\(\int \frac {(a+b x^2)^3}{(c+d x^2)^2 (e+f x^2)^2} \, dx\) [250]

Optimal result

Integrand size = 28, antiderivative size = 299 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=-\frac {\left (6 a^2 b c d^2 e f^2-3 a b^2 c d e f (d e+c f)-a^3 d^2 f^2 (d e+c f)+b^3 \left (c d^2 e^3+c^3 e f^2\right )\right ) x}{2 c d^2 e f (d e-c f)^2 \left (e+f x^2\right )}-\frac {(b c-a d)^3 x}{2 c d^2 (d e-c f) \left (c+d x^2\right ) \left (e+f x^2\right )}+\frac {(b c-a d)^2 (a d (d e-5 c f)+b c (5 d e-c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{3/2} (d e-c f)^3}+\frac {(b e-a f)^2 (b e (d e-5 c f)+a f (5 d e-c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} f^{3/2} (d e-c f)^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {(b c-a d)^3 x}{c d (d e-c f)^2 \left (c+d x^2\right )}-\frac {(b e-a f)^3 x}{e f (d e-c f)^2 \left (e+f x^2\right )}+\frac {(b c-a d)^2 (b c (-5 d e+c f)+a d (-d e+5 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} d^{3/2} (-d e+c f)^3}+\frac {(b e-a f)^2 (b e (d e-5 c f)+a f (5 d e-c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{3/2} f^{3/2} (d e-c f)^3}\right ) \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(639\) vs. \(2(299)=598\).

Time = 1.04 (sec) , antiderivative size = 639, normalized size of antiderivative = 2.14, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {425, 419, 25, 299, 218, 401, 27, 299, 218, 425, 402, 25, 397, 218, 402, 27, 397, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 419

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(b*(-((d*(b*c - a*d)*((b*x)/d - ((b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/ 
(Sqrt[c]*d^(3/2))))/(d*e - c*f)^2) + (((b*e - a*f)*(d*e - c*f)*x*(a + b*x^ 
2))/(2*e*(e + f*x^2)) - ((b*(b*e*(d*e - 3*c*f) + a*f*(d*e + c*f))*x)/f - ( 
(b*e - a*f)*(b*e*(d*e - 3*c*f) + a*f*(3*d*e - c*f))*ArcTan[(Sqrt[f]*x)/Sqr 
t[e]])/(Sqrt[e]*f^(3/2)))/(2*e))/(d*e - c*f)^2))/d - ((b*c - a*d)*((b*(((b 
*e - a*f)*x)/(2*e*(d*e - c*f)*(e + f*x^2)) - ((2*Sqrt[d]*(b*c - a*d)*e*Arc 
Tan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(d*e - c*f)) + ((a*f*(3*d*e - c*f) - b* 
e*(d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]*(d*e - c*f))) 
/(2*e*(d*e - c*f))))/d - ((b*c - a*d)*(-1/2*((b*c - a*d)*x)/(c*(d*e - c*f) 
*(c + d*x^2)*(e + f*x^2)) + (-((f*(2*b*c*e - a*d*e - a*c*f)*x)/(e*(d*e - c 
*f)*(e + f*x^2))) + ((Sqrt[d]*e*(a*d*(d*e - 5*c*f) + b*c*(d*e + 3*c*f))*Ar 
cTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(d*e - c*f)) + (c*Sqrt[f]*(a*f*(5*d*e 
- c*f) - b*e*(3*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(d*e - c 
*f)))/(e*(d*e - c*f)))/(2*c*(d*e - c*f))))/d))/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 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 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 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 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 = 0.82 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.50

method	result	size
default	\(\frac {\frac {\left (a^{3} c \,f^{4}-a^{3} d e \,f^{3}-3 a^{2} b c e \,f^{3}+3 a^{2} b d \,e^{2} f^{2}+3 a \,b^{2} c \,e^{2} f^{2}-3 a \,b^{2} d \,e^{3} f -b^{3} c \,e^{3} f +e^{4} b^{3} d \right ) x}{2 e f \left (f \,x^{2}+e \right )}+\frac {\left (a^{3} c \,f^{4}-5 a^{3} d e \,f^{3}+3 a^{2} b c e \,f^{3}+9 a^{2} b d \,e^{2} f^{2}-9 a \,b^{2} c \,e^{2} f^{2}-3 a \,b^{2} d \,e^{3} f +5 b^{3} c \,e^{3} f -e^{4} b^{3} d \right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 e f \sqrt {e f}}}{\left (c f -d e \right )^{3}}+\frac {\frac {\left (a^{3} c f \,d^{3}-a^{3} e \,d^{4}-3 a^{2} b \,c^{2} d^{2} f +3 a^{2} b c \,d^{3} e +3 a \,b^{2} c^{3} d f -3 a \,b^{2} c^{2} d^{2} e -b^{3} c^{4} f +b^{3} c^{3} d e \right ) x}{2 c d \left (x^{2} d +c \right )}+\frac {\left (5 a^{3} c f \,d^{3}-a^{3} e \,d^{4}-9 a^{2} b \,c^{2} d^{2} f -3 a^{2} b c \,d^{3} e +3 a \,b^{2} c^{3} d f +9 a \,b^{2} c^{2} d^{2} e +b^{3} c^{4} f -5 b^{3} c^{3} d e \right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{2 c d \sqrt {c d}}}{\left (c f -d e \right )^{3}}\)	\(448\)
risch	\(\text {Expression too large to display}\)	\(1606\)

Input:

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

Output:

1/(c*f-d*e)^3*(1/2*(a^3*c*f^4-a^3*d*e*f^3-3*a^2*b*c*e*f^3+3*a^2*b*d*e^2*f^ 
2+3*a*b^2*c*e^2*f^2-3*a*b^2*d*e^3*f-b^3*c*e^3*f+b^3*d*e^4)/e/f*x/(f*x^2+e) 
+1/2*(a^3*c*f^4-5*a^3*d*e*f^3+3*a^2*b*c*e*f^3+9*a^2*b*d*e^2*f^2-9*a*b^2*c* 
e^2*f^2-3*a*b^2*d*e^3*f+5*b^3*c*e^3*f-b^3*d*e^4)/e/f/(e*f)^(1/2)*arctan(f* 
x/(e*f)^(1/2)))+1/(c*f-d*e)^3*(1/2*(a^3*c*d^3*f-a^3*d^4*e-3*a^2*b*c^2*d^2* 
f+3*a^2*b*c*d^3*e+3*a*b^2*c^3*d*f-3*a*b^2*c^2*d^2*e-b^3*c^4*f+b^3*c^3*d*e) 
/c/d*x/(d*x^2+c)+1/2*(5*a^3*c*d^3*f-a^3*d^4*e-9*a^2*b*c^2*d^2*f-3*a^2*b*c* 
d^3*e+3*a*b^2*c^3*d*f+9*a*b^2*c^2*d^2*e+b^3*c^4*f-5*b^3*c^3*d*e)/c/d/(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. 1287 vs. \(2 (275) = 550\).

Time = 146.92 (sec) , antiderivative size = 5227, normalized size of antiderivative = 17.48 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (275) = 550\).

Time = 0.13 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {{\left (5 \, b^{3} c^{3} d e - 9 \, a b^{2} c^{2} d^{2} e + 3 \, a^{2} b c d^{3} e + a^{3} d^{4} e - b^{3} c^{4} f - 3 \, a b^{2} c^{3} d f + 9 \, a^{2} b c^{2} d^{2} f - 5 \, a^{3} c d^{3} f\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (c d^{4} e^{3} - 3 \, c^{2} d^{3} e^{2} f + 3 \, c^{3} d^{2} e f^{2} - c^{4} d f^{3}\right )} \sqrt {c d}} + \frac {{\left (b^{3} d e^{4} - 5 \, b^{3} c e^{3} f + 3 \, a b^{2} d e^{3} f + 9 \, a b^{2} c e^{2} f^{2} - 9 \, a^{2} b d e^{2} f^{2} - 3 \, a^{2} b c e f^{3} + 5 \, a^{3} d e f^{3} - a^{3} c f^{4}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \, {\left (d^{3} e^{4} f - 3 \, c d^{2} e^{3} f^{2} + 3 \, c^{2} d e^{2} f^{3} - c^{3} e f^{4}\right )} \sqrt {e f}} - \frac {b^{3} c d^{2} e^{3} x^{3} - 3 \, a b^{2} c d^{2} e^{2} f x^{3} + b^{3} c^{3} e f^{2} x^{3} - 3 \, a b^{2} c^{2} d e f^{2} x^{3} + 6 \, a^{2} b c d^{2} e f^{2} x^{3} - a^{3} d^{3} e f^{2} x^{3} - a^{3} c d^{2} f^{3} x^{3} + b^{3} c^{2} d e^{3} x + b^{3} c^{3} e^{2} f x - 6 \, a b^{2} c^{2} d e^{2} f x + 3 \, a^{2} b c d^{2} e^{2} f x - a^{3} d^{3} e^{2} f x + 3 \, a^{2} b c^{2} d e f^{2} x - a^{3} c^{2} d f^{3} x}{2 \, {\left (c d^{3} e^{3} f - 2 \, c^{2} d^{2} e^{2} f^{2} + c^{3} d e f^{3}\right )} {\left (d f x^{4} + d e x^{2} + c f x^{2} + c e\right )}} \] Input:

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

Output:

1/2*(5*b^3*c^3*d*e - 9*a*b^2*c^2*d^2*e + 3*a^2*b*c*d^3*e + a^3*d^4*e - b^3 
*c^4*f - 3*a*b^2*c^3*d*f + 9*a^2*b*c^2*d^2*f - 5*a^3*c*d^3*f)*arctan(d*x/s 
qrt(c*d))/((c*d^4*e^3 - 3*c^2*d^3*e^2*f + 3*c^3*d^2*e*f^2 - c^4*d*f^3)*sqr 
t(c*d)) + 1/2*(b^3*d*e^4 - 5*b^3*c*e^3*f + 3*a*b^2*d*e^3*f + 9*a*b^2*c*e^2 
*f^2 - 9*a^2*b*d*e^2*f^2 - 3*a^2*b*c*e*f^3 + 5*a^3*d*e*f^3 - a^3*c*f^4)*ar 
ctan(f*x/sqrt(e*f))/((d^3*e^4*f - 3*c*d^2*e^3*f^2 + 3*c^2*d*e^2*f^3 - c^3* 
e*f^4)*sqrt(e*f)) - 1/2*(b^3*c*d^2*e^3*x^3 - 3*a*b^2*c*d^2*e^2*f*x^3 + b^3 
*c^3*e*f^2*x^3 - 3*a*b^2*c^2*d*e*f^2*x^3 + 6*a^2*b*c*d^2*e*f^2*x^3 - a^3*d 
^3*e*f^2*x^3 - a^3*c*d^2*f^3*x^3 + b^3*c^2*d*e^3*x + b^3*c^3*e^2*f*x - 6*a 
*b^2*c^2*d*e^2*f*x + 3*a^2*b*c*d^2*e^2*f*x - a^3*d^3*e^2*f*x + 3*a^2*b*c^2 
*d*e*f^2*x - a^3*c^2*d*f^3*x)/((c*d^3*e^3*f - 2*c^2*d^2*e^2*f^2 + c^3*d*e* 
f^3)*(d*f*x^4 + d*e*x^2 + c*f*x^2 + c*e))
 

Mupad [B] (verification not implemented)

Time = 17.26 (sec) , antiderivative size = 142283, normalized size of antiderivative = 475.86 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((160*a^3*c^2*d^10*e^8*f^4 - 640*a^3*c^3*d^9*e^7*f^5 + 1376*a^3*c^4 
*d^8*e^6*f^6 - 1760*a^3*c^5*d^7*e^5*f^7 + 1376*a^3*c^6*d^6*e^4*f^8 - 640*a 
^3*c^7*d^5*e^3*f^9 + 160*a^3*c^8*d^4*e^2*f^10 - 16*b^3*c^3*d^9*e^10*f^2 + 
80*b^3*c^4*d^8*e^9*f^3 - 144*b^3*c^5*d^7*e^8*f^4 + 80*b^3*c^6*d^6*e^7*f^5 
+ 80*b^3*c^7*d^5*e^6*f^6 - 144*b^3*c^8*d^4*e^5*f^7 + 80*b^3*c^9*d^3*e^4*f^ 
8 - 16*b^3*c^10*d^2*e^3*f^9 - 16*a^3*c*d^11*e^9*f^3 - 16*a^3*c^9*d^3*e*f^1 
1 + 96*a*b^2*c^3*d^9*e^9*f^3 - 576*a*b^2*c^4*d^8*e^8*f^4 + 1440*a*b^2*c^5* 
d^7*e^7*f^5 - 1920*a*b^2*c^6*d^6*e^6*f^6 + 1440*a*b^2*c^7*d^5*e^5*f^7 - 57 
6*a*b^2*c^8*d^4*e^4*f^8 + 96*a*b^2*c^9*d^3*e^3*f^9 - 48*a^2*b*c^2*d^10*e^9 
*f^3 + 240*a^2*b*c^3*d^9*e^8*f^4 - 432*a^2*b*c^4*d^8*e^7*f^5 + 240*a^2*b*c 
^5*d^7*e^6*f^6 + 240*a^2*b*c^6*d^6*e^5*f^7 - 432*a^2*b*c^7*d^5*e^4*f^8 + 2 
40*a^2*b*c^8*d^4*e^3*f^9 - 48*a^2*b*c^9*d^3*e^2*f^10)/(8*(c^2*d^7*e^8*f + 
c^8*d*e^2*f^7 - 6*c^3*d^6*e^7*f^2 + 15*c^4*d^5*e^6*f^3 - 20*c^5*d^4*e^5*f^ 
4 + 15*c^6*d^3*e^4*f^5 - 6*c^7*d^2*e^3*f^6)) - (x*((2360*a^6*c^3*d^11*e^8* 
f^6 - 8*a^6*c^11*d^3*f^14 - 8*b^6*c^3*d^11*e^14 - 8*a^6*d^14*e^11*f^3 - 8* 
b^6*c^14*e^3*f^11 - 800*a^6*c^2*d^12*e^9*f^5 - ((4720*a^6*c^3*d^11*e^8*f^6 
 - 16*b^6*c^3*d^11*e^14 - 16*a^6*d^14*e^11*f^3 - 16*b^6*c^14*e^3*f^11 - 16 
00*a^6*c^2*d^12*e^9*f^5 - 16*a^6*c^11*d^3*f^14 - 6880*a^6*c^4*d^10*e^7*f^7 
 + 3520*a^6*c^5*d^9*e^6*f^8 + 3520*a^6*c^6*d^8*e^5*f^9 - 6880*a^6*c^7*d^7* 
e^4*f^10 + 4720*a^6*c^8*d^6*e^3*f^11 - 1600*a^6*c^9*d^5*e^2*f^12 - 1600...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 2451, normalized size of antiderivative = 8.20 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(5*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*c**2*d**3*e**3*f**3 
+ 5*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*c**2*d**3*e**2*f**4 
*x**2 - sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*c*d**4*e**4*f** 
2 + 4*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*c*d**4*e**3*f**3* 
x**2 + 5*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*c*d**4*e**2*f* 
*4*x**4 - sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*d**5*e**4*f** 
2*x**2 - sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*d**5*e**3*f**3 
*x**4 - 9*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c**3*d**2*e 
**3*f**3 - 9*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c**3*d** 
2*e**2*f**4*x**2 - 3*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b* 
c**2*d**3*e**4*f**2 - 12*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a** 
2*b*c**2*d**3*e**3*f**3*x**2 - 9*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt( 
c)))*a**2*b*c**2*d**3*e**2*f**4*x**4 - 3*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt( 
d)*sqrt(c)))*a**2*b*c*d**4*e**4*f**2*x**2 - 3*sqrt(d)*sqrt(c)*atan((d*x)/( 
sqrt(d)*sqrt(c)))*a**2*b*c*d**4*e**3*f**3*x**4 + 3*sqrt(d)*sqrt(c)*atan((d 
*x)/(sqrt(d)*sqrt(c)))*a*b**2*c**4*d*e**3*f**3 + 3*sqrt(d)*sqrt(c)*atan((d 
*x)/(sqrt(d)*sqrt(c)))*a*b**2*c**4*d*e**2*f**4*x**2 + 9*sqrt(d)*sqrt(c)*at 
an((d*x)/(sqrt(d)*sqrt(c)))*a*b**2*c**3*d**2*e**4*f**2 + 12*sqrt(d)*sqrt(c 
)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b**2*c**3*d**2*e**3*f**3*x**2 + 3*sqrt(d 
)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b**2*c**3*d**2*e**2*f**4*x**4...