\(\int \frac {a+b x^2}{(c+d x^2)^2 (e+f x^2)^3} \, dx\) [223]

Optimal result

Integrand size = 26, antiderivative size = 313 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=-\frac {f (3 b c e-2 a d e-a c f) x}{4 c e (d e-c f)^2 \left (e+f x^2\right )^2}-\frac {(b c-a d) x}{2 c (d e-c f) \left (c+d x^2\right ) \left (e+f x^2\right )^2}-\frac {f \left (b c e (11 d e+c f)-a \left (4 d^2 e^2+11 c d e f-3 c^2 f^2\right )\right ) x}{8 c e^2 (d e-c f)^3 \left (e+f x^2\right )}+\frac {d^{3/2} (a d (d e-7 c f)+b c (d e+5 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (d e-c f)^4}-\frac {\sqrt {f} \left (b e \left (15 d^2 e^2+10 c d e f-c^2 f^2\right )-a f \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} (d e-c f)^4} \] Output:

-1/4*f*(-a*c*f-2*a*d*e+3*b*c*e)*x/c/e/(-c*f+d*e)^2/(f*x^2+e)^2-1/2*(-a*d+b 
*c)*x/c/(-c*f+d*e)/(d*x^2+c)/(f*x^2+e)^2-1/8*f*(b*c*e*(c*f+11*d*e)-a*(-3*c 
^2*f^2+11*c*d*e*f+4*d^2*e^2))*x/c/e^2/(-c*f+d*e)^3/(f*x^2+e)+1/2*d^(3/2)*( 
a*d*(-7*c*f+d*e)+b*c*(5*c*f+d*e))*arctan(d^(1/2)*x/c^(1/2))/c^(3/2)/(-c*f+ 
d*e)^4-1/8*f^(1/2)*(b*e*(-c^2*f^2+10*c*d*e*f+15*d^2*e^2)-a*f*(3*c^2*f^2-14 
*c*d*e*f+35*d^2*e^2))*arctan(f^(1/2)*x/e^(1/2))/e^(5/2)/(-c*f+d*e)^4
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.86 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (\frac {4 d^2 (b c-a d) x}{c (-d e+c f)^3 \left (c+d x^2\right )}+\frac {2 f (-b e+a f) x}{e (d e-c f)^2 \left (e+f x^2\right )^2}-\frac {f (b e (7 d e+c f)+a f (-11 d e+3 c f)) x}{e^2 (d e-c f)^3 \left (e+f x^2\right )}+\frac {4 d^{3/2} (a d (d e-7 c f)+b c (d e+5 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (d e-c f)^4}+\frac {\sqrt {f} \left (b e \left (-15 d^2 e^2-10 c d e f+c^2 f^2\right )+a f \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{5/2} (d e-c f)^4}\right ) \] Input:

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

Output:

((4*d^2*(b*c - a*d)*x)/(c*(-(d*e) + c*f)^3*(c + d*x^2)) + (2*f*(-(b*e) + a 
*f)*x)/(e*(d*e - c*f)^2*(e + f*x^2)^2) - (f*(b*e*(7*d*e + c*f) + a*f*(-11* 
d*e + 3*c*f))*x)/(e^2*(d*e - c*f)^3*(e + f*x^2)) + (4*d^(3/2)*(a*d*(d*e - 
7*c*f) + b*c*(d*e + 5*c*f))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/2)*(d*e - c 
*f)^4) + (Sqrt[f]*(b*e*(-15*d^2*e^2 - 10*c*d*e*f + c^2*f^2) + a*f*(35*d^2* 
e^2 - 14*c*d*e*f + 3*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(e^(5/2)*(d*e 
- c*f)^4))/8
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {402, 25, 402, 27, 402, 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.

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\frac {\int \frac {-d f \left (b c e (11 d e+c f)-a \left (4 d^2 e^2+11 c d f e-3 c^2 f^2\right )\right ) x^2+b c e \left (4 d^2 e^2+9 c d f e-c^2 f^2\right )+a \left (4 d^3 e^3-24 c d^2 f e^2+11 c^2 d f^2 e-3 c^3 f^3\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{2 e (d e-c f)}-\frac {f x \left (b c e (c f+11 d e)-a \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {\frac {\frac {4 d^2 e^2 (a d (d e-7 c f)+b c (5 c f+d e)) \int \frac {1}{d x^2+c}dx}{d e-c f}-\frac {c f \left (b e \left (-c^2 f^2+10 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2-14 c d e f+35 d^2 e^2\right )\right ) \int \frac {1}{f x^2+e}dx}{d e-c f}}{2 e (d e-c f)}-\frac {f x \left (b c e (c f+11 d e)-a \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\frac {4 d^{3/2} e^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) (a d (d e-7 c f)+b c (5 c f+d e))}{\sqrt {c} (d e-c f)}-\frac {c \sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2+10 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2-14 c d e f+35 d^2 e^2\right )\right )}{\sqrt {e} (d e-c f)}}{2 e (d e-c f)}-\frac {f x \left (b c e (c f+11 d e)-a \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\)

Input:

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

Output:

-1/2*((b*c - a*d)*x)/(c*(d*e - c*f)*(c + d*x^2)*(e + f*x^2)^2) + (-1/2*(f* 
(3*b*c*e - 2*a*d*e - a*c*f)*x)/(e*(d*e - c*f)*(e + f*x^2)^2) + (-1/2*(f*(b 
*c*e*(11*d*e + c*f) - a*(4*d^2*e^2 + 11*c*d*e*f - 3*c^2*f^2))*x)/(e*(d*e - 
 c*f)*(e + f*x^2)) + ((4*d^(3/2)*e^2*(a*d*(d*e - 7*c*f) + b*c*(d*e + 5*c*f 
))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(d*e - c*f)) - (c*Sqrt[f]*(b*e*(1 
5*d^2*e^2 + 10*c*d*e*f - c^2*f^2) - a*f*(35*d^2*e^2 - 14*c*d*e*f + 3*c^2*f 
^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(d*e - c*f)))/(2*e*(d*e - c*f)) 
)/(2*e*(d*e - c*f)))/(2*c*(d*e - c*f))
 

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]
 

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {f \left (\frac {\frac {f \left (3 a \,c^{2} f^{3}-14 a c d e \,f^{2}+11 a \,d^{2} e^{2} f +b \,c^{2} e \,f^{2}+6 b c d \,e^{2} f -7 e^{3} b \,d^{2}\right ) x^{3}}{8 e^{2}}+\frac {\left (5 a \,c^{2} f^{3}-18 a c d e \,f^{2}+13 a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}+10 b c d \,e^{2} f -9 e^{3} b \,d^{2}\right ) x}{8 e}}{\left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a \,c^{2} f^{3}-14 a c d e \,f^{2}+35 a \,d^{2} e^{2} f +b \,c^{2} e \,f^{2}-10 b c d \,e^{2} f -15 e^{3} b \,d^{2}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 e^{2} \sqrt {e f}}\right )}{\left (c f -d e \right )^{4}}-\frac {d^{2} \left (\frac {\left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) x}{2 c \left (x^{2} d +c \right )}+\frac {\left (7 a c d f -a \,d^{2} e -5 b \,c^{2} f -b c d e \right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{2 c \sqrt {c d}}\right )}{\left (c f -d e \right )^{4}}\)	\(331\)
risch	\(\text {Expression too large to display}\)	\(21547\)

Input:

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

Output:

f/(c*f-d*e)^4*((1/8*f*(3*a*c^2*f^3-14*a*c*d*e*f^2+11*a*d^2*e^2*f+b*c^2*e*f 
^2+6*b*c*d*e^2*f-7*b*d^2*e^3)/e^2*x^3+1/8*(5*a*c^2*f^3-18*a*c*d*e*f^2+13*a 
*d^2*e^2*f-b*c^2*e*f^2+10*b*c*d*e^2*f-9*b*d^2*e^3)/e*x)/(f*x^2+e)^2+1/8*(3 
*a*c^2*f^3-14*a*c*d*e*f^2+35*a*d^2*e^2*f+b*c^2*e*f^2-10*b*c*d*e^2*f-15*b*d 
^2*e^3)/e^2/(e*f)^(1/2)*arctan(f*x/(e*f)^(1/2)))-d^2/(c*f-d*e)^4*(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*(7*a*c*d*f-a*d^2*e-5*b*c^2 
*f-b*c*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. 1166 vs. \(2 (287) = 574\).

Time = 17.64 (sec) , antiderivative size = 4763, normalized size of antiderivative = 15.22 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x^2+a)/(d*x^2+c)^2/(f*x^2+e)^3,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 = 443, normalized size of antiderivative = 1.42 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\frac {{\left (b c d^{3} e + a d^{4} e + 5 \, b c^{2} d^{2} f - 7 \, a c d^{3} f\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (c d^{4} e^{4} - 4 \, c^{2} d^{3} e^{3} f + 6 \, c^{3} d^{2} e^{2} f^{2} - 4 \, c^{4} d e f^{3} + c^{5} f^{4}\right )} \sqrt {c d}} - \frac {{\left (15 \, b d^{2} e^{3} f + 10 \, b c d e^{2} f^{2} - 35 \, a d^{2} e^{2} f^{2} - b c^{2} e f^{3} + 14 \, a c d e f^{3} - 3 \, a c^{2} f^{4}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \, {\left (d^{4} e^{6} - 4 \, c d^{3} e^{5} f + 6 \, c^{2} d^{2} e^{4} f^{2} - 4 \, c^{3} d e^{3} f^{3} + c^{4} e^{2} f^{4}\right )} \sqrt {e f}} - \frac {b c d^{2} x - a d^{3} x}{2 \, {\left (c d^{3} e^{3} - 3 \, c^{2} d^{2} e^{2} f + 3 \, c^{3} d e f^{2} - c^{4} f^{3}\right )} {\left (d x^{2} + c\right )}} - \frac {7 \, b d e^{2} f^{2} x^{3} + b c e f^{3} x^{3} - 11 \, a d e f^{3} x^{3} + 3 \, a c f^{4} x^{3} + 9 \, b d e^{3} f x - b c e^{2} f^{2} x - 13 \, a d e^{2} f^{2} x + 5 \, a c e f^{3} x}{8 \, {\left (d^{3} e^{5} - 3 \, c d^{2} e^{4} f + 3 \, c^{2} d e^{3} f^{2} - c^{3} e^{2} f^{3}\right )} {\left (f x^{2} + e\right )}^{2}} \] Input:

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

Output:

1/2*(b*c*d^3*e + a*d^4*e + 5*b*c^2*d^2*f - 7*a*c*d^3*f)*arctan(d*x/sqrt(c* 
d))/((c*d^4*e^4 - 4*c^2*d^3*e^3*f + 6*c^3*d^2*e^2*f^2 - 4*c^4*d*e*f^3 + c^ 
5*f^4)*sqrt(c*d)) - 1/8*(15*b*d^2*e^3*f + 10*b*c*d*e^2*f^2 - 35*a*d^2*e^2* 
f^2 - b*c^2*e*f^3 + 14*a*c*d*e*f^3 - 3*a*c^2*f^4)*arctan(f*x/sqrt(e*f))/(( 
d^4*e^6 - 4*c*d^3*e^5*f + 6*c^2*d^2*e^4*f^2 - 4*c^3*d*e^3*f^3 + c^4*e^2*f^ 
4)*sqrt(e*f)) - 1/2*(b*c*d^2*x - a*d^3*x)/((c*d^3*e^3 - 3*c^2*d^2*e^2*f + 
3*c^3*d*e*f^2 - c^4*f^3)*(d*x^2 + c)) - 1/8*(7*b*d*e^2*f^2*x^3 + b*c*e*f^3 
*x^3 - 11*a*d*e*f^3*x^3 + 3*a*c*f^4*x^3 + 9*b*d*e^3*f*x - b*c*e^2*f^2*x - 
13*a*d*e^2*f^2*x + 5*a*c*e*f^3*x)/((d^3*e^5 - 3*c*d^2*e^4*f + 3*c^2*d*e^3* 
f^2 - c^3*e^2*f^3)*(f*x^2 + e)^2)
 

Mupad [B] (verification not implemented)

Time = 8.39 (sec) , antiderivative size = 17499, normalized size of antiderivative = 55.91 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((x^5*(3*a*c^2*d*f^4 - 4*a*d^3*e^2*f^2 - 11*a*c*d^2*e*f^3 + b*c^2*d*e*f^3 
+ 11*b*c*d^2*e^2*f^2))/(8*c*e^2*(c^3*f^3 - d^3*e^3 + 3*c*d^2*e^2*f - 3*c^2 
*d*e*f^2)) + (x*(5*a*c^3*f^3 - 4*a*d^3*e^3 + 4*b*c*d^2*e^3 - b*c^3*e*f^2 - 
 13*a*c^2*d*e*f^2 + 9*b*c^2*d*e^2*f))/(8*c*e*(c*f - d*e)*(c^2*f^2 + d^2*e^ 
2 - 2*c*d*e*f)) + (f*x^3*(3*a*c^3*f^3 - 8*a*d^3*e^3 + 17*b*c*d^2*e^3 + b*c 
^3*e*f^2 - 13*a*c*d^2*e^2*f - 6*a*c^2*d*e*f^2 + 6*b*c^2*d*e^2*f))/(8*c*e^2 
*(c*f - d*e)*(c^2*f^2 + d^2*e^2 - 2*c*d*e*f)))/(c*e^2 + x^2*(d*e^2 + 2*c*e 
*f) + x^4*(c*f^2 + 2*d*e*f) + d*f^2*x^6) - atan(((((256*a*c*d^13*e^13*f^2 
- 3584*a*c^2*d^12*e^12*f^3 + 20160*a*c^3*d^11*e^11*f^4 - 63168*a*c^4*d^10* 
e^10*f^5 + 125184*a*c^5*d^9*e^9*f^6 - 166656*a*c^6*d^8*e^8*f^7 + 153216*a* 
c^7*d^7*e^7*f^8 - 97920*a*c^8*d^6*e^6*f^9 + 43008*a*c^9*d^5*e^5*f^10 - 125 
44*a*c^10*d^4*e^4*f^11 + 2240*a*c^11*d^3*e^3*f^12 - 192*a*c^12*d^2*e^2*f^1 
3 + 256*b*c^2*d^12*e^13*f^2 - 1472*b*c^3*d^11*e^12*f^3 + 2496*b*c^4*d^10*e 
^11*f^4 + 2304*b*c^5*d^9*e^10*f^5 - 16128*b*c^6*d^8*e^9*f^6 + 29568*b*c^7* 
d^7*e^8*f^7 - 29568*b*c^8*d^6*e^7*f^8 + 17664*b*c^9*d^5*e^6*f^9 - 6144*b*c 
^10*d^4*e^5*f^10 + 1088*b*c^11*d^3*e^4*f^11 - 64*b*c^12*d^2*e^3*f^12)/(128 
*(c^2*d^9*e^13 - c^11*e^4*f^9 - 9*c^3*d^8*e^12*f + 9*c^10*d*e^5*f^8 + 36*c 
^4*d^7*e^11*f^2 - 84*c^5*d^6*e^10*f^3 + 126*c^6*d^5*e^9*f^4 - 126*c^7*d^4* 
e^8*f^5 + 84*c^8*d^3*e^7*f^6 - 36*c^9*d^2*e^6*f^7)) - (x*(-(9*a^2*c^4*f^7 
+ 225*b^2*d^4*e^6*f + 1225*a^2*d^4*e^4*f^3 + b^2*c^4*e^2*f^5 + 6*a*b*c^...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 2169, normalized size of antiderivative = 6.93 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 28*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*c**2*d**2*e**5*f - 
56*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*c**2*d**2*e**4*f**2*x** 
2 - 28*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*c**2*d**2*e**3*f**3 
*x**4 + 4*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*c*d**3*e**6 - 20 
*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*c*d**3*e**5*f*x**2 - 52*s 
qrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*c*d**3*e**4*f**2*x**4 - 28* 
sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*c*d**3*e**3*f**3*x**6 + 4* 
sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*d**4*e**6*x**2 + 8*sqrt(d) 
*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*d**4*e**5*f*x**4 + 4*sqrt(d)*sqrt 
(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*d**4*e**4*f**2*x**6 + 20*sqrt(d)*sqrt( 
c)*atan((d*x)/(sqrt(d)*sqrt(c)))*b*c**3*d*e**5*f + 40*sqrt(d)*sqrt(c)*atan 
((d*x)/(sqrt(d)*sqrt(c)))*b*c**3*d*e**4*f**2*x**2 + 20*sqrt(d)*sqrt(c)*ata 
n((d*x)/(sqrt(d)*sqrt(c)))*b*c**3*d*e**3*f**3*x**4 + 4*sqrt(d)*sqrt(c)*ata 
n((d*x)/(sqrt(d)*sqrt(c)))*b*c**2*d**2*e**6 + 28*sqrt(d)*sqrt(c)*atan((d*x 
)/(sqrt(d)*sqrt(c)))*b*c**2*d**2*e**5*f*x**2 + 44*sqrt(d)*sqrt(c)*atan((d* 
x)/(sqrt(d)*sqrt(c)))*b*c**2*d**2*e**4*f**2*x**4 + 20*sqrt(d)*sqrt(c)*atan 
((d*x)/(sqrt(d)*sqrt(c)))*b*c**2*d**2*e**3*f**3*x**6 + 4*sqrt(d)*sqrt(c)*a 
tan((d*x)/(sqrt(d)*sqrt(c)))*b*c*d**3*e**6*x**2 + 8*sqrt(d)*sqrt(c)*atan(( 
d*x)/(sqrt(d)*sqrt(c)))*b*c*d**3*e**5*f*x**4 + 4*sqrt(d)*sqrt(c)*atan((d*x 
)/(sqrt(d)*sqrt(c)))*b*c*d**3*e**4*f**2*x**6 + 3*sqrt(f)*sqrt(e)*atan((...