\(\int \frac {(a+b x^2)^2 (c+d x^2)^3}{(e+f x^2)^2} \, dx\) [233]

Optimal result

Integrand size = 28, antiderivative size = 275 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=-\frac {\left (a^2 d^2 f^2 (2 d e-3 c f)-6 a b d f (d e-c f)^2+b^2 (d e-c f)^2 (4 d e-c f)\right ) x}{f^5}+\frac {d \left (a^2 d^2 f^2-2 a b d f (2 d e-3 c f)+3 b^2 (d e-c f)^2\right ) x^3}{3 f^4}-\frac {b d^2 (2 b d e-3 b c f-2 a d f) x^5}{5 f^3}+\frac {b^2 d^3 x^7}{7 f^2}-\frac {(b e-a f)^2 (d e-c f)^3 x}{2 e f^5 \left (e+f x^2\right )}+\frac {(b e-a f) (d e-c f)^2 (3 b e (3 d e-c f)-a f (5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} f^{11/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\frac {\left (6 a b d f (d e-c f)^2-b^2 (d e-c f)^2 (4 d e-c f)+a^2 d^2 f^2 (-2 d e+3 c f)\right ) x}{f^5}+\frac {d \left (a^2 d^2 f^2+3 b^2 (d e-c f)^2+2 a b d f (-2 d e+3 c f)\right ) x^3}{3 f^4}-\frac {b d^2 (2 b d e-3 b c f-2 a d f) x^5}{5 f^3}+\frac {b^2 d^3 x^7}{7 f^2}-\frac {(b e-a f)^2 (d e-c f)^3 x}{2 e f^5 \left (e+f x^2\right )}+\frac {(b e-a f) (d e-c f)^2 (3 b e (3 d e-c f)-a f (5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} f^{11/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.90, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {425, 401, 25, 403, 25, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {b \left (\frac {\int -\frac {\left (d x^2+c\right )^2 \left ((7 b d e-6 b c f-7 a d f) x^2+c (b e-7 a f)\right )}{f x^2+e}dx}{7 f}+\frac {b x \left (c+d x^2\right )^3}{7 f}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {\int -\frac {\left (d x^2+c\right ) \left (d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x^2+c (b e (7 d e-5 c f)-5 a f (d e+c f))\right )}{f x^2+e}dx}{5 f}+\frac {d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{5 f}}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\int \frac {\left (d x^2+c\right )^2 \left ((7 b d e-6 b c f-7 a d f) x^2+c (b e-7 a f)\right )}{f x^2+e}dx}{7 f}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{5 f}-\frac {\int \frac {\left (d x^2+c\right ) \left (d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x^2+c (b e (7 d e-5 c f)-5 a f (d e+c f))\right )}{f x^2+e}dx}{5 f}}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\right )}{f}\)

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(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))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(3/2)))/( 
3*f))/(5*f))/(2*e*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 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. \(613\) vs. \(2(257)=514\).

Time = 0.68 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.23

method	result	size
default	\(\frac {\frac {1}{7} b^{2} d^{3} x^{7} f^{3}+\frac {2}{5} a b \,d^{3} f^{3} x^{5}+\frac {3}{5} b^{2} c \,d^{2} f^{3} x^{5}-\frac {2}{5} b^{2} d^{3} e \,f^{2} x^{5}+\frac {1}{3} a^{2} d^{3} f^{3} x^{3}+2 a b c \,d^{2} f^{3} x^{3}-\frac {4}{3} a b \,d^{3} e \,f^{2} x^{3}+b^{2} c^{2} d \,f^{3} x^{3}-2 b^{2} c \,d^{2} e \,f^{2} x^{3}+b^{2} d^{3} e^{2} f \,x^{3}+3 a^{2} c \,d^{2} f^{3} x -2 a^{2} d^{3} e \,f^{2} x +6 a b \,c^{2} d \,f^{3} x -12 a b c \,d^{2} e \,f^{2} x +6 a b \,d^{3} e^{2} f x +b^{2} c^{3} f^{3} x -6 b^{2} c^{2} d e \,f^{2} x +9 b^{2} c \,d^{2} e^{2} f x -4 b^{2} d^{3} e^{3} x}{f^{5}}+\frac {\frac {\left (a^{2} c^{3} f^{5}-3 a^{2} c^{2} d e \,f^{4}+3 a^{2} c \,d^{2} e^{2} f^{3}-a^{2} d^{3} e^{3} f^{2}-2 a b \,c^{3} e \,f^{4}+6 a b \,c^{2} d \,e^{2} f^{3}-6 a b c \,d^{2} e^{3} f^{2}+2 a b \,d^{3} e^{4} f +b^{2} c^{3} e^{2} f^{3}-3 b^{2} c^{2} d \,e^{3} f^{2}+3 b^{2} c \,d^{2} e^{4} f -b^{2} d^{3} e^{5}\right ) x}{2 e \left (f \,x^{2}+e \right )}+\frac {\left (a^{2} c^{3} f^{5}+3 a^{2} c^{2} d e \,f^{4}-9 a^{2} c \,d^{2} e^{2} f^{3}+5 a^{2} d^{3} e^{3} f^{2}+2 a b \,c^{3} e \,f^{4}-18 a b \,c^{2} d \,e^{2} f^{3}+30 a b c \,d^{2} e^{3} f^{2}-14 a b \,d^{3} e^{4} f -3 b^{2} c^{3} e^{2} f^{3}+15 b^{2} c^{2} d \,e^{3} f^{2}-21 b^{2} c \,d^{2} e^{4} f +9 b^{2} d^{3} e^{5}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 e \sqrt {e f}}}{f^{5}}\)	\(614\)
risch	\(\text {Expression too large to display}\)	\(1171\)

Input:

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

Output:

1/f^5*(1/7*b^2*d^3*x^7*f^3+2/5*a*b*d^3*f^3*x^5+3/5*b^2*c*d^2*f^3*x^5-2/5*b 
^2*d^3*e*f^2*x^5+1/3*a^2*d^3*f^3*x^3+2*a*b*c*d^2*f^3*x^3-4/3*a*b*d^3*e*f^2 
*x^3+b^2*c^2*d*f^3*x^3-2*b^2*c*d^2*e*f^2*x^3+b^2*d^3*e^2*f*x^3+3*a^2*c*d^2 
*f^3*x-2*a^2*d^3*e*f^2*x+6*a*b*c^2*d*f^3*x-12*a*b*c*d^2*e*f^2*x+6*a*b*d^3* 
e^2*f*x+b^2*c^3*f^3*x-6*b^2*c^2*d*e*f^2*x+9*b^2*c*d^2*e^2*f*x-4*b^2*d^3*e^ 
3*x)+1/f^5*(1/2*(a^2*c^3*f^5-3*a^2*c^2*d*e*f^4+3*a^2*c*d^2*e^2*f^3-a^2*d^3 
*e^3*f^2-2*a*b*c^3*e*f^4+6*a*b*c^2*d*e^2*f^3-6*a*b*c*d^2*e^3*f^2+2*a*b*d^3 
*e^4*f+b^2*c^3*e^2*f^3-3*b^2*c^2*d*e^3*f^2+3*b^2*c*d^2*e^4*f-b^2*d^3*e^5)/ 
e*x/(f*x^2+e)+1/2*(a^2*c^3*f^5+3*a^2*c^2*d*e*f^4-9*a^2*c*d^2*e^2*f^3+5*a^2 
*d^3*e^3*f^2+2*a*b*c^3*e*f^4-18*a*b*c^2*d*e^2*f^3+30*a*b*c*d^2*e^3*f^2-14* 
a*b*d^3*e^4*f-3*b^2*c^3*e^2*f^3+15*b^2*c^2*d*e^3*f^2-21*b^2*c*d^2*e^4*f+9* 
b^2*d^3*e^5)/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. 711 vs. \(2 (257) = 514\).

Time = 0.10 (sec) , antiderivative size = 1442, normalized size of antiderivative = 5.24 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/420*(60*b^2*d^3*e^2*f^5*x^9 - 12*(9*b^2*d^3*e^3*f^4 - 7*(3*b^2*c*d^2 + 
2*a*b*d^3)*e^2*f^5)*x^7 + 28*(9*b^2*d^3*e^4*f^3 - 7*(3*b^2*c*d^2 + 2*a*b*d 
^3)*e^3*f^4 + 5*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^2*f^5)*x^5 - 140*( 
9*b^2*d^3*e^5*f^2 - 7*(3*b^2*c*d^2 + 2*a*b*d^3)*e^4*f^3 + 5*(3*b^2*c^2*d + 
 6*a*b*c*d^2 + a^2*d^3)*e^3*f^4 - 3*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)* 
e^2*f^5)*x^3 - 105*(9*b^2*d^3*e^6 + a^2*c^3*e*f^5 - 7*(3*b^2*c*d^2 + 2*a*b 
*d^3)*e^5*f + 5*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^4*f^2 - 3*(b^2*c^3 
 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^3*f^3 + (2*a*b*c^3 + 3*a^2*c^2*d)*e^2*f^4 
+ (9*b^2*d^3*e^5*f + a^2*c^3*f^6 - 7*(3*b^2*c*d^2 + 2*a*b*d^3)*e^4*f^2 + 5 
*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^3*f^3 - 3*(b^2*c^3 + 6*a*b*c^2*d 
+ 3*a^2*c*d^2)*e^2*f^4 + (2*a*b*c^3 + 3*a^2*c^2*d)*e*f^5)*x^2)*sqrt(-e*f)* 
log((f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)) - 210*(9*b^2*d^3*e^6*f - a^2 
*c^3*e*f^6 - 7*(3*b^2*c*d^2 + 2*a*b*d^3)*e^5*f^2 + 5*(3*b^2*c^2*d + 6*a*b* 
c*d^2 + a^2*d^3)*e^4*f^3 - 3*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^3*f^4 
 + (2*a*b*c^3 + 3*a^2*c^2*d)*e^2*f^5)*x)/(e^2*f^7*x^2 + e^3*f^6), 1/210*(3 
0*b^2*d^3*e^2*f^5*x^9 - 6*(9*b^2*d^3*e^3*f^4 - 7*(3*b^2*c*d^2 + 2*a*b*d^3) 
*e^2*f^5)*x^7 + 14*(9*b^2*d^3*e^4*f^3 - 7*(3*b^2*c*d^2 + 2*a*b*d^3)*e^3*f^ 
4 + 5*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^2*f^5)*x^5 - 70*(9*b^2*d^3*e 
^5*f^2 - 7*(3*b^2*c*d^2 + 2*a*b*d^3)*e^4*f^3 + 5*(3*b^2*c^2*d + 6*a*b*c*d^ 
2 + a^2*d^3)*e^3*f^4 - 3*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^2*f^5)...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1096 vs. \(2 (264) = 528\).

Time = 5.69 (sec) , antiderivative size = 1096, normalized size of antiderivative = 3.99 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

b**2*d**3*x**7/(7*f**2) + x**5*(2*a*b*d**3/(5*f**2) + 3*b**2*c*d**2/(5*f** 
2) - 2*b**2*d**3*e/(5*f**3)) + x**3*(a**2*d**3/(3*f**2) + 2*a*b*c*d**2/f** 
2 - 4*a*b*d**3*e/(3*f**3) + b**2*c**2*d/f**2 - 2*b**2*c*d**2*e/f**3 + b**2 
*d**3*e**2/f**4) + x*(3*a**2*c*d**2/f**2 - 2*a**2*d**3*e/f**3 + 6*a*b*c**2 
*d/f**2 - 12*a*b*c*d**2*e/f**3 + 6*a*b*d**3*e**2/f**4 + b**2*c**3/f**2 - 6 
*b**2*c**2*d*e/f**3 + 9*b**2*c*d**2*e**2/f**4 - 4*b**2*d**3*e**3/f**5) + x 
*(a**2*c**3*f**5 - 3*a**2*c**2*d*e*f**4 + 3*a**2*c*d**2*e**2*f**3 - a**2*d 
**3*e**3*f**2 - 2*a*b*c**3*e*f**4 + 6*a*b*c**2*d*e**2*f**3 - 6*a*b*c*d**2* 
e**3*f**2 + 2*a*b*d**3*e**4*f + b**2*c**3*e**2*f**3 - 3*b**2*c**2*d*e**3*f 
**2 + 3*b**2*c*d**2*e**4*f - b**2*d**3*e**5)/(2*e**2*f**5 + 2*e*f**6*x**2) 
 - sqrt(-1/(e**3*f**11))*(a*f - b*e)*(c*f - d*e)**2*(a*c*f**2 + 5*a*d*e*f 
+ 3*b*c*e*f - 9*b*d*e**2)*log(-e**2*f**5*sqrt(-1/(e**3*f**11))*(a*f - b*e) 
*(c*f - d*e)**2*(a*c*f**2 + 5*a*d*e*f + 3*b*c*e*f - 9*b*d*e**2)/(a**2*c**3 
*f**5 + 3*a**2*c**2*d*e*f**4 - 9*a**2*c*d**2*e**2*f**3 + 5*a**2*d**3*e**3* 
f**2 + 2*a*b*c**3*e*f**4 - 18*a*b*c**2*d*e**2*f**3 + 30*a*b*c*d**2*e**3*f* 
*2 - 14*a*b*d**3*e**4*f - 3*b**2*c**3*e**2*f**3 + 15*b**2*c**2*d*e**3*f**2 
 - 21*b**2*c*d**2*e**4*f + 9*b**2*d**3*e**5) + x)/4 + sqrt(-1/(e**3*f**11) 
)*(a*f - b*e)*(c*f - d*e)**2*(a*c*f**2 + 5*a*d*e*f + 3*b*c*e*f - 9*b*d*e** 
2)*log(e**2*f**5*sqrt(-1/(e**3*f**11))*(a*f - b*e)*(c*f - d*e)**2*(a*c*f** 
2 + 5*a*d*e*f + 3*b*c*e*f - 9*b*d*e**2)/(a**2*c**3*f**5 + 3*a**2*c**2*d...
 

Maxima [F(-2)]

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

integrate((b*x^2+a)^2*(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. 638 vs. \(2 (257) = 514\).

Time = 0.12 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\frac {{\left (9 \, b^{2} d^{3} e^{5} - 21 \, b^{2} c d^{2} e^{4} f - 14 \, a b d^{3} e^{4} f + 15 \, b^{2} c^{2} d e^{3} f^{2} + 30 \, a b c d^{2} e^{3} f^{2} + 5 \, a^{2} d^{3} e^{3} f^{2} - 3 \, b^{2} c^{3} e^{2} f^{3} - 18 \, a b c^{2} d e^{2} f^{3} - 9 \, a^{2} c d^{2} e^{2} f^{3} + 2 \, a b c^{3} e f^{4} + 3 \, a^{2} c^{2} d e f^{4} + a^{2} c^{3} f^{5}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \, \sqrt {e f} e f^{5}} - \frac {b^{2} d^{3} e^{5} x - 3 \, b^{2} c d^{2} e^{4} f x - 2 \, a b d^{3} e^{4} f x + 3 \, b^{2} c^{2} d e^{3} f^{2} x + 6 \, a b c d^{2} e^{3} f^{2} x + a^{2} d^{3} e^{3} f^{2} x - b^{2} c^{3} e^{2} f^{3} x - 6 \, a b c^{2} d e^{2} f^{3} x - 3 \, a^{2} c d^{2} e^{2} f^{3} x + 2 \, a b c^{3} e f^{4} x + 3 \, a^{2} c^{2} d e f^{4} x - a^{2} c^{3} f^{5} x}{2 \, {\left (f x^{2} + e\right )} e f^{5}} + \frac {15 \, b^{2} d^{3} f^{12} x^{7} - 42 \, b^{2} d^{3} e f^{11} x^{5} + 63 \, b^{2} c d^{2} f^{12} x^{5} + 42 \, a b d^{3} f^{12} x^{5} + 105 \, b^{2} d^{3} e^{2} f^{10} x^{3} - 210 \, b^{2} c d^{2} e f^{11} x^{3} - 140 \, a b d^{3} e f^{11} x^{3} + 105 \, b^{2} c^{2} d f^{12} x^{3} + 210 \, a b c d^{2} f^{12} x^{3} + 35 \, a^{2} d^{3} f^{12} x^{3} - 420 \, b^{2} d^{3} e^{3} f^{9} x + 945 \, b^{2} c d^{2} e^{2} f^{10} x + 630 \, a b d^{3} e^{2} f^{10} x - 630 \, b^{2} c^{2} d e f^{11} x - 1260 \, a b c d^{2} e f^{11} x - 210 \, a^{2} d^{3} e f^{11} x + 105 \, b^{2} c^{3} f^{12} x + 630 \, a b c^{2} d f^{12} x + 315 \, a^{2} c d^{2} f^{12} x}{105 \, f^{14}} \] Input:

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

Output:

1/2*(9*b^2*d^3*e^5 - 21*b^2*c*d^2*e^4*f - 14*a*b*d^3*e^4*f + 15*b^2*c^2*d* 
e^3*f^2 + 30*a*b*c*d^2*e^3*f^2 + 5*a^2*d^3*e^3*f^2 - 3*b^2*c^3*e^2*f^3 - 1 
8*a*b*c^2*d*e^2*f^3 - 9*a^2*c*d^2*e^2*f^3 + 2*a*b*c^3*e*f^4 + 3*a^2*c^2*d* 
e*f^4 + a^2*c^3*f^5)*arctan(f*x/sqrt(e*f))/(sqrt(e*f)*e*f^5) - 1/2*(b^2*d^ 
3*e^5*x - 3*b^2*c*d^2*e^4*f*x - 2*a*b*d^3*e^4*f*x + 3*b^2*c^2*d*e^3*f^2*x 
+ 6*a*b*c*d^2*e^3*f^2*x + a^2*d^3*e^3*f^2*x - b^2*c^3*e^2*f^3*x - 6*a*b*c^ 
2*d*e^2*f^3*x - 3*a^2*c*d^2*e^2*f^3*x + 2*a*b*c^3*e*f^4*x + 3*a^2*c^2*d*e* 
f^4*x - a^2*c^3*f^5*x)/((f*x^2 + e)*e*f^5) + 1/105*(15*b^2*d^3*f^12*x^7 - 
42*b^2*d^3*e*f^11*x^5 + 63*b^2*c*d^2*f^12*x^5 + 42*a*b*d^3*f^12*x^5 + 105* 
b^2*d^3*e^2*f^10*x^3 - 210*b^2*c*d^2*e*f^11*x^3 - 140*a*b*d^3*e*f^11*x^3 + 
 105*b^2*c^2*d*f^12*x^3 + 210*a*b*c*d^2*f^12*x^3 + 35*a^2*d^3*f^12*x^3 - 4 
20*b^2*d^3*e^3*f^9*x + 945*b^2*c*d^2*e^2*f^10*x + 630*a*b*d^3*e^2*f^10*x - 
 630*b^2*c^2*d*e*f^11*x - 1260*a*b*c*d^2*e*f^11*x - 210*a^2*d^3*e*f^11*x + 
 105*b^2*c^3*f^12*x + 630*a*b*c^2*d*f^12*x + 315*a^2*c*d^2*f^12*x)/f^14
 

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.67 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=x\,\left (\frac {3\,a^2\,c\,d^2+6\,a\,b\,c^2\,d+b^2\,c^3}{f^2}-\frac {2\,e\,\left (\frac {a^2\,d^3+6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d}{f^2}+\frac {2\,e\,\left (\frac {2\,b^2\,d^3\,e}{f^3}-\frac {b\,d^2\,\left (2\,a\,d+3\,b\,c\right )}{f^2}\right )}{f}-\frac {b^2\,d^3\,e^2}{f^4}\right )}{f}+\frac {e^2\,\left (\frac {2\,b^2\,d^3\,e}{f^3}-\frac {b\,d^2\,\left (2\,a\,d+3\,b\,c\right )}{f^2}\right )}{f^2}\right )-x^5\,\left (\frac {2\,b^2\,d^3\,e}{5\,f^3}-\frac {b\,d^2\,\left (2\,a\,d+3\,b\,c\right )}{5\,f^2}\right )+x^3\,\left (\frac {a^2\,d^3+6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d}{3\,f^2}+\frac {2\,e\,\left (\frac {2\,b^2\,d^3\,e}{f^3}-\frac {b\,d^2\,\left (2\,a\,d+3\,b\,c\right )}{f^2}\right )}{3\,f}-\frac {b^2\,d^3\,e^2}{3\,f^4}\right )+\frac {b^2\,d^3\,x^7}{7\,f^2}+\frac {x\,\left (a^2\,c^3\,f^5-3\,a^2\,c^2\,d\,e\,f^4+3\,a^2\,c\,d^2\,e^2\,f^3-a^2\,d^3\,e^3\,f^2-2\,a\,b\,c^3\,e\,f^4+6\,a\,b\,c^2\,d\,e^2\,f^3-6\,a\,b\,c\,d^2\,e^3\,f^2+2\,a\,b\,d^3\,e^4\,f+b^2\,c^3\,e^2\,f^3-3\,b^2\,c^2\,d\,e^3\,f^2+3\,b^2\,c\,d^2\,e^4\,f-b^2\,d^3\,e^5\right )}{2\,e\,\left (f^6\,x^2+e\,f^5\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x\,\left (a\,f-b\,e\right )\,{\left (c\,f-d\,e\right )}^2\,\left (a\,c\,f^2-9\,b\,d\,e^2+5\,a\,d\,e\,f+3\,b\,c\,e\,f\right )}{\sqrt {e}\,\left (a^2\,c^3\,f^5+3\,a^2\,c^2\,d\,e\,f^4-9\,a^2\,c\,d^2\,e^2\,f^3+5\,a^2\,d^3\,e^3\,f^2+2\,a\,b\,c^3\,e\,f^4-18\,a\,b\,c^2\,d\,e^2\,f^3+30\,a\,b\,c\,d^2\,e^3\,f^2-14\,a\,b\,d^3\,e^4\,f-3\,b^2\,c^3\,e^2\,f^3+15\,b^2\,c^2\,d\,e^3\,f^2-21\,b^2\,c\,d^2\,e^4\,f+9\,b^2\,d^3\,e^5\right )}\right )\,\left (a\,f-b\,e\right )\,{\left (c\,f-d\,e\right )}^2\,\left (a\,c\,f^2-9\,b\,d\,e^2+5\,a\,d\,e\,f+3\,b\,c\,e\,f\right )}{2\,e^{3/2}\,f^{11/2}} \] Input:

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

Output:

x*((b^2*c^3 + 3*a^2*c*d^2 + 6*a*b*c^2*d)/f^2 - (2*e*((a^2*d^3 + 3*b^2*c^2* 
d + 6*a*b*c*d^2)/f^2 + (2*e*((2*b^2*d^3*e)/f^3 - (b*d^2*(2*a*d + 3*b*c))/f 
^2))/f - (b^2*d^3*e^2)/f^4))/f + (e^2*((2*b^2*d^3*e)/f^3 - (b*d^2*(2*a*d + 
 3*b*c))/f^2))/f^2) - x^5*((2*b^2*d^3*e)/(5*f^3) - (b*d^2*(2*a*d + 3*b*c)) 
/(5*f^2)) + x^3*((a^2*d^3 + 3*b^2*c^2*d + 6*a*b*c*d^2)/(3*f^2) + (2*e*((2* 
b^2*d^3*e)/f^3 - (b*d^2*(2*a*d + 3*b*c))/f^2))/(3*f) - (b^2*d^3*e^2)/(3*f^ 
4)) + (b^2*d^3*x^7)/(7*f^2) + (x*(a^2*c^3*f^5 - b^2*d^3*e^5 - a^2*d^3*e^3* 
f^2 + b^2*c^3*e^2*f^3 - 2*a*b*c^3*e*f^4 + 2*a*b*d^3*e^4*f - 3*a^2*c^2*d*e* 
f^4 + 3*b^2*c*d^2*e^4*f + 3*a^2*c*d^2*e^2*f^3 - 3*b^2*c^2*d*e^3*f^2 - 6*a* 
b*c*d^2*e^3*f^2 + 6*a*b*c^2*d*e^2*f^3))/(2*e*(e*f^5 + f^6*x^2)) + (atan((f 
^(1/2)*x*(a*f - b*e)*(c*f - d*e)^2*(a*c*f^2 - 9*b*d*e^2 + 5*a*d*e*f + 3*b* 
c*e*f))/(e^(1/2)*(a^2*c^3*f^5 + 9*b^2*d^3*e^5 + 5*a^2*d^3*e^3*f^2 - 3*b^2* 
c^3*e^2*f^3 + 2*a*b*c^3*e*f^4 - 14*a*b*d^3*e^4*f + 3*a^2*c^2*d*e*f^4 - 21* 
b^2*c*d^2*e^4*f - 9*a^2*c*d^2*e^2*f^3 + 15*b^2*c^2*d*e^3*f^2 + 30*a*b*c*d^ 
2*e^3*f^2 - 18*a*b*c^2*d*e^2*f^3)))*(a*f - b*e)*(c*f - d*e)^2*(a*c*f^2 - 9 
*b*d*e^2 + 5*a*d*e*f + 3*b*c*e*f))/(2*e^(3/2)*f^(11/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1264, normalized size of antiderivative = 4.60 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(105*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**3*e*f**5 + 105* 
sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**3*f**6*x**2 + 315*sq 
rt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*d*e**2*f**4 + 315*sq 
rt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*d*e*f**5*x**2 - 945* 
sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d**2*e**3*f**3 - 945* 
sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d**2*e**2*f**4*x**2 + 
 525*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**3*e**4*f**2 + 5 
25*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**3*e**3*f**3*x**2 
+ 210*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**3*e**2*f**4 + 2 
10*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**3*e*f**5*x**2 - 18 
90*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*d*e**3*f**3 - 18 
90*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*d*e**2*f**4*x**2 
 + 3150*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c*d**2*e**4*f**2 
 + 3150*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c*d**2*e**3*f**3 
*x**2 - 1470*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*d**3*e**5*f 
 - 1470*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*d**3*e**4*f**2*x 
**2 - 315*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*b**2*c**3*e**3*f** 
3 - 315*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*b**2*c**3*e**2*f**4* 
x**2 + 1575*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*b**2*c**2*d*e**4 
*f**2 + 1575*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*b**2*c**2*d*...