\(\int \frac {(a+b x^2)^3}{(c+d x^2)^2 (e+f x^2)^3} \, dx\) [251]

Optimal result

Integrand size = 28, antiderivative size = 505 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=-\frac {\left (9 a^2 b c d^2 e f^2-a^3 d^2 f^2 (2 d e+c f)-3 a b^2 c d e f (d e+2 c f)+b^3 \left (c d^2 e^3+2 c^3 e f^2\right )\right ) x}{4 c d^2 e f (d e-c f)^2 \left (e+f x^2\right )^2}-\frac {(b c-a d)^3 x}{2 c d^2 (d e-c f) \left (c+d x^2\right ) \left (e+f x^2\right )^2}-\frac {\left (3 a^2 b c d e f^2 (11 d e+c f)-9 a b^2 c d e^2 f (d e+3 c f)-b^3 c e^2 \left (d^2 e^2-9 c d e f-4 c^2 f^2\right )-a^3 d f^2 \left (4 d^2 e^2+11 c d e f-3 c^2 f^2\right )\right ) x}{8 c d e^2 f (d e-c f)^3 \left (e+f x^2\right )}+\frac {(b c-a d)^2 (a d (d e-7 c f)+b c (5 d e+c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d} (d e-c f)^4}+\frac {(b e-a f) \left (b^2 e^2 \left (d^2 e^2-10 c d e f-15 c^2 f^2\right )+2 a b e f \left (5 d^2 e^2+22 c d e f-3 c^2 f^2\right )-a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} f^{3/2} (d e-c f)^4} \] Output:

-1/4*(9*a^2*b*c*d^2*e*f^2-a^3*d^2*f^2*(c*f+2*d*e)-3*a*b^2*c*d*e*f*(2*c*f+d 
*e)+b^3*(2*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)^2-1/2* 
(-a*d+b*c)^3*x/c/d^2/(-c*f+d*e)/(d*x^2+c)/(f*x^2+e)^2-1/8*(3*a^2*b*c*d*e*f 
^2*(c*f+11*d*e)-9*a*b^2*c*d*e^2*f*(3*c*f+d*e)-b^3*c*e^2*(-4*c^2*f^2-9*c*d* 
e*f+d^2*e^2)-a^3*d*f^2*(-3*c^2*f^2+11*c*d*e*f+4*d^2*e^2))*x/c/d/e^2/f/(-c* 
f+d*e)^3/(f*x^2+e)+1/2*(-a*d+b*c)^2*(a*d*(-7*c*f+d*e)+b*c*(c*f+5*d*e))*arc 
tan(d^(1/2)*x/c^(1/2))/c^(3/2)/d^(1/2)/(-c*f+d*e)^4+1/8*(-a*f+b*e)*(b^2*e^ 
2*(-15*c^2*f^2-10*c*d*e*f+d^2*e^2)+2*a*b*e*f*(-3*c^2*f^2+22*c*d*e*f+5*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^(3/2)/(-c*f+d*e)^4
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (\frac {4 (b c-a d)^3 x}{c (-d e+c f)^3 \left (c+d x^2\right )}-\frac {2 (b e-a f)^3 x}{e f (d e-c f)^2 \left (e+f x^2\right )^2}+\frac {(b e-a f)^2 (b e (d e-9 c f)+a f (11 d e-3 c f)) x}{e^2 f (d e-c f)^3 \left (e+f x^2\right )}+\frac {4 (b c-a d)^2 (a d (d e-7 c f)+b c (5 d e+c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} \sqrt {d} (d e-c f)^4}+\frac {(b e-a f) \left (b^2 e^2 \left (d^2 e^2-10 c d e f-15 c^2 f^2\right )+a^2 f^2 \left (-35 d^2 e^2+14 c d e f-3 c^2 f^2\right )+2 a b e f \left (5 d^2 e^2+22 c d e f-3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{5/2} f^{3/2} (d e-c f)^4}\right ) \] Input:

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

Output:

((4*(b*c - a*d)^3*x)/(c*(-(d*e) + c*f)^3*(c + d*x^2)) - (2*(b*e - a*f)^3*x 
)/(e*f*(d*e - c*f)^2*(e + f*x^2)^2) + ((b*e - a*f)^2*(b*e*(d*e - 9*c*f) + 
a*f*(11*d*e - 3*c*f))*x)/(e^2*f*(d*e - c*f)^3*(e + f*x^2)) + (4*(b*c - a*d 
)^2*(a*d*(d*e - 7*c*f) + b*c*(5*d*e + c*f))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/( 
c^(3/2)*Sqrt[d]*(d*e - c*f)^4) + ((b*e - a*f)*(b^2*e^2*(d^2*e^2 - 10*c*d*e 
*f - 15*c^2*f^2) + a^2*f^2*(-35*d^2*e^2 + 14*c*d*e*f - 3*c^2*f^2) + 2*a*b* 
e*f*(5*d^2*e^2 + 22*c*d*e*f - 3*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(e^ 
(5/2)*f^(3/2)*(d*e - c*f)^4))/8
 

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 953, normalized size of antiderivative = 1.89, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.679, Rules used = {425, 419, 25, 397, 218, 401, 27, 298, 218, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 419

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 298

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(b*(-((d*(b*c - a*d)*(-(((b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c] 
*Sqrt[d]*(d*e - c*f))) + ((b*e - a*f)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e 
]*Sqrt[f]*(d*e - c*f))))/(d*e - c*f)^2) + (((b*e - a*f)*(d*e - c*f)*x*(a + 
 b*x^2))/(4*e*(e + f*x^2)^2) - (-1/2*((b*e - a*f)*(a*f*(7*d*e - 3*c*f) - b 
*e*(d*e + 3*c*f))*x)/(e*f*(e + f*x^2)) + ((a^2*f^2*(7*d*e - 3*c*f) + 2*a*b 
*e*f*(d*e - c*f) - b^2*e^2*(d*e + 3*c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2* 
e^(3/2)*f^(3/2)))/(4*e))/(d*e - c*f)^2))/d - ((b*c - a*d)*((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*Ar 
cTan[(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))))...
 

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 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 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 = 0.89 (sec) , antiderivative size = 752, normalized size of antiderivative = 1.49

method	result	size
default	\(\frac {\frac {\frac {\left (3 a^{3} c^{2} f^{5}-14 a^{3} c d e \,f^{4}+11 a^{3} d^{2} e^{2} f^{3}+3 a^{2} b \,c^{2} e \,f^{4}+18 a^{2} b c d \,e^{2} f^{3}-21 a^{2} b \,d^{2} e^{3} f^{2}-15 a \,b^{2} c^{2} e^{2} f^{3}+6 a \,b^{2} c d \,e^{3} f^{2}+9 a \,b^{2} d^{2} e^{4} f +9 b^{3} c^{2} e^{3} f^{2}-10 b^{3} c d \,e^{4} f +b^{3} d^{2} e^{5}\right ) x^{3}}{8 e^{2}}+\frac {\left (5 a^{3} c^{2} f^{5}-18 a^{3} c d e \,f^{4}+13 a^{3} d^{2} e^{2} f^{3}-3 a^{2} b \,c^{2} e \,f^{4}+30 a^{2} b c d \,e^{2} f^{3}-27 a^{2} b \,d^{2} e^{3} f^{2}-9 a \,b^{2} c^{2} e^{2} f^{3}-6 a \,b^{2} c d \,e^{3} f^{2}+15 a \,b^{2} d^{2} e^{4} f +7 b^{3} c^{2} e^{3} f^{2}-6 b^{3} c d \,e^{4} f -b^{3} d^{2} e^{5}\right ) x}{8 e f}}{\left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a^{3} c^{2} f^{5}-14 a^{3} c d e \,f^{4}+35 a^{3} d^{2} e^{2} f^{3}+3 a^{2} b \,c^{2} e \,f^{4}-30 a^{2} b c d \,e^{2} f^{3}-45 a^{2} b \,d^{2} e^{3} f^{2}+9 a \,b^{2} c^{2} e^{2} f^{3}+54 a \,b^{2} c d \,e^{3} f^{2}+9 a \,b^{2} d^{2} e^{4} f -15 b^{3} c^{2} e^{3} f^{2}-10 b^{3} c d \,e^{4} f +b^{3} d^{2} e^{5}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 e^{2} f \sqrt {e f}}}{\left (c f -d e \right )^{4}}-\frac {\frac {\left (a^{3} c f \,d^{3}-a^{3} e \,d^{4}-3 a^{2} b \,c^{2} d^{2} f +3 a^{2} b c \,d^{3} e +3 a \,b^{2} c^{3} d f -3 a \,b^{2} c^{2} d^{2} e -b^{3} c^{4} f +b^{3} c^{3} d e \right ) x}{2 c \left (x^{2} d +c \right )}+\frac {\left (7 a^{3} c f \,d^{3}-a^{3} e \,d^{4}-15 a^{2} b \,c^{2} d^{2} f -3 a^{2} b c \,d^{3} e +9 a \,b^{2} c^{3} d f +9 a \,b^{2} c^{2} d^{2} e -b^{3} c^{4} f -5 b^{3} c^{3} d e \right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{2 c \sqrt {c d}}}{\left (c f -d e \right )^{4}}\)	\(752\)
risch	\(\text {Expression too large to display}\)	\(2378\)

Input:

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

Output:

1/(c*f-d*e)^4*((1/8*(3*a^3*c^2*f^5-14*a^3*c*d*e*f^4+11*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^3-21*a^2*b*d^2*e^3*f^2-15*a*b^2*c^2*e^2* 
f^3+6*a*b^2*c*d*e^3*f^2+9*a*b^2*d^2*e^4*f+9*b^3*c^2*e^3*f^2-10*b^3*c*d*e^4 
*f+b^3*d^2*e^5)/e^2*x^3+1/8*(5*a^3*c^2*f^5-18*a^3*c*d*e*f^4+13*a^3*d^2*e^2 
*f^3-3*a^2*b*c^2*e*f^4+30*a^2*b*c*d*e^2*f^3-27*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+15*a*b^2*d^2*e^4*f+7*b^3*c^2*e^3*f^2-6*b^3* 
c*d*e^4*f-b^3*d^2*e^5)/e/f*x)/(f*x^2+e)^2+1/8*(3*a^3*c^2*f^5-14*a^3*c*d*e* 
f^4+35*a^3*d^2*e^2*f^3+3*a^2*b*c^2*e*f^4-30*a^2*b*c*d*e^2*f^3-45*a^2*b*d^2 
*e^3*f^2+9*a*b^2*c^2*e^2*f^3+54*a*b^2*c*d*e^3*f^2+9*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+b^3*d^2*e^5)/e^2/f/(e*f)^(1/2)*arctan(f*x/(e 
*f)^(1/2)))-1/(c*f-d*e)^4*(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*x 
/(d*x^2+c)+1/2*(7*a^3*c*d^3*f-a^3*d^4*e-15*a^2*b*c^2*d^2*f-3*a^2*b*c*d^3*e 
+9*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/(c*d)^(1/2)* 
arctan(x*d/(c*d)^(1/2)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 750, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\frac {{\left (5 \, b^{3} c^{3} d e - 9 \, a b^{2} c^{2} d^{2} e + 3 \, a^{2} b c d^{3} e + a^{3} d^{4} e + b^{3} c^{4} f - 9 \, a b^{2} c^{3} d f + 15 \, a^{2} b c^{2} d^{2} f - 7 \, a^{3} c d^{3} f\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (c d^{4} e^{4} - 4 \, c^{2} d^{3} e^{3} f + 6 \, c^{3} d^{2} e^{2} f^{2} - 4 \, c^{4} d e f^{3} + c^{5} f^{4}\right )} \sqrt {c d}} + \frac {{\left (b^{3} d^{2} e^{5} - 10 \, b^{3} c d e^{4} f + 9 \, a b^{2} d^{2} e^{4} f - 15 \, b^{3} c^{2} e^{3} f^{2} + 54 \, a b^{2} c d e^{3} f^{2} - 45 \, a^{2} b d^{2} e^{3} f^{2} + 9 \, a b^{2} c^{2} e^{2} f^{3} - 30 \, a^{2} b c d e^{2} f^{3} + 35 \, a^{3} d^{2} e^{2} f^{3} + 3 \, a^{2} b c^{2} e f^{4} - 14 \, a^{3} c d e f^{4} + 3 \, a^{3} c^{2} f^{5}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \, {\left (d^{4} e^{6} f - 4 \, c d^{3} e^{5} f^{2} + 6 \, c^{2} d^{2} e^{4} f^{3} - 4 \, c^{3} d e^{3} f^{4} + c^{4} e^{2} f^{5}\right )} \sqrt {e f}} - \frac {b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{2 \, {\left (c d^{3} e^{3} - 3 \, c^{2} d^{2} e^{2} f + 3 \, c^{3} d e f^{2} - c^{4} f^{3}\right )} {\left (d x^{2} + c\right )}} + \frac {b^{3} d e^{4} f x^{3} - 9 \, b^{3} c e^{3} f^{2} x^{3} + 9 \, a b^{2} d e^{3} f^{2} x^{3} + 15 \, a b^{2} c e^{2} f^{3} x^{3} - 21 \, a^{2} b d e^{2} f^{3} x^{3} - 3 \, a^{2} b c e f^{4} x^{3} + 11 \, a^{3} d e f^{4} x^{3} - 3 \, a^{3} c f^{5} x^{3} - b^{3} d e^{5} x - 7 \, b^{3} c e^{4} f x + 15 \, a b^{2} d e^{4} f x + 9 \, a b^{2} c e^{3} f^{2} x - 27 \, a^{2} b d e^{3} f^{2} x + 3 \, a^{2} b c e^{2} f^{3} x + 13 \, a^{3} d e^{2} f^{3} x - 5 \, a^{3} c e f^{4} x}{8 \, {\left (d^{3} e^{5} f - 3 \, c d^{2} e^{4} f^{2} + 3 \, c^{2} d e^{3} f^{3} - c^{3} e^{2} f^{4}\right )} {\left (f x^{2} + e\right )}^{2}} \] Input:

integrate((b*x^2+a)^3/(d*x^2+c)^2/(f*x^2+e)^3,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 - 9*a*b^2*c^3*d*f + 15*a^2*b*c^2*d^2*f - 7*a^3*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*(b^3*d^2*e^5 - 10*b^3*c*d*e^4*f + 9*a*b^2*d 
^2*e^4*f - 15*b^3*c^2*e^3*f^2 + 54*a*b^2*c*d*e^3*f^2 - 45*a^2*b*d^2*e^3*f^ 
2 + 9*a*b^2*c^2*e^2*f^3 - 30*a^2*b*c*d*e^2*f^3 + 35*a^3*d^2*e^2*f^3 + 3*a^ 
2*b*c^2*e*f^4 - 14*a^3*c*d*e*f^4 + 3*a^3*c^2*f^5)*arctan(f*x/sqrt(e*f))/(( 
d^4*e^6*f - 4*c*d^3*e^5*f^2 + 6*c^2*d^2*e^4*f^3 - 4*c^3*d*e^3*f^4 + c^4*e^ 
2*f^5)*sqrt(e*f)) - 1/2*(b^3*c^3*x - 3*a*b^2*c^2*d*x + 3*a^2*b*c*d^2*x - a 
^3*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*(b^3*d*e^4*f*x^3 - 9*b^3*c*e^3*f^2*x^3 + 9*a*b^2*d*e^3*f^2*x^3 
 + 15*a*b^2*c*e^2*f^3*x^3 - 21*a^2*b*d*e^2*f^3*x^3 - 3*a^2*b*c*e*f^4*x^3 + 
 11*a^3*d*e*f^4*x^3 - 3*a^3*c*f^5*x^3 - b^3*d*e^5*x - 7*b^3*c*e^4*f*x + 15 
*a*b^2*d*e^4*f*x + 9*a*b^2*c*e^3*f^2*x - 27*a^2*b*d*e^3*f^2*x + 3*a^2*b*c* 
e^2*f^3*x + 13*a^3*d*e^2*f^3*x - 5*a^3*c*e*f^4*x)/((d^3*e^5*f - 3*c*d^2*e^ 
4*f^2 + 3*c^2*d*e^3*f^3 - c^3*e^2*f^4)*(f*x^2 + e)^2)
 

Mupad [B] (verification not implemented)

Time = 20.84 (sec) , antiderivative size = 184741, normalized size of antiderivative = 365.82 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((20160*a^3*c^3*d^11*e^11*f^5 - 3584*a^3*c^2*d^12*e^12*f^4 - 63168* 
a^3*c^4*d^10*e^10*f^6 + 125184*a^3*c^5*d^9*e^9*f^7 - 166656*a^3*c^6*d^8*e^ 
8*f^8 + 153216*a^3*c^7*d^7*e^7*f^9 - 97920*a^3*c^8*d^6*e^6*f^10 + 43008*a^ 
3*c^9*d^5*e^5*f^11 - 12544*a^3*c^10*d^4*e^4*f^12 + 2240*a^3*c^11*d^3*e^3*f 
^13 - 192*a^3*c^12*d^2*e^2*f^14 + 64*b^3*c^3*d^11*e^14*f^2 + 192*b^3*c^4*d 
^10*e^13*f^3 - 3840*b^3*c^5*d^9*e^12*f^4 + 16128*b^3*c^6*d^8*e^11*f^5 - 34 
944*b^3*c^7*d^7*e^10*f^6 + 45696*b^3*c^8*d^6*e^9*f^7 - 37632*b^3*c^9*d^5*e 
^8*f^8 + 19200*b^3*c^10*d^4*e^7*f^9 - 5568*b^3*c^11*d^3*e^6*f^10 + 704*b^3 
*c^12*d^2*e^5*f^11 + 256*a^3*c*d^13*e^13*f^3 - 1728*a*b^2*c^3*d^11*e^13*f^ 
3 + 13248*a*b^2*c^4*d^10*e^12*f^4 - 43776*a*b^2*c^5*d^9*e^11*f^5 + 80640*a 
*b^2*c^6*d^8*e^10*f^6 - 88704*a*b^2*c^7*d^7*e^9*f^7 + 56448*a*b^2*c^8*d^6* 
e^8*f^8 - 16128*a*b^2*c^9*d^5*e^7*f^9 - 2304*a*b^2*c^10*d^4*e^6*f^10 + 288 
0*a*b^2*c^11*d^3*e^5*f^11 - 576*a*b^2*c^12*d^2*e^4*f^12 + 768*a^2*b*c^2*d^ 
12*e^13*f^3 - 4416*a^2*b*c^3*d^11*e^12*f^4 + 7488*a^2*b*c^4*d^10*e^11*f^5 
+ 6912*a^2*b*c^5*d^9*e^10*f^6 - 48384*a^2*b*c^6*d^8*e^9*f^7 + 88704*a^2*b* 
c^7*d^7*e^8*f^8 - 88704*a^2*b*c^8*d^6*e^7*f^9 + 52992*a^2*b*c^9*d^5*e^6*f^ 
10 - 18432*a^2*b*c^10*d^4*e^5*f^11 + 3264*a^2*b*c^11*d^3*e^4*f^12 - 192*a^ 
2*b*c^12*d^2*e^3*f^13)/(128*(c^11*e^4*f^10 - c^2*d^9*e^13*f - 9*c^10*d*e^5 
*f^9 + 9*c^3*d^8*e^12*f^2 - 36*c^4*d^7*e^11*f^3 + 84*c^5*d^6*e^10*f^4 - 12 
6*c^6*d^5*e^9*f^5 + 126*c^7*d^4*e^8*f^6 - 84*c^8*d^3*e^7*f^7 + 36*c^9*d...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 4530, normalized size of antiderivative = 8.97 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^3/(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**3*c**2*d**3*e**5*f 
**3 - 56*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*c**2*d**3*e**4 
*f**4*x**2 - 28*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*c**2*d* 
*3*e**3*f**5*x**4 + 4*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*c 
*d**4*e**6*f**2 - 20*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*c* 
d**4*e**5*f**3*x**2 - 52*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a** 
3*c*d**4*e**4*f**4*x**4 - 28*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c))) 
*a**3*c*d**4*e**3*f**5*x**6 + 4*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c 
)))*a**3*d**5*e**6*f**2*x**2 + 8*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt( 
c)))*a**3*d**5*e**5*f**3*x**4 + 4*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt 
(c)))*a**3*d**5*e**4*f**4*x**6 + 60*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sq 
rt(c)))*a**2*b*c**3*d**2*e**5*f**3 + 120*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt( 
d)*sqrt(c)))*a**2*b*c**3*d**2*e**4*f**4*x**2 + 60*sqrt(d)*sqrt(c)*atan((d* 
x)/(sqrt(d)*sqrt(c)))*a**2*b*c**3*d**2*e**3*f**5*x**4 + 12*sqrt(d)*sqrt(c) 
*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c**2*d**3*e**6*f**2 + 84*sqrt(d)*sqr 
t(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c**2*d**3*e**5*f**3*x**2 + 132*s 
qrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c**2*d**3*e**4*f**4*x* 
*4 + 60*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c**2*d**3*e** 
3*f**5*x**6 + 12*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c*d* 
*4*e**6*f**2*x**2 + 24*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*...