\(\int \frac {(a+b x^2)^3 (c+d x^2)^3}{e+f x^2} \, dx\) [243]

Optimal result

Integrand size = 28, antiderivative size = 428 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{e+f x^2} \, dx=-\frac {\left (b^3 e^2 (d e-c f)^3-3 a b^2 e f (d e-c f)^3+3 a^2 b f^2 (d e-c f)^3-a^3 d f^3 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x}{f^6}-\frac {\left (a^3 d^2 f^3 (d e-3 c f)-b^3 e (d e-c f)^3+3 a b^2 f (d e-c f)^3-3 a^2 b d f^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^3}{3 f^5}+\frac {\left (a^3 d^3 f^3-3 a^2 b d^2 f^2 (d e-3 c f)-b^3 (d e-c f)^3+3 a b^2 d f \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^5}{5 f^4}+\frac {b d \left (3 a^2 d^2 f^2-3 a b d f (d e-3 c f)+b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^7}{7 f^3}-\frac {b^2 d^2 (b d e-3 b c f-3 a d f) x^9}{9 f^2}+\frac {b^3 d^3 x^{11}}{11 f}+\frac {(b e-a f)^3 (d e-c f)^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{13/2}} \] Output:

-(b^3*e^2*(-c*f+d*e)^3-3*a*b^2*e*f*(-c*f+d*e)^3+3*a^2*b*f^2*(-c*f+d*e)^3-a 
^3*d*f^3*(3*c^2*f^2-3*c*d*e*f+d^2*e^2))*x/f^6-1/3*(a^3*d^2*f^3*(-3*c*f+d*e 
)-b^3*e*(-c*f+d*e)^3+3*a*b^2*f*(-c*f+d*e)^3-3*a^2*b*d*f^2*(3*c^2*f^2-3*c*d 
*e*f+d^2*e^2))*x^3/f^5+1/5*(a^3*d^3*f^3-3*a^2*b*d^2*f^2*(-3*c*f+d*e)-b^3*( 
-c*f+d*e)^3+3*a*b^2*d*f*(3*c^2*f^2-3*c*d*e*f+d^2*e^2))*x^5/f^4+1/7*b*d*(3* 
a^2*d^2*f^2-3*a*b*d*f*(-3*c*f+d*e)+b^2*(3*c^2*f^2-3*c*d*e*f+d^2*e^2))*x^7/ 
f^3-1/9*b^2*d^2*(-3*a*d*f-3*b*c*f+b*d*e)*x^9/f^2+1/11*b^3*d^3*x^11/f+(-a*f 
+b*e)^3*(-c*f+d*e)^3*arctan(f^(1/2)*x/e^(1/2))/e^(1/2)/f^(13/2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{e+f x^2} \, dx=\frac {\left (-b^3 e^2 (d e-c f)^3+3 a b^2 e f (d e-c f)^3+3 a^2 b f^2 (-d e+c f)^3+a^3 d f^3 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x}{f^6}+\frac {\left (b^3 e (d e-c f)^3+3 a b^2 f (-d e+c f)^3+a^3 d^2 f^3 (-d e+3 c f)+3 a^2 b d f^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^3}{3 f^5}+\frac {\left (a^3 d^3 f^3-3 a^2 b d^2 f^2 (d e-3 c f)-b^3 (d e-c f)^3+3 a b^2 d f \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^5}{5 f^4}+\frac {b d \left (3 a^2 d^2 f^2-3 a b d f (d e-3 c f)+b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^7}{7 f^3}-\frac {b^2 d^2 (b d e-3 b c f-3 a d f) x^9}{9 f^2}+\frac {b^3 d^3 x^{11}}{11 f}+\frac {(b e-a f)^3 (d e-c f)^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{13/2}} \] Input:

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

Output:

((-(b^3*e^2*(d*e - c*f)^3) + 3*a*b^2*e*f*(d*e - c*f)^3 + 3*a^2*b*f^2*(-(d* 
e) + c*f)^3 + a^3*d*f^3*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x)/f^6 + ((b^3* 
e*(d*e - c*f)^3 + 3*a*b^2*f*(-(d*e) + c*f)^3 + a^3*d^2*f^3*(-(d*e) + 3*c*f 
) + 3*a^2*b*d*f^2*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x^3)/(3*f^5) + ((a^3* 
d^3*f^3 - 3*a^2*b*d^2*f^2*(d*e - 3*c*f) - b^3*(d*e - c*f)^3 + 3*a*b^2*d*f* 
(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x^5)/(5*f^4) + (b*d*(3*a^2*d^2*f^2 - 3* 
a*b*d*f*(d*e - 3*c*f) + b^2*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x^7)/(7*f^3 
) - (b^2*d^2*(b*d*e - 3*b*c*f - 3*a*d*f)*x^9)/(9*f^2) + (b^3*d^3*x^11)/(11 
*f) + ((b*e - a*f)^3*(d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f 
^(13/2))
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {420, 290, 420, 290, 403, 25, 403, 25, 403, 299, 218, 2009}

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.

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 290

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 290

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {b \int \left (b^2 d^3 x^{10}+b d^2 (3 b c+2 a d) x^8+d \left (3 b^2 c^2+6 a b d c+a^2 d^2\right ) x^6+c \left (b^2 c^2+6 a b d c+3 a^2 d^2\right ) x^4+a c^2 (2 b c+3 a d) x^2+a^2 c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {\int -\frac {\left (d x^2+c\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))-\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x^2\right )}{f x^2+e}dx}{5 f}+\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}}{7 f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \int \left (b^2 d^3 x^{10}+b d^2 (3 b c+2 a d) x^8+d \left (3 b^2 c^2+6 a b d c+a^2 d^2\right ) x^6+c \left (b^2 c^2+6 a b d c+3 a^2 d^2\right ) x^4+a c^2 (2 b c+3 a d) x^2+a^2 c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))-\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x^2\right )}{f x^2+e}dx}{5 f}}{7 f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b \int \left (b^2 d^3 x^{10}+b d^2 (3 b c+2 a d) x^8+d \left (3 b^2 c^2+6 a b d c+a^2 d^2\right ) x^6+c \left (b^2 c^2+6 a b d c+3 a^2 d^2\right ) x^4+a c^2 (2 b c+3 a d) x^2+a^2 c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\int \frac {\left (7 a d f \left (15 d^2 e^2-40 c d f e+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 f e^2+231 c^2 d f^2 e-48 c^3 f^3\right )\right ) x^2+c \left (7 a f \left (5 d^2 e^2-12 c d f e+15 c^2 f^2\right )-b e \left (35 d^2 e^2-84 c d f e+57 c^2 f^2\right )\right )}{f x^2+e}dx}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b \int \left (b^2 d^3 x^{10}+b d^2 (3 b c+2 a d) x^8+d \left (3 b^2 c^2+6 a b d c+a^2 d^2\right ) x^6+c \left (b^2 c^2+6 a b d c+3 a^2 d^2\right ) x^4+a c^2 (2 b c+3 a d) x^2+a^2 c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) (d e-c f)^3 \int \frac {1}{f x^2+e}dx}{f}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \int \left (b^2 d^3 x^{10}+b d^2 (3 b c+2 a d) x^8+d \left (3 b^2 c^2+6 a b d c+a^2 d^2\right ) x^6+c \left (b^2 c^2+6 a b d c+3 a^2 d^2\right ) x^4+a c^2 (2 b c+3 a d) x^2+a^2 c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (\frac {1}{7} d x^7 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{5} c x^5 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac {1}{3} a c^2 x^3 (3 a d+2 b c)+\frac {1}{9} b d^2 x^9 (2 a d+3 b c)+\frac {1}{11} b^2 d^3 x^{11}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {1}{3} c^2 x^3 (3 a d+b c)+\frac {1}{7} d^2 x^7 (a d+3 b c)+\frac {3}{5} c d x^5 (a d+b c)+a c^3 x+\frac {1}{9} b d^3 x^9\right )}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}\)

Input:

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

Output:

(b*(a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^3)/3 + (c*(b^2*c^2 + 6*a*b*c*d + 
3*a^2*d^2)*x^5)/5 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^7)/7 + (b*d^2*( 
3*b*c + 2*a*d)*x^9)/9 + (b^2*d^3*x^11)/11))/f - ((b*e - a*f)*((b*(a*c^3*x 
+ (c^2*(b*c + 3*a*d)*x^3)/3 + (3*c*d*(b*c + a*d)*x^5)/5 + (d^2*(3*b*c + a* 
d)*x^7)/7 + (b*d^3*x^9)/9))/f - ((b*e - a*f)*((b*x*(c + d*x^2)^3)/(7*f) - 
(((7*b*d*e - 6*b*c*f - 7*a*d*f)*x*(c + d*x^2)^2)/(5*f) - (-1/3*((7*a*d*f*( 
5*d*e - 9*c*f) - b*(35*d^2*e^2 - 63*c*d*e*f + 24*c^2*f^2))*x*(c + d*x^2))/ 
f + (((7*a*d*f*(15*d^2*e^2 - 40*c*d*e*f + 33*c^2*f^2) - b*(105*d^3*e^3 - 2 
80*c*d^2*e^2*f + 231*c^2*d*e*f^2 - 48*c^3*f^3))*x)/f + (105*(b*e - a*f)*(d 
*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(3/2)))/(3*f))/(5*f))/ 
(7*f)))/f))/f
 

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 290

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I 
nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d 
}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
 

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 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 420

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(926\) vs. \(2(410)=820\).

Time = 0.68 (sec) , antiderivative size = 927, normalized size of antiderivative = 2.17

method	result	size
default	\(\frac {\frac {b^{3} d^{3} x^{11} f^{5}}{11}+\frac {\left (\left (a d f +b c f -b d e \right ) b^{2} d^{2} f^{4}+b d f \left (2 a b \,d^{2} f^{4}+2 b^{2} c d \,f^{4}\right )\right ) x^{9}}{9}+\frac {\left (\left (a d f +b c f -b d e \right ) \left (2 a b \,d^{2} f^{4}+2 b^{2} c d \,f^{4}\right )+b d f \left (a^{2} d^{2} f^{4}+5 a b c d \,f^{4}-a b \,d^{2} e \,f^{3}+b^{2} c^{2} f^{4}-b^{2} c d e \,f^{3}+b^{2} d^{2} e^{2} f^{2}\right )\right ) x^{7}}{7}+\frac {\left (\left (a d f +b c f -b d e \right ) \left (a^{2} d^{2} f^{4}+5 a b c d \,f^{4}-a b \,d^{2} e \,f^{3}+b^{2} c^{2} f^{4}-b^{2} c d e \,f^{3}+b^{2} d^{2} e^{2} f^{2}\right )+b d f \left (3 a^{2} c d \,f^{4}-a^{2} d^{2} e \,f^{3}+3 a b \,c^{2} f^{4}-2 a b c d e \,f^{3}+a b \,d^{2} e^{2} f^{2}-b^{2} c^{2} e \,f^{3}+b^{2} c d \,e^{2} f^{2}\right )\right ) x^{5}}{5}+\frac {\left (\left (a d f +b c f -b d e \right ) \left (3 a^{2} c d \,f^{4}-a^{2} d^{2} e \,f^{3}+3 a b \,c^{2} f^{4}-2 a b c d e \,f^{3}+a b \,d^{2} e^{2} f^{2}-b^{2} c^{2} e \,f^{3}+b^{2} c d \,e^{2} f^{2}\right )+b d f \left (3 a^{2} c^{2} f^{4}-3 a^{2} c d e \,f^{3}+a^{2} d^{2} e^{2} f^{2}-3 a b \,c^{2} e \,f^{3}+5 a b c d \,e^{2} f^{2}-2 a b \,d^{2} e^{3} f +b^{2} c^{2} e^{2} f^{2}-2 b^{2} c d \,e^{3} f +b^{2} d^{2} e^{4}\right )\right ) x^{3}}{3}+\left (a d f +b c f -b d e \right ) \left (3 a^{2} c^{2} f^{4}-3 a^{2} c d e \,f^{3}+a^{2} d^{2} e^{2} f^{2}-3 a b \,c^{2} e \,f^{3}+5 a b c d \,e^{2} f^{2}-2 a b \,d^{2} e^{3} f +b^{2} c^{2} e^{2} f^{2}-2 b^{2} c d \,e^{3} f +b^{2} d^{2} e^{4}\right ) x}{f^{6}}+\frac {\left (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}-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}-9 a^{2} b c \,d^{2} e^{3} f^{3}+3 a^{2} b \,d^{3} e^{4} f^{2}+3 a \,b^{2} c^{3} e^{2} f^{4}-9 a \,b^{2} c^{2} d \,e^{3} f^{3}+9 a \,b^{2} c \,d^{2} e^{4} f^{2}-3 a \,b^{2} d^{3} e^{5} f -b^{3} c^{3} e^{3} f^{3}+3 b^{3} c^{2} d \,e^{4} f^{2}-3 b^{3} c \,d^{2} e^{5} f +b^{3} d^{3} e^{6}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{f^{6} \sqrt {e f}}\)	\(927\)
risch	\(\text {Expression too large to display}\)	\(1779\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 1604, normalized size of antiderivative = 3.75 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{e+f x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/6930*(630*b^3*d^3*e*f^6*x^11 - 770*(b^3*d^3*e^2*f^5 - 3*(b^3*c*d^2 + a* 
b^2*d^3)*e*f^6)*x^9 + 990*(b^3*d^3*e^3*f^4 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2 
*f^5 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e*f^6)*x^7 - 1386*(b^3*d^ 
3*e^4*f^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^3*f^4 + 3*(b^3*c^2*d + 3*a*b^2*c*d 
^2 + a^2*b*d^3)*e^2*f^5 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d 
^3)*e*f^6)*x^5 + 2310*(b^3*d^3*e^5*f^2 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^4*f^3 
 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^3*f^4 - (b^3*c^3 + 9*a*b^2* 
c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^2*f^5 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + 
a^3*c*d^2)*e*f^6)*x^3 - 3465*(b^3*d^3*e^6 + a^3*c^3*f^6 - 3*(b^3*c*d^2 + a 
*b^2*d^3)*e^5*f + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^4*f^2 - (b^3 
*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^3*f^3 + 3*(a*b^2*c^3 + 3 
*a^2*b*c^2*d + a^3*c*d^2)*e^2*f^4 - 3*(a^2*b*c^3 + a^3*c^2*d)*e*f^5)*sqrt( 
-e*f)*log((f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)) - 6930*(b^3*d^3*e^6*f 
- 3*(b^3*c*d^2 + a*b^2*d^3)*e^5*f^2 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b 
*d^3)*e^4*f^3 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^3*f^ 
4 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^2*f^5 - 3*(a^2*b*c^3 + a^3 
*c^2*d)*e*f^6)*x)/(e*f^7), 1/3465*(315*b^3*d^3*e*f^6*x^11 - 385*(b^3*d^3*e 
^2*f^5 - 3*(b^3*c*d^2 + a*b^2*d^3)*e*f^6)*x^9 + 495*(b^3*d^3*e^3*f^4 - 3*( 
b^3*c*d^2 + a*b^2*d^3)*e^2*f^5 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3) 
*e*f^6)*x^7 - 693*(b^3*d^3*e^4*f^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^3*f^4 ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1379 vs. \(2 (428) = 856\).

Time = 3.79 (sec) , antiderivative size = 1379, normalized size of antiderivative = 3.22 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{e+f x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

b**3*d**3*x**11/(11*f) + x**9*(a*b**2*d**3/(3*f) + b**3*c*d**2/(3*f) - b** 
3*d**3*e/(9*f**2)) + x**7*(3*a**2*b*d**3/(7*f) + 9*a*b**2*c*d**2/(7*f) - 3 
*a*b**2*d**3*e/(7*f**2) + 3*b**3*c**2*d/(7*f) - 3*b**3*c*d**2*e/(7*f**2) + 
 b**3*d**3*e**2/(7*f**3)) + x**5*(a**3*d**3/(5*f) + 9*a**2*b*c*d**2/(5*f) 
- 3*a**2*b*d**3*e/(5*f**2) + 9*a*b**2*c**2*d/(5*f) - 9*a*b**2*c*d**2*e/(5* 
f**2) + 3*a*b**2*d**3*e**2/(5*f**3) + b**3*c**3/(5*f) - 3*b**3*c**2*d*e/(5 
*f**2) + 3*b**3*c*d**2*e**2/(5*f**3) - b**3*d**3*e**3/(5*f**4)) + x**3*(a* 
*3*c*d**2/f - a**3*d**3*e/(3*f**2) + 3*a**2*b*c**2*d/f - 3*a**2*b*c*d**2*e 
/f**2 + a**2*b*d**3*e**2/f**3 + a*b**2*c**3/f - 3*a*b**2*c**2*d*e/f**2 + 3 
*a*b**2*c*d**2*e**2/f**3 - a*b**2*d**3*e**3/f**4 - b**3*c**3*e/(3*f**2) + 
b**3*c**2*d*e**2/f**3 - b**3*c*d**2*e**3/f**4 + b**3*d**3*e**4/(3*f**5)) + 
 x*(3*a**3*c**2*d/f - 3*a**3*c*d**2*e/f**2 + a**3*d**3*e**2/f**3 + 3*a**2* 
b*c**3/f - 9*a**2*b*c**2*d*e/f**2 + 9*a**2*b*c*d**2*e**2/f**3 - 3*a**2*b*d 
**3*e**3/f**4 - 3*a*b**2*c**3*e/f**2 + 9*a*b**2*c**2*d*e**2/f**3 - 9*a*b** 
2*c*d**2*e**3/f**4 + 3*a*b**2*d**3*e**4/f**5 + b**3*c**3*e**2/f**3 - 3*b** 
3*c**2*d*e**3/f**4 + 3*b**3*c*d**2*e**4/f**5 - b**3*d**3*e**5/f**6) - sqrt 
(-1/(e*f**13))*(a*f - b*e)**3*(c*f - d*e)**3*log(-e*f**6*sqrt(-1/(e*f**13) 
)*(a*f - b*e)**3*(c*f - d*e)**3/(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 - 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 - 9*a**2*b*c*d**2*e**3*f**3 + 3*a**2*b*d**3*e**4*...
 

Maxima [F(-2)]

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

integrate((b*x^2+a)^3*(d*x^2+c)^3/(f*x^2+e),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. 1013 vs. \(2 (410) = 820\).

Time = 0.14 (sec) , antiderivative size = 1013, normalized size of antiderivative = 2.37 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{e+f x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(b^3*d^3*e^6 - 3*b^3*c*d^2*e^5*f - 3*a*b^2*d^3*e^5*f + 3*b^3*c^2*d*e^4*f^2 
 + 9*a*b^2*c*d^2*e^4*f^2 + 3*a^2*b*d^3*e^4*f^2 - b^3*c^3*e^3*f^3 - 9*a*b^2 
*c^2*d*e^3*f^3 - 9*a^2*b*c*d^2*e^3*f^3 - 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 + a^3*c^3*f^6)*arctan(f*x/sqrt(e*f))/(sqrt(e*f)*f^6) + 1/ 
3465*(315*b^3*d^3*f^10*x^11 - 385*b^3*d^3*e*f^9*x^9 + 1155*b^3*c*d^2*f^10* 
x^9 + 1155*a*b^2*d^3*f^10*x^9 + 495*b^3*d^3*e^2*f^8*x^7 - 1485*b^3*c*d^2*e 
*f^9*x^7 - 1485*a*b^2*d^3*e*f^9*x^7 + 1485*b^3*c^2*d*f^10*x^7 + 4455*a*b^2 
*c*d^2*f^10*x^7 + 1485*a^2*b*d^3*f^10*x^7 - 693*b^3*d^3*e^3*f^7*x^5 + 2079 
*b^3*c*d^2*e^2*f^8*x^5 + 2079*a*b^2*d^3*e^2*f^8*x^5 - 2079*b^3*c^2*d*e*f^9 
*x^5 - 6237*a*b^2*c*d^2*e*f^9*x^5 - 2079*a^2*b*d^3*e*f^9*x^5 + 693*b^3*c^3 
*f^10*x^5 + 6237*a*b^2*c^2*d*f^10*x^5 + 6237*a^2*b*c*d^2*f^10*x^5 + 693*a^ 
3*d^3*f^10*x^5 + 1155*b^3*d^3*e^4*f^6*x^3 - 3465*b^3*c*d^2*e^3*f^7*x^3 - 3 
465*a*b^2*d^3*e^3*f^7*x^3 + 3465*b^3*c^2*d*e^2*f^8*x^3 + 10395*a*b^2*c*d^2 
*e^2*f^8*x^3 + 3465*a^2*b*d^3*e^2*f^8*x^3 - 1155*b^3*c^3*e*f^9*x^3 - 10395 
*a*b^2*c^2*d*e*f^9*x^3 - 10395*a^2*b*c*d^2*e*f^9*x^3 - 1155*a^3*d^3*e*f^9* 
x^3 + 3465*a*b^2*c^3*f^10*x^3 + 10395*a^2*b*c^2*d*f^10*x^3 + 3465*a^3*c*d^ 
2*f^10*x^3 - 3465*b^3*d^3*e^5*f^5*x + 10395*b^3*c*d^2*e^4*f^6*x + 10395*a* 
b^2*d^3*e^4*f^6*x - 10395*b^3*c^2*d*e^3*f^7*x - 31185*a*b^2*c*d^2*e^3*f^7* 
x - 10395*a^2*b*d^3*e^3*f^7*x + 3465*b^3*c^3*e^2*f^8*x + 31185*a*b^2*c^...
 

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 838, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{e+f x^2} \, dx=x\,\left (\frac {e\,\left (\frac {e\,\left (\frac {a^3\,d^3+9\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d+b^3\,c^3}{f}-\frac {e\,\left (\frac {e\,\left (\frac {b^3\,d^3\,e}{f^2}-\frac {3\,b^2\,d^2\,\left (a\,d+b\,c\right )}{f}\right )}{f}+\frac {3\,b\,d\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{f}\right )}{f}\right )}{f}-\frac {3\,a\,c\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{f}\right )}{f}+\frac {3\,a^2\,c^2\,\left (a\,d+b\,c\right )}{f}\right )-x^3\,\left (\frac {e\,\left (\frac {a^3\,d^3+9\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d+b^3\,c^3}{f}-\frac {e\,\left (\frac {e\,\left (\frac {b^3\,d^3\,e}{f^2}-\frac {3\,b^2\,d^2\,\left (a\,d+b\,c\right )}{f}\right )}{f}+\frac {3\,b\,d\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{f}\right )}{f}\right )}{3\,f}-\frac {a\,c\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{f}\right )+x^5\,\left (\frac {a^3\,d^3+9\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d+b^3\,c^3}{5\,f}-\frac {e\,\left (\frac {e\,\left (\frac {b^3\,d^3\,e}{f^2}-\frac {3\,b^2\,d^2\,\left (a\,d+b\,c\right )}{f}\right )}{f}+\frac {3\,b\,d\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{f}\right )}{5\,f}\right )+x^7\,\left (\frac {e\,\left (\frac {b^3\,d^3\,e}{f^2}-\frac {3\,b^2\,d^2\,\left (a\,d+b\,c\right )}{f}\right )}{7\,f}+\frac {3\,b\,d\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{7\,f}\right )-x^9\,\left (\frac {b^3\,d^3\,e}{9\,f^2}-\frac {b^2\,d^2\,\left (a\,d+b\,c\right )}{3\,f}\right )+\frac {b^3\,d^3\,x^{11}}{11\,f}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x\,{\left (a\,f-b\,e\right )}^3\,{\left (c\,f-d\,e\right )}^3}{\sqrt {e}\,\left (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-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^4-9\,a^2\,b\,c\,d^2\,e^3\,f^3+3\,a^2\,b\,d^3\,e^4\,f^2+3\,a\,b^2\,c^3\,e^2\,f^4-9\,a\,b^2\,c^2\,d\,e^3\,f^3+9\,a\,b^2\,c\,d^2\,e^4\,f^2-3\,a\,b^2\,d^3\,e^5\,f-b^3\,c^3\,e^3\,f^3+3\,b^3\,c^2\,d\,e^4\,f^2-3\,b^3\,c\,d^2\,e^5\,f+b^3\,d^3\,e^6\right )}\right )\,{\left (a\,f-b\,e\right )}^3\,{\left (c\,f-d\,e\right )}^3}{\sqrt {e}\,f^{13/2}} \] Input:

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

Output:

x*((e*((e*((a^3*d^3 + b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2)/f - (e*((e* 
((b^3*d^3*e)/f^2 - (3*b^2*d^2*(a*d + b*c))/f))/f + (3*b*d*(a^2*d^2 + b^2*c 
^2 + 3*a*b*c*d))/f))/f))/f - (3*a*c*(a^2*d^2 + b^2*c^2 + 3*a*b*c*d))/f))/f 
 + (3*a^2*c^2*(a*d + b*c))/f) - x^3*((e*((a^3*d^3 + b^3*c^3 + 9*a*b^2*c^2* 
d + 9*a^2*b*c*d^2)/f - (e*((e*((b^3*d^3*e)/f^2 - (3*b^2*d^2*(a*d + b*c))/f 
))/f + (3*b*d*(a^2*d^2 + b^2*c^2 + 3*a*b*c*d))/f))/f))/(3*f) - (a*c*(a^2*d 
^2 + b^2*c^2 + 3*a*b*c*d))/f) + x^5*((a^3*d^3 + b^3*c^3 + 9*a*b^2*c^2*d + 
9*a^2*b*c*d^2)/(5*f) - (e*((e*((b^3*d^3*e)/f^2 - (3*b^2*d^2*(a*d + b*c))/f 
))/f + (3*b*d*(a^2*d^2 + b^2*c^2 + 3*a*b*c*d))/f))/(5*f)) + x^7*((e*((b^3* 
d^3*e)/f^2 - (3*b^2*d^2*(a*d + b*c))/f))/(7*f) + (3*b*d*(a^2*d^2 + b^2*c^2 
 + 3*a*b*c*d))/(7*f)) - x^9*((b^3*d^3*e)/(9*f^2) - (b^2*d^2*(a*d + b*c))/( 
3*f)) + (b^3*d^3*x^11)/(11*f) + (atan((f^(1/2)*x*(a*f - b*e)^3*(c*f - d*e) 
^3)/(e^(1/2)*(a^3*c^3*f^6 + b^3*d^3*e^6 - a^3*d^3*e^3*f^3 - b^3*c^3*e^3*f^ 
3 - 3*a^2*b*c^3*e*f^5 - 3*a*b^2*d^3*e^5*f - 3*a^3*c^2*d*e*f^5 - 3*b^3*c*d^ 
2*e^5*f + 3*a*b^2*c^3*e^2*f^4 + 3*a^2*b*d^3*e^4*f^2 + 3*a^3*c*d^2*e^2*f^4 
+ 3*b^3*c^2*d*e^4*f^2 + 9*a*b^2*c*d^2*e^4*f^2 - 9*a*b^2*c^2*d*e^3*f^3 - 9* 
a^2*b*c*d^2*e^3*f^3 + 9*a^2*b*c^2*d*e^2*f^4)))*(a*f - b*e)^3*(c*f - d*e)^3 
)/(e^(1/2)*f^(13/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1293, normalized size of antiderivative = 3.02 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{e+f x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(3465*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**3*f**6 - 10395 
*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**2*d*e*f**5 + 10395* 
sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c*d**2*e**2*f**4 - 3465 
*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*d**3*e**3*f**3 - 10395 
*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*b*c**3*e*f**5 + 31185* 
sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*b*c**2*d*e**2*f**4 - 31 
185*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*b*c*d**2*e**3*f**3 
+ 10395*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*b*d**3*e**4*f** 
2 + 10395*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b**2*c**3*e**2*f 
**4 - 31185*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b**2*c**2*d*e* 
*3*f**3 + 31185*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b**2*c*d** 
2*e**4*f**2 - 10395*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b**2*d 
**3*e**5*f - 3465*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*b**3*c**3* 
e**3*f**3 + 10395*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*b**3*c**2* 
d*e**4*f**2 - 10395*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*b**3*c*d 
**2*e**5*f + 3465*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*b**3*d**3* 
e**6 + 10395*a**3*c**2*d*e*f**6*x - 10395*a**3*c*d**2*e**2*f**5*x + 3465*a 
**3*c*d**2*e*f**6*x**3 + 3465*a**3*d**3*e**3*f**4*x - 1155*a**3*d**3*e**2* 
f**5*x**3 + 693*a**3*d**3*e*f**6*x**5 + 10395*a**2*b*c**3*e*f**6*x - 31185 
*a**2*b*c**2*d*e**2*f**5*x + 10395*a**2*b*c**2*d*e*f**6*x**3 + 31185*a*...