\(\int \frac {1}{(a+b x^2) (c+d x^2)^2 (e+f x^2)^3} \, dx\) [257]

Optimal result

Integrand size = 28, antiderivative size = 400 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=-\frac {d^4 x}{2 c (b c-a d) (d e-c f)^3 \left (c+d x^2\right )}-\frac {f^3 x}{4 e (b e-a f) (d e-c f)^2 \left (e+f x^2\right )^2}-\frac {f^3 (b e (15 d e-7 c f)-a f (11 d e-3 c f)) x}{8 e^2 (b e-a f)^2 (d e-c f)^3 \left (e+f x^2\right )}+\frac {b^{9/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)^2 (b e-a f)^3}+\frac {d^{7/2} (a d (d e-7 c f)-3 b c (d e-3 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^2 (d e-c f)^4}-\frac {f^{5/2} \left (a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )-2 a b e f \left (45 d^2 e^2-26 c d e f+5 c^2 f^2\right )+3 b^2 e^2 \left (21 d^2 e^2-18 c d e f+5 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} (b e-a f)^3 (d e-c f)^4} \] Output:

-1/2*d^4*x/c/(-a*d+b*c)/(-c*f+d*e)^3/(d*x^2+c)-1/4*f^3*x/e/(-a*f+b*e)/(-c* 
f+d*e)^2/(f*x^2+e)^2-1/8*f^3*(b*e*(-7*c*f+15*d*e)-a*f*(-3*c*f+11*d*e))*x/e 
^2/(-a*f+b*e)^2/(-c*f+d*e)^3/(f*x^2+e)+b^(9/2)*arctan(b^(1/2)*x/a^(1/2))/a 
^(1/2)/(-a*d+b*c)^2/(-a*f+b*e)^3+1/2*d^(7/2)*(a*d*(-7*c*f+d*e)-3*b*c*(-3*c 
*f+d*e))*arctan(d^(1/2)*x/c^(1/2))/c^(3/2)/(-a*d+b*c)^2/(-c*f+d*e)^4-1/8*f 
^(5/2)*(a^2*f^2*(3*c^2*f^2-14*c*d*e*f+35*d^2*e^2)-2*a*b*e*f*(5*c^2*f^2-26* 
c*d*e*f+45*d^2*e^2)+3*b^2*e^2*(5*c^2*f^2-18*c*d*e*f+21*d^2*e^2))*arctan(f^ 
(1/2)*x/e^(1/2))/e^(5/2)/(-a*f+b*e)^3/(-c*f+d*e)^4
 

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (\frac {4 d^4 x}{c (b c-a d) (-d e+c f)^3 \left (c+d x^2\right )}-\frac {2 f^3 x}{e (b e-a f) (d e-c f)^2 \left (e+f x^2\right )^2}-\frac {f^3 (b e (15 d e-7 c f)+a f (-11 d e+3 c f)) x}{e^2 (b e-a f)^2 (d e-c f)^3 \left (e+f x^2\right )}-\frac {8 b^{9/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)^2 (-b e+a f)^3}+\frac {4 d^{7/2} (a d (d e-7 c f)+3 b c (-d e+3 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^2 (d e-c f)^4}-\frac {f^{5/2} \left (a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )-2 a b e f \left (45 d^2 e^2-26 c d e f+5 c^2 f^2\right )+3 b^2 e^2 \left (21 d^2 e^2-18 c d e f+5 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{5/2} (b e-a f)^3 (d e-c f)^4}\right ) \] Input:

Integrate[1/((a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^3),x]
 

Output:

((4*d^4*x)/(c*(b*c - a*d)*(-(d*e) + c*f)^3*(c + d*x^2)) - (2*f^3*x)/(e*(b* 
e - a*f)*(d*e - c*f)^2*(e + f*x^2)^2) - (f^3*(b*e*(15*d*e - 7*c*f) + a*f*( 
-11*d*e + 3*c*f))*x)/(e^2*(b*e - a*f)^2*(d*e - c*f)^3*(e + f*x^2)) - (8*b^ 
(9/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d)^2*(-(b*e) + a*f)^3 
) + (4*d^(7/2)*(a*d*(d*e - 7*c*f) + 3*b*c*(-(d*e) + 3*c*f))*ArcTan[(Sqrt[d 
]*x)/Sqrt[c]])/(c^(3/2)*(b*c - a*d)^2*(d*e - c*f)^4) - (f^(5/2)*(a^2*f^2*( 
35*d^2*e^2 - 14*c*d*e*f + 3*c^2*f^2) - 2*a*b*e*f*(45*d^2*e^2 - 26*c*d*e*f 
+ 5*c^2*f^2) + 3*b^2*e^2*(21*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2))*ArcTan[(Sq 
rt[f]*x)/Sqrt[e]])/(e^(5/2)*(b*e - a*f)^3*(d*e - c*f)^4))/8
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.56, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {421, 316, 402, 25, 397, 218, 402, 27, 402, 25, 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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {b^2 \left (\frac {\frac {\frac {8 b^{5/2} e^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b e-a f)}-\frac {\sqrt {f} \left (3 a^2 f^2-10 a b e f+15 b^2 e^2\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}}{2 e (b e-a f)}-\frac {f x (7 b e-3 a f)}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}-\frac {d \left (\frac {\frac {\frac {\int -\frac {d f \left (a d \left (4 d^2 e^2+11 c d f e-3 c^2 f^2\right )+3 b c \left (d^2 e^2-7 c d f e+2 c^2 f^2\right )\right ) x^2+a d \left (4 d^3 e^3-24 c d^2 f e^2+11 c^2 d f^2 e-3 c^3 f^3\right )-3 b c \left (4 d^3 e^3-13 c d^2 f e^2+7 c^2 d f^2 e-2 c^3 f^3\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{2 e (d e-c f)}-\frac {f x \left (a d \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )+3 b c \left (2 c^2 f^2-7 c d e f+d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (a d (c f+2 d e)-b c (2 c f+d e))}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}+\frac {d x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{(b c-a d)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b^2 \left (\frac {\frac {\frac {8 b^{5/2} e^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b e-a f)}-\frac {\sqrt {f} \left (3 a^2 f^2-10 a b e f+15 b^2 e^2\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}}{2 e (b e-a f)}-\frac {f x (7 b e-3 a f)}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}-\frac {d \left (\frac {\frac {-\frac {\int \frac {d f \left (a d \left (4 d^2 e^2+11 c d f e-3 c^2 f^2\right )+3 b c \left (d^2 e^2-7 c d f e+2 c^2 f^2\right )\right ) x^2+a d \left (4 d^3 e^3-24 c d^2 f e^2+11 c^2 d f^2 e-3 c^3 f^3\right )-3 b c \left (4 d^3 e^3-13 c d^2 f e^2+7 c^2 d f^2 e-2 c^3 f^3\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{2 e (d e-c f)}-\frac {f x \left (a d \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )+3 b c \left (2 c^2 f^2-7 c d e f+d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (a d (c f+2 d e)-b c (2 c f+d e))}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}+\frac {d x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{(b c-a d)^2}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {b^2 \left (\frac {\frac {\frac {8 b^{5/2} e^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b e-a f)}-\frac {\sqrt {f} \left (3 a^2 f^2-10 a b e f+15 b^2 e^2\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}}{2 e (b e-a f)}-\frac {f x (7 b e-3 a f)}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}-\frac {d \left (\frac {\frac {-\frac {\frac {c f \left (a d f \left (3 c^2 f^2-14 c d e f+35 d^2 e^2\right )+3 b \left (-2 c^3 f^3+9 c^2 d e f^2-20 c d^2 e^2 f+5 d^3 e^3\right )\right ) \int \frac {1}{f x^2+e}dx}{d e-c f}+\frac {4 d^3 e^2 (a d (d e-7 c f)-3 b c (d e-3 c f)) \int \frac {1}{d x^2+c}dx}{d e-c f}}{2 e (d e-c f)}-\frac {f x \left (a d \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )+3 b c \left (2 c^2 f^2-7 c d e f+d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (a d (c f+2 d e)-b c (2 c f+d e))}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}+\frac {d x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{(b c-a d)^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b^2 \left (\frac {\frac {\frac {8 b^{5/2} e^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b e-a f)}-\frac {\sqrt {f} \left (3 a^2 f^2-10 a b e f+15 b^2 e^2\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}}{2 e (b e-a f)}-\frac {f x (7 b e-3 a f)}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}-\frac {d \left (\frac {\frac {-\frac {\frac {c \sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a d f \left (3 c^2 f^2-14 c d e f+35 d^2 e^2\right )+3 b \left (-2 c^3 f^3+9 c^2 d e f^2-20 c d^2 e^2 f+5 d^3 e^3\right )\right )}{\sqrt {e} (d e-c f)}+\frac {4 d^{5/2} e^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) (a d (d e-7 c f)-3 b c (d e-3 c f))}{\sqrt {c} (d e-c f)}}{2 e (d e-c f)}-\frac {f x \left (a d \left (-3 c^2 f^2+11 c d e f+4 d^2 e^2\right )+3 b c \left (2 c^2 f^2-7 c d e f+d^2 e^2\right )\right )}{2 e \left (e+f x^2\right ) (d e-c f)}}{2 e (d e-c f)}-\frac {f x (a d (c f+2 d e)-b c (2 c f+d e))}{2 e \left (e+f x^2\right )^2 (d e-c f)}}{2 c (d e-c f)}+\frac {d x (b c-a d)}{2 c \left (c+d x^2\right ) \left (e+f x^2\right )^2 (d e-c f)}\right )}{(b c-a d)^2}\)

Input:

Int[1/((a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^3),x]
 

Output:

(b^2*(-1/4*(f*x)/(e*(b*e - a*f)*(e + f*x^2)^2) + (-1/2*(f*(7*b*e - 3*a*f)* 
x)/(e*(b*e - a*f)*(e + f*x^2)) + ((8*b^(5/2)*e^2*ArcTan[(Sqrt[b]*x)/Sqrt[a 
]])/(Sqrt[a]*(b*e - a*f)) - (Sqrt[f]*(15*b^2*e^2 - 10*a*b*e*f + 3*a^2*f^2) 
*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(b*e - a*f)))/(2*e*(b*e - a*f)))/(4 
*e*(b*e - a*f))))/(b*c - a*d)^2 - (d*((d*(b*c - a*d)*x)/(2*c*(d*e - c*f)*( 
c + d*x^2)*(e + f*x^2)^2) + (-1/2*(f*(a*d*(2*d*e + c*f) - b*c*(d*e + 2*c*f 
))*x)/(e*(d*e - c*f)*(e + f*x^2)^2) + (-1/2*(f*(a*d*(4*d^2*e^2 + 11*c*d*e* 
f - 3*c^2*f^2) + 3*b*c*(d^2*e^2 - 7*c*d*e*f + 2*c^2*f^2))*x)/(e*(d*e - c*f 
)*(e + f*x^2)) - ((4*d^(5/2)*e^2*(a*d*(d*e - 7*c*f) - 3*b*c*(d*e - 3*c*f)) 
*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(d*e - c*f)) + (c*Sqrt[f]*(a*d*f*(3 
5*d^2*e^2 - 14*c*d*e*f + 3*c^2*f^2) + 3*b*(5*d^3*e^3 - 20*c*d^2*e^2*f + 9* 
c^2*d*e*f^2 - 2*c^3*f^3))*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))))/(b*c - a*d)^ 
2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 316

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

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 421

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

Maple [A] (verified)

Time = 3.01 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.36

method	result	size
default	\(-\frac {b^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right )^{2} \left (a f -b e \right )^{3} \sqrt {a b}}+\frac {f^{3} \left (\frac {\frac {f \left (3 a^{2} c^{2} f^{4}-14 a^{2} c d e \,f^{3}+11 a^{2} d^{2} e^{2} f^{2}-10 a b \,c^{2} e \,f^{3}+36 a b c d \,e^{2} f^{2}-26 a b \,d^{2} e^{3} f +7 b^{2} c^{2} e^{2} f^{2}-22 b^{2} c d \,e^{3} f +15 b^{2} d^{2} e^{4}\right ) x^{3}}{8 e^{2}}+\frac {\left (5 a^{2} c^{2} f^{4}-18 a^{2} c d e \,f^{3}+13 a^{2} d^{2} e^{2} f^{2}-14 a b \,c^{2} e \,f^{3}+44 a b c d \,e^{2} f^{2}-30 a b \,d^{2} e^{3} f +9 b^{2} c^{2} e^{2} f^{2}-26 b^{2} c d \,e^{3} f +17 b^{2} d^{2} e^{4}\right ) x}{8 e}}{\left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a^{2} c^{2} f^{4}-14 a^{2} c d e \,f^{3}+35 a^{2} d^{2} e^{2} f^{2}-10 a b \,c^{2} e \,f^{3}+52 a b c d \,e^{2} f^{2}-90 a b \,d^{2} e^{3} f +15 b^{2} c^{2} e^{2} f^{2}-54 b^{2} c d \,e^{3} f +63 b^{2} d^{2} e^{4}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 e^{2} \sqrt {e f}}\right )}{\left (c f -d e \right )^{4} \left (a f -b e \right )^{3}}-\frac {d^{4} \left (\frac {\left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) x}{2 c \left (x^{2} d +c \right )}+\frac {\left (7 a c d f -a \,d^{2} e -9 b \,c^{2} f +3 b c d e \right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{2 c \sqrt {c d}}\right )}{\left (a d -b c \right )^{2} \left (c f -d e \right )^{4}}\)	\(544\)
risch	\(\text {Expression too large to display}\)	\(683174\)

Input:

int(1/(b*x^2+a)/(d*x^2+c)^2/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
 

Output:

-b^5/(a*d-b*c)^2/(a*f-b*e)^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))+f^3/(c*f- 
d*e)^4/(a*f-b*e)^3*((1/8*f*(3*a^2*c^2*f^4-14*a^2*c*d*e*f^3+11*a^2*d^2*e^2* 
f^2-10*a*b*c^2*e*f^3+36*a*b*c*d*e^2*f^2-26*a*b*d^2*e^3*f+7*b^2*c^2*e^2*f^2 
-22*b^2*c*d*e^3*f+15*b^2*d^2*e^4)/e^2*x^3+1/8*(5*a^2*c^2*f^4-18*a^2*c*d*e* 
f^3+13*a^2*d^2*e^2*f^2-14*a*b*c^2*e*f^3+44*a*b*c*d*e^2*f^2-30*a*b*d^2*e^3* 
f+9*b^2*c^2*e^2*f^2-26*b^2*c*d*e^3*f+17*b^2*d^2*e^4)/e*x)/(f*x^2+e)^2+1/8* 
(3*a^2*c^2*f^4-14*a^2*c*d*e*f^3+35*a^2*d^2*e^2*f^2-10*a*b*c^2*e*f^3+52*a*b 
*c*d*e^2*f^2-90*a*b*d^2*e^3*f+15*b^2*c^2*e^2*f^2-54*b^2*c*d*e^3*f+63*b^2*d 
^2*e^4)/e^2/(e*f)^(1/2)*arctan(f*x/(e*f)^(1/2)))-d^4/(a*d-b*c)^2/(c*f-d*e) 
^4*(1/2*(a*c*d*f-a*d^2*e-b*c^2*f+b*c*d*e)/c*x/(d*x^2+c)+1/2*(7*a*c*d*f-a*d 
^2*e-9*b*c^2*f+3*b*c*d*e)/c/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^2+a)/(d*x^2+c)^2/(f*x^2+e)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x**2+a)/(d*x**2+c)**2/(f*x**2+e)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(b*x^2+a)/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1240 vs. \(2 (367) = 734\).

Time = 0.15 (sec) , antiderivative size = 1240, normalized size of antiderivative = 3.10 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(1/(b*x^2+a)/(d*x^2+c)^2/(f*x^2+e)^3,x, algorithm="giac")
 

Output:

b^5*arctan(b*x/sqrt(a*b))/((b^5*c^2*e^3 - 2*a*b^4*c*d*e^3 + a^2*b^3*d^2*e^ 
3 - 3*a*b^4*c^2*e^2*f + 6*a^2*b^3*c*d*e^2*f - 3*a^3*b^2*d^2*e^2*f + 3*a^2* 
b^3*c^2*e*f^2 - 6*a^3*b^2*c*d*e*f^2 + 3*a^4*b*d^2*e*f^2 - a^3*b^2*c^2*f^3 
+ 2*a^4*b*c*d*f^3 - a^5*d^2*f^3)*sqrt(a*b)) - 1/2*d^4*x/((b*c^2*d^3*e^3 - 
a*c*d^4*e^3 - 3*b*c^3*d^2*e^2*f + 3*a*c^2*d^3*e^2*f + 3*b*c^4*d*e*f^2 - 3* 
a*c^3*d^2*e*f^2 - b*c^5*f^3 + a*c^4*d*f^3)*(d*x^2 + c)) - 1/2*(3*b*c*d^5*e 
 - a*d^6*e - 9*b*c^2*d^4*f + 7*a*c*d^5*f)*arctan(d*x/sqrt(c*d))/((b^2*c^3* 
d^4*e^4 - 2*a*b*c^2*d^5*e^4 + a^2*c*d^6*e^4 - 4*b^2*c^4*d^3*e^3*f + 8*a*b* 
c^3*d^4*e^3*f - 4*a^2*c^2*d^5*e^3*f + 6*b^2*c^5*d^2*e^2*f^2 - 12*a*b*c^4*d 
^3*e^2*f^2 + 6*a^2*c^3*d^4*e^2*f^2 - 4*b^2*c^6*d*e*f^3 + 8*a*b*c^5*d^2*e*f 
^3 - 4*a^2*c^4*d^3*e*f^3 + b^2*c^7*f^4 - 2*a*b*c^6*d*f^4 + a^2*c^5*d^2*f^4 
)*sqrt(c*d)) - 1/8*(63*b^2*d^2*e^4*f^3 - 54*b^2*c*d*e^3*f^4 - 90*a*b*d^2*e 
^3*f^4 + 15*b^2*c^2*e^2*f^5 + 52*a*b*c*d*e^2*f^5 + 35*a^2*d^2*e^2*f^5 - 10 
*a*b*c^2*e*f^6 - 14*a^2*c*d*e*f^6 + 3*a^2*c^2*f^7)*arctan(f*x/sqrt(e*f))/( 
(b^3*d^4*e^9 - 4*b^3*c*d^3*e^8*f - 3*a*b^2*d^4*e^8*f + 6*b^3*c^2*d^2*e^7*f 
^2 + 12*a*b^2*c*d^3*e^7*f^2 + 3*a^2*b*d^4*e^7*f^2 - 4*b^3*c^3*d*e^6*f^3 - 
18*a*b^2*c^2*d^2*e^6*f^3 - 12*a^2*b*c*d^3*e^6*f^3 - a^3*d^4*e^6*f^3 + b^3* 
c^4*e^5*f^4 + 12*a*b^2*c^3*d*e^5*f^4 + 18*a^2*b*c^2*d^2*e^5*f^4 + 4*a^3*c* 
d^3*e^5*f^4 - 3*a*b^2*c^4*e^4*f^5 - 12*a^2*b*c^3*d*e^4*f^5 - 6*a^3*c^2*d^2 
*e^4*f^5 + 3*a^2*b*c^4*e^3*f^6 + 4*a^3*c^3*d*e^3*f^6 - a^3*c^4*e^2*f^7)...
 

Mupad [B] (verification not implemented)

Time = 33.86 (sec) , antiderivative size = 212642, normalized size of antiderivative = 531.60 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(1/((a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^3),x)
 

Output:

symsum(log(- root(61018521600*a^13*b^8*c^11*d^16*e^21*f^12*z^6 + 610185216 
00*a^9*b^12*c^19*d^8*e^17*f^16*z^6 - 839516160*a^14*b^7*c^22*d^5*e^9*f^24* 
z^6 - 839516160*a^8*b^13*c^8*d^19*e^29*f^4*z^6 - 22304522240*a^17*b^4*c^14 
*d^13*e^14*f^19*z^6 - 22304522240*a^16*b^5*c^15*d^12*e^14*f^19*z^6 - 22304 
522240*a^6*b^15*c^15*d^12*e^24*f^9*z^6 - 22304522240*a^5*b^16*c^16*d^11*e^ 
24*f^9*z^6 + 35188113408*a^14*b^7*c^13*d^14*e^18*f^15*z^6 + 35188113408*a^ 
8*b^13*c^17*d^10*e^20*f^13*z^6 + 314353385472*a^12*b^9*c^16*d^11*e^17*f^16 
*z^6 + 314353385472*a^10*b^11*c^14*d^13*e^21*f^12*z^6 + 818413568*a^12*b^9 
*c^22*d^5*e^11*f^22*z^6 + 818413568*a^10*b^11*c^8*d^19*e^27*f^6*z^6 - 8016 
36352*a^12*b^9*c^6*d^21*e^27*f^6*z^6 - 801636352*a^10*b^11*c^24*d^3*e^11*f 
^22*z^6 + 750059520*a^15*b^6*c^21*d^6*e^9*f^24*z^6 + 750059520*a^7*b^14*c^ 
9*d^18*e^29*f^4*z^6 + 39332085760*a^11*b^10*c^20*d^7*e^14*f^19*z^6 + 39332 
085760*a^11*b^10*c^10*d^17*e^24*f^9*z^6 - 9264955392*a^17*b^4*c^16*d^11*e^ 
12*f^21*z^6 - 9264955392*a^16*b^5*c^16*d^11*e^13*f^20*z^6 - 9264955392*a^6 
*b^15*c^14*d^13*e^25*f^8*z^6 - 9264955392*a^5*b^16*c^14*d^13*e^26*f^7*z^6 
+ 620232704*a^2*b^19*c^22*d^5*e^21*f^12*z^6 - 619970560*a^16*b^5*c^5*d^22* 
e^24*f^9*z^6 - 619970560*a^6*b^15*c^25*d^2*e^14*f^19*z^6 + 13473546240*a^1 
2*b^9*c^8*d^19*e^25*f^8*z^6 + 13473546240*a^10*b^11*c^22*d^5*e^13*f^20*z^6 
 - 583532544*a^18*b^3*c^9*d^18*e^18*f^15*z^6 - 583532544*a^4*b^17*c^21*d^6 
*e^20*f^13*z^6 - 129415249920*a^15*b^6*c^12*d^15*e^18*f^15*z^6 - 129415...
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 10641, normalized size of antiderivative = 26.60 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(b*x^2+a)/(d*x^2+c)^2/(f*x^2+e)^3,x)
 

Output:

( - 8*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**7*e**5*f**4 - 
16*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**7*e**4*f**5*x**2 
- 8*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**7*e**3*f**6*x**4 
 + 32*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**6*d*e**6*f**3 
+ 56*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**6*d*e**5*f**4*x 
**2 + 16*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**6*d*e**4*f* 
*5*x**4 - 8*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**6*d*e**3 
*f**6*x**6 - 48*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**5*d* 
*2*e**7*f**2 - 64*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**5* 
d**2*e**6*f**3*x**2 + 16*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b** 
4*c**5*d**2*e**5*f**4*x**4 + 32*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a 
)))*b**4*c**5*d**2*e**4*f**5*x**6 + 32*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b) 
*sqrt(a)))*b**4*c**4*d**3*e**8*f + 16*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)* 
sqrt(a)))*b**4*c**4*d**3*e**7*f**2*x**2 - 64*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*b**4*c**4*d**3*e**6*f**3*x**4 - 48*sqrt(b)*sqrt(a)*atan(( 
b*x)/(sqrt(b)*sqrt(a)))*b**4*c**4*d**3*e**5*f**4*x**6 - 8*sqrt(b)*sqrt(a)* 
atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**3*d**4*e**9 + 16*sqrt(b)*sqrt(a)*ata 
n((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**3*d**4*e**8*f*x**2 + 56*sqrt(b)*sqrt(a) 
*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**3*d**4*e**7*f**2*x**4 + 32*sqrt(b)* 
sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*c**3*d**4*e**6*f**3*x**6 - 8...