\(\int \frac {(a+b x^2)^3}{(c+d x^2)^3 (e+f x^2)^3} \, dx\) [252]

Optimal result

Integrand size = 28, antiderivative size = 621 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\frac {(b c-a d)^2 (a d (3 d e-11 c f)+b c (9 d e-c f)) x}{8 c^2 (d e-c f)^4 \left (c+d x^2\right )}+\frac {\left (6 a^2 b c d^2 e f^2-3 a b^2 c d e f (d e+c f)-a^3 d^2 f^2 (d e+c f)+b^3 \left (c d^2 e^3+c^3 e f^2\right )\right ) x}{4 c d^2 e (d e-c f)^3 \left (e+f x^2\right )^2}-\frac {(b c-a d)^3 x}{4 c d^2 (d e-c f) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2}+\frac {\left (3 a^2 b c d e f^2 (15 d e+c f)-3 a b^2 c d e^2 f (7 d e+9 c f)-a^3 d f^2 \left (4 d^2 e^2+15 c d e f-3 c^2 f^2\right )+b^3 c e^2 \left (3 d^2 e^2+9 c d e f+4 c^2 f^2\right )\right ) x}{8 c d e^2 (d e-c f)^4 \left (e+f x^2\right )}-\frac {3 (b c-a d) \left (2 a b c d \left (d^2 e^2-10 c d e f-7 c^2 f^2\right )+b^2 c^2 \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )+a^2 d^2 \left (d^2 e^2-6 c d e f+21 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d} (d e-c f)^5}-\frac {3 (b e-a f) \left (2 a b e f \left (7 d^2 e^2+10 c d e f-c^2 f^2\right )-a^2 f^2 \left (21 d^2 e^2-6 c d e f+c^2 f^2\right )-b^2 e^2 \left (d^2 e^2+10 c d e f+5 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} \sqrt {f} (d e-c f)^5} \] Output:

1/8*(-a*d+b*c)^2*(a*d*(-11*c*f+3*d*e)+b*c*(-c*f+9*d*e))*x/c^2/(-c*f+d*e)^4 
/(d*x^2+c)+1/4*(6*a^2*b*c*d^2*e*f^2-3*a*b^2*c*d*e*f*(c*f+d*e)-a^3*d^2*f^2* 
(c*f+d*e)+b^3*(c^3*e*f^2+c*d^2*e^3))*x/c/d^2/e/(-c*f+d*e)^3/(f*x^2+e)^2-1/ 
4*(-a*d+b*c)^3*x/c/d^2/(-c*f+d*e)/(d*x^2+c)^2/(f*x^2+e)^2+1/8*(3*a^2*b*c*d 
*e*f^2*(c*f+15*d*e)-3*a*b^2*c*d*e^2*f*(9*c*f+7*d*e)-a^3*d*f^2*(-3*c^2*f^2+ 
15*c*d*e*f+4*d^2*e^2)+b^3*c*e^2*(4*c^2*f^2+9*c*d*e*f+3*d^2*e^2))*x/c/d/e^2 
/(-c*f+d*e)^4/(f*x^2+e)-3/8*(-a*d+b*c)*(2*a*b*c*d*(-7*c^2*f^2-10*c*d*e*f+d 
^2*e^2)+b^2*c^2*(c^2*f^2+10*c*d*e*f+5*d^2*e^2)+a^2*d^2*(21*c^2*f^2-6*c*d*e 
*f+d^2*e^2))*arctan(d^(1/2)*x/c^(1/2))/c^(5/2)/d^(1/2)/(-c*f+d*e)^5-3/8*(- 
a*f+b*e)*(2*a*b*e*f*(-c^2*f^2+10*c*d*e*f+7*d^2*e^2)-a^2*f^2*(c^2*f^2-6*c*d 
*e*f+21*d^2*e^2)-b^2*e^2*(5*c^2*f^2+10*c*d*e*f+d^2*e^2))*arctan(f^(1/2)*x/ 
e^(1/2))/e^(5/2)/f^(1/2)/(-c*f+d*e)^5
 

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 453, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (\frac {2 (b c-a d)^3 x}{c (-d e+c f)^3 \left (c+d x^2\right )^2}+\frac {3 (b c-a d)^2 (a d (d e-5 c f)+b c (3 d e+c f)) x}{c^2 (d e-c f)^4 \left (c+d x^2\right )}+\frac {2 (b e-a f)^3 x}{e (d e-c f)^3 \left (e+f x^2\right )^2}+\frac {3 (b e-a f)^2 (a f (-5 d e+c f)+b e (d e+3 c f)) x}{e^2 (d e-c f)^4 \left (e+f x^2\right )}+\frac {3 (b c-a d) \left (b^2 c^2 \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )-2 a b c d \left (-d^2 e^2+10 c d e f+7 c^2 f^2\right )+a^2 d^2 \left (d^2 e^2-6 c d e f+21 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} \sqrt {d} (-d e+c f)^5}+\frac {3 (b e-a f) \left (-2 a b e f \left (7 d^2 e^2+10 c d e f-c^2 f^2\right )+a^2 f^2 \left (21 d^2 e^2-6 c d e f+c^2 f^2\right )+b^2 e^2 \left (d^2 e^2+10 c d e f+5 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{5/2} \sqrt {f} (d e-c f)^5}\right ) \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1540\) vs. \(2(621)=1242\).

Time = 2.20 (sec) , antiderivative size = 1540, normalized size of antiderivative = 2.48, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {425, 425, 402, 25, 402, 25, 27, 397, 218, 402, 27, 397, 218, 402, 27, 397, 218}

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

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 397

\(\displaystyle \frac {b \left (\frac {b \left (\frac {(b e-a f) x}{4 e (d e-c f) \left (f x^2+e\right )^2}-\frac {\frac {(a f (7 d e-3 c f)-b e (3 d e+c f)) x}{2 e (d e-c f) \left (f x^2+e\right )}+\frac {\frac {8 d^2 (b c-a d) e^2 \int \frac {1}{d x^2+c}dx}{d e-c f}-\frac {\left (b e \left (3 d^2 e^2+6 c d f e-c^2 f^2\right )-a f \left (15 d^2 e^2-10 c d f e+3 c^2 f^2\right )\right ) \int \frac {1}{f x^2+e}dx}{d e-c f}}{2 e (d e-c f)}}{4 e (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\int \frac {-3 d f (3 b c e-2 a d e-a c f) x^2+b c e (2 d e+c f)+a \left (2 d^2 e^2-8 c d f e+3 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{2 e (d e-c f)}-\frac {f (3 b c e-2 a d e-a c f) x}{2 e (d e-c f) \left (f x^2+e\right )^2}}{2 c (d e-c f)}-\frac {(b c-a d) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}\right )}{d}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {\frac {\int \frac {-3 d f (3 b c e-2 a d e-a c f) x^2+b c e (2 d e+c f)+a \left (2 d^2 e^2-8 c d f e+3 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{2 e (d e-c f)}-\frac {f (3 b c e-2 a d e-a c f) x}{2 e (d e-c f) \left (f x^2+e\right )^2}}{2 c (d e-c f)}-\frac {(b c-a d) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {(a d (3 d e-11 c f)+b c (d e+7 c f)) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}+\frac {\int \frac {5 f (a d (3 d e-11 c f)+b c (d e+7 c f)) x^2+b c e (d e-9 c f)+a \left (3 d^2 e^2-3 c d f e+8 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{2 c (d e-c f)}}{4 c (d e-c f)}-\frac {(b c-a d) x}{4 c (d e-c f) \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {b \left (\frac {(b e-a f) x}{4 e (d e-c f) \left (f x^2+e\right )^2}-\frac {\frac {(a f (7 d e-3 c f)-b e (3 d e+c f)) x}{2 e (d e-c f) \left (f x^2+e\right )}+\frac {\frac {8 d^{3/2} (b c-a d) e^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (d e-c f)}-\frac {\left (b e \left (3 d^2 e^2+6 c d f e-c^2 f^2\right )-a f \left (15 d^2 e^2-10 c d f e+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f} (d e-c f)}}{2 e (d e-c f)}}{4 e (d e-c f)}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {\int \frac {-3 d f (3 b c e-2 a d e-a c f) x^2+b c e (2 d e+c f)+a \left (2 d^2 e^2-8 c d f e+3 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{2 e (d e-c f)}-\frac {f (3 b c e-2 a d e-a c f) x}{2 e (d e-c f) \left (f x^2+e\right )^2}}{2 c (d e-c f)}-\frac {(b c-a d) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}\right )}{d}\right )}{d}-\frac {(b c-a d) \left (\frac {b \left (\frac {\frac {\int \frac {-3 d f (3 b c e-2 a d e-a c f) x^2+b c e (2 d e+c f)+a \left (2 d^2 e^2-8 c d f e+3 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^2}dx}{2 e (d e-c f)}-\frac {f (3 b c e-2 a d e-a c f) x}{2 e (d e-c f) \left (f x^2+e\right )^2}}{2 c (d e-c f)}-\frac {(b c-a d) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}\right )}{d}-\frac {(b c-a d) \left (\frac {\frac {(a d (3 d e-11 c f)+b c (d e+7 c f)) x}{2 c (d e-c f) \left (d x^2+c\right ) \left (f x^2+e\right )^2}+\frac {\int \frac {5 f (a d (3 d e-11 c f)+b c (d e+7 c f)) x^2+b c e (d e-9 c f)+a \left (3 d^2 e^2-3 c d f e+8 c^2 f^2\right )}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{2 c (d e-c f)}}{4 c (d e-c f)}-\frac {(b c-a d) x}{4 c (d e-c f) \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(b*((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*ArcTan[(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)/Sq 
rt[c]])/(Sqrt[c]*(d*e - c*f)) - (c*Sqrt[f]*(b*e*(15*d^2*e^2 + 10*c*d*e*f - 
 c^2*f^2) - a*f*(35*d^2*e^2 - 14*c*d*e*f + 3*c^2*f^2))*ArcTan[(Sqrt[f]*x)/ 
Sqrt[e]])/(Sqrt[e]*(d*e - c*f)))/(2*e*(d*e - c*f)))/(2*e*(d*e - c*f)))/(2* 
c*(d*e - c*f))))/d))/d - ((b*c - a*d)*((b*(-1/2*((b*c - a*d)*x)/(c*(d*e - 
c*f)*(c + d*x^2)*(e + f*x^2)^2) + (-1/2*(f*(3*b*c*e - 2*a*d*e - a*c*f)*x)/ 
(e*(d*e - c*f)*(e + f*x^2)^2) + (-1/2*(f*(b*c*e*(11*d*e + c*f) - a*(4*d^2* 
e^2 + 11*c*d*e*f - 3*c^2*f^2))*x)/(e*(d*e - c*f)*(e + f*x^2)) + ((4*d^(3/2 
)*e^2*(a*d*(d*e - 7*c*f) + b*c*(d*e + 5*c*f))*ArcTan[(Sqrt[d]*x)/Sqrt[c]]) 
/(Sqrt[c]*(d*e - c*f)) - (c*Sqrt[f]*(b*e*(15*d^2*e^2 + 10*c*d*e*f - c^2*f^ 
2) - a*f*(35*d^2*e^2 - 14*c*d*e*f + 3*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt...
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 218

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

rule 397

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

rule 402

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

rule 425

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

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 1051, normalized size of antiderivative = 1.69

method	result	size
default	\(\text {Expression too large to display}\)	\(1051\)
risch	\(\text {Expression too large to display}\)	\(3271\)

Input:

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

Output:

1/(c*f-d*e)^5*((3/8*f*(a^3*c^2*f^5-6*a^3*c*d*e*f^4+5*a^3*d^2*e^2*f^3+a^2*b 
*c^2*e*f^4+10*a^2*b*c*d*e^2*f^3-11*a^2*b*d^2*e^3*f^2-5*a*b^2*c^2*e^2*f^3-2 
*a*b^2*c*d*e^3*f^2+7*a*b^2*d^2*e^4*f+3*b^3*c^2*e^3*f^2-2*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^5-22*a^3*c*d*e*f^4+17*a^3*d^2*e^2*f^3-3 
*a^2*b*c^2*e*f^4+42*a^2*b*c*d*e^2*f^3-39*a^2*b*d^2*e^3*f^2-9*a*b^2*c^2*e^2 
*f^3-18*a*b^2*c*d*e^3*f^2+27*a*b^2*d^2*e^4*f+7*b^3*c^2*e^3*f^2-2*b^3*c*d*e 
^4*f-5*b^3*d^2*e^5)/e*x)/(f*x^2+e)^2+3/8*(a^3*c^2*f^5-6*a^3*c*d*e*f^4+21*a 
^3*d^2*e^2*f^3+a^2*b*c^2*e*f^4-14*a^2*b*c*d*e^2*f^3-35*a^2*b*d^2*e^3*f^2+3 
*a*b^2*c^2*e^2*f^3+30*a*b^2*c*d*e^3*f^2+15*a*b^2*d^2*e^4*f-5*b^3*c^2*e^3*f 
^2-10*b^3*c*d*e^4*f-b^3*d^2*e^5)/e^2/(e*f)^(1/2)*arctan(f*x/(e*f)^(1/2)))- 
1/(c*f-d*e)^5*((3/8*d*(5*a^3*c^2*d^3*f^2-6*a^3*c*d^4*e*f+a^3*d^5*e^2-11*a^ 
2*b*c^3*d^2*f^2+10*a^2*b*c^2*d^3*e*f+a^2*b*c*d^4*e^2+7*a*b^2*c^4*d*f^2-2*a 
*b^2*c^3*d^2*e*f-5*a*b^2*c^2*d^3*e^2-b^3*c^5*f^2-2*b^3*c^4*d*e*f+3*b^3*c^3 
*d^2*e^2)/c^2*x^3+1/8*(17*a^3*c^2*d^3*f^2-22*a^3*c*d^4*e*f+5*a^3*d^5*e^2-3 
9*a^2*b*c^3*d^2*f^2+42*a^2*b*c^2*d^3*e*f-3*a^2*b*c*d^4*e^2+27*a*b^2*c^4*d* 
f^2-18*a*b^2*c^3*d^2*e*f-9*a*b^2*c^2*d^3*e^2-5*b^3*c^5*f^2-2*b^3*c^4*d*e*f 
+7*b^3*c^3*d^2*e^2)/c*x)/(d*x^2+c)^2+3/8*(21*a^3*c^2*d^3*f^2-6*a^3*c*d^4*e 
*f+a^3*d^5*e^2-35*a^2*b*c^3*d^2*f^2-14*a^2*b*c^2*d^3*e*f+a^2*b*c*d^4*e^2+1 
5*a*b^2*c^4*d*f^2+30*a*b^2*c^3*d^2*e*f+3*a*b^2*c^2*d^3*e^2-b^3*c^5*f^2-10* 
b^3*c^4*d*e*f-5*b^3*c^3*d^2*e^2)/c^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2)...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1605 vs. \(2 (591) = 1182\).

Time = 0.13 (sec) , antiderivative size = 1605, normalized size of antiderivative = 2.58 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-3/8*(5*b^3*c^3*d^2*e^2 - 3*a*b^2*c^2*d^3*e^2 - a^2*b*c*d^4*e^2 - a^3*d^5* 
e^2 + 10*b^3*c^4*d*e*f - 30*a*b^2*c^3*d^2*e*f + 14*a^2*b*c^2*d^3*e*f + 6*a 
^3*c*d^4*e*f + b^3*c^5*f^2 - 15*a*b^2*c^4*d*f^2 + 35*a^2*b*c^3*d^2*f^2 - 2 
1*a^3*c^2*d^3*f^2)*arctan(d*x/sqrt(c*d))/((c^2*d^5*e^5 - 5*c^3*d^4*e^4*f + 
 10*c^4*d^3*e^3*f^2 - 10*c^5*d^2*e^2*f^3 + 5*c^6*d*e*f^4 - c^7*f^5)*sqrt(c 
*d)) + 3/8*(b^3*d^2*e^5 + 10*b^3*c*d*e^4*f - 15*a*b^2*d^2*e^4*f + 5*b^3*c^ 
2*e^3*f^2 - 30*a*b^2*c*d*e^3*f^2 + 35*a^2*b*d^2*e^3*f^2 - 3*a*b^2*c^2*e^2* 
f^3 + 14*a^2*b*c*d*e^2*f^3 - 21*a^3*d^2*e^2*f^3 - a^2*b*c^2*e*f^4 + 6*a^3* 
c*d*e*f^4 - a^3*c^2*f^5)*arctan(f*x/sqrt(e*f))/((d^5*e^7 - 5*c*d^4*e^6*f + 
 10*c^2*d^3*e^5*f^2 - 10*c^3*d^2*e^4*f^3 + 5*c^4*d*e^3*f^4 - c^5*e^2*f^5)* 
sqrt(e*f)) + 1/8*(3*b^3*c^2*d^3*e^4*f*x^7 + 18*b^3*c^3*d^2*e^3*f^2*x^7 - 3 
6*a*b^2*c^2*d^3*e^3*f^2*x^7 + 3*a^2*b*c*d^4*e^3*f^2*x^7 + 3*a^3*d^5*e^3*f^ 
2*x^7 + 3*b^3*c^4*d*e^2*f^3*x^7 - 36*a*b^2*c^3*d^2*e^2*f^3*x^7 + 66*a^2*b* 
c^2*d^3*e^2*f^3*x^7 - 15*a^3*c*d^4*e^2*f^3*x^7 + 3*a^2*b*c^3*d^2*e*f^4*x^7 
 - 15*a^3*c^2*d^3*e*f^4*x^7 + 3*a^3*c^3*d^2*f^5*x^7 + 5*b^3*c^2*d^3*e^5*x^ 
5 + 31*b^3*c^3*d^2*e^4*f*x^5 - 57*a*b^2*c^2*d^3*e^4*f*x^5 + 6*a^2*b*c*d^4* 
e^4*f*x^5 + 6*a^3*d^5*e^4*f*x^5 + 31*b^3*c^4*d*e^3*f^2*x^5 - 102*a*b^2*c^3 
*d^2*e^3*f^2*x^5 + 102*a^2*b*c^2*d^3*e^3*f^2*x^5 - 25*a^3*c*d^4*e^3*f^2*x^ 
5 + 5*b^3*c^5*e^2*f^3*x^5 - 57*a*b^2*c^4*d*e^2*f^3*x^5 + 102*a^2*b*c^3*d^2 
*e^2*f^3*x^5 - 34*a^3*c^2*d^3*e^2*f^3*x^5 + 6*a^2*b*c^4*d*e*f^4*x^5 - 2...
 

Mupad [B] (verification not implemented)

Time = 24.53 (sec) , antiderivative size = 227222, normalized size of antiderivative = 365.90 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((x^3*(3*a^3*c^5*f^5 + 3*a^3*d^5*e^5 + 19*b^3*c^3*d^2*e^5 + 19*b^3*c^5*e^3 
*f^2 - 34*a^3*c^2*d^3*e^3*f^2 - 34*a^3*c^3*d^2*e^2*f^3 + 3*a^2*b*c*d^4*e^5 
 + 3*a^2*b*c^5*e*f^4 - 5*a^3*c*d^4*e^4*f - 5*a^3*c^4*d*e*f^4 + 34*b^3*c^4* 
d*e^4*f - 15*a*b^2*c^2*d^3*e^5 - 15*a*b^2*c^5*e^2*f^3 - 93*a*b^2*c^3*d^2*e 
^4*f - 93*a*b^2*c^4*d*e^3*f^2 + 27*a^2*b*c^2*d^3*e^4*f + 27*a^2*b*c^4*d*e^ 
2*f^3 + 156*a^2*b*c^3*d^2*e^3*f^2))/(8*c^2*e^2*(c^4*f^4 + d^4*e^4 + 6*c^2* 
d^2*e^2*f^2 - 4*c*d^3*e^3*f - 4*c^3*d*e*f^3)) - (x*(3*a^2*b*c*d^3*e^4 - 5* 
a^3*d^4*e^4 - 12*b^3*c^3*d*e^4 - 12*b^3*c^4*e^3*f - 5*a^3*c^4*f^4 + 3*a^2* 
b*c^4*e*f^3 + 17*a^3*c*d^3*e^3*f + 17*a^3*c^3*d*e*f^3 + 9*a*b^2*c^2*d^2*e^ 
4 + 9*a*b^2*c^4*e^2*f^2 - 39*a^2*b*c^2*d^2*e^3*f - 39*a^2*b*c^3*d*e^2*f^2 
+ 54*a*b^2*c^3*d*e^3*f))/(8*c*e*(c^4*f^4 + d^4*e^4 + 6*c^2*d^2*e^2*f^2 - 4 
*c*d^3*e^3*f - 4*c^3*d*e*f^3)) + (x^5*(6*a^3*c^4*d*f^5 + 6*a^3*d^5*e^4*f + 
 5*b^3*c^2*d^3*e^5 + 5*b^3*c^5*e^2*f^3 - 34*a^3*c^2*d^3*e^2*f^3 - 25*a^3*c 
*d^4*e^3*f^2 - 25*a^3*c^3*d^2*e*f^4 + 31*b^3*c^3*d^2*e^4*f + 31*b^3*c^4*d* 
e^3*f^2 - 57*a*b^2*c^2*d^3*e^4*f - 57*a*b^2*c^4*d*e^2*f^3 - 102*a*b^2*c^3* 
d^2*e^3*f^2 + 102*a^2*b*c^2*d^3*e^3*f^2 + 102*a^2*b*c^3*d^2*e^2*f^3 + 6*a^ 
2*b*c*d^4*e^4*f + 6*a^2*b*c^4*d*e*f^4))/(8*c^2*e^2*(c^4*f^4 + d^4*e^4 + 6* 
c^2*d^2*e^2*f^2 - 4*c*d^3*e^3*f - 4*c^3*d*e*f^3)) + (3*d*f*x^7*(a^3*c^3*d* 
f^4 + a^3*d^4*e^3*f + b^3*c^2*d^2*e^4 + b^3*c^4*e^2*f^2 + 6*b^3*c^3*d*e^3* 
f - 5*a^3*c*d^3*e^2*f^2 - 5*a^3*c^2*d^2*e*f^3 - 12*a*b^2*c^2*d^2*e^3*f ...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 7120, normalized size of antiderivative = 11.47 \[ \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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