\(\int \frac {(a+b x^2)^3}{(c+d x^2) (e+f x^2)^3} \, dx\) [249]

Optimal result

Integrand size = 28, antiderivative size = 286 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\frac {(b e-a f)^3 x}{4 e f^2 (d e-c f) \left (e+f x^2\right )^2}-\frac {(b e-a f)^2 (b e (5 d e-9 c f)+a f (7 d e-3 c f)) x}{8 e^2 f^2 (d e-c f)^2 \left (e+f x^2\right )}-\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} (d e-c f)^3}+\frac {(b e-a f) \left (2 a b e f \left (3 d^2 e^2-14 c d e f+3 c^2 f^2\right )+a^2 f^2 \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )+b^2 e^2 \left (3 d^2 e^2-10 c d e f+15 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} f^{5/2} (d e-c f)^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (\frac {2 (b e-a f)^3 x}{e f^2 (d e-c f) \left (e+f x^2\right )^2}-\frac {(b e-a f)^2 (b e (5 d e-9 c f)+a f (7 d e-3 c f)) x}{e^2 f^2 (d e-c f)^2 \left (e+f x^2\right )}+\frac {8 (b c-a d)^3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} (-d e+c f)^3}+\frac {(b e-a f) \left (2 a b e f \left (3 d^2 e^2-14 c d e f+3 c^2 f^2\right )+a^2 f^2 \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )+b^2 e^2 \left (3 d^2 e^2-10 c d e f+15 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{5/2} f^{5/2} (d e-c f)^3}\right ) \] Input:

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

Output:

((2*(b*e - a*f)^3*x)/(e*f^2*(d*e - c*f)*(e + f*x^2)^2) - ((b*e - a*f)^2*(b 
*e*(5*d*e - 9*c*f) + a*f*(7*d*e - 3*c*f))*x)/(e^2*f^2*(d*e - c*f)^2*(e + f 
*x^2)) + (8*(b*c - a*d)^3*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*Sqrt[d]*(- 
(d*e) + c*f)^3) + ((b*e - a*f)*(2*a*b*e*f*(3*d^2*e^2 - 14*c*d*e*f + 3*c^2* 
f^2) + a^2*f^2*(15*d^2*e^2 - 10*c*d*e*f + 3*c^2*f^2) + b^2*e^2*(3*d^2*e^2 
- 10*c*d*e*f + 15*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(e^(5/2)*f^(5/2)* 
(d*e - c*f)^3))/8
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.52, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {419, 25, 401, 27, 401, 25, 299, 218, 420, 299, 218, 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.

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

\(\Big \downarrow \) 419

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {x \left (a+b x^2\right )^2 (b e-a f) (d e-c f)}{4 e \left (e+f x^2\right )^2}-\frac {\frac {\frac {b x \left (-a^2 f^2 (7 d e-3 c f)+4 a b e f (c f+3 d e)+3 b^2 e^2 (d e-5 c f)\right )}{f}-\frac {(b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a^2 f^2 (7 d e-3 c f)+2 a b e f (7 d e-3 c f)+3 b^2 e^2 (d e-5 c f)\right )}{\sqrt {e} f^{3/2}}}{2 e f}-\frac {x \left (a+b x^2\right ) (b e-a f) (a f (7 d e-3 c f)+b e (d e-5 c f))}{2 e f \left (e+f x^2\right )}}{4 e}}{(d e-c f)^2}-\frac {d (b c-a d) \left (\frac {b \left (\frac {b x}{f}-\frac {(b e-a f) \int \frac {1}{f x^2+e}dx}{f}\right )}{d}-\frac {(b c-a d) \int \frac {b x^2+a}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {x \left (a+b x^2\right )^2 (b e-a f) (d e-c f)}{4 e \left (e+f x^2\right )^2}-\frac {\frac {\frac {b x \left (-a^2 f^2 (7 d e-3 c f)+4 a b e f (c f+3 d e)+3 b^2 e^2 (d e-5 c f)\right )}{f}-\frac {(b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a^2 f^2 (7 d e-3 c f)+2 a b e f (7 d e-3 c f)+3 b^2 e^2 (d e-5 c f)\right )}{\sqrt {e} f^{3/2}}}{2 e f}-\frac {x \left (a+b x^2\right ) (b e-a f) (a f (7 d e-3 c f)+b e (d e-5 c f))}{2 e f \left (e+f x^2\right )}}{4 e}}{(d e-c f)^2}-\frac {d (b c-a d) \left (\frac {b \left (\frac {b x}{f}-\frac {(b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}\right )}{d}-\frac {(b c-a d) \left (\frac {(b e-a f) \int \frac {1}{f x^2+e}dx}{d e-c f}-\frac {(b c-a d) \int \frac {1}{d x^2+c}dx}{d e-c f}\right )}{d}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 218

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

Input:

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

Output:

-((d*(b*c - a*d)*((b*((b*x)/f - ((b*e - a*f)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/ 
(Sqrt[e]*f^(3/2))))/d - ((b*c - a*d)*(-(((b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sq 
rt[c]])/(Sqrt[c]*Sqrt[d]*(d*e - c*f))) + ((b*e - a*f)*ArcTan[(Sqrt[f]*x)/S 
qrt[e]])/(Sqrt[e]*Sqrt[f]*(d*e - c*f))))/d))/(d*e - c*f)^2) + (((b*e - a*f 
)*(d*e - c*f)*x*(a + b*x^2)^2)/(4*e*(e + f*x^2)^2) - (-1/2*((b*e - a*f)*(b 
*e*(d*e - 5*c*f) + a*f*(7*d*e - 3*c*f))*x*(a + b*x^2))/(e*f*(e + f*x^2)) + 
 ((b*(3*b^2*e^2*(d*e - 5*c*f) - a^2*f^2*(7*d*e - 3*c*f) + 4*a*b*e*f*(3*d*e 
 + c*f))*x)/f - ((b*e - a*f)*(3*b^2*e^2*(d*e - 5*c*f) + 2*a*b*e*f*(7*d*e - 
 3*c*f) + a^2*f^2*(7*d*e - 3*c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f 
^(3/2)))/(2*e*f))/(4*e))/(d*e - c*f)^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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

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 401

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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 419

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

rule 420

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(598\) vs. \(2(264)=528\).

Time = 0.90 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.09

method	result	size
default	\(\frac {\frac {\frac {\left (3 a^{3} c^{2} f^{5}-10 a^{3} c d e \,f^{4}+7 a^{3} d^{2} e^{2} f^{3}+3 a^{2} b \,c^{2} e \,f^{4}+6 a^{2} b c d \,e^{2} f^{3}-9 a^{2} b \,d^{2} e^{3} f^{2}-15 a \,b^{2} c^{2} e^{2} f^{3}+18 a \,b^{2} c d \,e^{3} f^{2}-3 a \,b^{2} d^{2} e^{4} f +9 b^{3} c^{2} e^{3} f^{2}-14 b^{3} c d \,e^{4} f +5 b^{3} d^{2} e^{5}\right ) x^{3}}{8 e^{2} f}+\frac {\left (5 a^{3} c^{2} f^{5}-14 a^{3} c d e \,f^{4}+9 a^{3} d^{2} e^{2} f^{3}-3 a^{2} b \,c^{2} e \,f^{4}+18 a^{2} b c d \,e^{2} f^{3}-15 a^{2} b \,d^{2} e^{3} f^{2}-9 a \,b^{2} c^{2} e^{2} f^{3}+6 a \,b^{2} c d \,e^{3} f^{2}+3 a \,b^{2} d^{2} e^{4} f +7 b^{3} c^{2} e^{3} f^{2}-10 b^{3} c d \,e^{4} f +3 b^{3} d^{2} e^{5}\right ) x}{8 e \,f^{2}}}{\left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a^{3} c^{2} f^{5}-10 a^{3} c d e \,f^{4}+15 a^{3} d^{2} e^{2} f^{3}+3 a^{2} b \,c^{2} e \,f^{4}-18 a^{2} b c d \,e^{2} f^{3}-9 a^{2} b \,d^{2} e^{3} f^{2}+9 a \,b^{2} c^{2} e^{2} f^{3}+18 a \,b^{2} c d \,e^{3} f^{2}-3 a \,b^{2} d^{2} e^{4} f -15 b^{3} c^{2} e^{3} f^{2}+10 b^{3} c d \,e^{4} f -3 b^{3} d^{2} e^{5}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 e^{2} f^{2} \sqrt {e f}}}{\left (c f -d e \right )^{3}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{\left (c f -d e \right )^{3} \sqrt {c d}}\)	\(599\)
risch	\(\text {Expression too large to display}\)	\(1645\)

Input:

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

Output:

1/(c*f-d*e)^3*((1/8*(3*a^3*c^2*f^5-10*a^3*c*d*e*f^4+7*a^3*d^2*e^2*f^3+3*a^ 
2*b*c^2*e*f^4+6*a^2*b*c*d*e^2*f^3-9*a^2*b*d^2*e^3*f^2-15*a*b^2*c^2*e^2*f^3 
+18*a*b^2*c*d*e^3*f^2-3*a*b^2*d^2*e^4*f+9*b^3*c^2*e^3*f^2-14*b^3*c*d*e^4*f 
+5*b^3*d^2*e^5)/e^2/f*x^3+1/8*(5*a^3*c^2*f^5-14*a^3*c*d*e*f^4+9*a^3*d^2*e^ 
2*f^3-3*a^2*b*c^2*e*f^4+18*a^2*b*c*d*e^2*f^3-15*a^2*b*d^2*e^3*f^2-9*a*b^2* 
c^2*e^2*f^3+6*a*b^2*c*d*e^3*f^2+3*a*b^2*d^2*e^4*f+7*b^3*c^2*e^3*f^2-10*b^3 
*c*d*e^4*f+3*b^3*d^2*e^5)/e/f^2*x)/(f*x^2+e)^2+1/8*(3*a^3*c^2*f^5-10*a^3*c 
*d*e*f^4+15*a^3*d^2*e^2*f^3+3*a^2*b*c^2*e*f^4-18*a^2*b*c*d*e^2*f^3-9*a^2*b 
*d^2*e^3*f^2+9*a*b^2*c^2*e^2*f^3+18*a*b^2*c*d*e^3*f^2-3*a*b^2*d^2*e^4*f-15 
*b^3*c^2*e^3*f^2+10*b^3*c*d*e^4*f-3*b^3*d^2*e^5)/e^2/f^2/(e*f)^(1/2)*arcta 
n(f*x/(e*f)^(1/2)))+(-a^3*d^3+3*a^2*b*c*d^2-3*a*b^2*c^2*d+b^3*c^3)/(c*f-d* 
e)^3/(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. 1191 vs. \(2 (264) = 528\).

Time = 112.85 (sec) , antiderivative size = 4843, normalized size of antiderivative = 16.93 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x^2+a)^3/(d*x^2+c)/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (264) = 528\).

Time = 0.13 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=-\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3}\right )} \sqrt {c d}} + \frac {{\left (3 \, b^{3} d^{2} e^{5} - 10 \, b^{3} c d e^{4} f + 3 \, a b^{2} d^{2} e^{4} f + 15 \, b^{3} c^{2} e^{3} f^{2} - 18 \, a b^{2} c d e^{3} f^{2} + 9 \, a^{2} b d^{2} e^{3} f^{2} - 9 \, a b^{2} c^{2} e^{2} f^{3} + 18 \, a^{2} b c d e^{2} f^{3} - 15 \, a^{3} d^{2} e^{2} f^{3} - 3 \, a^{2} b c^{2} e f^{4} + 10 \, a^{3} c d e f^{4} - 3 \, a^{3} c^{2} f^{5}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \, {\left (d^{3} e^{5} f^{2} - 3 \, c d^{2} e^{4} f^{3} + 3 \, c^{2} d e^{3} f^{4} - c^{3} e^{2} f^{5}\right )} \sqrt {e f}} - \frac {5 \, b^{3} d e^{4} f x^{3} - 9 \, b^{3} c e^{3} f^{2} x^{3} - 3 \, a b^{2} d e^{3} f^{2} x^{3} + 15 \, a b^{2} c e^{2} f^{3} x^{3} - 9 \, a^{2} b d e^{2} f^{3} x^{3} - 3 \, a^{2} b c e f^{4} x^{3} + 7 \, a^{3} d e f^{4} x^{3} - 3 \, a^{3} c f^{5} x^{3} + 3 \, b^{3} d e^{5} x - 7 \, b^{3} c e^{4} f x + 3 \, a b^{2} d e^{4} f x + 9 \, a b^{2} c e^{3} f^{2} x - 15 \, a^{2} b d e^{3} f^{2} x + 3 \, a^{2} b c e^{2} f^{3} x + 9 \, a^{3} d e^{2} f^{3} x - 5 \, a^{3} c e f^{4} x}{8 \, {\left (d^{2} e^{4} f^{2} - 2 \, c d e^{3} f^{3} + c^{2} e^{2} f^{4}\right )} {\left (f x^{2} + e\right )}^{2}} \] Input:

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

Output:

-(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(d*x/sqrt(c*d)) 
/((d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*sqrt(c*d)) + 1/8*(3* 
b^3*d^2*e^5 - 10*b^3*c*d*e^4*f + 3*a*b^2*d^2*e^4*f + 15*b^3*c^2*e^3*f^2 - 
18*a*b^2*c*d*e^3*f^2 + 9*a^2*b*d^2*e^3*f^2 - 9*a*b^2*c^2*e^2*f^3 + 18*a^2* 
b*c*d*e^2*f^3 - 15*a^3*d^2*e^2*f^3 - 3*a^2*b*c^2*e*f^4 + 10*a^3*c*d*e*f^4 
- 3*a^3*c^2*f^5)*arctan(f*x/sqrt(e*f))/((d^3*e^5*f^2 - 3*c*d^2*e^4*f^3 + 3 
*c^2*d*e^3*f^4 - c^3*e^2*f^5)*sqrt(e*f)) - 1/8*(5*b^3*d*e^4*f*x^3 - 9*b^3* 
c*e^3*f^2*x^3 - 3*a*b^2*d*e^3*f^2*x^3 + 15*a*b^2*c*e^2*f^3*x^3 - 9*a^2*b*d 
*e^2*f^3*x^3 - 3*a^2*b*c*e*f^4*x^3 + 7*a^3*d*e*f^4*x^3 - 3*a^3*c*f^5*x^3 + 
 3*b^3*d*e^5*x - 7*b^3*c*e^4*f*x + 3*a*b^2*d*e^4*f*x + 9*a*b^2*c*e^3*f^2*x 
 - 15*a^2*b*d*e^3*f^2*x + 3*a^2*b*c*e^2*f^3*x + 9*a^3*d*e^2*f^3*x - 5*a^3* 
c*e*f^4*x)/((d^2*e^4*f^2 - 2*c*d*e^3*f^3 + c^2*e^2*f^4)*(f*x^2 + e)^2)
 

Mupad [B] (verification not implemented)

Time = 7.07 (sec) , antiderivative size = 18014, normalized size of antiderivative = 62.99 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((x^3*(3*a^3*c*f^4 - 5*b^3*d*e^4 - 7*a^3*d*e*f^3 + 9*b^3*c*e^3*f + 3*a^2*b 
*c*e*f^3 + 3*a*b^2*d*e^3*f - 15*a*b^2*c*e^2*f^2 + 9*a^2*b*d*e^2*f^2))/(8*e 
^2*(c^2*f^3 + d^2*e^2*f - 2*c*d*e*f^2)) - (x*(3*b^3*d*e^4 - 5*a^3*c*f^4 + 
9*a^3*d*e*f^3 - 7*b^3*c*e^3*f + 3*a^2*b*c*e*f^3 + 3*a*b^2*d*e^3*f + 9*a*b^ 
2*c*e^2*f^2 - 15*a^2*b*d*e^2*f^2))/(8*e*f*(c^2*f^3 + d^2*e^2*f - 2*c*d*e*f 
^2)))/(e^2 + f^2*x^4 + 2*e*f*x^2) + (atan((((-c*d)^(1/2)*((x*(9*b^6*d^7*e^ 
10 + 9*a^6*c^4*d^3*f^10 + 289*a^6*d^7*e^4*f^6 + 63*a^2*b^4*d^7*e^8*f^2 - 3 
6*a^3*b^3*d^7*e^7*f^3 - 9*a^4*b^2*d^7*e^6*f^4 + 190*a^6*c^2*d^5*e^2*f^8 + 
190*b^6*c^2*d^5*e^8*f^2 - 300*b^6*c^3*d^4*e^7*f^3 + 225*b^6*c^4*d^3*e^6*f^ 
4 + 18*a*b^5*d^7*e^9*f - 60*b^6*c*d^6*e^9*f - 270*a^5*b*d^7*e^5*f^5 - 300* 
a^6*c*d^6*e^3*f^7 - 60*a^6*c^3*d^4*e*f^9 + 64*b^6*c^6*d*e^4*f^6 - 168*a*b^ 
5*c*d^6*e^8*f^2 - 744*a^5*b*c*d^6*e^4*f^6 + 18*a^5*b*c^4*d^3*e*f^9 + 396*a 
*b^5*c^2*d^5*e^7*f^3 - 360*a*b^5*c^3*d^4*e^6*f^4 - 270*a*b^5*c^4*d^3*e^5*f 
^5 - 384*a*b^5*c^5*d^2*e^4*f^6 - 180*a^2*b^4*c*d^6*e^7*f^3 + 144*a^3*b^3*c 
*d^6*e^6*f^4 + 924*a^4*b^2*c*d^6*e^5*f^5 + 396*a^5*b*c^2*d^5*e^3*f^7 - 168 
*a^5*b*c^3*d^4*e^2*f^8 + 162*a^2*b^4*c^2*d^5*e^6*f^4 + 924*a^2*b^4*c^3*d^4 
*e^5*f^5 + 951*a^2*b^4*c^4*d^3*e^4*f^6 - 1496*a^3*b^3*c^2*d^5*e^5*f^5 - 11 
36*a^3*b^3*c^3*d^4*e^4*f^6 - 36*a^3*b^3*c^4*d^3*e^3*f^7 + 1122*a^4*b^2*c^2 
*d^5*e^4*f^6 - 180*a^4*b^2*c^3*d^4*e^3*f^7 + 63*a^4*b^2*c^4*d^3*e^2*f^8))/ 
(32*(c^4*e^4*f^7 + d^4*e^8*f^3 - 4*c*d^3*e^7*f^4 - 4*c^3*d*e^5*f^6 + 6*...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 2204, normalized size of antiderivative = 7.71 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 8*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*d**3*e**5*f**3 - 
16*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*d**3*e**4*f**4*x**2 
- 8*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*d**3*e**3*f**5*x**4 
 + 24*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c*d**2*e**5*f** 
3 + 48*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c*d**2*e**4*f* 
*4*x**2 + 24*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c*d**2*e 
**3*f**5*x**4 - 24*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b**2*c* 
*2*d*e**5*f**3 - 48*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b**2*c 
**2*d*e**4*f**4*x**2 - 24*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a* 
b**2*c**2*d*e**3*f**5*x**4 + 8*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c) 
))*b**3*c**3*e**5*f**3 + 16*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))* 
b**3*c**3*e**4*f**4*x**2 + 8*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c))) 
*b**3*c**3*e**3*f**5*x**4 + 3*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)) 
)*a**3*c**3*d*e**2*f**5 + 6*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))* 
a**3*c**3*d*e*f**6*x**2 + 3*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))* 
a**3*c**3*d*f**7*x**4 - 10*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a 
**3*c**2*d**2*e**3*f**4 - 20*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e))) 
*a**3*c**2*d**2*e**2*f**5*x**2 - 10*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sq 
rt(e)))*a**3*c**2*d**2*e*f**6*x**4 + 15*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f 
)*sqrt(e)))*a**3*c*d**3*e**4*f**3 + 30*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt...