\(\int \frac {(a+b x^2)^2}{(c+d x^2)^3 (e+f x^2)^3} \, dx\) [241]

Optimal result

Integrand size = 28, antiderivative size = 588 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=-\frac {f \left (b^2 c^2 e (7 d e+5 c f)-2 a b c d e (d e+11 c f)-a^2 d \left (3 d^2 e^2-13 c d e f-2 c^2 f^2\right )\right ) x}{8 c^2 d e (d e-c f)^3 \left (e+f x^2\right )^2}+\frac {(b c-a d)^2 x}{4 c d (d e-c f) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2}-\frac {(b c-a d) (a d (3 d e-11 c f)+b c (5 d e+3 c f)) x}{8 c^2 d (d e-c f)^2 \left (c+d x^2\right ) \left (e+f x^2\right )^2}-\frac {f \left (12 b^2 c^2 e^2 (d e+c f)-2 a b c e \left (d^2 e^2+22 c d e f+c^2 f^2\right )-3 a^2 \left (d^3 e^3-5 c d^2 e^2 f-5 c^2 d e f^2+c^3 f^3\right )\right ) x}{8 c^2 e^2 (d e-c f)^4 \left (e+f x^2\right )}+\frac {\sqrt {d} \left (2 a b c d \left (d^2 e^2-14 c d e f-35 c^2 f^2\right )+3 b^2 c^2 \left (d^2 e^2+10 c d e f+5 c^2 f^2\right )+3 a^2 d^2 \left (d^2 e^2-6 c d e f+21 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} (d e-c f)^5}+\frac {\sqrt {f} \left (2 a b e f \left (35 d^2 e^2+14 c d e f-c^2 f^2\right )-3 a^2 f^2 \left (21 d^2 e^2-6 c d e f+c^2 f^2\right )-3 b^2 e^2 \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} (d e-c f)^5} \] Output:

-1/8*f*(b^2*c^2*e*(5*c*f+7*d*e)-2*a*b*c*d*e*(11*c*f+d*e)-a^2*d*(-2*c^2*f^2 
-13*c*d*e*f+3*d^2*e^2))*x/c^2/d/e/(-c*f+d*e)^3/(f*x^2+e)^2+1/4*(-a*d+b*c)^ 
2*x/c/d/(-c*f+d*e)/(d*x^2+c)^2/(f*x^2+e)^2-1/8*(-a*d+b*c)*(a*d*(-11*c*f+3* 
d*e)+b*c*(3*c*f+5*d*e))*x/c^2/d/(-c*f+d*e)^2/(d*x^2+c)/(f*x^2+e)^2-1/8*f*( 
12*b^2*c^2*e^2*(c*f+d*e)-2*a*b*c*e*(c^2*f^2+22*c*d*e*f+d^2*e^2)-3*a^2*(c^3 
*f^3-5*c^2*d*e*f^2-5*c*d^2*e^2*f+d^3*e^3))*x/c^2/e^2/(-c*f+d*e)^4/(f*x^2+e 
)+1/8*d^(1/2)*(2*a*b*c*d*(-35*c^2*f^2-14*c*d*e*f+d^2*e^2)+3*b^2*c^2*(5*c^2 
*f^2+10*c*d*e*f+d^2*e^2)+3*a^2*d^2*(21*c^2*f^2-6*c*d*e*f+d^2*e^2))*arctan( 
d^(1/2)*x/c^(1/2))/c^(5/2)/(-c*f+d*e)^5+1/8*f^(1/2)*(2*a*b*e*f*(-c^2*f^2+1 
4*c*d*e*f+35*d^2*e^2)-3*a^2*f^2*(c^2*f^2-6*c*d*e*f+21*d^2*e^2)-3*b^2*e^2*( 
c^2*f^2+10*c*d*e*f+5*d^2*e^2))*arctan(f^(1/2)*x/e^(1/2))/e^(5/2)/(-c*f+d*e 
)^5
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (-\frac {2 d (b c-a d)^2 x}{c (-d e+c f)^3 \left (c+d x^2\right )^2}+\frac {d (-b c+a d) (3 a d (d e-5 c f)+b c (5 d e+7 c f)) x}{c^2 (d e-c f)^4 \left (c+d x^2\right )}-\frac {2 f (b e-a f)^2 x}{e (d e-c f)^3 \left (e+f x^2\right )^2}+\frac {f (-b e+a f) (3 a f (-5 d e+c f)+b e (7 d e+5 c f)) x}{e^2 (d e-c f)^4 \left (e+f x^2\right )}-\frac {\sqrt {d} \left (2 a b c d \left (d^2 e^2-14 c d e f-35 c^2 f^2\right )+3 b^2 c^2 \left (d^2 e^2+10 c d e f+5 c^2 f^2\right )+3 a^2 d^2 \left (d^2 e^2-6 c d e f+21 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} (-d e+c f)^5}-\frac {\sqrt {f} \left (2 a b e f \left (-35 d^2 e^2-14 c d e f+c^2 f^2\right )+3 a^2 f^2 \left (21 d^2 e^2-6 c d e f+c^2 f^2\right )+3 b^2 e^2 \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{5/2} (d e-c f)^5}\right ) \] Input:

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

Output:

((-2*d*(b*c - a*d)^2*x)/(c*(-(d*e) + c*f)^3*(c + d*x^2)^2) + (d*(-(b*c) + 
a*d)*(3*a*d*(d*e - 5*c*f) + b*c*(5*d*e + 7*c*f))*x)/(c^2*(d*e - c*f)^4*(c 
+ d*x^2)) - (2*f*(b*e - a*f)^2*x)/(e*(d*e - c*f)^3*(e + f*x^2)^2) + (f*(-( 
b*e) + a*f)*(3*a*f*(-5*d*e + c*f) + b*e*(7*d*e + 5*c*f))*x)/(e^2*(d*e - c* 
f)^4*(e + f*x^2)) - (Sqrt[d]*(2*a*b*c*d*(d^2*e^2 - 14*c*d*e*f - 35*c^2*f^2 
) + 3*b^2*c^2*(d^2*e^2 + 10*c*d*e*f + 5*c^2*f^2) + 3*a^2*d^2*(d^2*e^2 - 6* 
c*d*e*f + 21*c^2*f^2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(5/2)*(-(d*e) + c*f 
)^5) - (Sqrt[f]*(2*a*b*e*f*(-35*d^2*e^2 - 14*c*d*e*f + c^2*f^2) + 3*a^2*f^ 
2*(21*d^2*e^2 - 6*c*d*e*f + c^2*f^2) + 3*b^2*e^2*(5*d^2*e^2 + 10*c*d*e*f + 
 c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(e^(5/2)*(d*e - c*f)^5))/8
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 882, normalized size of antiderivative = 1.50, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {425, 402, 25, 402, 25, 27, 402, 27, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {\frac {\int \frac {2 \left (-3 d f (3 b c e-2 a d e-a c f) x^2+b c e (2 d e+c f)+a \left (2 d^2 e^2-8 c d f e+3 c^2 f^2\right )\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{4 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {x (a d (3 d e-11 c f)+b c (7 c f+d e))}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}-\frac {\int -\frac {5 f (a d (3 d e-11 c f)+b c (d e+7 c f)) x^2+b c e (d e-9 c f)+a \left (3 d^2 e^2-3 c d f e+8 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{2 c (d e-c f)}}{4 c (d e-c f)}-\frac {x (b c-a d)}{4 c \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {\frac {\int \frac {2 \left (-3 d f (3 b c e-2 a d e-a c f) x^2+b c e (2 d e+c f)+a \left (2 d^2 e^2-8 c d f e+3 c^2 f^2\right )\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{4 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\int \frac {5 f (a d (3 d e-11 c f)+b c (d e+7 c f)) x^2+b c e (d e-9 c f)+a \left (3 d^2 e^2-3 c d f e+8 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{2 c (d e-c f)}+\frac {x (a d (3 d e-11 c f)+b c (7 c f+d e))}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}}{4 c (d e-c f)}-\frac {x (b c-a d)}{4 c \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {\frac {\int \frac {-3 d f (3 b c e-2 a d e-a c f) x^2+b c e (2 d e+c f)+a \left (2 d^2 e^2-8 c d f e+3 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\int \frac {5 f (a d (3 d e-11 c f)+b c (d e+7 c f)) x^2+b c e (d e-9 c f)+a \left (3 d^2 e^2-3 c d f e+8 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{2 c (d e-c f)}+\frac {x (a d (3 d e-11 c f)+b c (7 c f+d e))}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}}{4 c (d e-c f)}-\frac {x (b c-a d)}{4 c \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {\frac {\frac {\int \frac {-d f \left (b c e (11 d e+c f)-a \left (4 d^2 e^2+11 c d f e-3 c^2 f^2\right )\right ) x^2+b c e \left (4 d^2 e^2+9 c d f e-c^2 f^2\right )+a \left (4 d^3 e^3-24 c d^2 f e^2+11 c^2 d f^2 e-3 c^3 f^3\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{2 e (d e-c f)}-\frac {f x \left (b c e (c f+11 d e)-a \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\frac {\int \frac {4 \left (3 d f \left (b c e (d e+11 c f)+a \left (3 d^2 e^2-13 c d f e-2 c^2 f^2\right )\right ) x^2+b c e \left (d^2 e^2-11 c d f e-2 c^2 f^2\right )+3 a \left (d^3 e^3-3 c d^2 f e^2+8 c^2 d f^2 e-2 c^3 f^3\right )\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{4 e (d e-c f)}+\frac {f x \left (a \left (-2 c^2 f^2-13 c d e f+3 d^2 e^2\right )+b c e (11 c f+d e)\right )}{e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}+\frac {x (a d (3 d e-11 c f)+b c (7 c f+d e))}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}}{4 c (d e-c f)}-\frac {x (b c-a d)}{4 c \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {\frac {\frac {\int \frac {-d f \left (b c e (11 d e+c f)-a \left (4 d^2 e^2+11 c d f e-3 c^2 f^2\right )\right ) x^2+b c e \left (4 d^2 e^2+9 c d f e-c^2 f^2\right )+a \left (4 d^3 e^3-24 c d^2 f e^2+11 c^2 d f^2 e-3 c^3 f^3\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{2 e (d e-c f)}-\frac {f x \left (b c e (c f+11 d e)-a \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\frac {\int \frac {3 d f \left (b c e (d e+11 c f)+a \left (3 d^2 e^2-13 c d f e-2 c^2 f^2\right )\right ) x^2+b c e \left (d^2 e^2-11 c d f e-2 c^2 f^2\right )+3 a \left (d^3 e^3-3 c d^2 f e^2+8 c^2 d f^2 e-2 c^3 f^3\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{e (d e-c f)}+\frac {f x \left (a \left (-2 c^2 f^2-13 c d e f+3 d^2 e^2\right )+b c e (11 c f+d e)\right )}{e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}+\frac {x (a d (3 d e-11 c f)+b c (7 c f+d e))}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}}{4 c (d e-c f)}-\frac {x (b c-a d)}{4 c \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {b \left (\frac {\frac {\frac {\frac {4 d^2 e^2 (a d (d e-7 c f)+b c (5 c f+d e)) \int \frac {1}{d x^2+c}dx}{d e-c f}-\frac {c f \left (b e \left (-c^2 f^2+10 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2-14 c d e f+35 d^2 e^2\right )\right ) \int \frac {1}{f x^2+e}dx}{d e-c f}}{2 e (d e-c f)}-\frac {f x \left (b c e (c f+11 d e)-a \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\frac {\int \frac {3 d f \left (b c e (d e+11 c f)+a \left (3 d^2 e^2-13 c d f e-2 c^2 f^2\right )\right ) x^2+b c e \left (d^2 e^2-11 c d f e-2 c^2 f^2\right )+3 a \left (d^3 e^3-3 c d^2 f e^2+8 c^2 d f^2 e-2 c^3 f^3\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{e (d e-c f)}+\frac {f x \left (a \left (-2 c^2 f^2-13 c d e f+3 d^2 e^2\right )+b c e (11 c f+d e)\right )}{e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}+\frac {x (a d (3 d e-11 c f)+b c (7 c f+d e))}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}}{4 c (d e-c f)}-\frac {x (b c-a d)}{4 c \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {\frac {\frac {\frac {4 d^{3/2} e^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) (a d (d e-7 c f)+b c (5 c f+d e))}{\sqrt {c} (d e-c f)}-\frac {c \sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2+10 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2-14 c d e f+35 d^2 e^2\right )\right )}{\sqrt {e} (d e-c f)}}{2 e (d e-c f)}-\frac {f x \left (b c e (c f+11 d e)-a \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (-a c f-2 a d e+3 b c e)}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}-\frac {x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\frac {\int \frac {3 d f \left (b c e (d e+11 c f)+a \left (3 d^2 e^2-13 c d f e-2 c^2 f^2\right )\right ) x^2+b c e \left (d^2 e^2-11 c d f e-2 c^2 f^2\right )+3 a \left (d^3 e^3-3 c d^2 f e^2+8 c^2 d f^2 e-2 c^3 f^3\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{e (d e-c f)}+\frac {f x \left (a \left (-2 c^2 f^2-13 c d e f+3 d^2 e^2\right )+b c e (11 c f+d e)\right )}{e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}+\frac {x (a d (3 d e-11 c f)+b c (7 c f+d e))}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}}{4 c (d e-c f)}-\frac {x (b c-a d)}{4 c \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 (d e-c f)}\right )}{d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {\frac {\frac {\frac {4 d^{3/2} e^2 (a d (d e-7 c f)+b c (d e+5 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {c \sqrt {f} \left (b e \left (15 d^2 e^2+10 c d f e-c^2 f^2\right )-a f \left (35 d^2 e^2-14 c d f e+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (d e-c f)}}{2 e (d e-c f)}-\frac {f \left (b c e (11 d e+c f)-a \left (4 d^2 e^2+11 c d f e-3 c^2 f^2\right )\right ) x}{2 e (d e-c f) \left (f x^2+e\right )}}{2 e (d e-c f)}-\frac {f (3 b c e-2 a d e-a c f) x}{2 e (d e-c f) \left (f x^2+e\right )^2}}{2 c (d e-c f)}-\frac {(b c-a d) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {(a d (3 d e-11 c f)+b c (d e+7 c f)) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}+\frac {\frac {f \left (b c e (d e+11 c f)+a \left (3 d^2 e^2-13 c d f e-2 c^2 f^2\right )\right ) x}{e (d e-c f) \left (f x^2+e\right )^2}+\frac {\frac {f \left (b c e \left (d^2 e^2+22 c d f e+c^2 f^2\right )+3 a \left (d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+c^3 f^3\right )\right ) x}{e (d e-c f) \left (f x^2+e\right )}+\frac {\int \frac {2 \left (d f \left (b c e \left (d^2 e^2+22 c d f e+c^2 f^2\right )+3 a \left (d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+c^3 f^3\right )\right ) x^2+b c e \left (d^3 e^3-13 c d^2 f e^2-13 c^2 d f^2 e+c^3 f^3\right )+3 a \left (d^4 e^4-5 c d^3 f e^3+16 c^2 d^2 f^2 e^2-5 c^3 d f^3 e+c^4 f^4\right )\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{2 e (d e-c f)}}{e (d e-c f)}}{2 c (d e-c f)}}{4 c (d e-c f)}-\frac {(b c-a d) x}{4 c (d e-c f) \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {\frac {\frac {\frac {4 d^{3/2} e^2 (a d (d e-7 c f)+b c (d e+5 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {c \sqrt {f} \left (b e \left (15 d^2 e^2+10 c d f e-c^2 f^2\right )-a f \left (35 d^2 e^2-14 c d f e+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (d e-c f)}}{2 e (d e-c f)}-\frac {f \left (b c e (11 d e+c f)-a \left (4 d^2 e^2+11 c d f e-3 c^2 f^2\right )\right ) x}{2 e (d e-c f) \left (f x^2+e\right )}}{2 e (d e-c f)}-\frac {f (3 b c e-2 a d e-a c f) x}{2 e (d e-c f) \left (f x^2+e\right )^2}}{2 c (d e-c f)}-\frac {(b c-a d) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {(a d (3 d e-11 c f)+b c (d e+7 c f)) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}+\frac {\frac {f \left (b c e (d e+11 c f)+a \left (3 d^2 e^2-13 c d f e-2 c^2 f^2\right )\right ) x}{e (d e-c f) \left (f x^2+e\right )^2}+\frac {\frac {f \left (b c e \left (d^2 e^2+22 c d f e+c^2 f^2\right )+3 a \left (d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+c^3 f^3\right )\right ) x}{e (d e-c f) \left (f x^2+e\right )}+\frac {\int \frac {d f \left (b c e \left (d^2 e^2+22 c d f e+c^2 f^2\right )+3 a \left (d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+c^3 f^3\right )\right ) x^2+b c e \left (d^3 e^3-13 c d^2 f e^2-13 c^2 d f^2 e+c^3 f^3\right )+3 a \left (d^4 e^4-5 c d^3 f e^3+16 c^2 d^2 f^2 e^2-5 c^3 d f^3 e+c^4 f^4\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{e (d e-c f)}}{e (d e-c f)}}{2 c (d e-c f)}}{4 c (d e-c f)}-\frac {(b c-a d) x}{4 c (d e-c f) \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}\right )}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {b \left (\frac {\frac {\frac {\frac {4 d^{3/2} e^2 (a d (d e-7 c f)+b c (d e+5 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {c \sqrt {f} \left (b e \left (15 d^2 e^2+10 c d f e-c^2 f^2\right )-a f \left (35 d^2 e^2-14 c d f e+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (d e-c f)}}{2 e (d e-c f)}-\frac {f \left (b c e (11 d e+c f)-a \left (4 d^2 e^2+11 c d f e-3 c^2 f^2\right )\right ) x}{2 e (d e-c f) \left (f x^2+e\right )}}{2 e (d e-c f)}-\frac {f (3 b c e-2 a d e-a c f) x}{2 e (d e-c f) \left (f x^2+e\right )^2}}{2 c (d e-c f)}-\frac {(b c-a d) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {(a d (3 d e-11 c f)+b c (d e+7 c f)) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}+\frac {\frac {f \left (b c e (d e+11 c f)+a \left (3 d^2 e^2-13 c d f e-2 c^2 f^2\right )\right ) x}{e (d e-c f) \left (f x^2+e\right )^2}+\frac {\frac {f \left (b c e \left (d^2 e^2+22 c d f e+c^2 f^2\right )+3 a \left (d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+c^3 f^3\right )\right ) x}{e (d e-c f) \left (f x^2+e\right )}+\frac {\frac {d^2 \left (b c \left (d^2 e^2-14 c d f e-35 c^2 f^2\right )+3 a d \left (d^2 e^2-6 c d f e+21 c^2 f^2\right )\right ) \int \frac {1}{d x^2+c}dx e^2}{d e-c f}+\frac {c^2 f^2 \left (b e \left (35 d^2 e^2+14 c d f e-c^2 f^2\right )-3 a f \left (21 d^2 e^2-6 c d f e+c^2 f^2\right )\right ) \int \frac {1}{f x^2+e}dx}{d e-c f}}{e (d e-c f)}}{e (d e-c f)}}{2 c (d e-c f)}}{4 c (d e-c f)}-\frac {(b c-a d) x}{4 c (d e-c f) \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}\right )}{d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {\frac {\frac {\frac {4 d^{3/2} e^2 (a d (d e-7 c f)+b c (d e+5 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {c \sqrt {f} \left (b e \left (15 d^2 e^2+10 c d f e-c^2 f^2\right )-a f \left (35 d^2 e^2-14 c d f e+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (d e-c f)}}{2 e (d e-c f)}-\frac {f \left (b c e (11 d e+c f)-a \left (4 d^2 e^2+11 c d f e-3 c^2 f^2\right )\right ) x}{2 e (d e-c f) \left (f x^2+e\right )}}{2 e (d e-c f)}-\frac {f (3 b c e-2 a d e-a c f) x}{2 e (d e-c f) \left (f x^2+e\right )^2}}{2 c (d e-c f)}-\frac {(b c-a d) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {(a d (3 d e-11 c f)+b c (d e+7 c f)) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}+\frac {\frac {f \left (b c e (d e+11 c f)+a \left (3 d^2 e^2-13 c d f e-2 c^2 f^2\right )\right ) x}{e (d e-c f) \left (f x^2+e\right )^2}+\frac {\frac {f \left (b c e \left (d^2 e^2+22 c d f e+c^2 f^2\right )+3 a \left (d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+c^3 f^3\right )\right ) x}{e (d e-c f) \left (f x^2+e\right )}+\frac {\frac {f^{3/2} \left (b e \left (35 d^2 e^2+14 c d f e-c^2 f^2\right )-3 a f \left (21 d^2 e^2-6 c d f e+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) c^2}{\sqrt {e} (d e-c f)}+\frac {d^{3/2} e^2 \left (b c \left (d^2 e^2-14 c d f e-35 c^2 f^2\right )+3 a d \left (d^2 e^2-6 c d f e+21 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{(d e-c f) \sqrt {c}}}{e (d e-c f)}}{e (d e-c f)}}{2 c (d e-c f)}}{4 c (d e-c f)}-\frac {(b c-a d) x}{4 c (d e-c f) \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}\right )}{d}\)

Input:

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

Output:

(b*(-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 - ((b*c - a*d)*(-1/4*((b*c 
- a*d)*x)/(c*(d*e - c*f)*(c + d*x^2)^2*(e + f*x^2)^2) + (((a*d*(3*d*e - 11 
*c*f) + b*c*(d*e + 7*c*f))*x)/(2*c*(d*e - c*f)*(c + d*x^2)*(e + f*x^2)^2) 
+ ((f*(b*c*e*(d*e + 11*c*f) + a*(3*d^2*e^2 - 13*c*d*e*f - 2*c^2*f^2))*x)/( 
e*(d*e - c*f)*(e + f*x^2)^2) + ((f*(b*c*e*(d^2*e^2 + 22*c*d*e*f + c^2*f^2) 
 + 3*a*(d^3*e^3 - 5*c*d^2*e^2*f - 5*c^2*d*e*f^2 + c^3*f^3))*x)/(e*(d*e - c 
*f)*(e + f*x^2)) + ((d^(3/2)*e^2*(b*c*(d^2*e^2 - 14*c*d*e*f - 35*c^2*f^2) 
+ 3*a*d*(d^2*e^2 - 6*c*d*e*f + 21*c^2*f^2))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/( 
Sqrt[c]*(d*e - c*f)) + (c^2*f^(3/2)*(b*e*(35*d^2*e^2 + 14*c*d*e*f - c^2*f^ 
2) - 3*a*f*(21*d^2*e^2 - 6*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]] 
)/(Sqrt[e]*(d*e - c*f)))/(e*(d*e - c*f)))/(e*(d*e - c*f)))/(2*c*(d*e - c*f 
)))/(4*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 = 0.93 (sec) , antiderivative size = 761, normalized size of antiderivative = 1.29

method	result	size
default	\(\frac {f \left (\frac {\frac {f \left (3 a^{2} c^{2} f^{4}-18 a^{2} c d e \,f^{3}+15 a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}+20 a b c d \,e^{2} f^{2}-22 a b \,d^{2} e^{3} f -5 b^{2} c^{2} e^{2} f^{2}-2 b^{2} c d \,e^{3} f +7 b^{2} d^{2} e^{4}\right ) x^{3}}{8 e^{2}}+\frac {\left (5 a^{2} c^{2} f^{4}-22 a^{2} c d e \,f^{3}+17 a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}+28 a b c d \,e^{2} f^{2}-26 a b \,d^{2} e^{3} f -3 b^{2} c^{2} e^{2} f^{2}-6 b^{2} c d \,e^{3} f +9 b^{2} d^{2} e^{4}\right ) x}{8 e}}{\left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a^{2} c^{2} f^{4}-18 a^{2} c d e \,f^{3}+63 a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}-28 a b c d \,e^{2} f^{2}-70 a b \,d^{2} e^{3} f +3 b^{2} c^{2} e^{2} f^{2}+30 b^{2} c d \,e^{3} f +15 b^{2} d^{2} e^{4}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 e^{2} \sqrt {e f}}\right )}{\left (c f -d e \right )^{5}}-\frac {d \left (\frac {\frac {d \left (15 a^{2} c^{2} f^{2} d^{2}-18 a^{2} c e f \,d^{3}+3 a^{2} e^{2} d^{4}-22 a b \,c^{3} d \,f^{2}+20 a b \,c^{2} d^{2} e f +2 a b c \,d^{3} e^{2}+7 b^{2} c^{4} f^{2}-2 b^{2} c^{3} d e f -5 b^{2} c^{2} d^{2} e^{2}\right ) x^{3}}{8 c^{2}}+\frac {\left (17 a^{2} c^{2} f^{2} d^{2}-22 a^{2} c e f \,d^{3}+5 a^{2} e^{2} d^{4}-26 a b \,c^{3} d \,f^{2}+28 a b \,c^{2} d^{2} e f -2 a b c \,d^{3} e^{2}+9 b^{2} c^{4} f^{2}-6 b^{2} c^{3} d e f -3 b^{2} c^{2} d^{2} e^{2}\right ) x}{8 c}}{\left (x^{2} d +c \right )^{2}}+\frac {\left (63 a^{2} c^{2} f^{2} d^{2}-18 a^{2} c e f \,d^{3}+3 a^{2} e^{2} d^{4}-70 a b \,c^{3} d \,f^{2}-28 a b \,c^{2} d^{2} e f +2 a b c \,d^{3} e^{2}+15 b^{2} c^{4} f^{2}+30 b^{2} c^{3} d e f +3 b^{2} c^{2} d^{2} e^{2}\right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{8 c^{2} \sqrt {c d}}\right )}{\left (c f -d e \right )^{5}}\)	\(761\)
risch	\(\text {Expression too large to display}\)	\(78959\)

Input:

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

Output:

f/(c*f-d*e)^5*((1/8*f*(3*a^2*c^2*f^4-18*a^2*c*d*e*f^3+15*a^2*d^2*e^2*f^2+2 
*a*b*c^2*e*f^3+20*a*b*c*d*e^2*f^2-22*a*b*d^2*e^3*f-5*b^2*c^2*e^2*f^2-2*b^2 
*c*d*e^3*f+7*b^2*d^2*e^4)/e^2*x^3+1/8*(5*a^2*c^2*f^4-22*a^2*c*d*e*f^3+17*a 
^2*d^2*e^2*f^2-2*a*b*c^2*e*f^3+28*a*b*c*d*e^2*f^2-26*a*b*d^2*e^3*f-3*b^2*c 
^2*e^2*f^2-6*b^2*c*d*e^3*f+9*b^2*d^2*e^4)/e*x)/(f*x^2+e)^2+1/8*(3*a^2*c^2* 
f^4-18*a^2*c*d*e*f^3+63*a^2*d^2*e^2*f^2+2*a*b*c^2*e*f^3-28*a*b*c*d*e^2*f^2 
-70*a*b*d^2*e^3*f+3*b^2*c^2*e^2*f^2+30*b^2*c*d*e^3*f+15*b^2*d^2*e^4)/e^2/( 
e*f)^(1/2)*arctan(f*x/(e*f)^(1/2)))-d/(c*f-d*e)^5*((1/8*d*(15*a^2*c^2*d^2* 
f^2-18*a^2*c*d^3*e*f+3*a^2*d^4*e^2-22*a*b*c^3*d*f^2+20*a*b*c^2*d^2*e*f+2*a 
*b*c*d^3*e^2+7*b^2*c^4*f^2-2*b^2*c^3*d*e*f-5*b^2*c^2*d^2*e^2)/c^2*x^3+1/8* 
(17*a^2*c^2*d^2*f^2-22*a^2*c*d^3*e*f+5*a^2*d^4*e^2-26*a*b*c^3*d*f^2+28*a*b 
*c^2*d^2*e*f-2*a*b*c*d^3*e^2+9*b^2*c^4*f^2-6*b^2*c^3*d*e*f-3*b^2*c^2*d^2*e 
^2)/c*x)/(d*x^2+c)^2+1/8*(63*a^2*c^2*d^2*f^2-18*a^2*c*d^3*e*f+3*a^2*d^4*e^ 
2-70*a*b*c^3*d*f^2-28*a*b*c^2*d^2*e*f+2*a*b*c*d^3*e^2+15*b^2*c^4*f^2+30*b^ 
2*c^3*d*e*f+3*b^2*c^2*d^2*e^2)/c^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x^2+a)^2/(d*x^2+c)^3/(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. 1271 vs. \(2 (560) = 1120\).

Time = 0.13 (sec) , antiderivative size = 1271, normalized size of antiderivative = 2.16 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*(3*b^2*c^2*d^3*e^2 + 2*a*b*c*d^4*e^2 + 3*a^2*d^5*e^2 + 30*b^2*c^3*d^2* 
e*f - 28*a*b*c^2*d^3*e*f - 18*a^2*c*d^4*e*f + 15*b^2*c^4*d*f^2 - 70*a*b*c^ 
3*d^2*f^2 + 63*a^2*c^2*d^3*f^2)*arctan(d*x/sqrt(c*d))/((c^2*d^5*e^5 - 5*c^ 
3*d^4*e^4*f + 10*c^4*d^3*e^3*f^2 - 10*c^5*d^2*e^2*f^3 + 5*c^6*d*e*f^4 - c^ 
7*f^5)*sqrt(c*d)) - 1/8*(15*b^2*d^2*e^4*f + 30*b^2*c*d*e^3*f^2 - 70*a*b*d^ 
2*e^3*f^2 + 3*b^2*c^2*e^2*f^3 - 28*a*b*c*d*e^2*f^3 + 63*a^2*d^2*e^2*f^3 + 
2*a*b*c^2*e*f^4 - 18*a^2*c*d*e*f^4 + 3*a^2*c^2*f^5)*arctan(f*x/sqrt(e*f))/ 
((d^5*e^7 - 5*c*d^4*e^6*f + 10*c^2*d^3*e^5*f^2 - 10*c^3*d^2*e^4*f^3 + 5*c^ 
4*d*e^3*f^4 - c^5*e^2*f^5)*sqrt(e*f)) - 1/8*(12*b^2*c^2*d^3*e^3*f^2*x^7 - 
2*a*b*c*d^4*e^3*f^2*x^7 - 3*a^2*d^5*e^3*f^2*x^7 + 12*b^2*c^3*d^2*e^2*f^3*x 
^7 - 44*a*b*c^2*d^3*e^2*f^3*x^7 + 15*a^2*c*d^4*e^2*f^3*x^7 - 2*a*b*c^3*d^2 
*e*f^4*x^7 + 15*a^2*c^2*d^3*e*f^4*x^7 - 3*a^2*c^3*d^2*f^5*x^7 + 19*b^2*c^2 
*d^3*e^4*f*x^5 - 4*a*b*c*d^4*e^4*f*x^5 - 6*a^2*d^5*e^4*f*x^5 + 34*b^2*c^3* 
d^2*e^3*f^2*x^5 - 68*a*b*c^2*d^3*e^3*f^2*x^5 + 25*a^2*c*d^4*e^3*f^2*x^5 + 
19*b^2*c^4*d*e^2*f^3*x^5 - 68*a*b*c^3*d^2*e^2*f^3*x^5 + 34*a^2*c^2*d^3*e^2 
*f^3*x^5 - 4*a*b*c^4*d*e*f^4*x^5 + 25*a^2*c^3*d^2*e*f^4*x^5 - 6*a^2*c^4*d* 
f^5*x^5 + 5*b^2*c^2*d^3*e^5*x^3 - 2*a*b*c*d^4*e^5*x^3 - 3*a^2*d^5*e^5*x^3 
+ 31*b^2*c^3*d^2*e^4*f*x^3 - 18*a*b*c^2*d^3*e^4*f*x^3 + 5*a^2*c*d^4*e^4*f* 
x^3 + 31*b^2*c^4*d*e^3*f^2*x^3 - 104*a*b*c^3*d^2*e^3*f^2*x^3 + 34*a^2*c^2* 
d^3*e^3*f^2*x^3 + 5*b^2*c^5*e^2*f^3*x^3 - 18*a*b*c^4*d*e^2*f^3*x^3 + 34...
 

Mupad [B] (verification not implemented)

Time = 20.55 (sec) , antiderivative size = 161006, normalized size of antiderivative = 273.82 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan(((((768*a^2*c^2*d^16*e^16*f^2 - 11520*a^2*c^3*d^15*e^15*f^3 + 85248 
*a^2*c^4*d^14*e^14*f^4 - 391680*a^2*c^5*d^13*e^13*f^5 + 1214208*a^2*c^6*d^ 
12*e^12*f^6 - 2654976*a^2*c^7*d^11*e^11*f^7 + 4204800*a^2*c^8*d^10*e^10*f^ 
8 - 4893696*a^2*c^9*d^9*e^9*f^9 + 4204800*a^2*c^10*d^8*e^8*f^10 - 2654976* 
a^2*c^11*d^7*e^7*f^11 + 1214208*a^2*c^12*d^6*e^6*f^12 - 391680*a^2*c^13*d^ 
5*e^5*f^13 + 85248*a^2*c^14*d^4*e^4*f^14 - 11520*a^2*c^15*d^3*e^3*f^15 + 7 
68*a^2*c^16*d^2*e^2*f^16 + 768*b^2*c^4*d^14*e^16*f^2 - 3072*b^2*c^5*d^13*e 
^15*f^3 - 10752*b^2*c^6*d^12*e^14*f^4 + 107520*b^2*c^7*d^11*e^13*f^5 - 357 
120*b^2*c^8*d^10*e^12*f^6 + 681984*b^2*c^9*d^9*e^11*f^7 - 838656*b^2*c^10* 
d^8*e^10*f^8 + 681984*b^2*c^11*d^7*e^9*f^9 - 357120*b^2*c^12*d^6*e^8*f^10 
+ 107520*b^2*c^13*d^5*e^7*f^11 - 10752*b^2*c^14*d^4*e^6*f^12 - 3072*b^2*c^ 
15*d^3*e^5*f^13 + 768*b^2*c^16*d^2*e^4*f^14 + 512*a*b*c^3*d^15*e^16*f^2 - 
11776*a*b*c^4*d^14*e^15*f^3 + 82944*a*b*c^5*d^13*e^14*f^4 - 293888*a*b*c^6 
*d^12*e^13*f^5 + 601600*a*b*c^7*d^11*e^12*f^6 - 705024*a*b*c^8*d^10*e^11*f 
^7 + 325632*a*b*c^9*d^9*e^10*f^8 + 325632*a*b*c^10*d^8*e^9*f^9 - 705024*a* 
b*c^11*d^7*e^8*f^10 + 601600*a*b*c^12*d^6*e^7*f^11 - 293888*a*b*c^13*d^5*e 
^6*f^12 + 82944*a*b*c^14*d^4*e^5*f^13 - 11776*a*b*c^15*d^3*e^4*f^14 + 512* 
a*b*c^16*d^2*e^3*f^15)/(512*(c^4*d^12*e^16 + c^16*e^4*f^12 - 12*c^5*d^11*e 
^15*f - 12*c^15*d*e^5*f^11 + 66*c^6*d^10*e^14*f^2 - 220*c^7*d^9*e^13*f^3 + 
 495*c^8*d^8*e^12*f^4 - 792*c^9*d^7*e^11*f^5 + 924*c^10*d^6*e^10*f^6 - ...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 5234, normalized size of antiderivative = 8.90 \[ \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 63*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c**4*d**2*e**5*f 
**2 - 126*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c**4*d**2*e** 
4*f**3*x**2 - 63*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c**4*d 
**2*e**3*f**4*x**4 + 18*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2 
*c**3*d**3*e**6*f - 90*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2* 
c**3*d**3*e**5*f**2*x**2 - 234*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c) 
))*a**2*c**3*d**3*e**4*f**3*x**4 - 126*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d) 
*sqrt(c)))*a**2*c**3*d**3*e**3*f**4*x**6 - 3*sqrt(d)*sqrt(c)*atan((d*x)/(s 
qrt(d)*sqrt(c)))*a**2*c**2*d**4*e**7 + 30*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt 
(d)*sqrt(c)))*a**2*c**2*d**4*e**6*f*x**2 + 6*sqrt(d)*sqrt(c)*atan((d*x)/(s 
qrt(d)*sqrt(c)))*a**2*c**2*d**4*e**5*f**2*x**4 - 90*sqrt(d)*sqrt(c)*atan(( 
d*x)/(sqrt(d)*sqrt(c)))*a**2*c**2*d**4*e**4*f**3*x**6 - 63*sqrt(d)*sqrt(c) 
*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c**2*d**4*e**3*f**4*x**8 - 6*sqrt(d)*s 
qrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c*d**5*e**7*x**2 + 6*sqrt(d)*sqr 
t(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c*d**5*e**6*f*x**4 + 30*sqrt(d)*sq 
rt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c*d**5*e**5*f**2*x**6 + 18*sqrt(d 
)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*c*d**5*e**4*f**3*x**8 - 3*sqr 
t(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*d**6*e**7*x**4 - 6*sqrt(d) 
*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*d**6*e**6*f*x**6 - 3*sqrt(d)*s 
qrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*d**6*e**5*f**2*x**8 + 70*sqrt...