\(\int \frac {(a+b x^2)^3 (c+d x^2)^3}{(e+f x^2)^4} \, dx\) [246]

Optimal result

Integrand size = 28, antiderivative size = 554 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\frac {b d \left (3 a^2 d^2 f^2-3 a b d f (4 d e-3 c f)+b^2 \left (10 d^2 e^2-12 c d e f+3 c^2 f^2\right )\right ) x}{f^6}-\frac {b^2 d^2 (4 b d e-3 b c f-3 a d f) x^3}{3 f^5}+\frac {b^3 d^3 x^5}{5 f^4}+\frac {(b e-a f)^3 (d e-c f)^3 x}{6 e f^6 \left (e+f x^2\right )^3}-\frac {(b e-a f)^2 (d e-c f)^2 (b e (31 d e-13 c f)-a f (13 d e+5 c f)) x}{24 e^2 f^6 \left (e+f x^2\right )^2}-\frac {(b e-a f) (d e-c f) \left (4 a b e f \left (19 d^2 e^2-5 c d e f-2 c^2 f^2\right )-a^2 f^2 \left (11 d^2 e^2+8 c d e f+5 c^2 f^2\right )-b^2 e^2 \left (89 d^2 e^2-76 c d e f+11 c^2 f^2\right )\right ) x}{16 e^3 f^6 \left (e+f x^2\right )}-\frac {\left (b^3 e^3 \left (231 d^3 e^3-315 c d^2 e^2 f+105 c^2 d e f^2-5 c^3 f^3\right )+3 a^2 b e f^2 \left (35 d^3 e^3-15 c d^2 e^2 f-3 c^2 d e f^2-c^3 f^3\right )-3 a b^2 e^2 f \left (105 d^3 e^3-105 c d^2 e^2 f+15 c^2 d e f^2+c^3 f^3\right )-a^3 f^3 \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{16 e^{7/2} f^{13/2}} \] Output:

b*d*(3*a^2*d^2*f^2-3*a*b*d*f*(-3*c*f+4*d*e)+b^2*(3*c^2*f^2-12*c*d*e*f+10*d 
^2*e^2))*x/f^6-1/3*b^2*d^2*(-3*a*d*f-3*b*c*f+4*b*d*e)*x^3/f^5+1/5*b^3*d^3* 
x^5/f^4+1/6*(-a*f+b*e)^3*(-c*f+d*e)^3*x/e/f^6/(f*x^2+e)^3-1/24*(-a*f+b*e)^ 
2*(-c*f+d*e)^2*(b*e*(-13*c*f+31*d*e)-a*f*(5*c*f+13*d*e))*x/e^2/f^6/(f*x^2+ 
e)^2-1/16*(-a*f+b*e)*(-c*f+d*e)*(4*a*b*e*f*(-2*c^2*f^2-5*c*d*e*f+19*d^2*e^ 
2)-a^2*f^2*(5*c^2*f^2+8*c*d*e*f+11*d^2*e^2)-b^2*e^2*(11*c^2*f^2-76*c*d*e*f 
+89*d^2*e^2))*x/e^3/f^6/(f*x^2+e)-1/16*(b^3*e^3*(-5*c^3*f^3+105*c^2*d*e*f^ 
2-315*c*d^2*e^2*f+231*d^3*e^3)+3*a^2*b*e*f^2*(-c^3*f^3-3*c^2*d*e*f^2-15*c* 
d^2*e^2*f+35*d^3*e^3)-3*a*b^2*e^2*f*(c^3*f^3+15*c^2*d*e*f^2-105*c*d^2*e^2* 
f+105*d^3*e^3)-a^3*f^3*(5*c^3*f^3+3*c^2*d*e*f^2+3*c*d^2*e^2*f+5*d^3*e^3))* 
arctan(f^(1/2)*x/e^(1/2))/e^(7/2)/f^(13/2)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\frac {b d \left (3 a^2 d^2 f^2+3 a b d f (-4 d e+3 c f)+b^2 \left (10 d^2 e^2-12 c d e f+3 c^2 f^2\right )\right ) x}{f^6}-\frac {b^2 d^2 (4 b d e-3 b c f-3 a d f) x^3}{3 f^5}+\frac {b^3 d^3 x^5}{5 f^4}+\frac {(b e-a f)^3 (d e-c f)^3 x}{6 e f^6 \left (e+f x^2\right )^3}-\frac {(b e-a f)^2 (d e-c f)^2 (b e (31 d e-13 c f)-a f (13 d e+5 c f)) x}{24 e^2 f^6 \left (e+f x^2\right )^2}+\frac {(b e-a f) (d e-c f) \left (4 a b e f \left (-19 d^2 e^2+5 c d e f+2 c^2 f^2\right )+a^2 f^2 \left (11 d^2 e^2+8 c d e f+5 c^2 f^2\right )+b^2 e^2 \left (89 d^2 e^2-76 c d e f+11 c^2 f^2\right )\right ) x}{16 e^3 f^6 \left (e+f x^2\right )}-\frac {\left (b^3 e^3 \left (231 d^3 e^3-315 c d^2 e^2 f+105 c^2 d e f^2-5 c^3 f^3\right )+3 a^2 b e f^2 \left (35 d^3 e^3-15 c d^2 e^2 f-3 c^2 d e f^2-c^3 f^3\right )-3 a b^2 e^2 f \left (105 d^3 e^3-105 c d^2 e^2 f+15 c^2 d e f^2+c^3 f^3\right )-a^3 f^3 \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{16 e^{7/2} f^{13/2}} \] Input:

Integrate[((a + b*x^2)^3*(c + d*x^2)^3)/(e + f*x^2)^4,x]
 

Output:

(b*d*(3*a^2*d^2*f^2 + 3*a*b*d*f*(-4*d*e + 3*c*f) + b^2*(10*d^2*e^2 - 12*c* 
d*e*f + 3*c^2*f^2))*x)/f^6 - (b^2*d^2*(4*b*d*e - 3*b*c*f - 3*a*d*f)*x^3)/( 
3*f^5) + (b^3*d^3*x^5)/(5*f^4) + ((b*e - a*f)^3*(d*e - c*f)^3*x)/(6*e*f^6* 
(e + f*x^2)^3) - ((b*e - a*f)^2*(d*e - c*f)^2*(b*e*(31*d*e - 13*c*f) - a*f 
*(13*d*e + 5*c*f))*x)/(24*e^2*f^6*(e + f*x^2)^2) + ((b*e - a*f)*(d*e - c*f 
)*(4*a*b*e*f*(-19*d^2*e^2 + 5*c*d*e*f + 2*c^2*f^2) + a^2*f^2*(11*d^2*e^2 + 
 8*c*d*e*f + 5*c^2*f^2) + b^2*e^2*(89*d^2*e^2 - 76*c*d*e*f + 11*c^2*f^2))* 
x)/(16*e^3*f^6*(e + f*x^2)) - ((b^3*e^3*(231*d^3*e^3 - 315*c*d^2*e^2*f + 1 
05*c^2*d*e*f^2 - 5*c^3*f^3) + 3*a^2*b*e*f^2*(35*d^3*e^3 - 15*c*d^2*e^2*f - 
 3*c^2*d*e*f^2 - c^3*f^3) - 3*a*b^2*e^2*f*(105*d^3*e^3 - 105*c*d^2*e^2*f + 
 15*c^2*d*e*f^2 + c^3*f^3) - a^3*f^3*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d* 
e*f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(13/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1308\) vs. \(2(554)=1108\).

Time = 1.91 (sec) , antiderivative size = 1308, normalized size of antiderivative = 2.36, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {425, 425, 401, 25, 401, 25, 401, 299, 218, 403, 25, 299, 218, 403, 299, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \frac {\left (d x^2+c\right )^2 \left (d (7 b e-5 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {\int \frac {\left (d x^2+c\right ) \left (c (3 a f (d e-c f)-b e (7 d e+c f))-d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e (7 d e-c f)-3 a f (c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {\int \frac {\left (d x^2+c\right ) \left (c (3 a f (d e-c f)-b e (7 d e+c f))-d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e (7 d e-c f)-3 a f (c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {\int -\frac {\left (d x^2+c\right ) \left (d (b e (35 d e-c f)-5 a f (d e+c f)) x^2+c (d e (7 b e-a f)+3 c f (b e+5 a f))\right )}{\left (f x^2+e\right )^2}dx}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e (7 d e-c f)-a f (5 c f+d e))}{4 e f \left (e+f x^2\right )^2}}{6 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{6 e f \left (e+f x^2\right )^3}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \frac {\left (d x^2+c\right )^2 \left (d (7 b e-5 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {\int \frac {\left (d x^2+c\right ) \left (c (3 a f (d e-c f)-b e (7 d e+c f))-d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e (7 d e-c f)-3 a f (c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {\int \frac {\left (d x^2+c\right ) \left (c (3 a f (d e-c f)-b e (7 d e+c f))-d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^2 (b e (7 d e-c f)-3 a f (c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {\int \frac {\left (d x^2+c\right ) \left (d (b e (35 d e-c f)-5 a f (d e+c f)) x^2+c (d e (7 b e-a f)+3 c f (b e+5 a f))\right )}{\left (f x^2+e\right )^2}dx}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e (7 d e-c f)-a f (5 c f+d e))}{4 e f \left (e+f x^2\right )^2}}{6 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{6 e f \left (e+f x^2\right )^3}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \frac {\left (d x^2+c\right )^2 \left (d (7 b e-5 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (3 a f (d e-c f)-b e (7 d e+c f))-d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{f x^2+e}dx}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (3 a f (d e-c f)-b e (7 d e+c f))-d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{f x^2+e}dx}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {-\frac {\left (b e \left (35 d^2 e^2-8 c d f e-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d f e+15 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{2 e f \left (f x^2+e\right )}-\frac {\int \frac {c \left (a f \left (5 d^2 e^2+6 c d f e-15 c^2 f^2\right )-b e \left (35 d^2 e^2+6 c d f e+3 c^2 f^2\right )\right )-d \left (b e \left (105 d^2 e^2-10 c d f e-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d f e+15 c^2 f^2\right )\right ) x^2}{f x^2+e}dx}{2 e f}}{4 e f}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (d x^2+c\right )^2}{4 e f \left (f x^2+e\right )^2}}{6 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{6 e f \left (f x^2+e\right )^3}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \frac {\left (d x^2+c\right )^2 \left (d (7 b e-5 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (3 a f (d e-c f)-b e (7 d e+c f))-d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{f x^2+e}dx}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (3 a f (d e-c f)-b e (7 d e+c f))-d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{f x^2+e}dx}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {-\frac {\left (b e \left (35 d^2 e^2-8 c d f e-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d f e+15 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{2 e f \left (f x^2+e\right )}-\frac {\frac {3 \left (b e \left (35 d^3 e^3-15 c d^2 f e^2-3 c^2 d f^2 e-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 f e^2+3 c^2 d f^2 e+5 c^3 f^3\right )\right ) \int \frac {1}{f x^2+e}dx}{f}-\frac {d \left (b e \left (105 d^2 e^2-10 c d f e-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d f e+15 c^2 f^2\right )\right ) x}{f}}{2 e f}}{4 e f}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (d x^2+c\right )^2}{4 e f \left (f x^2+e\right )^2}}{6 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{6 e f \left (f x^2+e\right )^3}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \frac {\left (d x^2+c\right )^2 \left (d (7 b e-5 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (3 a f (d e-c f)-b e (7 d e+c f))-d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{f x^2+e}dx}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (3 a f (d e-c f)-b e (7 d e+c f))-d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{f x^2+e}dx}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {-\frac {\left (b e \left (35 d^2 e^2-8 c d f e-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d f e+15 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{2 e f \left (f x^2+e\right )}-\frac {\frac {3 \left (b e \left (35 d^3 e^3-15 c d^2 f e^2-3 c^2 d f^2 e-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 f e^2+3 c^2 d f^2 e+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}-\frac {d \left (b e \left (105 d^2 e^2-10 c d f e-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d f e+15 c^2 f^2\right )\right ) x}{f}}{2 e f}}{4 e f}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (d x^2+c\right )^2}{4 e f \left (f x^2+e\right )^2}}{6 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{6 e f \left (f x^2+e\right )^3}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {d (7 b e-5 a f) x \left (d x^2+c\right )^2}{5 f}+\frac {\int -\frac {\left (d x^2+c\right ) \left (d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x^2+c (b e (7 d e-5 c f)-5 a f (d e+c f))\right )}{f x^2+e}dx}{5 f}}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {\int \frac {c \left (b e \left (35 d^2 e^2-24 c d f e-3 c^2 f^2\right )-3 a f \left (5 d^2 e^2+3 c^2 f^2\right )\right )-d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x^2}{f x^2+e}dx}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {\int \frac {c \left (b e \left (35 d^2 e^2-24 c d f e-3 c^2 f^2\right )-3 a f \left (5 d^2 e^2+3 c^2 f^2\right )\right )-d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x^2}{f x^2+e}dx}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {-\frac {\left (b e \left (35 d^2 e^2-8 c d f e-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d f e+15 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{2 e f \left (f x^2+e\right )}-\frac {\frac {3 \left (b e \left (35 d^3 e^3-15 c d^2 f e^2-3 c^2 d f^2 e-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 f e^2+3 c^2 d f^2 e+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}-\frac {d \left (b e \left (105 d^2 e^2-10 c d f e-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d f e+15 c^2 f^2\right )\right ) x}{f}}{2 e f}}{4 e f}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (d x^2+c\right )^2}{4 e f \left (f x^2+e\right )^2}}{6 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{6 e f \left (f x^2+e\right )^3}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {d (7 b e-5 a f) x \left (d x^2+c\right )^2}{5 f}-\frac {\int \frac {\left (d x^2+c\right ) \left (d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x^2+c (b e (7 d e-5 c f)-5 a f (d e+c f))\right )}{f x^2+e}dx}{5 f}}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {\int \frac {c \left (b e \left (35 d^2 e^2-24 c d f e-3 c^2 f^2\right )-3 a f \left (5 d^2 e^2+3 c^2 f^2\right )\right )-d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x^2}{f x^2+e}dx}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {\int \frac {c \left (b e \left (35 d^2 e^2-24 c d f e-3 c^2 f^2\right )-3 a f \left (5 d^2 e^2+3 c^2 f^2\right )\right )-d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x^2}{f x^2+e}dx}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {-\frac {\left (b e \left (35 d^2 e^2-8 c d f e-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d f e+15 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{2 e f \left (f x^2+e\right )}-\frac {\frac {3 \left (b e \left (35 d^3 e^3-15 c d^2 f e^2-3 c^2 d f^2 e-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 f e^2+3 c^2 d f^2 e+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}-\frac {d \left (b e \left (105 d^2 e^2-10 c d f e-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d f e+15 c^2 f^2\right )\right ) x}{f}}{2 e f}}{4 e f}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (d x^2+c\right )^2}{4 e f \left (f x^2+e\right )^2}}{6 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{6 e f \left (f x^2+e\right )^3}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {d (7 b e-5 a f) x \left (d x^2+c\right )^2}{5 f}-\frac {\int \frac {\left (d x^2+c\right ) \left (d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x^2+c (b e (7 d e-5 c f)-5 a f (d e+c f))\right )}{f x^2+e}dx}{5 f}}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {-\frac {d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x}{f}-\frac {3 (d e-c f) \left (b e \left (35 d^2 e^2-10 c d f e-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) \int \frac {1}{f x^2+e}dx}{f}}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {-\frac {d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x}{f}-\frac {3 (d e-c f) \left (b e \left (35 d^2 e^2-10 c d f e-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) \int \frac {1}{f x^2+e}dx}{f}}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {-\frac {\left (b e \left (35 d^2 e^2-8 c d f e-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d f e+15 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{2 e f \left (f x^2+e\right )}-\frac {\frac {3 \left (b e \left (35 d^3 e^3-15 c d^2 f e^2-3 c^2 d f^2 e-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 f e^2+3 c^2 d f^2 e+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}-\frac {d \left (b e \left (105 d^2 e^2-10 c d f e-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d f e+15 c^2 f^2\right )\right ) x}{f}}{2 e f}}{4 e f}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (d x^2+c\right )^2}{4 e f \left (f x^2+e\right )^2}}{6 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{6 e f \left (f x^2+e\right )^3}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {d (7 b e-5 a f) x \left (d x^2+c\right )^2}{5 f}-\frac {\int \frac {\left (d x^2+c\right ) \left (d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x^2+c (b e (7 d e-5 c f)-5 a f (d e+c f))\right )}{f x^2+e}dx}{5 f}}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {-\frac {d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x}{f}-\frac {3 (d e-c f) \left (b e \left (35 d^2 e^2-10 c d f e-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {-\frac {d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x}{f}-\frac {3 (d e-c f) \left (b e \left (35 d^2 e^2-10 c d f e-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {-\frac {\left (b e \left (35 d^2 e^2-8 c d f e-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d f e+15 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{2 e f \left (f x^2+e\right )}-\frac {\frac {3 \left (b e \left (35 d^3 e^3-15 c d^2 f e^2-3 c^2 d f^2 e-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 f e^2+3 c^2 d f^2 e+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}-\frac {d \left (b e \left (105 d^2 e^2-10 c d f e-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d f e+15 c^2 f^2\right )\right ) x}{f}}{2 e f}}{4 e f}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (d x^2+c\right )^2}{4 e f \left (f x^2+e\right )^2}}{6 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{6 e f \left (f x^2+e\right )^3}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {d (7 b e-5 a f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x \left (d x^2+c\right )}{3 f}+\frac {\int \frac {d \left (5 a f \left (15 d^2 e^2-22 c d f e+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d f e+81 c^2 f^2\right )\right ) x^2+c \left (5 a f \left (5 d^2 e^2-6 c d f e-3 c^2 f^2\right )-b e \left (35 d^2 e^2-54 c d f e+15 c^2 f^2\right )\right )}{f x^2+e}dx}{3 f}}{5 f}}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {-\frac {d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x}{f}-\frac {3 (d e-c f) \left (b e \left (35 d^2 e^2-10 c d f e-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {-\frac {d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x}{f}-\frac {3 (d e-c f) \left (b e \left (35 d^2 e^2-10 c d f e-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {-\frac {\left (b e \left (35 d^2 e^2-8 c d f e-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d f e+15 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{2 e f \left (f x^2+e\right )}-\frac {\frac {3 \left (b e \left (35 d^3 e^3-15 c d^2 f e^2-3 c^2 d f^2 e-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 f e^2+3 c^2 d f^2 e+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}-\frac {d \left (b e \left (105 d^2 e^2-10 c d f e-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d f e+15 c^2 f^2\right )\right ) x}{f}}{2 e f}}{4 e f}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (d x^2+c\right )^2}{4 e f \left (f x^2+e\right )^2}}{6 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{6 e f \left (f x^2+e\right )^3}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {d (7 b e-5 a f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x \left (d x^2+c\right )}{3 f}+\frac {\frac {15 (b e (7 d e-c f)-a f (5 d e+c f)) \int \frac {1}{f x^2+e}dx (d e-c f)^2}{f}+\frac {d \left (5 a f \left (15 d^2 e^2-22 c d f e+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d f e+81 c^2 f^2\right )\right ) x}{f}}{3 f}}{5 f}}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {-\frac {d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x}{f}-\frac {3 (d e-c f) \left (b e \left (35 d^2 e^2-10 c d f e-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {-\frac {d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x}{f}-\frac {3 (d e-c f) \left (b e \left (35 d^2 e^2-10 c d f e-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {-\frac {\left (b e \left (35 d^2 e^2-8 c d f e-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d f e+15 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{2 e f \left (f x^2+e\right )}-\frac {\frac {3 \left (b e \left (35 d^3 e^3-15 c d^2 f e^2-3 c^2 d f^2 e-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 f e^2+3 c^2 d f^2 e+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}-\frac {d \left (b e \left (105 d^2 e^2-10 c d f e-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d f e+15 c^2 f^2\right )\right ) x}{f}}{2 e f}}{4 e f}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (d x^2+c\right )^2}{4 e f \left (f x^2+e\right )^2}}{6 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{6 e f \left (f x^2+e\right )^3}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {d (7 b e-5 a f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x \left (d x^2+c\right )}{3 f}+\frac {\frac {15 (b e (7 d e-c f)-a f (5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^2}{\sqrt {e} f^{3/2}}+\frac {d \left (5 a f \left (15 d^2 e^2-22 c d f e+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d f e+81 c^2 f^2\right )\right ) x}{f}}{3 f}}{5 f}}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {-\frac {d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x}{f}-\frac {3 (d e-c f) \left (b e \left (35 d^2 e^2-10 c d f e-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (d x^2+c\right )^2}{2 e f \left (f x^2+e\right )}-\frac {\frac {-\frac {d \left (3 a f \left (15 d^2 e^2-4 c d f e-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d f e+3 c^2 f^2\right )\right ) x}{f}-\frac {3 (d e-c f) \left (b e \left (35 d^2 e^2-10 c d f e-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}}{3 f}-\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (d x^2+c\right )}{3 f}}{2 e f}}{4 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{4 e f \left (f x^2+e\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {-\frac {\left (b e \left (35 d^2 e^2-8 c d f e-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d f e+15 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{2 e f \left (f x^2+e\right )}-\frac {\frac {3 \left (b e \left (35 d^3 e^3-15 c d^2 f e^2-3 c^2 d f^2 e-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 f e^2+3 c^2 d f^2 e+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{3/2}}-\frac {d \left (b e \left (105 d^2 e^2-10 c d f e-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d f e+15 c^2 f^2\right )\right ) x}{f}}{2 e f}}{4 e f}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (d x^2+c\right )^2}{4 e f \left (f x^2+e\right )^2}}{6 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{6 e f \left (f x^2+e\right )^3}\right )}{f}\right )}{f}\)

Input:

Int[((a + b*x^2)^3*(c + d*x^2)^3)/(e + f*x^2)^4,x]
 

Output:

(b*((b*(-1/2*((b*e - a*f)*x*(c + d*x^2)^3)/(e*f*(e + f*x^2)) + ((d*(7*b*e 
- 5*a*f)*x*(c + d*x^2)^2)/(5*f) - ((d*(b*e*(35*d*e - 33*c*f) - 5*a*f*(5*d* 
e - 3*c*f))*x*(c + d*x^2))/(3*f) + ((d*(5*a*f*(15*d^2*e^2 - 22*c*d*e*f + 3 
*c^2*f^2) - b*e*(105*d^2*e^2 - 190*c*d*e*f + 81*c^2*f^2))*x)/f + (15*(d*e 
- c*f)^2*(b*e*(7*d*e - c*f) - a*f*(5*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e 
]])/(Sqrt[e]*f^(3/2)))/(3*f))/(5*f))/(2*e*f)))/f - ((b*e - a*f)*(-1/4*((b* 
e - a*f)*x*(c + d*x^2)^3)/(e*f*(e + f*x^2)^2) + (-1/2*((b*e*(7*d*e - c*f) 
- 3*a*f*(d*e + c*f))*x*(c + d*x^2)^2)/(e*f*(e + f*x^2)) - (-1/3*(d*(b*e*(3 
5*d*e - 3*c*f) - 3*a*f*(5*d*e + 3*c*f))*x*(c + d*x^2))/f + (-((d*(3*a*f*(1 
5*d^2*e^2 - 4*c*d*e*f - 3*c^2*f^2) - b*e*(105*d^2*e^2 - 100*c*d*e*f + 3*c^ 
2*f^2))*x)/f) - (3*(d*e - c*f)*(b*e*(35*d^2*e^2 - 10*c*d*e*f - c^2*f^2) - 
3*a*f*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqr 
t[e]*f^(3/2)))/(3*f))/(2*e*f))/(4*e*f)))/f))/f - ((b*e - a*f)*((b*(-1/4*(( 
b*e - a*f)*x*(c + d*x^2)^3)/(e*f*(e + f*x^2)^2) + (-1/2*((b*e*(7*d*e - c*f 
) - 3*a*f*(d*e + c*f))*x*(c + d*x^2)^2)/(e*f*(e + f*x^2)) - (-1/3*(d*(b*e* 
(35*d*e - 3*c*f) - 3*a*f*(5*d*e + 3*c*f))*x*(c + d*x^2))/f + (-((d*(3*a*f* 
(15*d^2*e^2 - 4*c*d*e*f - 3*c^2*f^2) - b*e*(105*d^2*e^2 - 100*c*d*e*f + 3* 
c^2*f^2))*x)/f) - (3*(d*e - c*f)*(b*e*(35*d^2*e^2 - 10*c*d*e*f - c^2*f^2) 
- 3*a*f*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(S 
qrt[e]*f^(3/2)))/(3*f))/(2*e*f))/(4*e*f)))/f - ((b*e - a*f)*(-1/6*((b*e...
 

Defintions of rubi rules used

rule 25

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

rule 218

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

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 401

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

rule 403

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1094\) vs. \(2(534)=1068\).

Time = 0.62 (sec) , antiderivative size = 1095, normalized size of antiderivative = 1.98

method	result	size
default	\(\text {Expression too large to display}\)	\(1095\)
risch	\(\text {Expression too large to display}\)	\(1861\)

Input:

int((b*x^2+a)^3*(d*x^2+c)^3/(f*x^2+e)^4,x,method=_RETURNVERBOSE)
 

Output:

b*d/f^6*(1/5*f^2*x^5*b^2*d^2+a*b*d^2*f^2*x^3+b^2*c*d*f^2*x^3-4/3*b^2*d^2*e 
*f*x^3+3*a^2*d^2*f^2*x+9*a*b*c*d*f^2*x-12*a*b*d^2*e*f*x+3*b^2*c^2*f^2*x-12 
*b^2*c*d*e*f*x+10*b^2*d^2*e^2*x)+1/f^6*((1/16*f^2*(5*a^3*c^3*f^6+3*a^3*c^2 
*d*e*f^5+3*a^3*c*d^2*e^2*f^4-11*a^3*d^3*e^3*f^3+3*a^2*b*c^3*e*f^5+9*a^2*b* 
c^2*d*e^2*f^4-99*a^2*b*c*d^2*e^3*f^3+87*a^2*b*d^3*e^4*f^2+3*a*b^2*c^3*e^2* 
f^4-99*a*b^2*c^2*d*e^3*f^3+261*a*b^2*c*d^2*e^4*f^2-165*a*b^2*d^3*e^5*f-11* 
b^3*c^3*e^3*f^3+87*b^3*c^2*d*e^4*f^2-165*b^3*c*d^2*e^5*f+89*b^3*d^3*e^6)/e 
^3*x^5+1/6*f*(5*a^3*c^3*f^6+3*a^3*c^2*d*e*f^5-3*a^3*c*d^2*e^2*f^4-5*a^3*d^ 
3*e^3*f^3+3*a^2*b*c^3*e*f^5-9*a^2*b*c^2*d*e^2*f^4-45*a^2*b*c*d^2*e^3*f^3+5 
1*a^2*b*d^3*e^4*f^2-3*a*b^2*c^3*e^2*f^4-45*a*b^2*c^2*d*e^3*f^3+153*a*b^2*c 
*d^2*e^4*f^2-105*a*b^2*d^3*e^5*f-5*b^3*c^3*e^3*f^3+51*b^3*c^2*d*e^4*f^2-10 
5*b^3*c*d^2*e^5*f+59*b^3*d^3*e^6)/e^2*x^3+1/16*(11*a^3*c^3*f^6-3*a^3*c^2*d 
*e*f^5-3*a^3*c*d^2*e^2*f^4-5*a^3*d^3*e^3*f^3-3*a^2*b*c^3*e*f^5-9*a^2*b*c^2 
*d*e^2*f^4-45*a^2*b*c*d^2*e^3*f^3+57*a^2*b*d^3*e^4*f^2-3*a*b^2*c^3*e^2*f^4 
-45*a*b^2*c^2*d*e^3*f^3+171*a*b^2*c*d^2*e^4*f^2-123*a*b^2*d^3*e^5*f-5*b^3* 
c^3*e^3*f^3+57*b^3*c^2*d*e^4*f^2-123*b^3*c*d^2*e^5*f+71*b^3*d^3*e^6)/e*x)/ 
(f*x^2+e)^3+1/16*(5*a^3*c^3*f^6+3*a^3*c^2*d*e*f^5+3*a^3*c*d^2*e^2*f^4+5*a^ 
3*d^3*e^3*f^3+3*a^2*b*c^3*e*f^5+9*a^2*b*c^2*d*e^2*f^4+45*a^2*b*c*d^2*e^3*f 
^3-105*a^2*b*d^3*e^4*f^2+3*a*b^2*c^3*e^2*f^4+45*a*b^2*c^2*d*e^3*f^3-315*a* 
b^2*c*d^2*e^4*f^2+315*a*b^2*d^3*e^5*f+5*b^3*c^3*e^3*f^3-105*b^3*c^2*d*e...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1558 vs. \(2 (532) = 1064\).

Time = 0.35 (sec) , antiderivative size = 3136, normalized size of antiderivative = 5.66 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^3*(d*x^2+c)^3/(f*x^2+e)^4,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**3*(d*x**2+c)**3/(f*x**2+e)**4,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x^2+a)^3*(d*x^2+c)^3/(f*x^2+e)^4,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. 1230 vs. \(2 (532) = 1064\).

Time = 0.12 (sec) , antiderivative size = 1230, normalized size of antiderivative = 2.22 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^3*(d*x^2+c)^3/(f*x^2+e)^4,x, algorithm="giac")
 

Output:

-1/16*(231*b^3*d^3*e^6 - 315*b^3*c*d^2*e^5*f - 315*a*b^2*d^3*e^5*f + 105*b 
^3*c^2*d*e^4*f^2 + 315*a*b^2*c*d^2*e^4*f^2 + 105*a^2*b*d^3*e^4*f^2 - 5*b^3 
*c^3*e^3*f^3 - 45*a*b^2*c^2*d*e^3*f^3 - 45*a^2*b*c*d^2*e^3*f^3 - 5*a^3*d^3 
*e^3*f^3 - 3*a*b^2*c^3*e^2*f^4 - 9*a^2*b*c^2*d*e^2*f^4 - 3*a^3*c*d^2*e^2*f 
^4 - 3*a^2*b*c^3*e*f^5 - 3*a^3*c^2*d*e*f^5 - 5*a^3*c^3*f^6)*arctan(f*x/sqr 
t(e*f))/(sqrt(e*f)*e^3*f^6) + 1/48*(267*b^3*d^3*e^6*f^2*x^5 - 495*b^3*c*d^ 
2*e^5*f^3*x^5 - 495*a*b^2*d^3*e^5*f^3*x^5 + 261*b^3*c^2*d*e^4*f^4*x^5 + 78 
3*a*b^2*c*d^2*e^4*f^4*x^5 + 261*a^2*b*d^3*e^4*f^4*x^5 - 33*b^3*c^3*e^3*f^5 
*x^5 - 297*a*b^2*c^2*d*e^3*f^5*x^5 - 297*a^2*b*c*d^2*e^3*f^5*x^5 - 33*a^3* 
d^3*e^3*f^5*x^5 + 9*a*b^2*c^3*e^2*f^6*x^5 + 27*a^2*b*c^2*d*e^2*f^6*x^5 + 9 
*a^3*c*d^2*e^2*f^6*x^5 + 9*a^2*b*c^3*e*f^7*x^5 + 9*a^3*c^2*d*e*f^7*x^5 + 1 
5*a^3*c^3*f^8*x^5 + 472*b^3*d^3*e^7*f*x^3 - 840*b^3*c*d^2*e^6*f^2*x^3 - 84 
0*a*b^2*d^3*e^6*f^2*x^3 + 408*b^3*c^2*d*e^5*f^3*x^3 + 1224*a*b^2*c*d^2*e^5 
*f^3*x^3 + 408*a^2*b*d^3*e^5*f^3*x^3 - 40*b^3*c^3*e^4*f^4*x^3 - 360*a*b^2* 
c^2*d*e^4*f^4*x^3 - 360*a^2*b*c*d^2*e^4*f^4*x^3 - 40*a^3*d^3*e^4*f^4*x^3 - 
 24*a*b^2*c^3*e^3*f^5*x^3 - 72*a^2*b*c^2*d*e^3*f^5*x^3 - 24*a^3*c*d^2*e^3* 
f^5*x^3 + 24*a^2*b*c^3*e^2*f^6*x^3 + 24*a^3*c^2*d*e^2*f^6*x^3 + 40*a^3*c^3 
*e*f^7*x^3 + 213*b^3*d^3*e^8*x - 369*b^3*c*d^2*e^7*f*x - 369*a*b^2*d^3*e^7 
*f*x + 171*b^3*c^2*d*e^6*f^2*x + 513*a*b^2*c*d^2*e^6*f^2*x + 171*a^2*b*d^3 
*e^6*f^2*x - 15*b^3*c^3*e^5*f^3*x - 135*a*b^2*c^2*d*e^5*f^3*x - 135*a^2...
 

Mupad [B] (verification not implemented)

Time = 2.35 (sec) , antiderivative size = 1132, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx =\text {Too large to display} \] Input:

int(((a + b*x^2)^3*(c + d*x^2)^3)/(e + f*x^2)^4,x)
 

Output:

x*((4*e*((4*b^3*d^3*e)/f^5 - (3*b^2*d^2*(a*d + b*c))/f^4))/f - (6*b^3*d^3* 
e^2)/f^6 + (3*b*d*(a^2*d^2 + b^2*c^2 + 3*a*b*c*d))/f^4) - ((x^3*(5*a^3*d^3 
*e^3*f^4 - 59*b^3*d^3*e^6*f - 5*a^3*c^3*f^7 + 5*b^3*c^3*e^3*f^4 - 3*a^2*b* 
c^3*e*f^6 - 3*a^3*c^2*d*e*f^6 + 3*a*b^2*c^3*e^2*f^5 + 105*a*b^2*d^3*e^5*f^ 
2 - 51*a^2*b*d^3*e^4*f^3 + 3*a^3*c*d^2*e^2*f^5 + 105*b^3*c*d^2*e^5*f^2 - 5 
1*b^3*c^2*d*e^4*f^3 - 153*a*b^2*c*d^2*e^4*f^3 + 45*a*b^2*c^2*d*e^3*f^4 + 4 
5*a^2*b*c*d^2*e^3*f^4 + 9*a^2*b*c^2*d*e^2*f^5))/(6*e^2) - (x^5*(5*a^3*c^3* 
f^8 - 11*a^3*d^3*e^3*f^5 - 11*b^3*c^3*e^3*f^5 + 89*b^3*d^3*e^6*f^2 + 3*a^2 
*b*c^3*e*f^7 + 3*a^3*c^2*d*e*f^7 + 3*a*b^2*c^3*e^2*f^6 - 165*a*b^2*d^3*e^5 
*f^3 + 87*a^2*b*d^3*e^4*f^4 + 3*a^3*c*d^2*e^2*f^6 - 165*b^3*c*d^2*e^5*f^3 
+ 87*b^3*c^2*d*e^4*f^4 + 261*a*b^2*c*d^2*e^4*f^4 - 99*a*b^2*c^2*d*e^3*f^5 
- 99*a^2*b*c*d^2*e^3*f^5 + 9*a^2*b*c^2*d*e^2*f^6))/(16*e^3) + (x*(5*a^3*d^ 
3*e^3*f^3 - 71*b^3*d^3*e^6 - 11*a^3*c^3*f^6 + 5*b^3*c^3*e^3*f^3 + 3*a^2*b* 
c^3*e*f^5 + 123*a*b^2*d^3*e^5*f + 3*a^3*c^2*d*e*f^5 + 123*b^3*c*d^2*e^5*f 
+ 3*a*b^2*c^3*e^2*f^4 - 57*a^2*b*d^3*e^4*f^2 + 3*a^3*c*d^2*e^2*f^4 - 57*b^ 
3*c^2*d*e^4*f^2 - 171*a*b^2*c*d^2*e^4*f^2 + 45*a*b^2*c^2*d*e^3*f^3 + 45*a^ 
2*b*c*d^2*e^3*f^3 + 9*a^2*b*c^2*d*e^2*f^4))/(16*e))/(e^3*f^6 + f^9*x^6 + 3 
*e*f^8*x^4 + 3*e^2*f^7*x^2) - x^3*((4*b^3*d^3*e)/(3*f^5) - (b^2*d^2*(a*d + 
 b*c))/f^4) + (b^3*d^3*x^5)/(5*f^4) + (atan((f^(1/2)*x)/e^(1/2))*(5*a^3*c^ 
3*f^6 - 231*b^3*d^3*e^6 + 5*a^3*d^3*e^3*f^3 + 5*b^3*c^3*e^3*f^3 + 3*a^2...
 

Reduce [B] (verification not implemented)

Time = 4.22 (sec) , antiderivative size = 3156, normalized size of antiderivative = 5.70 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^3*(d*x^2+c)^3/(f*x^2+e)^4,x)
 

Output:

(75*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**3*e**3*f**6 + 22 
5*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**3*e**2*f**7*x**2 + 
 225*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**3*e*f**8*x**4 + 
 75*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**3*f**9*x**6 + 45 
*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**2*d*e**4*f**5 + 135 
*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**2*d*e**3*f**6*x**2 
+ 135*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**2*d*e**2*f**7* 
x**4 + 45*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**2*d*e*f**8 
*x**6 + 45*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c*d**2*e**5* 
f**4 + 135*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c*d**2*e**4* 
f**5*x**2 + 135*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c*d**2* 
e**3*f**6*x**4 + 45*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c*d 
**2*e**2*f**7*x**6 + 75*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3 
*d**3*e**6*f**3 + 225*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*d 
**3*e**5*f**4*x**2 + 225*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a** 
3*d**3*e**4*f**5*x**4 + 75*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a 
**3*d**3*e**3*f**6*x**6 + 45*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e))) 
*a**2*b*c**3*e**4*f**5 + 135*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e))) 
*a**2*b*c**3*e**3*f**6*x**2 + 135*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt 
(e)))*a**2*b*c**3*e**2*f**7*x**4 + 45*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(...