\(\int \frac {(a+b x^2)^3 (c+d x^2)^3}{(e+f x^2)^2} \, dx\) [244]

Optimal result

Integrand size = 28, antiderivative size = 387 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=-\frac {\left (a^3 d^2 f^3 (2 d e-3 c f)-9 a^2 b d f^2 (d e-c f)^2-b^3 e (5 d e-2 c f) (d e-c f)^2+3 a b^2 f (d e-c f)^2 (4 d e-c f)\right ) x}{f^6}-\frac {(b d e-b c f-a d f) \left (a^2 d^2 f^2-a b d f (5 d e-8 c f)+b^2 \left (4 d^2 e^2-5 c d e f+c^2 f^2\right )\right ) x^3}{3 f^5}+\frac {3 b d \left (a^2 d^2 f^2-a b d f (2 d e-3 c f)+b^2 (d e-c f)^2\right ) x^5}{5 f^4}-\frac {b^2 d^2 (2 b d e-3 b c f-3 a d f) x^7}{7 f^3}+\frac {b^3 d^3 x^9}{9 f^2}+\frac {(b e-a f)^3 (d e-c f)^3 x}{2 e f^6 \left (e+f x^2\right )}-\frac {(b e-a f)^2 (d e-c f)^2 (b e (11 d e-5 c f)-a f (5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} f^{13/2}} \] Output:

-(a^3*d^2*f^3*(-3*c*f+2*d*e)-9*a^2*b*d*f^2*(-c*f+d*e)^2-b^3*e*(-2*c*f+5*d* 
e)*(-c*f+d*e)^2+3*a*b^2*f*(-c*f+d*e)^2*(-c*f+4*d*e))*x/f^6-1/3*(-a*d*f-b*c 
*f+b*d*e)*(a^2*d^2*f^2-a*b*d*f*(-8*c*f+5*d*e)+b^2*(c^2*f^2-5*c*d*e*f+4*d^2 
*e^2))*x^3/f^5+3/5*b*d*(a^2*d^2*f^2-a*b*d*f*(-3*c*f+2*d*e)+b^2*(-c*f+d*e)^ 
2)*x^5/f^4-1/7*b^2*d^2*(-3*a*d*f-3*b*c*f+2*b*d*e)*x^7/f^3+1/9*b^3*d^3*x^9/ 
f^2+1/2*(-a*f+b*e)^3*(-c*f+d*e)^3*x/e/f^6/(f*x^2+e)-1/2*(-a*f+b*e)^2*(-c*f 
+d*e)^2*(b*e*(-5*c*f+11*d*e)-a*f*(c*f+5*d*e))*arctan(f^(1/2)*x/e^(1/2))/e^ 
(3/2)/f^(13/2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\frac {\left (9 a^2 b d f^2 (d e-c f)^2+b^3 e (5 d e-2 c f) (d e-c f)^2+3 a b^2 f (d e-c f)^2 (-4 d e+c f)+a^3 d^2 f^3 (-2 d e+3 c f)\right ) x}{f^6}+\frac {\left (a^3 d^3 f^3+9 a b^2 d f (d e-c f)^2-b^3 (d e-c f)^2 (4 d e-c f)+3 a^2 b d^2 f^2 (-2 d e+3 c f)\right ) x^3}{3 f^5}+\frac {3 b d \left (a^2 d^2 f^2+b^2 (d e-c f)^2+a b d f (-2 d e+3 c f)\right ) x^5}{5 f^4}-\frac {b^2 d^2 (2 b d e-3 b c f-3 a d f) x^7}{7 f^3}+\frac {b^3 d^3 x^9}{9 f^2}+\frac {(b e-a f)^3 (d e-c f)^3 x}{2 e f^6 \left (e+f x^2\right )}-\frac {(b e-a f)^2 (d e-c f)^2 (b e (11 d e-5 c f)-a f (5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} f^{13/2}} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(880\) vs. \(2(387)=774\).

Time = 1.64 (sec) , antiderivative size = 880, normalized size of antiderivative = 2.27, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.821, Rules used = {425, 420, 290, 403, 25, 403, 25, 403, 299, 218, 425, 401, 25, 403, 25, 403, 25, 299, 218, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 290

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {b \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {\int -\frac {\left (d x^2+c\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))-\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x^2\right )}{f x^2+e}dx}{5 f}+\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}}{7 f}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{5 f}-\frac {\frac {\int \frac {d \left (5 a f \left (15 d^2 e^2-22 c d f e+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d f e+81 c^2 f^2\right )\right ) x^2+c \left (5 a f \left (5 d^2 e^2-6 c d f e-3 c^2 f^2\right )-b e \left (35 d^2 e^2-54 c d f e+15 c^2 f^2\right )\right )}{f x^2+e}dx}{3 f}+\frac {d x \left (c+d x^2\right ) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{3 f}}{5 f}}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))-\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x^2\right )}{f x^2+e}dx}{5 f}}{7 f}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{5 f}-\frac {\frac {\int \frac {d \left (5 a f \left (15 d^2 e^2-22 c d f e+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d f e+81 c^2 f^2\right )\right ) x^2+c \left (5 a f \left (5 d^2 e^2-6 c d f e-3 c^2 f^2\right )-b e \left (35 d^2 e^2-54 c d f e+15 c^2 f^2\right )\right )}{f x^2+e}dx}{3 f}+\frac {d x \left (c+d x^2\right ) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{3 f}}{5 f}}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))-\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x^2\right )}{f x^2+e}dx}{5 f}}{7 f}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{5 f}-\frac {\frac {\frac {15 (d e-c f)^2 (b e (7 d e-c f)-a f (c f+5 d e)) \int \frac {1}{f x^2+e}dx}{f}+\frac {d x \left (5 a f \left (3 c^2 f^2-22 c d e f+15 d^2 e^2\right )-b e \left (81 c^2 f^2-190 c d e f+105 d^2 e^2\right )\right )}{f}}{3 f}+\frac {d x \left (c+d x^2\right ) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{3 f}}{5 f}}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))-\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x^2\right )}{f x^2+e}dx}{5 f}}{7 f}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{5 f}-\frac {\frac {\frac {15 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^2 (b e (7 d e-c f)-a f (c f+5 d e))}{\sqrt {e} f^{3/2}}+\frac {d x \left (5 a f \left (3 c^2 f^2-22 c d e f+15 d^2 e^2\right )-b e \left (81 c^2 f^2-190 c d e f+105 d^2 e^2\right )\right )}{f}}{3 f}+\frac {d x \left (c+d x^2\right ) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{3 f}}{5 f}}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (d x^2+c\right )^3}{7 f}-\frac {\frac {(7 b d e-6 b c f-7 a d f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {\left (7 a d f \left (15 d^2 e^2-40 c d f e+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 f e^2+231 c^2 d f^2 e-48 c^3 f^3\right )\right ) x}{f}}{3 f}-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b x \left (d x^2+c\right )^3}{7 f}-\frac {\frac {(7 b d e-6 b c f-7 a d f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {\int \frac {\left (7 a d f \left (15 d^2 e^2-40 c d f e+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 f e^2+231 c^2 d f^2 e-48 c^3 f^3\right )\right ) x^2+c \left (7 a f \left (5 d^2 e^2-12 c d f e+15 c^2 f^2\right )-b e \left (35 d^2 e^2-84 c d f e+57 c^2 f^2\right )\right )}{f x^2+e}dx}{3 f}-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{3 f}}{5 f}}{7 f}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {d (7 b e-5 a f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x \left (d x^2+c\right )}{3 f}+\frac {\frac {15 (b e (7 d e-c f)-a f (5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^2}{\sqrt {e} f^{3/2}}+\frac {d \left (5 a f \left (15 d^2 e^2-22 c d f e+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d f e+81 c^2 f^2\right )\right ) x}{f}}{3 f}}{5 f}}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (d x^2+c\right )^3}{7 f}-\frac {\frac {(7 b d e-6 b c f-7 a d f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {\left (7 a d f \left (15 d^2 e^2-40 c d f e+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 f e^2+231 c^2 d f^2 e-48 c^3 f^3\right )\right ) x}{f}}{3 f}-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b x \left (d x^2+c\right )^3}{7 f}-\frac {\frac {(7 b d e-6 b c f-7 a d f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {\frac {105 (b e-a f) \int \frac {1}{f x^2+e}dx (d e-c f)^3}{f}+\frac {\left (7 a d f \left (15 d^2 e^2-40 c d f e+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 f e^2+231 c^2 d f^2 e-48 c^3 f^3\right )\right ) x}{f}}{3 f}-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{3 f}}{5 f}}{7 f}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {d (7 b e-5 a f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x \left (d x^2+c\right )}{3 f}+\frac {\frac {15 (b e (7 d e-c f)-a f (5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^2}{\sqrt {e} f^{3/2}}+\frac {d \left (5 a f \left (15 d^2 e^2-22 c d f e+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d f e+81 c^2 f^2\right )\right ) x}{f}}{3 f}}{5 f}}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \left (\frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (d x^2+c\right )^3}{7 f}-\frac {\frac {(7 b d e-6 b c f-7 a d f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {\left (7 a d f \left (15 d^2 e^2-40 c d f e+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 f e^2+231 c^2 d f^2 e-48 c^3 f^3\right )\right ) x}{f}}{3 f}-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b x \left (d x^2+c\right )^3}{7 f}-\frac {\frac {(7 b d e-6 b c f-7 a d f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {\left (7 a d f \left (15 d^2 e^2-40 c d f e+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 f e^2+231 c^2 d f^2 e-48 c^3 f^3\right )\right ) x}{f}}{3 f}-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{3 f}}{5 f}}{7 f}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {d (7 b e-5 a f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x \left (d x^2+c\right )}{3 f}+\frac {\frac {15 (b e (7 d e-c f)-a f (5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^2}{\sqrt {e} f^{3/2}}+\frac {d \left (5 a f \left (15 d^2 e^2-22 c d f e+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d f e+81 c^2 f^2\right )\right ) x}{f}}{3 f}}{5 f}}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (\frac {b \left (\frac {1}{9} b d^3 x^9+\frac {1}{7} d^2 (3 b c+a d) x^7+\frac {3}{5} c d (b c+a d) x^5+\frac {1}{3} c^2 (b c+3 a d) x^3+a c^3 x\right )}{f}-\frac {(b e-a f) \left (\frac {b x \left (d x^2+c\right )^3}{7 f}-\frac {\frac {(7 b d e-6 b c f-7 a d f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {\left (7 a d f \left (15 d^2 e^2-40 c d f e+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 f e^2+231 c^2 d f^2 e-48 c^3 f^3\right )\right ) x}{f}}{3 f}-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{3 f}}{5 f}}{7 f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b x \left (d x^2+c\right )^3}{7 f}-\frac {\frac {(7 b d e-6 b c f-7 a d f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {\left (7 a d f \left (15 d^2 e^2-40 c d f e+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 f e^2+231 c^2 d f^2 e-48 c^3 f^3\right )\right ) x}{f}}{3 f}-\frac {\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x \left (d x^2+c\right )}{3 f}}{5 f}}{7 f}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {d (7 b e-5 a f) x \left (d x^2+c\right )^2}{5 f}-\frac {\frac {d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x \left (d x^2+c\right )}{3 f}+\frac {\frac {15 (b e (7 d e-c f)-a f (5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^2}{\sqrt {e} f^{3/2}}+\frac {d \left (5 a f \left (15 d^2 e^2-22 c d f e+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d f e+81 c^2 f^2\right )\right ) x}{f}}{3 f}}{5 f}}{2 e f}-\frac {(b e-a f) x \left (d x^2+c\right )^3}{2 e f \left (f x^2+e\right )}\right )}{f}\right )}{f}\)

Input:

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

Output:

(b*((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 - 280*c*d^2*e^2*f + 231*c^2*d*e*f^2 - 48*c^3*f^3))*x)/f + (10 
5*(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 - ((b*e - a*f)*((b*((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 
 - 280*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 - ((b*e - a*f)*(-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))*Arc 
Tan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(3/2)))/(3*f))/(5*f))/(2*e*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 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 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 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]
 

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. \(995\) vs. \(2(367)=734\).

Time = 0.62 (sec) , antiderivative size = 996, normalized size of antiderivative = 2.57

method	result	size
default	\(\frac {-\frac {6}{5} a \,b^{2} d^{3} e \,f^{3} x^{5}+\frac {1}{3} b^{3} c^{3} f^{4} x^{3}+3 a \,b^{2} c^{3} f^{4} x +\frac {9}{5} a \,b^{2} c \,d^{2} f^{4} x^{5}+9 a^{2} b \,c^{2} d \,f^{4} x +9 a^{2} b \,d^{3} e^{2} f^{2} x +\frac {3}{5} a^{2} b \,d^{3} f^{4} x^{5}+\frac {3}{5} b^{3} c^{2} d \,f^{4} x^{5}+\frac {3}{5} b^{3} d^{3} e^{2} f^{2} x^{5}-\frac {4}{3} b^{3} d^{3} e^{3} f \,x^{3}+3 a^{3} c \,d^{2} f^{4} x -2 a^{3} d^{3} e \,f^{3} x +\frac {1}{3} a^{3} d^{3} f^{4} x^{3}-6 a \,b^{2} c \,d^{2} e \,f^{3} x^{3}-18 a^{2} b c \,d^{2} e \,f^{3} x -18 a \,b^{2} c^{2} d e \,f^{3} x +27 a \,b^{2} c \,d^{2} e^{2} f^{2} x -\frac {6}{5} b^{3} c \,d^{2} e \,f^{3} x^{5}+3 a^{2} b c \,d^{2} f^{4} x^{3}-2 a^{2} b \,d^{3} e \,f^{3} x^{3}-2 b^{3} c^{2} d e \,f^{3} x^{3}+3 b^{3} c \,d^{2} e^{2} f^{2} x^{3}+5 b^{3} d^{3} e^{4} x -12 a \,b^{2} d^{3} e^{3} f x +9 b^{3} c^{2} d \,e^{2} f^{2} x -12 b^{3} c \,d^{2} e^{3} f x +\frac {1}{9} d^{3} x^{9} b^{3} f^{4}+\frac {3}{7} a \,b^{2} d^{3} f^{4} x^{7}+\frac {3}{7} b^{3} c \,d^{2} f^{4} x^{7}-\frac {2}{7} b^{3} d^{3} e \,f^{3} x^{7}+3 a \,b^{2} c^{2} d \,f^{4} x^{3}+3 a \,b^{2} d^{3} e^{2} f^{2} x^{3}-2 b^{3} c^{3} e \,f^{3} x}{f^{6}}+\frac {\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 ) x}{2 e \left (f \,x^{2}+e \right )}+\frac {\left (a^{3} c^{3} f^{6}+3 a^{3} c^{2} d e \,f^{5}-9 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}-27 a^{2} b \,c^{2} d \,e^{2} f^{4}+45 a^{2} b c \,d^{2} e^{3} f^{3}-21 a^{2} b \,d^{3} e^{4} f^{2}-9 a \,b^{2} c^{3} e^{2} f^{4}+45 a \,b^{2} c^{2} d \,e^{3} f^{3}-63 a \,b^{2} c \,d^{2} e^{4} f^{2}+27 a \,b^{2} d^{3} e^{5} f +5 b^{3} c^{3} e^{3} f^{3}-21 b^{3} c^{2} d \,e^{4} f^{2}+27 b^{3} c \,d^{2} e^{5} f -11 b^{3} d^{3} e^{6}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 e \sqrt {e f}}}{f^{6}}\)	\(996\)
risch	\(\text {Expression too large to display}\)	\(1763\)

Input:

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

Output:

1/f^6*(-6/5*a*b^2*d^3*e*f^3*x^5+1/3*b^3*c^3*f^4*x^3+3*a*b^2*c^3*f^4*x+9/5* 
a*b^2*c*d^2*f^4*x^5+9*a^2*b*c^2*d*f^4*x+9*a^2*b*d^3*e^2*f^2*x+3/5*a^2*b*d^ 
3*f^4*x^5+3/5*b^3*c^2*d*f^4*x^5+3/5*b^3*d^3*e^2*f^2*x^5-4/3*b^3*d^3*e^3*f* 
x^3+3*a^3*c*d^2*f^4*x-2*a^3*d^3*e*f^3*x+1/3*a^3*d^3*f^4*x^3-6*a*b^2*c*d^2* 
e*f^3*x^3-18*a^2*b*c*d^2*e*f^3*x-18*a*b^2*c^2*d*e*f^3*x+27*a*b^2*c*d^2*e^2 
*f^2*x-6/5*b^3*c*d^2*e*f^3*x^5+3*a^2*b*c*d^2*f^4*x^3-2*a^2*b*d^3*e*f^3*x^3 
-2*b^3*c^2*d*e*f^3*x^3+3*b^3*c*d^2*e^2*f^2*x^3+5*b^3*d^3*e^4*x-12*a*b^2*d^ 
3*e^3*f*x+9*b^3*c^2*d*e^2*f^2*x-12*b^3*c*d^2*e^3*f*x+1/9*d^3*x^9*b^3*f^4+3 
/7*a*b^2*d^3*f^4*x^7+3/7*b^3*c*d^2*f^4*x^7-2/7*b^3*d^3*e*f^3*x^7+3*a*b^2*c 
^2*d*f^4*x^3+3*a*b^2*d^3*e^2*f^2*x^3-2*b^3*c^3*e*f^3*x)+1/f^6*(1/2*(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)/e*x/( 
f*x^2+e)+1/2*(a^3*c^3*f^6+3*a^3*c^2*d*e*f^5-9*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^5-27*a^2*b*c^2*d*e^2*f^4+45*a^2*b*c*d^2*e^3*f^3-21 
*a^2*b*d^3*e^4*f^2-9*a*b^2*c^3*e^2*f^4+45*a*b^2*c^2*d*e^3*f^3-63*a*b^2*c*d 
^2*e^4*f^2+27*a*b^2*d^3*e^5*f+5*b^3*c^3*e^3*f^3-21*b^3*c^2*d*e^4*f^2+27*b^ 
3*c*d^2*e^5*f-11*b^3*d^3*e^6)/e/(e*f)^(1/2)*arctan(f*x/(e*f)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1028 vs. \(2 (367) = 734\).

Time = 0.12 (sec) , antiderivative size = 2076, normalized size of antiderivative = 5.36 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/1260*(140*b^3*d^3*e^2*f^6*x^11 - 20*(11*b^3*d^3*e^3*f^5 - 27*(b^3*c*d^2 
 + a*b^2*d^3)*e^2*f^6)*x^9 + 36*(11*b^3*d^3*e^4*f^4 - 27*(b^3*c*d^2 + a*b^ 
2*d^3)*e^3*f^5 + 21*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^2*f^6)*x^7 - 
 84*(11*b^3*d^3*e^5*f^3 - 27*(b^3*c*d^2 + a*b^2*d^3)*e^4*f^4 + 21*(b^3*c^2 
*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^3*f^5 - 5*(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^6)*x^5 + 420*(11*b^3*d^3*e^6*f^2 - 27*(b^3*c*d 
^2 + a*b^2*d^3)*e^5*f^3 + 21*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^4*f 
^4 - 5*(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^5 + 9*(a* 
b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^2*f^6)*x^3 + 315*(11*b^3*d^3*e^7 - 
a^3*c^3*e*f^6 - 27*(b^3*c*d^2 + a*b^2*d^3)*e^6*f + 21*(b^3*c^2*d + 3*a*b^2 
*c*d^2 + a^2*b*d^3)*e^5*f^2 - 5*(b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + 
 a^3*d^3)*e^4*f^3 + 9*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^3*f^4 - 3* 
(a^2*b*c^3 + a^3*c^2*d)*e^2*f^5 + (11*b^3*d^3*e^6*f - a^3*c^3*f^7 - 27*(b^ 
3*c*d^2 + a*b^2*d^3)*e^5*f^2 + 21*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)* 
e^4*f^3 - 5*(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 + 
9*(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^2)*sqrt(-e*f)*log((f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)) + 
 630*(11*b^3*d^3*e^7*f + a^3*c^3*e*f^7 - 27*(b^3*c*d^2 + a*b^2*d^3)*e^6*f^ 
2 + 21*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^5*f^3 - 5*(b^3*c^3 + 9*a* 
b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^4*f^4 + 9*(a*b^2*c^3 + 3*a^2*b*c...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1590 vs. \(2 (379) = 758\).

Time = 13.48 (sec) , antiderivative size = 1590, normalized size of antiderivative = 4.11 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

b**3*d**3*x**9/(9*f**2) + x**7*(3*a*b**2*d**3/(7*f**2) + 3*b**3*c*d**2/(7* 
f**2) - 2*b**3*d**3*e/(7*f**3)) + x**5*(3*a**2*b*d**3/(5*f**2) + 9*a*b**2* 
c*d**2/(5*f**2) - 6*a*b**2*d**3*e/(5*f**3) + 3*b**3*c**2*d/(5*f**2) - 6*b* 
*3*c*d**2*e/(5*f**3) + 3*b**3*d**3*e**2/(5*f**4)) + x**3*(a**3*d**3/(3*f** 
2) + 3*a**2*b*c*d**2/f**2 - 2*a**2*b*d**3*e/f**3 + 3*a*b**2*c**2*d/f**2 - 
6*a*b**2*c*d**2*e/f**3 + 3*a*b**2*d**3*e**2/f**4 + b**3*c**3/(3*f**2) - 2* 
b**3*c**2*d*e/f**3 + 3*b**3*c*d**2*e**2/f**4 - 4*b**3*d**3*e**3/(3*f**5)) 
+ x*(3*a**3*c*d**2/f**2 - 2*a**3*d**3*e/f**3 + 9*a**2*b*c**2*d/f**2 - 18*a 
**2*b*c*d**2*e/f**3 + 9*a**2*b*d**3*e**2/f**4 + 3*a*b**2*c**3/f**2 - 18*a* 
b**2*c**2*d*e/f**3 + 27*a*b**2*c*d**2*e**2/f**4 - 12*a*b**2*d**3*e**3/f**5 
 - 2*b**3*c**3*e/f**3 + 9*b**3*c**2*d*e**2/f**4 - 12*b**3*c*d**2*e**3/f**5 
 + 5*b**3*d**3*e**4/f**6) + 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)/(2*e**2*f**6 + 2*e*f**7*x**2 
) - sqrt(-1/(e**3*f**13))*(a*f - b*e)**2*(c*f - d*e)**2*(a*c*f**2 + 5*a*d* 
e*f + 5*b*c*e*f - 11*b*d*e**2)*log(-e**2*f**6*sqrt(-1/(e**3*f**13))*(a*f - 
 b*e)**2*(c*f - d*e)**2*(a*c*f**2 + 5*a*d*e*f + 5*b*c*e*f - 11*b*d*e**2...
 

Maxima [F(-2)]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1022 vs. \(2 (367) = 734\).

Time = 0.13 (sec) , antiderivative size = 1022, normalized size of antiderivative = 2.64 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(11*b^3*d^3*e^6 - 27*b^3*c*d^2*e^5*f - 27*a*b^2*d^3*e^5*f + 21*b^3*c^ 
2*d*e^4*f^2 + 63*a*b^2*c*d^2*e^4*f^2 + 21*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 + 9*a*b^2*c^3*e^2*f^4 + 27*a^2*b*c^2*d*e^2*f^4 + 9*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)*e*f^6) + 1/2*(b^3*d^3*e^6*x - 3*b^3*c*d^2*e^5*f*x - 3*a*b^2*d^3 
*e^5*f*x + 3*b^3*c^2*d*e^4*f^2*x + 9*a*b^2*c*d^2*e^4*f^2*x + 3*a^2*b*d^3*e 
^4*f^2*x - b^3*c^3*e^3*f^3*x - 9*a*b^2*c^2*d*e^3*f^3*x - 9*a^2*b*c*d^2*e^3 
*f^3*x - a^3*d^3*e^3*f^3*x + 3*a*b^2*c^3*e^2*f^4*x + 9*a^2*b*c^2*d*e^2*f^4 
*x + 3*a^3*c*d^2*e^2*f^4*x - 3*a^2*b*c^3*e*f^5*x - 3*a^3*c^2*d*e*f^5*x + a 
^3*c^3*f^6*x)/((f*x^2 + e)*e*f^6) + 1/315*(35*b^3*d^3*f^16*x^9 - 90*b^3*d^ 
3*e*f^15*x^7 + 135*b^3*c*d^2*f^16*x^7 + 135*a*b^2*d^3*f^16*x^7 + 189*b^3*d 
^3*e^2*f^14*x^5 - 378*b^3*c*d^2*e*f^15*x^5 - 378*a*b^2*d^3*e*f^15*x^5 + 18 
9*b^3*c^2*d*f^16*x^5 + 567*a*b^2*c*d^2*f^16*x^5 + 189*a^2*b*d^3*f^16*x^5 - 
 420*b^3*d^3*e^3*f^13*x^3 + 945*b^3*c*d^2*e^2*f^14*x^3 + 945*a*b^2*d^3*e^2 
*f^14*x^3 - 630*b^3*c^2*d*e*f^15*x^3 - 1890*a*b^2*c*d^2*e*f^15*x^3 - 630*a 
^2*b*d^3*e*f^15*x^3 + 105*b^3*c^3*f^16*x^3 + 945*a*b^2*c^2*d*f^16*x^3 + 94 
5*a^2*b*c*d^2*f^16*x^3 + 105*a^3*d^3*f^16*x^3 + 1575*b^3*d^3*e^4*f^12*x - 
3780*b^3*c*d^2*e^3*f^13*x - 3780*a*b^2*d^3*e^3*f^13*x + 2835*b^3*c^2*d*e^2 
*f^14*x + 8505*a*b^2*c*d^2*e^2*f^14*x + 2835*a^2*b*d^3*e^2*f^14*x - 630...
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 1177, normalized size of antiderivative = 3.04 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1885, normalized size of antiderivative = 4.87 \[ \int \frac {\left (a+b x^2\right )^3 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(315*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**3*e*f**6 + 315* 
sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**3*f**7*x**2 + 945*sq 
rt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**2*d*e**2*f**5 + 945*sq 
rt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c**2*d*e*f**6*x**2 - 2835 
*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c*d**2*e**3*f**4 - 283 
5*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*c*d**2*e**2*f**5*x**2 
 + 1575*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*d**3*e**4*f**3 
+ 1575*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*d**3*e**3*f**4*x 
**2 + 945*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*b*c**3*e**2*f 
**5 + 945*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*b*c**3*e*f**6 
*x**2 - 8505*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*b*c**2*d*e 
**3*f**4 - 8505*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*b*c**2* 
d*e**2*f**5*x**2 + 14175*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a** 
2*b*c*d**2*e**4*f**3 + 14175*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e))) 
*a**2*b*c*d**2*e**3*f**4*x**2 - 6615*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*s 
qrt(e)))*a**2*b*d**3*e**5*f**2 - 6615*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)* 
sqrt(e)))*a**2*b*d**3*e**4*f**3*x**2 - 2835*sqrt(f)*sqrt(e)*atan((f*x)/(sq 
rt(f)*sqrt(e)))*a*b**2*c**3*e**3*f**4 - 2835*sqrt(f)*sqrt(e)*atan((f*x)/(s 
qrt(f)*sqrt(e)))*a*b**2*c**3*e**2*f**5*x**2 + 14175*sqrt(f)*sqrt(e)*atan(( 
f*x)/(sqrt(f)*sqrt(e)))*a*b**2*c**2*d*e**4*f**3 + 14175*sqrt(f)*sqrt(e)...