\(\int \frac {(a+b x^2)^2}{(c+d x^2)^2 (e+f x^2)^3} \, dx\) [240]

Optimal result

Integrand size = 28, antiderivative size = 409 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=-\frac {\left (6 a b c d e f-a^2 d f (2 d e+c f)-b^2 c e (d e+2 c f)\right ) x}{4 c d e (d e-c f)^2 \left (e+f x^2\right )^2}+\frac {(b c-a d)^2 x}{2 c d (d e-c f) \left (c+d x^2\right ) \left (e+f x^2\right )^2}-\frac {\left (2 a b c e f (11 d e+c f)-3 b^2 c e^2 (d e+3 c f)-a^2 f \left (4 d^2 e^2+11 c d e f-3 c^2 f^2\right )\right ) x}{8 c e^2 (d e-c f)^3 \left (e+f x^2\right )}-\frac {\sqrt {d} (b c-a d) (a d (d e-7 c f)+3 b c (d e+c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (d e-c f)^4}-\frac {\left (2 a b e f \left (15 d^2 e^2+10 c d e f-c^2 f^2\right )-3 b^2 e^2 \left (d^2 e^2+6 c d e f+c^2 f^2\right )-a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} \sqrt {f} (d e-c f)^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (-\frac {4 d (b c-a d)^2 x}{c (-d e+c f)^3 \left (c+d x^2\right )}+\frac {2 (b e-a f)^2 x}{e (d e-c f)^2 \left (e+f x^2\right )^2}+\frac {(b e-a f) (a f (-11 d e+3 c f)+b e (3 d e+5 c f)) x}{e^2 (d e-c f)^3 \left (e+f x^2\right )}+\frac {4 \sqrt {d} (-b c+a d) (a d (d e-7 c f)+3 b c (d e+c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (d e-c f)^4}+\frac {\left (2 a b e f \left (-15 d^2 e^2-10 c d e f+c^2 f^2\right )+3 b^2 e^2 \left (d^2 e^2+6 c d e f+c^2 f^2\right )+a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{5/2} \sqrt {f} (d e-c f)^4}\right ) \] Input:

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

Output:

((-4*d*(b*c - a*d)^2*x)/(c*(-(d*e) + c*f)^3*(c + d*x^2)) + (2*(b*e - a*f)^ 
2*x)/(e*(d*e - c*f)^2*(e + f*x^2)^2) + ((b*e - a*f)*(a*f*(-11*d*e + 3*c*f) 
 + b*e*(3*d*e + 5*c*f))*x)/(e^2*(d*e - c*f)^3*(e + f*x^2)) + (4*Sqrt[d]*(- 
(b*c) + a*d)*(a*d*(d*e - 7*c*f) + 3*b*c*(d*e + c*f))*ArcTan[(Sqrt[d]*x)/Sq 
rt[c]])/(c^(3/2)*(d*e - c*f)^4) + ((2*a*b*e*f*(-15*d^2*e^2 - 10*c*d*e*f + 
c^2*f^2) + 3*b^2*e^2*(d^2*e^2 + 6*c*d*e*f + c^2*f^2) + a^2*f^2*(35*d^2*e^2 
 - 14*c*d*e*f + 3*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(e^(5/2)*Sqrt[f]* 
(d*e - c*f)^4))/8
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.56, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {425, 402, 25, 402, 27, 397, 218, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}-\frac {\frac {\int \frac {d (a f (7 d e-3 c f)-b e (3 d e+c f)) x^2+b c e (5 d e-c f)-a \left (8 d^2 e^2-7 c d f e+3 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{2 e (d e-c f)}+\frac {x (a f (7 d e-3 c f)-b e (c f+3 d e))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\int \frac {2 \left (-3 d f (3 b c e-2 a d e-a c f) x^2+b c e (2 d e+c f)+a \left (2 d^2 e^2-8 c d f e+3 c^2 f^2\right )\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{4 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}-\frac {\frac {\int \frac {d (a f (7 d e-3 c f)-b e (3 d e+c f)) x^2+b c e (5 d e-c f)-a \left (8 d^2 e^2-7 c d f e+3 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{2 e (d e-c f)}+\frac {x (a f (7 d e-3 c f)-b e (c f+3 d e))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\int \frac {-3 d f (3 b c e-2 a d e-a c f) x^2+b c e (2 d e+c f)+a \left (2 d^2 e^2-8 c d f e+3 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {b \left (\frac {x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}-\frac {\frac {\frac {8 d^2 e^2 (b c-a d) \int \frac {1}{d x^2+c}dx}{d e-c f}-\frac {\left (b e \left (-c^2 f^2+6 c d e f+3 d^2 e^2\right )-a f \left (3 c^2 f^2-10 c d e f+15 d^2 e^2\right )\right ) \int \frac {1}{f x^2+e}dx}{d e-c f}}{2 e (d e-c f)}+\frac {x (a f (7 d e-3 c f)-b e (c f+3 d e))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\int \frac {-3 d f (3 b c e-2 a d e-a c f) x^2+b c e (2 d e+c f)+a \left (2 d^2 e^2-8 c d f e+3 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}-\frac {\frac {\frac {8 d^{3/2} e^2 (b c-a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2+6 c d e f+3 d^2 e^2\right )-a f \left (3 c^2 f^2-10 c d e f+15 d^2 e^2\right )\right )}{\sqrt {e} \sqrt {f} (d e-c f)}}{2 e (d e-c f)}+\frac {x (a f (7 d e-3 c f)-b e (c f+3 d e))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\int \frac {-3 d f (3 b c e-2 a d e-a c f) x^2+b c e (2 d e+c f)+a \left (2 d^2 e^2-8 c d f e+3 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}-\frac {\frac {\frac {8 d^{3/2} e^2 (b c-a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2+6 c d e f+3 d^2 e^2\right )-a f \left (3 c^2 f^2-10 c d e f+15 d^2 e^2\right )\right )}{\sqrt {e} \sqrt {f} (d e-c f)}}{2 e (d e-c f)}+\frac {x (a f (7 d e-3 c f)-b e (c f+3 d e))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\frac {\int \frac {-d f \left (b c e (11 d e+c f)-a \left (4 d^2 e^2+11 c d f e-3 c^2 f^2\right )\right ) x^2+b c e \left (4 d^2 e^2+9 c d f e-c^2 f^2\right )+a \left (4 d^3 e^3-24 c d^2 f e^2+11 c^2 d f^2 e-3 c^3 f^3\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{2 e (d e-c f)}-\frac {f x \left (b c e (c f+11 d e)-a \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {b \left (\frac {x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}-\frac {\frac {\frac {8 d^{3/2} e^2 (b c-a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2+6 c d e f+3 d^2 e^2\right )-a f \left (3 c^2 f^2-10 c d e f+15 d^2 e^2\right )\right )}{\sqrt {e} \sqrt {f} (d e-c f)}}{2 e (d e-c f)}+\frac {x (a f (7 d e-3 c f)-b e (c f+3 d e))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\frac {\frac {4 d^2 e^2 (a d (d e-7 c f)+b c (5 c f+d e)) \int \frac {1}{d x^2+c}dx}{d e-c f}-\frac {c f \left (b e \left (-c^2 f^2+10 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2-14 c d e f+35 d^2 e^2\right )\right ) \int \frac {1}{f x^2+e}dx}{d e-c f}}{2 e (d e-c f)}-\frac {f x \left (b c e (c f+11 d e)-a \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {x (b e-a f)}{4 e \left (e+f x^2\right )^2 (d e-c f)}-\frac {\frac {\frac {8 d^{3/2} e^2 (b c-a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2+6 c d e f+3 d^2 e^2\right )-a f \left (3 c^2 f^2-10 c d e f+15 d^2 e^2\right )\right )}{\sqrt {e} \sqrt {f} (d e-c f)}}{2 e (d e-c f)}+\frac {x (a f (7 d e-3 c f)-b e (c f+3 d e))}{2 e \left (e+f x^2\right ) (d e-c f)}}{4 e (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\frac {\frac {4 d^{3/2} e^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) (a d (d e-7 c f)+b c (5 c f+d e))}{\sqrt {c} (d e-c f)}-\frac {c \sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2+10 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2-14 c d e f+35 d^2 e^2\right )\right )}{\sqrt {e} (d e-c f)}}{2 e (d e-c f)}-\frac {f x \left (b c e (c f+11 d e)-a \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

Input:

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

Output:

(b*(((b*e - a*f)*x)/(4*e*(d*e - c*f)*(e + f*x^2)^2) - (((a*f*(7*d*e - 3*c* 
f) - b*e*(3*d*e + c*f))*x)/(2*e*(d*e - c*f)*(e + f*x^2)) + ((8*d^(3/2)*(b* 
c - a*d)*e^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(d*e - c*f)) - ((b*e*(3 
*d^2*e^2 + 6*c*d*e*f - c^2*f^2) - a*f*(15*d^2*e^2 - 10*c*d*e*f + 3*c^2*f^2 
))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]*(d*e - c*f)))/(2*e*(d*e - 
 c*f)))/(4*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)^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*(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]])/(Sqrt[e]*(d*e - c*f)))/(2*e*(d*e - c*f)))/(2*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 = 0.85 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {\frac {\frac {f \left (3 a^{2} c^{2} f^{4}-14 a^{2} c d e \,f^{3}+11 a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}+12 a b c d \,e^{2} f^{2}-14 a b \,d^{2} e^{3} f -5 b^{2} c^{2} e^{2} f^{2}+2 b^{2} c d \,e^{3} f +3 b^{2} d^{2} e^{4}\right ) x^{3}}{8 e^{2}}+\frac {\left (5 a^{2} c^{2} f^{4}-18 a^{2} c d e \,f^{3}+13 a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}+20 a b c d \,e^{2} f^{2}-18 a b \,d^{2} e^{3} f -3 b^{2} c^{2} e^{2} f^{2}-2 b^{2} c d \,e^{3} f +5 b^{2} d^{2} e^{4}\right ) x}{8 e}}{\left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a^{2} c^{2} f^{4}-14 a^{2} c d e \,f^{3}+35 a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}-20 a b c d \,e^{2} f^{2}-30 a b \,d^{2} e^{3} f +3 b^{2} c^{2} e^{2} f^{2}+18 b^{2} c d \,e^{3} f +3 b^{2} d^{2} e^{4}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 e^{2} \sqrt {e f}}}{\left (c f -d e \right )^{4}}-\frac {d \left (\frac {\left (a^{2} c f \,d^{2}-a^{2} d^{3} e -2 a b \,c^{2} d f +2 a b c \,d^{2} e +b^{2} c^{3} f -b^{2} c^{2} d e \right ) x}{2 c \left (x^{2} d +c \right )}+\frac {\left (7 a^{2} c f \,d^{2}-a^{2} d^{3} e -10 a b \,c^{2} d f -2 a b c \,d^{2} e +3 b^{2} c^{3} f +3 b^{2} c^{2} d e \right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{2 c \sqrt {c d}}\right )}{\left (c f -d e \right )^{4}}\)	\(540\)
risch	\(\text {Expression too large to display}\)	\(21164\)

Input:

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

Output:

1/(c*f-d*e)^4*((1/8*f*(3*a^2*c^2*f^4-14*a^2*c*d*e*f^3+11*a^2*d^2*e^2*f^2+2 
*a*b*c^2*e*f^3+12*a*b*c*d*e^2*f^2-14*a*b*d^2*e^3*f-5*b^2*c^2*e^2*f^2+2*b^2 
*c*d*e^3*f+3*b^2*d^2*e^4)/e^2*x^3+1/8*(5*a^2*c^2*f^4-18*a^2*c*d*e*f^3+13*a 
^2*d^2*e^2*f^2-2*a*b*c^2*e*f^3+20*a*b*c*d*e^2*f^2-18*a*b*d^2*e^3*f-3*b^2*c 
^2*e^2*f^2-2*b^2*c*d*e^3*f+5*b^2*d^2*e^4)/e*x)/(f*x^2+e)^2+1/8*(3*a^2*c^2* 
f^4-14*a^2*c*d*e*f^3+35*a^2*d^2*e^2*f^2+2*a*b*c^2*e*f^3-20*a*b*c*d*e^2*f^2 
-30*a*b*d^2*e^3*f+3*b^2*c^2*e^2*f^2+18*b^2*c*d*e^3*f+3*b^2*d^2*e^4)/e^2/(e 
*f)^(1/2)*arctan(f*x/(e*f)^(1/2)))-d/(c*f-d*e)^4*(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*(7 
*a^2*c*d^2*f-a^2*d^3*e-10*a*b*c^2*d*f-2*a*b*c*d^2*e+3*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. 1679 vs. \(2 (381) = 762\).

Time = 64.86 (sec) , antiderivative size = 6809, normalized size of antiderivative = 16.65 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x^2+a)^2/(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 = 0.14 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=-\frac {{\left (3 \, b^{2} c^{2} d^{2} e - 2 \, a b c d^{3} e - a^{2} d^{4} e + 3 \, b^{2} c^{3} d f - 10 \, a b c^{2} d^{2} f + 7 \, a^{2} c d^{3} f\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (c d^{4} e^{4} - 4 \, c^{2} d^{3} e^{3} f + 6 \, c^{3} d^{2} e^{2} f^{2} - 4 \, c^{4} d e f^{3} + c^{5} f^{4}\right )} \sqrt {c d}} + \frac {{\left (3 \, b^{2} d^{2} e^{4} + 18 \, b^{2} c d e^{3} f - 30 \, a b d^{2} e^{3} f + 3 \, b^{2} c^{2} e^{2} f^{2} - 20 \, a b c d e^{2} f^{2} + 35 \, a^{2} d^{2} e^{2} f^{2} + 2 \, a b c^{2} e f^{3} - 14 \, a^{2} c d e f^{3} + 3 \, a^{2} c^{2} f^{4}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \, {\left (d^{4} e^{6} - 4 \, c d^{3} e^{5} f + 6 \, c^{2} d^{2} e^{4} f^{2} - 4 \, c^{3} d e^{3} f^{3} + c^{4} e^{2} f^{4}\right )} \sqrt {e f}} + \frac {b^{2} c^{2} d x - 2 \, a b c d^{2} x + a^{2} d^{3} x}{2 \, {\left (c d^{3} e^{3} - 3 \, c^{2} d^{2} e^{2} f + 3 \, c^{3} d e f^{2} - c^{4} f^{3}\right )} {\left (d x^{2} + c\right )}} + \frac {3 \, b^{2} d e^{3} f x^{3} + 5 \, b^{2} c e^{2} f^{2} x^{3} - 14 \, a b d e^{2} f^{2} x^{3} - 2 \, a b c e f^{3} x^{3} + 11 \, a^{2} d e f^{3} x^{3} - 3 \, a^{2} c f^{4} x^{3} + 5 \, b^{2} d e^{4} x + 3 \, b^{2} c e^{3} f x - 18 \, a b d e^{3} f x + 2 \, a b c e^{2} f^{2} x + 13 \, a^{2} d e^{2} f^{2} x - 5 \, a^{2} c e f^{3} x}{8 \, {\left (d^{3} e^{5} - 3 \, c d^{2} e^{4} f + 3 \, c^{2} d e^{3} f^{2} - c^{3} e^{2} f^{3}\right )} {\left (f x^{2} + e\right )}^{2}} \] Input:

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

Output:

-1/2*(3*b^2*c^2*d^2*e - 2*a*b*c*d^3*e - a^2*d^4*e + 3*b^2*c^3*d*f - 10*a*b 
*c^2*d^2*f + 7*a^2*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*(3*b 
^2*d^2*e^4 + 18*b^2*c*d*e^3*f - 30*a*b*d^2*e^3*f + 3*b^2*c^2*e^2*f^2 - 20* 
a*b*c*d*e^2*f^2 + 35*a^2*d^2*e^2*f^2 + 2*a*b*c^2*e*f^3 - 14*a^2*c*d*e*f^3 
+ 3*a^2*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^2*c^2*d*x 
- 2*a*b*c*d^2*x + a^2*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*(3*b^2*d*e^3*f*x^3 + 5*b^2*c*e^2*f^2*x^3 - 
14*a*b*d*e^2*f^2*x^3 - 2*a*b*c*e*f^3*x^3 + 11*a^2*d*e*f^3*x^3 - 3*a^2*c*f^ 
4*x^3 + 5*b^2*d*e^4*x + 3*b^2*c*e^3*f*x - 18*a*b*d*e^3*f*x + 2*a*b*c*e^2*f 
^2*x + 13*a^2*d*e^2*f^2*x - 5*a^2*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 = 17.00 (sec) , antiderivative size = 127501, normalized size of antiderivative = 311.74 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((3584*a^2*c^2*d^12*e^12*f^3 - 20160*a^2*c^3*d^11*e^11*f^4 + 63168* 
a^2*c^4*d^10*e^10*f^5 - 125184*a^2*c^5*d^9*e^9*f^6 + 166656*a^2*c^6*d^8*e^ 
8*f^7 - 153216*a^2*c^7*d^7*e^7*f^8 + 97920*a^2*c^8*d^6*e^6*f^9 - 43008*a^2 
*c^9*d^5*e^5*f^10 + 12544*a^2*c^10*d^4*e^4*f^11 - 2240*a^2*c^11*d^3*e^3*f^ 
12 + 192*a^2*c^12*d^2*e^2*f^13 + 576*b^2*c^3*d^11*e^13*f^2 - 4416*b^2*c^4* 
d^10*e^12*f^3 + 14592*b^2*c^5*d^9*e^11*f^4 - 26880*b^2*c^6*d^8*e^10*f^5 + 
29568*b^2*c^7*d^7*e^9*f^6 - 18816*b^2*c^8*d^6*e^8*f^7 + 5376*b^2*c^9*d^5*e 
^7*f^8 + 768*b^2*c^10*d^4*e^6*f^9 - 960*b^2*c^11*d^3*e^5*f^10 + 192*b^2*c^ 
12*d^2*e^4*f^11 - 256*a^2*c*d^13*e^13*f^2 - 512*a*b*c^2*d^12*e^13*f^2 + 29 
44*a*b*c^3*d^11*e^12*f^3 - 4992*a*b*c^4*d^10*e^11*f^4 - 4608*a*b*c^5*d^9*e 
^10*f^5 + 32256*a*b*c^6*d^8*e^9*f^6 - 59136*a*b*c^7*d^7*e^8*f^7 + 59136*a* 
b*c^8*d^6*e^7*f^8 - 35328*a*b*c^9*d^5*e^6*f^9 + 12288*a*b*c^10*d^4*e^5*f^1 
0 - 2176*a*b*c^11*d^3*e^4*f^11 + 128*a*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^1 
1*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*(-(72*a^4*c^15*f^16 - ((144 
*a^4*c^15*f^16 + 256*a^4*d^15*e^15*f + 144*b^4*c^3*d^12*e^16 + 144*b^4*c^1 
5*e^4*f^12 + 352*a^2*b^2*c^15*e^2*f^14 + 48384*a^4*c^2*d^13*e^13*f^3 - 195 
440*a^4*c^3*d^12*e^12*f^4 + 397376*a^4*c^4*d^11*e^11*f^5 - 286944*a^4*c^5* 
d^10*e^10*f^6 - 504000*a^4*c^6*d^9*e^9*f^7 + 1638000*a^4*c^7*d^8*e^8*f^...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 3303, normalized size of antiderivative = 8.08 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^2/(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**2*c**2*d**2*e**5*f 
**2 - 56*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c**2*d**2*e**4 
*f**3*x**2 - 28*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c**2*d* 
*2*e**3*f**4*x**4 + 4*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c 
*d**3*e**6*f - 20*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c*d** 
3*e**5*f**2*x**2 - 52*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c 
*d**3*e**4*f**3*x**4 - 28*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a* 
*2*c*d**3*e**3*f**4*x**6 + 4*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c))) 
*a**2*d**4*e**6*f*x**2 + 8*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a 
**2*d**4*e**5*f**2*x**4 + 4*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))* 
a**2*d**4*e**4*f**3*x**6 + 40*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)) 
)*a*b*c**3*d*e**5*f**2 + 80*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))* 
a*b*c**3*d*e**4*f**3*x**2 + 40*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c) 
))*a*b*c**3*d*e**3*f**4*x**4 + 8*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt( 
c)))*a*b*c**2*d**2*e**6*f + 56*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c) 
))*a*b*c**2*d**2*e**5*f**2*x**2 + 88*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*s 
qrt(c)))*a*b*c**2*d**2*e**4*f**3*x**4 + 40*sqrt(d)*sqrt(c)*atan((d*x)/(sqr 
t(d)*sqrt(c)))*a*b*c**2*d**2*e**3*f**4*x**6 + 8*sqrt(d)*sqrt(c)*atan((d*x) 
/(sqrt(d)*sqrt(c)))*a*b*c*d**3*e**6*f*x**2 + 16*sqrt(d)*sqrt(c)*atan((d*x) 
/(sqrt(d)*sqrt(c)))*a*b*c*d**3*e**5*f**2*x**4 + 8*sqrt(d)*sqrt(c)*atan(...