\(\int \frac {1}{(a+b x^2)^2 (c+d x^2)^2 (e+f x^2)^2} \, dx\) [259]

Optimal result

Integrand size = 28, antiderivative size = 338 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {b^4 x}{2 a (b c-a d)^2 (b e-a f)^2 \left (a+b x^2\right )}+\frac {d^4 x}{2 c (b c-a d)^2 (d e-c f)^2 \left (c+d x^2\right )}+\frac {f^4 x}{2 e (b e-a f)^2 (d e-c f)^2 \left (e+f x^2\right )}+\frac {b^{7/2} \left (b^2 c e+9 a^2 d f-5 a b (d e+c f)\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} (b c-a d)^3 (b e-a f)^3}+\frac {d^{7/2} (b c (5 d e-9 c f)-a d (d e-5 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^3 (d e-c f)^3}+\frac {f^{7/2} (b e (9 d e-5 c f)-a f (5 d e-c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} (b e-a f)^3 (d e-c f)^3} \] Output:

1/2*b^4*x/a/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x^2+a)+1/2*d^4*x/c/(-a*d+b*c)^2/( 
-c*f+d*e)^2/(d*x^2+c)+1/2*f^4*x/e/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*x^2+e)+1/2* 
b^(7/2)*(b^2*c*e+9*a^2*d*f-5*a*b*(c*f+d*e))*arctan(b^(1/2)*x/a^(1/2))/a^(3 
/2)/(-a*d+b*c)^3/(-a*f+b*e)^3+1/2*d^(7/2)*(b*c*(-9*c*f+5*d*e)-a*d*(-5*c*f+ 
d*e))*arctan(d^(1/2)*x/c^(1/2))/c^(3/2)/(-a*d+b*c)^3/(-c*f+d*e)^3+1/2*f^(7 
/2)*(b*e*(-5*c*f+9*d*e)-a*f*(-c*f+5*d*e))*arctan(f^(1/2)*x/e^(1/2))/e^(3/2 
)/(-a*f+b*e)^3/(-c*f+d*e)^3
 

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {b^4 x}{a (b c-a d)^2 (b e-a f)^2 \left (a+b x^2\right )}+\frac {d^4 x}{c (b c-a d)^2 (d e-c f)^2 \left (c+d x^2\right )}+\frac {f^4 x}{e (b e-a f)^2 (d e-c f)^2 \left (e+f x^2\right )}+\frac {b^{7/2} \left (b^2 c e+9 a^2 d f-5 a b (d e+c f)\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} (b c-a d)^3 (b e-a f)^3}+\frac {d^{7/2} (a d (d e-5 c f)+b c (-5 d e+9 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^3 (-d e+c f)^3}+\frac {f^{7/2} (b e (9 d e-5 c f)+a f (-5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{3/2} (b e-a f)^3 (d e-c f)^3}\right ) \] Input:

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

Output:

((b^4*x)/(a*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x^2)) + (d^4*x)/(c*(b*c - a 
*d)^2*(d*e - c*f)^2*(c + d*x^2)) + (f^4*x)/(e*(b*e - a*f)^2*(d*e - c*f)^2* 
(e + f*x^2)) + (b^(7/2)*(b^2*c*e + 9*a^2*d*f - 5*a*b*(d*e + c*f))*ArcTan[( 
Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(b*c - a*d)^3*(b*e - a*f)^3) + (d^(7/2)*(a*d 
*(d*e - 5*c*f) + b*c*(-5*d*e + 9*c*f))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(3/ 
2)*(b*c - a*d)^3*(-(d*e) + c*f)^3) + (f^(7/2)*(b*e*(9*d*e - 5*c*f) + a*f*( 
-5*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(e^(3/2)*(b*e - a*f)^3*(d*e - 
c*f)^3))/2
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(850\) vs. \(2(338)=676\).

Time = 1.19 (sec) , antiderivative size = 850, normalized size of antiderivative = 2.51, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {426, 421, 25, 316, 397, 218, 402, 25, 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.

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 397

\(\displaystyle \frac {b \left (\frac {\left (\frac {\frac {2 d^{3/2} e \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {\sqrt {f} (3 d e-c f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (d e-c f)}}{2 e (d e-c f)}-\frac {f x}{2 e (d e-c f) \left (f x^2+e\right )}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{2 a (b e-a f) \left (b x^2+a\right ) \left (f x^2+e\right )}+\frac {\frac {f \left (-2 d f a^2+b c f a+b^2 c e\right ) x}{e (b e-a f) \left (f x^2+e\right )}+\frac {\frac {e \left (7 d f a^2-3 b d e a-5 b c f a+b^2 c e\right ) \int \frac {1}{b x^2+a}dx b^2}{b e-a f}+\frac {a f \left (e (3 d e+5 c f) b^2-a f (9 d e+c f) b+2 a^2 d f^2\right ) \int \frac {1}{f x^2+e}dx}{b e-a f}}{e (b e-a f)}}{2 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b^2 \left (\frac {\frac {2 b^{3/2} e \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b e-a f)}-\frac {\sqrt {f} (3 b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}}{2 e (b e-a f)}-\frac {f x}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(b c-a d)^2}-\frac {d \left (\frac {d (b c-a d) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )}+\frac {\frac {f \left (2 b c^2 f-a d (d e+c f)\right ) x}{e (d e-c f) \left (f x^2+e\right )}-\frac {\frac {c f \left (a d f (5 d e-c f)+b \left (3 d^2 e^2-9 c d f e+2 c^2 f^2\right )\right ) \int \frac {1}{f x^2+e}dx}{d e-c f}-\frac {d^2 e (b c (3 d e-7 c f)-a d (d e-5 c f)) \int \frac {1}{d x^2+c}dx}{d e-c f}}{e (d e-c f)}}{2 c (d e-c f)}\right )}{(b c-a d)^2}\right )}{b c-a d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {\left (\frac {\frac {2 d^{3/2} e \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {\sqrt {f} (3 d e-c f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (d e-c f)}}{2 e (d e-c f)}-\frac {f x}{2 e (d e-c f) \left (f x^2+e\right )}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{2 a (b e-a f) \left (b x^2+a\right ) \left (f x^2+e\right )}+\frac {\frac {f \left (-2 d f a^2+b c f a+b^2 c e\right ) x}{e (b e-a f) \left (f x^2+e\right )}+\frac {\frac {e \left (7 d f a^2-3 b d e a-5 b c f a+b^2 c e\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) b^{3/2}}{\sqrt {a} (b e-a f)}+\frac {a \sqrt {f} \left (e (3 d e+5 c f) b^2-a f (9 d e+c f) b+2 a^2 d f^2\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}}{e (b e-a f)}}{2 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b^2 \left (\frac {\frac {2 b^{3/2} e \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b e-a f)}-\frac {\sqrt {f} (3 b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (b e-a f)}}{2 e (b e-a f)}-\frac {f x}{2 e (b e-a f) \left (f x^2+e\right )}\right )}{(b c-a d)^2}-\frac {d \left (\frac {d (b c-a d) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )}+\frac {\frac {f \left (2 b c^2 f-a d (d e+c f)\right ) x}{e (d e-c f) \left (f x^2+e\right )}-\frac {\frac {c \sqrt {f} \left (a d f (5 d e-c f)+b \left (3 d^2 e^2-9 c d f e+2 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} (d e-c f)}-\frac {d^{3/2} e (b c (3 d e-7 c f)-a d (d e-5 c f)) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}}{e (d e-c f)}}{2 c (d e-c f)}\right )}{(b c-a d)^2}\right )}{b c-a d}\)

Input:

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

Output:

(b*((d^2*(-1/2*(f*x)/(e*(d*e - c*f)*(e + f*x^2)) + ((2*d^(3/2)*e*ArcTan[(S 
qrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(d*e - c*f)) - (Sqrt[f]*(3*d*e - c*f)*ArcTan[ 
(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(d*e - c*f)))/(2*e*(d*e - c*f))))/(b*c - a* 
d)^2 + (b*((b*(b*c - a*d)*x)/(2*a*(b*e - a*f)*(a + b*x^2)*(e + f*x^2)) + ( 
(f*(b^2*c*e + a*b*c*f - 2*a^2*d*f)*x)/(e*(b*e - a*f)*(e + f*x^2)) + ((b^(3 
/2)*e*(b^2*c*e - 3*a*b*d*e - 5*a*b*c*f + 7*a^2*d*f)*ArcTan[(Sqrt[b]*x)/Sqr 
t[a]])/(Sqrt[a]*(b*e - a*f)) + (a*Sqrt[f]*(2*a^2*d*f^2 - a*b*f*(9*d*e + c* 
f) + b^2*e*(3*d*e + 5*c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(b*e - a 
*f)))/(e*(b*e - a*f)))/(2*a*(b*e - a*f))))/(b*c - a*d)^2))/(b*c - a*d) - ( 
d*((b^2*(-1/2*(f*x)/(e*(b*e - a*f)*(e + f*x^2)) + ((2*b^(3/2)*e*ArcTan[(Sq 
rt[b]*x)/Sqrt[a]])/(Sqrt[a]*(b*e - a*f)) - (Sqrt[f]*(3*b*e - a*f)*ArcTan[( 
Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(b*e - a*f)))/(2*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)) + (( 
f*(2*b*c^2*f - a*d*(d*e + c*f))*x)/(e*(d*e - c*f)*(e + f*x^2)) - (-((d^(3/ 
2)*e*(b*c*(3*d*e - 7*c*f) - a*d*(d*e - 5*c*f))*ArcTan[(Sqrt[d]*x)/Sqrt[c]] 
)/(Sqrt[c]*(d*e - c*f))) + (c*Sqrt[f]*(a*d*f*(5*d*e - c*f) + b*(3*d^2*e^2 
- 9*c*d*e*f + 2*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*(d*e - c*f 
)))/(e*(d*e - c*f)))/(2*c*(d*e - c*f))))/(b*c - a*d)^2))/(b*c - a*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 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]
 

rule 426

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ 
)^2)^(r_), x_Symbol] :> Simp[b/(b*c - a*d)   Int[(a + b*x^2)^p*(c + d*x^2)^ 
(q + 1)*(e + f*x^2)^r, x], x] - Simp[d/(b*c - a*d)   Int[(a + b*x^2)^(p + 1 
)*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && 
 ILtQ[p, 0] && LeQ[q, -1]
 

Maple [A] (verified)

Time = 2.65 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {b^{4} \left (\frac {\left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) x}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (9 a^{2} d f -5 a b c f -5 a b d e +c e \,b^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{\left (a d -b c \right )^{3} \left (a f -b e \right )^{3}}+\frac {f^{4} \left (\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) x}{2 e \left (f \,x^{2}+e \right )}+\frac {\left (a c \,f^{2}-5 a d e f -5 b c e f +9 b d \,e^{2}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 e \sqrt {e f}}\right )}{\left (a f -b e \right )^{3} \left (c f -d e \right )^{3}}+\frac {d^{4} \left (\frac {\left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) x}{2 c \left (x^{2} d +c \right )}+\frac {\left (5 a c d f -a \,d^{2} e -9 b \,c^{2} f +5 b c d e \right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{2 c \sqrt {c d}}\right )}{\left (a d -b c \right )^{3} \left (c f -d e \right )^{3}}\)	\(333\)
risch	\(\text {Expression too large to display}\)	\(555134\)

Input:

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

Output:

b^4/(a*d-b*c)^3/(a*f-b*e)^3*(1/2*(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/a*x/(b* 
x^2+a)+1/2*(9*a^2*d*f-5*a*b*c*f-5*a*b*d*e+b^2*c*e)/a/(a*b)^(1/2)*arctan(b* 
x/(a*b)^(1/2)))+f^4/(a*f-b*e)^3/(c*f-d*e)^3*(1/2*(a*c*f^2-a*d*e*f-b*c*e*f+ 
b*d*e^2)/e*x/(f*x^2+e)+1/2*(a*c*f^2-5*a*d*e*f-5*b*c*e*f+9*b*d*e^2)/e/(e*f) 
^(1/2)*arctan(f*x/(e*f)^(1/2)))+d^4/(a*d-b*c)^3/(c*f-d*e)^3*(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*(5*a*c*d*f-a*d^2*e-9*b*c^2*f+5* 
b*c*d*e)/c/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(1/(b*x^2+a)^2/(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. 1680 vs. \(2 (302) = 604\).

Time = 0.19 (sec) , antiderivative size = 1680, normalized size of antiderivative = 4.97 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(b^6*c*e - 5*a*b^5*d*e - 5*a*b^5*c*f + 9*a^2*b^4*d*f)*arctan(b*x/sqrt( 
a*b))/((a*b^6*c^3*e^3 - 3*a^2*b^5*c^2*d*e^3 + 3*a^3*b^4*c*d^2*e^3 - a^4*b^ 
3*d^3*e^3 - 3*a^2*b^5*c^3*e^2*f + 9*a^3*b^4*c^2*d*e^2*f - 9*a^4*b^3*c*d^2* 
e^2*f + 3*a^5*b^2*d^3*e^2*f + 3*a^3*b^4*c^3*e*f^2 - 9*a^4*b^3*c^2*d*e*f^2 
+ 9*a^5*b^2*c*d^2*e*f^2 - 3*a^6*b*d^3*e*f^2 - a^4*b^3*c^3*f^3 + 3*a^5*b^2* 
c^2*d*f^3 - 3*a^6*b*c*d^2*f^3 + a^7*d^3*f^3)*sqrt(a*b)) + 1/2*(5*b*c*d^5*e 
 - a*d^6*e - 9*b*c^2*d^4*f + 5*a*c*d^5*f)*arctan(d*x/sqrt(c*d))/((b^3*c^4* 
d^3*e^3 - 3*a*b^2*c^3*d^4*e^3 + 3*a^2*b*c^2*d^5*e^3 - a^3*c*d^6*e^3 - 3*b^ 
3*c^5*d^2*e^2*f + 9*a*b^2*c^4*d^3*e^2*f - 9*a^2*b*c^3*d^4*e^2*f + 3*a^3*c^ 
2*d^5*e^2*f + 3*b^3*c^6*d*e*f^2 - 9*a*b^2*c^5*d^2*e*f^2 + 9*a^2*b*c^4*d^3* 
e*f^2 - 3*a^3*c^3*d^4*e*f^2 - b^3*c^7*f^3 + 3*a*b^2*c^6*d*f^3 - 3*a^2*b*c^ 
5*d^2*f^3 + a^3*c^4*d^3*f^3)*sqrt(c*d)) + 1/2*(9*b*d*e^2*f^4 - 5*b*c*e*f^5 
 - 5*a*d*e*f^5 + a*c*f^6)*arctan(f*x/sqrt(e*f))/((b^3*d^3*e^7 - 3*b^3*c*d^ 
2*e^6*f - 3*a*b^2*d^3*e^6*f + 3*b^3*c^2*d*e^5*f^2 + 9*a*b^2*c*d^2*e^5*f^2 
+ 3*a^2*b*d^3*e^5*f^2 - b^3*c^3*e^4*f^3 - 9*a*b^2*c^2*d*e^4*f^3 - 9*a^2*b* 
c*d^2*e^4*f^3 - a^3*d^3*e^4*f^3 + 3*a*b^2*c^3*e^3*f^4 + 9*a^2*b*c^2*d*e^3* 
f^4 + 3*a^3*c*d^2*e^3*f^4 - 3*a^2*b*c^3*e^2*f^5 - 3*a^3*c^2*d*e^2*f^5 + a^ 
3*c^3*e*f^6)*sqrt(e*f)) + 1/2*(b^4*c*d^3*e^3*f*x^5 + a*b^3*d^4*e^3*f*x^5 - 
 2*b^4*c^2*d^2*e^2*f^2*x^5 - 2*a^2*b^2*d^4*e^2*f^2*x^5 + b^4*c^3*d*e*f^3*x 
^5 + a^3*b*d^4*e*f^3*x^5 + a*b^3*c^3*d*f^4*x^5 - 2*a^2*b^2*c^2*d^2*f^4*...
 

Mupad [B] (verification not implemented)

Time = 39.11 (sec) , antiderivative size = 221098, normalized size of antiderivative = 654.14 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log(- root(838582272*a^23*b^4*c^12*d^15*e^10*f^17*z^6 + 838582272*a 
^23*b^4*c^10*d^17*e^12*f^15*z^6 + 838582272*a^20*b^7*c^18*d^9*e^7*f^20*z^6 
 + 838582272*a^20*b^7*c^7*d^20*e^18*f^9*z^6 + 838582272*a^18*b^9*c^20*d^7* 
e^7*f^20*z^6 + 838582272*a^18*b^9*c^7*d^20*e^20*f^7*z^6 + 838582272*a^12*b 
^15*c^23*d^4*e^10*f^17*z^6 + 838582272*a^12*b^15*c^10*d^17*e^23*f^4*z^6 + 
838582272*a^10*b^17*c^23*d^4*e^12*f^15*z^6 + 838582272*a^10*b^17*c^12*d^15 
*e^23*f^4*z^6 + 838582272*a^7*b^20*c^20*d^7*e^18*f^9*z^6 + 838582272*a^7*b 
^20*c^18*d^9*e^20*f^7*z^6 + 13566541824*a^19*b^8*c^14*d^13*e^12*f^15*z^6 + 
 13566541824*a^19*b^8*c^12*d^15*e^14*f^13*z^6 + 13566541824*a^18*b^9*c^16* 
d^11*e^11*f^16*z^6 + 13566541824*a^18*b^9*c^11*d^16*e^16*f^11*z^6 + 135665 
41824*a^16*b^11*c^18*d^9*e^11*f^16*z^6 + 13566541824*a^16*b^11*c^11*d^16*e 
^18*f^9*z^6 + 13566541824*a^14*b^13*c^19*d^8*e^12*f^15*z^6 + 13566541824*a 
^14*b^13*c^12*d^15*e^19*f^8*z^6 + 13566541824*a^12*b^15*c^19*d^8*e^14*f^13 
*z^6 + 13566541824*a^12*b^15*c^14*d^13*e^19*f^8*z^6 + 13566541824*a^11*b^1 
6*c^18*d^9*e^16*f^11*z^6 + 13566541824*a^11*b^16*c^16*d^11*e^18*f^9*z^6 - 
510935040*a^23*b^4*c^14*d^13*e^8*f^19*z^6 - 510935040*a^23*b^4*c^8*d^19*e^ 
14*f^13*z^6 - 510935040*a^22*b^5*c^16*d^11*e^7*f^20*z^6 - 510935040*a^22*b 
^5*c^7*d^20*e^16*f^11*z^6 - 510935040*a^16*b^11*c^22*d^5*e^7*f^20*z^6 - 51 
0935040*a^16*b^11*c^7*d^20*e^22*f^5*z^6 - 510935040*a^14*b^13*c^23*d^4*e^8 
*f^19*z^6 - 510935040*a^14*b^13*c^8*d^19*e^23*f^4*z^6 - 510935040*a^8*b...
 

Reduce [B] (verification not implemented)

Time = 9.33 (sec) , antiderivative size = 10200, normalized size of antiderivative = 30.18 \[ \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(9*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**3*c**6*d*e**3*f** 
4 + 9*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**3*c**6*d*e**2* 
f**5*x**2 - 27*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**3*c** 
5*d**2*e**4*f**3 - 18*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b 
**3*c**5*d**2*e**3*f**4*x**2 + 9*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt( 
a)))*a**3*b**3*c**5*d**2*e**2*f**5*x**4 + 27*sqrt(b)*sqrt(a)*atan((b*x)/(s 
qrt(b)*sqrt(a)))*a**3*b**3*c**4*d**3*e**5*f**2 - 27*sqrt(b)*sqrt(a)*atan(( 
b*x)/(sqrt(b)*sqrt(a)))*a**3*b**3*c**4*d**3*e**3*f**4*x**4 - 9*sqrt(b)*sqr 
t(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**3*c**3*d**4*e**6*f + 18*sqrt(b) 
*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**3*c**3*d**4*e**5*f**2*x**2 
+ 27*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**3*c**3*d**4*e** 
4*f**3*x**4 - 9*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*b**3*c* 
*2*d**5*e**6*f*x**2 - 9*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3 
*b**3*c**2*d**5*e**5*f**2*x**4 - 5*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqr 
t(a)))*a**2*b**4*c**7*e**3*f**4 - 5*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*a**2*b**4*c**7*e**2*f**5*x**2 + 10*sqrt(b)*sqrt(a)*atan((b*x)/(sqr 
t(b)*sqrt(a)))*a**2*b**4*c**6*d*e**4*f**3 + 14*sqrt(b)*sqrt(a)*atan((b*x)/ 
(sqrt(b)*sqrt(a)))*a**2*b**4*c**6*d*e**3*f**4*x**2 + 4*sqrt(b)*sqrt(a)*ata 
n((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**4*c**6*d*e**2*f**5*x**4 - 17*sqrt(b)*sq 
rt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b**4*c**5*d**2*e**4*f**3*x**2 ...