e*f)^(1/2)*arctan(f*x/(e*f)^(1 
/2)))+1/(c*f-d*e)^3*(1/2*(a^2*c*d^2*f-a^2*d^3*e-2*a*b*c^2*d*f+2*a*b*c*d^2* 
e+b^2*c^3*f-b^2*c^2*d*e)/c*x/(d*x^2+c)+1/2*(5*a^2*c*d^2*f-a^2*d^3*e-6*a*b* 
c^2*d*f-2*a*b*c*d^2*e+b^2*c^3*f+3*b^2*c^2*d*e)/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. 945 vs. \(2 (234) = 468\).

Time = 19.59 (sec) , antiderivative size = 3861, normalized size of antiderivative = 14.97 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x^2+a)^2/(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 [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.53 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=-\frac {{\left (3 \, b^{2} c^{2} d e - 2 \, a b c d^{2} e - a^{2} d^{3} e + b^{2} c^{3} f - 6 \, a b c^{2} d f + 5 \, a^{2} c d^{2} f\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (c d^{3} e^{3} - 3 \, c^{2} d^{2} e^{2} f + 3 \, c^{3} d e f^{2} - c^{4} f^{3}\right )} \sqrt {c d}} + \frac {{\left (b^{2} d e^{3} + 3 \, b^{2} c e^{2} f - 6 \, a b d e^{2} f - 2 \, a b c e f^{2} + 5 \, a^{2} d e f^{2} - a^{2} c f^{3}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \, {\left (d^{3} e^{4} - 3 \, c d^{2} e^{3} f + 3 \, c^{2} d e^{2} f^{2} - c^{3} e f^{3}\right )} \sqrt {e f}} + \frac {b^{2} c d e^{2} x^{3} + b^{2} c^{2} e f x^{3} - 4 \, a b c d e f x^{3} + a^{2} d^{2} e f x^{3} + a^{2} c d f^{2} x^{3} + 2 \, b^{2} c^{2} e^{2} x - 2 \, a b c d e^{2} x + a^{2} d^{2} e^{2} x - 2 \, a b c^{2} e f x + a^{2} c^{2} f^{2} x}{2 \, {\left (c d^{2} e^{3} - 2 \, c^{2} d e^{2} f + c^{3} e f^{2}\right )} {\left (d f x^{4} + d e x^{2} + c f x^{2} + c e\right )}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 6.44 (sec) , antiderivative size = 12581, normalized size of antiderivative = 48.76 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((x^3*(a^2*c*d*f^2 + b^2*c*d*e^2 + a^2*d^2*e*f + b^2*c^2*e*f - 4*a*b*c*d*e 
*f))/(2*c*e*(c^2*f^2 + d^2*e^2 - 2*c*d*e*f)) + (x*(a^2*c^2*f^2 + a^2*d^2*e 
^2 + 2*b^2*c^2*e^2 - 2*a*b*c*d*e^2 - 2*a*b*c^2*e*f))/(2*c*e*(c^2*f^2 + d^2 
*e^2 - 2*c*d*e*f)))/(c*e + x^2*(c*f + d*e) + d*f*x^4) + (atan((((-e^3*f)^( 
1/2)*(a*f - b*e)*((x*(a^4*c^4*d^3*f^7 + a^4*d^7*e^4*f^3 + 50*a^4*c^2*d^5*e 
^2*f^5 + 6*b^4*c^3*d^4*e^5*f^2 + 18*b^4*c^4*d^3*e^4*f^3 + 6*b^4*c^5*d^2*e^ 
3*f^4 - 10*a^4*c*d^6*e^3*f^4 - 10*a^4*c^3*d^4*e*f^6 + b^4*c^2*d^5*e^6*f + 
b^4*c^6*d*e^2*f^5 + 4*a^3*b*c*d^6*e^4*f^3 + 4*a^3*b*c^4*d^3*e*f^6 - 12*a*b 
^3*c^2*d^5*e^5*f^2 - 52*a*b^3*c^3*d^4*e^4*f^3 - 52*a*b^3*c^4*d^3*e^3*f^4 - 
 12*a*b^3*c^5*d^2*e^2*f^5 - 68*a^3*b*c^2*d^5*e^3*f^4 - 68*a^3*b*c^3*d^4*e^ 
2*f^5 + 44*a^2*b^2*c^2*d^5*e^4*f^3 + 104*a^2*b^2*c^3*d^4*e^3*f^4 + 44*a^2* 
b^2*c^4*d^3*e^2*f^5))/(2*(c^2*d^4*e^6 + c^6*e^2*f^4 - 4*c^3*d^3*e^5*f - 4* 
c^5*d*e^3*f^3 + 6*c^4*d^2*e^4*f^2)) - (((20*a^2*c^2*d^9*e^8*f^3 - 80*a^2*c 
^3*d^8*e^7*f^4 + 172*a^2*c^4*d^7*e^6*f^5 - 220*a^2*c^5*d^6*e^5*f^6 + 172*a 
^2*c^6*d^5*e^4*f^7 - 80*a^2*c^7*d^4*e^3*f^8 + 20*a^2*c^8*d^3*e^2*f^9 + 4*b 
^2*c^3*d^8*e^9*f^2 - 24*b^2*c^4*d^7*e^8*f^3 + 60*b^2*c^5*d^6*e^7*f^4 - 80* 
b^2*c^6*d^5*e^6*f^5 + 60*b^2*c^7*d^4*e^5*f^6 - 24*b^2*c^8*d^3*e^4*f^7 + 4* 
b^2*c^9*d^2*e^3*f^8 - 2*a^2*c*d^10*e^9*f^2 - 2*a^2*c^9*d^2*e*f^10 - 4*a*b* 
c^2*d^9*e^9*f^2 + 20*a*b*c^3*d^8*e^8*f^3 - 36*a*b*c^4*d^7*e^7*f^4 + 20*a*b 
*c^5*d^6*e^6*f^5 + 20*a*b*c^6*d^5*e^5*f^6 - 36*a*b*c^7*d^4*e^4*f^7 + 20...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1759, normalized size of antiderivative = 6.82 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(5*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c**2*d**2*e**3*f**2 
+ 5*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c**2*d**2*e**2*f**3 
*x**2 - sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c*d**3*e**4*f + 
 4*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c*d**3*e**3*f**2*x** 
2 + 5*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c*d**3*e**2*f**3* 
x**4 - sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*d**4*e**4*f*x**2 
 - sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*d**4*e**3*f**2*x**4 
- 6*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b*c**3*d*e**3*f**2 - 6 
*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b*c**3*d*e**2*f**3*x**2 - 
 2*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b*c**2*d**2*e**4*f - 8* 
sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b*c**2*d**2*e**3*f**2*x**2 
 - 6*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b*c**2*d**2*e**2*f**3 
*x**4 - 2*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b*c*d**3*e**4*f* 
x**2 - 2*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b*c*d**3*e**3*f** 
2*x**4 + sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**2*c**4*e**3*f**2 
 + sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**2*c**4*e**2*f**3*x**2 
+ 3*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**2*c**3*d*e**4*f + 4*s 
qrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**2*c**3*d*e**3*f**2*x**2 + 
sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**2*c**3*d*e**2*f**3*x**4 + 
 3*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b**2*c**2*d**2*e**4*f*...